+2 college entrance (after SEE) Trinity International College +2 Entrance (after SEE): Mathematics

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2083 · Management Practice Set A · 1 mark
2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set B · 1 mark

If $n(A - B) = 327$ , $n(B - A) = 217$ and $n(A \cup B) = 600$ , find $n(A \cap B)$ . (a) 56 (b) 46 (c) 66 (d) 36

Answer

(a) 56

The union splits into three disjoint parts:

$n(A \cup B) = n(A - B) + n(B - A) + n(A \cap B)$

$600 = 327 + 217 + n(A \cap B) \Rightarrow n(A \cap B) = 600 - 544 = 56$

Answer checked by Claude Sonnet · Oct 21, 2026

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2083 · Management Practice Set A · 1 mark
2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set D · 1 mark

If $y$ is the measure of an acute angle such that $\sin y = \frac{a}{5}$ , then $\tan y =$ (a) $\frac{\sqrt{25-a^2}}{5}$ (b) $\frac{a}{\sqrt{25-a^2}}$ (c) $\frac{1}{\sqrt{a^2+25}}$ (d) $\frac{5}{\sqrt{25-a^2}}$

Answer

(b) $\frac{a}{\sqrt{25-a^2}}$

With hypotenuse $5$ and opposite side $a$ , the adjacent side is $\sqrt{25-a^2}$ , so

$\tan y = \frac{a}{\sqrt{25-a^2}}$

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2083 · Management Practice Set A · 1 mark
2083 · Management Practice Set B · 1 mark
2083 · Science Practice Set A · 1 mark

If the graphs of the equations $2x + 5y = 7$ and $6x + cy = 11$ are parallel lines, then what is $c$ ? (a) 3 (b) 5 (c) 10 (d) 15

Answer

(d) 15

Parallel lines have equal slopes, so the coefficients are proportional: $\frac{6}{2}=\frac{c}{5}$ , giving $c=15$ . The constants are not in the same ratio ( $\frac{11}{7}\neq 3$ ), so the lines are distinct and truly parallel.

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2083 · Management Practice Set A · 1 mark
2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set D · 1 mark

If $m\%$ of 300 is 150, what is $m\%$ of 180? (a) 45 (b) 90 (c) 120 (d) 360

Answer

(b) 90

$\frac{m}{100}\times 300=150$ gives $m=50$ . Then $50\%$ of $180=90$ .

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2083 · Management Practice Set A · 1 mark
2083 · Management Practice Set B · 1 mark
2083 · Management Practice Set C · 1 mark

If $\frac{1}{x} + y = 2$ and $x + \frac{1}{y} = 3$ , the ratio of $x$ to $y$ is equal to (a) 1:2 (b) 2:3 (c) 3:1 (d) 3:2

Answer

(d) 3:2

From the first equation $y=\frac{2x-1}{x}$ . Substituting in the second: $x+\frac{x}{2x-1}=3$ , which simplifies to $2x^2-6x+3=0$ , so $x^2=3x-\frac{3}{2}$ .

Then $\frac{x}{y}=\frac{x^2}{2x-1}=\frac{3x-\frac{3}{2}}{2x-1}=\frac{3}{2}.$ So $x:y=3:2$ .

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2083 · Management Practice Set B · 1 mark
2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set B · 1 mark

The population of a town increases at the rate of 5% per annum and the present population of the town is 185220. What was the population of the town 3 years ago? (a) 160000 (b) 170000 (c) 180000 (d) 150000

Answer

(a) 160000

$P\left(1+\tfrac{5}{100}\right)^3=185220\Rightarrow P=\frac{185220}{1.157625}=160000$

(Check: $\frac{21}{20}$ cubed is $\frac{9261}{8000}$ , and $185220\times\frac{8000}{9261}=160000$ .)

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2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set C · 1 mark
2083 · Science Practice Set D · 1 mark

If the volume of a cone is $24\pi$ cubic feet and its height is 2 feet, what is the diameter of the base in feet? (a) 3 (b) 6 (c) 12 (d) 24

Answer

(c) 12

$24\pi=\tfrac13\pi r^2(2) \implies r^2=36 \implies r=6,$

so the diameter is $2r=12$ feet.

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2083 · Management Practice Set A · 1 mark
2083 · Management Practice Set C · 1 mark

A man can do a job in $h$ hours alone and his son can do it in $2h$ hours alone. Together, how many hours will it take them to do the job? (a) $3h$ (b) $\frac{h}{3}$ (c) $\frac{3h}{4}$ (d) $\frac{2h}{3}$

Similar questions: A man can do a job in h hours alone and his son (2083 · Science Practice Set D)

Answer

(d) $\frac{2h}{3}$

Combined rate:

$\frac{1}{h} + \frac{1}{2h} = \frac{3}{2h}$

Time together $= \dfrac{2h}{3}$ hours.

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2083 · Management Practice Set C · 1 mark

If $x$ is a positive acute angle and $\tan x = p$ , then what is $\cos x$ ? (a) $\frac{1}{\sqrt{p^2+1}}$ (b) $\frac{p}{\sqrt{p^2+1}}$ (c) $\frac{p}{\sqrt{p^2-1}}$ (d) $\sqrt{p^2+1}$

Similar questions: If x is a positive acute angle and tan x = p, (2083 · Management Practice Set B)

Answer

(a) $\frac{1}{\sqrt{p^2+1}}$

Since $\tan x = p/1$ , take opposite $= p$ and adjacent $= 1$ , so hypotenuse $= \sqrt{p^2+1}$ .

$\cos x = \frac{1}{\sqrt{p^2+1}}$

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2083 · Management Practice Set A · 1 mark
2083 · Management Practice Set C · 1 mark

If the radius of a circle is decreased by 30%, by what percent will the area of the circular region be decreased? (a) 70% (b) 60% (c) 51% (d) 49%

Similar questions: If the radius of a circle is decreased by 30%, by (2083 · Science Practice Set D)

Answer

(c) 51%

New radius $=0.7r$ , so new area $=\pi(0.7r)^2=0.49\pi r^2$ . The area decreases by $100-49=51\%$ .

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2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set C · 1 mark

If $n$ is an odd integer, which one of the following is an even integer? (a) $2n + n$ (b) $2n + 1$ (c) $n^3$ (d) $n(n+3)$

Similar questions: If n is an even integer, which one of the (2083 · Management Practice Set B)

Answer

(d) $n(n+3)$

For odd $n$ : $2n+n=3n$ is odd, $2n+1$ is odd, and $n^3$ is odd. But $n+3$ is even (odd plus odd), so $n(n+3)$ is even.

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2083 · Management Practice Set B · 1 mark

If $x$ is a positive acute angle and $\tan x = p$ , then what is $\cos x$ ? (a) $\frac{p}{\sqrt{p^2+1}}$ (b) $\frac{1}{\sqrt{p^2+1}}$ (c) $\frac{p}{\sqrt{p^2-1}}$ (d) $\sqrt{p^2+1}$

Similar questions: If x is a positive acute angle and tan x = p, (2083 · Management Practice Set A)

Answer

(b) $\frac{1}{\sqrt{p^2+1}}$

$\sec^2x=1+\tan^2x=1+p^2$ , and $\cos x>0$ for an acute angle, so $\cos x=\frac{1}{\sqrt{p^2+1}}$ .

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2083 · Management Practice Set B · 1 mark

If $n$ is an even integer, which one of the following must be an odd integer? (a) $3n - 2$ (b) $3(n+1)$ (c) $n - 2$ (d) $\frac{n}{3}$

Similar questions: If n is an odd integer, which one of the (2083 · Management Practice Set C)

Answer

(b) $3(n+1)$

If $n$ is even, $n+1$ is odd, and 3 times an odd number is odd. $3n-2$ and $n-2$ are even, and $\frac n3$ need not even be an integer.

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2083 · Science Practice Set D · 1 mark

A man can do a job in $h$ hours alone and his son can do it in $2h$ hours alone. Together, how many hours will it take them to do the job? (a) $\frac{1}{3h}$ (b) $\frac{h}{3}$ (c) $\frac{3h}{4}$ (d) $\frac{2h}{3}$

Similar questions: A man can do a job in h hours alone and his son (2083 · Management Practice Set A)

Answer

(d) $\frac{2h}{3}$

Their combined rate per hour is $\frac1h+\frac1{2h}=\frac{3}{2h}$ , so together they take $\frac{2h}{3}$ hours.

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2083 · Science Practice Set D · 1 mark

If the radius of a circle is decreased by 30%, by what percent will the area of the circular region be decreased? (a) 70% (b) 60% (c) 49% (d) 51%

Similar questions: If the radius of a circle is decreased by 30%, by (2083 · Management Practice Set A)

Answer

(d) 51%

The new radius is $0.7r$ , so the new area is $\pi(0.7r)^2=0.49\pi r^2$ . The area decreases by $100-49=51\%$ .

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2083 · Management Practice Set A · 1 mark
2083 · Management Practice Set C · 1 mark

A survey of students in a class shows that 45 students like mathematics and 41 students like English. If 12 students like both subjects, how many students like either Mathematics or English? (a) 66 (b) 75 (c) 98 (d) 74

Answer

(d) 74

$n(M \cup E) = n(M) + n(E) - n(M \cap E) = 45 + 41 - 12 = 74$

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2083 · Management Practice Set C · 1 mark

How many numbers in the set $\{-8, -5, 0, 10, 20\}$ satisfy the condition $|x - 5| < 11$ ? (a) One (b) Two (c) Three (d) Four

Answer

(c) Three

$|x-5|<11$ means $-11 < x-5 < 11$ , i.e. $-6 < x < 16$ .

Checking the set: $-8$ fails, $-5$ passes, $0$ passes, $10$ passes, $20$ fails. So three numbers satisfy it.

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2083 · Management Practice Set C · 1 mark

What is the simplified expression for $\frac{2x-4}{4x+20}\cdot\frac{x^2-25}{x^2-7x+10}$ ? (a) $\frac{1}{4}$ (b) $\frac{1}{2}$ (c) 2 (d) 1

Answer

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

Factorise each part:

$\frac{2(x-2)}{4(x+5)} \cdot \frac{(x-5)(x+5)}{(x-5)(x-2)} = \frac{2}{4} = \frac{1}{2}$

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2083 · Management Practice Set C · 1 mark

The legs of a right-angled triangle are in the ratio of 1 : 2. If the area of the triangle is 25, what is the hypotenuse? (a) $5\sqrt{3}$ (b) $5\sqrt{5}$ (c) $10\sqrt{3}$ (d) $10\sqrt{5}$

Answer

(b) $5\sqrt{5}$

Let the legs be $k$ and $2k$ . Area: $\tfrac12 \cdot k \cdot 2k = k^2 = 25$ , so $k = 5$ and the legs are $5$ and $10$ .

$\text{Hypotenuse} = \sqrt{5^2 + 10^2} = \sqrt{125} = 5\sqrt{5}$

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2083 · Management Practice Set C · 1 mark

What is the solution of the equation $p = \frac{ry}{r+y}$ for $y$ ? (a) $\frac{r+p}{rp}$ (b) $\frac{rp}{r-p}$ (c) $\frac{r+p}{r-p}$ (d) $\frac{r-p}{rp}$

Answer

(b) $\frac{rp}{r-p}$

$p(r+y) = ry \Rightarrow pr + py = ry \Rightarrow pr = y(r-p) \Rightarrow y = \frac{rp}{r-p}$

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2083 · Management Practice Set A · 1 mark
2083 · Science Practice Set A · 1 mark

In the xy-plane, line $l$ is perpendicular to the line $4x - y = 1$ and contains the point $(0, 2)$ . Which one of the following is the equation of line $l$ ? (a) $x + 4y = 8$ (b) $4x - y = 2$ (c) $4x + y = -2$ (d) $4x + y = 8$

Answer

(a) $x + 4y = 8$

The line $4x-y=1$ has slope $4$ , so a perpendicular line has slope $-\frac{1}{4}$ . Through $(0,2)$ : $y=-\frac{1}{4}x+2 \Rightarrow x+4y=8.$

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In the xy-plane, the lines with the equations $x + y = 15$ and $y = 3x - 1$ intersect at $(a, b)$ . What is the value of $a - b$ ? (a) 11 (b) 4 (c) $-7$ (d) $-6$

Answer

(c) $-7$

Substitute $y=3x-1$ into $x+y=15$ : $4x-1=15$ , so $x=4$ and $y=11$ . Hence $a-b=4-11=-7$ .

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2083 · Management Practice Set C · 1 mark

20% of what number is equal to $\frac{2}{3}$ of 90? (a) 30 (b) 120 (c) 300 (d) 600

Answer

(c) 300

$\frac{2}{3}\times 90=60$ . Let the number be $N$ : $0.2N=60$ , so $N=300$ .

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2083 · Management Practice Set C · 1 mark

If $4x + 16 = 24$ , what is the value of $x + 8$ ? (a) 14 (b) 12 (c) 10 (d) 8

Answer

(c) 10

$4x=24-16=8$ , so $x=2$ and $x+8=10$ .

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2083 · Management Practice Set C · 1 mark

A couple plans to have three children. What is the probability that exactly two will be boys? (a) $\frac{2}{3}$ (b) $\frac{3}{8}$ (c) $\frac{2}{8}$ (d) 18

Answer

(b) $\frac{3}{8}$

There are $2^3=8$ equally likely outcomes. Exactly two boys can occur in $\binom{3}{2}=3$ ways (BBG, BGB, GBB), so the probability is $\frac{3}{8}$ .

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A bag contains 60 marbles. A marble is to be chosen from the bag. If the probability of a blue marble being chosen is 0.35, how many marbles in the bag are blue? (a) 21 (b) 25 (c) 35 (d) 39

Answer

(a) 21

Number of blue marbles $=0.35\times 60=21$ .

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2083 · Management Practice Set C · 1 mark

If $\frac{1}{3}, y_1, y_2, 9$ are in G.P., then what is the value of $y_2$ ? (a) 1 (b) 3 (c) 6 (d) $\frac{15}{2}$

Answer

(b) 3

Let the common ratio be $r$ . The fourth term is $\frac{1}{3}r^3=9$ , so $r^3=27$ and $r=3$ . Then $y_1=1$ and $y_2=1\times 3=3$ .

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2083 · Management Practice Set C · 1 mark

The sum of the roots of $3x^2 - 4x + 4 = 0$ is (a) $\frac{4}{3}$ (b) $\frac{3}{4}$ (c) $-\frac{4}{3}$ (d) 4

Answer

(a) $\frac{4}{3}$

For $ax^2+bx+c=0$ the sum of the roots is $-\frac{b}{a}=-\frac{-4}{3}=\frac{4}{3}$ .

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2083 · Management Practice Set C · 1 mark

If $x = 1 + \sqrt{2}$ , then $x^2 - 2x + 1$ is equal to (a) $1 + \sqrt{2}$ (b) $-1$ (c) $\sqrt{2}$ (d) 2

Answer

(d) 2

$x^2-2x+1=(x-1)^2=(\sqrt{2})^2=2$ .

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2083 · Science Practice Set D · 1 mark

The average of 34 students in a class is 13 years. If the age of a teacher is included, the average is increased by 1 year. What is the age of the teacher in years? (a) 40 (b) 48 (c) 55 (d) 65

Answer

(b) 48

Total age of the students $=34\times 13=442$ . With the teacher there are 35 people with average 14, so total $=35\times 14=490$ . Teacher's age $=490-442=48$ years.

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2083 · Management Practice Set B · 1 mark

If 6 Japanese yen = 5 Swedish kroner and 4 Swedish kroner = 3 Danish kroner, how many Japanese yen can be exchanged for 48 Danish kroner? (a) 70.8 Yen (b) 76.8 Yen (c) 78.8 Yen (d) 77.8 Yen

Answer

(b) 76.8 Yen

Convert Danish kroner to Swedish kroner: $48 \times \frac{4}{3} = 64$ Swedish kroner. Then convert to yen: $64 \times \frac{6}{5} = 76.8$ yen.

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2083 · Management Practice Set B · 1 mark
2083 · Science Practice Set C · 1 mark

A fraction becomes $\frac{4}{5}$ if 2 is added to the numerator. If 1 is subtracted from the denominator, the fraction becomes $\frac{3}{4}$ . What was the original fraction? (a) $\frac{18}{25}$ (b) $\frac{20}{25}$ (c) $\frac{24}{25}$ (d) $\frac{25}{18}$

Answer

(a) $\frac{18}{25}$

Let the fraction be $\frac{x}{y}$ . Then $5(x+2)=4y$ and $4x=3(y-1)$ , i.e. $5x+10=4y$ and $4x=3y-3$ .

From the first, $y=\frac{5x+10}{4}$ . Substituting: $16x=15x+30-12$ , so $x=18$ and $y=25$ .

Check: $\frac{20}{25}=\frac45$ and $\frac{18}{24}=\frac34$ .

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2083 · Management Practice Set B · 1 mark
2083 · Science Practice Set C · 1 mark

If $p$ lb of salt are dissolved in $q$ lb of water, the percent of salt in the resulting solution is (a) $\frac{100p}{p+q}$ (b) $\frac{p}{p+q}$ (c) $\frac{100\,p}{q}$ (d) $\frac{100\,q}{pq}$

Answer

(a) $\frac{100p}{p+q}$

The solution weighs $p+q$ lb and contains $p$ lb of salt, so the percentage is $\frac{p}{p+q}\times100$ .

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2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set B · 1 mark

$2x + 3y = 1200$ , $3x + 2y = 1300$ . Based on the system of equations above, what is the value of $5x + 5y$ ? (a) 500 (b) 1000 (c) 2300 (d) 2500

Answer

(d) 2500

Add the two equations:

$(2x+3y)+(3x+2y)=1200+1300 \implies 5x+5y=2500.$

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2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set B · 1 mark

If $u + t = 5$ and $u - t = 2$ , what is the value of $(u - t)(u^2 - t^2)$ ? (a) 10 (b) 20 (c) 25 (d) 50

Answer

(b) 20

$u^2-t^2=(u+t)(u-t)=5\times 2=10$ , so $(u-t)(u^2-t^2)=2\times 10=20$ .

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2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set B · 1 mark

A right circular cone has a volume of $24\pi$ cubic inches. If the height of the cone is 2 inches, what is the radius, in inches, of the base of the cone? (a) $2\sqrt{3}$ (b) 6 (c) 12 (d) 36

Answer

(b) 6

$V=\tfrac13\pi r^2h \implies 24\pi=\tfrac13\pi r^2(2) \implies r^2=36 \implies r=6.$

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2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set B · 1 mark

Which of the following gives the radius of the sphere in terms of the volume of the sphere? (a) $\sqrt{\frac{V}{4\pi}}$ (b) $\sqrt{\frac{3V}{4\pi}}$ (c) $\sqrt[3]{\frac{3V}{4\pi}}$ (d) $\frac{V}{\sqrt[3]{4\pi}}$

Answer

(c) $\sqrt[3]{\frac{3V}{4\pi}}$

From $V=\dfrac{4}{3}\pi r^3$ we get $r^3=\dfrac{3V}{4\pi}$ , so $r=\sqrt[3]{\dfrac{3V}{4\pi}}$ .

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2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set B · 1 mark

Circles O and O' are internally tangent to each other. Circle O' passes through the center of O. If the area of circle O is 16, the area of circle O' is (a) $2\sqrt{2}$ (b) 2 (c) $2\sqrt{\pi}$ (d) 4

Answer

(d) 4

Since $O'$ is internally tangent to $O$ and passes through the centre of $O$ , the diameter of $O'$ equals the radius $R$ of $O$ , so its radius is $R/2$ . Area scales with the square of radius, so area of $O'$ is $\tfrac14$ of the area of $O$ : $\tfrac14\times 16=4$ .

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2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set B · 1 mark

Given $\triangle PQR$ , $\angle P = 70^\circ$ , QS bisects $\angle Q$ , and RS bisects $\angle R$ (S is the point where the two bisectors meet). $\angle S =$ (a) $125^\circ$ (b) $120^\circ$ (c) $115^\circ$ (d) $110^\circ$

Answer

(a) $125^\circ$

In $\triangle QSR$ : $\angle S=180^\circ-\tfrac12(\angle Q+\angle R)$ . Since $\angle Q+\angle R=180^\circ-70^\circ=110^\circ$ ,

$\angle S=180^\circ-55^\circ=125^\circ.$

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2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set B · 1 mark

If $f(x) = 5x - 3$ and $f(t) = 7$ , what is the value of $t$ ? (a) 0 (b) 1 (c) 2 (d) 32

Answer

(c) 2

$f(t)=5t-3=7 \implies 5t=10 \implies t=2$ .

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2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set B · 1 mark

Let $x$ be the angle formed by the diagonal of the face of a cube and the diagonal of the cube, drawn from the same vertex. $\cos^2 x =$ (a) $\frac{2}{3}$ (b) $\frac{1}{3}$ (c) $\frac{1}{2}$ (d) 1

Answer

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

Take a cube of side 1. The face diagonal has length $\sqrt2$ and the cube diagonal has length $\sqrt3$ . The face diagonal, the edge perpendicular to that face and the cube diagonal form a right triangle (right angle at the end of the face diagonal), so

$\cos x=\frac{\sqrt2}{\sqrt3}\implies \cos^2x=\frac23.$

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2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set B · 1 mark

A flask contains 40 grams of solution that is 25% sugar by weight. How much sugar must be added to the flask so that the resulting solution is 40% sugar by weight? (a) 6 (b) 8 (c) 10 (d) 15

Answer

(c) 10

The flask has $0.25\times40=10$ g sugar. Adding $x$ g of sugar:

$\frac{10+x}{40+x}=0.4 \implies 10+x=16+0.4x \implies 0.6x=6 \implies x=10.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set C · 1 mark

ABCD is a parallelogram with $A(4, 5)$ , $D(2, -3)$ and $C(6, -3)$ . What is the x-coordinate of point B? (a) 2 (b) 5 (c) 6 (d) 8

Answer

(d) 8

The diagonals of a parallelogram bisect each other, so the midpoint of $AC$ equals the midpoint of $BD$ : $B=A+C-D$ .

$B=(4+6-2,\;5-3+3)=(8,5).$

The x-coordinate is $8$ .

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2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set C · 1 mark

ABCD is a trapezoid (AB parallel to DC, BC perpendicular to AB) with area 50 square units. If the length of AB is 6 units and the length of BC is 5 units, what is the length of CD? (a) 4 (b) 7 (c) 10 (d) 14

Answer

(d) 14

$BC$ is perpendicular to the parallel sides, so it is the height. Then

$\tfrac12(AB+CD)\cdot BC=50 \implies \tfrac12(6+CD)(5)=50 \implies 6+CD=20 \implies CD=14.$

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2083 · Management Practice Set C · 1 mark
2083 · Science Practice Set B · 1 mark

In the xy-plane, the graph of the function $f(x) = x^2 + 5x + 4$ has two x-intercepts. What is the distance between the x-intercepts? (a) 1 (b) 2 (c) 3 (d) 4

Answer

(c) 3

Setting $f(x)=0$ : $x^2+5x+4=(x+1)(x+4)=0$ , so the x-intercepts are $x=-1$ and $x=-4$ . The distance between them is $|-1-(-4)|=3$ .

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2083 · Management Practice Set B · 1 mark

The selling price of a TV is Rs. 18000. If there is a discount of 10%, find the marked price. (a) Rs. 19000 (b) Rs. 20000 (c) Rs. 17000 (d) Rs. 21000

Similar questions: The selling price of a TV is Rs. 18000. If there (2083 · Science Practice Set C)

Answer

(b) Rs. 20000

SP is 90% of MP, so

$MP=\frac{18000}{0.9}=\text{Rs. }20000$

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2083 · Management Practice Set B · 1 mark

A car drives a distance of $d$ miles at 30 mph and returns at 60 mph. What is the average rate for the round trip? (a) 40 mph (b) 43 mph (c) $\frac{2d}{35}$ (d) $\frac{35}{2d}$

Similar questions: A car drives a distance of d miles at 30 mph and (2083 · Science Practice Set A)

Answer

(a) 40 mph

Average speed $=\dfrac{\text{total distance}}{\text{total time}}=\dfrac{2d}{\frac{d}{30}+\frac{d}{60}}=\dfrac{2d}{\frac{d}{20}}=40$ mph.

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2083 · Management Practice Set C · 1 mark

A book was on sale at 40% off its original price. If the sale price of the book was Rs. 18, what was the original price of the book? (a) Rs. 7.20 (b) Rs. 10.80 (c) Rs. 30.00 (d) Rs. 45.00

Similar questions: A book was on sale for 40% off its original (2083 · Science Practice Set B)

Answer

(c) Rs. 30.00

The sale price is $60\%$ of the original price $P$ : $0.6P=18$ , so $P=\dfrac{18}{0.6}=30$ .

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2083 · Management Practice Set C · 1 mark

A boy grew one year from a height of $x$ inches to a height of $y$ inches. The percent of increase was (a) $\frac{100(y-x)}{y}$ (b) $\frac{100(y-x)}{x}$ (c) $\frac{y-x}{x}$ (d) $\frac{y-x}{100}$

Similar questions: A boy grew in one year from a height of x inches (2083 · Science Practice Set B)

Answer

(b) $\frac{100(y-x)}{x}$

Percent increase is the increase divided by the original value, times 100: $\dfrac{y-x}{x}\times100$ .

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2083 · Science Practice Set A · 1 mark

Let $\vec{a}$ and $\vec{b}$ be two given vectors such that $|\vec{a}| = \sqrt{3}$ , $|\vec{b}| = 2$ and $\vec{a}\cdot\vec{b} = \sqrt{6}$ , then what is the angle between the vectors? (a) $\frac{\pi}{3}$ (b) $\frac{\pi}{4}$ (c) $\frac{\pi}{2}$ (d) 0

Similar questions: Let a and b be two given vectors such that |a| = (2011 · Science Entrance Sample Paper)

Answer

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

$\cos\theta=\frac{\vec a\cdot\vec b}{|\vec a||\vec b|}=\frac{\sqrt6}{2\sqrt3}=\frac{\sqrt2}{2}$

So $\theta=\frac{\pi}{4}$ .

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2083 · Science Practice Set A · 1 mark

A car drives a distance of $d$ miles at 30 mph and returns at 60 mph. What is its average rate for the round trip? (a) 40 mph (b) 43 mph (c) $\frac{2d}{35}$ mph (d) $\frac{35}{2d}$ mph

Similar questions: A car drives a distance of d miles at 30 mph and (2083 · Management Practice Set B)

Answer

(a) 40 mph

$\text{Average speed}=\frac{2d}{\frac d{30}+\frac d{60}}=\frac{2d}{\frac{3d}{60}}=40\text{ mph}$

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2083 · Science Practice Set B · 1 mark

A book was on sale for 40% off its original price. If the sale price of the book was Rs 18, what was the original price of the book? (a) Rs. 7.20 (b) Rs. 10.80 (c) Rs. 30.00 (d) Rs. 45.00

Similar questions: A book was on sale at 40% off its original price. (2083 · Management Practice Set C)

Answer

(c) Rs 30.00

The sale price is $60\%$ of the original price $P$ .

$0.6P = 18 \Rightarrow P = \frac{18}{0.6} = 30$

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2083 · Science Practice Set B · 1 mark

A boy grew in one year from a height of $x$ inches to a height of $y$ inches. The percent of increase was: (a) $\frac{100(y-x)}{y}$ (b) $\frac{100(y-x)}{x}$ (c) $\frac{y-x}{x}$ (d) $\frac{y-x}{100}$

Similar questions: A boy grew one year from a height of x inches to (2083 · Management Practice Set C)

Answer

(b) $\frac{100(y-x)}{x}$

Percent increase is the increase divided by the original value, times $100$ . The increase is $y-x$ and the original height is $x$ , so the percent increase is $\frac{100(y-x)}{x}$ .

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2083 · Science Practice Set C · 1 mark

The selling price of a TV is Rs. 18000. If there is a discount of 10%, what is the marked price? (a) Rs. 19000 (b) Rs. 20000 (c) Rs. 17000 (d) Rs. 21000

Similar questions: The selling price of a TV is Rs. 18000. If there (2083 · Management Practice Set B)

Answer

(b) Rs. 20000

Selling price $=90\%$ of marked price, so $MP=\frac{18000}{0.9}=\text{Rs. }20000.$

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2011 · Science Entrance Sample Paper · 1 mark

Let $\vec{a}$ and $\vec{b}$ be two given vectors such that $|\vec{a}| = \sqrt{3}$ , $|\vec{b}| = 2$ and $\vec{a}\cdot\vec{b} = \sqrt{6}$ , then the angle between $\vec{a}$ and $\vec{b}$ is (a) $\frac{\pi}{3}$ (b) $\frac{\pi}{4}$ (c) $\frac{\pi}{2}$ (d) 0

Similar questions: Let a and b be two given vectors such that |a| = (2083 · Science Practice Set A)

Answer

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

$\cos\theta = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|} = \frac{\sqrt{6}}{\sqrt{3}\cdot 2} = \frac{\sqrt{2}}{2},$

so $\theta = \frac{\pi}{4}$ .

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2083 · Management Practice Set A · 1 mark

If $n$ is a positive integer, which of the following is always odd? (a) $19n + 6$ (b) $19n + 5$ (c) $19n^2 + 5$ (d) $18n + 5$

Answer

(d) $18n + 5$

$18n$ is always even, and even + 5 is always odd. The others can be even: for odd $n$ , $19n+5$ and $19n^2+5$ are even, and for even $n$ , $19n+6$ is even.

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2083 · Management Practice Set A · 1 mark

$2x + 7y = 9$ , $8x + 28y = a$ . In the given system of equations, $a$ is a constant. If the system has infinitely many solutions, what is the value of $a$ ? (a) 4 (b) 9 (c) 36 (d) 54

Answer

(c) 36

Infinitely many solutions need the second equation to be a multiple of the first. Multiplying $2x+7y=9$ by 4 gives $8x+28y=36$ , so $a=36$ .

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2083 · Management Practice Set A · 1 mark

The equation $y = 7^x - 1$ is graphed in the xy-plane. What is the y-intercept of the graph? (a) $(0, -1)$ (b) $(0, 7)$ (c) $(0, 6)$ (d) $(0, 0)$

Answer

(d) $(0, 0)$

Put $x=0$ : $y=7^0-1=1-1=0$ , so the y-intercept is $(0,0)$ .

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2083 · Management Practice Set A · 1 mark

At what time does a sum of money double itself at the rate of 8% p.a.? (a) 8 years (b) 10 years (c) 12.5 years (d) 16 years

Answer

(c) 12.5 years

For simple interest, doubling means the interest equals the principal: $P=\frac{P\times 8\times T}{100}$ , so $T=\frac{100}{8}=12.5$ years.

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2083 · Management Practice Set A · 1 mark

Which one of the following is the compound interest of amount $P$ after $n$ years at the rate of $r\%$ ? (a) $P\left[\left(1+\frac{r}{100}\right)^n - 1\right]$ (b) $P\left[\left(1+\frac{r}{100}\right)^n + 1\right]$ (c) $\left[\left(1+\frac{r}{100}\right)^n - 1\right]$ (d) $P\left[\left(1-\frac{r}{100}\right)^n + 1\right]$

Answer

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

The amount is $A=P\left(1+\frac{r}{100}\right)^n$ , and compound interest is the amount minus the principal, $A-P$ .

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2083 · Management Practice Set A · 1 mark

If $x = -8$ , the value of $x^{2/3} + 2x^0$ is (a) $-2$ (b) 4 (c) 6 (d) $-6$

Answer

(c) 6

$x^{2/3}=\left((-8)^{1/3}\right)^2=(-2)^2=4$ , and $2x^0=2$ . The sum is $4+2=6$ .

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2083 · Management Practice Set A · 1 mark

If $2^{3x+10} = \left(\frac{1}{4}\right)^x$ , then $x =$ (a) $-2$ (b) 1 (c) $-1$ (d) 2

Answer

(a) $-2$

Write $\left(\frac{1}{4}\right)^x=2^{-2x}$ . Then $3x+10=-2x$ , so $5x=-10$ and $x=-2$ .

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2083 · Management Practice Set A · 1 mark

If $3x = 4y$ , the value of $(x+y)^2 : (x-y)^2$ is (a) 4 (b) 9 (c) 49 (d) 7

Answer

(c) 49

Let $x=4k$ , $y=3k$ . Then $(x+y)^2:(x-y)^2=(7k)^2:(k)^2=49:1$ , so the value is $49$ .

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2083 · Management Practice Set A · 1 mark

How many cubes with side 4 cm can be made from a cube with 16 cm in length? (a) 4 (b) 16 (c) 64 (d) 404

Answer

(c) 64

Number of cubes $=\frac{16^3}{4^3}=\left(\frac{16}{4}\right)^3=4^3=64$ .

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2083 · Management Practice Set A · 1 mark

If $d$ is the diameter of a sphere, what is the surface area of the sphere? (a) $4\pi d^2$ (b) $2\pi d^2$ (c) $\frac{\pi d^2}{4}$ (d) $\pi d^2$

Answer

(d) $\pi d^2$

Surface area $=4\pi r^2$ with $r=\frac{d}{2}$ , so $4\pi\frac{d^2}{4}=\pi d^2$ .

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2083 · Management Practice Set A · 1 mark

Two identical solid cubes of side $a$ are joined end to end. Then what is the volume of the resulting cuboid? (a) $a^3$ (b) $2a^3$ (c) $8a^3$ (d) $10a^2$

Answer

(b) $2a^3$

Each cube has volume $a^3$ , and joining them preserves total volume: $a^3+a^3=2a^3$ (a cuboid $2a\times a\times a$ ).

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2083 · Management Practice Set A · 1 mark

If $3^{n-2} = \frac{1}{81}$ , then the value of $n$ is equal to (a) 0 (b) 1 (c) 2 (d) $-2$

Answer

(d) $-2$

$\frac{1}{81}=3^{-4}$ , so $n-2=-4$ and $n=-2$ .

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2083 · Management Practice Set B · 1 mark

If $n(A) = 40$ , $n(B) = 50$ , $n(U) = 200$ , $n(A \cap B) = 20$ , then $n(\overline{A \cup B}) =$ (a) 20 (b) 130 (c) 40 (d) 100

Answer

(b) 130

$n(A\cup B)=40+50-20=70$

$n(\overline{A\cup B})=200-70=130$

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2083 · Management Practice Set B · 1 mark

Which one of the following is the defined statement for $A \cup B$ ? (a) $\{x : x \in A \text{ and } x \in B\}$ (b) $\{x : x \notin A \text{ and } x \notin B\}$ (c) $\{x : x \in A \text{ and } x \notin B\}$ (d) $\{x : x \in A \text{ or } x \in B\}$

Answer

(d) $\{x : x \in A \text{ or } x \in B\}$

The union contains the elements that belong to at least one of the two sets.

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2083 · Management Practice Set B · 1 mark

When $k$ is subtracted from 10, and the difference is divided by 2, the result is 3. What is the value of $k$ ? (a) 3 (b) 4 (c) 6 (d) 10

Answer

(b) 4

$\frac{10-k}{2}=3 \Rightarrow 10-k=6 \Rightarrow k=4$

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2083 · Management Practice Set B · 1 mark

The sum of the digits of a two-digit number is 8. If 18 is subtracted from the number the digits are reversed. What is the original number? (a) 35 (b) 44 (c) 53 (d) 62

Answer

(c) 53

Let the digits be $a$ (tens) and $b$ (units), so $a+b=8$ . Reversing the digits gives $(10a+b)-(10b+a)=9(a-b)=18$ , so $a-b=2$ . Hence $a=5$ , $b=3$ and the number is 53 (check: $53-18=35$ ).

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2083 · Management Practice Set B · 1 mark

The area of an isosceles triangle is $12\ \text{cm}^2$ and its base is 8 cm. Find the length of the equal sides of the triangle. (a) 6 cm (b) 8 cm (c) 5 cm (d) 7 cm

Answer

(c) 5 cm

$\tfrac12\times 8\times h=12 \Rightarrow h=3\text{ cm}$

The height bisects the base, so each equal side is $\sqrt{4^2+3^2}=5$ cm.

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2083 · Management Practice Set B · 1 mark

The perimeter of rectangle ABCD (with length $y$ and width $x$ ) is equal to $p$ , and $x = \frac{2}{3}y$ . What is the value of $y$ in terms of $p$ ? (a) $\frac{p}{10}$ (b) $\frac{3p}{10}$ (c) $\frac{p}{3}$ (d) $\frac{2p}{5}$

Answer

(b) $\frac{3p}{10}$

$p=2(x+y)=2\left(\tfrac{2}{3}y+y\right)=\tfrac{10y}{3}\Rightarrow y=\tfrac{3p}{10}$

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2083 · Management Practice Set B · 1 mark

If the average measure of two angles in a parallelogram is $y^\circ$ , what is the average degree measure of the other two angles? (a) $180 - y$ (b) $180 - \frac{y}{2}$ (c) $360 - 2y$ (d) $360 + y$

Answer

(a) $180 - y$

The four angles of a parallelogram add up to $360^\circ$ . The two given angles total $2y$ , so the other two total $360-2y$ , with average $180-y$ .

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2083 · Management Practice Set B · 1 mark

If 0.1% of $m$ is equal to 10% of $n$ , then $m$ is what percent of $10n$ ? (a) $\frac{1}{1000}\%$ (b) 1% (c) 10% (d) 1000%

Answer

(d) 1000%

$\frac{0.1}{100}m=\frac{10}{100}n\Rightarrow m=100n$

$\frac{m}{10n}\times100=\frac{100n}{10n}\times100=1000\%$

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2083 · Management Practice Set B · 1 mark

A conical tent is 5 m tall and its diameter is 14 m. What is the cost of cloth required to make the tent at Rs. 200 per square meter? (a) Rs. 37850 (b) Rs. 37800 (c) Rs. 37875 (d) Rs. 37880

Answer

(a) Rs. 37850

Radius $r=7$ m, height $h=5$ m, so slant height $l=\sqrt{49+25}=\sqrt{74}\approx 8.602$ m.

$\text{CSA}=\pi r l=\tfrac{22}{7}\times7\times8.602\approx189.25\ \text{m}^2$

$\text{Cost}=189.25\times200\approx\text{Rs. }37850$

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2083 · Management Practice Set B · 1 mark

If $a + 2b = 10$ and $b - a = 2$ , what is the value of $b$ ? (a) 10 (b) 8 (c) 6 (d) 4

Answer

(d) 4

From $b-a=2$ , $a=b-2$ . Then $(b-2)+2b=10\Rightarrow 3b=12\Rightarrow b=4$ .

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2083 · Management Practice Set B · 1 mark

If $a > b$ and $a^2 - 2ab + b^2 = 169$ , what is the value of $a - b$ ? (a) 11 (b) 12 (c) 13 (d) 15

Answer

(c) 13

$a^2-2ab+b^2=(a-b)^2=169$ , so $a-b=\pm13$ . Since $a>b$ , $a-b=13$ .

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2083 · Management Practice Set B · 1 mark

Which of the following is a factor of $3x^3 - 11x^2 - 42x$ ? (a) $x - 7$ (b) $x$ (c) $3x - 6$ (d) $x + 6$

Answer

(b) $x$

$3x^3-11x^2-42x = x(3x^2-11x-42) = x(3x+7)(x-6)$ , so $x$ is a factor. None of $x-7$ , $3x-6$ , $x+6$ divides it.

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2083 · Management Practice Set B · 1 mark

If the perimeter of a square is 28, what is the length of the diagonal? (a) $2\sqrt{14}$ (b) $7\sqrt{2}$ (c) $7\sqrt{3}$ (d) 14

Answer

(b) $7\sqrt{2}$

Side $=28/4=7$ , and the diagonal of a square is $\text{side}\times\sqrt2 = 7\sqrt2$ .

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2083 · Management Practice Set B · 1 mark

If $a^m = a^n$ , then what is the relation between $m$ and $n$ ? (a) $m + n = 1$ (b) $m - n = 0$ (c) $mn = 0$ (d) $\frac{m}{n} = 0$

Answer

(b) $m - n = 0$

Equal powers of the same base give equal exponents, so $m=n$ , i.e. $m-n=0$ (taking $a\ne 0,\pm1$ ).

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2083 · Management Practice Set B · 1 mark

If $10^{3y-3} = \frac{1}{0.001}$ , then what will be the value of $y$ ? (a) 3 (b) 4 (c) 1 (d) 2

Answer

(d) 2

$\frac{1}{0.001}=1000=10^3$ , so $3y-3=3$ , giving $y=2$ .

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2083 · Management Practice Set B · 1 mark

Which one of the following is the simplest form for $\frac{2^{n+4} - 2(2^n)}{2(2^{n+3})}$ ? (a) $\frac{1}{2}$ (b) $\frac{1}{4}$ (c) $\frac{7}{8}$ (d) $\frac{5}{8}$

Answer

(c) $\frac{7}{8}$

$\frac{2^{n+4}-2\cdot2^n}{2\cdot2^{n+3}}=\frac{2^n(16-2)}{2^{n}\cdot 16}=\frac{14}{16}=\frac78$

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2083 · Management Practice Set B · 1 mark

What is the total surface area of a cone of radius 7 cm and height 24 cm? (a) $710\ \text{cm}^2$ (b) $704\ \text{cm}^2$ (c) $700\ \text{cm}^2$ (d) $725\ \text{cm}^2$

Answer

(b) $704\ \text{cm}^2$

Slant height $l=\sqrt{7^2+24^2}=25$ cm.

$\text{TSA}=\pi r(l+r)=\frac{22}{7}\times7\times(25+7)=22\times32=704\ \text{cm}^2$

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2083 · Management Practice Set B · 1 mark

1 litre = --------- cubic centimeter. (a) 1000 (b) 100 (c) 10 (d) 1

Answer

(a) 1000

$1\ \text{L}=1\ \text{dm}^3=(10\ \text{cm})^3=1000\ \text{cm}^3$ .

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2083 · Management Practice Set B · 1 mark

If the radius of a sphere is doubled, what is the percent increase in the surface area? (a) 100 (b) 200 (c) 300 (d) 400

Answer

(c) 300

Surface area $=4\pi r^2$ . With radius $2r$ it becomes $16\pi r^2$ , which is 4 times the original, so the increase is $\frac{16-4}{4}\times100=300\%$ .

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2083 · Management Practice Set B · 1 mark

In a cube, the ratio of the longest diagonal to a diagonal of a base is (a) $\sqrt{2} : \sqrt{3}$ (b) $\sqrt{3} : \sqrt{2}$ (c) $2 : 1$ (d) $\sqrt{6} : \sqrt{2}$

Answer

(b) $\sqrt{3} : \sqrt{2}$

For edge $a$ , the longest (space) diagonal is $a\sqrt3$ and a base diagonal is $a\sqrt2$ , so the ratio is $\sqrt3:\sqrt2$ .

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2083 · Management Practice Set B · 1 mark

In a bag there are 7 red and 5 blue colour balls. What is the probability that two different coloured balls are selected when the balls are drawn without replacement? (a) $\frac{35}{132}$ (b) $\frac{35}{66}$ (c) $\frac{12}{35}$ (d) $\frac{5}{11}$

Answer

(b) $\frac{35}{66}$

$P=\frac{7}{12}\cdot\frac{5}{11}+\frac{5}{12}\cdot\frac{7}{11}=\frac{70}{132}=\frac{35}{66}$

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2083 · Management Practice Set B · 1 mark

A laptop which is priced at Rs. 45000 is sold at a discount of 12%. If the selling price of the laptop including VAT is Rs. 44748, what is the rate of VAT? (a) 13% (b) 14% (c) 15% (d) 12%

Answer

(a) 13%

Price after discount $=45000\times0.88=\text{Rs }39600$ . VAT amount $=44748-39600=\text{Rs }5148$ .

$\text{VAT rate}=\frac{5148}{39600}\times100=13\%$

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2083 · Management Practice Set B · 1 mark

If the graphs of $x - ay = 10$ and $2x - y = 3$ are perpendicular lines, then the value of $a =$ (a) 2 (b) $-1$ (c) $-2$ (d) 1

Answer

(c) $-2$

Slope of $x-ay=10$ is $\frac1a$ ; slope of $2x-y=3$ is $2$ . For perpendicular lines the product is $-1$ : $\frac{2}{a}=-1$ , so $a=-2$ .

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2083 · Management Practice Set B · 1 mark

Opposite angles of a cyclic quadrilateral are (a) equal (b) complementary (c) supplementary (d) double of each other

Answer

(c) supplementary

In a cyclic quadrilateral the opposite angles add up to $180^\circ$ .

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2083 · Management Practice Set B · 1 mark

The radius of the circle passing through the point $(6, 2)$ whose two diameters are $x + y = 6$ and $x + 2y = 4$ is (a) $2\sqrt{5}$ (b) 6 (c) 4 (d) 10

Answer

(a) $2\sqrt{5}$

The centre is the intersection of the diameters. Subtracting $x+y=6$ from $x+2y=4$ gives $y=-2$ , so $x=8$ ; centre $(8,-2)$ .

$r=\sqrt{(8-6)^2+(-2-2)^2}=\sqrt{4+16}=2\sqrt5$

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2083 · Management Practice Set B · 1 mark

A circle of radius $x$ has an area twice that of a square of side $a$ . The equation used to find the radius of the circle is (a) $2\pi x^2 = a^2\pi$ (b) $\pi x^2 = 2a^2$ (c) $a^2 = 2x^2$ (d) $\pi a^2 = 4x^2$

Answer

(b) $\pi x^2 = 2a^2$

Circle area $\pi x^2$ is twice the square area $a^2$ , so $\pi x^2=2a^2$ .

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2083 · Management Practice Set B · 1 mark

If the graph of the equation $x + y - 8 + 4k = 0$ passes through the origin, then the value of $k$ is equal to (a) $-2$ (b) 2 (c) 0 (d) 1

Answer

(b) 2

Putting $(0,0)$ in $x+y-8+4k=0$ gives $-8+4k=0$ , so $k=2$ .

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2083 · Management Practice Set B · 1 mark

If the $n^{th}$ term of a sequence is $b^{3n} - 3$ and the second term of the sequence is 61, which one of the following could be the value of $b$ ? (a) 1 (b) 2 (c) 3 (d) 4

Answer

(b) 2

Second term: $b^{6}-3=61$ , so $b^6=64$ and $b=2$ (for positive $b$ ; $2^6=64$ ).

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2083 · Management Practice Set C · 1 mark

If $t_1, t_2, t_3$ are in AP, then $\frac{t_1}{k}, \frac{t_2}{k}, \frac{t_3}{k}, \dots$ are in (a) AP (b) GP (c) HP (d) none

Answer

(a) AP

If the common difference of the original AP is $d$ , then dividing every term by the constant $k$ gives consecutive differences $\dfrac{t_2}{k}-\dfrac{t_1}{k}=\dfrac dk$ , which are equal. So the new sequence is an AP.

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2083 · Science Practice Set A · 1 mark

If $n(A) = 40$ , $n(B) = 50$ , $n(U) = 200$ and $n(A \cap B) = 20$ , then $n(\overline{A \cup B}) =$ (a) 20 (b) 130 (c) 40 (d) 100

Answer

(b) 130

$n(A\cup B)=40+50-20=70$

$n(\overline{A\cup B})=n(U)-n(A\cup B)=200-70=130$

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2083 · Science Practice Set A · 1 mark

In a week a company produced 10,000 computer disks, of which 30 are found to be defective. At this rate, if the factory produced 1,000,000 computer disks, how many would be defective? (a) 30 (b) 300 (c) 3,000 (d) 30,000

Answer

(c) 3,000

The rate is $\frac{30}{10000}$ . The new number is 100 times larger, so the defective disks are $30\times100=3000$ .

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2083 · Science Practice Set A · 1 mark

A bag contains 4 red balls and 10 blue balls. If two balls are removed from the bag at random and no ball is returned to the bag after removal, what is the probability that both balls will be blue? (a) $\frac{25}{49}$ (b) $\frac{24}{49}$ (c) $\frac{45}{98}$ (d) $\frac{45}{91}$

Answer

(d) $\frac{45}{91}$

There are 14 balls in total. Without replacement:

$P(\text{both blue})=\frac{10}{14}\times\frac{9}{13}=\frac{90}{182}=\frac{45}{91}$

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2083 · Science Practice Set A · 1 mark

If $f(x) = x^2$ and $g(x) = x - 1$ , what is the value of $f\circ g(2x+1)$ ? (a) $2x^2$ (b) $4x^2 + 4x$ (c) $4x^2$ (d) $(2x-1)^2$

Answer

(c) $4x^2$

$g(2x+1)=(2x+1)-1=2x$

$f\circ g(2x+1)=f(2x)=(2x)^2=4x^2$

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2083 · Science Practice Set A · 1 mark

What is the radius of the circle whose equation is $x^2 + y^2 - 6x + 8y = 0$ ? (a) 1 (b) 2 (c) 3 (d) 5

Answer

(d) 5

Complete the squares:

$(x-3)^2+(y+4)^2=9+16=25$

So the centre is $(3,-4)$ and the radius is $\sqrt{25}=5$ .

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2083 · Science Practice Set A · 1 mark

AC and AB are radii of the circle with centre A, and $AB \perp AC$ . If $\triangle ABC$ has area 18, what is the circumference of the circle? (a) $6\pi$ (b) $9\pi$ (c) $12\pi$ (d) $18\pi$

Answer

(c) $12\pi$

$\triangle ABC$ is right-angled at $A$ with legs $r$ , so $\frac12 r^2=18$ , giving $r=6$ .

$C=2\pi r=12\pi$

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2083 · Science Practice Set A · 1 mark

A cylindrical log of wood is cut into two equal halves vertically from its diameter. If its length and diameter are 200 cm and 140 cm respectively, calculate the total surface area of one half. (a) $87000\ \text{cm}^2$ (b) $87200\ \text{cm}^2$ (c) $87400\ \text{cm}^2$ (d) $87600\ \text{cm}^2$

Answer

(c) $87400\ \text{cm}^2$

Radius $r=70$ cm, length $l=200$ cm, using $\pi=\frac{22}{7}$ . One half has a curved half-surface, two half-circular ends and one flat rectangular face.

$\text{Curved}=\pi r l=\frac{22}{7}\times70\times200=44000$

$\text{Two half-ends}=\pi r^2=\frac{22}{7}\times4900=15400$

$\text{Flat face}=2r\times l=140\times200=28000$

$\text{Total}=44000+15400+28000=87400\ \text{cm}^2$

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2083 · Science Practice Set A · 1 mark

A researcher found that the number of bacteria in a certain sample doubles every hour. If there were 6 bacteria in the sample at the beginning, how many bacteria were there after 9 hours? (a) 54 (b) 512 (c) 1536 (d) 3072

Answer

(d) 3072

Doubling each hour gives $6\times2^9=6\times512=3072$ .

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2083 · Science Practice Set A · 1 mark

A value that satisfies the equation $\cos^2 x - 2\cos x = 0$ in degrees is (a) 0 (b) 30 (c) 60 (d) 90

Answer

(d) 90

$\cos x(\cos x-2)=0$

Since $\cos x=2$ is impossible, $\cos x=0$ , so $x=90^\circ$ .

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2083 · Science Practice Set A · 1 mark

$\cot\theta - \frac{\cos 2\theta}{\sin\theta\cos\theta}$ is equal to (a) $\sin\theta$ (b) $\cos\theta$ (c) $\tan\theta$ (d) $\sec\theta$

Answer

(c) $\tan\theta$

$\cot\theta-\frac{\cos2\theta}{\sin\theta\cos\theta}=\frac{\cos^2\theta-(\cos^2\theta-\sin^2\theta)}{\sin\theta\cos\theta}=\frac{\sin^2\theta}{\sin\theta\cos\theta}=\tan\theta$

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2083 · Science Practice Set A · 1 mark

If $m > 1$ , the maximum value of $2m\sin 2x$ is equal to (a) 2 (b) $m$ (c) $2m$ (d) $4m$

Answer

(c) $2m$

The maximum of $\sin2x$ is 1, so the maximum of $2m\sin2x$ is $2m$ .

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2083 · Science Practice Set A · 1 mark

If $A$ is a matrix of $3 \times 4$ , $B$ is another matrix such that $AB$ and $BA$ are both possible, the order of matrix $B$ is (a) $3 \times 3$ (b) $3 \times 4$ (c) $4 \times 3$ (d) $4 \times 4$

Answer

(c) $4\times3$

$AB$ exists only if $B$ has 4 rows, and $BA$ exists only if the columns of $B$ equal the 3 rows of $A$ . So $B$ is $4\times3$ .

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2083 · Science Practice Set A · 1 mark

$\sin(135^\circ + x)\cdot\sin(135^\circ - x)$ equals (a) $\sqrt{2}\sin x$ (b) $\sqrt{2}\cos x$ (c) $2\cos 2x$ (d) $\frac{\cos 2x}{2}$

Answer

(d) $\frac{\cos 2x}{2}$

Using $\sin A\sin B=\frac12[\cos(A-B)-\cos(A+B)]$ :

$\sin(135^\circ+x)\sin(135^\circ-x)=\frac12[\cos2x-\cos270^\circ]=\frac{\cos2x}{2}$

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2083 · Science Practice Set A · 1 mark

The first, second and last terms of an A.P. are $a$ , $b$ and $2a$ , then the number of terms in the series is (a) $\frac{b}{b-a}$ (b) $\frac{b}{b+a}$ (c) $\frac{a}{b-a}$ (d) $\frac{b}{b+a}$

Answer

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

The common difference is $d=b-a$ . The last term gives $2a=a+(n-1)(b-a)$ , so $n-1=\frac{a}{b-a}$ .

$n=\frac{a}{b-a}+1=\frac{b}{b-a}$

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2083 · Science Practice Set B · 1 mark

$-3x + y = 6$ , $ax + 2y = 4$ . In the system of equations above, $a$ is a constant. For which of the following values of $a$ does the system have no solution? (a) $-6$ (b) $-3$ (c) 3 (d) 6

Answer

(a) $-6$

Write each equation as $y = mx + c$ . The first gives $y = 3x + 6$ with slope $3$ . The second gives $y = -\frac{a}{2}x + 2$ with slope $-\frac{a}{2}$ .

No solution requires parallel distinct lines, so $-\frac{a}{2} = 3$ , which gives $a = -6$ . The intercepts $6$ and $2$ differ, so the lines are distinct and the system has no solution.

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2083 · Science Practice Set B · 1 mark

$\tan\frac{A}{2} + \cot\frac{A}{2}$ is equivalent to (a) $2\sin A$ (b) $2\cos A$ (c) $2\sec A$ (d) $2\csc A$

Answer

(d) $2\csc A$

$\tan\frac{A}{2}+\cot\frac{A}{2}=\frac{\sin^2\frac{A}{2}+\cos^2\frac{A}{2}}{\sin\frac{A}{2}\cos\frac{A}{2}}=\frac{1}{\sin\frac{A}{2}\cos\frac{A}{2}}=\frac{2}{\sin A}=2\csc A$

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2083 · Science Practice Set B · 1 mark

For $0 \le x \le \pi$ and $\cos x = -1$ , what is the value of $\cos\frac{x}{2}$ ? (a) $-\frac{\sqrt{3}}{2}$ (b) $-\frac{1}{2}$ (c) 0 (d) $\frac{\sqrt{3}}{2}$

Answer

(c) $0$

In $0 \le x \le \pi$ , $\cos x = -1$ gives $x = \pi$ . Then $\cos\frac{x}{2} = \cos\frac{\pi}{2} = 0$ .

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2083 · Science Practice Set B · 1 mark

If two unbiased dice are rolled once, what is the probability that the number of dots on the two faces will be 4? (a) $\frac{1}{4}$ (b) $\frac{1}{12}$ (c) $\frac{1}{18}$ (d) $\frac{1}{36}$

Answer

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

Taking the total on the two faces to be $4$ , the favourable outcomes are $(1,3),(2,2),(3,1)$ , which is $3$ of the $36$ equally likely outcomes.

$P=\frac{3}{36}=\frac{1}{12}$

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2083 · Science Practice Set B · 1 mark

If $t_1, t_2, t_3$ are in AP, then $\frac{t_1}{k}, \frac{t_2}{k}, \frac{t_3}{k}, \dots$ are in (a) AP (b) GP (c) HP (d) None of them

Answer

(a) AP

If $t_2 - t_1 = t_3 - t_2 = d$ , then dividing each term by $k$ gives a common difference $\frac{d}{k}$ between consecutive terms. The new sequence is therefore also an AP.

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2083 · Science Practice Set B · 1 mark

If $A = \begin{pmatrix} 2 & 1 \\ 4 & -1 \end{pmatrix}$ , then $A^2 - A$ is equal to (a) $I$ (b) $2I$ (c) $6I$ (d) $O$

Answer

(c) $6I$

$A^2=\begin{pmatrix}2&1\\4&-1\end{pmatrix}\begin{pmatrix}2&1\\4&-1\end{pmatrix}=\begin{pmatrix}8&1\\4&5\end{pmatrix}$

$A^2-A=\begin{pmatrix}8-2&1-1\\4-4&5+1\end{pmatrix}=\begin{pmatrix}6&0\\0&6\end{pmatrix}=6I$

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2083 · Science Practice Set C · 1 mark

If $A$ and $B$ are any two sets such that $n(A) = 17$ , $n(A \cup B) = 38$ , $n(A \cap B) = 2$ , then $n(A - B) =$ (a) 21 (b) 15 (c) 23 (d) 51

Answer

(b) 15

$n(A-B)=n(A)-n(A\cap B)=17-2=15$

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2083 · Science Practice Set C · 1 mark

What is the area of the circle whose equation is $(x-3)^2 + (y+5)^2 = 18$ ? (a) $9\pi$ (b) $18\pi$ (c) $72\pi$ (d) $81\pi$

Answer

(b) $18\pi$

The equation is of the form $(x-h)^2+(y-k)^2=r^2$ , so $r^2=18$ .

$\text{Area}=\pi r^2=18\pi$

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2083 · Science Practice Set C · 1 mark

A sphere is inscribed inside a cube. What is the probability that a point that is inside the cube is also inside the sphere? (a) $\frac{1}{\pi}$ (b) $\frac{\pi}{8}$ (c) $\frac{\pi}{6}$ (d) $\frac{\pi}{4}$

Answer

(c) $\frac{\pi}{6}$

Let the cube have side $2r$ , so the sphere has radius $r$ .

$P=\frac{\frac43\pi r^3}{(2r)^3}=\frac{4\pi r^3/3}{8r^3}=\frac{\pi}{6}$

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2083 · Science Practice Set C · 1 mark

John's six test scores are given. What is the positive difference between the mean of the scores and the median of the scores: 86, 72, 92, 62, 99, 93? (a) 5 (b) 6 (c) 7 (d) 37

Answer

(a) 5

Mean:

$\frac{86+72+92+62+99+93}{6}=\frac{504}{6}=84$

Ordered scores: 62, 72, 86, 92, 93, 99, so the median is $\frac{86+92}{2}=89$ .

Difference: $89-84=5$ .

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2083 · Science Practice Set C · 1 mark

If $\tan\alpha = \frac{1}{2}$ , $\tan\beta = \frac{1}{3}$ , then $\alpha + \beta$ is equal to (a) $\pi/2$ (b) $\pi/3$ (c) $\pi/4$ (d) $\pi/6$

Answer

(c) $\pi/4$

$\tan(\alpha+\beta)=\frac{\frac12+\frac13}{1-\frac12\cdot\frac13}=\frac{5/6}{5/6}=1$

Hence $\alpha+\beta=\pi/4$ .

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2083 · Science Practice Set C · 1 mark

Two poles stand on opposite sides of a level street that is 12 meters wide. If one pole is 5 meters taller than the other, what is the angle of elevation between the tops of the poles? (a) $\sin^{-1}\frac{5}{12}$ (b) $\sin^{-1}\frac{1}{5}$ (c) $\tan^{-1}\frac{5}{12}$ (d) $\tan^{-1}\frac{5}{13}$

Answer

(c) $\tan^{-1}\frac{5}{12}$

Joining the two tops gives a right triangle with horizontal side 12 m (the street) and vertical side 5 m (the difference in heights). So

$\tan\theta=\frac{5}{12}\Rightarrow \theta=\tan^{-1}\frac{5}{12}$

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2083 · Science Practice Set C · 1 mark

If $f(x) = 3x + 5$ and $f(g(x)) = 6x - 4$ , what is $g(x)$ ? (a) $2x - 9$ (b) $2x - 3$ (c) $3x - 9$ (d) $3x - 3$

Answer

(b) $2x-3$

$f(g(x))=3g(x)+5=6x-4\Rightarrow g(x)=\frac{6x-9}{3}=2x-3$

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2083 · Science Practice Set C · 1 mark

What is the tenth term of the arithmetic sequence whose first term is $x$ and its third term is $x + 6a$ ? (a) $33a$ (b) $x + 24a$ (c) $x + 27a$ (d) $x + 30a$

Answer

(c) $x+27a$

$a_3=x+2d=x+6a$ , so $d=3a$ .

$a_{10}=x+9d=x+27a$

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2083 · Science Practice Set C · 1 mark

If $A = \begin{bmatrix} K & 1 \\ 4 & K \end{bmatrix}$ is a singular matrix, then the value of $K$ is (a) 1 (b) 2 (c) 3 (d) 4

Answer

(b) 2

A matrix is singular when its determinant is zero: $K\cdot K-1\cdot 4=K^2-4=0\Rightarrow K=\pm 2.$ Of the options given, $K=2$ is the one listed.

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2083 · Science Practice Set C · 1 mark

$\frac{5(x+1)}{(x-1)(x+4)} = \frac{A}{x-1} + \frac{3}{x+4}$ , the value of $A$ is (a) 1 (b) 2 (c) 3 (d) $-3$

Answer

(b) 2

Multiply through by $(x-1)(x+4)$ : $5(x+1)=A(x+4)+3(x-1)$ .

Put $x=1$ : $10=5A$ , so $A=2$ . Check with $x=-4$ : $-15=-5\cdot 3$ , which holds.

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2083 · Science Practice Set C · 1 mark

If $2^{x+4} = 32$ , what is the value of $3^{-x}$ ? (a) $\frac{1}{3}$ (b) 3 (c) $\frac{1}{9}$ (d) 1

Answer

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

$2^{x+4}=32=2^5$ , so $x+4=5$ and $x=1$ . Then $3^{-x}=3^{-1}=\frac{1}{3}$ .

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2083 · Science Practice Set C · 1 mark

If OACB is a parallelogram with $\overrightarrow{OC} = \vec{a}$ and $\overrightarrow{AB} = \vec{b}$ , then $\overrightarrow{OA} =$ … (a) $\vec{a} + \vec{b}$ (b) $\vec{a} - \vec{b}$ (c) $\frac{1}{2}(\vec{a} + \vec{b})$ (d) $\frac{1}{2}(\vec{a} - \vec{b})$

Answer

(d) $\frac{1}{2}(\vec{a}-\vec{b})$

In parallelogram OACB, $\overrightarrow{OC}$ and $\overrightarrow{AB}$ are the diagonals. Then $\overrightarrow{OC}=\overrightarrow{OA}+\overrightarrow{OB}=\vec a$ and $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=\vec b$ .

Subtracting: $2\overrightarrow{OA}=\vec a-\vec b$ , so $\overrightarrow{OA}=\frac12(\vec a-\vec b)$ .

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2083 · Science Practice Set C · 1 mark

If Rs 300 is invested at 20%, compounded continuously, how long (to the nearest year) will it take for the money to double? (a) 3 (b) 4 (c) 5 (d) 6

Answer

(a) 3

Doubling: $300e^{0.2t}=600$ , so $e^{0.2t}=2$ and $t=\frac{\ln 2}{0.2}=\frac{0.6931}{0.2}\approx 3.47\approx 3\text{ years}.$

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2083 · Science Practice Set D · 1 mark

If $A$ and $B$ are any two non-empty sets, then $A \cap (A \cup B)'$ is equal to (a) $B$ (b) $A$ (c) $\phi$ (d) $A \cap B$

Answer

(c) $\phi$

By De Morgan's law, $(A\cup B)'=A'\cap B'$ . So $A\cap(A\cup B)'=A\cap A'\cap B'=\phi\cap B'=\phi.$

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2083 · Science Practice Set D · 1 mark

If $a = b + \frac{1}{2}$ , $b = 2c + \frac{1}{2}$ and $c = 3d + \frac{1}{2}$ , which of the following is an expression for $d$ in terms of $a$ ? (a) $\frac{a-2}{6}$ (b) $\frac{2a-3}{6}$ (c) $\frac{2a-3}{12}$ (d) $\frac{3a-2}{18}$

Answer

(a) $\frac{a-2}{6}$

From $a=b+\frac12$ we get $b=a-\frac12$ . Then $c=\frac{b-\frac12}{2}=\frac{a-1}{2}$ . Then $d=\frac{c-\frac12}{3}=\frac{\frac{a-1}{2}-\frac12}{3}=\frac{a-2}{6}.$

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2083 · Science Practice Set D · 1 mark

The relationship between the area $A$ of a circle and its circumference is given by the formula $A = kC^2$ , where $k$ is a constant. What is the value of $k$ ? (a) $\frac{1}{4\pi}$ (b) $\frac{1}{2\pi}$ (c) $\frac{1}{4}$ (d) $4\pi^2$

Answer

(a) $\frac{1}{4\pi}$

With $A=\pi r^2$ and $C=2\pi r$ , we have $C^2=4\pi^2r^2$ , so $k=\frac{A}{C^2}=\frac{\pi r^2}{4\pi^2r^2}=\frac{1}{4\pi}.$

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2083 · Science Practice Set D · 1 mark

What is the ratio of the area of a square to the area of a circle if the square of 10 unit length is inscribed in the circle? (a) $\pi : 2$ (b) $1 : \pi$ (c) $1 : 2$ (d) $2 : \pi$

Answer

(d) $2 : \pi$

The square has side 10, so its area is $100$ . Its diagonal $10\sqrt2$ is the circle's diameter, so $r=5\sqrt2$ and the circle's area is $\pi\cdot 50=50\pi$ . The ratio is $100:50\pi=2:\pi$ .

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2083 · Science Practice Set D · 1 mark

If the area of the triangle with vertices $(0, 0)$ , $(5, 0)$ and $(0, y)$ is 30 square units, what is the value of $y$ ? (a) 5 (b) 6 (c) 10 (d) 12

Answer

(d) 12

The base along the x-axis is 5 and the height is $y$ , so $\tfrac12\cdot5\cdot y=30\Rightarrow y=12.$

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2083 · Science Practice Set D · 1 mark

If $f(2x) = x + 5$ and $f(g(6)) = 13$ , then $2g(6) =$ (a) 6 (b) 16 (c) 32 (d) 36

Answer

(c) 32

Let $t=2x$ . Then $f(t)=\frac t2+5$ . Since $f(g(6))=13$ , we get $\frac{g(6)}{2}+5=13$ , so $g(6)=16$ and $2g(6)=32$ .

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2083 · Science Practice Set D · 1 mark

If the sum of the first 6 terms of a G.P. is 9 times the sum of its first three terms, then the common ratio of the series is (a) 1 (b) $\frac{3}{2}$ (c) 2 (d) $-2$

Answer

(c) 2

$S_6=9S_3$ gives $\frac{a(r^6-1)}{r-1}=9\cdot\frac{a(r^3-1)}{r-1}$ . Dividing by $\frac{a(r^3-1)}{r-1}$ gives $r^3+1=9$ , so $r^3=8$ and $r=2$ .

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2083 · Science Practice Set D · 1 mark

If $6, 9, 14, 2a - 8, 2a + 10, 38, 47, 50$ are in ascending order and their median is 25, then the value of $a$ will be (a) 13 (b) 12 (c) 11 (d) 14

Answer

(b) 12

There are 8 terms, so the median is the mean of the 4th and 5th terms: $\frac{(2a-8)+(2a+10)}{2}=2a+1=25\Rightarrow a=12.$ Check: the terms are $6,9,14,16,34,38,47,50$ , which are ascending.

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2083 · Science Practice Set D · 1 mark

A bag contains 4 white and 6 black balls. Two balls are drawn in succession. Find the probability that both the balls are differently coloured if the first ball is not replaced. (a) $\frac{8}{15}$ (b) $\frac{3}{15}$ (c) $\frac{4}{15}$ (d) $\frac{1}{24}$

Answer

(a) $\frac{8}{15}$

$P=\frac{4}{10}\cdot\frac{6}{9}+\frac{6}{10}\cdot\frac{4}{9}=\frac{24+24}{90}=\frac{8}{15}.$

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2083 · Science Practice Set D · 1 mark

The value of $\sqrt{\frac{1-\sin x}{1+\sin x}}$ is equal to (a) $\tan\left(\frac{x}{2}+\frac{\pi}{4}\right)$ (b) $\tan\left(\frac{x}{2}-\frac{\pi}{4}\right)$ (c) $\cot\left(\frac{x}{2}+\pi\right)$ (d) $\tan\left(\frac{\pi}{4}-\frac{x}{2}\right)$

Answer

(d) $\tan\left(\frac{\pi}{4}-\frac{x}{2}\right)$

Write $1-\sin x=\left(\cos\frac x2-\sin\frac x2\right)^2$ and $1+\sin x=\left(\cos\frac x2+\sin\frac x2\right)^2$ . Then $\sqrt{\frac{1-\sin x}{1+\sin x}}=\frac{\cos\frac x2-\sin\frac x2}{\cos\frac x2+\sin\frac x2}=\frac{1-\tan\frac x2}{1+\tan\frac x2}=\tan\left(\frac{\pi}{4}-\frac{x}{2}\right).$

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2083 · Science Practice Set D · 1 mark

The value of $4\sin A\cos^3 A - 4\cos A\sin^3 A =$ (a) $\cos 8A$ (b) $\sin 2A$ (c) $\cos 4A$ (d) $\sin 4A$

Answer

(d) $\sin 4A$

$4\sin A\cos A(\cos^2A-\sin^2A)=2\sin2A\cos2A=\sin4A.$

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2083 · Science Practice Set D · 1 mark

If $3^{2x+1} = 81$ , then $x =$ (a) 1 (b) $\frac{3}{2}$ (c) 2 (d) $\frac{5}{2}$

Answer

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

Since $81=3^4$ , we have $2x+1=4$ , so $x=\frac32$ .

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2083 · Science Practice Set D · 1 mark

Which of the following matrices does not have an inverse matrix? (a) $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ (b) $\begin{pmatrix} 5 & 3 \\ 1 & 7 \end{pmatrix}$ (c) $\begin{pmatrix} 2 & -4 \\ 3 & -5 \end{pmatrix}$ (d) $\begin{pmatrix} 2 & 3 \\ 4 & 6 \end{pmatrix}$

Answer

(d) $\begin{pmatrix} 2 & 3 \\ 4 & 6 \end{pmatrix}$

A matrix has no inverse when its determinant is zero. Here $2\cdot6-3\cdot4=0$ . The other determinants are $1$ , $35-3=32$ and $-10+12=2$ , all non-zero.

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2083 · Science Practice Set D · 1 mark

PQ and PT are tangent to circle O. If angle P measures $70^\circ$ , how many degrees are in the minor arc QT? (a) 110 (b) 120 (c) 125 (d) 140

Answer

(a) 110

Let the minor arc $QT$ be $m$ . Then the major arc is $360^\circ-m$ , and the angle between two tangents is half the difference of the arcs: $70=\tfrac12\big((360-m)-m\big)\Rightarrow 140=360-2m\Rightarrow m=110^\circ.$

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2011 · Science Entrance Sample Paper · 1 mark

Grace has 16 jellybeans in her pocket. She has 8 red ones, 4 green ones, and 4 blue ones. What is the minimum number of jellybeans she must take out of her pocket to ensure that she has one of each color? (a) 4 (b) 8 (c) 12 (d) 13

Answer

(d) 13

In the worst case she first takes all 8 red and all 4 green jellybeans, which is 12 and still has no blue one. The 13th jellybean must be blue, so 13 guarantees one of each colour.

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2011 · Science Entrance Sample Paper · 1 mark

The function $f(x) = -3x + 12$ on $\mathbb{R}$ is (a) Increasing (b) Decreasing (c) Constant (d) None of these

Answer

(b) Decreasing

$f'(x) = -3 < 0$ for all $x \in \mathbb{R}$ , so the function always decreases as $x$ increases.

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2011 · Science Entrance Sample Paper · 1 mark

The system of equations $x + 2y = -1$ , $-2x - 4y = 2$ has (a) Only one solution (b) Finitely many solutions (c) No solution (d) Infinitely many solutions

Answer

(d) Infinitely many solutions

Multiplying the first equation by $-2$ gives $-2x - 4y = 2$ , which is exactly the second equation. The two equations represent the same line, so every point on it is a solution.

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2011 · Science Entrance Sample Paper · 1 mark

The area of the triangle formed by the lines $y - x = 0$ , $y + x = 0$ and $x = c$ is (a) $c$ sq. units (b) $c^2$ sq. units (c) $\frac{1}{2}c$ sq. units (d) $\frac{1}{2}c^2$ sq. units

Answer

(b) $c^2$ sq. units

The lines $y = x$ and $y = -x$ meet at the origin, and $x = c$ cuts them at $(c, c)$ and $(c, -c)$ . The base along $x = c$ is $2c$ and the height is $c$ :

$\text{Area} = \tfrac{1}{2}(2c)(c) = c^2.$

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2011 · Science Entrance Sample Paper · 1 mark

If $A$ is an $m \times n$ matrix and $B$ is a matrix given in such a way that $AB$ and $BA$ are both defined, then the size of the $B$ matrix is (a) $m \times n$ (b) $n \times n$ (c) $m \times n$ (d) $n \times m$

Answer

(d) $n \times m$

$AB$ is defined only if the number of columns of $A$ ( $n$ ) equals the number of rows of $B$ . $BA$ is defined only if the number of columns of $B$ equals the number of rows of $A$ ( $m$ ). So $B$ is $n \times m$ .

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2011 · Science Entrance Sample Paper · 1 mark

The value of $p$ for which the vectors $\vec{a} = p\vec{i} + \vec{j} - 2\vec{k}$ and $\vec{b} = \vec{i} + \vec{j} - 3\vec{k}$ are perpendicular to each other is (a) 9 (b) $-9$ (c) 7 (d) $-7$

Answer

(d) $-7$

Perpendicular vectors have zero dot product:

$\vec{a}\cdot\vec{b} = p + 1 + 6 = 0 \Rightarrow p = -7.$

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2011 · Science Entrance Sample Paper · 1 mark

A can solve 90% of the problems given in a book and B can solve 70%. What is the probability that at least one of them will solve a problem selected at random from the book? (a) 0.63 (b) 0.16 (c) 0.97 (d) 1.60

Answer

(c) 0.97

$P(A \cup B) = P(A) + P(B) - P(A)P(B) = 0.9 + 0.7 - 0.63 = 0.97.$

Check: $1 - (0.1)(0.3) = 0.97$ .

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2011 · Science Entrance Sample Paper · 1 mark

The number of non-empty proper subsets of order 6 is (a) 64 (b) 63 (c) 62 (d) 60

Answer

(c) 62

A set of 6 elements has $2^6 = 64$ subsets. Removing the empty set and the set itself leaves $64 - 2 = 62$ non-empty proper subsets.

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2011 · Science Entrance Sample Paper · 1 mark

If the common difference is 5, then the total number of arithmetic means between 10 and 50 is (a) 8 (b) 9 (c) 5 (d) 7

Answer

(d) 7

Let there be $n$ arithmetic means between 10 and 50. Then 50 is the $(n+2)$ th term:

$50 = 10 + (n+1)\cdot 5 \Rightarrow n + 1 = 8 \Rightarrow n = 7.$

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2011 · Science Entrance Sample Paper · 1 mark

The area of the triangle formed by the line $2x + 3y = 12$ with the co-ordinate axes is (a) 24 (b) 12 (c) 6 (d) 3

Answer

(b) 12

The intercepts of $2x + 3y = 12$ are $x = 6$ and $y = 4$ .

$\text{Area} = \tfrac{1}{2}\cdot 6 \cdot 4 = 12.$

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2011 · Science Entrance Sample Paper · 1 mark

In a race of 1 km, A beats B by 100 m, B beats C by 50 m, C beats D by 40 m, then A beats D by (a) 190 m (b) 820.8 m (c) 179.2 m (d) none

Answer

(c) 179.2 m

When A runs 1000 m, B runs 900 m. When B runs 1000 m, C runs 950 m. When C runs 1000 m, D runs 960 m.

So when A runs 1000 m, B runs 900 m, C runs $900 \times \frac{950}{1000} = 855$ m, and D runs $855 \times \frac{960}{1000} = 820.8$ m. A beats D by $1000 - 820.8 = 179.2$ m.

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2011 · Science Entrance Sample Paper · 1 mark

If the $m^{th}$ term of an AP is $\frac{1}{n}$ and its $n^{th}$ term is $\frac{1}{m}$ , then its $(mn)^{th}$ term is (a) $mn$ (b) $\frac{1}{mn}$ (c) 1 (d) $\frac{m}{n}$

Answer

(c) 1

Let the first term be $a$ and common difference $d$ . Then $a + (m-1)d = \frac{1}{n}$ and $a + (n-1)d = \frac{1}{m}$ . Subtracting:

$(m-n)d = \frac{1}{n} - \frac{1}{m} = \frac{m-n}{mn} \Rightarrow d = \frac{1}{mn}.$

Then $a = \frac{1}{n} - \frac{m-1}{mn} = \frac{1}{mn}$ . Hence

$T_{mn} = a + (mn-1)d = \frac{1}{mn} + \frac{mn-1}{mn} = 1.$

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2011 · Science Entrance Sample Paper · 1 mark

If the AM and GM of the roots of a quadratic equation are 8 and 5 respectively, then the equation is (a) $x^2 + 6x + 25 = 0$ (b) $x^2 - 16x + 25 = 0$ (c) $x^2 - 16x - 25 = 0$ (d) $x^2 - 13x + 40 = 0$

Answer

(b) $x^2 - 16x + 25 = 0$

AM $= 8$ gives sum of roots $= 16$ . GM $= 5$ gives product of roots $= 25$ . The equation $x^2 - (\text{sum})x + \text{product} = 0$ is

$x^2 - 16x + 25 = 0.$

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2011 · Science Entrance Sample Paper · 1 mark

The value of $\frac{\cos 11^\circ + \sin 11^\circ}{\cos 11^\circ - \sin 11^\circ}$ is equal to the value of (a) $\tan 56^\circ$ (b) $\sin 56^\circ$ (c) $\cot 56^\circ$ (d) $\tan 79^\circ$

Answer

(a) $\tan 56^\circ$

Divide numerator and denominator by $\cos 11^\circ$ :

$\frac{1 + \tan 11^\circ}{1 - \tan 11^\circ} = \frac{\tan 45^\circ + \tan 11^\circ}{1 - \tan 45^\circ \tan 11^\circ} = \tan(45^\circ + 11^\circ) = \tan 56^\circ.$

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2011 · Science Entrance Sample Paper · 1 mark

Two cyclists start biking from a trail's start 3 hours apart. The second cyclist travels at 10 miles per hour and starts 3 hours after the first cyclist who is traveling at 6 miles per hour. How much time will pass before the second cyclist catches up with the first from the time the second cyclist started biking? (a) 2 hours (b) $4\frac{1}{2}$ hours (c) $5\frac{3}{4}$ hours (d) 6 hours

Answer

(b) $4\frac{1}{2}$ hours

When the second cyclist starts, the first is already $6 \times 3 = 18$ miles ahead. The second gains $10 - 6 = 4$ miles per hour, so

$t = \frac{18}{4} = 4\tfrac{1}{2} \text{ hours}.$

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2011 · Science Entrance Sample Paper · 1 mark

The sum of the first $n$ terms of the series $\frac{n-1}{n}, \frac{n-2}{n}, \frac{n-3}{n}, \dots$ is (a) $\frac{n-1}{2n}$ (b) $\frac{n-1}{2}$ (c) $\frac{n(n+1)}{2}$ (d) $\frac{n}{2}$

Answer

(b) $\frac{n-1}{2}$

The series is an AP with first term $\frac{n-1}{n}$ , common difference $-\frac{1}{n}$ and last ( $n$ th) term $\frac{n-n}{n} = 0$ .

$S_n = \frac{n}{2}\left(\frac{n-1}{n} + 0\right) = \frac{n-1}{2}.$

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2011 · Science Entrance Sample Paper · 1 mark

If $(a+b+c)(a-b+c) = 3ac$ , then angle $B$ is equal to (a) $\frac{\pi}{9}$ (b) $\frac{\pi}{2}$ (c) $\frac{\pi}{3}$ (d) $\frac{\pi}{4}$

Answer

(c) $\frac{\pi}{3}$

$(a+c+b)(a+c-b) = (a+c)^2 - b^2 = 3ac$ , so $a^2 + c^2 - b^2 = ac$ . By the cosine rule,

$\cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{ac}{2ac} = \frac{1}{2} \Rightarrow B = \frac{\pi}{3}.$

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2011 · Science Entrance Sample Paper · 1 mark

The circles $x^2 + y^2 + 2ax + c = 0$ and $x^2 + y^2 + 2by + c = 0$ will touch each other if (a) $\frac{1}{a^2} - \frac{1}{b^2} = \frac{1}{c^2}$ (b) $\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} = 0$ (c) $\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2}$ (d) $\frac{1}{a} + \frac{1}{b} = \frac{1}{c}$

Answer

(c) $\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2}$

The centres are $(-a, 0)$ and $(0, -b)$ , so the distance between them satisfies $d^2 = a^2 + b^2$ . Writing the constant term as $c^2$ (the standard form of this problem), the radii satisfy $r_1^2 = a^2 - c^2$ and $r_2^2 = b^2 - c^2$ .

For the circles to touch, $d^2 = (r_1 \pm r_2)^2$ , which gives $r_1 r_2 = \pm c^2$ . Squaring:

$(a^2 - c^2)(b^2 - c^2) = c^4 \Rightarrow a^2 b^2 = c^2(a^2 + b^2).$

Dividing by $a^2 b^2 c^2$ :

$\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2}.$

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Past questions and answers

1.If n(A−B)=327n(A - B) = 327n(A−B)=327, n(B−A)=217n(B - A) = 217n(B−A)=217 and n(A∪B)=600n(A \cup B) = 600n(A∪B)=600, find n(A∩B)n(A \cap B)n(A∩B). (a) 56 (b) 46 (c) 66 (d) 36

3.If the graphs of the equations 2x+5y=72x + 5y = 72x+5y=7 and 6x+cy=116x + cy = 116x+cy=11 are parallel lines, then what is ccc? (a) 3 (b) 5 (c) 10 (d) 15

4.If m%m\%m% of 300 is 150, what is m%m\%m% of 180? (a) 45 (b) 90 (c) 120 (d) 360

5.If 1x+y=2\frac{1}{x} + y = 2x1​+y=2 and x+1y=3x + \frac{1}{y} = 3x+y1​=3, the ratio of xxx to yyy is equal to (a) 1:2 (b) 2:3 (c) 3:1 (d) 3:2

6.The population of a town increases at the rate of 5% per annum and the present population of the town is 185220. What was the population of the town 3 years ago? (a) 160000 (b) 170000 (c) 180000 (d) 150000

7.If the volume of a cone is 24π24\pi24π cubic feet and its height is 2 feet, what is the diameter of the base in feet? (a) 3 (b) 6 (c) 12 (d) 24

8.A man can do a job in hhh hours alone and his son can do it in 2h2h2h hours alone. Together, how many hours will it take them to do the job? (a) 3h3h3h (b) h3\frac{h}{3}3h​ (c) 3h4\frac{3h}{4}43h​ (d) 2h3\frac{2h}{3}32h​

9.If xxx is a positive acute angle and tan⁡x=p\tan x = ptanx=p, then what is cos⁡x\cos xcosx? (a) 1p2+1\frac{1}{\sqrt{p^2+1}}p2+1​1​ (b) pp2+1\frac{p}{\sqrt{p^2+1}}p2+1​p​ (c) pp2−1\frac{p}{\sqrt{p^2-1}}p2−1​p​ (d) p2+1\sqrt{p^2+1}p2+1​

10.If the radius of a circle is decreased by 30%, by what percent will the area of the circular region be decreased? (a) 70% (b) 60% (c) 51% (d) 49%

11.If nnn is an odd integer, which one of the following is an even integer? (a) 2n+n2n + n2n+n (b) 2n+12n + 12n+1 (c) n3n^3n3 (d) n(n+3)n(n+3)n(n+3)

12.If xxx is a positive acute angle and tan⁡x=p\tan x = ptanx=p, then what is cos⁡x\cos xcosx? (a) pp2+1\frac{p}{\sqrt{p^2+1}}p2+1​p​ (b) 1p2+1\frac{1}{\sqrt{p^2+1}}p2+1​1​ (c) pp2−1\frac{p}{\sqrt{p^2-1}}p2−1​p​ (d) p2+1\sqrt{p^2+1}p2+1​

13.If nnn is an even integer, which one of the following must be an odd integer? (a) 3n−23n - 23n−2 (b) 3(n+1)3(n+1)3(n+1) (c) n−2n - 2n−2 (d) n3\frac{n}{3}3n​

14.A man can do a job in hhh hours alone and his son can do it in 2h2h2h hours alone. Together, how many hours will it take them to do the job? (a) 13h\frac{1}{3h}3h1​ (b) h3\frac{h}{3}3h​ (c) 3h4\frac{3h}{4}43h​ (d) 2h3\frac{2h}{3}32h​

15.If the radius of a circle is decreased by 30%, by what percent will the area of the circular region be decreased? (a) 70% (b) 60% (c) 49% (d) 51%

16.A survey of students in a class shows that 45 students like mathematics and 41 students like English. If 12 students like both subjects, how many students like either Mathematics or English? (a) 66 (b) 75 (c) 98 (d) 74

17.How many numbers in the set {−8,−5,0,10,20}\{-8, -5, 0, 10, 20\}{−8,−5,0,10,20} satisfy the condition ∣x−5∣<11|x - 5| < 11∣x−5∣<11? (a) One (b) Two (c) Three (d) Four

18.What is the simplified expression for 2x−44x+20⋅x2−25x2−7x+10\frac{2x-4}{4x+20}\cdot\frac{x^2-25}{x^2-7x+10}4x+202x−4​⋅x2−7x+10x2−25​? (a) 14\frac{1}{4}41​ (b) 12\frac{1}{2}21​ (c) 2 (d) 1

19.The legs of a right-angled triangle are in the ratio of 1 : 2. If the area of the triangle is 25, what is the hypotenuse? (a) 535\sqrt{3}53​ (b) 555\sqrt{5}55​ (c) 10310\sqrt{3}103​ (d) 10510\sqrt{5}105​

20.What is the solution of the equation p=ryr+yp = \frac{ry}{r+y}p=r+yry​ for yyy? (a) r+prp\frac{r+p}{rp}rpr+p​ (b) rpr−p\frac{rp}{r-p}r−prp​ (c) r+pr−p\frac{r+p}{r-p}r−pr+p​ (d) r−prp\frac{r-p}{rp}rpr−p​

22.In the xy-plane, the lines with the equations x+y=15x + y = 15x+y=15 and y=3x−1y = 3x - 1y=3x−1 intersect at (a,b)(a, b)(a,b). What is the value of a−ba - ba−b? (a) 11 (b) 4 (c) −7-7−7 (d) −6-6−6

23.20% of what number is equal to 23\frac{2}{3}32​ of 90? (a) 30 (b) 120 (c) 300 (d) 600

24.If 4x+16=244x + 16 = 244x+16=24, what is the value of x+8x + 8x+8? (a) 14 (b) 12 (c) 10 (d) 8

25.A couple plans to have three children. What is the probability that exactly two will be boys? (a) 23\frac{2}{3}32​ (b) 38\frac{3}{8}83​ (c) 28\frac{2}{8}82​ (d) 18

26.A bag contains 60 marbles. A marble is to be chosen from the bag. If the probability of a blue marble being chosen is 0.35, how many marbles in the bag are blue? (a) 21 (b) 25 (c) 35 (d) 39

27.If 13,y1,y2,9\frac{1}{3}, y_1, y_2, 931​,y1​,y2​,9 are in G.P., then what is the value of y2y_2y2​? (a) 1 (b) 3 (c) 6 (d) 152\frac{15}{2}215​

28.The sum of the roots of 3x2−4x+4=03x^2 - 4x + 4 = 03x2−4x+4=0 is (a) 43\frac{4}{3}34​ (b) 34\frac{3}{4}43​ (c) −43-\frac{4}{3}−34​ (d) 4

29.If x=1+2x = 1 + \sqrt{2}x=1+2​, then x2−2x+1x^2 - 2x + 1x2−2x+1 is equal to (a) 1+21 + \sqrt{2}1+2​ (b) −1-1−1 (c) 2\sqrt{2}2​ (d) 2

30.The average of 34 students in a class is 13 years. If the age of a teacher is included, the average is increased by 1 year. What is the age of the teacher in years? (a) 40 (b) 48 (c) 55 (d) 65

31.If 6 Japanese yen = 5 Swedish kroner and 4 Swedish kroner = 3 Danish kroner, how many Japanese yen can be exchanged for 48 Danish kroner? (a) 70.8 Yen (b) 76.8 Yen (c) 78.8 Yen (d) 77.8 Yen

33.If ppp lb of salt are dissolved in qqq lb of water, the percent of salt in the resulting solution is (a) 100pp+q\frac{100p}{p+q}p+q100p​ (b) pp+q\frac{p}{p+q}p+qp​ (c) 100 pq\frac{100\,p}{q}q100p​ (d) 100 qpq\frac{100\,q}{pq}pq100q​

34.2x+3y=12002x + 3y = 12002x+3y=1200, 3x+2y=13003x + 2y = 13003x+2y=1300. Based on the system of equations above, what is the value of 5x+5y5x + 5y5x+5y? (a) 500 (b) 1000 (c) 2300 (d) 2500

35.If u+t=5u + t = 5u+t=5 and u−t=2u - t = 2u−t=2, what is the value of (u−t)(u2−t2)(u - t)(u^2 - t^2)(u−t)(u2−t2)? (a) 10 (b) 20 (c) 25 (d) 50

36.A right circular cone has a volume of 24π24\pi24π cubic inches. If the height of the cone is 2 inches, what is the radius, in inches, of the base of the cone? (a) 232\sqrt{3}23​ (b) 6 (c) 12 (d) 36

37.Which of the following gives the radius of the sphere in terms of the volume of the sphere? (a) V4π\sqrt{\frac{V}{4\pi}}4πV​​ (b) 3V4π\sqrt{\frac{3V}{4\pi}}4π3V​​ (c) 3V4π3\sqrt[3]{\frac{3V}{4\pi}}34π3V​​ (d) V4π3\frac{V}{\sqrt[3]{4\pi}}34π​V​

38.Circles O and O' are internally tangent to each other. Circle O' passes through the center of O. If the area of circle O is 16, the area of circle O' is (a) 222\sqrt{2}22​ (b) 2 (c) 2π2\sqrt{\pi}2π​ (d) 4

40.If f(x)=5x−3f(x) = 5x - 3f(x)=5x−3 and f(t)=7f(t) = 7f(t)=7, what is the value of ttt? (a) 0 (b) 1 (c) 2 (d) 32

41.Let xxx be the angle formed by the diagonal of the face of a cube and the diagonal of the cube, drawn from the same vertex. cos⁡2x=\cos^2 x = cos2x= (a) 23\frac{2}{3}32​ (b) 13\frac{1}{3}31​ (c) 12\frac{1}{2}21​ (d) 1

42.A flask contains 40 grams of solution that is 25% sugar by weight. How much sugar must be added to the flask so that the resulting solution is 40% sugar by weight? (a) 6 (b) 8 (c) 10 (d) 15

43.ABCD is a parallelogram with A(4,5)A(4, 5)A(4,5), D(2,−3)D(2, -3)D(2,−3) and C(6,−3)C(6, -3)C(6,−3). What is the x-coordinate of point B? (a) 2 (b) 5 (c) 6 (d) 8

44.ABCD is a trapezoid (AB parallel to DC, BC perpendicular to AB) with area 50 square units. If the length of AB is 6 units and the length of BC is 5 units, what is the length of CD? (a) 4 (b) 7 (c) 10 (d) 14

45.In the xy-plane, the graph of the function f(x)=x2+5x+4f(x) = x^2 + 5x + 4f(x)=x2+5x+4 has two x-intercepts. What is the distance between the x-intercepts? (a) 1 (b) 2 (c) 3 (d) 4

46.The selling price of a TV is Rs. 18000. If there is a discount of 10%, find the marked price. (a) Rs. 19000 (b) Rs. 20000 (c) Rs. 17000 (d) Rs. 21000

47.A car drives a distance of ddd miles at 30 mph and returns at 60 mph. What is the average rate for the round trip? (a) 40 mph (b) 43 mph (c) 2d35\frac{2d}{35}352d​ (d) 352d\frac{35}{2d}2d35​

48.A book was on sale at 40% off its original price. If the sale price of the book was Rs. 18, what was the original price of the book? (a) Rs. 7.20 (b) Rs. 10.80 (c) Rs. 30.00 (d) Rs. 45.00

49.A boy grew one year from a height of xxx inches to a height of yyy inches. The percent of increase was (a) 100(y−x)y\frac{100(y-x)}{y}y100(y−x)​ (b) 100(y−x)x\frac{100(y-x)}{x}x100(y−x)​ (c) y−xx\frac{y-x}{x}xy−x​ (d) y−x100\frac{y-x}{100}100y−x​

51.A car drives a distance of ddd miles at 30 mph and returns at 60 mph. What is its average rate for the round trip? (a) 40 mph (b) 43 mph (c) 2d35\frac{2d}{35}352d​ mph (d) 352d\frac{35}{2d}2d35​ mph

52.A book was on sale for 40% off its original price. If the sale price of the book was Rs 18, what was the original price of the book? (a) Rs. 7.20 (b) Rs. 10.80 (c) Rs. 30.00 (d) Rs. 45.00

53.A boy grew in one year from a height of xxx inches to a height of yyy inches. The percent of increase was: (a) 100(y−x)y\frac{100(y-x)}{y}y100(y−x)​ (b) 100(y−x)x\frac{100(y-x)}{x}x100(y−x)​ (c) y−xx\frac{y-x}{x}xy−x​ (d) y−x100\frac{y-x}{100}100y−x​

54.The selling price of a TV is Rs. 18000. If there is a discount of 10%, what is the marked price? (a) Rs. 19000 (b) Rs. 20000 (c) Rs. 17000 (d) Rs. 21000

56.If nnn is a positive integer, which of the following is always odd? (a) 19n+619n + 619n+6 (b) 19n+519n + 519n+5 (c) 19n2+519n^2 + 519n2+5 (d) 18n+518n + 518n+5

57.2x+7y=92x + 7y = 92x+7y=9, 8x+28y=a8x + 28y = a8x+28y=a. In the given system of equations, aaa is a constant. If the system has infinitely many solutions, what is the value of aaa? (a) 4 (b) 9 (c) 36 (d) 54

58.The equation y=7x−1y = 7^x - 1y=7x−1 is graphed in the xy-plane. What is the y-intercept of the graph? (a) (0,−1)(0, -1)(0,−1) (b) (0,7)(0, 7)(0,7) (c) (0,6)(0, 6)(0,6) (d) (0,0)(0, 0)(0,0)

59.At what time does a sum of money double itself at the rate of 8% p.a.? (a) 8 years (b) 10 years (c) 12.5 years (d) 16 years

61.If x=−8x = -8x=−8, the value of x2/3+2x0x^{2/3} + 2x^0x2/3+2x0 is (a) −2-2−2 (b) 4 (c) 6 (d) −6-6−6

62.If 23x+10=(14)x2^{3x+10} = \left(\frac{1}{4}\right)^x23x+10=(41​)x, then x=x = x= (a) −2-2−2 (b) 1 (c) −1-1−1 (d) 2

63.If 3x=4y3x = 4y3x=4y, the value of (x+y)2:(x−y)2(x+y)^2 : (x-y)^2(x+y)2:(x−y)2 is (a) 4 (b) 9 (c) 49 (d) 7

64.How many cubes with side 4 cm can be made from a cube with 16 cm in length? (a) 4 (b) 16 (c) 64 (d) 404

65.If ddd is the diameter of a sphere, what is the surface area of the sphere? (a) 4πd24\pi d^24πd2 (b) 2πd22\pi d^22πd2 (c) πd24\frac{\pi d^2}{4}4πd2​ (d) πd2\pi d^2πd2

66.Two identical solid cubes of side aaa are joined end to end. Then what is the volume of the resulting cuboid? (a) a3a^3a3 (b) 2a32a^32a3 (c) 8a38a^38a3 (d) 10a210a^210a2

67.If 3n−2=1813^{n-2} = \frac{1}{81}3n−2=811​, then the value of nnn is equal to (a) 0 (b) 1 (c) 2 (d) −2-2−2

68.If n(A)=40n(A) = 40n(A)=40, n(B)=50n(B) = 50n(B)=50, n(U)=200n(U) = 200n(U)=200, n(A∩B)=20n(A \cap B) = 20n(A∩B)=20, then n(A∪B‾)=n(\overline{A \cup B}) = n(A∪B)= (a) 20 (b) 130 (c) 40 (d) 100

70.When kkk is subtracted from 10, and the difference is divided by 2, the result is 3. What is the value of kkk? (a) 3 (b) 4 (c) 6 (d) 10

71.The sum of the digits of a two-digit number is 8. If 18 is subtracted from the number the digits are reversed. What is the original number? (a) 35 (b) 44 (c) 53 (d) 62

72.The area of an isosceles triangle is 12 cm212\ \text{cm}^212 cm2 and its base is 8 cm. Find the length of the equal sides of the triangle. (a) 6 cm (b) 8 cm (c) 5 cm (d) 7 cm

73.The perimeter of rectangle ABCD (with length yyy and width xxx) is equal to ppp, and x=23yx = \frac{2}{3}yx=32​y. What is the value of yyy in terms of ppp? (a) p10\frac{p}{10}10p​ (b) 3p10\frac{3p}{10}103p​ (c) p3\frac{p}{3}3p​ (d) 2p5\frac{2p}{5}52p​

74.If the average measure of two angles in a parallelogram is y∘y^\circy∘, what is the average degree measure of the other two angles? (a) 180−y180 - y180−y (b) 180−y2180 - \frac{y}{2}180−2y​ (c) 360−2y360 - 2y360−2y (d) 360+y360 + y360+y

75.If 0.1% of mmm is equal to 10% of nnn, then mmm is what percent of 10n10n10n? (a) 11000%\frac{1}{1000}\%10001​% (b) 1% (c) 10% (d) 1000%

76.A conical tent is 5 m tall and its diameter is 14 m. What is the cost of cloth required to make the tent at Rs. 200 per square meter? (a) Rs. 37850 (b) Rs. 37800 (c) Rs. 37875 (d) Rs. 37880

77.If a+2b=10a + 2b = 10a+2b=10 and b−a=2b - a = 2b−a=2, what is the value of bbb? (a) 10 (b) 8 (c) 6 (d) 4

78.If a>ba > ba>b and a2−2ab+b2=169a^2 - 2ab + b^2 = 169a2−2ab+b2=169, what is the value of a−ba - ba−b? (a) 11 (b) 12 (c) 13 (d) 15

79.Which of the following is a factor of 3x3−11x2−42x3x^3 - 11x^2 - 42x3x3−11x2−42x? (a) x−7x - 7x−7 (b) xxx (c) 3x−63x - 63x−6 (d) x+6x + 6x+6

80.If the perimeter of a square is 28, what is the length of the diagonal? (a) 2142\sqrt{14}214​ (b) 727\sqrt{2}72​ (c) 737\sqrt{3}73​ (d) 14

81.If am=ana^m = a^nam=an, then what is the relation between mmm and nnn? (a) m+n=1m + n = 1m+n=1 (b) m−n=0m - n = 0m−n=0 (c) mn=0mn = 0mn=0 (d) mn=0\frac{m}{n} = 0nm​=0

82.If 103y−3=10.00110^{3y-3} = \frac{1}{0.001}103y−3=0.0011​, then what will be the value of yyy? (a) 3 (b) 4 (c) 1 (d) 2

83.Which one of the following is the simplest form for 2n+4−2(2n)2(2n+3)\frac{2^{n+4} - 2(2^n)}{2(2^{n+3})}2(2n+3)2n+4−2(2n)​? (a) 12\frac{1}{2}21​ (b) 14\frac{1}{4}41​ (c) 78\frac{7}{8}87​ (d) 58\frac{5}{8}85​

84.What is the total surface area of a cone of radius 7 cm and height 24 cm? (a) 710 cm2710\ \text{cm}^2710 cm2 (b) 704 cm2704\ \text{cm}^2704 cm2 (c) 700 cm2700\ \text{cm}^2700 cm2 (d) 725 cm2725\ \text{cm}^2725 cm2

85.1 litre = --------- cubic centimeter. (a) 1000 (b) 100 (c) 10 (d) 1

86.If the radius of a sphere is doubled, what is the percent increase in the surface area? (a) 100 (b) 200 (c) 300 (d) 400

87.In a cube, the ratio of the longest diagonal to a diagonal of a base is (a) 2:3\sqrt{2} : \sqrt{3}2​:3​ (b) 3:2\sqrt{3} : \sqrt{2}3​:2​ (c) 2:12 : 12:1 (d) 6:2\sqrt{6} : \sqrt{2}6​:2​

If $n(A - B) = 327$ , $n(B - A) = 217$ and $n(A \cup B) = 600$ , find $n(A \cap B)$ . (a) 56 (b) 46 (c) 66 (d) 36

If the graphs of the equations $2x + 5y = 7$ and $6x + cy = 11$ are parallel lines, then what is $c$ ? (a) 3 (b) 5 (c) 10 (d) 15

If $m\%$ of 300 is 150, what is $m\%$ of 180? (a) 45 (b) 90 (c) 120 (d) 360

If $\frac{1}{x} + y = 2$ and $x + \frac{1}{y} = 3$ , the ratio of $x$ to $y$ is equal to (a) 1:2 (b) 2:3 (c) 3:1 (d) 3:2

The population of a town increases at the rate of 5% per annum and the present population of the town is 185220. What was the population of the town 3 years ago? (a) 160000 (b) 170000 (c) 180000 (d) 150000

If the volume of a cone is $24\pi$ cubic feet and its height is 2 feet, what is the diameter of the base in feet? (a) 3 (b) 6 (c) 12 (d) 24

A man can do a job in $h$ hours alone and his son can do it in $2h$ hours alone. Together, how many hours will it take them to do the job? (a) $3h$ (b) $\frac{h}{3}$ (c) $\frac{3h}{4}$ (d) $\frac{2h}{3}$

If $x$ is a positive acute angle and $\tan x = p$ , then what is $\cos x$ ? (a) $\frac{1}{\sqrt{p^2+1}}$ (b) $\frac{p}{\sqrt{p^2+1}}$ (c) $\frac{p}{\sqrt{p^2-1}}$ (d) $\sqrt{p^2+1}$

If the radius of a circle is decreased by 30%, by what percent will the area of the circular region be decreased? (a) 70% (b) 60% (c) 51% (d) 49%

If $n$ is an odd integer, which one of the following is an even integer? (a) $2n + n$ (b) $2n + 1$ (c) $n^3$ (d) $n(n+3)$

If $x$ is a positive acute angle and $\tan x = p$ , then what is $\cos x$ ? (a) $\frac{p}{\sqrt{p^2+1}}$ (b) $\frac{1}{\sqrt{p^2+1}}$ (c) $\frac{p}{\sqrt{p^2-1}}$ (d) $\sqrt{p^2+1}$

If $n$ is an even integer, which one of the following must be an odd integer? (a) $3n - 2$ (b) $3(n+1)$ (c) $n - 2$ (d) $\frac{n}{3}$

A man can do a job in $h$ hours alone and his son can do it in $2h$ hours alone. Together, how many hours will it take them to do the job? (a) $\frac{1}{3h}$ (b) $\frac{h}{3}$ (c) $\frac{3h}{4}$ (d) $\frac{2h}{3}$

If the radius of a circle is decreased by 30%, by what percent will the area of the circular region be decreased? (a) 70% (b) 60% (c) 49% (d) 51%

A survey of students in a class shows that 45 students like mathematics and 41 students like English. If 12 students like both subjects, how many students like either Mathematics or English? (a) 66 (b) 75 (c) 98 (d) 74

How many numbers in the set $\{-8, -5, 0, 10, 20\}$ satisfy the condition $|x - 5| < 11$ ? (a) One (b) Two (c) Three (d) Four

What is the simplified expression for $\frac{2x-4}{4x+20}\cdot\frac{x^2-25}{x^2-7x+10}$ ? (a) $\frac{1}{4}$ (b) $\frac{1}{2}$ (c) 2 (d) 1

The legs of a right-angled triangle are in the ratio of 1 : 2. If the area of the triangle is 25, what is the hypotenuse? (a) $5\sqrt{3}$ (b) $5\sqrt{5}$ (c) $10\sqrt{3}$ (d) $10\sqrt{5}$

What is the solution of the equation $p = \frac{ry}{r+y}$ for $y$ ? (a) $\frac{r+p}{rp}$ (b) $\frac{rp}{r-p}$ (c) $\frac{r+p}{r-p}$ (d) $\frac{r-p}{rp}$

In the xy-plane, the lines with the equations $x + y = 15$ and $y = 3x - 1$ intersect at $(a, b)$ . What is the value of $a - b$ ? (a) 11 (b) 4 (c) $-7$ (d) $-6$

20% of what number is equal to $\frac{2}{3}$ of 90? (a) 30 (b) 120 (c) 300 (d) 600

If $4x + 16 = 24$ , what is the value of $x + 8$ ? (a) 14 (b) 12 (c) 10 (d) 8

A couple plans to have three children. What is the probability that exactly two will be boys? (a) $\frac{2}{3}$ (b) $\frac{3}{8}$ (c) $\frac{2}{8}$ (d) 18

A bag contains 60 marbles. A marble is to be chosen from the bag. If the probability of a blue marble being chosen is 0.35, how many marbles in the bag are blue? (a) 21 (b) 25 (c) 35 (d) 39

If $\frac{1}{3}, y_1, y_2, 9$ are in G.P., then what is the value of $y_2$ ? (a) 1 (b) 3 (c) 6 (d) $\frac{15}{2}$

The sum of the roots of $3x^2 - 4x + 4 = 0$ is (a) $\frac{4}{3}$ (b) $\frac{3}{4}$ (c) $-\frac{4}{3}$ (d) 4

If $x = 1 + \sqrt{2}$ , then $x^2 - 2x + 1$ is equal to (a) $1 + \sqrt{2}$ (b) $-1$ (c) $\sqrt{2}$ (d) 2

The average of 34 students in a class is 13 years. If the age of a teacher is included, the average is increased by 1 year. What is the age of the teacher in years? (a) 40 (b) 48 (c) 55 (d) 65

If 6 Japanese yen = 5 Swedish kroner and 4 Swedish kroner = 3 Danish kroner, how many Japanese yen can be exchanged for 48 Danish kroner? (a) 70.8 Yen (b) 76.8 Yen (c) 78.8 Yen (d) 77.8 Yen

If $p$ lb of salt are dissolved in $q$ lb of water, the percent of salt in the resulting solution is (a) $\frac{100p}{p+q}$ (b) $\frac{p}{p+q}$ (c) $\frac{100\,p}{q}$ (d) $\frac{100\,q}{pq}$

$2x + 3y = 1200$ , $3x + 2y = 1300$ . Based on the system of equations above, what is the value of $5x + 5y$ ? (a) 500 (b) 1000 (c) 2300 (d) 2500

If $u + t = 5$ and $u - t = 2$ , what is the value of $(u - t)(u^2 - t^2)$ ? (a) 10 (b) 20 (c) 25 (d) 50

A right circular cone has a volume of $24\pi$ cubic inches. If the height of the cone is 2 inches, what is the radius, in inches, of the base of the cone? (a) $2\sqrt{3}$ (b) 6 (c) 12 (d) 36

Which of the following gives the radius of the sphere in terms of the volume of the sphere? (a) $\sqrt{\frac{V}{4\pi}}$ (b) $\sqrt{\frac{3V}{4\pi}}$ (c) $\sqrt[3]{\frac{3V}{4\pi}}$ (d) $\frac{V}{\sqrt[3]{4\pi}}$

Circles O and O' are internally tangent to each other. Circle O' passes through the center of O. If the area of circle O is 16, the area of circle O' is (a) $2\sqrt{2}$ (b) 2 (c) $2\sqrt{\pi}$ (d) 4

If $f(x) = 5x - 3$ and $f(t) = 7$ , what is the value of $t$ ? (a) 0 (b) 1 (c) 2 (d) 32

Let $x$ be the angle formed by the diagonal of the face of a cube and the diagonal of the cube, drawn from the same vertex. $\cos^2 x =$ (a) $\frac{2}{3}$ (b) $\frac{1}{3}$ (c) $\frac{1}{2}$ (d) 1

A flask contains 40 grams of solution that is 25% sugar by weight. How much sugar must be added to the flask so that the resulting solution is 40% sugar by weight? (a) 6 (b) 8 (c) 10 (d) 15

ABCD is a parallelogram with $A(4, 5)$ , $D(2, -3)$ and $C(6, -3)$ . What is the x-coordinate of point B? (a) 2 (b) 5 (c) 6 (d) 8

ABCD is a trapezoid (AB parallel to DC, BC perpendicular to AB) with area 50 square units. If the length of AB is 6 units and the length of BC is 5 units, what is the length of CD? (a) 4 (b) 7 (c) 10 (d) 14

In the xy-plane, the graph of the function $f(x) = x^2 + 5x + 4$ has two x-intercepts. What is the distance between the x-intercepts? (a) 1 (b) 2 (c) 3 (d) 4

The selling price of a TV is Rs. 18000. If there is a discount of 10%, find the marked price. (a) Rs. 19000 (b) Rs. 20000 (c) Rs. 17000 (d) Rs. 21000

A car drives a distance of $d$ miles at 30 mph and returns at 60 mph. What is the average rate for the round trip? (a) 40 mph (b) 43 mph (c) $\frac{2d}{35}$ (d) $\frac{35}{2d}$

A book was on sale at 40% off its original price. If the sale price of the book was Rs. 18, what was the original price of the book? (a) Rs. 7.20 (b) Rs. 10.80 (c) Rs. 30.00 (d) Rs. 45.00

A boy grew one year from a height of $x$ inches to a height of $y$ inches. The percent of increase was (a) $\frac{100(y-x)}{y}$ (b) $\frac{100(y-x)}{x}$ (c) $\frac{y-x}{x}$ (d) $\frac{y-x}{100}$

A car drives a distance of $d$ miles at 30 mph and returns at 60 mph. What is its average rate for the round trip? (a) 40 mph (b) 43 mph (c) $\frac{2d}{35}$ mph (d) $\frac{35}{2d}$ mph

A book was on sale for 40% off its original price. If the sale price of the book was Rs 18, what was the original price of the book? (a) Rs. 7.20 (b) Rs. 10.80 (c) Rs. 30.00 (d) Rs. 45.00

A boy grew in one year from a height of $x$ inches to a height of $y$ inches. The percent of increase was: (a) $\frac{100(y-x)}{y}$ (b) $\frac{100(y-x)}{x}$ (c) $\frac{y-x}{x}$ (d) $\frac{y-x}{100}$

The selling price of a TV is Rs. 18000. If there is a discount of 10%, what is the marked price? (a) Rs. 19000 (b) Rs. 20000 (c) Rs. 17000 (d) Rs. 21000

If $n$ is a positive integer, which of the following is always odd? (a) $19n + 6$ (b) $19n + 5$ (c) $19n^2 + 5$ (d) $18n + 5$

$2x + 7y = 9$ , $8x + 28y = a$ . In the given system of equations, $a$ is a constant. If the system has infinitely many solutions, what is the value of $a$ ? (a) 4 (b) 9 (c) 36 (d) 54

The equation $y = 7^x - 1$ is graphed in the xy-plane. What is the y-intercept of the graph? (a) $(0, -1)$ (b) $(0, 7)$ (c) $(0, 6)$ (d) $(0, 0)$

At what time does a sum of money double itself at the rate of 8% p.a.? (a) 8 years (b) 10 years (c) 12.5 years (d) 16 years

If $x = -8$ , the value of $x^{2/3} + 2x^0$ is (a) $-2$ (b) 4 (c) 6 (d) $-6$

If $2^{3x+10} = \left(\frac{1}{4}\right)^x$ , then $x =$ (a) $-2$ (b) 1 (c) $-1$ (d) 2

If $3x = 4y$ , the value of $(x+y)^2 : (x-y)^2$ is (a) 4 (b) 9 (c) 49 (d) 7

How many cubes with side 4 cm can be made from a cube with 16 cm in length? (a) 4 (b) 16 (c) 64 (d) 404

If $d$ is the diameter of a sphere, what is the surface area of the sphere? (a) $4\pi d^2$ (b) $2\pi d^2$ (c) $\frac{\pi d^2}{4}$ (d) $\pi d^2$

Two identical solid cubes of side $a$ are joined end to end. Then what is the volume of the resulting cuboid? (a) $a^3$ (b) $2a^3$ (c) $8a^3$ (d) $10a^2$

If $3^{n-2} = \frac{1}{81}$ , then the value of $n$ is equal to (a) 0 (b) 1 (c) 2 (d) $-2$

If $n(A) = 40$ , $n(B) = 50$ , $n(U) = 200$ , $n(A \cap B) = 20$ , then $n(\overline{A \cup B}) =$ (a) 20 (b) 130 (c) 40 (d) 100

When $k$ is subtracted from 10, and the difference is divided by 2, the result is 3. What is the value of $k$ ? (a) 3 (b) 4 (c) 6 (d) 10

The sum of the digits of a two-digit number is 8. If 18 is subtracted from the number the digits are reversed. What is the original number? (a) 35 (b) 44 (c) 53 (d) 62

The area of an isosceles triangle is $12\ \text{cm}^2$ and its base is 8 cm. Find the length of the equal sides of the triangle. (a) 6 cm (b) 8 cm (c) 5 cm (d) 7 cm

The perimeter of rectangle ABCD (with length $y$ and width $x$ ) is equal to $p$ , and $x = \frac{2}{3}y$ . What is the value of $y$ in terms of $p$ ? (a) $\frac{p}{10}$ (b) $\frac{3p}{10}$ (c) $\frac{p}{3}$ (d) $\frac{2p}{5}$

If the average measure of two angles in a parallelogram is $y^\circ$ , what is the average degree measure of the other two angles? (a) $180 - y$ (b) $180 - \frac{y}{2}$ (c) $360 - 2y$ (d) $360 + y$

If 0.1% of $m$ is equal to 10% of $n$ , then $m$ is what percent of $10n$ ? (a) $\frac{1}{1000}\%$ (b) 1% (c) 10% (d) 1000%

A conical tent is 5 m tall and its diameter is 14 m. What is the cost of cloth required to make the tent at Rs. 200 per square meter? (a) Rs. 37850 (b) Rs. 37800 (c) Rs. 37875 (d) Rs. 37880

If $a + 2b = 10$ and $b - a = 2$ , what is the value of $b$ ? (a) 10 (b) 8 (c) 6 (d) 4

If $a > b$ and $a^2 - 2ab + b^2 = 169$ , what is the value of $a - b$ ? (a) 11 (b) 12 (c) 13 (d) 15

Which of the following is a factor of $3x^3 - 11x^2 - 42x$ ? (a) $x - 7$ (b) $x$ (c) $3x - 6$ (d) $x + 6$

If the perimeter of a square is 28, what is the length of the diagonal? (a) $2\sqrt{14}$ (b) $7\sqrt{2}$ (c) $7\sqrt{3}$ (d) 14

If $a^m = a^n$ , then what is the relation between $m$ and $n$ ? (a) $m + n = 1$ (b) $m - n = 0$ (c) $mn = 0$ (d) $\frac{m}{n} = 0$

If $10^{3y-3} = \frac{1}{0.001}$ , then what will be the value of $y$ ? (a) 3 (b) 4 (c) 1 (d) 2

Which one of the following is the simplest form for $\frac{2^{n+4} - 2(2^n)}{2(2^{n+3})}$ ? (a) $\frac{1}{2}$ (b) $\frac{1}{4}$ (c) $\frac{7}{8}$ (d) $\frac{5}{8}$

What is the total surface area of a cone of radius 7 cm and height 24 cm? (a) $710\ \text{cm}^2$ (b) $704\ \text{cm}^2$ (c) $700\ \text{cm}^2$ (d) $725\ \text{cm}^2$

1 litre = --------- cubic centimeter. (a) 1000 (b) 100 (c) 10 (d) 1

If the radius of a sphere is doubled, what is the percent increase in the surface area? (a) 100 (b) 200 (c) 300 (d) 400

In a cube, the ratio of the longest diagonal to a diagonal of a base is (a) $\sqrt{2} : \sqrt{3}$ (b) $\sqrt{3} : \sqrt{2}$ (c) $2 : 1$ (d) $\sqrt{6} : \sqrt{2}$

In a bag there are 7 red and 5 blue colour balls. What is the probability that two different coloured balls are selected when the balls are drawn without replacement? (a) $\frac{35}{132}$ (b) $\frac{35}{66}$ (c) $\frac{12}{35}$ (d) $\frac{5}{11}$