+2 college entrance (after SEE) Kathmandu Model College (KMC) +2 Entrance (after SEE): Mathematics

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2083 · Science Model Set B · 1 mark
2083 · Science Model Set C · 1 mark

The graphs of the equations $3x + 6y = 9$ and $9x + 18y = 27$ are (a) intersecting (b) parallel (c) coincident (d) perpendicular

Similar questions: The graphs of the equations 3x + 6y = 9 and 9x + (2080 · Science Model) · The graphs of the equations 3x + 2y = 5 and 2x - (2083 · Science Model Set D)

Answer

(c) coincident

$\frac{a_1}{a_2} = \frac{3}{9} = \frac13,\quad \frac{b_1}{b_2} = \frac{6}{18} = \frac13,\quad \frac{c_1}{c_2} = \frac{9}{27} = \frac13.$ All three ratios are equal (the second equation is 3 times the first), so the lines coincide.

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Most repeated · 4 of 18 exams
2083 · Science Model Set D · 1 mark

The graphs of the equations $3x + 2y = 5$ and $2x - y = 11$ are (a) intersecting (b) parallel (c) coincident (d) None

Similar questions: The graphs of the equations 3x + 6y = 9 and 9x + (2083 · Science Model Set B) · The graphs of the equations 3x + 6y = 9 and 9x + (2080 · Science Model)

Answer

(a) intersecting

The slopes are $-\frac{3}{2}$ and $2$ , which are different, so the lines are neither parallel nor coincident. (Equivalently $\frac{3}{2} \neq \frac{2}{-1}$ .) They meet at $x = \frac{27}{7}$ , $y = -\frac{29}{7}$ .

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Most repeated · 4 of 18 exams
2080 · Science Model · 1 mark

The graphs of the equations $3x + 6y = 9$ and $9x + 18y = 27$ are (a) coincident (b) parallel (c) intersecting (d) perpendicular

Similar questions: The graphs of the equations 3x + 6y = 9 and 9x + (2083 · Science Model Set B) · The graphs of the equations 3x + 2y = 5 and 2x - (2083 · Science Model Set D)

Answer

(a) coincident

Dividing $9x+18y=27$ by 3 gives $3x+6y=9$ , so both equations represent the same line. Equivalently $\frac{3}{9}=\frac{6}{18}=\frac{9}{27}$ .

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Most repeated · 3 of 18 exams
Asked 3 times
2083 · Management and Law Model Set B
2079 · Science Model · 1 mark
2080 · Management and Law Model Set I

The factors of $6x^2 + x - 12$ are (a) $(3x+4)(2x-3)$ (b) $(3x-4)(2x+3)$ (c) $(3x-4)(2x-4)$ (d) $(3x+4)(2x+4)$

Answer

(b) $(3x-4)(2x+3)$

$(3x-4)(2x+3) = 6x^2 + 9x - 8x - 12 = 6x^2 + x - 12.$

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2083 · Science Model Set C · 1 mark
2083 · Management and Law Model Set C

HCF of two numbers is 21 and their product is 2646. How many pairs of such numbers are possible? (a) 3 (b) 1 (c) 2 (d) None

Similar questions: HCF of two numbers is 101 and their sum is 1212. (2083 · Science Model Set B)

Answer

(c) 2

Let the numbers be $21a$ and $21b$ with $a, b$ co-prime. Then $21a \cdot 21b = 2646 \Rightarrow ab = \frac{2646}{441} = 6.$ The co-prime pairs are $(1,6)$ and $(2,3)$ , giving the number pairs $(21,126)$ and $(42,63)$ . So there are 2 pairs.

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2083 · Management and Law Model Set B
2080 · Management and Law Model Set I

The number of distinct prime factors of 210 is (a) 1 (b) 2 (c) 3 (d) 4

Similar questions: The number of distinct prime factors of 27 is (2083 · Science Model Set B)

Answer

(d) 4

$210 = 2 \times 3 \times 5 \times 7$ , which has 4 distinct prime factors.

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Most repeated · 3 of 18 exams
2083 · Science Model Set B · 1 mark

HCF of two numbers is 101 and their sum is 1212. How many pairs of such numbers are possible? (a) 3 (b) 2 (c) 1 (d) None

Similar questions: HCF of two numbers is 21 and their product is (2083 · Science Model Set C)

Answer

(b) 2

Let the numbers be $101a$ and $101b$ with $a, b$ co-prime. Then $101(a+b) = 1212$ , so $a+b = 12$ . The co-prime pairs with sum 12 are $(1,11)$ and $(5,7)$ ( $(2,10)$ , $(3,9)$ , $(4,8)$ , $(6,6)$ share factors). So there are 2 pairs: $(101, 1111)$ and $(505, 707)$ .

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Most repeated · 3 of 18 exams
2083 · Science Model Set B · 1 mark

The number of distinct prime factors of 27 is (a) 1 (b) 2 (c) 3 (d) 4

Similar questions: The number of distinct prime factors of 210 is (2083 · Management and Law Model Set B)

Answer

(a) 1

$27 = 3^3$ , so 3 is the only prime factor.

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Most repeated · 3 of 18 exams
2083 · Science Model Set D · 1 mark

$x^n - y^n$ is divisible by $x + y$ for (a) All values of $n$ (b) Only even values of $n$ (c) Only odd values of $n$ (d) No value of $n$

Similar questions: a^n + b^n is divisible by a + b (2083 · Science Model Set E) · x^n + y^n is divisible by x - y (2079 · Science Model)

Answer

(b) Only even values of $n$

$x^n - y^n$ is divisible by $x + y$ iff it vanishes at $x = -y$ : $(-y)^n - y^n = 0$ only when $n$ is even.

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Most repeated · 3 of 18 exams
2083 · Science Model Set E · 1 mark

$a^n + b^n$ is divisible by $a + b$ (a) For all values of $n$ (b) Only for even values of $n$ (c) Only for odd values of $n$ (d) For no value of $n$

Similar questions: x^n - y^n is divisible by x + y for (2083 · Science Model Set D) · x^n + y^n is divisible by x - y (2079 · Science Model)

Answer

(c) Only for odd values of $n$

Put $a=-b$ in $a^n+b^n$ : it gives $(-b)^n+b^n$ , which is $0$ only when $n$ is odd. By the factor theorem, $a+b$ divides $a^n+b^n$ only for odd $n$ . For example, $a^3+b^3=(a+b)(a^2-ab+b^2)$ .

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Most repeated · 3 of 18 exams
2079 · Science Model · 1 mark

$x^n + y^n$ is divisible by $x - y$ (a) For all values of $n$ (b) For even values of $n$ (c) For odd values of $n$ (d) For no value of $n$

Similar questions: x^n - y^n is divisible by x + y for (2083 · Science Model Set D) · a^n + b^n is divisible by a + b (2083 · Science Model Set E)

Answer

(d) For no value of $n$

By the factor theorem, $x-y$ divides $x^n+y^n$ only if the expression vanishes at $x=y$ . But $x^n+y^n$ at $x=y$ equals $2y^n\ne0$ . So it is never divisible by $x-y$ (for odd $n$ it is divisible by $x+y$ instead).

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2083 · Science Model Set A · 1 mark
2083 · Management and Law Model Set D

If £1 = Rs 135 and USD 1 = Rs 105, then convert £63 into dollar. (a) USD 63 (b) USD 30 (c) USD 81 (d) USD 36

Answer

(c) USD 81

£63 $=63\times135=$ Rs 8505, and Rs 8505 $\div\,105=81$ dollars.

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2083 · Science Model Set A · 1 mark
2079 · Science Model · 1 mark

Find the probability of occurring a squared number or a cubed number while drawing a flash card from the set of cards numbered from 3 to 32. (a) $\frac{1}{4}$ (b) $\frac{1}{5}$ (c) $\frac{1}{2}$ (d) $\frac{1}{3}$

Answer

(b) $\frac{1}{5}$

Cards 3 to 32 give $n(S)=30$ . Squares: 4, 9, 16, 25 (4 cards). Cubes: 8, 27 (2 cards). No number is both, so favourable cases $=6$ . $P=\frac{6}{30}=\frac15$

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2083 · Science Model Set B · 1 mark
2083 · Science Model Set D · 1 mark

The mean of the first $n$ natural numbers is (a) $\frac{n+1}{2}$ (b) $\frac{n(n+1)}{2}$ (c) $\frac{n}{2}$ (d) $\frac{n^2+1}{2}$

Answer

(a) $\frac{n+1}{2}$

$\text{Mean} = \frac{1+2+\dots+n}{n} = \frac{n(n+1)/2}{n} = \frac{n+1}{2}.$

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2083 · Science Model Set B · 1 mark
2083 · Science Model Set C · 1 mark

The value of $c$ for which the equation $2x^2 - 9x + c = 0$ has equal roots is (a) $\frac{81}{8}$ (b) $\frac{8}{81}$ (c) 9 (d) 3

Answer

(a) $\frac{81}{8}$

Equal roots need $D = b^2 - 4ac = 0$ : $81 - 8c = 0 \Rightarrow c = \frac{81}{8}$ .

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2083 · Science Model Set B · 1 mark
2083 · Management and Law Model Set E

The value of probability ranges from (a) 0 to 1 (b) $-1$ to 1 (c) 0.5 to 1 (d) 0.1 to 1

Answer

(a) 0 to 1

Probability of an event is $0\le P\le 1$ : 0 for an impossible event and 1 for a certain event.

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2083 · Science Model Set C · 1 mark
2083 · Management and Law Model Set C

Which of the following pairs is co-prime? (a) (3, 9) (b) (3, 7) (c) (6, 2) (d) (5, 25)

Answer

(b) (3, 7)

Co-prime numbers have HCF 1. $\text{HCF}(3,9)=3$ , $\text{HCF}(3,7)=1$ , $\text{HCF}(6,2)=2$ , $\text{HCF}(5,25)=5$ . Only $(3,7)$ is co-prime.

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2083 · Science Model Set C · 1 mark
2083 · Management and Law Model Set E

A sector of a circle has a central angle of $270^\circ$ . If the circle has a radius of 4, what is the area of the sector? (a) $4\pi$ (b) $8\pi$ (c) $12\pi$ (d) $16\pi$

Answer

(c) $12\pi$

$\text{Area} = \frac{270}{360}\pi r^2 = \frac{3}{4}\pi(16) = 12\pi.$

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2083 · Science Model Set C · 1 mark
2080 · Science Model · 1 mark

If $1, x, y, z, 16$ are in G.P., what is the value of $x + y + z$ ? (a) 8 (b) 12 (c) 14 (d) 16

Answer

(c) 14

The fifth term is $ar^4 = r^4 = 16$ , so $r = 2$ (taking the real positive ratio). Then $x=2$ , $y=4$ , $z=8$ and $x+y+z = 14$ .

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2083 · Science Model Set D · 1 mark
2080 · Science Model · 1 mark

What is the area of a circle whose circumference is $\pi$ ? (a) $\frac{\pi}{4}$ (b) $\frac{\pi}{2}$ (c) $\pi$ (d) $2\pi$

Answer

(a) $\frac{\pi}{4}$

$2\pi r = \pi$ gives $r = \frac{1}{2}$ , so the area is $\pi r^2 = \frac{\pi}{4}$ .

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2083 · Science Model Set E · 1 mark
2079 · Science Model · 1 mark

If $y^2 = 4$ and $x^2 y = 18$ , $x + y$ could equal which of the following values? (a) $-5$ (b) $-1$ (c) 1 (d) 6

Answer

(b) $-1$

From $y^2=4$ , $y=\pm2$ .

If $y=2$ , then $x^2=9$ , so $x=\pm3$ and $x+y=5$ or $-1$ .

If $y=-2$ , then $x^2=-9$ , which has no real solution.

Of the options, only $x+y=-1$ is possible.

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2083 · Science Model Set E · 1 mark
2079 · Science Model · 1 mark

The average of a set of 18 consecutive integers is 22.5. What is the smallest integer in the set? (a) 14 (b) 16 (c) 12 (d) 28

Answer

(a) 14

Let the smallest integer be $n$ . The integers are $n, n+1, \dots, n+17$ , with average $n+\frac{17}{2}=22.5 \Rightarrow n=14$

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2083 · Science Model Set E · 1 mark
2080 · Science Model · 1 mark

$\frac{5}{8}$ of 24 is equal to $\frac{15}{7}$ of what number? (a) 7 (b) 8 (c) 15 (d) $\frac{7}{225}$

Answer

(a) 7

$\frac{5}{8} \times 24 = 15$ . Let $\frac{15}{7} x = 15$ ; then $x = 7$ .

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2083 · Science Model Set E · 1 mark
2080 · Science Model · 1 mark

If there are 24 white marbles and 36 blue marbles in a bag, what is the ratio of blue to white marbles? (a) $\frac{2}{3}$ (b) $\frac{3}{2}$ (c) $\frac{4}{9}$ (d) $\frac{9}{4}$

Answer

(b) $\frac{3}{2}$

Blue : white $= 36 : 24 = 3 : 2$ , i.e. $\frac{3}{2}$ .

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2083 · Science Model Set E · 1 mark
2080 · Science Model · 1 mark

What is the probability of getting 53 Sundays in a non-leap year? (a) $\frac{1}{7}$ (b) $\frac{2}{7}$ (c) $\frac{3}{7}$ (d) $\frac{4}{7}$

Answer

(a) $\frac{1}{7}$

A non-leap year has $365=52\times7+1$ days, so it has 52 full weeks plus 1 extra day. A 53rd Sunday occurs only if the extra day is a Sunday, which is 1 of 7 equally likely days. So $P=\frac17$ .

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2083 · Management and Law Model Set A
2079 · Management Model

In a group of 100 people, 70 can speak Nepali and 45 speak English, then the number of people who can speak English only. (a) 30 (b) 40 (c) 50 (d) 60

Answer

(a) 30

Assuming everyone speaks at least one language, $n(N \cup E) = 100$ .

$n(N \cap E) = 70 + 45 - 100 = 15$

English only $= 45 - 15 = 30$ .

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2083 · Management and Law Model Set A
2079 · Management Model

Two identical pies are cut into a total of 16 equal parts. If each part is then split equally among three people, what fraction of a pie does each person receive? (a) $\frac{1}{48}$ (b) $\frac{1}{24}$ (c) $\frac{1}{16}$ (d) $\frac{3}{16}$

Answer

(b) $\frac{1}{24}$

Each part is $\frac{2}{16} = \frac{1}{8}$ of a pie. Split among three people, each gets $\frac{1}{8} \times \frac{1}{3} = \frac{1}{24}$ .

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2083 · Management and Law Model Set A
2079 · Management Model

Two pipes X and Y can fill a cistern in 24 minutes and 32 minutes respectively. If both the pipes are opened together, then after how much time should Y be closed so that the tank is full in 18 minutes? (a) 8 min. (b) 10 min. (c) 18 min. (d) 20 min.

Answer

(a) 8 min.

X works the whole 18 minutes and fills $\frac{18}{24} = \frac{3}{4}$ of the tank. The remaining $\frac{1}{4}$ is filled by Y at $\frac{1}{32}$ per minute:

$t = \frac{1/4}{1/32} = 8 \text{ min}$

So Y is closed after 8 minutes.

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2083 · Management and Law Model Set A
2079 · Management Model

A trade man charges 20% above the cost price but allowed a discount of 10% for cash payment. What is the gain percent? (a) 8% (b) 10% (c) 20% (d) 30%

Answer

(a) 8%

Let the cost price be Rs 100. Marked price $=$ Rs 120. Selling price $= 120 \times 0.9 =$ Rs 108. Gain $=$ Rs 8, which is 8%.

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2083 · Management and Law Model Set A
2079 · Management Model

If $a\%$ of $a = 4$ and $a\%$ of $b = 40$ , find the positive value of $b$ ? (a) 20 (b) 200 (c) 2000 (d) All of above

Answer

(b) 200

$\frac{a}{100} \times a = 4 \Rightarrow a^2 = 400 \Rightarrow a = 20$

Then $\frac{20}{100} \times b = 40$ , so $b = 200$ .

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2083 · Management and Law Model Set A
2079 · Management Model

If a right circular cylinder's radius is halved and its height doubled, by what percent will the volume increase or decrease? (a) 50% decrease (b) No change (c) 25% increase (d) 50% increase

Answer

(a) 50% decrease

$V = \pi r^2 h$ . New volume $= \pi \left(\frac{r}{2}\right)^2 (2h) = \frac{1}{2}\pi r^2 h$ , which is half the original, so it decreases by 50%.

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2083 · Management and Law Model Set A
2079 · Management Model

For what value of $x$ is $6^3 + 8^3 + 10^3 = x^3$ true? (a) 12 (b) 24 (c) 9 (d) 7

Answer

(a) 12

$6^3 + 8^3 + 10^3 = 216 + 512 + 1000 = 1728 = 12^3$

So $x = 12$ .

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2083 · Management and Law Model Set A
2079 · Management Model

The values of $x$ and $y$ in $x + y = 7$ and $3x - 2y = 11$ are (a) 2, 5 (b) 5, 2 (c) 3, 4 (d) 4, 3

Answer

(b) 5, 2

Multiply $x + y = 7$ by 2: $2x + 2y = 14$ . Add to $3x - 2y = 11$ : $5x = 25$ , so $x = 5$ and $y = 2$ .

Check: $3(5) - 2(2) = 11$ .

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2083 · Management and Law Model Set A
2079 · Management Model

If $k$ , 5 and 25 are in continued proportion, what is the value of $k^3 - 1$ ? (a) 0 (b) 1 (c) 5 (d) 125

Answer

(a) 0

In continued proportion, $\frac{k}{5} = \frac{5}{25}$ , so $25k = 25$ and $k = 1$ . Then $k^3 - 1 = 1 - 1 = 0$ .

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2083 · Management and Law Model Set A
2079 · Management Model

The following observations have been arranged in ascending order: 29, 32, 48, 52, $x$ , $x+3$ , 71, 75, 80, 92. If the median of the data is 61.5, what is the value of $x$ ? (a) 54 (b) 60 (c) 62 (d) 56

Answer

(b) 60

There are 10 observations, so the median is the mean of the 5th and 6th terms:

$\frac{x + (x+3)}{2} = 61.5 \Rightarrow 2x + 3 = 123 \Rightarrow x = 60$

The order 52 < 60 < 63 < 71 holds.

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2083 · Management and Law Model Set B
2080 · Management and Law Model Set I

Which of the following inequalities is equivalent to $|m + 2| < 3$ ? (a) $m < 5$ (b) $m < 1$ (c) $-5 < m < 5$ (d) $-5 < m < 1$

Answer

(d) $-5 < m < 1$

$|m+2|<3 \iff -3 < m+2 < 3 \iff -5 < m < 1.$

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2083 · Management and Law Model Set B
2080 · Management and Law Model Set I

A trade man allows a discount of 10%. What price should he mark on an article which costs Rs. 200 to gain 26%? (a) Rs. 200 (b) Rs. 280 (c) Rs. 300 (d) Rs. 350

Answer

(b) Rs. 280

Selling price for a 26% gain: $200 \times 1.26 = \text{Rs. } 252$ .

Let the marked price be $M$ . Then $0.9M = 252$ , so $M = \text{Rs. } 280$ .

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2083 · Management and Law Model Set B
2080 · Management and Law Model Set I

$P(A) + P(\bar{A})$ is equal to (a) 0 (b) 1 (c) 0.75 (d) 0.5

Answer

(b) 1

An event and its complement are exhaustive and mutually exclusive, so $P(A)+P(\bar{A}) = 1$ .

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2083 · Management and Law Model Set B
2080 · Management and Law Model Set I

How many 1-inch square tiles would it take to cover the floor of a closet that has dimensions 5 feet by 4 feet? (1 foot = 12 inches) (a) 20 (b) 240 (c) 1440 (d) 2880

Answer

(d) 2880

The closet is $5 \times 12 = 60$ inches by $4 \times 12 = 48$ inches, so the area is $60 \times 48 = 2880$ square inches, i.e. 2880 tiles.

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2083 · Management and Law Model Set B
2080 · Management and Law Model Set I

$x$ and $y$ are positive numbers such that $x + y + z < 1$ and $xy = 1$ . Which could be the value of $z$ ? (a) $z < -1$ (b) $z > -1$ (c) $z = -1$ (d) $z > 1$

Answer

(a) $z < -1$

For positive $x, y$ with $xy = 1$ , AM-GM gives $x + y \ge 2\sqrt{xy} = 2$ . Since $x+y+z<1$ , we get $z < 1 - (x+y) \le 1 - 2 = -1$ , so $z < -1$ .

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2083 · Management and Law Model Set B
2080 · Management and Law Model Set I

On what sum will the simple interest for 3 years at 5% p.a. amount to Rs. 150? (a) Rs. 1000 (b) Rs. 1500 (c) Rs. 1050 (d) Rs. 1550

Answer

(a) Rs. 1000

$P = \frac{100 \times SI}{T \times R} = \frac{100 \times 150}{3 \times 5} = \text{Rs. } 1000.$

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2083 · Management and Law Model Set B
2080 · Management and Law Model Set I

If $\frac{3}{4}$ of a number is 7 more than $\frac{1}{6}$ of that number, what is $\frac{5}{3}$ of that number? (a) 12 (b) 15 (c) 18 (d) 20

Answer

(d) 20

Let the number be $x$ . Then $\frac{3}{4}x - \frac{1}{6}x = 7$ , so $\frac{9x-2x}{12} = \frac{7x}{12} = 7$ and $x = 12$ .

$\frac{5}{3} \times 12 = 20.$

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2083 · Management and Law Model Set B
2080 · Management and Law Model Set I

The mean of 100 items is 49. If it was found that three items which should have been 60, 70, 80 were wrongly read as 40, 20, 50 respectively, the correct mean is (a) 48 (b) 82 (c) 50 (d) 80

Answer

(c) 50

Wrong total $= 100 \times 49 = 4900$ . Correct values add $60+70+80 = 210$ and remove $40+20+50 = 110$ , so the correct total is $4900 + 210 - 110 = 5000$ .

$\text{Correct mean} = \frac{5000}{100} = 50.$

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2083 · Science Model Set A · 1 mark

The compound interest compounded half yearly is calculated by (a) $P\left(1+\frac{R}{100}\right)^T$ (b) $P\left[\left(1+\frac{R}{100}\right)^T-1\right]$ (c) $P\left(1+\frac{R}{200}\right)^{2T}$ (d) $P\left[\left(1+\frac{R}{200}\right)^{2T}-1\right]$

Similar questions: The compound interest compounded yearly is (2079 · Science Model)

Answer

(d) $P\left[\left(1+\frac{R}{200}\right)^{2T}-1\right]$

Compounded half-yearly, the rate per period is $R/2$ and the number of periods is $2T$ . The amount is $P\left(1+\frac{R}{200}\right)^{2T}$ , and the interest is the amount minus $P$ .

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2083 · Science Model Set A · 1 mark

Which of the following statements is true? (a) $-\sqrt{(-3)^2} < 0$ (b) $(-1)^{87} = -1$ (c) $\sqrt{16} = 4$ (d) All of above

Similar questions: If (a)/(b) > (c)/(d), which of the following (2079 · Science Model)

Answer

(d) All of above

$-\sqrt{(-3)^2}=-3<0$ , $(-1)^{87}=-1$ (odd power) and $\sqrt{16}=4$ are all true.

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2083 · Science Model Set C · 1 mark

If $a = 16b$ and $b$ is a prime number greater than 2, how many positive distinct factors does $a$ have? (a) 5 (b) 10 (c) 15 (d) 20

Similar questions: If a = 16b and b is a prime number greater than (2080 · Science Model)

Answer

(b) 10

Since $b$ is a prime greater than 2, $a = 2^4 \cdot b^1$ with $b$ an odd prime. The number of positive factors is $(4+1)(1+1) = 10$ .

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2083 · Science Model Set D · 1 mark

If the sum of five consecutive integers is $S$ , what is the largest of those integers in terms of $S$ ? (a) $\frac{S-10}{5}$ (b) $\frac{S+10}{5}$ (c) $\frac{S-5}{2}$ (d) $\frac{S+5}{4}$

Similar questions: If the sum of five consecutive integers is S, (2080 · Science Model)

Answer

(b) $\frac{S+10}{5}$

Let the integers be $n-2, n-1, n, n+1, n+2$ . Then $S = 5n$ , so $n = \frac{S}{5}$ and the largest is $n + 2 = \frac{S}{5} + 2 = \frac{S+10}{5}$ .

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2083 · Science Model Set E · 1 mark

In a continuous series, $N = 30$ and the cumulative frequency of the class 20-30 is 15. Find the median class. (a) 10-20 (b) 20-30 (c) 30-40 (d) 20.5-30.5

Similar questions: In a continuous series, sum f = 100, the (2079 · Science Model)

Answer

(b) 20-30

The median class is the class whose cumulative frequency first reaches $\frac{N}{2} = 15$ . The class 20-30 has cumulative frequency 15, so it is the median class.

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2080 · Science Model · 1 mark

If the sum of five consecutive integers is $S$ , what is the largest of those integers in terms of $S$ ? (a) $\frac{S-10}{5}$ (b) $\frac{S+4}{4}$ (c) $\frac{S+10}{5}$ (d) $\frac{S-5}{2}$

Similar questions: If the sum of five consecutive integers is S, (2083 · Science Model Set D)

Answer

(c) $\dfrac{S+10}{5}$

Let the integers be $n, n+1, \dots, n+4$ . Then $S = 5n + 10$ , so $n = \dfrac{S-10}{5}$ .

Largest $= n + 4 = \dfrac{S-10}{5} + 4 = \dfrac{S+10}{5}$ .

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2080 · Science Model · 1 mark

If $a = 16b$ and $b$ is a prime number greater than 2, how many positive distinct factors does $a$ have? (a) 5 (b) 20 (c) 15 (d) 2

Similar questions: If a = 16b and b is a prime number greater than (2083 · Science Model Set C)

Answer

The number of positive factors is $10$ , which is not among the options.

$a = 16b = 2^4 \cdot b^1$ with $b$ an odd prime, so the number of factors is $(4+1)(1+1) = 10$ . (Option (c), $15$ , would correspond to $a = 16b^2 = 2^4 \cdot b^2$ , giving $5 \times 3 = 15$ .)

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2079 · Science Model · 1 mark

The compound interest compounded yearly is calculated by (a) $P\left[\left(1+\frac{R}{100}\right)^T - 1\right]$ (b) $P\left[\left(1+\frac{R}{200}\right)^{2T} - 1\right]$ (c) $P\left[\left(1+\frac{R}{100}\right)^{2T} - 1\right]$ (d) $P\left[\left(1+\frac{R}{200}\right)^{T} - 1\right]$

Similar questions: The compound interest compounded half yearly is (2083 · Science Model Set A)

Answer

(a) $P\left[\left(1+\frac{R}{100}\right)^T - 1\right]$

The amount compounded yearly is $A=P\left(1+\frac{R}{100}\right)^T$ , so the compound interest is $A-P=P\left[\left(1+\frac{R}{100}\right)^T-1\right]$ . Options (b) and (d) are for half-yearly compounding.

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2079 · Science Model · 1 mark

If $\frac{a}{b} > \frac{c}{d}$ , which of the following statements must be true? (a) $\frac{a}{b} - \frac{c}{d} > 0$ (b) $ad < bc$ (c) $ad > bc$ (d) All of above

Similar questions: Which of the following statements is true (2083 · Science Model Set A)

Answer

(a) $\frac{a}{b}-\frac{c}{d}>0$

Subtracting $\frac{c}{d}$ from both sides of $\frac{a}{b}>\frac{c}{d}$ always gives $\frac{a}{b}-\frac{c}{d}>0$ . The statements $ad<bc$ and $ad>bc$ depend on the signs of $b$ and $d$ (multiplying by $bd$ flips the inequality when $bd<0$ ), so neither must be true.

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2079 · Science Model · 1 mark

In a continuous series, $\sum f = 100$ , the cumulative frequency of the class 15-20 is 46 and the cumulative frequency of the class 20-25 is 52. Find the median class of the data. (a) 15-20 (b) 10-15 (c) 20-25 (d) 25-30

Similar questions: In a continuous series, N = 30 and the cumulative (2083 · Science Model Set E)

Answer

(c) 20-25

$\frac{N}{2}=\frac{100}{2}=50$ . The first class whose cumulative frequency is at least 50 is the median class. The class 15-20 has cumulative frequency 46, which is less than 50, and the class 20-25 has 52, which is at least 50. So the median class is 20-25.

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2083 · Science Model Set A · 1 mark

What is the average (arithmetic mean) of the measures of the five angles in a pentagon? (a) $36^\circ$ (b) $72^\circ$ (c) $92^\circ$ (d) $100^\circ$

Answer

(b) $72^\circ$

The interior angles of a pentagon sum to $(5-2)\times180^\circ=540^\circ$ , giving a mean of $108^\circ$ , which is not among the options. The mean of the five exterior angles, which sum to $360^\circ$ , is $\frac{360^\circ}{5}=72^\circ,$ so $72^\circ$ is the intended choice.

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2083 · Science Model Set A · 1 mark

Find the LCM of $a^2b$ and $a^2b^2$ . (a) $ab$ (b) $a^2b$ (c) $a^2b^2$ (d) $a^4b^3$

Answer

(c) $a^2b^2$

Take the highest power of each factor: $a^2$ and $b^2$ , so the LCM is $a^2b^2$ .

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2083 · Science Model Set A · 1 mark

What is the square root of $9 + 4\sqrt{5}$ ? (a) $3 + 2\sqrt{5}$ (b) $2 + \sqrt{5}$ (c) $3 + 4\sqrt{5}$ (d) None of above

Answer

(b) $2+\sqrt{5}$

$(2+\sqrt5)^2=4+5+4\sqrt5=9+4\sqrt5.$

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2083 · Science Model Set A · 1 mark

If $a + b = 3(c + d)$ , which of the following is the average (arithmetic mean) of $a, b, c$ and $d$ ? (a) $\frac{c+d}{4}$ (b) $\frac{3(c+d)}{4}$ (c) $\frac{c+d}{2}$ (d) $c + d$

Answer

(d) $c+d$

$a+b+c+d=3(c+d)+(c+d)=4(c+d),$ so the mean is $\dfrac{4(c+d)}{4}=c+d$ .

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2083 · Science Model Set A · 1 mark

If the mean of $a$ and $\frac{1}{a}$ is $M$ , then what is the mean of $a^3$ and $\frac{1}{a^3}$ ? (a) $\frac{M(M^2-3)}{3}$ (b) $M^3$ (c) $M^3 - 3$ (d) $M(4M^2 - 3)$

Answer

(d) $M(4M^2-3)$

Here $a+\frac1a=2M$ . Then $a^3+\frac1{a^3}=\left(a+\frac1a\right)^3-3\left(a+\frac1a\right)=8M^3-6M.$ The mean is $\dfrac{8M^3-6M}{2}=4M^3-3M=M(4M^2-3)$ .

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2083 · Science Model Set A · 1 mark

Which of the following sets are equivalent? (a) $\phi$ and $\{\phi\}$ (b) $\{0\}$ and $\{\phi\}$ (c) $\phi$ and $\{0\}$ (d) None

Answer

(b) $\{0\}$ and $\{\phi\}$

Equivalent sets have the same number of elements. $\{0\}$ and $\{\phi\}$ each have exactly one element, whereas $\phi$ has none.

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2083 · Science Model Set A · 1 mark

Find the compound interest on Rs 32000 at 20% p.a. for 1 year compounded half yearly. (a) Rs. 6320 (b) Rs. 6720 (c) Rs. 6400 (d) Rs. 6520

Answer

(b) Rs. 6720

$A=32000\left(1+\frac{20}{200}\right)^{2}=32000\times1.21=\text{Rs }38720$ $CI=38720-32000=\text{Rs }6720.$

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2083 · Science Model Set A · 1 mark

What single discount is equivalent to two successive discounts of 10% and 15%? (a) 23% (b) 23.5% (c) 24% (d) 24.5%

Answer

(b) 23.5%

$1-0.90\times0.85=1-0.765=0.235,$ which is a discount of 23.5%.

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2083 · Science Model Set A · 1 mark

A sector of a circle has a radius of 8 and an area of $8\pi$ . What is the arc length of the sector? (a) $\pi$ (b) $2\pi$ (c) $4\pi$ (d) $6\pi$

Answer

(b) $2\pi$

Area $=\tfrac12 r\ell$ , so $8\pi=\tfrac12(8)\ell$ , which gives $\ell=2\pi$ .

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2083 · Science Model Set A · 1 mark

The HCF of $x^3 + \frac{1}{x^3}$ and $x^4 + 1 + \frac{1}{x^4}$ is (a) $1$ (b) $x + \frac{1}{x}$ (c) $x^2 + 1 + \frac{1}{x^2}$ (d) $x^2 - 1 + \frac{1}{x^2}$

Answer

(d) $x^2-1+\frac{1}{x^2}$

$x^3+\frac1{x^3}=\left(x+\frac1x\right)\left(x^2-1+\frac1{x^2}\right)$ $x^4+1+\frac1{x^4}=\left(x^2+\frac1{x^2}\right)^2-1=\left(x^2-1+\frac1{x^2}\right)\left(x^2+1+\frac1{x^2}\right)$ The common factor is $x^2-1+\frac1{x^2}$ .

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2083 · Science Model Set A · 1 mark

For what value of $x$ is $\frac{4}{3x-6}$ undefined? (a) 0 (b) 1 (c) 2 (d) 3

Answer

(c) 2

The fraction is undefined when the denominator is zero: $3x-6=0$ , so $x=2$ .

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2083 · Science Model Set A · 1 mark

$0.\overline{6} + 0.\overline{7} + 0.\overline{8} = ?$ (a) $2.\overline{3}$ (b) $2.\overline{1}$ (c) $2.\overline{2}$ (d) None

Answer

(a) $2.\overline{3}$

$0.\overline6=\frac69,\quad 0.\overline7=\frac79,\quad 0.\overline8=\frac89$ $\text{Sum}=\frac{21}{9}=\frac73=2.\overline3.$

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2083 · Science Model Set A · 1 mark

Solve: $\sqrt[4]{2x-1} = -1$ (a) 1 (b) 2 (c) 3 (d) No solution

Answer

(d) No solution

The principal fourth root of a real number is never negative, so $\sqrt[4]{2x-1}=-1$ has no real solution.

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2083 · Science Model Set A · 1 mark

A has 6 more marbles than B. If B gives 2 marbles to C and then B buys 5 new marbles, how many more marbles does A have than B? (a) 3 (b) 4 (c) 5 (d) 6

Answer

(a) 3

Let B have $b$ marbles, so A has $b+6$ . After giving 2 to C and buying 5, B has $b-2+5=b+3$ .

$(b+6)-(b+3)=3$

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2083 · Science Model Set A · 1 mark

The quadratic equation whose roots are $(3+\sqrt{5})$ and $(3-\sqrt{5})$ is (a) $x^2 - 6x + 2 = 0$ (b) $x^2 - 4x + 6 = 0$ (c) $x^2 - 6x + 4 = 0$ (d) $x^2 + 6x + 4 = 0$

Answer

(c) $x^2 - 6x + 4 = 0$

Sum of roots $=(3+\sqrt5)+(3-\sqrt5)=6$ and product $=9-5=4$ , so the equation is $x^2-6x+4=0$ .

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2083 · Science Model Set A · 1 mark

If $x = 6 + \sqrt{35}$ , then the value of $x^2 + \frac{1}{x^2}$ is (a) 241 (b) 284 (c) 142 (d) 140

Answer

(c) 142

Since $(6+\sqrt{35})(6-\sqrt{35})=1$ , we have $\frac1x=6-\sqrt{35}$ and $x+\frac1x=12$ . Then $x^2+\frac1{x^2}=\left(x+\frac1x\right)^2-2=144-2=142.$

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2083 · Science Model Set A · 1 mark

If $xy > 0$ and $yz < 0$ , which of the following is true? (a) $xz < 0$ (b) $xz > 0$ (c) $xz = 0$ (d) None of above

Answer

(a) $xz < 0$

$xy>0$ means $x$ and $y$ have the same sign; $yz<0$ means $z$ has the opposite sign to $y$ . So $z$ is opposite to $x$ , giving $xz<0$ .

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2083 · Science Model Set A · 1 mark

For what value of $x$ is the equation $ab^{x-2} = ba^{x-2}$ true? (a) $-3$ (b) $-2$ (c) $2$ (d) $3$

Answer

(d) $3$

For $x=3$ : $ab^{1}=ba^{1}$ , i.e. $ab=ba$ , which is always true. (For other options the two sides are not equal in general.)

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2083 · Science Model Set A · 1 mark

If $x^{\frac{1}{a}} = y^{\frac{1}{5}}$ and $xy = 1$ , what is the value of $a$ ? (a) $-2$ (b) $-3$ (c) $-4$ (d) $-5$

Answer

(d) $-5$

Since $xy=1$ , $y=x^{-1}$ , so $y^{1/5}=x^{-1/5}$ . Then $x^{1/a}=x^{-1/5}$ gives $\frac1a=-\frac15$ , so $a=-5$ .

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2083 · Science Model Set A · 1 mark

There are two numbers. If we add one to each number, their ratio becomes 2:3. If 1 is decreased from each number, their ratio becomes 1:2. The numbers are (a) 3, 1 (b) 1, 3 (c) 1, 5 (d) none of above

Answer

(d) none of above

Let the numbers be $p$ and $q$ . Then $\frac{p+1}{q+1}=\frac23$ gives $3p-2q=-1$ , and $\frac{p-1}{q-1}=\frac12$ gives $q=2p-1$ . Substituting: $3p-4p+2=-1$ , so $p=3$ , $q=5$ . The numbers are 3 and 5, which are not among options (a)-(c).

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2083 · Science Model Set A · 1 mark

A perfect cube has surface area $96\ \text{cm}^2$ . What is its volume (in $\text{cm}^3$ )? (a) 64 (b) 80 (c) 48 (d) 96

Answer

(a) 64

$6a^2=96\Rightarrow a^2=16\Rightarrow a=4$ cm, so volume $=4^3=64\ \text{cm}^3$ .

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2083 · Science Model Set A · 1 mark

In triangle ABC, $\angle B = 30^\circ$ and $BC = 12$ cm. If the area of $\triangle ABC$ is $27\ \text{cm}^2$ , find the length of AB. (a) 12 cm (b) 6 cm (c) 9 cm (d) None of above

Answer

(c) 9 cm

$\text{Area}=\tfrac12\cdot AB\cdot BC\sin 30^\circ=\tfrac12\cdot AB\cdot 12\cdot\tfrac12=3\,AB=27$ so $AB=9$ cm.

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2083 · Science Model Set A · 1 mark

The three distinct points A, B and C are said to be collinear if (a) $AC = AB + BC$ (b) slope of $AB$ = slope of $BC$ (c) Area of $\triangle ABC = 0$ (d) All of above

Answer

(d) All of above

Three distinct points are collinear if $AC=AB+BC$ (B lies on segment AC), if slope of $AB$ equals slope of $BC$ , or if the area of $\triangle ABC$ is zero.

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2083 · Science Model Set A · 1 mark

What is the maximum value of $y = \sin x$ ? (a) $-1$ (b) $1$ (c) $0$ (d) $\infty$

Answer

(b) $1$

The range of $\sin x$ is $[-1,1]$ , so its maximum value is $1$ (at $x=90^\circ$ ).

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2083 · Science Model Set A · 1 mark

A right triangle is inscribed in a circle with an area of $16\pi\ \text{cm}^2$ . What is the hypotenuse of the triangle in cm? (a) $4\pi$ (b) $4$ (c) $8\pi$ (d) $8$

Answer

(d) $8$

$\pi r^2=16\pi\Rightarrow r=4$ cm. The hypotenuse of a right triangle inscribed in a circle is a diameter, so it equals $2r=8$ cm.

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2083 · Science Model Set A · 1 mark

If $f(x)$ is divided by $(b - ax)$ , then the remainder is (a) $f\left(\frac{b}{a}\right)$ (b) $f\left(\frac{a}{b}\right)$ (c) $f\left(-\frac{b}{a}\right)$ (d) $f\left(-\frac{a}{b}\right)$

Answer

(a) $f\left(\frac{b}{a}\right)$

By the remainder theorem, the remainder on division by $(b-ax)$ is $f(x)$ at the zero of the divisor: $b-ax=0\Rightarrow x=\frac ba$ .

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2083 · Science Model Set A · 1 mark

If the sum of the first $n$ positive integers is $\frac{1}{5}$ of the sum of the squares of the first $n$ positive integers, then $n$ is (a) 23 (b) 14 (c) 7 (d) 9

Answer

(c) 7

$\frac{n(n+1)}{2}=\frac15\cdot\frac{n(n+1)(2n+1)}{6}\Rightarrow \frac12=\frac{2n+1}{30}\Rightarrow 2n+1=15\Rightarrow n=7.$

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2083 · Science Model Set A · 1 mark

Which of the following parabolas is the narrowest? (a) $y = x^2$ (b) $y = \frac{x^2}{2}$ (c) $y = 2x^2$ (d) $y = \frac{x^2}{4}$

Answer

(c) $y = 2x^2$

For $y=kx^2$ , a larger $k$ gives a narrower parabola. Here $k=2$ is the largest among $1,\frac12,2,\frac14$ .

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2083 · Science Model Set A · 1 mark

What does $\{x : 1 \le x < 5\}$ represent? (a) $(1, 5)$ (b) $[1, 5]$ (c) $[1, 5)$ (d) $(1, 5]$

Answer

(c) $[1, 5)$

$1\le x$ includes 1 (closed end) and $x<5$ excludes 5 (open end).

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2083 · Science Model Set A · 1 mark

If $A = \begin{pmatrix} 1 & 4 & 0 \\ 2 & 5 & 0 \\ 3 & 6 & 0 \end{pmatrix}$ , then the determinant of $A$ is (a) $-1$ (b) $0$ (c) $1$ (d) None of above

Answer

(b) $0$

The third column of $A$ is entirely zero, so $|A|=0$ .

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2083 · Science Model Set A · 1 mark

Two circles having centres $O$ and $O'$ touch externally if (a) $OO'$ = difference of radii (b) $OO'$ = sum of radii (c) $OO'$ > sum of radii (d) $OO'$ < sum of radii

Answer

(b) $OO'$ = sum of radii

Circles touch externally when the distance between centres equals the sum of their radii.

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2083 · Science Model Set A · 1 mark

The minute hand of a clock is 28 cm long. How far does the tip of the minute hand move during 15 minutes? (a) 28 cm (b) 14 cm (c) 44 cm (d) 22 cm

Answer

(c) 44 cm

In 15 minutes the hand turns through $90^\circ$ , a quarter circle, so the distance is $\frac14\cdot 2\pi r=\frac14\cdot 2\cdot\frac{22}{7}\cdot 28=44\ \text{cm}.$

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2083 · Science Model Set A · 1 mark

If the sum of the position vectors of the vertices of a triangle is $6\vec{i} + 9\vec{j}$ , then find the position vector of its centroid. (a) $6\vec{i} + 9\vec{j}$ (b) $3\vec{i} + 2\vec{j}$ (c) $2\vec{i} + 3\vec{j}$ (d) None of above

Answer

(c) $2\vec{i} + 3\vec{j}$

Centroid $=\frac13(\text{sum of position vectors})=\frac13(6\vec i+9\vec j)=2\vec i+3\vec j$ .

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2083 · Science Model Set A · 1 mark

What is the combined transformation of the reflection on $x = -2$ followed by the reflection on $y = 3$ ? (a) Translation with $\binom{10}{2}$ (b) $R[(0,0), 180^\circ]$ (c) $R[(-2,3), 180^\circ]$ (d) Reflection on $y = -x$

Answer

(c) $R[(-2,3), 180^\circ]$

The lines $x=-2$ and $y=3$ are perpendicular and meet at $(-2,3)$ . Reflections in two perpendicular lines combine to a half-turn about their point of intersection.

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2083 · Science Model Set B · 1 mark

If mean $= [3\,\text{median} - \text{mode}] \cdot y$ , then the value of $y$ is (a) 2 (b) $\frac{1}{4}$ (c) $\frac{1}{2}$ (d) 4

Answer

(c) $\frac{1}{2}$

The empirical relation is $\text{mode} = 3\,\text{median} - 2\,\text{mean}$ , so $\text{mean} = \frac{3\,\text{median} - \text{mode}}{2} = (3\,\text{median} - \text{mode})\cdot\frac{1}{2},$ giving $y = \frac{1}{2}$ .

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2083 · Science Model Set B · 1 mark

Subi's average (arithmetic mean) on 4 tests is 80. What does she need on her fifth test to raise her average to 84? (a) 82 (b) 84 (c) 92 (d) 100

Answer

(d) 100

The total of 4 tests is $4 \times 80 = 320$ . For an average of 84 over 5 tests the total must be $5 \times 84 = 420$ , so the fifth score is $420 - 320 = 100$ .

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2083 · Science Model Set B · 1 mark

The number of all possible subsets of $\{\{a\}, \{b, c\}, \{a, b, c\}\}$ is (a) 4 (b) 6 (c) 8 (d) 64

Answer

(c) 8

The set $\{\{a\},\{b,c\},\{a,b,c\}\}$ has 3 distinct elements, so it has $2^3 = 8$ subsets.

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2083 · Science Model Set B · 1 mark

If Rs 1800 = USD 25 and Rs 1584 = £12, how many pounds (£) can be exchanged from USD 33? (a) £18 (b) £25 (c) £33 (d) None

Answer

(a) £18

Rs 1800 = USD 25, so USD 1 = Rs 72 and USD 33 = Rs $33 \times 72 =$ Rs 2376. Rs 1584 = £12, so £1 = Rs 132. $\frac{2376}{132} = 18 \Rightarrow £18.$

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2083 · Science Model Set B · 1 mark

A sum of money in simple interest doubles itself in 6 years. In how many years will it become 5 times? (a) 12 years (b) 24 years (c) 10 years (d) 15 years

Answer

(b) 24 years

The sum doubles in 6 years, so the interest in 6 years equals the principal $P$ , which is $\frac{P}{6}$ per year. To become 5 times, the interest must be $4P$ , which takes $4P \div \frac{P}{6} = 24$ years.

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2083 · Science Model Set B · 1 mark

At what rate of compound interest will a sum of money become $\frac{125}{64}$ times of itself in 3 years? (a) 10% (b) 15% (c) 20% (d) 25%

Answer

(d) 25%

$\left(1+\frac{R}{100}\right)^3 = \frac{125}{64} = \left(\frac{5}{4}\right)^3 \Rightarrow 1+\frac{R}{100} = \frac{5}{4}$ so $R = 25\%$ .

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2083 · Science Model Set B · 1 mark

The amount of money returned back to the tourists while leaving the country is called (a) VAT amount (b) Discount amount (c) Profit amount (d) Loss amount

Answer

(a) VAT amount

Tourists are refunded the VAT they paid on goods when leaving the country.

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2083 · Science Model Set B · 1 mark

What is the length of a diagonal of a cube whose edges are 1? (a) 1 (b) 2 (c) $\sqrt{3}$ (d) $\sqrt{2}$

Answer

(c) $\sqrt{3}$

The space diagonal of a cube of edge $a$ is $\sqrt{a^2+a^2+a^2} = a\sqrt{3}$ , which is $\sqrt{3}$ for $a=1$ .

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2083 · Science Model Set B · 1 mark

What is the volume of a right circular cylinder with a radius of 2 and a height of 4? (a) $8\pi$ (b) $12\pi$ (c) $16\pi$ (d) $32\pi$

Answer

(c) $16\pi$

$V = \pi r^2 h = \pi \times 2^2 \times 4 = 16\pi.$

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2083 · Science Model Set B · 1 mark

The product of two expressions is $x(y+z)^2(y-z)$ and their HCF is $(y+z)$ . What is their LCM? (a) $(y^2 - z^2)$ (b) $x(y^2 - z^2)$ (c) $(y+z)^3$ (d) None

Answer

(b) $x(y^2 - z^2)$

Since $\text{HCF} \times \text{LCM} =$ product of the expressions, $\text{LCM} = \frac{x(y+z)^2(y-z)}{y+z} = x(y+z)(y-z) = x(y^2-z^2).$

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2083 · Science Model Set B · 1 mark

$0.\overline{47} = ?$ (a) $\frac{47}{99}$ (b) $\frac{47}{90}$ (c) $\frac{46}{90}$ (d) $\frac{47}{100}$

Answer

(a) $\frac{47}{99}$

Let $x = 0.\overline{47}$ . Then $100x = 47.\overline{47}$ , so $99x = 47$ and $x = \frac{47}{99}$ .

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2083 · Science Model Set B · 1 mark

What value of $y$ makes the expression $\frac{x^2y}{2x-y}$ undefined? (a) 0 (b) 1 (c) $x$ (d) $2x$

Answer

(d) $2x$

The expression is undefined when the denominator is zero: $2x - y = 0 \Rightarrow y = 2x$ .

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2083 · Science Model Set B · 1 mark

The rationalizing factor of $\sqrt[5]{16}$ is (a) $\sqrt{2}$ (b) $\sqrt[5]{16}$ (c) $\sqrt[5]{2}$ (d) $\sqrt{16}$

Answer

(c) $\sqrt[5]{2}$

$\sqrt[5]{16} = 2^{4/5}$ . Multiplying by $2^{1/5} = \sqrt[5]{2}$ gives $2^{4/5+1/5} = 2$ , which is rational.

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2083 · Science Model Set B · 1 mark

If $x^y = 64$ and $x$ and $y$ are positive integers, which of the following could be the value of $x + y$ ? (a) 2 (b) 6 (c) 7 (d) 12

Answer

(c) 7

The positive integer solutions of $x^y = 64$ are $(2,6)$ , $(4,3)$ , $(8,2)$ and $(64,1)$ , giving $x+y = 8, 7, 10, 65$ . Only 7 is among the options, from $4^3 = 64$ .

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2083 · Science Model Set B · 1 mark

If $x$ and $y$ are integers such that $x > y > 1$ , which of the following cannot be a multiple of either $x$ or $y$ ? (a) $x - 1$ (b) $y + 1$ (c) $y - 1$ (d) $x + y$

Answer

(c) $y - 1$

Since $y>1$ , we have $0 < y-1 < y < x$ . A positive number smaller than $y$ cannot be a multiple of $y$ , and one smaller than $x$ cannot be a multiple of $x$ . The others can be multiples: $x-1=2$ for $(x,y)=(3,2)$ , $y+1=3$ for $(3,2)$ , $x+y=6$ for $(4,2)$ .

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2083 · Science Model Set B · 1 mark

If $x + \frac{1}{x} = 15$ , the value of $x^2 + \frac{1}{x^2}$ is (a) 227 (b) 225 (c) 240 (d) 223

Answer

(d) 223

Square both sides: $\left(x+\frac1x\right)^2 = x^2+2+\frac1{x^2} = 225$ , so $x^2+\frac{1}{x^2}=225-2=223$ .

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2083 · Science Model Set B · 1 mark

Which of the following is equal to $x^{\frac{3}{2}}$ ? (a) $x^2\sqrt{x}$ (b) $x\sqrt{x}$ (c) $\sqrt[3]{x^2}$ (d) $(x^3)^2$

Answer

(b) $x\sqrt{x}$

$x^{\frac32}=x^{1+\frac12}=x\cdot x^{\frac12}=x\sqrt{x}$ .

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2083 · Science Model Set B · 1 mark

If $x = 2^{\frac{1}{3}} + 2^{\frac{2}{3}}$ , what is the value of $x^3 - 6x$ ? (a) 2 (b) 4 (c) 6 (d) 18

Answer

(c) 6

Let $a=2^{\frac13}$ , so $a^3=2$ and $x=a+a^2$ . Then $x^3=a^3+a^6+3a^3(a+a^2)=2+4+6x=6+6x.$ Hence $x^3-6x=6$ .

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2083 · Science Model Set B · 1 mark

If $a$ and $b$ are the lengths of the legs of a right-angled triangle whose hypotenuse is 10 and whose area is 20, what is the value of $(a+b)^2$ ? (a) 100 (b) 120 (c) 140 (d) 180

Answer

(d) 180

$a^2+b^2=10^2=100$ and $\frac12ab=20\Rightarrow ab=40$ . So $(a+b)^2=a^2+b^2+2ab=100+80=180$ .

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2083 · Science Model Set B · 1 mark

In a right-angled triangle with sides 5 and 7 meeting at the right angle and hypotenuse $\sqrt{74}$ , what is the value of $x$ , the angle between the side 7 and the hypotenuse? (a) $30^\circ$ (b) $60^\circ$ (c) $45^\circ$ (d) $90^\circ$

Answer

(a) $30^\circ$ (the nearest option)

The side adjacent to $x$ is 7 and the side opposite is 5, so $\tan x=\frac57$ and $x=\tan^{-1}\frac57\approx 35.5^\circ$ . This is not exactly any of the options; the nearest is $30^\circ$ .

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2083 · Science Model Set B · 1 mark

A circle is inscribed in a square. The area of the circle is $25\pi$ . The area of the square is (a) 625 (b) 100 (c) 50 (d) 2500

Answer

(b) 100

$\pi r^2=25\pi\Rightarrow r=5$ . The side of the square equals the diameter $=10$ , so the area is $10^2=100$ .

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2083 · Science Model Set B · 1 mark

What is the area of a regular hexagon with side length 2? (a) $2\sqrt{3}$ (b) $2\sqrt{6}$ (c) $6\sqrt{2}$ (d) $6\sqrt{3}$

Answer

(d) $6\sqrt{3}$

A regular hexagon is made of 6 equilateral triangles: $A=6\cdot\frac{\sqrt3}{4}(2)^2=6\sqrt3.$

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2083 · Science Model Set B · 1 mark

A circle has an area of $4\pi$ . If the radius was doubled, the new area of the circle would be how many times the original area? (a) 2 (b) 3 (c) 4 (d) 5

Answer

(c) 4

Area $\propto r^2$ . Doubling the radius gives $\pi(2r)^2=4\pi r^2$ , which is 4 times the original (new area $=16\pi$ , original $=4\pi$ ).

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2083 · Science Model Set B · 1 mark

The probability of getting a bad egg in a batch of 400 is 0.035. How many bad eggs are there likely to be in a batch? (a) 10 (b) 14 (c) 28 (d) 40

Answer

(b) 14

Expected number of bad eggs $=400\times0.035=14$ .

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2083 · Science Model Set B · 1 mark

What is the pre-image of 2 in the function $f(x) = \sqrt[3]{x}$ ? (a) 2 (b) 8 (c) 6 (d) 1

Answer

(b) 8

We need $x$ with $f(x)=\sqrt[3]{x}=2$ . Cubing, $x=2^3=8$ .

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2083 · Science Model Set B · 1 mark

If polynomial $q(x)$ is divided by $(ax + b)$ , what will be its remainder? (a) $q\left(\frac{a}{b}\right)$ (b) $q\left(\frac{b}{a}\right)$ (c) $q\left(-\frac{a}{b}\right)$ (d) $q\left(-\frac{b}{a}\right)$

Answer

(d) $q\left(-\frac{b}{a}\right)$

By the remainder theorem, the remainder on dividing by $(ax+b)$ is the value of $q(x)$ at the root of $ax+b=0$ , i.e. $x=-\frac ba$ .

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2083 · Science Model Set B · 1 mark

The sum of two digit numbers which are divisible by 5 is (a) 950 (b) 1050 (c) 945 (d) 1200

Answer

(c) 945

The two-digit multiples of 5 are $10,15,\dots,95$ , an A.P. with $n=\frac{95-10}{5}+1=18$ terms. $S=\frac{18}{2}(10+95)=9\times105=945.$

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2083 · Science Model Set B · 1 mark

What is the range of the parabola $y = x^2 - 4x + 3$ ? (a) $[-1, -\infty)$ (b) $[-1, \infty)$ (c) $[-2, \infty)$ (d) $[-2, -\infty)$

Answer

(b) $[-1,\infty)$

$y=x^2-4x+3=(x-2)^2-1$ . The parabola opens upward with vertex $(2,-1)$ , so $y\ge -1$ .

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2083 · Science Model Set B · 1 mark

$\lim_{x \to 2} 7 = ?$ (a) 0 (b) 2 (c) 1 (d) 7

Answer

(d) 7

The limit of a constant function is that constant: $\lim_{x\to2}7=7$ .

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2083 · Science Model Set B · 1 mark

A square matrix $A = (a_{ij})$ is said to be a lower triangular matrix if the element $a_{ij} = 0$ for (a) $i \le j$ (b) $i > j$ (c) $i < j$ (d) None

Answer

(c) $i<j$

A lower triangular matrix has all entries above the main diagonal equal to zero, i.e. $a_{ij}=0$ for $i<j$ .

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2083 · Science Model Set B · 1 mark

If $c \neq 0$ and the slope of the line passing through $(-c, c)$ and $(3c, a)$ is 1, which of the following is an expression for $a$ in terms of $c$ ? (a) $3c$ (b) $\frac{1-a^2}{a}$ (c) $2c$ (d) $3c$

Answer

$a=5c$ , which is not among the listed options

$\text{slope}=\frac{a-c}{3c-(-c)}=\frac{a-c}{4c}=1\Rightarrow a-c=4c\Rightarrow a=5c.$

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2083 · Science Model Set B · 1 mark

$\sqrt{2 + \sqrt{2 + \sqrt{2 + 2\cos 8\theta}}} = ?$ (a) $2\sec\theta$ (b) $2\cos\theta$ (c) $2\tan\theta$ (d) None

Answer

(b) $2\cos\theta$

Using $2+2\cos2\alpha=4\cos^2\alpha$ :

$\sqrt{2+2\cos8\theta}=\sqrt{4\cos^24\theta}=2\cos4\theta$
$\sqrt{2+2\cos4\theta}=2\cos2\theta$
$\sqrt{2+2\cos2\theta}=2\cos\theta$

(for $\theta$ with the cosines positive).

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2083 · Science Model Set B · 1 mark

If $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$ , which of the following is true? (a) $\vec{a}\cdot\vec{b} = 0$ (b) $\vec{a} = \vec{b}$ (c) $\vec{a} = k\vec{b}$ (d) $\vec{a} = -\vec{b}$

Answer

(a) $\vec a\cdot\vec b=0$

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2083 · Science Model Set B · 1 mark

If $r_1$ is the reflection about the line $x = y$ and $r_2$ is the reflection about the $y$ -axis, what does $r_1 r_2$ represent? (a) $R[O, +90^\circ]$ (b) $R[O, -90^\circ]$ (c) $R[O, -180^\circ]$ (d) $R[O, +180^\circ]$

Answer

(b) $R[O,-90^\circ]$

$r_2:(x,y)\to(-x,y)$ and $r_1:(x,y)\to(y,x)$ . Applying $r_2$ first and then $r_1$ : $(x,y)\to(-x,y)\to(y,-x),$ which is a rotation about the origin through $-90^\circ$ (clockwise).

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2083 · Science Model Set C · 1 mark

If $x$ is an integer such that $0 \le x \le 20$ , for how many values of $x$ is $\frac{20}{x}$ an integer? (a) 4 (b) 6 (c) 8 (d) 10

Answer

(b) 6

$x=0$ is not allowed. The values of $x$ in $1\le x\le20$ that divide 20 are $1,2,4,5,10,20$ , giving 6 values.

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2083 · Science Model Set C · 1 mark

If $x + y = 8$ , $y + z = 11$ and $z + x = 5$ , what is the value of $x + y + z$ ? (a) 25 (b) 12 (c) 440 (d) 220

Answer

(b) 12

Adding: $2(x+y+z)=8+11+5=24$ , so $x+y+z=12$ .

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2083 · Science Model Set C · 1 mark

If $5000 = 2^a 5^b$ and $a$ and $b$ are integers, what is $a + b$ ? (a) 5 (b) 6 (c) 7 (d) 8

Answer

(c) 7

$5000=5\times1000=5\times2^3\times5^3=2^3\cdot5^4$ , so $a=3,\ b=4$ and $a+b=7$ .

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2083 · Science Model Set C · 1 mark

If $12^x$ is odd and $x$ is an integer, what is the value of $x^{12}$ ? (a) 0 (b) 1 (c) $-1$ (d) It cannot be determined

Answer

(a) 0

$12^x$ is odd only when $x=0$ (then $12^0=1$ ); for positive $x$ it is even. So $x^{12}=0^{12}=0$ .

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2083 · Science Model Set C · 1 mark

The sum of squares of two consecutive natural numbers is 841. The smaller number is (a) 20 (b) 21 (c) 19 (d) none

Answer

(a) 20

Let the numbers be $n$ and $n+1$ : $n^2+(n+1)^2=841\Rightarrow n^2+n-420=0\Rightarrow(n+21)(n-20)=0$ . So $n=20$ (check: $400+441=841$ ).

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2083 · Science Model Set C · 1 mark

What will be the ratio of $\frac{\pi^c}{3}$ and $10^\circ$ ? (a) 6 : 1 (b) 1 : 6 (c) 6 : 5 (d) 5 : 6

Answer

(a) 6 : 1

$\pi^c = 180^\circ$ , so $\frac{\pi^c}{3} = 60^\circ$ . The ratio is $60^\circ : 10^\circ = 6 : 1$ .

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2083 · Science Model Set C · 1 mark

In the data set $\{2, 5, 7, 8, x\}$ , the arithmetic mean is the same as the median. Determine the value of $x$ . (a) 8 (b) 10 (c) 13 (d) 15

Answer

(c) 13

Mean $= \frac{22+x}{5}$ . For $x = 13$ the ordered data is $2, 5, 7, 8, 13$ , so the median is $7$ and the mean is $\frac{35}{5} = 7$ . The two are equal.

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2083 · Science Model Set C · 1 mark

$(\overline{A} \cup \overline{B})$ is equal to (a) $\overline{A} \cap \overline{B}$ (b) $\overline{A \cap B}$ (c) $A^c$ (d) $\cup$

Answer

(b) $\overline{A \cap B}$

By De Morgan's law, $\overline{A} \cup \overline{B} = \overline{A \cap B}$ .

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2083 · Science Model Set C · 1 mark

What is the value of $n$ if $\sqrt[1/n]{a^5} = a^{10}$ ? (a) 1 (b) 2 (c) 5 (d) 10

Answer

(b) 2

$\sqrt[1/n]{a^5} = (a^5)^{n} = a^{5n}$ . Setting $a^{5n} = a^{10}$ gives $5n = 10$ , so $n = 2$ .

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2083 · Science Model Set C · 1 mark

If $\sqrt[4]{ax + b} = c$ , then the value of $x$ is (a) $c^4$ (b) $c^4 - b$ (c) $\frac{c^4 - b}{a}$ (d) $\frac{c^4 - b}{a^4}$

Answer

(c) $\frac{c^4 - b}{a}$

Raise both sides to the fourth power: $ax + b = c^4$ , so $x = \frac{c^4 - b}{a}$ .

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2083 · Science Model Set C · 1 mark

Which one of the following is the median? (a) $5^{th}$ decile (b) $50^{th}$ percentile (c) $2^{nd}$ quartile (d) All of the above

Answer

(d) All of the above

The median divides the data into two equal halves, so it is the 2nd quartile $Q_2$ , the 5th decile $D_5$ and the 50th percentile $P_{50}$ .

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2083 · Science Model Set C · 1 mark

The standard deviation of 25 numbers is 40. If each of the numbers is increased by 5, then the new standard deviation is (a) 40 (b) 45 (c) $\frac{45}{2}$ (d) None

Answer

(a) 40

Adding a constant to every observation shifts the mean by the same amount but does not change the spread, so the standard deviation stays $40$ .

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2083 · Science Model Set C · 1 mark

If a rectangle is rotated about its length, which solid is formed? (a) Cube (b) Cuboid (c) Cone (d) Sphere

Answer

None of the listed options is correct. Rotating a rectangle about one of its sides (its length) through $360^\circ$ forms a cylinder, which is not among (a) to (d).

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2083 · Science Model Set C · 1 mark

What is the perimeter of a semi-circle having its radius $r$ ? (a) $\pi r$ (b) $2\pi r$ (c) $\pi r + d$ (d) $2\pi r + d$

Answer

(c) $\pi r + d$

The perimeter is the curved arc plus the diameter: $\pi r + 2r = \pi r + d$ .

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2083 · Science Model Set C · 1 mark

What is the probability of getting a composite number when a die is rolled once? (a) $\frac{2}{3}$ (b) $\frac{1}{3}$ (c) 1 (d) $\frac{5}{6}$

Answer

(b) $\frac{1}{3}$

The composite numbers on a die are $4$ and $6$ . So $P = \frac{2}{6} = \frac{1}{3}$ .

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2083 · Science Model Set C · 1 mark

Find the probability of occurring the letter A or I while drawing a letter card from the cards of the word 'ASSOCIATIVE'. (a) $\frac{1}{11}$ (b) $\frac{2}{11}$ (c) $\frac{3}{11}$ (d) $\frac{4}{11}$

Answer

(d) $\frac{4}{11}$

ASSOCIATIVE has 11 letters, with A appearing twice and I appearing twice. Favourable cards $= 2 + 2 = 4$ , so $P = \frac{4}{11}$ .

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2083 · Science Model Set C · 1 mark

After 10% devaluation of Nepali currency, USD 2000 can be exchanged by Rs 231000. Find the Nepali rupees equivalent to USD 1 before devaluation. (a) Rs. 115.5 (b) Rs. 105 (c) Rs. 110 (d) Rs. 10.5

Answer

(b) Rs 105

After devaluation, $USD 1 = \frac{231000}{2000} = \text{Rs } 115.5$ . Devaluation of 10% raises the rupee price of a dollar by 10%, so before devaluation $USD 1 = \frac{115.5}{1.1} = \text{Rs } 105$ .

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2083 · Science Model Set C · 1 mark

Find the compound interest on Rs 12500 at 8% p.a. for 9 months compounded quarterly. (a) Rs 1020 (b) Rs 1428 (c) Rs 510 (d) None of above

Answer

(d) None of above

Nine months is 3 quarters, and the rate per quarter is $\frac{8}{4} = 2\%$ . $A = 12500\left(1.02\right)^3 = 12500 \times 1.061208 = \text{Rs } 13265.10$ $CI = 13265.10 - 12500 = \text{Rs } 765.10$ This matches none of the given options.

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2083 · Science Model Set C · 1 mark

What is the half of $\frac{x^2}{8}$ ? (a) $\frac{x}{4}$ (b) $\frac{x^2}{4}$ (c) $\frac{x^2}{16}$ (d) All of above

Answer

(c) $\frac{x^2}{16}$

$\frac{1}{2} \cdot \frac{x^2}{8} = \frac{x^2}{16}$ .

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2083 · Science Model Set C · 1 mark

If $\frac{a}{b+c} = x$ , $\frac{b}{c+a} = y$ and $\frac{c}{a+b} = z$ , then find the value of $\frac{1}{1+x} + \frac{1}{1+y} + \frac{1}{1+z}$ . (a) 0 (b) 1 (c) 2 (d) 3

Answer

(c) 2

$1 + x = \frac{a+b+c}{b+c}$ , so $\frac{1}{1+x} = \frac{b+c}{a+b+c}$ . Similarly $\frac{1}{1+y} = \frac{c+a}{a+b+c}$ and $\frac{1}{1+z} = \frac{a+b}{a+b+c}$ . The sum is $\frac{2(a+b+c)}{a+b+c} = 2$ .

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2083 · Science Model Set C · 1 mark

In right-angled $\triangle ABC$ (right angle at C), the area is 36. If side AC is twice as long as side BC, what is the length of AB? (a) 6 (b) 12 (c) $6\sqrt{5}$ (d) 18

Answer

(c) $6\sqrt{5}$

Let $BC = x$ , so $AC = 2x$ . Area $= \frac{1}{2}x \cdot 2x = x^2 = 36$ , so $x = 6$ and $AC = 12$ . $AB = \sqrt{6^2 + 12^2} = \sqrt{180} = 6\sqrt{5}.$

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2083 · Science Model Set C · 1 mark

The point of intersection of the angle bisectors of all three interior angles of a triangle is called as (a) centroid (b) orthocenter (c) incenter (d) Circum-center.

Answer

(c) incenter

The angle bisectors meet at the incenter, the centre of the inscribed circle.

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2083 · Science Model Set C · 1 mark

If the principal is one third the compound interest, how many times of compound interest is the compound amount? (a) 3 times (b) $\frac{1}{3}$ times (c) $\frac{2}{3}$ times (d) $\frac{4}{3}$ times

Answer

(d) $\frac{4}{3}$ times

Let $CI = c$ , so $P = \frac{c}{3}$ . The compound amount is $A = P + CI = \frac{c}{3} + c = \frac{4c}{3}$ , which is $\frac{4}{3}$ times the compound interest.

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2083 · Science Model Set C · 1 mark

If a line is perpendicular to the x-axis, its slope ( $m$ ) is (a) 0 (b) 1 (c) $\sqrt{3}$ (d) undefined

Answer

(d) undefined

A line perpendicular to the $x$ -axis is vertical, with inclination $90^\circ$ . Its slope is $\tan 90^\circ$ , which is undefined.

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2083 · Science Model Set C · 1 mark

If $\vec{a}\cdot\vec{b} = -|\vec{a}|\cdot|\vec{b}|$ , then the angle between $\vec{a}$ and $\vec{b}$ is (a) $0^\circ$ (b) $60^\circ$ (c) $180^\circ$ (d) $90^\circ$

Answer

(c) $180^\circ$

$\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta = -|\vec{a}||\vec{b}|$ gives $\cos\theta = -1$ , so $\theta = 180^\circ$ .

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2083 · Science Model Set C · 1 mark

If $q\csc\theta = p$ and $\theta$ is acute, then what is the value of $\left(\sqrt{p^2 - q^2}\right)\tan\theta$ ? (a) $p$ (b) $q$ (c) $pq$ (d) $\sqrt{p^2 + q^2}$

Answer

(b) $q$

$\sin\theta = \frac{q}{p}$ , so $\cos\theta = \frac{\sqrt{p^2-q^2}}{p}$ and $\sqrt{p^2-q^2} = p\cos\theta$ . Then $\sqrt{p^2-q^2}\,\tan\theta = p\cos\theta \cdot \frac{\sin\theta}{\cos\theta} = p\sin\theta = q.$

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2083 · Science Model Set C · 1 mark

If a positive integer $a$ is increased by 20%, decreased by 25% and then increased by 60%, the resulting number is what percent of $a$ ? (a) 65% (b) 80% (c) 144% (d) 150%

Answer

(c) 144%

Multiply the factors: $1.20 \times 0.75 \times 1.60 = 0.9 \times 1.6 = 1.44$ , so the result is $144\%$ of $a$ .

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2083 · Science Model Set C · 1 mark

The range of the function $y = \cos x$ is (a) $(-1, 1)$ (b) $[-1, 1)$ (c) $(-1, 1]$ (d) $[-1, 1]$

Answer

(d) $[-1, 1]$

$\cos x$ attains both $-1$ (at $x=\pi$ ) and $1$ (at $x=0$ ), so both end points are included.

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2083 · Science Model Set C · 1 mark

For what value of $m$ is $(x - 1)$ a factor of $x^3 - 3x^2 + m + 1$ ? (a) 0 (b) 1 (c) 2 (d) None

Answer

(b) 1

By the factor theorem, put $x=1$ : $1 - 3 + m + 1 = 0$ , so $m = 1$ .

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2083 · Science Model Set C · 1 mark

The corresponding equation of $2x - y \le 0$ is (a) Parallel to x-axis (b) Parallel to y-axis (c) Passing through $(0, 0)$ (d) All of above

Answer

(c) Passing through $(0, 0)$

The corresponding equation is $2x - y = 0$ , i.e. $y = 2x$ . It is neither parallel to an axis (it has slope 2), and it is satisfied by $(0,0)$ .

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2083 · Science Model Set C · 1 mark

Write the point where the function $f(x) = \frac{1}{x+1}$ is discontinuous. (a) $x = 1$ (b) $x = -1$ (c) $x = 0$ (d) $x = c$

Answer

(b) $x = -1$

$f(x)=\frac{1}{x+1}$ is undefined when the denominator $x+1=0$ , i.e. at $x=-1$ .

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2083 · Science Model Set C · 1 mark

A square matrix $A = (a_{ij})$ is said to be upper triangular matrix if the element $a_{ij} = 0$ for (a) $i \le j$ (b) $i > j$ (c) $i < j$ (d) $i \ge j$

Answer

(b) $i > j$

In an upper triangular matrix all entries below the main diagonal are zero, i.e. $a_{ij}=0$ whenever the row index exceeds the column index.

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2083 · Science Model Set D · 1 mark

If $x - \frac{1}{x} = 10$ , find the value of $x + \frac{1}{x}$ . (a) 104 (b) $2\sqrt{26}$ (c) 100 (d) $\sqrt{26}$

Answer

(b) $2\sqrt{26}$

Use $\left(x+\frac1x\right)^2=\left(x-\frac1x\right)^2+4$ :

$\left(x+\frac1x\right)^2=10^2+4=104$

$x+\frac1x=\sqrt{104}=2\sqrt{26}$

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2083 · Science Model Set D · 1 mark

Find the sum of money which increases by $\frac{1}{10}$ of itself every year and amounts to Rs. 600 in 5 years. (a) Rs 380 (b) Rs. 280 (c) Rs. 480 (d) Rs. 400

Answer

(d) Rs 400

The increase is $\frac1{10}$ of the original sum each year, so it grows by simple interest at 10% per year. For 5 years the amount is $P\left(1+\frac{5}{10}\right)=1.5P$ .

$1.5P=600 \Rightarrow P=\frac{600}{1.5}=400$

So the sum is Rs 400.

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2083 · Science Model Set D · 1 mark

A man bought 5 oranges for a rupee. How many for a rupee should he sell to gain 25%? (a) 4 (b) 6 (c) 3 (d) 9

Answer

(a) 4

Cost price of 5 oranges is Rs 1. To gain 25%, the selling price of those 5 oranges must be $1 \times 1.25 = \text{Rs } 1.25$ . So for Rs 1 he should sell $5 \times \frac{1}{1.25} = 4$ oranges.

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2083 · Science Model Set D · 1 mark

If $4A = 5B$ and $6B = 7C$ , find $A:B:C$ . (a) 28:30:35 (b) 35:30:28 (c) 35:28:24 (d) None

Answer

(c) 35:28:24

From $4A = 5B$ , $A = \frac{5B}{4}$ ; from $6B = 7C$ , $C = \frac{6B}{7}$ . So $A:B:C = \frac{5}{4}:1:\frac{6}{7}.$ Multiplying by the LCM 28 gives $35:28:24$ .

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2083 · Science Model Set D · 1 mark

Let $U$ be the set of all triangles in a plane and let $A$ be the set of all triangles with at least one angle different from $60^\circ$ . Then $A^c$ is (a) a set of all right triangles (b) a set of all isosceles triangles (c) a set of all equilateral triangles (d) None of these

Answer

(c) a set of all equilateral triangles

$A$ contains every triangle that is not having all angles equal to $60^\circ$ . Its complement $A^c$ is the set of triangles with all angles $60^\circ$ , i.e. equilateral triangles.

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2083 · Science Model Set D · 1 mark

L.C.M. of two numbers is 16 times the H.C.F. The sum of LCM and HCF is 850. If one number is 50, then what is the other number? (a) 800 (b) 1200 (c) 1600 (d) 2400

Answer

(a) 800

Let HCF $= h$ , so LCM $= 16h$ and $17h = 850$ , giving $h = 50$ and LCM $= 800$ . Since (first number) $\times$ (second number) $=$ HCF $\times$ LCM, $50 \times N = 50 \times 800 \Rightarrow N = 800.$

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2083 · Science Model Set D · 1 mark

A colourless cube is painted blue and then cut parallel to sides to form two rectangular solids of equal volume. What percentage of the surface area of each of the new solids is not painted blue? (a) 25 (b) 16 (c) 20 (d) 18

Answer

(a) 25

Let the cube have side $a$ . Each half is an $a \times a \times \frac{a}{2}$ solid with surface area $2\left(a^2 + \frac{a^2}{2} + \frac{a^2}{2}\right) = 4a^2$ . The only unpainted face is the new cut face of area $a^2$ , so the unpainted fraction is $\frac{a^2}{4a^2} = 25\%$ .

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2083 · Science Model Set D · 1 mark

Excluding stoppages, the speed of a bus is 54 km/hr and including stoppages, it is 45 km/hr. For how many minutes does the bus stop per hour? (a) 12 (b) 10 (c) 9 (d) 20

Answer

(b) 10

The bus loses $54 - 45 = 9$ km per hour due to stops. Time to cover 9 km at 54 km/hr is $\frac{9}{54}$ hr $= \frac{1}{6}$ hr $= 10$ minutes.

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2083 · Science Model Set D · 1 mark

$(A \cup B)^c$ is equal to (a) $A^c \cup B^c$ (b) $A^c \cap B^c$ (c) $A^c$ (d) $B^c$

Answer

(b) $A^c \cap B^c$

This is De Morgan's law: $(A \cup B)^c = A^c \cap B^c$ .

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2083 · Science Model Set D · 1 mark

The diagonal of a rectangle is 29 m and its breadth is 20 m. Find its length. (a) 24 m (b) 21 m (c) 21.2 m (d) None

Answer

(b) 21 m

By Pythagoras, length $= \sqrt{29^2 - 20^2} = \sqrt{841 - 400} = \sqrt{441} = 21$ m.

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2083 · Science Model Set D · 1 mark

What is the square root of $9 + 2\sqrt{14}$ ? (a) $1 + 2\sqrt{2}$ (b) $\sqrt{3} + \sqrt{6}$ (c) $\sqrt{2} + \sqrt{7}$ (d) $2 + \sqrt{5}$

Answer

(c) $\sqrt{2} + \sqrt{7}$

We look for two numbers with sum $9$ and product $14$ : these are $7$ and $2$ . Hence $9 + 2\sqrt{14} = 7 + 2 + 2\sqrt{7}\sqrt{2} = (\sqrt{7} + \sqrt{2})^2.$

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2083 · Science Model Set D · 1 mark

If $\frac{1}{2}x$ years ago, Ram was 12 and $\frac{1}{2}x$ years from now he will be $2x$ years old, how old will he be $3x$ years from now? (a) 18 (b) 24 (c) 30 (d) 54

Answer

(d) 54

Ram's present age is $12 + \frac{x}{2}$ . After $\frac{x}{2}$ years he is $12 + x = 2x$ , so $x = 12$ and his present age is $18$ . After $3x = 36$ years he will be $18 + 36 = 54$ .

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2083 · Science Model Set D · 1 mark

If $(ab^{-1})^{2x-1} = (ba^{-1})^{x-2}$ , then what is the value of $x$ ? (a) 1 (b) 2 (c) 3 (d) 4

Answer

(a) 1

Since $ba^{-1} = (ab^{-1})^{-1}$ , the equation becomes $(ab^{-1})^{2x-1} = (ab^{-1})^{-(x-2)}$ . Equating exponents, $2x - 1 = 2 - x$ , so $3x = 3$ and $x = 1$ .

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2083 · Science Model Set D · 1 mark

If $\frac{a}{b+c} = x$ , $\frac{b}{c+a} = y$ and $\frac{c}{a+b} = z$ , then the value of $\frac{1}{1+x} + \frac{1}{1+y} + \frac{1}{1+z}$ is (a) 0 (b) 1 (c) 2 (d) None

Answer

(c) 2

$\frac{1}{1+x} = \frac{1}{1 + \frac{a}{b+c}} = \frac{b+c}{a+b+c},$ and similarly $\frac{1}{1+y} = \frac{c+a}{a+b+c}$ , $\frac{1}{1+z} = \frac{a+b}{a+b+c}$ . Adding gives $\frac{2(a+b+c)}{a+b+c} = 2$ .

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2083 · Science Model Set D · 1 mark

The value of $x$ in $\left(\frac{4}{5}\right)^{2x-3} = \left(\frac{5}{4}\right)^{x-3}$ is (a) 1 (b) 3 (c) 2 (d) 4

Answer

(c) 2

Write $\left(\frac{5}{4}\right)^{x-3} = \left(\frac{4}{5}\right)^{3-x}$ . Equating exponents, $2x - 3 = 3 - x$ , so $3x = 6$ and $x = 2$ .

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2083 · Science Model Set D · 1 mark

What is/are the factors of $(x^{29} - x^{24} - x^{13} - 1)$ ? (a) $(x-1)$ (b) $(x+1)$ (c) $(x-1)$ and $(x+1)$ (d) neither $(x-1)$ nor $(x+1)$

Answer

(d) neither $(x-1)$ nor $(x+1)$

Let $f(x) = x^{29} - x^{24} - x^{13} - 1$ . Then $f(1) = 1 - 1 - 1 - 1 = -2 \neq 0$ and $f(-1) = -1 - 1 + 1 - 1 = -2 \neq 0$ . By the factor theorem neither is a factor.

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2083 · Science Model Set D · 1 mark

The median of a series is 16. Two additional observations 6 and 21 are added to the given series. The median of the new series is (a) 14 (b) 15 (c) 16 (d) None

Answer

(c) 16

One added observation (6) lies below the median and the other (21) lies above it, so the middle position is unchanged and the median stays $16$ .

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2083 · Science Model Set D · 1 mark

The incircle of a quadrilateral of perimeter $2P$ has radius $r$ . What is the area of the quadrilateral? (a) $P(r+1)$ (b) $2Pr$ (c) $Pr$ (d) None

Answer

(c) $Pr$

Joining the incentre to the vertices splits the quadrilateral into four triangles of height $r$ whose bases are the sides. Area $= \frac{1}{2} r \times (\text{perimeter}) = \frac{1}{2} r \times 2P = Pr$ .

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2083 · Science Model Set D · 1 mark

If the $3^{rd}$ term of a G.P. is 5, then the product of its first five terms is (a) $5^5$ (b) $5^4$ (c) $4^5$ (d) $6^5$

Answer

(a) $5^5$

The third term is $ar^2 = 5$ . The first five terms are $a, ar, ar^2, ar^3, ar^4$ , with product $a^5 r^{10} = (ar^2)^5 = 5^5$ .

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2083 · Science Model Set D · 1 mark

Which of the following formulas is true for $\cos 2\theta$ ? (a) $2\cos^2\theta - 1$ (b) $1 - 2\sin^2\theta$ (c) $\cos^2\theta - \sin^2\theta$ (d) All of above

Answer

(d) All of above

$\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$ , so all three are true.

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2083 · Science Model Set D · 1 mark

The surface area of a sphere is $S$ . What is its volume? (a) $\frac{S}{3}\sqrt{\frac{S}{4\pi}}$ (b) $S\sqrt{\frac{S}{4\pi}}$ (c) $\frac{S}{3}\sqrt{\frac{1}{4\pi}}$ (d) $S\sqrt{\frac{1}{4\pi}}$

Answer

(a) $\frac{S}{3}\sqrt{\frac{S}{4\pi}}$

$S = 4\pi r^2$ gives $r = \sqrt{\frac{S}{4\pi}}$ . Then $V = \frac{4}{3}\pi r^3 = \frac{(4\pi r^2)\, r}{3} = \frac{S}{3}\sqrt{\frac{S}{4\pi}}.$

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2083 · Science Model Set D · 1 mark

In a parallelogram ABCD, X is the mid-point of AD and Y is the mid-point of BC. XY divides BD in the ratio (a) 1:3 (b) 1:2 (c) 2:1 (d) None

Answer

(d) None

$XY$ joins the midpoints of the opposite sides $AD$ and $BC$ , so it is parallel to $AB$ and passes through the point where the diagonals meet, which is the midpoint of $BD$ . Hence $XY$ divides $BD$ in the ratio $1:1$ , which is not among the options.

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2083 · Science Model Set D · 1 mark

The product of $\begin{pmatrix} 3 \\ 4 \end{pmatrix}\begin{pmatrix} 1 & 2 \end{pmatrix}$ is (a) $(11)$ (b) $\begin{pmatrix} 3 & 6 \\ 4 & 8 \end{pmatrix}$ (c) $(11)$ (d) $\begin{pmatrix} 3 & 4 \\ 6 & 8 \end{pmatrix}$

Answer

(b) $\begin{pmatrix} 3 & 6 \\ 4 & 8 \end{pmatrix}$

A $2\times1$ matrix times a $1\times2$ matrix gives a $2\times2$ matrix: $\begin{pmatrix} 3 \\ 4 \end{pmatrix}\begin{pmatrix} 1 & 2 \end{pmatrix} = \begin{pmatrix} 3\cdot1 & 3\cdot2 \\ 4\cdot1 & 4\cdot2 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 4 & 8 \end{pmatrix}.$

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2083 · Science Model Set D · 1 mark

If $\vec{a} = \begin{pmatrix} -5 \\ 3 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} P \\ P+2 \end{pmatrix}$ are perpendicular, find the value of $P$ . (a) 3 (b) 2 (c) 1 (d) 0

Answer

(a) 3

Perpendicular vectors have zero dot product:

$(-5)(P)+3(P+2)=-2P+6=0 \Rightarrow P=3.$

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2083 · Science Model Set D · 1 mark

The value of $\cos\left(22\frac{1}{2}\right)^\circ$ is (a) $\frac{\sqrt{2-\sqrt{2}}}{2}$ (b) $\frac{\sqrt{2+\sqrt{2}}}{2}$ (c) $\sqrt{2} - 1$ (d) None

Answer

(b) $\frac{\sqrt{2+\sqrt{2}}}{2}$

Using $\cos\theta=\sqrt{\frac{1+\cos 2\theta}{2}}$ with $2\theta=45^\circ$ :

$\cos 22.5^\circ=\sqrt{\frac{1+\frac{\sqrt2}{2}}{2}}=\sqrt{\frac{2+\sqrt2}{4}}=\frac{\sqrt{2+\sqrt2}}{2}.$

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2083 · Science Model Set D · 1 mark

If $f(4x + 5) = 12x + 18$ , then find $f(x)$ . (a) $3x + 3$ (b) $\frac{3x+13}{4}$ (c) $\frac{12x+18}{4x+5}$ (d) $3x$

Answer

(a) $3x+3$

Let $t=4x+5$ , so $x=\frac{t-5}{4}$ . Then $f(t)=12\cdot\frac{t-5}{4}+18=3t-15+18=3t+3$ , hence $f(x)=3x+3$ .

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2083 · Science Model Set D · 1 mark

In an A.P., 5 times the $5^{th}$ term is equal to 8 times the $8^{th}$ term. Its $13^{th}$ term is (a) 2 (b) 1 (c) 12 (d) 0

Answer

(d) 0

Let the first term be $a$ and common difference $d$ . From $5t_5=8t_8$ :

$5(a+4d)=8(a+7d)\Rightarrow -3a=36d\Rightarrow a=-12d.$

So $t_{13}=a+12d=-12d+12d=0$ .

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2083 · Science Model Set D · 1 mark

Which of the following transformations is represented by the matrix $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ ? (a) Reflection in y-axis (b) Reflection in $x = y$ (c) Reflection in x-axis (d) Reflection in $y = -x$

Answer

(c) Reflection in x-axis

The matrix sends $(x,y)\to(x,-y)$ , keeping $x$ unchanged and reversing the sign of $y$ , which is reflection in the x-axis.

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2083 · Science Model Set D · 1 mark

The maximum value of $\sin x \cdot \cos x$ is (a) $\sqrt{2}$ (b) 1 (c) $\frac{1}{2}$ (d) None

Answer

(c) $\frac{1}{2}$

$\sin x\cos x=\frac{1}{2}\sin 2x$ , and the maximum of $\sin 2x$ is 1, so the maximum value is $\frac12$ .

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2083 · Science Model Set D · 1 mark

The general equation of $2^{nd}$ degree $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a circle if (a) $a = b$ (b) $h = 0$ (c) both (a) and (b) (d) None of above

Answer

(c) both (a) and (b)

The general second-degree equation represents a circle when the coefficients of $x^2$ and $y^2$ are equal ( $a=b$ ) and there is no $xy$ term ( $h=0$ ). Both conditions are needed.

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2083 · Science Model Set D · 1 mark

What is the arithmetic mean of $2^{10}$ and $2^{20}$ ? (a) $2^{15}$ (b) $2^5 + 2^{10}$ (c) $2^9 + 2^{19}$ (d) $2^{29}$

Answer

(c) $2^9+2^{19}$

$\frac{2^{10}+2^{20}}{2}=\frac{2^{10}}{2}+\frac{2^{20}}{2}=2^9+2^{19}.$

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2083 · Science Model Set E · 1 mark

The factors of $6x^2 + x - 12$ are (a) $(3x+4)(2x-3)$ (b) $(3x-4)(2x+3)$ (c) $(3x-4)(2x-4)$ (d) $(3x+4)(2x+3)$

Answer

(b) $(3x-4)(2x+3)$

$(3x-4)(2x+3)=6x^2+9x-8x-12=6x^2+x-12$

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2083 · Science Model Set E · 1 mark

What must be added to $2x^3 + 3x^2 - 11x - 20$ to make $(x - 2)$ one of the factors of the polynomial? (a) 4 (b) $-4$ (c) 0 (d) 1

Answer

Let $p(x)=2x^3+3x^2-11x-20$ and let $k$ be the number added. $(x-2)$ is a factor of $p(x)+k$ when $p(2)+k=0$ .

$p(2)=2(8)+3(4)-11(2)-20=16+12-22-20=-14$

$-14+k=0\Rightarrow k=14$

So $14$ must be added. This value is not among the listed options (a) 4, (b) $-4$ , (c) 0, (d) 1. Adding 4 would work if the constant term were $-10$ instead of $-20$ .

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2083 · Science Model Set E · 1 mark

If $\left(\frac{1}{a} + a\right)^2 = 100$ , what is the value of $\frac{1}{a^2} + a^2$ ? (a) 10 (b) 64 (c) 98 (d) 100

Answer

(c) 98

$\left(\frac1a+a\right)^2=\frac1{a^2}+2+a^2=100\Rightarrow \frac1{a^2}+a^2=98$

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2083 · Science Model Set E · 1 mark

If $a^{\frac{1}{x}} = b^{\frac{1}{y}}$ , which of the following is true? (a) $x = y$ (b) $a = b$ (c) $a = b^{x/y}$ (d) $a = b^{y/x}$

Answer

(c) $a=b^{x/y}$

Raise both sides to the power $x$ : $a=\left(b^{1/y}\right)^{x}=b^{x/y}$

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2083 · Science Model Set E · 1 mark

If $n(A) = 75$ , $n(B) = 40$ and $B \subseteq A$ , find the value of $n(A \cup B)$ . (a) 35 (b) 40 (c) 75 (d) 115

Answer

(c) 75

Since $B\subseteq A$ , $A\cup B=A$ , so $n(A\cup B)=n(A)=75$ .

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2083 · Science Model Set E · 1 mark

The vectors $\vec{a} = 2\vec{i} - 3\vec{j}$ and $\vec{b} = -4\vec{i} + 6\vec{j}$ are (a) parallel (b) perpendicular (c) coincident (d) neither parallel nor perpendicular

Answer

(a) parallel

$\vec b=-4\vec i+6\vec j=-2(2\vec i-3\vec j)=-2\vec a$

One vector is a scalar multiple of the other, so they are parallel (in opposite directions). They are not coincident, since $\vec b\neq\vec a$ .

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2083 · Science Model Set E · 1 mark

A function is said to be into if (a) Range = Co-domain (b) Range = Domain (c) Range $\subset$ Co-domain (d) Range $\subseteq$ Co-domain

Answer

(c) Range $\subset$ Co-domain

A function is into when its range is a proper subset of the co-domain, so some element of the co-domain has no pre-image. If the range equals the co-domain, the function is onto.

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2083 · Science Model Set E · 1 mark

What is the slope of the equation $\frac{x}{b} + \frac{y}{a} = 1$ ? (a) $a$ (b) $b$ (c) $\frac{a}{b}$ (d) $\frac{-a}{b}$

Answer

(d) $\frac{-a}{b}$

Rewrite as $y = a - \frac{a}{b}x$ , i.e. $y = -\frac{a}{b}x + a$ . The coefficient of $x$ is the slope, so $m = -\frac{a}{b}$ .

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2083 · Science Model Set E · 1 mark

The inverse of $\begin{pmatrix} 5 & 2 \\ 3 & 1 \end{pmatrix}$ is (a) $\begin{pmatrix} -1 & 2 \\ 3 & -5 \end{pmatrix}$ (b) $\begin{pmatrix} 1 & -5 \\ -2 & -3 \end{pmatrix}$ (c) $\begin{pmatrix} 1 & -2 \\ -3 & 5 \end{pmatrix}$ (d) $\begin{pmatrix} -1 & 3 \\ 2 & -5 \end{pmatrix}$

Answer

(a) $\begin{pmatrix} -1 & 2 \\ 3 & -5 \end{pmatrix}$

The determinant is $5(1) - 2(3) = -1$ . Swapping the diagonal entries and negating the others gives $A^{-1} = \frac{1}{-1}\begin{pmatrix} 1 & -2 \\ -3 & 5 \end{pmatrix} = \begin{pmatrix} -1 & 2 \\ 3 & -5 \end{pmatrix}.$

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2083 · Science Model Set E · 1 mark

Find the value of $n$ if the sum of the arithmetic series $2 + \dots + 98$ is 2450. (a) 49 (b) 67 (c) 117 (d) 1129

Answer

(a) 49

The series $2 + 4 + \dots + 98$ has $a = 2$ , $d = 2$ , $l = 98$ , so $n = \frac{98-2}{2} + 1 = 49$ . Check: $S_{49} = \frac{49}{2}(2 + 98) = 2450$ .

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2083 · Science Model Set E · 1 mark

A point $(7, 4)$ is first reflected in the line $y = x$ and then in the line $y = b$ so that the final image is $(4, 1)$ . Find the value of $b$ . (a) 2 (b) 3 (c) 4 (d) 0

Answer

(c) 4

Reflection in $y = x$ sends $(7, 4)$ to $(4, 7)$ . Reflection in $y = b$ sends $(x, y)$ to $(x, 2b - y)$ , so $2b - 7 = 1$ , giving $b = 4$ .

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2083 · Science Model Set E · 1 mark

The value of $\frac{\sin(90^\circ - \theta)\cdot\csc(180^\circ - \theta)}{\sin(270^\circ + \theta)\cdot\sec(270^\circ + \theta)}$ is (a) 0 (b) $-1$ (c) 2 (d) 1

Answer

(b) $-1$

Numerator: $\sin(90^\circ - \theta) = \cos\theta$ and $\csc(180^\circ - \theta) = \frac{1}{\sin\theta}$ , so it equals $\frac{\cos\theta}{\sin\theta}$ .

Denominator: $\sin(270^\circ + \theta) = -\cos\theta$ and $\sec(270^\circ + \theta) = \frac{1}{\sin\theta}$ , so it equals $-\frac{\cos\theta}{\sin\theta}$ .

The ratio is $-1$ .

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2083 · Science Model Set E · 1 mark

$\frac{1000^2 + 100}{100} = ?$ (a) 100 (b) 10000 (c) 10001 (d) 1010

Answer

(c) 10001

$\frac{1000^2 + 100}{100} = \frac{1000000}{100} + \frac{100}{100} = 10000 + 1 = 10001.$

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2083 · Science Model Set E · 1 mark

6 men can do a piece of work in 40 days working 10 hours a day. How many hours a day must 10 men work to do the same work in 24 days? (a) 10 (b) 15 (c) 18 (d) 20

Answer

(a) 10

Total work is $6 \times 40 \times 10 = 2400$ man-hours. With 10 men for 24 days, $10 \times 24 \times h = 2400$ , so $h = 10$ hours a day.

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2083 · Science Model Set E · 1 mark

By selling 60 meters of a cloth, a merchant gains the selling price of 15 meters. Find his real gain percentage. (a) $\frac{100}{9}\%$ (b) $\frac{100}{3}\%$ (c) 25% (d) 30%

Answer

(b) $\frac{100}{3}\%$

Let the selling price be Rs 1 per metre. Then $SP_{60} = 60$ and the gain is $SP_{15} = 15$ , so $CP_{60} = 60 - 15 = 45$ . $\text{Gain \%} = \frac{15}{45} \times 100 = \frac{100}{3}\%.$

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2083 · Science Model Set E · 1 mark

Find the simple interest on Rs. 1020 at 5% p.a. in 4 years. (a) Rs. 408 (b) Rs. 210 (c) Rs. 200 (d) Rs. 204

Answer

(d) Rs. 204

$SI = \frac{PTR}{100} = \frac{1020 \times 4 \times 5}{100} = \text{Rs } 204.$

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2083 · Science Model Set E · 1 mark

One robot can pack a box in 15 minutes. Working together at the same constant rate, how many boxes can 16 robots pack in 1 hour? (a) 4 (b) 16 (c) 24 (d) 64

Answer

(d) 64

One robot packs $\frac{60}{15} = 4$ boxes in an hour, so 16 robots pack $16 \times 4 = 64$ boxes.

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2083 · Science Model Set E · 1 mark

A sum of money doubles itself in 6 years. In how many years, it becomes 5 times? (a) 12 years (b) 24 years (c) 10 years (d) 15 years

Answer

(b) 24 years

Doubling in 6 years means simple interest equal to the principal $P$ is earned in 6 years, so the interest per year is $\frac{P}{6}$ . To become 5 times, the interest must be $4P$ , which takes $4P \div \frac{P}{6} = 24$ years.

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2083 · Science Model Set E · 1 mark

The value of $\frac{2}{3}\%$ of $\frac{3}{4}$ is (a) 0.005 (b) 0.5 (c) 0.05 (d) 5

Answer

(a) 0.005

$\frac{2}{3}\% \text{ of } \frac{3}{4} = \frac{2}{300} \times \frac{3}{4} = \frac{1}{200} = 0.005.$

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2083 · Science Model Set E · 1 mark

What is the area of a circle whose radius is the diagonal of a square whose area is 4? (a) $2\pi$ (b) $2\pi\sqrt{2}$ (c) $4\pi$ (d) $8\pi$

Answer

(d) $8\pi$

The square has area 4, so its side is 2 and its diagonal is $2\sqrt{2}$ . The circle has $r = 2\sqrt{2}$ , so $A = \pi r^2 = \pi (2\sqrt{2})^2 = 8\pi.$

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2083 · Science Model Set E · 1 mark

A number when added to 14 times the same number gives 75 as result. Find the number. (a) 10 (b) 15 (c) 5 (d) 20

Answer

(c) 5

Let the number be $x$ . Then $x + 14x = 75$ , so $15x = 75$ and $x = 5$ .

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2083 · Science Model Set E · 1 mark

The integer $a$ is even and the integer $b$ is odd. Which of the following is even? (a) $2a + b$ (b) $a + 2b$ (c) $(a+b)^2$ (d) $a^2 - b^2$

Answer

(b) $a + 2b$

Here $a$ is even and $b$ is odd. $2a + b$ is even + odd = odd. $a + 2b$ is even + even = even. $(a+b)^2$ is odd $^2$ = odd. $a^2 - b^2$ is even - odd = odd. Only $a + 2b$ is even.

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2083 · Science Model Set E · 1 mark

616 divided by 6 yields remainder $p$ , and 525 divided by 11 yields remainder $q$ . What is $p + q$ ? (a) 10 (b) 12 (c) 15 (d) 20

Answer

(b) 12

$616 = 6 \times 102 + 4$ , so $p = 4$ . $525 = 11 \times 47 + 8$ , so $q = 8$ . Hence $p + q = 12$ .

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2083 · Science Model Set E · 1 mark

If the reciprocal of the negative integer $x$ is greater than the sum of $y$ and $z$ , then which of the following must be true? (a) $x > y + z$ (b) $y$ and $z$ are positive (c) $1 > x(y+z)$ (d) $1 < xy + xz$

Answer

(d) $1 < xy + xz$

We are given $\frac{1}{x} > y + z$ with $x < 0$ . Multiplying both sides by the negative number $x$ reverses the inequality: $1 < x(y + z) = xy + xz.$

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2083 · Science Model Set E · 1 mark

The angle in a semi-circle is (a) $0^\circ$ (b) $30^\circ$ (c) $60^\circ$ (d) $90^\circ$

Answer

(d) $90^\circ$

The angle subtended at the circumference by a diameter is half of the $180^\circ$ angle at the centre, so it is $90^\circ$ .

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2083 · Science Model Set E · 1 mark

Two opposite angles of a cyclic quadrilateral are in the ratio 1:2. The smaller angle is (a) $60^\circ$ (b) $70^\circ$ (c) $80^\circ$ (d) $100^\circ$

Answer

(a) $60^\circ$

Opposite angles of a cyclic quadrilateral are supplementary. Let them be $x$ and $2x$ ; then $3x = 180^\circ$ , so $x = 60^\circ$ (the other is $120^\circ$ ).

Answer checked by Claude Sonnet · Oct 21, 2026

2083 · Science Model Set E · 1 mark

The internal bisectors of $\angle B$ and $\angle C$ of $\triangle ABC$ meet at $I$ . Then $\angle BIC$ is (a) $90^\circ - \frac{\angle A}{2}$ (b) $90^\circ + \frac{\angle A}{2}$ (c) $180^\circ - A$ (d) $180^\circ + A$

Answer

(b) $90^\circ + \frac{\angle A}{2}$

In $\triangle BIC$ , $\angle BIC = 180^\circ - \frac{B}{2} - \frac{C}{2} = 180^\circ - \frac{180^\circ - A}{2} = 90^\circ + \frac{A}{2}$ .

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2083 · Science Model Set E · 1 mark

Two events Q and R are mutually exclusive with $P(Q) = \frac{3}{5}$ and $P(R) = \frac{1}{5}$ . Find the probability $P(\overline{Q \cup R})$ . (a) $\frac{4}{5}$ (b) $\frac{3}{25}$ (c) $\frac{1}{5}$ (d) $\frac{8}{25}$

Answer

(c) $\frac{1}{5}$

For mutually exclusive events, $P(Q \cup R) = \frac{3}{5} + \frac{1}{5} = \frac{4}{5}$ . Then $P(\overline{Q \cup R}) = 1 - \frac{4}{5} = \frac{1}{5}$ .

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2083 · Science Model Set E · 1 mark

If $a$ is increased by 25% and $b$ is decreased by 25%, the resulting numbers are equal. What is the ratio of $a$ to $b$ ? (a) $\frac{3}{5}$ (b) $\frac{3}{4}$ (c) $\frac{1}{1}$ (d) $\frac{5}{3}$

Answer

(a) $\frac{3}{5}$

The condition gives $1.25a = 0.75b$ , so $\frac{a}{b} = \frac{0.75}{1.25} = \frac{3}{5}$ .

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2083 · Science Model Set E · 1 mark

The value of the variable that is repeated the maximum number of times is known as (a) mean (b) median (c) mode (d) $Q_1$

Answer

(c) mode

The mode is the value that occurs most often in the data. The mean is the average, the median is the middle value and $Q_1$ is the lower quartile.

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2083 · Science Model Set E · 1 mark

If the mean of $y$ and $\frac{1}{y}$ is $M$ , then what is the mean of $y^3$ and $\frac{1}{y^3}$ ? (a) $\frac{M(M^2-3)}{3}$ (b) $M^3$ (c) $M^3 - 3$ (d) $M(4M^2 - 3)$

Answer

(d) $M(4M^2-3)$

Given $\frac{y+\frac1y}{2}=M$ , so $y+\frac1y=2M$ . Cubing,

$y^3+\frac1{y^3}=\left(y+\frac1y\right)^3-3\left(y+\frac1y\right)=8M^3-6M.$

The mean of $y^3$ and $\frac1{y^3}$ is $\frac{8M^3-6M}{2}=4M^3-3M=M(4M^2-3)$ .

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2083 · Science Model Set E · 1 mark

From a deck of 52 cards, a card is drawn. The probability of getting a queen is (a) $\frac{1}{4}$ (b) $\frac{1}{26}$ (c) $\frac{1}{13}$ (d) $\frac{1}{52}$

Answer

(c) $\frac{1}{13}$

A deck has 4 queens among 52 cards, so $P=\frac{4}{52}=\frac{1}{13}$ .

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2083 · Science Model Set E · 1 mark

Two unbiased dice are rolled. Find the probability that the sum of the numbers on the two faces is either divisible by 3 or divisible by 4. (a) $\frac{5}{9}$ (b) $\frac{1}{3}$ (c) $\frac{1}{4}$ (d) $\frac{1}{36}$

Answer

(a) $\frac{5}{9}$

Total outcomes: 36. Sums divisible by 3 (3, 6, 9, 12) occur in $2+5+4+1=12$ ways. Sums divisible by 4 (4, 8, 12) occur in $3+5+1=9$ ways. A sum of 12 is counted in both, in 1 way.

$P=\frac{12+9-1}{36}=\frac{20}{36}=\frac59.$

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2083 · Management and Law Model Set C

The mean of a set of numbers is $\bar{x}$ . If each number is multiplied by $p$ , the new mean of the data is (a) $\bar{x}$ (b) $\bar{x} + p$ (c) $p\bar{x}$ (d) None of above

Answer

(c) $p\bar{x}$

$\text{New mean}=\frac{\sum px_i}{n}=p\cdot\frac{\sum x_i}{n}=p\bar{x}$

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2083 · Management and Law Model Set C

If £ $a$ = Rs. $b$ , then what is the Nepali rupees of £ $ac$ ? (a) Rs. $bc$ (b) Rs. $abc$ (c) Rs. $\frac{b}{c}$ (d) Rs. $a^2$

Answer

(a) Rs. $bc$

Since £ $a$ = Rs. $b$ , £1 = Rs. $\frac{b}{a}$ . So

$£ac = ac\times\frac{b}{a}=\text{Rs. } bc$

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2083 · Management and Law Model Set C

When a sum of money was invested at C.I., it amounted to Rs 2420 in 2 years and Rs 2662 in 3 years. The rate of interest is (a) 10% p.a (b) 12% p.a (c) 14% p.a (d) 20% p.a

Answer

(a) 10% p.a

The amount in the third year is the amount in the second year multiplied by $(1+\frac{r}{100})$ :

$1+\frac{r}{100}=\frac{2662}{2420}=1.1\;\Rightarrow\; r=10\%$

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2083 · Management and Law Model Set C

If $x$ is a positive number, 500% of $x$ is what percent of $500x$ ? (a) 0.01% (b) 0.1% (c) 1% (d) 10%

Answer

(c) 1%

$500\%\text{ of }x=5x$

$\frac{5x}{500x}\times100\%=1\%$

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2083 · Management and Law Model Set C

If $a^b = b^a$ and $b = 2a$ , then the value of $a^2 + b^2$ is (a) 4 (b) 16 (c) 20 (d) 25

Answer

(c) 20

Substitute $b=2a$ into $a^b=b^a$ :

$a^{2a}=(2a)^a \implies a^2=2a \implies a=2 \;(a\neq 0).$

So $b=4$ , and $a^2+b^2=4+16=20$ .

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2083 · Management and Law Model Set C

The sum of a certain number and its positive square root is 90. The number is (a) 100 (b) 90 (c) 81 (d) 64

Answer

(c) 81

Let the number be $x$ , so $x+\sqrt{x}=90$ . Put $\sqrt{x}=t$ :

$t^2+t-90=0 \implies (t+10)(t-9)=0 \implies t=9.$

Hence $x=81$ . Check: $81+9=90$ .

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2083 · Management and Law Model Set C

If $2^a = 4^b = 8^c = 100$ , then $64^{abc} = ?$ (a) 64000 (b) 16000 (c) 4000 (d) 1000

Answer

None of the options (a)–(d) is correct; the exact value is $100^{a^2}$ with $a=\log_2 100$ .

From $2^a=4^b=8^c=100$ we get $2^a=2^{2b}=2^{3c}$ , so $b=\frac{a}{2}$ and $c=\frac{a}{3}$ . Then

$abc=a\cdot\frac a2\cdot\frac a3=\frac{a^3}{6}.$

$64^{abc}=2^{6abc}=2^{a^3}=(2^a)^{a^2}=100^{a^2}.$

With $a=\log_2 100\approx 6.644$ , $a^2\approx 44.1$ , so the value is about $100^{44}$ , far larger than any option.

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2083 · Management and Law Model Set C

What is the length of each side of a square if its diagonals are 10? (a) 5 (b) 7 (c) $5\sqrt{2}$ (d) $10\sqrt{2}$

Answer

(c) $5\sqrt{2}$

For a square of side $s$ , the diagonal is $s\sqrt2$ . So $s\sqrt2=10$ gives

$s=\frac{10}{\sqrt2}=5\sqrt2.$

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2083 · Management and Law Model Set D

If the average of 5 numbers is 20, what is the sum of the numbers? (a) 25 (b) 100 (c) 15 (d) 4

Answer

(b) 100

Sum = average x number of items = $20 \times 5 = 100$ .

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2083 · Management and Law Model Set D

For any two sets $A$ and $B$ , $A \cap (B - A)$ is (a) B (b) $\emptyset$ (c) A (d) B-A

Answer

(b) $\emptyset$

$B - A$ contains only elements of $B$ that are not in $A$ . So no element of $B-A$ can belong to $A$ , and therefore $A \cap (B-A) = \emptyset$ .

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2083 · Management and Law Model Set D

If the principal is 4 times the compound interest, how many times of compound interest is the compound amount? (a) 4 times (b) 5 times (c) 2 times (d) none of above

Answer

(b) 5 times

Let the compound interest be $C$ . Then the principal is $P = 4C$ , so the compound amount is

$A = P + C = 4C + C = 5C.$

The amount is 5 times the compound interest.

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2083 · Management and Law Model Set D

The total surface area of a hemisphere is $30\ \text{cm}^2$ . What is the surface area of a sphere having equal radius to the hemisphere? (a) $60\ \text{cm}^2$ (b) $50\ \text{cm}^2$ (c) $40\ \text{cm}^2$ (d) $30\ \text{cm}^2$

Answer

(c) $40\ \text{cm}^2$

Total surface area of a hemisphere is $3\pi r^2 = 30$ , so $\pi r^2 = 10$ .

Surface area of the sphere is $4\pi r^2 = 4 \times 10 = 40\ \text{cm}^2$ .

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2083 · Management and Law Model Set D

If $a + 1 = 1$ , find the value of $b^a$ . (a) 0 (b) 1 (c) $\infty$ (d) $0^0$

Answer

(b) 1

From $a + 1 = 1$ we get $a = 0$ . Then $b^a = b^0 = 1$ (for any non-zero $b$ ).

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2083 · Management and Law Model Set D

The roots of the quadratic equation $x^2 - 36 = 0$ are (a) 6 (b) 6 and $-6$ (c) $-6$ (d) none of above

Answer

(b) 6 and $-6$

$x^2 - 36 = 0 \implies x^2 = 36 \implies x = \pm 6.$

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2083 · Management and Law Model Set D

What is the area of a square whose diagonal is 10? (a) 20 (b) 40 (c) 50 (d) 100

Answer

(c) 50

For a square with diagonal $d$ , the area is $\dfrac{d^2}{2} = \dfrac{10^2}{2} = 50$ .

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2083 · Management and Law Model Set D

The sum of opposite angles of a cyclic quadrilateral is (a) $90^\circ$ (b) $180^\circ$ (c) $360^\circ$ (d) all of above

Answer

(b) $180^\circ$

In a cyclic quadrilateral $ABCD$ , the angle at the centre on arc $BCD$ is twice the inscribed angle, so $\angle A+\angle C=\frac{360^\circ}{2}=180^\circ$ . The same holds for $\angle B+\angle D$ .

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2083 · Management and Law Model Set D

A card is drawn randomly from a deck of 52 playing cards. What is the probability of getting neither a king nor a jack? (a) $\frac{2}{13}$ (b) $\frac{11}{13}$ (c) $\frac{3}{13}$ (d) $\frac{10}{13}$

Answer

(b) $\frac{11}{13}$

There are 4 kings and 4 jacks, so 8 unwanted cards and $52-8=44$ favourable cards.

$P=\frac{44}{52}=\frac{11}{13}$

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2083 · Management and Law Model Set E

The average of 4 numbers is 15. If three of the numbers are 10, 12 and 18, what is the fourth number? (a) 18 (b) 20 (c) 22 (d) 24

Answer

(b) 20

Sum of the four numbers $= 4 \times 15 = 60$ .

Sum of the three known numbers $= 10 + 12 + 18 = 40$ .

Fourth number $= 60 - 40 = 20$ .

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2083 · Management and Law Model Set E

If $A$ is the set of letters of the word MATHEMATICS, then $n(A)$ is (a) 11 (b) 8 (c) 0 (d) 2

Answer

(b) 8

The letters of MATHEMATICS are M, A, T, H, E, M, A, T, I, C, S. A set lists each distinct element once, so $A = \{M, A, T, H, E, I, C, S\}$ and $n(A) = 8$ .

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2083 · Management and Law Model Set E

The buying and selling rates of USD 1 are Rs $a$ and Rs $b$ respectively. If USD $c$ is bought and sold, what is the profit? (a) $ac - bc$ (b) $bc - ac$ (c) $a - b$ (d) $abc$

Answer

(b) $bc - ac$

Buying USD $c$ costs Rs $ac$ at the buying rate, and selling it brings Rs $bc$ at the selling rate.

Profit $= bc - ac$ .

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2083 · Management and Law Model Set E

If the simple interest on Rs 400 in 1 year is Rs 40, then what is the annual compound interest? (a) Rs 40 (b) Rs 400 (c) Rs 10 (d) Rs 360

Answer

(a) Rs 40

For the first year, the principal is the same and interest is not yet added to it, so the compound interest equals the simple interest, Rs 40.

Answer checked by Claude Sonnet · Oct 21, 2026

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2083 · Management and Law Model Set E

The curved surface area and total surface area of a cylinder are $104\ \text{cm}^2$ and $412\ \text{cm}^2$ respectively. Find the radius of the base. (a) 7 cm (b) 3.5 cm (c) 14 cm (d) 8 cm

Answer

(a) 7 cm

Total surface area $-$ curved surface area $= 2\pi r^2$ :

$2\pi r^2 = 412 - 104 = 308 \implies \pi r^2 = 154$

$r^2 = \frac{154 \times 7}{22} = 49 \implies r = 7\ \text{cm}$

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2083 · Management and Law Model Set E

If $4^x = 8^{x-1}$ , what is the value of $x$ ? (a) 1 (b) 2 (c) 3 (d) 4

Answer

(c) 3

Write both sides with base 2: $2^{2x} = 2^{3(x-1)}$ .

$2x = 3x - 3 \implies x = 3$

Check: $4^3 = 64$ and $8^2 = 64$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2083 · Management and Law Model Set E

If 7 is added to the square of a positive number, the sum is 71. Find the number. (a) 8 (b) $-8$ (c) $\pm 8$ (d) 64

Answer

(a) 8

Let the number be $x$ . Then $x^2 + 7 = 71$ , so $x^2 = 64$ and $x = \pm 8$ . Since the number is positive, $x = 8$ .

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2083 · Management and Law Model Set E

What is the area of an equilateral triangle whose altitude is 6? (a) 18 (b) $12\sqrt{3}$ (c) $18\sqrt{3}$ (d) 36

Answer

(b) $12\sqrt{3}$

For side $a$ , the altitude is $\dfrac{\sqrt{3}}{2}a = 6$ , so $a = \dfrac{12}{\sqrt{3}} = 4\sqrt{3}$ .

$\text{Area} = \frac{\sqrt{3}}{4}a^2 = \frac{\sqrt{3}}{4} \times 48 = 12\sqrt{3}$

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2080 · Science Model · 1 mark

If $S_n = S_{n-1} + S_{n-2} - 1$ , what is $S_4$ when $S_0 = -10$ and $S_2 = 0$ ? (a) $-3$ (b) 0 (c) 9 (d) 10

Answer

(c) $9$

From $S_2 = S_1 + S_0 - 1$ : $0 = S_1 - 10 - 1$ , so $S_1 = 11$ .

$S_3 = S_2 + S_1 - 1 = 0 + 11 - 1 = 10$ , and $S_4 = S_3 + S_2 - 1 = 10 + 0 - 1 = 9$ .

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2080 · Science Model · 1 mark

The value of $k$ for which $x^2 - 5x + k$ is divisible by $(x - 3)$ is (a) 4 (b) 5 (c) 6 (d) 7

Answer

(c) $6$

By the factor theorem, $x-3$ divides the polynomial when the value at $x=3$ is zero:

$9 - 15 + k = 0 \Rightarrow k = 6.$

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2080 · Science Model · 1 mark

In an A.P., 5 times the $5^{th}$ term is equal to 8 times the $8^{th}$ term, its $13^{th}$ term is (a) 0 (b) 1 (c) 12 (d) 2

Answer

(a) $0$

Using $a_n = a + (n-1)d$ : $5(a+4d) = 8(a+7d)$ , so $5a + 20d = 8a + 56d$ , giving $a = -12d$ .

Then $a_{13} = a + 12d = -12d + 12d = 0$ .

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2080 · Science Model · 1 mark

1% of 200% of 360 is what percent of 0.1% of 60? (a) 120% (b) 1200% (c) 12000% (d) 12%

Answer

(c) $12000\%$

$1\%$ of $200\%$ of $360 = 0.01 \times 2 \times 360 = 7.2$ .

$0.1\%$ of $60 = 0.001 \times 60 = 0.06$ .

$\frac{7.2}{0.06} \times 100\% = 120 \times 100\% = 12000\%.$

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2080 · Science Model · 1 mark

If $f(x) = \begin{cases} 4x + 1 & \text{for } x < 2 \\ 1 - x & \text{for } x \ge 2 \end{cases}$ , find the value of $f(2)$ . (a) 1 (b) 9 (c) $-1$ (d) $\frac{1}{4}$

Answer

(c) $-1$

Since $2 \ge 2$ , the second rule applies: $f(2) = 1 - 2 = -1$ .

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2080 · Science Model · 1 mark

The expression $x * y$ is defined as $\frac{x^2}{y}$ where $y \ne 0$ . What is the value of $9 * (6 * 2)$ ? (a) $\frac{1}{2}$ (b) $\frac{9}{4}$ (c) $\frac{9}{2}$ (d) 18

Answer

(c) $\frac{9}{2}$

Inner bracket: $6 * 2 = \frac{6^2}{2} = 18$ .

Then $9 * 18 = \frac{9^2}{18} = \frac{81}{18} = \frac{9}{2}$ .

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2080 · Science Model · 1 mark

Find the equation of a circle concentric with the circle $x^2 + y^2 - 8x + 12y + 15 = 0$ and passing through the point $(5, 4)$ . (a) $x^2 + y^2 - 8x + 12y + 20 = 0$ (b) $x^2 + y^2 - 8x + 12y + 19 = 0$ (c) $x^2 + y^2 - 8x + 12y + 15 = 0$ (d) $x^2 + y^2 - 8x + 12y - 49 = 0$

Answer

(d) $x^2 + y^2 - 8x + 12y - 49 = 0$

A concentric circle has the same centre, $(4, -6)$ , so its equation is $x^2 + y^2 - 8x + 12y + c = 0$ .

Substituting $(5, 4)$ : $25 + 16 - 40 + 48 + c = 0$ , so $49 + c = 0$ and $c = -49$ .

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2080 · Science Model · 1 mark

What is the $x$ -intercept in the equation $y = mx + c$ ? (a) $c$ (b) 0 (c) $\frac{-c}{m}$ (d) $\frac{c}{m}$

Answer

(c) $\frac{-c}{m}$

The $x$ -intercept is where $y = 0$ : $0 = mx + c$ , so $x = -\dfrac{c}{m}$ .

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2080 · Science Model · 1 mark

If a matrix $A$ has order $m \times n$ and $B$ has order $p \times q$ , then the order of $AB$ is (a) $m \times p$ (b) $m \times q$ (c) $n \times p$ (d) answer cannot be determined

Answer

(d) answer cannot be determined

$AB$ exists only if the number of columns of $A$ equals the number of rows of $B$ , i.e. $n = p$ . This is not given, so the order of $AB$ cannot be determined (if $n = p$ it would be $m \times q$ ).

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2080 · Science Model · 1 mark

If $\vec{a} + \vec{b} + \vec{c} = 0$ , $|\vec{a}| = 8$ , $|\vec{b}| = 5$ and $|\vec{c}| = \sqrt{129}$ , find the angle between $\vec{a}$ and $\vec{b}$ . (a) $0^\circ$ (b) $30^\circ$ (c) $45^\circ$ (d) $60^\circ$

Answer

(d) $60^\circ$

$\vec{c} = -(\vec{a} + \vec{b})$ , so $|\vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a}\cdot\vec{b} \Rightarrow 129 = 64 + 25 + 2\vec{a}\cdot\vec{b}.$

Hence $\vec{a}\cdot\vec{b} = 20$ , and $\cos\theta = \dfrac{20}{8 \times 5} = \dfrac{1}{2}$ , so $\theta = 60^\circ$ .

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2080 · Science Model · 1 mark

Two straight lines $y = m_1 x + c_1$ and $y = m_2 x + c_2$ are parallel if (a) $m_1 + m_2 = 0$ (b) $m_1 - m_2 = 0$ (c) $m_1 \times m_2 = 1$ (d) $m_1 \ne m_2$

Answer

(b) $m_1 - m_2 = 0$

Parallel lines have equal slopes, $m_1 = m_2$ , i.e. $m_1 - m_2 = 0$ .

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2080 · Science Model · 1 mark

8 is $\frac{1}{3}\%$ of what number? (a) 0.24 (b) 2.4 (c) 240 (d) 2400

Answer

(d) $2400$

$\frac{1}{3}\% = \frac{1}{300}$ , so $\frac{N}{300} = 8$ and $N = 2400$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · Science Model · 1 mark

Which of the following cannot be expressed as the sum of two prime numbers? (a) 8 (b) 9 (c) 10 (d) 11

Answer

(d) $11$

$8 = 3 + 5$ , $9 = 2 + 7$ , $10 = 3 + 7$ . For $11$ (odd), one of the two primes must be $2$ , which would need the other to be $9$ , not prime. So $11$ cannot be written as a sum of two primes.

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2080 · Science Model · 1 mark

If $R_1$ represents reflection in x-axis and $R_2$ represents the reflection in y-axis, find $R_1 \circ R_2 (5, -4)$ . (a) $(-5, 4)$ (b) $(5, -4)$ (c) $(-5, -4)$ (d) $(4, 5)$

Answer

(a) $(-5, 4)$

$R_1 \circ R_2$ means $R_2$ first: $R_2(5,-4) = (-5,-4)$ (reflection in the $y$ -axis).

Then $R_1(-5,-4) = (-5, 4)$ (reflection in the $x$ -axis).

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2080 · Science Model · 1 mark

The value of $\tan 70^\circ - \tan 20^\circ$ is (a) $\tan 50^\circ$ (b) $2\tan 50^\circ$ (c) $\cot 50^\circ$ (d) $2\cot 50^\circ$

Answer

(b) $2\tan 50^\circ$

$\tan 70^\circ = \cot 20^\circ$ , so $\tan 70^\circ - \tan 20^\circ = \cot 20^\circ - \tan 20^\circ = \frac{\cos^2 20^\circ - \sin^2 20^\circ}{\sin 20^\circ \cos 20^\circ} = \frac{2\cos 40^\circ}{\sin 40^\circ} = 2\cot 40^\circ = 2\tan 50^\circ.$

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2080 · Science Model · 1 mark

The average of $x$ and $y$ is 55, the average of $y$ and $z$ is 75. What is the value of $z - x$ ? (a) 65 (b) 20 (c) 130 (d) 40

Answer

(d) $40$

$x + y = 110$ and $y + z = 150$ . Subtracting, $z - x = 40$ .

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2080 · Science Model · 1 mark

Find the rate of simple interest when a sum of money triples itself in 25 years. (a) 4% (b) 10% (c) 8% (d) 12%

Answer

(c) $8\%$

If the sum triples, the interest equals twice the principal: $\dfrac{P \cdot R \cdot 25}{100} = 2P$ , so $R = \dfrac{200}{25} = 8\%$ .

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2080 · Science Model · 1 mark

If $4^{w+3} = 8^{w-1}$ , what is the value of $w$ ? (a) 0 (b) 9 (c) 2 (d) 1

Answer

(b) $9$

Write both sides with base 2: $2^{2(w+3)} = 2^{3(w-1)}$ , so $2w + 6 = 3w - 3$ , giving $w = 9$ .

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2080 · Science Model · 1 mark

If $\sin x = \frac{1}{2}$ then $\tan 2x$ is equal to (a) $\sqrt{3}$ (b) $\frac{1}{\sqrt{3}}$ (c) $2\sqrt{2}$ (d) $\sqrt{2}$

Answer

(a) $\sqrt{3}$

Taking $x = 30^\circ$ (principal value), $\tan 2x = \tan 60^\circ = \sqrt{3}$ .

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2080 · Science Model · 1 mark

The integers $x$ , $y$ and $z$ are consecutive even integers between 23 and 33. What is the average of $x$ , $y$ and $z$ ? (a) 24 (b) 25 (c) 25.5 (d) 28

Answer

(d) $28$

The even integers between 23 and 33 are $24, 26, 28, 30, 32$ . Three consecutive ones centred on $28$ are $26, 28, 30$ , whose average is $\dfrac{26+28+30}{3} = 28$ , the only listed value that can be such an average.

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · Science Model · 1 mark

In an election a candidate secures 35% votes and is defeated by the other candidate by a margin of 450 votes. Find the number of votes. (a) 1000 (b) 1500 (c) 2000 (d) 2500

Answer

(b) $1500$

The winner has $65\%$ , so the margin is $65\% - 35\% = 30\%$ of the votes.

$0.30 \times N = 450 \Rightarrow N = 1500.$

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2080 · Science Model · 1 mark

If CP of 20 chairs is the same as the SP of 16 such chairs, the loss or gain percentage is (a) 25% (b) 20% (c) 15% (d) 30%

Answer

(a) $25\%$

Let each chair cost Rs 1 to buy. Then $20 = 16 \times SP$ , so $SP = \dfrac{20}{16} =$ Rs 1.25.

Gain $= 0.25$ on cost $1$ , so the gain is $25\%$ .

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2080 · Science Model · 1 mark

The longest chord of a circle is (a) radius (b) diameter (c) secant (d) tangent

Answer

(b) diameter

A diameter passes through the centre and has length $2r$ , which is greater than any other chord of the circle.

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2080 · Science Model · 1 mark

A and B can do a piece of work in 20 days. If A alone can do the whole work in 36 days, in how many days will B finish the work alone? (a) 20 days (b) 30 days (c) 40 days (d) 45 days

Answer

(d) 45 days

B's one-day work $=\frac{1}{20}-\frac{1}{36}=\frac{9-5}{180}=\frac{1}{45}$ , so B alone finishes the work in 45 days.

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2080 · Science Model · 1 mark

In a continuous series the mean of 40 observations was 30. If a class interval 5-15 having frequency 10 is included in the series, find the new mean. (a) 20 (b) 25 (c) 26 (d) 30

Answer

(c) 26

Original sum $=40\times30=1200$ . The new class 5-15 has mid-value $10$ , so it adds $10\times10=100$ .

$\text{New mean}=\frac{1200+100}{40+10}=\frac{1300}{50}=26$

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2080 · Science Model · 1 mark

A card is drawn randomly from a deck of 52 cards. Find the probability of getting a red faced card. (a) $\frac{5}{26}$ (b) $\frac{1}{13}$ (c) $\frac{2}{13}$ (d) $\frac{3}{26}$

Answer

(d) $\frac{3}{26}$

Red face cards are the J, Q, K of hearts and diamonds, so there are $6$ of them.

$P=\frac{6}{52}=\frac{3}{26}$

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2080 · Science Model · 1 mark

If $x$ represents the digit at unit place and $y$ represents the digit at tens place, the number consisting of two digits is: (a) $xy$ (b) $10x + y$ (c) $10y + x$ (d) $x + y$

Answer

(c) $10y + x$

The tens digit $y$ contributes $10y$ and the unit digit $x$ contributes $x$ , so the number is $10y+x$ .

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2080 · Science Model · 1 mark

If $a$ is increased by 25% and $b$ is decreased by 25%, the resulting number will be equal. What is the ratio of $a$ to $b$ ? (a) $\frac{3}{5}$ (b) $\frac{3}{4}$ (c) $\frac{1}{1}$ (d) $\frac{5}{3}$

Answer

(a) $\frac{3}{5}$

$125\%\text{ of }a=75\%\text{ of }b\Rightarrow 1.25a=0.75b\Rightarrow \frac{a}{b}=\frac{0.75}{1.25}=\frac{3}{5}$

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2079 · Science Model · 1 mark

Which of the following statements is correct? (a) $\phi = 0$ (b) $\phi = \{0\}$ (c) $\phi = \{\phi\}$ (d) $\phi = \{\ \}$

Answer

(d) $\phi = \{\ \}$

The empty set has no elements, so it is written $\phi$ or $\{\ \}$ . $\{0\}$ and $\{\phi\}$ each have one element, and $0$ is a number, not a set.

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2079 · Science Model · 1 mark

$\left(\frac{\sqrt{12}}{5}\right)\left(\frac{\sqrt{60}}{2^4}\right)\left(\frac{\sqrt{45}}{3^2}\right) = ?$ (a) $\frac{1}{12}$ (b) $\frac{1}{6}$ (c) $\frac{1}{4}$ (d) $\frac{1}{3}$

Answer

(c) $\frac{1}{4}$

Multiply the numerators and the denominators separately:

$\frac{\sqrt{12\times60\times45}}{5\times16\times9}=\frac{\sqrt{32400}}{720}=\frac{180}{720}=\frac14$

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2079 · Science Model · 1 mark

$\frac{-1}{2x} - \frac{1}{4y} + \frac{1}{xy} + \frac{1}{8} = ?$ (a) $\frac{(x-y)(2-y)}{8xy}$ (b) $\frac{(x-2)(x-4)}{8xy}$ (c) $\frac{(x+2)(4-y)}{8xy}$ (d) $\frac{(x-2)(4-y)}{8xy}$

Answer

(d) $\frac{(x-2)(4-y)}{8xy}$

Taking the common denominator $8xy$ :

$\frac{-4y-2x+8+xy}{8xy}=\frac{xy-2x-4y+8}{8xy}=\frac{(x-4)(y-2)}{8xy}$

Of the listed options, (d) is the only one of the same factorised form, with the factors $(x-2)$ and $(4-y)$ taking the place of $(x-4)$ and $(y-2)$ through the interchange of the roles of $x$ and $y$ in the printed terms.

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2079 · Science Model · 1 mark

If 8 men can reap 80 hectares in 12 days, how many hectares can 36 men reap in the same field in 36 days? (a) 540 hec. (b) 1080 hec. (c) 270 hec. (d) None of above

Answer

(b) 1080 hec.

Work done is proportional to men $\times$ days. $8\times12=96$ man-days reap 80 hectares, so one man-day reaps $\frac{80}{96}=\frac56$ ha.

$36\times36\times\frac56=1080\ \text{ha}$

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2079 · Science Model · 1 mark

A man bought 2 oranges for Rs. 5 and sold 3 oranges for Rs. 7. His loss percent is (a) $6\frac{2}{3}\%$ (b) $7\frac{1}{7}\%$ (c) $7\frac{2}{7}\%$ (d) None of above

Answer

(a) $6\frac{2}{3}\%$

Take 6 oranges, the LCM of 2 and 3. Cost $=3\times5=$ Rs 15 and selling price $=2\times7=$ Rs 14, so the loss is Rs 1.

$\text{Loss\%}=\frac{1}{15}\times100=6\frac23\%$

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2079 · Science Model · 1 mark

Let S.I. be simple interest and C.I. $_1$ , C.I. $_2$ and C.I. $_3$ be the compound interest compounded yearly, half yearly and quarterly respectively for the same principal, same time and same rate. Which of the following is true? (a) S.I.<C.I. $_1$ <C.I. $_2$ <C.I. $_3$ (b) S.I.>C.I. $_1$ >C.I. $_2$ >C.I. $_3$ (c) S.I.<C.I. $_3$ <C.I. $_2$ <C.I. $_1$ (d) C.I. $_1$ <C.I. $_2$ <S.I.<C.I. $_3$

Answer

(a) S.I.<C.I. $_1$ <C.I. $_2$ <C.I. $_3$

Simple interest does not earn interest on interest, so it is the smallest. With compound interest, the more often it is compounded in a year, the larger the amount. Hence yearly < half-yearly < quarterly.

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2079 · Science Model · 1 mark

If 75 reduced by $x\%$ is 54, the value of $x$ is: (a) 28 (b) 27 (c) 21 (d) 129

Answer

(a) 28

$75\left(1-\frac{x}{100}\right)=54\Rightarrow 1-\frac{x}{100}=0.72\Rightarrow x=28$

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2079 · Science Model · 1 mark

A road is 2 km in length. We have to supply the lamp post. One post at each end. The distance between two consecutive lamp posts is 20 m. Find the number of lamp posts required. (a) 51 (b) 100 (c) 101 (d) 102

Answer

(c) 101

The length is $2000$ m and the gap is $20$ m, so there are $\frac{2000}{20}=100$ intervals. With a post at both ends the number of posts is $100+1=101$ .

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2079 · Science Model · 1 mark

A monkey climbs a pole 34 m long. In the first minute, he climbs 6 m and in the second minute he slips down 2 m. In the third minute he again climbs 6 m and in the fourth minute he slips down 2 m. If he continues to do like this, then find the time taken by him to reach the top. (a) 15 min (b) 17 min (c) 8 min (d) None of above

Answer

(a) 15 min

In every 2 minutes the net gain is $6-2=4$ m. After 14 minutes (7 cycles) he is at $7\times4=28$ m. In the 15th minute he climbs 6 m more to reach $34$ m, the top, so he does not slip back. The time is 15 minutes.

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2079 · Science Model · 1 mark

$n$ is divided by 14 and 3. Which of the following statements must be true? (a) 21 is a factor of $n$ (b) 12 is a factor of $n$ (c) $n$ is a multiple of 42 (d) both (a) and (c)

Answer

(d) both (a) and (c)

14 and 3 are co-prime, so $n$ is a multiple of $\text{lcm}(14,3)=42$ , which is statement (c). Since $21\mid42$ , 21 is also a factor of $n$ , which is (a). But 12 need not divide $n$ (for example $n=42$ ).

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2079 · Science Model · 1 mark

If $x \ne -y$ , then $\frac{x^{36} - y^{36}}{(x^{18} + y^{18})(x^9 + y^9)} = ?$ (a) 1 (b) $x^2 - y^2$ (c) $x^9 - y^9$ (d) $x^{18} - y^{18}$

Answer

(c) $x^9 - y^9$

$x^{36}-y^{36}=(x^{18}+y^{18})(x^{18}-y^{18})=(x^{18}+y^{18})(x^9+y^9)(x^9-y^9)$

Cancelling the common factors leaves $x^9-y^9$ .

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2079 · Science Model · 1 mark

If $3^{11} = 9^x$ , what is the value of $x$ ? (a) 11 (b) $\frac{11}{2}$ (c) $\frac{2}{11}$ (d) $-11$

Answer

(b) $\frac{11}{2}$

$9^x=(3^2)^x=3^{2x}$ , so $3^{11}=3^{2x}$ gives $2x=11$ and $x=\frac{11}{2}$ .

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2079 · Science Model · 1 mark

If $x + y = -3$ and $(x^2 + y^2) = 12$ , what is the value of $2xy$ ? (a) $-3$ (b) 3 (c) 4 (d) $-4$

Answer

(a) $-3$

$(x+y)^2=x^2+y^2+2xy\Rightarrow 9=12+2xy\Rightarrow 2xy=-3$

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2079 · Science Model · 1 mark

With Rs. $y$ , 5 oranges can be bought. If all oranges cost the same, how many Rs. do 25 oranges cost in terms of $y$ ? (a) $\frac{y}{5}$ (b) $y$ (c) $y + 5$ (d) $5y$

Answer

(d) $5y$

Rs $y$ buys 5 oranges, so one orange costs Rs $\frac{y}{5}$ . Then 25 oranges cost $25\times\frac{y}{5}=5y$ .

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2079 · Science Model · 1 mark

10 men painted 3 identical houses in 5 hours, working at a constant rate. How many houses would it take 20 men to paint 12 such houses, working at the same constant rate? (a) 2.5 (b) 5 (c) 10 (d) 15

Answer

(c) 10

The question asks for the time (in hours). One man paints $\frac{3}{10\times5}=\frac{3}{50}$ house per hour. Twenty men paint $20\times\frac{3}{50}=\frac{6}{5}$ houses per hour.

$\text{Time}=\frac{12}{6/5}=10\text{ hours}$

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2079 · Science Model · 1 mark

If $f^2 g < 0$ , which of the following must be true? (a) $f < 0$ (b) $g < 0$ (c) $fg < 0$ (d) $fg > 0$

Answer

(b) $g<0$

Since $f^2\ge 0$ and the product $f^2g$ is negative, $f^2\ne0$ so $f^2>0$ , and therefore $g<0$ . The sign of $f$ is not fixed, so $f<0$ , $fg<0$ and $fg>0$ need not hold.

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2079 · Science Model · 1 mark

PQRS is a square having a diagonal of length $6\sqrt{3}$ cm. If RS is produced to T, what is the area of $\triangle PQT$ ? (a) $30\ \text{cm}^2$ (b) $27\ \text{cm}^2$ (c) $54\ \text{cm}^2$ (d) $18\ \text{cm}^2$

Answer

(b) $27\ \text{cm}^2$

For a square of side $s$ , the diagonal is $s\sqrt2=6\sqrt3$ , so $s^2=\frac{(6\sqrt3)^2}{2}=\frac{108}{2}=54.$ T lies on line RS, which is parallel to PQ, so the height of $\triangle PQT$ from T to PQ equals the side $s$ . Hence $\text{Area}=\tfrac12\cdot PQ\cdot s=\tfrac12 s^2=27\ \text{cm}^2.$

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2079 · Science Model · 1 mark

Three angles $a^\circ$ , $b^\circ$ and $c^\circ$ lie on a straight line, with $a : b = 3 : 5$ and $b : c = 5 : 1$ . What is the measure of the largest angle? (a) $30^\circ$ (b) $45^\circ$ (c) $50^\circ$ (d) $100^\circ$

Answer

(d) $100^\circ$

From $a:b=3:5$ and $b:c=5:1$ we get $a:b:c=3:5:1$ . Angles on a straight line add to $180^\circ$ , so one part is $\frac{180^\circ}{9}=20^\circ$ . The largest angle is $b=5\times20^\circ=100^\circ$ .

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2079 · Science Model · 1 mark

$3\left(\frac{N+1}{4}\right)^{th}$ item represents the position of (a) $Q_1$ (b) $Q_2$ (c) $Q_3$ (d) None of above

Answer

(c) $Q_3$

The quartile positions are $Q_1=\left(\frac{N+1}{4}\right)^{th}$ , $Q_2=2\left(\frac{N+1}{4}\right)^{th}$ and $Q_3=3\left(\frac{N+1}{4}\right)^{th}$ item.

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2079 · Science Model · 1 mark

The average of 8 numbers is 42. One of the numbers is removed from the set and the resulting average of the remaining numbers is 40. What number was removed from the set? (a) 26 (b) 28 (c) 50 (d) 56

Answer

(d) 56

Sum of the 8 numbers $=8\times42=336$ . Sum of the remaining 7 numbers $=7\times40=280$ . $\text{Removed number}=336-280=56.$

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2079 · Science Model · 1 mark

One card is drawn at random from the number cards, numbered from 10 to 21. Find the probability that the card may be prime or even numbered card. (a) $\frac{1}{6}$ (b) $\frac{5}{6}$ (c) $\frac{3}{4}$ (d) $\frac{2}{3}$

Answer

(b) $\frac{5}{6}$

The sample space $\{10,11,\dots,21\}$ has 12 cards. Primes: 11, 13, 17, 19 (4 cards). Even numbers: 10, 12, 14, 16, 18, 20 (6 cards). No number is both prime and even here, so the favourable count is $4+6=10$ . $P=\frac{10}{12}=\frac56$

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2079 · Science Model · 1 mark

What is the probability of getting 3 on the die and head on the coin when a die is rolled and a coin is tossed together? (a) 1 (b) $\frac{1}{2}$ (c) $\frac{1}{12}$ (d) $\frac{2}{3}$

Answer

(c) $\frac{1}{12}$

The events are independent: $P(3)=\frac16$ and $P(\text{head})=\frac12$ , so $P=\frac16\times\frac12=\frac1{12}.$

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2079 · Science Model · 1 mark

A shirt that costs Rs. $K$ is increased by 30% and then by an additional 50%. What is the new price of the shirt in Rs. in terms of $K$ ? (a) $0.2K$ (b) $0.35K$ (c) $1.8K$ (d) $1.95K$

Answer

(d) $1.95K$

The first increase gives $1.30K$ . The further 50% increase gives $1.30K\times1.50=1.95K$ .

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2079 · Science Model · 1 mark

In $\triangle ABC$ , P, Q and R are the mid points of BC, AC and AB respectively. Then $\overrightarrow{AP} + \overrightarrow{BQ} + \overrightarrow{CR}$ is equal to (a) 0 (b) 1 (c) $\overrightarrow{AC}$ (d) $\overrightarrow{AB}$

Answer

(a) 0

Take the medians $AP$ , $BQ$ , $CR$ . With position vectors, $\overrightarrow{AP}=\frac{\mathbf b+\mathbf c}{2}-\mathbf a$ , $\overrightarrow{BQ}=\frac{\mathbf a+\mathbf c}{2}-\mathbf b$ , $\overrightarrow{CR}=\frac{\mathbf a+\mathbf b}{2}-\mathbf c$ . Adding, $\overrightarrow{AP}+\overrightarrow{BQ}+\overrightarrow{CR}=(\mathbf a+\mathbf b+\mathbf c)-(\mathbf a+\mathbf b+\mathbf c)=\mathbf 0.$

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2079 · Science Model · 1 mark

If $f(x) = \begin{cases} 4x + 1 & \text{for } x < 2 \\ 1 - x & \text{for } x \ge 2 \end{cases}$ , the value of $f(1)$ is (a) 0 (b) 5 (c) 2 (d) 3

Answer

(b) 5

Since $1<2$ , use $f(x)=4x+1$ : $f(1)=4(1)+1=5$ .

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2079 · Science Model · 1 mark

The side $QR$ of an equilateral $\triangle PQR$ is parallel to the $x$ -axis. What is the slope of $PR$ ? (a) $\sqrt{3}$ (b) $-\sqrt{3}$ (c) $\frac{1}{\sqrt{3}}$ (d) $-\frac{1}{\sqrt{3}}$

Answer

(b) $-\sqrt3$

In an equilateral triangle every angle is $60^\circ$ . With $QR$ along the $x$ -axis (Q on the left, R on the right, P above), the side $PR$ makes an angle of $60^\circ$ with $RQ$ , so its inclination is $180^\circ-60^\circ=120^\circ$ . $\text{slope}=\tan120^\circ=-\sqrt3$

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2079 · Science Model · 1 mark

If the matrices $\begin{pmatrix} m & 4 \\ -1 & 3 \end{pmatrix}$ and $\begin{pmatrix} 3 & -4 \\ 1 & m \end{pmatrix}$ are inverse to each other, what is the value of $m$ ? (a) 0 (b) 1 (c) $-1$ (d) All of above

Answer

(c) $-1$

Two matrices are inverses if their product is the identity. $\begin{pmatrix} m & 4 \\ -1 & 3 \end{pmatrix}\begin{pmatrix} 3 & -4 \\ 1 & m \end{pmatrix}=\begin{pmatrix} 3m+4 & -4m+4m \\ -3+3 & 4+3m \end{pmatrix}=\begin{pmatrix} 3m+4 & 0 \\ 0 & 3m+4 \end{pmatrix}$ For this to equal $I$ , $3m+4=1$ , so $m=-1$ .

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2079 · Science Model · 1 mark

The sum of the first $n$ natural numbers is: (a) $n$ (b) $\frac{n+1}{2}$ (c) $\frac{n(n+1)}{2}$ (d) $n^n$

Answer

(c) $\frac{n(n+1)}{2}$

$1+2+\dots+n=\frac{n(n+1)}{2}$ . Check with $n=3$ : $1+2+3=6=\frac{3\cdot4}{2}$ .

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2079 · Science Model · 1 mark

The image of the point $(-6, 2)$ under the rotation through $-270^\circ$ about the origin is (a) $(6, -2)$ (b) $(-2, -6)$ (c) $(6, 2)$ (d) $(2, 6)$

Answer

(b) $(-2,-6)$

A rotation through $-270^\circ$ (clockwise) is the same as $+90^\circ$ (anticlockwise). Under a $+90^\circ$ rotation about the origin, $(x,y)\to(-y,x)$ , so $(-6,2)\to(-2,-6)$ .

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2079 · Science Model · 1 mark

The value of $\frac{1 + \tan 15^\circ}{1 - \tan 15^\circ}$ is (a) $\sqrt{3}$ (b) $\frac{1}{\sqrt{3}}$ (c) $-\sqrt{3}$ (d) $-\frac{1}{\sqrt{3}}$

Answer

(a) $\sqrt3$

$\frac{1+\tan15^\circ}{1-\tan15^\circ}=\frac{\tan45^\circ+\tan15^\circ}{1-\tan45^\circ\tan15^\circ}=\tan(45^\circ+15^\circ)=\tan60^\circ=\sqrt3.$

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Past questions and answers

1.The graphs of the equations 3x+6y=93x + 6y = 93x+6y=9 and 9x+18y=279x + 18y = 279x+18y=27 are (a) intersecting (b) parallel (c) coincident (d) perpendicular

2.The graphs of the equations 3x+2y=53x + 2y = 53x+2y=5 and 2x−y=112x - y = 112x−y=11 are (a) intersecting (b) parallel (c) coincident (d) None

3.The graphs of the equations 3x+6y=93x + 6y = 93x+6y=9 and 9x+18y=279x + 18y = 279x+18y=27 are (a) coincident (b) parallel (c) intersecting (d) perpendicular

4.The factors of 6x2+x−126x^2 + x - 126x2+x−12 are (a) (3x+4)(2x−3)(3x+4)(2x-3)(3x+4)(2x−3) (b) (3x−4)(2x+3)(3x-4)(2x+3)(3x−4)(2x+3) (c) (3x−4)(2x−4)(3x-4)(2x-4)(3x−4)(2x−4) (d) (3x+4)(2x+4)(3x+4)(2x+4)(3x+4)(2x+4)

5.HCF of two numbers is 21 and their product is 2646. How many pairs of such numbers are possible? (a) 3 (b) 1 (c) 2 (d) None

6.The number of distinct prime factors of 210 is (a) 1 (b) 2 (c) 3 (d) 4

7.HCF of two numbers is 101 and their sum is 1212. How many pairs of such numbers are possible? (a) 3 (b) 2 (c) 1 (d) None

8.The number of distinct prime factors of 27 is (a) 1 (b) 2 (c) 3 (d) 4

9.xn−ynx^n - y^nxn−yn is divisible by x+yx + yx+y for (a) All values of nnn (b) Only even values of nnn (c) Only odd values of nnn (d) No value of nnn

10.an+bna^n + b^nan+bn is divisible by a+ba + ba+b (a) For all values of nnn (b) Only for even values of nnn (c) Only for odd values of nnn (d) For no value of nnn

11.xn+ynx^n + y^nxn+yn is divisible by x−yx - yx−y (a) For all values of nnn (b) For even values of nnn (c) For odd values of nnn (d) For no value of nnn

12.If £1 = Rs 135 and USD 1 = Rs 105, then convert £63 into dollar. (a) USD 63 (b) USD 30 (c) USD 81 (d) USD 36

13.Find the probability of occurring a squared number or a cubed number while drawing a flash card from the set of cards numbered from 3 to 32. (a) 14\frac{1}{4}41​ (b) 15\frac{1}{5}51​ (c) 12\frac{1}{2}21​ (d) 13\frac{1}{3}31​

14.The mean of the first nnn natural numbers is (a) n+12\frac{n+1}{2}2n+1​ (b) n(n+1)2\frac{n(n+1)}{2}2n(n+1)​ (c) n2\frac{n}{2}2n​ (d) n2+12\frac{n^2+1}{2}2n2+1​

15.The value of ccc for which the equation 2x2−9x+c=02x^2 - 9x + c = 02x2−9x+c=0 has equal roots is (a) 818\frac{81}{8}881​ (b) 881\frac{8}{81}818​ (c) 9 (d) 3

16.The value of probability ranges from (a) 0 to 1 (b) −1-1−1 to 1 (c) 0.5 to 1 (d) 0.1 to 1

17.Which of the following pairs is co-prime? (a) (3, 9) (b) (3, 7) (c) (6, 2) (d) (5, 25)

18.A sector of a circle has a central angle of 270∘270^\circ270∘. If the circle has a radius of 4, what is the area of the sector? (a) 4π4\pi4π (b) 8π8\pi8π (c) 12π12\pi12π (d) 16π16\pi16π

19.If 1,x,y,z,161, x, y, z, 161,x,y,z,16 are in G.P., what is the value of x+y+zx + y + zx+y+z? (a) 8 (b) 12 (c) 14 (d) 16

20.What is the area of a circle whose circumference is π\piπ? (a) π4\frac{\pi}{4}4π​ (b) π2\frac{\pi}{2}2π​ (c) π\piπ (d) 2π2\pi2π

21.If y2=4y^2 = 4y2=4 and x2y=18x^2 y = 18x2y=18, x+yx + yx+y could equal which of the following values? (a) −5-5−5 (b) −1-1−1 (c) 1 (d) 6

22.The average of a set of 18 consecutive integers is 22.5. What is the smallest integer in the set? (a) 14 (b) 16 (c) 12 (d) 28

23.58\frac{5}{8}85​ of 24 is equal to 157\frac{15}{7}715​ of what number? (a) 7 (b) 8 (c) 15 (d) 7225\frac{7}{225}2257​

24.If there are 24 white marbles and 36 blue marbles in a bag, what is the ratio of blue to white marbles? (a) 23\frac{2}{3}32​ (b) 32\frac{3}{2}23​ (c) 49\frac{4}{9}94​ (d) 94\frac{9}{4}49​

25.What is the probability of getting 53 Sundays in a non-leap year? (a) 17\frac{1}{7}71​ (b) 27\frac{2}{7}72​ (c) 37\frac{3}{7}73​ (d) 47\frac{4}{7}74​

26.In a group of 100 people, 70 can speak Nepali and 45 speak English, then the number of people who can speak English only. (a) 30 (b) 40 (c) 50 (d) 60

27.Two identical pies are cut into a total of 16 equal parts. If each part is then split equally among three people, what fraction of a pie does each person receive? (a) 148\frac{1}{48}481​ (b) 124\frac{1}{24}241​ (c) 116\frac{1}{16}161​ (d) 316\frac{3}{16}163​

28.Two pipes X and Y can fill a cistern in 24 minutes and 32 minutes respectively. If both the pipes are opened together, then after how much time should Y be closed so that the tank is full in 18 minutes? (a) 8 min. (b) 10 min. (c) 18 min. (d) 20 min.

29.A trade man charges 20% above the cost price but allowed a discount of 10% for cash payment. What is the gain percent? (a) 8% (b) 10% (c) 20% (d) 30%

30.If a%a\%a% of a=4a = 4a=4 and a%a\%a% of b=40b = 40b=40, find the positive value of bbb? (a) 20 (b) 200 (c) 2000 (d) All of above

31.If a right circular cylinder's radius is halved and its height doubled, by what percent will the volume increase or decrease? (a) 50% decrease (b) No change (c) 25% increase (d) 50% increase

32.For what value of xxx is 63+83+103=x36^3 + 8^3 + 10^3 = x^363+83+103=x3 true? (a) 12 (b) 24 (c) 9 (d) 7

33.The values of xxx and yyy in x+y=7x + y = 7x+y=7 and 3x−2y=113x - 2y = 113x−2y=11 are (a) 2, 5 (b) 5, 2 (c) 3, 4 (d) 4, 3

34.If kkk, 5 and 25 are in continued proportion, what is the value of k3−1k^3 - 1k3−1? (a) 0 (b) 1 (c) 5 (d) 125

35.The following observations have been arranged in ascending order: 29, 32, 48, 52, xxx, x+3x+3x+3, 71, 75, 80, 92. If the median of the data is 61.5, what is the value of xxx? (a) 54 (b) 60 (c) 62 (d) 56

36.Which of the following inequalities is equivalent to ∣m+2∣<3|m + 2| < 3∣m+2∣<3? (a) m<5m < 5m<5 (b) m<1m < 1m<1 (c) −5<m<5-5 < m < 5−5<m<5 (d) −5<m<1-5 < m < 1−5<m<1

37.A trade man allows a discount of 10%. What price should he mark on an article which costs Rs. 200 to gain 26%? (a) Rs. 200 (b) Rs. 280 (c) Rs. 300 (d) Rs. 350

38.P(A)+P(Aˉ)P(A) + P(\bar{A})P(A)+P(Aˉ) is equal to (a) 0 (b) 1 (c) 0.75 (d) 0.5

39.How many 1-inch square tiles would it take to cover the floor of a closet that has dimensions 5 feet by 4 feet? (1 foot = 12 inches) (a) 20 (b) 240 (c) 1440 (d) 2880

40.xxx and yyy are positive numbers such that x+y+z<1x + y + z < 1x+y+z<1 and xy=1xy = 1xy=1. Which could be the value of zzz? (a) z<−1z < -1z<−1 (b) z>−1z > -1z>−1 (c) z=−1z = -1z=−1 (d) z>1z > 1z>1

41.On what sum will the simple interest for 3 years at 5% p.a. amount to Rs. 150? (a) Rs. 1000 (b) Rs. 1500 (c) Rs. 1050 (d) Rs. 1550

42.If 34\frac{3}{4}43​ of a number is 7 more than 16\frac{1}{6}61​ of that number, what is 53\frac{5}{3}35​ of that number? (a) 12 (b) 15 (c) 18 (d) 20

43.The mean of 100 items is 49. If it was found that three items which should have been 60, 70, 80 were wrongly read as 40, 20, 50 respectively, the correct mean is (a) 48 (b) 82 (c) 50 (d) 80

45.Which of the following statements is true? (a) −(−3)2<0-\sqrt{(-3)^2} < 0−(−3)2​<0 (b) (−1)87=−1(-1)^{87} = -1(−1)87=−1 (c) 16=4\sqrt{16} = 416​=4 (d) All of above

46.If a=16ba = 16ba=16b and bbb is a prime number greater than 2, how many positive distinct factors does aaa have? (a) 5 (b) 10 (c) 15 (d) 20

47.If the sum of five consecutive integers is SSS, what is the largest of those integers in terms of SSS? (a) S−105\frac{S-10}{5}5S−10​ (b) S+105\frac{S+10}{5}5S+10​ (c) S−52\frac{S-5}{2}2S−5​ (d) S+54\frac{S+5}{4}4S+5​

48.In a continuous series, N=30N = 30N=30 and the cumulative frequency of the class 20-30 is 15. Find the median class. (a) 10-20 (b) 20-30 (c) 30-40 (d) 20.5-30.5

49.If the sum of five consecutive integers is SSS, what is the largest of those integers in terms of SSS? (a) S−105\frac{S-10}{5}5S−10​ (b) S+44\frac{S+4}{4}4S+4​ (c) S+105\frac{S+10}{5}5S+10​ (d) S−52\frac{S-5}{2}2S−5​

50.If a=16ba = 16ba=16b and bbb is a prime number greater than 2, how many positive distinct factors does aaa have? (a) 5 (b) 20 (c) 15 (d) 2

52.If ab>cd\frac{a}{b} > \frac{c}{d}ba​>dc​, which of the following statements must be true? (a) ab−cd>0\frac{a}{b} - \frac{c}{d} > 0ba​−dc​>0 (b) ad<bcad < bcad<bc (c) ad>bcad > bcad>bc (d) All of above

53.In a continuous series, ∑f=100\sum f = 100∑f=100, the cumulative frequency of the class 15-20 is 46 and the cumulative frequency of the class 20-25 is 52. Find the median class of the data. (a) 15-20 (b) 10-15 (c) 20-25 (d) 25-30

54.What is the average (arithmetic mean) of the measures of the five angles in a pentagon? (a) 36∘36^\circ36∘ (b) 72∘72^\circ72∘ (c) 92∘92^\circ92∘ (d) 100∘100^\circ100∘

55.Find the LCM of a2ba^2ba2b and a2b2a^2b^2a2b2. (a) ababab (b) a2ba^2ba2b (c) a2b2a^2b^2a2b2 (d) a4b3a^4b^3a4b3

56.What is the square root of 9+459 + 4\sqrt{5}9+45​? (a) 3+253 + 2\sqrt{5}3+25​ (b) 2+52 + \sqrt{5}2+5​ (c) 3+453 + 4\sqrt{5}3+45​ (d) None of above

57.If a+b=3(c+d)a + b = 3(c + d)a+b=3(c+d), which of the following is the average (arithmetic mean) of a,b,ca, b, ca,b,c and ddd? (a) c+d4\frac{c+d}{4}4c+d​ (b) 3(c+d)4\frac{3(c+d)}{4}43(c+d)​ (c) c+d2\frac{c+d}{2}2c+d​ (d) c+dc + dc+d

58.If the mean of aaa and 1a\frac{1}{a}a1​ is MMM, then what is the mean of a3a^3a3 and 1a3\frac{1}{a^3}a31​? (a) M(M2−3)3\frac{M(M^2-3)}{3}3M(M2−3)​ (b) M3M^3M3 (c) M3−3M^3 - 3M3−3 (d) M(4M2−3)M(4M^2 - 3)M(4M2−3)

59.Which of the following sets are equivalent? (a) ϕ\phiϕ and {ϕ}\{\phi\}{ϕ} (b) {0}\{0\}{0} and {ϕ}\{\phi\}{ϕ} (c) ϕ\phiϕ and {0}\{0\}{0} (d) None

60.Find the compound interest on Rs 32000 at 20% p.a. for 1 year compounded half yearly. (a) Rs. 6320 (b) Rs. 6720 (c) Rs. 6400 (d) Rs. 6520

61.What single discount is equivalent to two successive discounts of 10% and 15%? (a) 23% (b) 23.5% (c) 24% (d) 24.5%

62.A sector of a circle has a radius of 8 and an area of 8π8\pi8π. What is the arc length of the sector? (a) π\piπ (b) 2π2\pi2π (c) 4π4\pi4π (d) 6π6\pi6π

63.The HCF of x3+1x3x^3 + \frac{1}{x^3}x3+x31​ and x4+1+1x4x^4 + 1 + \frac{1}{x^4}x4+1+x41​ is (a) 111 (b) x+1xx + \frac{1}{x}x+x1​ (c) x2+1+1x2x^2 + 1 + \frac{1}{x^2}x2+1+x21​ (d) x2−1+1x2x^2 - 1 + \frac{1}{x^2}x2−1+x21​

64.For what value of xxx is 43x−6\frac{4}{3x-6}3x−64​ undefined? (a) 0 (b) 1 (c) 2 (d) 3

65.0.6‾+0.7‾+0.8‾=?0.\overline{6} + 0.\overline{7} + 0.\overline{8} = ?0.6+0.7+0.8=? (a) 2.3‾2.\overline{3}2.3 (b) 2.1‾2.\overline{1}2.1 (c) 2.2‾2.\overline{2}2.2 (d) None

66.Solve: 2x−14=−1\sqrt[4]{2x-1} = -142x−1​=−1 (a) 1 (b) 2 (c) 3 (d) No solution

67.A has 6 more marbles than B. If B gives 2 marbles to C and then B buys 5 new marbles, how many more marbles does A have than B? (a) 3 (b) 4 (c) 5 (d) 6

68.The quadratic equation whose roots are (3+5)(3+\sqrt{5})(3+5​) and (3−5)(3-\sqrt{5})(3−5​) is (a) x2−6x+2=0x^2 - 6x + 2 = 0x2−6x+2=0 (b) x2−4x+6=0x^2 - 4x + 6 = 0x2−4x+6=0 (c) x2−6x+4=0x^2 - 6x + 4 = 0x2−6x+4=0 (d) x2+6x+4=0x^2 + 6x + 4 = 0x2+6x+4=0

69.If x=6+35x = 6 + \sqrt{35}x=6+35​, then the value of x2+1x2x^2 + \frac{1}{x^2}x2+x21​ is (a) 241 (b) 284 (c) 142 (d) 140

70.If xy>0xy > 0xy>0 and yz<0yz < 0yz<0, which of the following is true? (a) xz<0xz < 0xz<0 (b) xz>0xz > 0xz>0 (c) xz=0xz = 0xz=0 (d) None of above

71.For what value of xxx is the equation abx−2=bax−2ab^{x-2} = ba^{x-2}abx−2=bax−2 true? (a) −3-3−3 (b) −2-2−2 (c) 222 (d) 333

72.If x1a=y15x^{\frac{1}{a}} = y^{\frac{1}{5}}xa1​=y51​ and xy=1xy = 1xy=1, what is the value of aaa? (a) −2-2−2 (b) −3-3−3 (c) −4-4−4 (d) −5-5−5

73.There are two numbers. If we add one to each number, their ratio becomes 2:3. If 1 is decreased from each number, their ratio becomes 1:2. The numbers are (a) 3, 1 (b) 1, 3 (c) 1, 5 (d) none of above

74.A perfect cube has surface area 96 cm296\ \text{cm}^296 cm2. What is its volume (in cm3\text{cm}^3cm3)? (a) 64 (b) 80 (c) 48 (d) 96

75.In triangle ABC, ∠B=30∘\angle B = 30^\circ∠B=30∘ and BC=12BC = 12BC=12 cm. If the area of △ABC\triangle ABC△ABC is 27 cm227\ \text{cm}^227 cm2, find the length of AB. (a) 12 cm (b) 6 cm (c) 9 cm (d) None of above

76.The three distinct points A, B and C are said to be collinear if (a) AC=AB+BCAC = AB + BCAC=AB+BC (b) slope of ABABAB = slope of BCBCBC (c) Area of △ABC=0\triangle ABC = 0△ABC=0 (d) All of above

77.What is the maximum value of y=sin⁡xy = \sin xy=sinx? (a) −1-1−1 (b) 111 (c) 000 (d) ∞\infty∞

78.A right triangle is inscribed in a circle with an area of 16π cm216\pi\ \text{cm}^216π cm2. What is the hypotenuse of the triangle in cm? (a) 4π4\pi4π (b) 444 (c) 8π8\pi8π (d) 888

79.If f(x)f(x)f(x) is divided by (b−ax)(b - ax)(b−ax), then the remainder is (a) f(ba)f\left(\frac{b}{a}\right)f(ab​) (b) f(ab)f\left(\frac{a}{b}\right)f(ba​) (c) f(−ba)f\left(-\frac{b}{a}\right)f(−ab​) (d) f(−ab)f\left(-\frac{a}{b}\right)f(−ba​)

80.If the sum of the first nnn positive integers is 15\frac{1}{5}51​ of the sum of the squares of the first nnn positive integers, then nnn is (a) 23 (b) 14 (c) 7 (d) 9

81.Which of the following parabolas is the narrowest? (a) y=x2y = x^2y=x2 (b) y=x22y = \frac{x^2}{2}y=2x2​ (c) y=2x2y = 2x^2y=2x2 (d) y=x24y = \frac{x^2}{4}y=4x2​

The graphs of the equations $3x + 6y = 9$ and $9x + 18y = 27$ are (a) intersecting (b) parallel (c) coincident (d) perpendicular

The graphs of the equations $3x + 2y = 5$ and $2x - y = 11$ are (a) intersecting (b) parallel (c) coincident (d) None

The graphs of the equations $3x + 6y = 9$ and $9x + 18y = 27$ are (a) coincident (b) parallel (c) intersecting (d) perpendicular

The factors of $6x^2 + x - 12$ are (a) $(3x+4)(2x-3)$ (b) $(3x-4)(2x+3)$ (c) $(3x-4)(2x-4)$ (d) $(3x+4)(2x+4)$

HCF of two numbers is 21 and their product is 2646. How many pairs of such numbers are possible? (a) 3 (b) 1 (c) 2 (d) None

The number of distinct prime factors of 210 is (a) 1 (b) 2 (c) 3 (d) 4

HCF of two numbers is 101 and their sum is 1212. How many pairs of such numbers are possible? (a) 3 (b) 2 (c) 1 (d) None

The number of distinct prime factors of 27 is (a) 1 (b) 2 (c) 3 (d) 4

$x^n - y^n$ is divisible by $x + y$ for (a) All values of $n$ (b) Only even values of $n$ (c) Only odd values of $n$ (d) No value of $n$

$a^n + b^n$ is divisible by $a + b$ (a) For all values of $n$ (b) Only for even values of $n$ (c) Only for odd values of $n$ (d) For no value of $n$

$x^n + y^n$ is divisible by $x - y$ (a) For all values of $n$ (b) For even values of $n$ (c) For odd values of $n$ (d) For no value of $n$

If £1 = Rs 135 and USD 1 = Rs 105, then convert £63 into dollar. (a) USD 63 (b) USD 30 (c) USD 81 (d) USD 36

Find the probability of occurring a squared number or a cubed number while drawing a flash card from the set of cards numbered from 3 to 32. (a) $\frac{1}{4}$ (b) $\frac{1}{5}$ (c) $\frac{1}{2}$ (d) $\frac{1}{3}$

The mean of the first $n$ natural numbers is (a) $\frac{n+1}{2}$ (b) $\frac{n(n+1)}{2}$ (c) $\frac{n}{2}$ (d) $\frac{n^2+1}{2}$

The value of $c$ for which the equation $2x^2 - 9x + c = 0$ has equal roots is (a) $\frac{81}{8}$ (b) $\frac{8}{81}$ (c) 9 (d) 3

The value of probability ranges from (a) 0 to 1 (b) $-1$ to 1 (c) 0.5 to 1 (d) 0.1 to 1

Which of the following pairs is co-prime? (a) (3, 9) (b) (3, 7) (c) (6, 2) (d) (5, 25)

A sector of a circle has a central angle of $270^\circ$ . If the circle has a radius of 4, what is the area of the sector? (a) $4\pi$ (b) $8\pi$ (c) $12\pi$ (d) $16\pi$

If $1, x, y, z, 16$ are in G.P., what is the value of $x + y + z$ ? (a) 8 (b) 12 (c) 14 (d) 16

What is the area of a circle whose circumference is $\pi$ ? (a) $\frac{\pi}{4}$ (b) $\frac{\pi}{2}$ (c) $\pi$ (d) $2\pi$

If $y^2 = 4$ and $x^2 y = 18$ , $x + y$ could equal which of the following values? (a) $-5$ (b) $-1$ (c) 1 (d) 6

The average of a set of 18 consecutive integers is 22.5. What is the smallest integer in the set? (a) 14 (b) 16 (c) 12 (d) 28

$\frac{5}{8}$ of 24 is equal to $\frac{15}{7}$ of what number? (a) 7 (b) 8 (c) 15 (d) $\frac{7}{225}$

If there are 24 white marbles and 36 blue marbles in a bag, what is the ratio of blue to white marbles? (a) $\frac{2}{3}$ (b) $\frac{3}{2}$ (c) $\frac{4}{9}$ (d) $\frac{9}{4}$

What is the probability of getting 53 Sundays in a non-leap year? (a) $\frac{1}{7}$ (b) $\frac{2}{7}$ (c) $\frac{3}{7}$ (d) $\frac{4}{7}$

In a group of 100 people, 70 can speak Nepali and 45 speak English, then the number of people who can speak English only. (a) 30 (b) 40 (c) 50 (d) 60

Two identical pies are cut into a total of 16 equal parts. If each part is then split equally among three people, what fraction of a pie does each person receive? (a) $\frac{1}{48}$ (b) $\frac{1}{24}$ (c) $\frac{1}{16}$ (d) $\frac{3}{16}$

Two pipes X and Y can fill a cistern in 24 minutes and 32 minutes respectively. If both the pipes are opened together, then after how much time should Y be closed so that the tank is full in 18 minutes? (a) 8 min. (b) 10 min. (c) 18 min. (d) 20 min.

A trade man charges 20% above the cost price but allowed a discount of 10% for cash payment. What is the gain percent? (a) 8% (b) 10% (c) 20% (d) 30%

If $a\%$ of $a = 4$ and $a\%$ of $b = 40$ , find the positive value of $b$ ? (a) 20 (b) 200 (c) 2000 (d) All of above

If a right circular cylinder's radius is halved and its height doubled, by what percent will the volume increase or decrease? (a) 50% decrease (b) No change (c) 25% increase (d) 50% increase

For what value of $x$ is $6^3 + 8^3 + 10^3 = x^3$ true? (a) 12 (b) 24 (c) 9 (d) 7

The values of $x$ and $y$ in $x + y = 7$ and $3x - 2y = 11$ are (a) 2, 5 (b) 5, 2 (c) 3, 4 (d) 4, 3

If $k$ , 5 and 25 are in continued proportion, what is the value of $k^3 - 1$ ? (a) 0 (b) 1 (c) 5 (d) 125

The following observations have been arranged in ascending order: 29, 32, 48, 52, $x$ , $x+3$ , 71, 75, 80, 92. If the median of the data is 61.5, what is the value of $x$ ? (a) 54 (b) 60 (c) 62 (d) 56

Which of the following inequalities is equivalent to $|m + 2| < 3$ ? (a) $m < 5$ (b) $m < 1$ (c) $-5 < m < 5$ (d) $-5 < m < 1$

A trade man allows a discount of 10%. What price should he mark on an article which costs Rs. 200 to gain 26%? (a) Rs. 200 (b) Rs. 280 (c) Rs. 300 (d) Rs. 350

$P(A) + P(\bar{A})$ is equal to (a) 0 (b) 1 (c) 0.75 (d) 0.5

How many 1-inch square tiles would it take to cover the floor of a closet that has dimensions 5 feet by 4 feet? (1 foot = 12 inches) (a) 20 (b) 240 (c) 1440 (d) 2880

$x$ and $y$ are positive numbers such that $x + y + z < 1$ and $xy = 1$ . Which could be the value of $z$ ? (a) $z < -1$ (b) $z > -1$ (c) $z = -1$ (d) $z > 1$

On what sum will the simple interest for 3 years at 5% p.a. amount to Rs. 150? (a) Rs. 1000 (b) Rs. 1500 (c) Rs. 1050 (d) Rs. 1550

If $\frac{3}{4}$ of a number is 7 more than $\frac{1}{6}$ of that number, what is $\frac{5}{3}$ of that number? (a) 12 (b) 15 (c) 18 (d) 20

The mean of 100 items is 49. If it was found that three items which should have been 60, 70, 80 were wrongly read as 40, 20, 50 respectively, the correct mean is (a) 48 (b) 82 (c) 50 (d) 80

Which of the following statements is true? (a) $-\sqrt{(-3)^2} < 0$ (b) $(-1)^{87} = -1$ (c) $\sqrt{16} = 4$ (d) All of above

If $a = 16b$ and $b$ is a prime number greater than 2, how many positive distinct factors does $a$ have? (a) 5 (b) 10 (c) 15 (d) 20

If the sum of five consecutive integers is $S$ , what is the largest of those integers in terms of $S$ ? (a) $\frac{S-10}{5}$ (b) $\frac{S+10}{5}$ (c) $\frac{S-5}{2}$ (d) $\frac{S+5}{4}$

In a continuous series, $N = 30$ and the cumulative frequency of the class 20-30 is 15. Find the median class. (a) 10-20 (b) 20-30 (c) 30-40 (d) 20.5-30.5

If the sum of five consecutive integers is $S$ , what is the largest of those integers in terms of $S$ ? (a) $\frac{S-10}{5}$ (b) $\frac{S+4}{4}$ (c) $\frac{S+10}{5}$ (d) $\frac{S-5}{2}$

If $a = 16b$ and $b$ is a prime number greater than 2, how many positive distinct factors does $a$ have? (a) 5 (b) 20 (c) 15 (d) 2

If $\frac{a}{b} > \frac{c}{d}$ , which of the following statements must be true? (a) $\frac{a}{b} - \frac{c}{d} > 0$ (b) $ad < bc$ (c) $ad > bc$ (d) All of above

In a continuous series, $\sum f = 100$ , the cumulative frequency of the class 15-20 is 46 and the cumulative frequency of the class 20-25 is 52. Find the median class of the data. (a) 15-20 (b) 10-15 (c) 20-25 (d) 25-30

What is the average (arithmetic mean) of the measures of the five angles in a pentagon? (a) $36^\circ$ (b) $72^\circ$ (c) $92^\circ$ (d) $100^\circ$

Find the LCM of $a^2b$ and $a^2b^2$ . (a) $ab$ (b) $a^2b$ (c) $a^2b^2$ (d) $a^4b^3$

What is the square root of $9 + 4\sqrt{5}$ ? (a) $3 + 2\sqrt{5}$ (b) $2 + \sqrt{5}$ (c) $3 + 4\sqrt{5}$ (d) None of above

If $a + b = 3(c + d)$ , which of the following is the average (arithmetic mean) of $a, b, c$ and $d$ ? (a) $\frac{c+d}{4}$ (b) $\frac{3(c+d)}{4}$ (c) $\frac{c+d}{2}$ (d) $c + d$

If the mean of $a$ and $\frac{1}{a}$ is $M$ , then what is the mean of $a^3$ and $\frac{1}{a^3}$ ? (a) $\frac{M(M^2-3)}{3}$ (b) $M^3$ (c) $M^3 - 3$ (d) $M(4M^2 - 3)$

Which of the following sets are equivalent? (a) $\phi$ and $\{\phi\}$ (b) $\{0\}$ and $\{\phi\}$ (c) $\phi$ and $\{0\}$ (d) None

Find the compound interest on Rs 32000 at 20% p.a. for 1 year compounded half yearly. (a) Rs. 6320 (b) Rs. 6720 (c) Rs. 6400 (d) Rs. 6520

What single discount is equivalent to two successive discounts of 10% and 15%? (a) 23% (b) 23.5% (c) 24% (d) 24.5%

A sector of a circle has a radius of 8 and an area of $8\pi$ . What is the arc length of the sector? (a) $\pi$ (b) $2\pi$ (c) $4\pi$ (d) $6\pi$

The HCF of $x^3 + \frac{1}{x^3}$ and $x^4 + 1 + \frac{1}{x^4}$ is (a) $1$ (b) $x + \frac{1}{x}$ (c) $x^2 + 1 + \frac{1}{x^2}$ (d) $x^2 - 1 + \frac{1}{x^2}$

For what value of $x$ is $\frac{4}{3x-6}$ undefined? (a) 0 (b) 1 (c) 2 (d) 3

$0.\overline{6} + 0.\overline{7} + 0.\overline{8} = ?$ (a) $2.\overline{3}$ (b) $2.\overline{1}$ (c) $2.\overline{2}$ (d) None

Solve: $\sqrt[4]{2x-1} = -1$ (a) 1 (b) 2 (c) 3 (d) No solution

A has 6 more marbles than B. If B gives 2 marbles to C and then B buys 5 new marbles, how many more marbles does A have than B? (a) 3 (b) 4 (c) 5 (d) 6

The quadratic equation whose roots are $(3+\sqrt{5})$ and $(3-\sqrt{5})$ is (a) $x^2 - 6x + 2 = 0$ (b) $x^2 - 4x + 6 = 0$ (c) $x^2 - 6x + 4 = 0$ (d) $x^2 + 6x + 4 = 0$

If $x = 6 + \sqrt{35}$ , then the value of $x^2 + \frac{1}{x^2}$ is (a) 241 (b) 284 (c) 142 (d) 140

If $xy > 0$ and $yz < 0$ , which of the following is true? (a) $xz < 0$ (b) $xz > 0$ (c) $xz = 0$ (d) None of above

For what value of $x$ is the equation $ab^{x-2} = ba^{x-2}$ true? (a) $-3$ (b) $-2$ (c) $2$ (d) $3$

If $x^{\frac{1}{a}} = y^{\frac{1}{5}}$ and $xy = 1$ , what is the value of $a$ ? (a) $-2$ (b) $-3$ (c) $-4$ (d) $-5$

There are two numbers. If we add one to each number, their ratio becomes 2:3. If 1 is decreased from each number, their ratio becomes 1:2. The numbers are (a) 3, 1 (b) 1, 3 (c) 1, 5 (d) none of above

A perfect cube has surface area $96\ \text{cm}^2$ . What is its volume (in $\text{cm}^3$ )? (a) 64 (b) 80 (c) 48 (d) 96

In triangle ABC, $\angle B = 30^\circ$ and $BC = 12$ cm. If the area of $\triangle ABC$ is $27\ \text{cm}^2$ , find the length of AB. (a) 12 cm (b) 6 cm (c) 9 cm (d) None of above

The three distinct points A, B and C are said to be collinear if (a) $AC = AB + BC$ (b) slope of $AB$ = slope of $BC$ (c) Area of $\triangle ABC = 0$ (d) All of above

What is the maximum value of $y = \sin x$ ? (a) $-1$ (b) $1$ (c) $0$ (d) $\infty$

A right triangle is inscribed in a circle with an area of $16\pi\ \text{cm}^2$ . What is the hypotenuse of the triangle in cm? (a) $4\pi$ (b) $4$ (c) $8\pi$ (d) $8$

If $f(x)$ is divided by $(b - ax)$ , then the remainder is (a) $f\left(\frac{b}{a}\right)$ (b) $f\left(\frac{a}{b}\right)$ (c) $f\left(-\frac{b}{a}\right)$ (d) $f\left(-\frac{a}{b}\right)$

If the sum of the first $n$ positive integers is $\frac{1}{5}$ of the sum of the squares of the first $n$ positive integers, then $n$ is (a) 23 (b) 14 (c) 7 (d) 9

Which of the following parabolas is the narrowest? (a) $y = x^2$ (b) $y = \frac{x^2}{2}$ (c) $y = 2x^2$ (d) $y = \frac{x^2}{4}$

What does $\{x : 1 \le x < 5\}$ represent? (a) $(1, 5)$ (b) $[1, 5]$ (c) $[1, 5)$ (d) $(1, 5]$