+2 college entrance (after SEE) Capital College & Research Center (CCRC) +2 Entrance (after SEE): Mathematics

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If the surface area of a spherical ball is $616\ \text{cm}^2$ , then its diameter is ………… (a) 14 cm (b) 7 cm (c) 21 cm (d) 28 cm

Similar questions: The surface area of a ball is 16pi cm^2, then (2083 · Management Model) · The surface area of a spherical ball is 616 cm^2. (2079 · Management and Humanities Model)

Answer

(a) 14 cm

$4\pi r^2=616\Rightarrow r^2=\frac{616\times7}{4\times22}=49\Rightarrow r=7\ \text{cm},$ so the diameter is $14$ cm.

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The greatest number of 3 digits, which is a perfect square is ……… (a) 981 (b) 999 (c) 951 (d) 961

Similar questions: The greatest number of 4 digits, which is a (2079 · Management and Humanities Model)

Answer

(d) 961

$31^2 = 961$ and $32^2 = 1024$ has four digits, so the greatest 3-digit perfect square is $961$ .

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If the side of a rhombus is 10 cm and one diagonal is 16 cm, the area of the rhombus is ……… (a) $24\ \text{cm}^2$ (b) $48\ \text{cm}^2$ (c) $96\ \text{cm}^2$ (d) $90\ \text{cm}^2$

Similar questions: The side of a rhombus is 10 cm and one diagonal (2079 · Science Model)

Answer

(c) $96\ \text{cm}^2$

The diagonals bisect each other at right angles. Half of the given diagonal is $8$ cm, so half of the other is $\sqrt{10^2 - 8^2} = 6$ cm, giving a diagonal of $12$ cm. $\text{Area} = \tfrac12 \times 16 \times 12 = 96\ \text{cm}^2.$

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The surface area of a ball is $16\pi\ \text{cm}^2$ , then what is its diameter? (a) 8 cm (b) 4 cm (c) 2 cm (d) 1 cm

Similar questions: If the surface area of a spherical ball is 616 (2081 · Science Model)

Answer

(b) 4 cm

Surface area of a sphere is $4\pi r^2$ , so $4\pi r^2 = 16\pi$ gives $r^2 = 4$ and $r = 2$ cm.

Diameter $= 2r = 4$ cm.

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The side of a rhombus is 10 cm and one diagonal is 16 cm. The area of the rhombus is ………. (a) $24\ \text{cm}^2$ (b) $48\ \text{cm}^2$ (c) $96\ \text{cm}^2$ (d) $90\ \text{cm}^2$

Similar questions: If the side of a rhombus is 10 cm and one (2083 · Management Model)

Answer

(c) $96\ \text{cm}^2$

The diagonals bisect each other at right angles. Half of the given diagonal is $8$ , so half of the other is $\sqrt{10^2-8^2}=6$ , making that diagonal $12$ cm.

$\text{Area}=\tfrac12\times16\times12=96\ \text{cm}^2$

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The greatest number of 4 digits, which is a perfect square is …….. (a) 9996 (b) 9801 (c) 9900 (d) 9604

Similar questions: The greatest number of 3 digits, which is a (2083 · Management Model)

Answer

(b) 9801

$99^2 = 9801$ and $100^2 = 10000$ has 5 digits, so 9801 is the greatest 4-digit perfect square.

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The quadratic equation whose roots are $\frac{1}{2}$ and $-\frac{1}{2}$ is …….. (a) $x^2 - 4 = 0$ (b) $4x^2 - 1 = 0$ (c) $4x^2 + 1 = 0$ (d) $x^2 - \frac{1}{2} = 0$

Answer

(b) $4x^2 - 1 = 0$

Sum of roots $=0$ , product $=-\frac14$ , so the equation is $x^2-0\cdot x-\frac14=0$ , i.e. $4x^2-1=0$ .

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Let $A = \{a, b, c\}$ , the possible number of subsets for the set $A$ is …… (a) 3 (b) 5 (c) 6 (d) 8

Similar questions: Let A = 1, 2, 3, the possible number of subsets (2079 · Management and Humanities Model)

Answer

(d) 8

A set with $n$ elements has $2^n$ subsets. Here $n=3$ , so the number of subsets is

$2^3 = 8.$

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The simple interest on Rs. 2500 for 3 years at $12\frac{1}{2}\%$ per annum is …….. (a) Rs. 900 (b) Rs. 937.50 (c) Rs. 1000 (d) Rs. 1050

Similar questions: The simple interest on Rs. 20,000 for 3 years at (2079 · Management and Humanities Model)

Answer

(b) Rs 937.50

$I = \frac{PTR}{100} = \frac{2500 \times 3 \times 12.5}{100} = \text{Rs } 937.50.$

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Rama bears a loss of 12% while selling apples at Rs. 1,320. What is her cost price? (a) Rs. 1500 (b) Rs. 1450 (c) Rs. 1400 (d) Rs. 1350

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Answer

(a) Rs. 1500

A loss of 12% means $SP = 88\%$ of $CP$ .

$CP = \frac{1320 \times 100}{88} = \text{Rs } 1500$

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A couple plans to have three children. What is the probability that at least two will be boys? (a) $\frac{3}{8}$ (b) $\frac{1}{2}$ (c) $\frac{2}{8}$ (d) $\frac{1}{8}$

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Answer

(b) $\frac12$

Of the $8$ equally likely outcomes, those with at least two boys are exactly $2$ boys ( $3$ ways) and $3$ boys ( $1$ way).

$P=\frac{3+1}{8}=\frac12.$

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How many cubes with 4 cm can be made from a cube with 16 cm in length? (a) 128 (b) 64 (c) 32 (d) 16

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Answer

(b) 64

Each edge holds $\frac{16}{4}=4$ small cubes, so the number is $4^3=64$ (equivalently $\frac{16^3}{4^3}$ ).

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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) $\frac{1}{8}$

Similar questions: A couple plans to have three children. What is (2079 · Science Model)

Answer

(b) $\dfrac38$

There are $2^3=8$ equally likely outcomes. Exactly two boys: BBG, BGB, GBB, which is $\binom32=3$ outcomes. So the probability is $\dfrac38$ .

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What is the probability that a leap year has 53 Sundays? (a) $\frac{1}{4}$ (b) $\frac{1}{3}$ (c) $\frac{1}{7}$ (d) $\frac{2}{7}$

Similar questions: What is the probability for 53 Sundays in a leap (2080 · Law Model)

Answer

(d) $\dfrac27$

A leap year has $366=52\times7+2$ days, so 52 full weeks plus 2 extra days. The pairs of extra days are (Sun,Mon), (Mon,Tue), (Tue,Wed), (Wed,Thu), (Thu,Fri), (Fri,Sat), (Sat,Sun). Two of these 7 contain Sunday, so the probability is $\dfrac27$ .

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

Similar questions: How many cubes with 4 cm can be made from a cube (2079 · Science Model)

Answer

(c) 64

$\frac{16^3}{4^3}=\left(\frac{16}{4}\right)^3=4^3=64$

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What is the probability for 53 Sundays in a leap year? (a) $\frac{6}{7}$ (b) $\frac{5}{7}$ (c) $\frac{2}{7}$ (d) $\frac{1}{7}$

Similar questions: What is the probability that a leap year has 53 (2081 · Science Model)

Answer

(c) $\frac{2}{7}$

A leap year has 366 days = 52 weeks + 2 extra days. The pair of extra days can be (Sun,Mon), (Mon,Tue), (Tue,Wed), (Wed,Thu), (Thu,Fri), (Fri,Sat), (Sat,Sun): 7 equally likely cases, of which 2 contain a Sunday.

$P = \frac{2}{7}$

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Let $A = \{1, 2, 3\}$ , the possible number of subsets for the set $A$ is ………. (a) 8 (b) 7 (c) 6 (d) 3

Similar questions: Let A = a, b, c, the possible number of subsets (2083 · Management Model)

Answer

(a) 8

A set with $n$ elements has $2^n$ subsets, so $2^3 = 8$ subsets (including $\emptyset$ and $A$ itself).

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The simple interest on Rs. 20,000 for 3 years at 13% per annum is …… (a) Rs. 7500 (b) Rs. 7800 (c) Rs. 8100 (d) Rs. 8500

Similar questions: The simple interest on Rs. 2500 for 3 years at (2083 · Management Model)

Answer

(b) Rs 7800

$I = \frac{PTR}{100} = \frac{20000 \times 3 \times 13}{100} = \text{Rs } 7800.$

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Rama bears a loss of 12% while selling a mobile at Rs. 1,320, then her cost price is ……… (a) Rs. 1,500 (b) Rs. 1,550 (c) Rs. 1,575 (d) Rs. 1,600

Similar questions: Rama bears a loss of 12% while selling apples at (2083 · Management Model)

Answer

(a) Rs. 1,500

Selling price is $100-12=88\%$ of the cost price.

$CP=\frac{1320}{88}\times 100=\text{Rs. }1{,}500$

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If $a + b + c = 0$ , then $a^3 + b^3 + c^3$ is equal to …… (a) $abc$ (b) 0 (c) $3abc$ (d) $9abc$

Answer

(c) $3abc$

$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=0$ , hence $a^3+b^3+c^3=3abc$ .

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If $x + \frac{1}{x} = 4$ , then the value of $x^2 + \frac{1}{x^2}$ is …… (a) 18 (b) 14 (c) 16 (d) 20

Answer

(b) 14

$x^2+\frac{1}{x^2}=\left(x+\frac1x\right)^2-2=16-2=14$ .

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If $x + y - z = 7$ , $x - y + z = 8$ and $-x + y + z = 6$ , then the average value of $x, y, z$ is ……….. (a) 21 (b) 15 (c) 9 (d) 7

Answer

(d) 7

Adding the three equations gives $x+y+z=7+8+6=21$ .

$\text{Average}=\frac{21}{3}=7$

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The area of a square field is $144\ \text{m}^2$ , then its perimeter is ……… (a) 12 m (b) 48 m (c) 144 m (d) 150 m

Answer

(b) 48 m

Side $=\sqrt{144}=12$ m, so the perimeter is $4\times12=48$ m.

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Ram does $\frac{3}{5}$ of a work in 12 days. He finishes the remaining work in 4 days with the help of Sanu. In how many days will Sanu do the work alone? (a) 18 days (b) 20 days (c) 24 days (d) 30 days

Answer

(b) 20 days

Ram does $\frac35$ in 12 days, so he does $\frac1{20}$ per day. The remaining $\frac25$ is done in 4 days by both, so together they do $\frac1{10}$ per day. Sanu does $\frac1{10}-\frac1{20}=\frac1{20}$ per day, so he needs 20 days alone.

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Which of the following is a rational number? (a) $\sqrt{2}$ (b) $\sqrt{14}$ (c) $\sqrt{24}$ (d) $\sqrt{36}$

Answer

(d) $\sqrt{36}$

$\sqrt{36}=6$ , which is rational. The numbers 2, 14 and 24 are not perfect squares, so $\sqrt{2}$ , $\sqrt{14}$ and $\sqrt{24}$ are irrational.

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If $A \subset B$ then $A \cup B$ is equal to ……. (a) $A$ (b) $B$ (c) both (a) and (b) (d) neither $A$ nor $B$

Answer

(b) $B$

Since every element of $A$ is already in $B$ , the union adds nothing new, so $A \cup B = B$ .

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If 2 American dollars (USD) = Rs. 156 and 2 Pounds (£) = Rs. 234, then 55 American dollars (USD) is equal to …….. (a) £ 36.67 (b) £ 63.67 (c) £ 67.36 (d) £ 67.60

Answer

(a) £ 36.67

$USD 1 = \text{Rs } 78$ , so $USD 55 = \text{Rs } 55 \times 78 = \text{Rs } 4290$ . Also £1 = Rs 117, so $\frac{4290}{117} = 36.67$ pounds.

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There are 40% men and 30% women in a village. If there are 321000 children, then the total population of the village is ……… (a) 1050000 (b) 1060000 (c) 1070000 (d) 1080000

Answer

(c) 1070000

Children are $100 - 40 - 30 = 30\%$ of the population, so $\text{Population} = \frac{321000 \times 100}{30} = 1070000.$

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One side of an equilateral triangle is 4 cm, its area is ….. (a) $8\sqrt{3}\ \text{cm}^2$ (b) $16\sqrt{3}\ \text{cm}^2$ (c) $4\sqrt{3}\ \text{cm}^2$ (d) $12\sqrt{3}\ \text{cm}^2$

Answer

(c) $4\sqrt{3}\ \text{cm}^2$

$\text{Area} = \frac{\sqrt{3}}{4}a^2 = \frac{\sqrt{3}}{4}\times 4^2 = 4\sqrt{3}\ \text{cm}^2.$

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$\sqrt[5]{\sqrt[4]{3^2}}$ equals ……….. (a) $3^{10}$ (b) $3^{\frac{1}{10}}$ (c) $3^{40}$ (d) $3^{\frac{1}{40}}$

Answer

(b) $3^{\frac{1}{10}}$

$\sqrt[5]{\sqrt[4]{3^2}} = \left(3^{2\cdot\frac14}\right)^{\frac15} = 3^{\frac12\cdot\frac15} = 3^{\frac{1}{10}}.$

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The polynomial $x$ is ……… (a) a monomial (b) a binomial (c) a trinomial (d) none of these

Answer

(a) a monomial

$x$ has exactly one term, so it is a monomial.

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In $\triangle ABC$ , if $\angle C > \angle B$ , then ………. (a) $BC > AC$ (b) $AB > AC$ (c) $AB < AC$ (d) $BC < AC$

Answer

(b) $AB > AC$

The larger angle has the longer opposite side. $\angle C > \angle B$ , and the side opposite $\angle C$ is $AB$ while the side opposite $\angle B$ is $AC$ , so $AB > AC$ .

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Signs of the abscissa and ordinate of a point in the third quadrant are respectively ………. (a) $+, 0$ (b) $-, -$ (c) $-, +$ (d) $+, +$

Answer

(b) $-, -$

In the third quadrant both $x < 0$ and $y < 0$ , so the abscissa and ordinate are both negative.

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The perpendicular distance of the point $P(7, 5)$ from the y-axis is …… (a) 5 (b) 12 (c) 7 (d) 2

Answer

(c) 7

The perpendicular distance of $(x, y)$ from the y-axis is $|x|$ , so for $P(7, 5)$ it is $7$ .

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Which of the lines represents a line parallel to the x-axis? (a) $5x + 3 = 4$ (b) $3y - 5 = y + 2$ (c) $x + y = 7$ (d) $x + 3 = 0$

Answer

(b) $3y - 5 = y + 2$

A line parallel to the x-axis has the form $y = k$ . Here $3y - y = 2 + 5$ gives $2y = 7$ , so $y = \tfrac72$ .

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If the diagonals of a quadrilateral bisect each other at right angles, then the quadrilateral is a …… (a) rhombus (b) rectangle (c) square (d) trapezium

Answer

(a) rhombus

A quadrilateral whose diagonals bisect each other is a parallelogram; if they also meet at right angles it is a rhombus. (A square also has this property, but it additionally needs equal diagonals, which is not given.)

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Two adjacent angles of a parallelogram are in the ratio of 4 : 5. The angles are …….. (a) $72^\circ, 108^\circ$ (b) $84^\circ, 96^\circ$ (c) $70^\circ, 110^\circ$ (d) $80^\circ, 100^\circ$

Answer

(d) $80^\circ, 100^\circ$

Adjacent angles of a parallelogram are supplementary. Let them be $4x$ and $5x$ : $4x + 5x = 180^\circ \Rightarrow x = 20^\circ,$ so the angles are $80^\circ$ and $100^\circ$ .

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In figure, PQ is a diameter of circle. ∠SQP is equal to ………. (a) 40° (b) 60° (c) 50° (d) 30°

Answer

(c) $50^\circ$

The angle in a semicircle is a right angle, so $\angle PSQ = 90^\circ$ . In $\triangle PSQ$ , with $\angle SPQ = 40^\circ$ , $\angle SQP = 180^\circ - 90^\circ - 40^\circ = 50^\circ.$

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The mean of ten numbers is 55. If one number is excluded, their mean becomes 50. The excluded number is ……. (a) 100 (b) 80 (c) 70 (d) 60

Answer

(a) 100

Sum of ten numbers $= 10 \times 55 = 550$ . Sum of the remaining nine $= 9 \times 50 = 450$ . $\text{Excluded number} = 550 - 450 = 100.$

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The median of the data 32, 35, 39, 31, 35, 36, 33 is ……… (a) 33 (b) 35 (c) 39 (d) 36

Answer

(b) 35

Arranged in order: 31, 32, 33, 35, 35, 36, 39. There are 7 values, so the median is the 4th value, $35$ .

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From a normal pack of 52 cards, a card is drawn at random. The probability of getting a king is ……… (a) $\frac{1}{4}$ (b) $\frac{1}{26}$ (c) $\frac{1}{13}$ (d) $\frac{1}{2}$

Answer

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

There are 4 kings among 52 cards, so $P = \dfrac{4}{52} = \dfrac{1}{13}$ .

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What is the probability of getting a 2 or a 6 in a roll of a die? (a) $\frac{1}{2}$ (b) $\frac{1}{3}$ (c) $\frac{1}{4}$ (d) $\frac{1}{6}$

Answer

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

Favourable outcomes are 2 and 6 (2 outcomes) out of 6, so $P = \dfrac{2}{6} = \dfrac{1}{3}$ .

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The value of $\sqrt{x^2\sqrt{x^{-2}}}$ is …….. (a) 1 (b) $x$ (c) $\sqrt{x}$ (d) $x^{-4}$

Answer

(c) $\sqrt{x}$

$\sqrt{x^2\sqrt{x^{-2}}} = \sqrt{x^2 \cdot x^{-1}} = \sqrt{x} \quad (x > 0).$

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The value of $\frac{\frac{x}{y} - 1}{1 - \frac{x}{y}}$ is ………. (a) 0 (b) 1 (c) $-1$ (d) $\frac{x}{y}$

Answer

(c) $-1$

The numerator $\frac{x}{y} - 1$ is the negative of the denominator $1 - \frac{x}{y}$ , so the fraction equals $-1$ (for $x \ne y$ ).

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The fourth proportional to 4, 6, 8 is …….. (a) 12 (b) 32 (c) 48 (d) 50

Answer

(a) 12

If $4 : 6 = 8 : x$ , then $x = \frac{6 \times 8}{4} = 12.$

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If $a : b = 3 : 4$ and $b : c = 1 : 3$ , then $a : c =$ ? (a) 1 : 4 (b) 3 : 4 (c) 4 : 1 (d) 1 : 1

Answer

(a) 1 : 4

Make $b$ equal in both: $a : b = 3 : 4$ and $b : c = 4 : 12$ , so $a : b : c = 3 : 4 : 12$ and $a : c = 3 : 12 = 1 : 4$ .

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A mobile of cost Rs. 25,000 is depreciated at 12% p.a., then what will be its value after 2 years? (a) Rs. 22,000 (b) Rs. 20,336 (c) Rs. 19,360 (d) Rs. 19,250

Answer

(c) Rs. 19,360

Value after $n$ years under reducing-balance depreciation: $V = P\left(1-\frac{r}{100}\right)^n$ .

$V = 25000\left(1-\frac{12}{100}\right)^2 = 25000 \times (0.88)^2 = 25000 \times 0.7744 = \text{Rs } 19{,}360$

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A can do a piece of work in 6 days and B does the same work in 3 days. How long will A and B take working together? (a) 4 days (b) 2 days (c) 1 day (d) $1\frac{1}{2}$ day

Answer

(b) 2 days

Work per day together: $\frac{1}{6}+\frac{1}{3}=\frac{1+2}{6}=\frac{1}{2}$ .

So the whole work takes $\frac{1}{1/2}=2$ days.

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H.C.F. of $a^3 - 1$ and $a^4 + a^2 + 1$ is …….. (a) $a - 1$ (b) $a^2 - a + 1$ (c) $a^2 + a + 1$ (d) $a^2 - a - 1$

Answer

(c) $a^2 + a + 1$

$a^3 - 1 = (a-1)(a^2+a+1)$

$a^4 + a^2 + 1 = (a^2+1)^2 - a^2 = (a^2+a+1)(a^2-a+1)$

The common factor is $a^2 + a + 1$ .

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The value of $\frac{12^7 \times 28^6}{21^7 \times 16^6}$ is ………. (a) $\frac{4}{7}$ (b) $\frac{7}{4}$ (c) $\frac{3}{4}$ (d) $\frac{4}{3}$

Answer

(a) $\frac{4}{7}$

Write in prime factors: $12^7 = 2^{14}3^7$ , $28^6 = 2^{12}7^6$ , $21^7 = 3^7 7^7$ , $16^6 = 2^{24}$ .

$\frac{2^{14}\cdot 3^7\cdot 2^{12}\cdot 7^6}{3^7\cdot 7^7\cdot 2^{24}} = \frac{2^{26}}{2^{24}\cdot 7} = \frac{4}{7}$

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If $N = 50$ and $\bar{X} = 15$ then $\sum fx$ is ……. (a) 600 (b) 750 (c) 500 (d) 450

Answer

(b) 750

Mean $\bar{X} = \frac{\sum fx}{N}$ , so $\sum fx = N\bar{X} = 50 \times 15 = 750$ .

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The sum of the digits of a two digit number is 7 and their product is 12. Then the numbers are ………. (a) 52 and 25 (b) 34 and 43 (c) 61 and 16 (d) 62 and 26

Answer

(b) 34 and 43

Let the digits be $x$ and $y$ with $x+y=7$ and $xy=12$ . They are the roots of $t^2 - 7t + 12 = 0$ , i.e. $(t-3)(t-4)=0$ , so the digits are 3 and 4.

The numbers are 34 and 43. Check: $3+4=7$ , $3\times4=12$ . The other options fail (52: product 10; 61: product 6; 62: sum 8).

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Ananda spends 70% of his income and saves Rs. 540. What is his income? (a) Rs. 1500 (b) Rs. 1650 (c) Rs. 1750 (d) Rs. 1800

Answer

(d) Rs. 1800

Savings are $100\% - 70\% = 30\%$ of income, so $30\%$ of income $=540$ .

$\text{Income} = \frac{540 \times 100}{30} = \text{Rs } 1800$

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If the circumference of a great circle of a metallic sphere is 44 cm, then the surface area of the sphere is ……. (a) $308\ \text{cm}^2$ (b) $416\ \text{cm}^2$ (c) $616\ \text{cm}^2$ (d) $710\ \text{cm}^2$

Answer

(c) $616\ \text{cm}^2$

For a great circle, $2\pi r = 44$ , so with $\pi=\frac{22}{7}$ : $r = \frac{44\times 7}{2\times 22} = 7$ cm.

$\text{Surface area} = 4\pi r^2 = 4 \times \frac{22}{7} \times 49 = 616\ \text{cm}^2$

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Three years ago, the sum of the ages of a father and his son was 48 years and three years hence the father's age will be three times that of his son. Then the present ages of the father and his son will be respectively ……… (a) 42 years and 12 years (b) 40 years and 8 years (c) 39 years and 9 years (d) 30 years and 18 years

Answer

(a) 42 years and 12 years

Let the present ages of father and son be $F$ and $S$ .

Three years ago: $(F-3)+(S-3)=48$ , so $F+S=54$ .

Three years hence: $F+3 = 3(S+3)$ , so $F = 3S+6$ .

Substituting: $4S + 6 = 54$ , so $S=12$ and $F=42$ . Check: $39+9=48$ and $45 = 3\times15$ .

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One of the factors of $r^7 + rs^6$ is …….. (a) $(r + s)$ (b) $r^2 + s^2$ (c) neither (a) nor (b) (d) both (a) and (b)

Answer

(b) $r^2 + s^2$

$r^7 + rs^6 = r(r^6+s^6) = r\left[(r^2)^3+(s^2)^3\right] = r(r^2+s^2)(r^4 - r^2s^2 + s^4)$

So $r^2+s^2$ is a factor. $(r+s)$ is not, since putting $r=-s$ gives $-s^7 - s^7 \ne 0$ .

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

Answer

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

$2\pi r = \pi$ gives $r = \frac{1}{2}$ .

$\text{Area} = \pi r^2 = \pi \times \frac{1}{4} = \frac{\pi}{4}$

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In a race of 1 km, if A beats B by 100 m and B beats C by 50 m, then by how much does A beat C? (a) 250 m (b) 150 m (c) 147.5 m (d) 145 m

Similar questions: In a race of 1 km, if A beats B by 50 m and B (2077 · Management Model)

Answer

(d) 145 m

When A runs 1000 m, B runs 900 m. When B runs 1000 m, C runs 950 m, so when B runs 900 m, C runs $900\times\frac{950}{1000}=855$ m.

A beats C by $1000-855=145$ m.

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

Similar questions: At what time does a sum of money double itself at (2080 · Science Model)

Answer

(c) 12.5 year

For the money to double, the simple interest must equal the principal: $I=P$ . $P=\frac{P\times 8\times T}{100}\Rightarrow T=\frac{100}{8}=12.5\text{ years}$

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2080 · Science Model · 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

Similar questions: At what time does a sum of money double itself at (2081 · Science Model)

Answer

(c) 12.5 years

For the sum to double, the interest equals the principal: $P=\dfrac{P\times8\times T}{100}$ , so

$T=\frac{100}{8}=12.5\text{ years}$

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2079 · Management and Humanities Model

The surface area of a spherical ball is $616\ \text{cm}^2$ . What is its diameter? (a) 7 cm (b) 14 cm (c) 21 cm (d) 28 cm

Similar questions: If the surface area of a spherical ball is 616 (2081 · Science Model)

Answer

(b) 14 cm

$4\pi r^2 = 616 \Rightarrow r^2 = \frac{616\times 7}{4\times 22}=49 \Rightarrow r = 7\text{ cm}$

Diameter $= 2r = 14$ cm.

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2077 · Management Model

In a race of 1 km, if A beats B by 50 m and B beats C by 50 m, then by how much does A beat C? (a) 90 m (b) 100 m (c) 97.5 m (d) none of these

Similar questions: In a race of 1 km, if A beats B by 100 m and B (2081 · Science Model)

Answer

(c) 97.5 m

When A runs 1000 m, B runs 950 m. When B runs 1000 m, C runs 950 m. So when B runs 950 m, C runs $950\times\frac{950}{1000}=902.5$ m.

Hence A beats C by $1000-902.5=97.5$ m.

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The cotangents of $\frac{\pi}{3}, \frac{\pi}{4}, \frac{\pi}{6}$ are in ……… (a) A. P. (b) G.P. (c) H.P. (d) None

Answer

(b) G.P.

$\cot\frac{\pi}{3}=\frac{1}{\sqrt3},\ \cot\frac{\pi}{4}=1,\ \cot\frac{\pi}{6}=\sqrt3$ . The ratios are $\frac{1}{1/\sqrt3}=\sqrt3$ and $\frac{\sqrt3}{1}=\sqrt3$ , which are equal, so the terms are in G.P. (the differences are not equal, so they are not in A.P.).

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Two vectors $\vec{a}$ and $\vec{b}$ are parallel if (a) $\vec{a} = k\vec{b}$ (b) $\vec{a} = 2\vec{b}$ (c) $\vec{a} = \vec{b}$ (d) All of these

Answer

(a) $\vec{a} = k\vec{b}$

Two non-zero vectors are parallel exactly when one is a scalar multiple of the other. Options (b) and (c) are only the special cases $k=2$ and $k=1$ .

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The sum of its $n$ -terms of the series $\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + \dots$ is ……. (a) $\sqrt{2}\,\frac{n(n+1)}{2}$ (b) $\sqrt{\frac{n(n+1)}{2}}$ (c) $\sqrt{\frac{n(n+1)n}{2}}$ (d) $\frac{n(n+1)}{2}$

Answer

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

$\sqrt8=2\sqrt2,\ \sqrt{18}=3\sqrt2,\ \sqrt{32}=4\sqrt2$ , so the series is $\sqrt2(1+2+3+\dots+n)=\sqrt2\,\frac{n(n+1)}{2}$ .

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The equations $3x - 4y = 5$ and $12x - 16y = 20$ have …. (a) no solution (b) exactly one solution (c) infinitely many solutions (d) none of above

Answer

(c) infinitely many solutions

The second equation is $4$ times the first: $12x-16y=4(3x-4y)=20$ . The two equations represent the same line, so every point on it is a solution.

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Which of the following statement is/are correct? i) every integer is a rational number. ii) every rational number is an integer. iii) a real number is either a rational or irrational number. iv) every whole number is a natural number. (a) (ii) (b) (iii) (c) (i) & (iii) (d) (i), (ii) & (iii)

Answer

(c) (i) & (iii)

(i) is true since every integer $n=\frac{n}{1}$ is rational. (ii) is false, e.g. $\frac12$ . (iii) is true. (iv) is false because $0$ is a whole number but not a natural number.

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The value of $2.999\ldots$ in the form of $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \ne 0$ , is ……… (a) $\frac{2999}{1000}$ (b) $\frac{19}{10}$ (c) 3 (d) $\frac{26}{9}$

Answer

(c) 3

Let $x=2.999\ldots$ . Then $10x=29.999\ldots$ , so $9x=27$ and $x=3$ .

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$\sqrt[5]{\sqrt[4]{3^2}}$ equals ………. (a) $3^{10}$ (b) $\frac{1}{3^{-10}}$ (c) $3^{\frac{1}{10}}$ (d) $3^{-10}$

Answer

(c) $3^{\frac{1}{10}}$

$\sqrt[5]{\sqrt[4]{3^2}}=\left(3^{2/4}\right)^{1/5}=3^{\frac{2}{20}}=3^{\frac{1}{10}}$ .

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If $x^{11} + 101$ is divided by $x + 1$ , the remainder is ……… (a) 0 (b) $-1$ (c) 100 (d) 102

Answer

(c) 100

By the remainder theorem, put $x=-1$ : $(-1)^{11}+101=-1+101=100$ .

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Which of the following polynomials has $(x + 1)$ as a factor? (a) $x^4 + x^3 + x^2 + x + 1$ (b) $x^3 + x^2 + x + 1$ (c) $x^3 + 2x^2 - 2x + 1$ (d) $x^4 + 3x^3 + 3x^2 + x + 1$

Answer

(b) $x^3 + x^2 + x + 1$

$(x+1)$ is a factor if the value at $x=-1$ is $0$ . For (b): $-1+1-1+1=0$ . The others give (a) $1$ , (c) $4$ , (d) $1$ , none zero.

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A polynomial of degree 4 in $y$ has at most ……… (a) 4 terms (b) 5 terms (c) 3 terms (d) 8 terms

Answer

(b) 5 terms

The terms can have powers $y^4,y^3,y^2,y,y^0$ , so there are at most $5$ terms.

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If the sides of a triangle are doubled then its area ……… (a) is doubled (b) remains the same (c) becomes three times (d) becomes four times

Answer

(d) becomes four times

Doubling all sides gives a similar triangle with ratio $2$ , so the area scales by $2^2=4$ .

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In parallelogram ABCD, the bisectors of angles A and B intersect each other at O. The value of $\angle AOB$ is (a) $30^\circ$ (b) $60^\circ$ (c) $90^\circ$ (d) cannot be determined

Answer

(c) $90^\circ$

Since $\angle A+\angle B=180^\circ$ (co-interior angles), in $\triangle AOB$ : $\angle OAB+\angle OBA=\frac{A}{2}+\frac{B}{2}=90^\circ$ , so $\angle AOB=90^\circ$ .

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The inverse of the matrix $\begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ is ……. (a) $\frac{1}{5}\begin{pmatrix} 4 & 1 \\ 3 & 2 \end{pmatrix}$ (b) $\frac{1}{5}\begin{pmatrix} 4 & -3 \\ -1 & 2 \end{pmatrix}$ (c) $\frac{1}{5}\begin{pmatrix} 4 & -1 \\ -3 & 2 \end{pmatrix}$ (d) $\frac{1}{5}\begin{pmatrix} 4 & 3 \\ 1 & 2 \end{pmatrix}$

Answer

(c) $\frac{1}{5}\begin{pmatrix} 4 & -1 \\ -3 & 2 \end{pmatrix}$

$\det=2\cdot4-1\cdot3=5$ . Swap the diagonal entries and negate the off-diagonal ones: $A^{-1}=\frac15\begin{pmatrix}4&-1\\-3&2\end{pmatrix}$ .

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If $\sin\frac{A}{2} = \frac{4}{5}$ , then the value of $\cos A$ is ……. (a) $\frac{3}{5}$ (b) $\frac{7}{25}$ (c) $\frac{-7}{25}$ (d) $\frac{-21}{25}$

Answer

(c) $\frac{-7}{25}$

$\cos A=1-2\sin^2\frac A2=1-2\cdot\frac{16}{25}=\frac{25-32}{25}=-\frac{7}{25}$ .

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The value of $\sin 18^\circ$ is ….. (a) $\frac{1+\sqrt{5}}{4}$ (b) $\frac{-1+\sqrt{5}}{4}$ (c) $\frac{1-\sqrt{5}}{4}$ (d) $\frac{-1+\sqrt{5}}{2}$

Answer

(b) $\frac{-1+\sqrt{5}}{4}$

Let $\theta=18^\circ$ , so $2\theta=90^\circ-3\theta$ and $\cos2\theta=\sin3\theta$ . This gives $1-2s^2=3s-4s^3$ , i.e. $4s^3-2s^2-3s+1=0$ , i.e. $(s-1)(4s^2+2s-1)=0$ . Taking the root with $0<s<1$ : $s=\frac{-1+\sqrt5}{4}\approx0.309$ .

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The value of $\sqrt{\frac{1+\sin\theta}{1-\sin\theta}}$ is …….. (a) $\tan\left(\frac{\pi}{4}+\frac{\theta}{2}\right)$ (b) $\tan\left(\frac{\pi}{4}-\frac{\theta}{2}\right)$ (c) $\tan\left(\frac{\pi}{4}+\theta\right)$ (d) $\tan\left(\frac{\pi}{4}-\theta\right)$

Answer

(a) $\tan\left(\frac{\pi}{4}+\frac{\theta}{2}\right)$

Write $1\pm\sin\theta=\left(\cos\frac\theta2\pm\sin\frac\theta2\right)^2$ . Then $\sqrt{\frac{1+\sin\theta}{1-\sin\theta}}=\frac{\cos\frac\theta2+\sin\frac\theta2}{\cos\frac\theta2-\sin\frac\theta2}=\frac{1+\tan\frac\theta2}{1-\tan\frac\theta2}=\tan\left(\frac\pi4+\frac\theta2\right).$

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The value of $x$ when $\cos 2x = \sin x$ is …….. (a) $0^\circ$ (b) $30^\circ$ (c) $60^\circ$ (d) $90^\circ$

Answer

(b) $30^\circ$

$1-2\sin^2x=\sin x\Rightarrow 2\sin^2x+\sin x-1=0\Rightarrow(2\sin x-1)(\sin x+1)=0$ . So $\sin x=\frac12$ , giving $x=30^\circ$ (the root $\sin x=-1$ is not among the options). Check: $\cos60^\circ=\frac12=\sin30^\circ$ .

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What is the value of $\theta$ when $\cos^2\frac{\theta}{2} - \cos\frac{\theta}{2} + \frac{1}{4} = 0$ ? (a) $60^\circ$ (b) $90^\circ$ (c) $120^\circ$ (d) $150^\circ$

Answer

(c) $120^\circ$

Let $c=\cos\frac\theta2$ . Then $c^2-c+\frac14=\left(c-\frac12\right)^2=0$ , so $\cos\frac\theta2=\frac12$ , $\frac\theta2=60^\circ$ and $\theta=120^\circ$ .

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Two circles $x^2 + y^2 + 2ax + c^2 = 0$ and $x^2 + y^2 + 2by + c^2 = 0$ touch externally if ……. (a) $a^2 + b^2 = c^2$ (b) $a^2 + b^2 + c^2 = 0$ (c) $\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} = 0$ (d) $\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2}$

Answer

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

Centres $(-a,0),(0,-b)$ ; radii $r_1=\sqrt{a^2-c^2},\ r_2=\sqrt{b^2-c^2}$ . External touching: $a^2+b^2=(r_1+r_2)^2=a^2+b^2-2c^2+2r_1r_2$ , so $c^2=r_1r_2$ and $c^4=(a^2-c^2)(b^2-c^2)$ . This gives $a^2b^2=c^2(a^2+b^2)$ , i.e. $\frac1{c^2}=\frac1{a^2}+\frac1{b^2}$ .

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A circle passes through $(0, 0)$ , $(a, 0)$ and $(0, b)$ . The co-ordinate of the centre is ……. (a) $(a, b)$ (b) $(b, a)$ (c) $(-a, -b)$ (d) $\left(\frac{a}{2}, \frac{b}{2}\right)$

Answer

(d) $\left(\frac{a}{2}, \frac{b}{2}\right)$

The points $(0,0),(a,0),(0,b)$ form a right triangle with the right angle at the origin, so the hypotenuse from $(a,0)$ to $(0,b)$ is a diameter. The centre is its midpoint $\left(\frac a2,\frac b2\right)$ .

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The value of $k$ such that the equation $2x^2 + 2y^2 - 6x + 8y + k = 0$ represents a point circle is …… (a) 0 (b) 25 (c) $\frac{25}{2}$ (d) $\frac{-25}{2}$

Answer

(c) $\frac{25}{2}$

Divide by $2$ : $x^2+y^2-3x+4y+\frac k2=0$ . The radius squared is $\frac94+4-\frac k2$ . For a point circle this is $0$ , so $\frac k2=\frac{25}{4}$ and $k=\frac{25}{2}$ .

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The general equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a circle if ……. (a) $a = b = h$ (b) $a = b$ and $h = 0$ (c) $g = f = c$ (d) $a = b$ and $g = f$

Answer

(b) $a = b$ and $h = 0$

A circle has equal coefficients of $x^2$ and $y^2$ and no $xy$ term.

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The value of $m$ such that $mx^2 - 8xy + 8y^2 = 0$ represents two coincident straight lines is ……. (a) $-1$ (b) $-3$ (c) 4 (d) 2

Answer

(d) 2

Two lines coincide when the discriminant condition $h^2 = ab$ holds. Here $a=m$ , $2h=-8$ so $h=-4$ , $b=8$ :

$16 = 8m \Rightarrow m = 2.$

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The angle between the lines $x\cos\alpha + y\sin\alpha = \beta$ and $x\cos\beta + y\sin\beta = p_1$ is ……. (a) $\alpha + \beta$ (b) $\alpha - \beta$ (c) $\alpha\beta$ (d) $\frac{\alpha}{\beta}, \beta \ne 0$

Answer

(b) $\alpha - \beta$

The line $x\cos\alpha + y\sin\alpha = p$ has normal making angle $\alpha$ with the $x$ -axis, and the second line has normal at angle $\beta$ . The angle between the lines equals the angle between their normals, $\alpha-\beta$ .

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The points $(a, 0)$ , $(0, b)$ and $(1, 1)$ are collinear if ……… (a) $a + b = ab$ (b) $a - b = ab$ (c) $b - a = ab$ (d) $a + b + ab = 0$

Answer

(a) $a + b = ab$

The line through $(a,0)$ and $(0,b)$ is $\frac{x}{a}+\frac{y}{b}=1$ . Passing through $(1,1)$ gives $\frac1a+\frac1b=1$ , so $a+b=ab$ .

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If A and B are square matrices of the same order then …… (a) both AB and BA are defined (b) $(AB)^T = B^T A^T$ (c) $(AB)^{-1} = B^{-1}A^{-1}$ (d) all of the above

Answer

(b) $(AB)^T = B^TA^T$

The reversal law for transposes holds for all matrices. Option (c) needs $A$ and $B$ to be invertible, so "all of the above" is not true in general.

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A square matrix $A = [a_{ij}]$ in which $a_{ij} = 0$ for $i \ne j$ and $a_{ij}$ = constant, is called …….. (a) diagonal matrix (b) scalar matrix (c) unit matrix (d) null matrix

Answer

(b) scalar matrix

All off-diagonal entries are $0$ and all diagonal entries equal the same constant, which is the definition of a scalar matrix (a diagonal matrix whose diagonal entries are equal).

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The sum of the series $1 + 4 + 9 + 16 + \dots$ to $n$ terms is ……. (a) $n^2$ (b) $\frac{n(n+1)}{2}$ (c) $\frac{n(n+1)(2n+1)}{6}$ (d) $\left[\frac{n(n+1)}{2}\right]^2$

Answer

(c) $\frac{n(n+1)(2n+1)}{6}$

The $k$ -th term is $k^2$ , so the sum is $\sum_{k=1}^n k^2=\frac{n(n+1)(2n+1)}{6}$ . Check $n=2$ : $\frac{2\cdot3\cdot5}{6}=5=1+4$ .

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If A and B are non-empty sets and $A \times B = B \times A$ then …… (a) A is a proper subset of B (b) A is a subset of B (c) B is a proper subset of A (d) A = B

Answer

(d) A = B

If $A\times B=B\times A$ , every $(a,b)$ with $a\in A,\ b\in B$ must also satisfy $a\in B,\ b\in A$ . Since both sets are non-empty, this gives $A\subseteq B$ and $B\subseteq A$ , so $A=B$ .

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A function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 3x + 5$ , $x \in \mathbb{R}$ is …… (a) one-one (b) onto (c) both (a) & (b) (d) neither (a) nor (b)

Answer

(c) both (a) & (b)

One-one: $3x_1+5=3x_2+5\Rightarrow x_1=x_2$ . Onto: for any $y\in\mathbb R$ , $x=\frac{y-5}{3}\in\mathbb R$ gives $f(x)=y$ .

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The curve of the function $y = -x^2 + 4x + 3$ ……….. (a) faced upward (b) faced downward (c) faced right side (d) faced left side

Answer

(b) faced downward

The coefficient of $x^2$ is $-1<0$ , so the parabola opens downward.

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If $(x + 3)^2 = 225$ , which of the following could be the value of $x - 1$ ? (a) 12 (b) $-12$ (c) $-16$ (d) $-19$

Answer

(d) $-19$

$x+3=\pm15$ , so $x=12$ or $x=-18$ . Then $x-1=11$ or $-19$ ; only $-19$ is listed.

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Triangle ABC has vertices $A(-2, -5)$ , $B(-7, 5)$ and $C(1, 1)$ . AD is a median. The gradient of AD is … (a) $-\frac{1}{2}$ (b) 2 (c) $-8$ (d) 8

Answer

(c) $-8$

The midpoint of $BC$ is $D\left(\frac{-7+1}{2},\frac{5+1}{2}\right)=(-3,3)$ .

$m_{AD}=\frac{3-(-5)}{-3-(-2)}=\frac{8}{-1}=-8.$

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If the sum of squares of 10 terms is 38500, variance is 825, then the coefficient of standard deviation is (a) 0.24 (b) 0.25 (c) 0.42 (d) 0.52

Answer

(d) 0.52

Mean of squares $=\frac{38500}{10}=3850$ , and variance $=3850-\bar x^2=825$ , so $\bar x^2=3025$ and $\bar x=55$ .

$\sigma=\sqrt{825}\approx28.72,\qquad \text{coefficient of SD}=\frac{\sigma}{\bar x}=\frac{28.72}{55}\approx0.52.$

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The standard deviation of a data is 15 and the coefficient of variation is 75%, then the mean of the data is; (a) 15 (b) 20 (c) 25 (d) 30

Answer

(b) 20

$CV=\frac{\sigma}{\bar x}\times100\Rightarrow 75=\frac{15}{\bar x}\times100\Rightarrow \bar x=20.$

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If $\vec{a} + \vec{b} + \vec{c} = 0$ , $|\vec{a}| = 3$ , $|\vec{b}| = 5$ and $|\vec{c}| = 7$ , then the angle between $\vec{a}$ and $\vec{b}$ is ………. (a) 0 (b) $\frac{\pi}{6}$ (c) $\frac{\pi}{3}$ (d) $\frac{\pi}{2}$

Answer

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

From $\vec a+\vec b=-\vec c$ , squaring: $9+25+2\vec a\cdot\vec b=49$ , so $\vec a\cdot\vec b=\frac{15}{2}$ .

$\cos\theta=\frac{15/2}{3\cdot5}=\frac12\Rightarrow\theta=\frac{\pi}{3}.$

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If a matrix $A$ is such that $A^2 = A$ , then $A$ is called ……… (a) Square matrix (b) Symmetric matrix (c) Idempotent matrix (d) Null matrix

Answer

(c) Idempotent matrix

A matrix satisfying $A^2=A$ is called idempotent.

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The probability that a non-leap year contains 53 Sundays is ……… (a) $\frac{1}{365}$ (b) $\frac{2}{7}$ (c) $\frac{2}{365}$ (d) $\frac{1}{7}$

Answer

(d) $\frac17$

A non-leap year has $52$ weeks plus $1$ extra day. That day must be a Sunday for $53$ Sundays, and it is equally likely to be any of $7$ days, so $P=\frac17$ .

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If $\cos\theta - \sin\theta = \sqrt{2}\sin\theta$ , then $\cos\theta + \sin\theta =$ ……… (a) $\sqrt{2}\cos\theta$ (b) $\sqrt{2}\sin\theta$ (c) $2\cos\theta$ (d) $-\sqrt{2}\sin\theta$

Answer

(a) $\sqrt2\cos\theta$

From $\cos\theta=(\sqrt2+1)\sin\theta$ , we get $\sin\theta=(\sqrt2-1)\cos\theta$ . Then

$\cos\theta+\sin\theta=\cos\theta+(\sqrt2-1)\cos\theta=\sqrt2\cos\theta.$

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What is 10% of 20% of 30%? (a) 0.006% (b) 0.06% (c) 0.6% (d) 6%

Answer

(c) 0.6%

$0.1\times0.2\times30\% = 0.02\times30\% = 0.6\%.$

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What is the measure, in degrees, of each interior angle in a regular decagon? (a) $102^\circ$ (b) $120^\circ$ (c) $144^\circ$ (d) $152^\circ$

Answer

(c) $144^\circ$

Each interior angle of a regular $n$ -gon is $\frac{(n-2)180^\circ}{n}$ . For $n=10$ : $\frac{8\cdot180^\circ}{10}=144^\circ$ .

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If $0 < x < 1$ , which of the following lists the numbers in increasing order? (a) $\sqrt{x}, x, x^2$ (b) $x^2, x, \sqrt{x}$ (c) $x^2, \sqrt{x}, x$ (d) $x, \sqrt{x}, x^2$

Answer

(b) $x^2, x, \sqrt{x}$

For $0<x<1$ , multiplying by $x$ shrinks a number, so $x^2<x$ ; and $x<\sqrt x$ . Check $x=\frac14$ : $\frac1{16}<\frac14<\frac12$ .

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In triangle ABC, the bisectors of $\angle B$ and $\angle C$ meet at O. Then $\angle BOC$ is …… (a) $90^\circ - \frac{1}{2}\angle A$ (b) $180^\circ - \frac{1}{2}\angle A$ (c) $90^\circ + \frac{1}{2}\angle A$ (d) $180^\circ + \frac{1}{2}\angle A$

Answer

(c) $90^\circ+\frac12\angle A$

In $\triangle BOC$ , $\angle BOC=180^\circ-\frac12(\angle B+\angle C)=180^\circ-\frac12(180^\circ-\angle A)=90^\circ+\frac12\angle A.$

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Axioms are assumed …… (a) universal truth specific to geometry (b) theorems (c) definition (d) universal truth in all branches of mathematics

Answer

(d) universal truth in all branches of mathematics

Euclid's axioms (common notions) are self-evident truths assumed without proof that apply across all of mathematics, unlike postulates, which are specific to geometry.

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Which of the following doesn't represent a function (graphs of the relations are shown) (a) $x = y$ (b) $y^2 = x$ (c) $y = x^2$ (d) $y = -x$

Answer

(b) $y^2 = x$

A relation is a function only if each $x$ gives exactly one $y$ . For $y^2 = x$ , $x = 4$ gives $y = 2$ and $y = -2$ , so a vertical line cuts the graph twice. The others, $y = x$ , $y = x^2$ and $y = -x$ , give one $y$ per $x$ .

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If $(x+3), 3x, (4x+1)$ are in AP then $x$ is equal to (a) 3 (b) 4 (c) 5 (d) 1

Answer

(b) 4

For an AP the common difference is equal, so $3x-(x+3)=(4x+1)-3x$ . This gives $2x-3=x+1$ , so $x=4$ . Check: the terms are 7, 12, 17.

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The sum of all two digits numbers is (a) 4895 (b) 4905 (c) 3065 (d) 2015

Answer

(b) 4905

The two-digit numbers 10, 11, ..., 99 form an AP with $n=90$ , $a=10$ , $l=99$ .

$S=\frac{90}{2}(10+99)=45\times109=4905$

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The value of $x^{\frac{1}{2}} \times x^{\frac{1}{4}} \times x^{\frac{1}{8}} \times \dots \infty$ is (a) $x$ (b) $x^2$ (c) $\frac{1}{x}$ (d) $x^3$

Answer

(a) $x$

The exponents add to a GP with $a=\tfrac12$ , $r=\tfrac12$ :

$\frac12+\frac14+\frac18+\dots=\frac{1/2}{1-1/2}=1$

so the product is $x^1=x$ .

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$A$ is a matrix of $3 \times 4$ . Both $A^T B$ and $BA^T$ are defined, then the order of $B$ is (a) $3 \times 4$ (b) $3 \times 4$ (c) $4 \times 3$ (d) $4 \times 4$

Answer

(a) $3\times4$ (option (b) is the same order)

$A^T$ is $4\times3$ . For $A^TB$ to be defined, $B$ must have 3 rows. For $BA^T$ to be defined, $B$ must have 4 columns. So $B$ is $3\times4$ .

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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

Answer

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

$\cos\theta=\frac{\vec a\cdot\vec b}{|\vec a||\vec b|}=\frac{\sqrt6}{2\sqrt3}=\frac{\sqrt2}{2}=\frac{1}{\sqrt2}$

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

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

Answer

(b) 12

Intercepts of $2x+3y=12$ : $x$ -intercept 6 (put $y=0$ ), $y$ -intercept 4 (put $x=0$ ).

$\text{Area}=\tfrac12\times6\times4=12$

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

Answer

(c) 51%

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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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) $\dfrac43$

For $ax^2+bx+c=0$ , the sum of roots is $-\dfrac ba=-\dfrac{-4}{3}=\dfrac43$ .

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The remainder when $4x^3 - 3x^2 + 2$ is divided by $x - 2$ is ……… (a) 20 (b) 22 (c) 16 (d) 24

Answer

(b) 22

By the remainder theorem, the remainder is $f(2)$ :

$4(8)-3(4)+2=32-12+2=22$

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The maximum value of $3\cos\theta + 4\sin\theta$ is ……. (a) 3 (b) 4 (c) 5 (d) 7

Answer

(c) 5

The maximum of $a\cos\theta+b\sin\theta$ is $\sqrt{a^2+b^2}=\sqrt{9+16}=5$ .

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If $\tan\theta + \sin\theta = m$ and $\tan\theta - \sin\theta = n$ , then $m^2 - n^2 =$ ………… (a) $4mn$ (b) $4\sqrt{mn}$ (c) $2\sqrt{mn}$ (d) $m^2 + n^2$

Answer

(b) $4\sqrt{mn}$

$m^2-n^2=(m+n)(m-n)=(2\tan\theta)(2\sin\theta)=4\tan\theta\sin\theta$

$mn=\tan^2\theta-\sin^2\theta=\tan^2\theta(1-\cos^2\theta)=\tan^2\theta\sin^2\theta$

so $\sqrt{mn}=\tan\theta\sin\theta$ and $m^2-n^2=4\sqrt{mn}$ .

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If $\cos\theta - \sin\theta = \sqrt{2}\sin\theta$ , then $\cos\theta + \sin\theta =$ ……….. (a) $\sqrt{2}\cos\theta$ (b) $\sqrt{2}\sin\theta$ (c) $2\cos\theta$ (d) $-\sqrt{2}\cos\theta$

Answer

(a) $\sqrt2\cos\theta$

From $\cos\theta-\sin\theta=\sqrt2\sin\theta$ we get $\cos\theta=(\sqrt2+1)\sin\theta$ , so

$\sin\theta=\frac{\cos\theta}{\sqrt2+1}=(\sqrt2-1)\cos\theta$

$\cos\theta+\sin\theta=\cos\theta+(\sqrt2-1)\cos\theta=\sqrt2\cos\theta$

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If $\frac{1}{3}, y_1, y_2, 9$ are in G.P. then the value of $y_2$ is ……… (a) 3 (b) 6 (c) 1 (d) None of above

Answer

(a) 3

The common ratio satisfies $r^3=\dfrac{9}{1/3}=27$ , so $r=3$ . Then $y_1=1$ and $y_2=\dfrac13\cdot3^2=3$ .

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The value of $\sin(A+B)\cdot\sin(A-B)$ is ……… (a) $\sin(A^2 - B^2)$ (b) $\sin^2(A-B)$ (c) $\sin^2 A - \sin^2 B$ (d) $\cos^2 A - \cos^2 B$

Answer

(c) $\sin^2A-\sin^2B$

$\sin(A+B)\sin(A-B)=(\sin A\cos B)^2-(\cos A\sin B)^2=\sin^2A(1-\sin^2B)-(1-\sin^2A)\sin^2B=\sin^2A-\sin^2B$

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If $a\cos\theta + b\sin\theta = m$ and $a\sin\theta - b\cos\theta = n$ then the value of $a^2 + b^2$ is ………. (a) $m + n$ (b) $mn$ (c) $m^2 + n^2$ (d) $\sqrt{mn}$

Answer

(c) $m^2+n^2$

$m^2+n^2=(a\cos\theta+b\sin\theta)^2+(a\sin\theta-b\cos\theta)^2=a^2(\cos^2\theta+\sin^2\theta)+b^2(\sin^2\theta+\cos^2\theta)=a^2+b^2$

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Which of the following matrices is a scalar matrix? (a) $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ (b) $\begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}$ (c) $\begin{pmatrix} 1 & 0 \\ 0 & 4 \end{pmatrix}$ (d) $\begin{pmatrix} 0 & 2 \\ 1 & 0 \end{pmatrix}$

Answer

(b) $\begin{pmatrix}5&0\\0&5\end{pmatrix}$

A scalar matrix is a diagonal matrix whose diagonal entries are all equal, and this one has both diagonal entries equal to 5 with zeros elsewhere.

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

The number of subsets of the set $\{2, 3, 5\}$ is …….. (a) 3 (b) 6 (c) 8 (d) 10

Answer

(c) 8

A set with $n$ elements has $2^n$ subsets, so $2^3=8$ .

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

If the price of milk is reduced from Rs. 22.50 per litre to Rs. 18 per litre, in what proportion is the water mixed? (a) 1 : 4 (b) 4 : 1 (c) 4 : 3 (d) 3 : 4

Answer

(a) 1 : 4

Let the mixture contain 1 litre of milk-and-water. Its price is Rs 18 against Rs 22.50 for pure milk, so the milk fraction is $\dfrac{18}{22.5}=\dfrac45$ and the water fraction is $\dfrac15$ .

Water : milk $=1:4$ .

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

If $A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 2 & 5 \\ 0 & 0 & 3 \end{pmatrix}$ , the name of this matrix is …… (a) Identity matrix (b) Diagonal matrix (c) Triangular matrix (d) Symmetric matrix

Answer

(c) Triangular matrix

All entries below the main diagonal are zero, so $A$ is an upper triangular matrix. It is not diagonal (the entries above the diagonal are non-zero), not an identity matrix, and not symmetric ( $a_{12}=2\ne a_{21}=0$ ).

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

If $A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ , then $A^4 =$ ……….. (a) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ (b) $\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}$ (c) $\begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix}$ (d) $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

Answer

(a) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

$A^2=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}=I$

so $A^4=(A^2)^2=I^2=I$ .

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

In an A.P. the $p^{th}$ term is $q$ and the $(p+q)^{th}$ term is 0, then the common difference is ……. (a) $-p$ (b) $q$ (c) $-1$ (d) 0

Answer

(c) $-1$

$a+(p-1)d=q$ and $a+(p+q-1)d=0$ . Subtracting the first from the second gives $qd=-q$ , so $d=-1$ .

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

The sum of the first $n$ natural numbers is $\frac{1}{5}$ times the sum of their squares, then $n =$ ……… (a) 5 (b) 6 (c) 7 (d) 8

Answer

(c) 7

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

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

The equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a circle if ……… (a) $h = 0$ (b) $a = b$ (c) $a = b$ and $h = 0$ (d) $a = b = h$

Answer

(c) $a = b$ and $h = 0$

A circle has equal coefficients of $x^2$ and $y^2$ and no $xy$ term, so $a=b$ and $h=0$ (together with a real, positive radius).

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

The circumcentre of the triangle whose vertices are $(0, 0)$ , $(6, 0)$ and $(0, 4)$ is ……… (a) $(2, 0)$ (b) $(3, 0)$ (c) $(0, 3)$ (d) $(3, 2)$

Answer

(d) $(3, 2)$

The triangle has a right angle at $(0,0)$ , so the circumcentre is the midpoint of the hypotenuse joining $(6,0)$ and $(0,4)$ : $\left(\frac{6+0}{2},\frac{0+4}{2}\right)=(3,2).$

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

$A, B, C$ are the vertices of a triangle, then $\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA}$ is equal to …………. (a) 0 (b) 1 (c) 3 (d) None

Answer

(a) 0

$\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$ , and $\overrightarrow{AC}+\overrightarrow{CA}=\vec 0$ . The sides taken round a closed triangle sum to the zero vector.

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors such that $\vec{a} + \vec{b} - \vec{c} = 0$ , then the angle between $\vec{a}$ and $\vec{b}$ is ……. (a) $\frac{\pi}{6}$ (b) $\frac{\pi}{3}$ (c) $\frac{\pi}{2}$ (d) $\frac{2\pi}{3}$

Answer

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

$\vec a+\vec b=\vec c$ , so $|\vec a+\vec b|^2=|\vec c|^2$ : $1+1+2\cos\theta=1\Rightarrow\cos\theta=-\frac12\Rightarrow\theta=\frac{2\pi}{3}.$

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2081 · Science Model · 1 mark

The normal form of the equation of the line given by $3x + 4y + 7 = 0$ is $x\cos\alpha + y\sin\alpha = p$ where ……… (a) $\tan\alpha = \frac{4}{3}, p = -\frac{7}{5}$ (b) $\tan\alpha = -\frac{3}{4}, p = -\frac{7}{5}$ (c) $\tan\alpha = \frac{3}{4}, p = -\frac{7}{5}$ (d) $\tan\alpha = \frac{3}{4}, p = \frac{7}{5}$

Answer

(a) $\tan\alpha = \frac{4}{3}, p = -\frac{7}{5}$

Write $3x+4y=-7$ and divide by $\sqrt{3^2+4^2}=5$ : $\frac35x+\frac45y=-\frac75.$ So $\cos\alpha=\frac35$ , $\sin\alpha=\frac45$ , giving $\tan\alpha=\frac43$ and $p=-\frac75$ , which matches option (a).

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

The minimum value of $3\cos\theta + 4\sin\theta + 8$ is ……… (a) 9 (b) 7 (c) 5 (d) 3

Answer

(d) 3

$3\cos\theta+4\sin\theta$ ranges over $[-5,5]$ since $\sqrt{3^2+4^2}=5$ . The minimum of the expression is $-5+8=3$ .

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

If $x$ and $y$ are rational numbers such that $\sqrt{xy}$ is irrational, then $\sqrt{x} + \sqrt{y}$ is ……. (a) Rational (b) Irrational (c) Non-real (d) None Of Above

Answer

(b) Irrational

Suppose $\sqrt x+\sqrt y=r$ were rational. Squaring, $x+y+2\sqrt{xy}=r^2$ , so $\sqrt{xy}=\frac{r^2-x-y}{2}$ would be rational, contradicting the given condition. Hence $\sqrt x+\sqrt y$ is irrational.

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

The L.C.M. of $x^3 + 2x^2 - x - 2$ and $x^3 + x^2 - 4x - 4$ is ……… (a) $(x+1)(x-2)$ (b) $(x+1)(x-1)(x+2)$ (c) $(x+1)(x+2)(x-2)$ (d) $(x^2-1)(x^2-4)$

Answer

(d) $(x^2-1)(x^2-4)$

$x^3+2x^2-x-2=x^2(x+2)-(x+2)=(x+2)(x-1)(x+1)$ $x^3+x^2-4x-4=x^2(x+1)-4(x+1)=(x+1)(x-2)(x+2)$ The L.C.M. is $(x+1)(x-1)(x+2)(x-2)=(x^2-1)(x^2-4)$ .

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

If $2^x = 3^y = 6^{-z}$ , then the value of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ equals ……… (a) 0 (b) 1 (c) $-1$ (d) 12

Answer

(a) 0

Let $2^x=3^y=6^{-z}=k$ . Then $2=k^{1/x}$ , $3=k^{1/y}$ , $6=k^{-1/z}$ . Since $6=2\cdot3$ , $-\frac1z=\frac1x+\frac1y\Rightarrow\frac1x+\frac1y+\frac1z=0.$

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

A number of two digits exceeds four times the sum of its digits by 3. If 36 is added to the number, the digits are reversed. Then the number is …….. (a) 27 (b) 63 (c) 59 (d) 72

Answer

(c) 59

Let the number be $10a+b$ . Then $10a+b=4(a+b)+3\Rightarrow 2a-b=1$ . Also $10a+b+36=10b+a\Rightarrow b-a=4$ . Solving, $a=5$ , $b=9$ .

Check: $4(14)+3=59$ and $59+36=95$ .

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

If the diagonals of a rhombus are 10 cm and 8 cm respectively, then the area of the rhombus is ……. (a) $80\ \text{cm}^2$ (b) $40\ \text{cm}^2$ (c) $36\ \text{cm}^2$ (d) $60\ \text{cm}^2$

Answer

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

$\text{Area}=\frac12d_1d_2=\frac12\times10\times8=40\ \text{cm}^2.$

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

If the mean of $2, 6, x, 10, 14$ is 8, then its median is ……….. (a) 8 (b) 8.5 (c) 9 (d) 10

Answer

(a) 8

$\frac{2+6+x+10+14}{5}=8\Rightarrow x+32=40\Rightarrow x=8$ . The ordered data $2,6,8,10,14$ has median $8$ .

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

Which of the following is a polynomial? (a) $\frac{x^3}{2} - \frac{5}{x^2}$ (b) $\sqrt{3}x - 1$ (c) $x^2 + \frac{4x^{3/2}}{\sqrt{x}}$ (d) $\frac{x+1}{x-1}$

Answer

(b) $\sqrt{3}x - 1$

A polynomial has only non-negative integer powers of $x$ . $\sqrt3x-1$ qualifies (the coefficient $\sqrt3$ is allowed). The others contain $x^{-2}$ , a fractional power, or a variable denominator. In (c), $\frac{4x^{3/2}}{\sqrt x}=4x$ simplifies to a polynomial term, but the expression as written is not in polynomial form, so (b) is the clear choice.

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

Zero of the zero polynomial is ……. (a) 1 (b) any real number (c) not defined (d) 0

Answer

(b) any real number

The zero polynomial is $p(x)=0$ , and $p(c)=0$ for every real $c$ , so every real number is a zero of it.

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

Mathematician Euclid belongs to the country ……….. (a) Babylonian (b) Greece (c) Germany (d) France

Answer

(b) Greece

Euclid was a Greek mathematician (of Alexandria), known as the "father of geometry" and author of The Elements.

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2080 · Science Model · 1 mark
2081 · Science Model · 1 mark

"Lines are parallel if they do not intersect" is stated in the term of …….. (a) Postulate (b) a definition (c) an axiom (d) a proof

Answer

(b) a definition

The statement explains what the term "parallel lines" means, so it is a definition, not a postulate, axiom or proof.

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2080 · Management Model · 1 mark
2083 · Management Model · 1 mark

If $x + \frac{1}{x} = 4$ , then the value of $x^2 + \frac{1}{x^2}$ is ……….. (a) 16 (b) 14 (c) 20 (d) 18

Answer

(b) 14

$x^2 + \frac{1}{x^2} = \left(x + \frac1x\right)^2 - 2 = 4^2 - 2 = 14.$

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2079 · Management and Humanities Model
2077 · Management Model

In a throw of a die, the probability of getting an even number is ……… (a) $\frac{1}{3}$ (b) $\frac{1}{2}$ (c) $\frac{2}{3}$ (d) $\frac{1}{6}$

Answer

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

Even numbers: 2, 4, 6, so $P = \frac{3}{6} = \frac12$ .

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2079 · Law Model · 1 mark
2080 · Law Model · 1 mark

What is 10% of 20% of 200? (a) 40 (b) 4 (c) 0.4 (d) 400

Answer

(b) 4

$20\% \text{ of } 200 = 40,\qquad 10\% \text{ of } 40 = 4$

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2079 · Law Model · 1 mark
2080 · Law Model · 1 mark

If mean is 16 and median is 15, what is the value of mode? (a) 15 (b) 1 (c) 14 (d) 4

Answer

(c) 14 (nearest option)

By the empirical relation, $\text{Mode} = 3\,\text{Median} - 2\,\text{Mean} = 3(15) - 2(16) = 45 - 32 = 13$ . The exact value 13 is not listed, so the closest option, 14, is the intended choice.

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2079 · Management and Humanities Model

The arithmetic mean of the first five natural numbers is …….. (a) 4 (b) 5 (c) 3 (d) 2

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Answer

(c) 3

$\text{Mean} = \frac{1+2+3+4+5}{5} = \frac{15}{5} = 3.$

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2079 · Management and Humanities Model

The length of each side of an equilateral triangle having area $16\sqrt{3}\ \text{cm}^2$ is ……. (a) 10 cm (b) 4 cm (c) 6 cm (d) 8 cm

Similar questions: Area of an equilateral triangle having side a is (2077 · Management Model)

Answer

(d) 8 cm

$\frac{\sqrt3}{4}a^2 = 16\sqrt3 \Rightarrow a^2 = 64 \Rightarrow a = 8 \text{ cm}.$

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2077 · Management Model

Area of an equilateral triangle having side $a$ is …… (a) $\frac{\sqrt{3}}{2}a^2$ (b) $\sqrt{\frac{3}{2}}\,a^2$ (c) $\sqrt{\frac{3}{4}}\,a^2$ (d) $\frac{\sqrt{3}}{4}a^2$

Similar questions: The length of each side of an equilateral (2079 · Management and Humanities Model)

Answer

(d) $\frac{\sqrt3}{4}a^2$

The height is $\frac{\sqrt3}{2}a$ , so the area is $\frac12\times a\times\frac{\sqrt3}{2}a=\frac{\sqrt3}{4}a^2$ .

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2077 · Management Model

What is the arithmetic mean of the first 10 natural numbers? (a) 5 (b) 4.5 (c) 5.5 (d) 6

Similar questions: The arithmetic mean of the first five natural (2079 · Management and Humanities Model)

Answer

(c) 5.5

$\bar x=\frac{1+2+\dots+10}{10}=\frac{55}{10}=5.5$

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2079 · Management and Humanities Model

Which of the following is an irrational number? (a) $\frac{22}{7}$ (b) $\pi$ (c) 0.333 (d) 4.5679

Answer

(b) $\pi$

$\pi$ is non-terminating and non-recurring, so it cannot be written as a ratio of integers. $\frac{22}{7}$ , $0.333$ and $4.5679$ are rational.

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2079 · Management and Humanities Model

If $A$ and $B$ are disjoint sets then $A - B =$ ……….. (a) $A$ (b) $B$ (c) $A \cap B$ (d) $A \cup B$

Answer

(a) $A$

Disjoint sets have no common element, so removing $B$ from $A$ removes nothing: $A - B = A$ .

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2079 · Management and Humanities Model

How many dollars shall we receive for Rs. 12,000 if Rs. 100 is equal to 136 marks and 400 marks is equal to 95 dollars? (a) USD 3786 (b) USD 8763 (c) USD 6783 (d) USD 3876

Answer

(d) USD 3876

Rs 100 = 136 marks, so Rs 12,000 = $120 \times 136 = 16320$ marks.

400 marks = 95 dollars, so $\frac{16320 \times 95}{400} = 40.8 \times 95 = 3876 \text{ dollars}.$

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2079 · Management and Humanities Model

What will be the population of a town in 3 years if its present population is 24,000 increasing at 5% annually? (a) 27,378 (b) 27,873 (c) 27,783 (d) 27,387

Answer

(c) 27,783

$P = 24000\left(1+\frac{5}{100}\right)^3 = 24000 \times (1.05)^3 = 24000 \times 1.157625 = 27783.$

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2079 · Management and Humanities Model

If the side of an equilateral triangle is $a$ unit, then the area of the triangle is ……… (a) $\sqrt{3}a^2$ sq. unit (b) $\frac{\sqrt{3}}{2}a^2$ sq. unit (c) $\frac{\sqrt{3}}{4}a^2$ sq. unit (d) $a^2$ sq. unit

Answer

(c) $\frac{\sqrt{3}}{4}a^2$ sq. unit

The height is $\sqrt{a^2 - \frac{a^2}{4}} = \frac{\sqrt{3}}{2}a$ , so area $= \frac12 \cdot a \cdot \frac{\sqrt3}{2}a = \frac{\sqrt3}{4}a^2$ .

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2079 · Management and Humanities Model

The value of $(81)^{0.16} \times (81)^{0.09}$ is equal to …… (a) 9 (b) 3 (c) 81.25 (d) 27

Answer

(b) 3

Add the powers: $(81)^{0.16+0.09} = (81)^{0.25} = (3^4)^{1/4} = 3$ .

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2079 · Management and Humanities Model

The value of the polynomial $4x^2 - 5x + 9$ , when $x = \frac{1}{2}$ is …….. (a) $\frac{13}{2}$ (b) $\frac{15}{2}$ (c) 8 (d) $\frac{1}{2}$

Answer

(b) $\frac{15}{2}$

$4\left(\tfrac12\right)^2 - 5\left(\tfrac12\right) + 9 = 1 - \tfrac52 + 9 = 10 - \tfrac52 = \tfrac{15}{2}.$

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2079 · Management and Humanities Model

In $\triangle ABC$ , if $\angle A > \angle B$ , then ………. (a) $AB > BC$ (b) $BC < AC$ (c) $AC > AB$ (d) $BC > AC$

Answer

(d) $BC > AC$

The larger angle has the longer opposite side. $BC$ is opposite $\angle A$ and $AC$ is opposite $\angle B$ , so $\angle A > \angle B$ gives $BC > AC$ .

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2079 · Management and Humanities Model

Abscissa of a point is negative in ………. (a) I and II quadrants (b) only II quadrant (c) only III quadrant (d) II and III quadrant

Answer

(d) II and III quadrant

The abscissa is the $x$ -coordinate, which is negative to the left of the $y$ -axis, that is in the II and III quadrants.

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2079 · Management and Humanities Model

The point which lies on the line $y = -3x$ is ……… (a) $(2, -7)$ (b) $(3, -6)$ (c) $(3, 9)$ (d) $(3, -9)$

Answer

(d) $(3, -9)$

A point lies on $y=-3x$ if its coordinates satisfy it. For $(3,-9)$ : $-3(3) = -9$ . The others fail: $(2,-7)$ needs $-6$ , $(3,-6)$ needs $-9$ , $(3,9)$ needs $-9$ .

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2079 · Management and Humanities Model

The equation of the y-axis is of the form ………. (a) $y = 0$ (b) $x = 0$ (c) $x = y$ (d) $x + y = 0$

Answer

(b) $x = 0$

Every point on the $y$ -axis has $x$ -coordinate $0$ .

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2079 · Management and Humanities Model

A quadrilateral which has exactly one pair of parallel sides is called ………. (a) a rectangle (b) a parallelogram (c) a trapezium (d) a rhombus

Answer

(c) a trapezium

A trapezium has exactly one pair of parallel sides; a rectangle, parallelogram and rhombus have two pairs.

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2079 · Management and Humanities Model

ABCD is a rhombus. If $\angle ACB = 50^\circ$ , then $\angle ADB$ is ………. (a) $50^\circ$ (b) $60^\circ$ (c) $45^\circ$ (d) $40^\circ$

Answer

(d) $40^\circ$

Diagonals of a rhombus bisect the vertex angles, so $\angle BCD = 2 \times 50^\circ = 100^\circ$ . Then $\angle BAD = 100^\circ$ (opposite angles). In $\triangle ABD$ , $AB = AD$ , so $\angle ADB = \frac{180^\circ - 100^\circ}{2} = 40^\circ.$

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2079 · Management and Humanities Model

The diameter of a circle is also ……. (a) secant (b) tangent (c) radius (d) chord

Answer

(d) chord

A diameter is a chord that passes through the centre (and it is the longest chord).

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2079 · Management and Humanities Model

The median of the data 12, 14, 15, 13, 11, 17, 10 is ……… (a) 13 (b) 12 (c) 14 (d) 15

Answer

(a) 13

Arranged in order: 10, 11, 12, 13, 14, 15, 17. There are 7 values, so the median is the 4th value, 13.

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2079 · Management and Humanities Model

A card is drawn from a normal pack of 52 cards. What is the probability that it is a face card? (a) $\frac{1}{4}$ (b) $\frac{3}{13}$ (c) $\frac{1}{13}$ (d) $\frac{4}{13}$

Answer

(b) $\frac{3}{13}$

There are $3 \times 4 = 12$ face cards (J, Q, K in each suit), so $P = \frac{12}{52} = \frac{3}{13}$ .

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2079 · Management and Humanities Model

If $x + \frac{1}{x} = 4$ , then the value of $x^3 + \frac{1}{x^3}$ is ……….. (a) 64 (b) 58 (c) 56 (d) 52

Answer

(d) 52

$x^3 + \frac1{x^3} = \left(x+\frac1x\right)^3 - 3\left(x+\frac1x\right) = 64 - 12 = 52.$

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2079 · Management and Humanities Model

The value of $\sqrt{x^{-1}\sqrt{x^3 \cdot x^{-4}}}$ is …….. (a) $x^{-1}$ (b) $x^{-3/4}$ (c) $x^{-4/3}$ (d) $x^{-1/4}$

Answer

(b) $x^{-3/4}$

Inside: $x^3 \cdot x^{-4} = x^{-1}$ , and $\sqrt{x^{-1}} = x^{-1/2}$ .

Then $x^{-1}\cdot x^{-1/2} = x^{-3/2}$ , and $\sqrt{x^{-3/2}} = x^{-3/4}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · Management and Humanities Model

$\frac{1}{1 + x^{m-n}} + \frac{1}{1 + x^{n-m}}$ is equal to ………. (a) 0 (b) 1 (c) $\frac{1}{2}$ (d) $a^{m+n}$

Answer

(b) 1

Let $t = x^{m-n}$ . Then $\frac{1}{1+t} + \frac{1}{1+\frac1t} = \frac{1}{1+t} + \frac{t}{t+1} = \frac{1+t}{1+t} = 1.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · Management and Humanities Model

If $5a + 2b : 7a + 3b = 9 : 13$ , then $a : b$ is ……… (a) 1 : 2 (b) 2 : 1 (c) 2 : 3 (d) 3 : 2

Answer

(a) 1 : 2

Cross-multiplying: $13(5a+2b) = 9(7a+3b)$ , so $65a + 26b = 63a + 27b$ , giving $2a = b$ . Hence $a : b = 1 : 2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · Management and Humanities Model

If $a = 21$ , $b = 13$ and $c = 20$ then the area of triangle ABC is ……… (a) 116 (b) 120 (c) 84 (d) 96

Answer

(b) 120 (nearest option; the exact area is 126 sq. unit)

By Heron's formula, $s = \frac{21+13+20}{2} = 27$ .

$\text{Area} = \sqrt{27(27-21)(27-13)(27-20)} = \sqrt{27 \cdot 6 \cdot 14 \cdot 7} = \sqrt{15876} = 126.$

Check: with base 21, height $h = 12$ gives feet at distances $5$ and $16$ ( $5+16=21$ ) and area $\frac12 \cdot 21 \cdot 12 = 126$ . None of the listed options equals 126, so the printed options appear to contain an error; 120 is the closest.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · Management and Humanities Model

A cycle of cost Rs. 1,000 is depreciated at 10% p.a., then what will be its value after 1 year? (a) Rs. 950 (b) Rs. 925 (c) Rs. 900 (d) Rs. 875

Answer

(c) Rs. 900

Depreciation is $10\%$ of Rs 1,000 $=$ Rs 100, so the value after 1 year is $1000 - 100 =$ Rs 900.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · Management and Humanities Model

What is the actual discount amount when 1.25% discount is given on a car marked at Rs. 5,00,000? (a) Rs. 6250 (b) Rs. 6000 (c) Rs. 5850 (d) Rs. 5500

Answer

(a) Rs. 6250

Discount $= 1.25\% \times 500000 = \dfrac{1.25}{100}\times 500000 =$ Rs 6,250.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · Management and Humanities Model

P and Q together can do a piece of work in 10 hours but P alone can do it in 15 hours. How many hours would Q alone take to do the same work? (a) 20 hours (b) 25 hours (c) 30 hours (d) 35 hours

Answer

(c) 30 hours

Q's one-hour work $= \dfrac{1}{10}-\dfrac{1}{15}=\dfrac{3-2}{30}=\dfrac{1}{30}$ , so Q alone takes 30 hours.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · Management and Humanities Model

H.C.F. of $x^3 - 1$ and $x^4 + x^2 + 1$ is …….. (a) $x - 1$ (b) $x^2 - x + 1$ (c) $x^2 + x + 1$ (d) $x^2 - x - 1$

Answer

(c) $x^2 + x + 1$

$x^3-1=(x-1)(x^2+x+1)$ $x^4+x^2+1=(x^2+1)^2-x^2=(x^2+x+1)(x^2-x+1)$

The common factor is $x^2+x+1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · Management and Humanities Model

The value of $\left(\frac{8}{27}\right)^{-\frac{2}{3}}$ is …….. (a) $\frac{9}{4}$ (b) $\frac{4}{9}$ (c) $\frac{3}{2}$ (d) $\frac{2}{3}$

Answer

(a) $\dfrac{9}{4}$

$\left(\frac{8}{27}\right)^{-\frac23}=\left(\frac{27}{8}\right)^{\frac23}=\left(\frac{3^3}{2^3}\right)^{\frac23}=\frac{3^2}{2^2}=\frac94$

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · Management and Humanities Model

If $N = 50$ and $\sum fx = 750$ , then the number of terms $(N) =$ …….. (a) 45 (b) 30 (c) 15 (d) 5

Answer

(c) 15

The quantity that can be found from the given data is the mean: $\bar{x}=\frac{\sum fx}{N}=\frac{750}{50}=15$

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · Management and Humanities Model

If 3 is subtracted from the square of a number, the remainder is 6, then the number is …….. (a) $\pm 6$ (b) $\pm 5$ (c) $\pm 4$ (d) $\pm 3$

Answer

(d) $\pm 3$

Let the number be $x$ . Then $x^2-3=6\Rightarrow x^2=9\Rightarrow x=\pm 3$ .

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2079 · Law Model · 1 mark

$\frac{1}{3}$ of a right angle is (a) $60^\circ$ (b) $45^\circ$ (c) $30^\circ$ (d) $90^\circ$

Answer

(c) $30^\circ$

A right angle is $90^\circ$ , so $\frac{1}{3}\times 90^\circ = 30^\circ$ .

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2079 · Law Model · 1 mark

The perimeter of a rectangle with length 50 cm and breadth 25 cm is (a) 1250 cm (b) 150 cm (c) 75 cm (d) none

Answer

(b) 150 cm

$P = 2(l+b) = 2(50+25) = 150 \text{ cm}$

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2079 · Law Model · 1 mark

By selling 12 books for the cost of 15 books, how much is the profit? (a) 20% (b) 25% (c) 24% (d) 15%

Answer

(b) 25%

Let the cost of one book be Rs 1. Then the cost of 12 books is Rs 12, and the selling price of 12 books equals the cost of 15 books, Rs 15.

$\text{Profit} = \frac{15-12}{12}\times 100\% = 25\%$

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · Law Model · 1 mark

If Indian Currency 100 is equivalent to Nepali Rupees 160, how much Indian Currency is equivalent to Rs. 100? (a) IC 62.5 (b) IC 67.5 (c) IC 65.2 (d) IC 58.5

Answer

(a) IC 62.5

Rs 160 = IC 100, so Rs 100 = $\frac{100}{160}\times 100 = 62.5$ IC.

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2077 · Management Model

$A - \overline{B}$ equals ……….. (a) $A$ (b) $A \cup B$ (c) $A \cap B$ (d) $\overline{A \cap B}$

Answer

(c) $A \cap B$

$A - \overline{B}$ means elements of $A$ that are not in $\overline{B}$ , that is, elements of $A$ that are in $B$ . So $A - \overline{B} = A \cap B$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

If $n(A) = 55\%$ , $n(B) = 65\%$ and $n(A \cap B) = 35\%$ then $n(\overline{A \cup B}) =$ ……….. (a) 5% (b) 10% (c) 15% (d) 20%

Answer

(c) 15%

$n(A\cup B)=55+65-35=85\%$

$n(\overline{A\cup B})=100-85=15\%$

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

The number of subsets of a set containing $n$ elements is …… (a) $n$ (b) $2n$ (c) $2^n$ (d) $2^{n-1}$

Answer

(c) $2^n$

Each of the $n$ elements is either in a subset or not, giving $2\times2\times\cdots\times2=2^n$ subsets.

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

The compound interest on Rs. 2000 in 3 years at 5% per annum is …….. (a) Rs. 315.25 (b) Rs. 300 (c) Rs. 328 (d) Rs. 351.50

Answer

(a) Rs. 315.25

$A=2000\left(1+\frac{5}{100}\right)^3=2000\times1.157625=\text{Rs. }2315.25$

$CI=A-P=2315.25-2000=\text{Rs. }315.25$

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

In $\triangle PQR$ , if $\angle P > \angle Q$ , then …….. (a) $QR < PR$ (b) $QR > PR$ (c) $PQ > QR$ (d) $PQ < QR$

Answer

(b) $QR > PR$

The larger angle has the larger side opposite it. $QR$ is opposite $\angle P$ and $PR$ is opposite $\angle Q$ , so $\angle P>\angle Q$ gives $QR>PR$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

The point which lies on the line $y = -3x$ is …….. (a) $(1, 3)$ (b) $(1, -3)$ (c) $(3, 1)$ (d) $(3, -1)$

Answer

(b) $(1,-3)$

Put $x=1$ : $y=-3(1)=-3$ , so $(1,-3)$ satisfies $y=-3x$ . The others do not: $(1,3)$ gives $-3\ne3$ , $(3,1)$ gives $-9\ne1$ , $(3,-1)$ gives $-9\ne-1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

If $x + \frac{1}{x} = 5$ , then the value of $x^2 + \frac{1}{x^2}$ is …….. (a) 25 (b) 20 (c) 23 (d) 27

Answer

(c) 23

$x^2+\frac1{x^2}=\left(x+\frac1x\right)^2-2=25-2=23$

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2077 · Management Model

The volume of a ball is $38808\ \text{cm}^3$ , then the radius is ……. (a) 7 cm (b) 14 cm (c) 21 cm (d) 28 cm

Answer

(c) 21 cm

$\frac43\pi r^3=38808\Rightarrow r^3=\frac{38808\times3\times7}{4\times22}=441\times21=9261$

$r=\sqrt[3]{9261}=21\text{ cm}$

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2077 · Management Model

The value of $\sqrt[3]{\sqrt{(64)^{-1}}} =$ ……. (a) 1 (b) $\frac{1}{2}$ (c) 2 (d) 4

Answer

(b) $\frac12$

$\sqrt{64^{-1}}=\frac18,\qquad \sqrt[3]{\frac18}=\frac12$

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2077 · Management Model

Which of the following numbers is an irrational? (a) $\sqrt{2}$ (b) $\pi$ (c) $e$ (d) all of these

Answer

(d) all of these

$\sqrt2$ , $\pi$ and $e$ are all non-terminating, non-repeating decimals, so each is irrational.

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

If a number is added to its square the result is 30, then the number is …… (a) 4 (b) 5 (c) 6 (d) 7

Answer

(b) 5

$n+n^2=30\Rightarrow n^2+n-30=0\Rightarrow(n+6)(n-5)=0$

Among the options, $n=5$ (check: $5+25=30$ ).

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

The area of the trapezium ABCD (AB parallel to DC, AB = 4 cm, DC = 6 cm, and the distance between the parallel sides 3 cm) is ………. (a) $15\ \text{cm}^2$ (b) $30\ \text{cm}^2$ (c) $60\ \text{cm}^2$ (d) none

Answer

(a) $15\ \text{cm}^2$

$A=\frac12(AB+DC)\times h=\frac12(4+6)\times3=15\text{ cm}^2$

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

The angle in a semi-circle is a ……. (a) complementary (b) supplementary (c) half of complementary (d) none of these

Answer

(d) none of these

The angle in a semi-circle is a right angle, $90^\circ$ . It is not supplementary ( $180^\circ$ ) and not half of complementary ( $45^\circ$ ), so none of (a) to (c) fits.

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

Every whole number is ……….. (a) real number (b) integers (c) rational number (d) all of these

Answer

(d) all of these

Whole numbers $0,1,2,\dots$ are integers, hence rational ( $n=\frac{n}{1}$ ), hence real.

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

The area of the triangle whose sides are 5 cm, 12 cm and 13 cm is ……. (a) $60\ \text{cm}^2$ (b) $40\ \text{cm}^2$ (c) $30\ \text{cm}^2$ (d) $20\ \text{cm}^2$

Answer

(c) $30\ \text{cm}^2$

Since $5^2+12^2=13^2$ , it is a right triangle, so area $=\frac12\times5\times12=30\ \text{cm}^2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

Two sides of a triangle are 4 cm and 1.5 cm, the length of the third side can't be …… (a) 2.6 cm (b) 2.7 cm (c) 2.8 cm (d) 2.4 cm

Answer

(d) 2.4 cm

The third side $c$ must satisfy $4-1.5<c<4+1.5$ , i.e. $2.5<c<5.5$ . Only 2.4 cm lies outside this range.

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

L.C.M. of $x^3 - y^3$ and $x^4 + x^2y^2 + y^4$ is …….. (a) $(x - y)$ (b) $x^2 + xy + y^2$ (c) $(x - y)(x^4 + x^2y^2 + y^4)$ (d) $x^2 - xy + y^2$

Answer

(c) $(x - y)(x^4 + x^2y^2 + y^4)$

$x^3-y^3=(x-y)(x^2+xy+y^2),\qquad x^4+x^2y^2+y^4=(x^2+xy+y^2)(x^2-xy+y^2)$

LCM $=(x-y)(x^2+xy+y^2)(x^2-xy+y^2)=(x-y)(x^4+x^2y^2+y^4)$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

Simplify: $\frac{3^{x+2} - 3^x}{8 \cdot 3^x}$ (a) 1 (b) $-1$ (c) 0 (d) none of these

Answer

(a) 1

$\frac{3^{x+2}-3^x}{8\cdot3^x}=\frac{3^x(9-1)}{8\cdot3^x}=1$

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2077 · Management Model

Which of the following represents the degree of $f(x, y) = x^3y^2 + 2x^2y + 4x^4$ ? (a) 4 (b) 5 (c) 3 (d) 11

Answer

(b) 5

The degrees of the terms are $3+2=5$ , $2+1=3$ and $4$ . The highest is 5.

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

A man earns Rs. 7,500 and spends Rs. 5,000 in a month. What percent does he save? (a) 25% (b) $\frac{100}{3}\%$ (c) $\frac{200}{3}\%$ (d) none of these

Answer

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

Savings $=$ Rs 7,500 $-$ Rs 5,000 $=$ Rs 2,500, so saving percent $=\frac{2500}{7500}\times100=\frac{100}{3}\%$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

Two types of rice costing Rs. 21 per kg and Rs. 30 per kg are mixed in the ratio of 4 : 5, then the cost per kg of the mixture is …… (a) Rs. 24 (b) Rs. 25 (c) Rs. 26 (d) Rs. 27

Answer

(c) Rs. 26

$\frac{4\times21+5\times30}{4+5}=\frac{84+150}{9}=\frac{234}{9}=26$

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

In what time will a sum of money double itself at 8% p.a.? (a) 10 yrs. (b) 12.5 yrs. (c) 15 yrs. (d) 17.5 yrs.

Answer

(b) 12.5 yrs.

To double, interest $=$ principal, so $P=\frac{P\times8\times T}{100}$ , giving $T=\frac{100}{8}=12.5$ years.

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

Given that Rs. 270 = £ 2 and £ 5 = USD 9.12, then USD 170 equals ….. (a) 12852.23 (b) 12258.23 (c) 12582.23 (d) none

Answer

(c) 12582.23

Rs 270 $=$ £2, so £1 $=$ Rs 135. Also £5 $=USD 9.12$ , so $USD 170=\frac{170\times5}{9.12}$ £ $=93.2018$ £.

Rs value $=93.2018\times135\approx12582.2$ , which matches option (c) 12582.23.

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

If $a : b = 2 : 3$ , then $(6a + 7b) : (9a + 8b)$ equals …….. (a) 14 : 11 (b) 11 : 14 (c) 10 : 13 (d) 13 : 10

Answer

(b) 11 : 14

Let $a=2k,\ b=3k$ . Then $6a+7b=12k+21k=33k$ and $9a+8b=18k+24k=42k$ .

$33k:42k=11:14$

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

The value of $x$ if $4 : x :: 7 : 42$ is ……. (a) 24 (b) 18 (c) 14 (d) 28

Answer

(a) 24

$4:x::7:42$ gives $\frac4x=\frac7{42}$ , so $x=\frac{4\times42}{7}=24$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

If Rs. 12,750 is left after paying an income tax at the rate of 15% without any allowances, find the income … (a) Rs. 14,000 (b) Rs. 14,500 (c) Rs. 15,000 (d) Rs. 15,500

Answer

(c) Rs. 15,000

After 15% tax, 85% of the income remains: $0.85I=12750$ , so $I=\frac{12750}{0.85}=$ Rs 15,000.

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

Ordinate of a point is negative in ……. (a) IInd and IIIrd quadrant (b) only in IInd quadrant (c) only in IIIrd quadrant (d) IIIrd and IVth quadrant

Answer

(d) IIIrd and IVth quadrant

The ordinate is the $y$ -coordinate, which is negative below the $x$ -axis, i.e. in the third and fourth quadrants.

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2077 · Management Model

If $N = 50$ and mean $= 15$ then $\sum fx$ equals ……… (a) 150 (b) 65 (c) 750 (d) 450

Answer

(c) 750

$\bar x=\frac{\sum fx}{N}\Rightarrow \sum fx=15\times50=750$

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2077 · Management Model

The value of $\cos 30^\circ$ is ….. (a) $\frac{1}{2}$ (b) $\frac{\sqrt{3}}{2}$ (c) $\frac{1}{\sqrt{2}}$ (d) $\frac{1}{\sqrt{3}}$

Answer

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

Standard value: $\cos30^\circ=\frac{\sqrt3}{2}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

In how many ways can 10 persons shake their hands without repetition? (a) 45 (b) 50 (c) 55 (d) 60

Answer

(a) 45

Each handshake involves a pair of persons: $\binom{10}{2}=\frac{10\times9}{2}=45$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

$(2.1)^2 - (1.9)^2$ equals ……. (a) 0.4 (b) 0.3 (c) 0.7 (d) 0.8

Answer

(d) 0.8

$(2.1)^2-(1.9)^2=(2.1+1.9)(2.1-1.9)=4\times0.2=0.8$

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · Management Model

What is the probability of getting a vowel letter from the word "EDUCATION" when one letter is selected? (a) $\frac{5}{8}$ (b) $\frac{4}{9}$ (c) $\frac{5}{9}$ (d) $\frac{6}{8}$

Answer

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

EDUCATION has 9 letters, of which E, U, A, I, O are vowels (5). Probability $=\frac59$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · Law Model · 1 mark

$\frac{1}{4}$ of a complete angle is (a) $60^\circ$ (b) $45^\circ$ (c) $30^\circ$ (d) $90^\circ$

Answer

(d) $90^\circ$

A complete angle is $360^\circ$ , so

$\frac{1}{4} \times 360^\circ = 90^\circ$

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · Law Model · 1 mark

The perimeter of a square with measuring side 25 cm is (a) 100 cm (b) 150 cm (c) 75 cm (d) none

Answer

(a) 100 cm

$P = 4 \times \text{side} = 4 \times 25 = 100 \text{ cm}$

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2080 · Law Model · 1 mark

What is the total sum of interior angles of a triangle? (a) 360 (b) 180 (c) 90 (d) 0

Answer

(b) 180

The sum of the interior angles of any triangle is $180^\circ$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · Law Model · 1 mark

If USD 1 is equivalent to Nepali Rupees 128, how much USD is equivalent to Rs. 100? (a) USD 0.78 (b) USD 12800 (c) USD 1.28 (d) none

Answer

(a) USD 0.78

$\text{USD} = \frac{100}{128} = 0.78125 \approx 0.78$

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · Law Model · 1 mark

The mean of the data 4, 10, 5, 9, 12 is (a) 8 (b) 10 (c) 9 (d) 15

Answer

(a) 8

$\text{Mean} = \frac{4+10+5+9+12}{5} = \frac{40}{5} = 8$

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · Law Model · 1 mark

Which of the following is not a sub-set of $A = \{1, 2, 3, 4, 5\}$ ? (a) $\{\ \}$ (b) $\{1, 2, 3, 4, 5, 6\}$ (c) both (d) none

Answer

(b) $\{1, 2, 3, 4, 5, 6\}$

A subset can contain only elements of $A$ . The set in (b) contains 6, which is not in $A$ . The empty set $\{\ \}$ is a subset of every set.

Answer checked by Claude Sonnet · Oct 21, 2026

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Past questions and answers

1.If the surface area of a spherical ball is 616 cm2616\ \text{cm}^2616 cm2, then its diameter is ………… (a) 14 cm (b) 7 cm (c) 21 cm (d) 28 cm

2.The greatest number of 3 digits, which is a perfect square is ……… (a) 981 (b) 999 (c) 951 (d) 961

3.If the side of a rhombus is 10 cm and one diagonal is 16 cm, the area of the rhombus is ……… (a) 24 cm224\ \text{cm}^224 cm2 (b) 48 cm248\ \text{cm}^248 cm2 (c) 96 cm296\ \text{cm}^296 cm2 (d) 90 cm290\ \text{cm}^290 cm2

4.The surface area of a ball is 16π cm216\pi\ \text{cm}^216π cm2, then what is its diameter? (a) 8 cm (b) 4 cm (c) 2 cm (d) 1 cm

5.The side of a rhombus is 10 cm and one diagonal is 16 cm. The area of the rhombus is ………. (a) 24 cm224\ \text{cm}^224 cm2 (b) 48 cm248\ \text{cm}^248 cm2 (c) 96 cm296\ \text{cm}^296 cm2 (d) 90 cm290\ \text{cm}^290 cm2

6.The greatest number of 4 digits, which is a perfect square is …….. (a) 9996 (b) 9801 (c) 9900 (d) 9604

7.The quadratic equation whose roots are 12\frac{1}{2}21​ and −12-\frac{1}{2}−21​ is …….. (a) x2−4=0x^2 - 4 = 0x2−4=0 (b) 4x2−1=04x^2 - 1 = 04x2−1=0 (c) 4x2+1=04x^2 + 1 = 04x2+1=0 (d) x2−12=0x^2 - \frac{1}{2} = 0x2−21​=0

8.Let A={a,b,c}A = \{a, b, c\}A={a,b,c}, the possible number of subsets for the set AAA is …… (a) 3 (b) 5 (c) 6 (d) 8

9.The simple interest on Rs. 2500 for 3 years at 1212%12\frac{1}{2}\%1221​% per annum is …….. (a) Rs. 900 (b) Rs. 937.50 (c) Rs. 1000 (d) Rs. 1050

10.Rama bears a loss of 12% while selling apples at Rs. 1,320. What is her cost price? (a) Rs. 1500 (b) Rs. 1450 (c) Rs. 1400 (d) Rs. 1350

11.A couple plans to have three children. What is the probability that at least two will be boys? (a) 38\frac{3}{8}83​ (b) 12\frac{1}{2}21​ (c) 28\frac{2}{8}82​ (d) 18\frac{1}{8}81​

12.How many cubes with 4 cm can be made from a cube with 16 cm in length? (a) 128 (b) 64 (c) 32 (d) 16

13.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\frac{1}{8}81​

14.What is the probability that a leap year has 53 Sundays? (a) 14\frac{1}{4}41​ (b) 13\frac{1}{3}31​ (c) 17\frac{1}{7}71​ (d) 27\frac{2}{7}72​

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

16.What is the probability for 53 Sundays in a leap year? (a) 67\frac{6}{7}76​ (b) 57\frac{5}{7}75​ (c) 27\frac{2}{7}72​ (d) 17\frac{1}{7}71​

17.Let A={1,2,3}A = \{1, 2, 3\}A={1,2,3}, the possible number of subsets for the set AAA is ………. (a) 8 (b) 7 (c) 6 (d) 3

18.The simple interest on Rs. 20,000 for 3 years at 13% per annum is …… (a) Rs. 7500 (b) Rs. 7800 (c) Rs. 8100 (d) Rs. 8500

19.Rama bears a loss of 12% while selling a mobile at Rs. 1,320, then her cost price is ……… (a) Rs. 1,500 (b) Rs. 1,550 (c) Rs. 1,575 (d) Rs. 1,600

20.If a+b+c=0a + b + c = 0a+b+c=0, then a3+b3+c3a^3 + b^3 + c^3a3+b3+c3 is equal to …… (a) abcabcabc (b) 0 (c) 3abc3abc3abc (d) 9abc9abc9abc

21.If x+1x=4x + \frac{1}{x} = 4x+x1​=4, then the value of x2+1x2x^2 + \frac{1}{x^2}x2+x21​ is …… (a) 18 (b) 14 (c) 16 (d) 20

22.If x+y−z=7x + y - z = 7x+y−z=7, x−y+z=8x - y + z = 8x−y+z=8 and −x+y+z=6-x + y + z = 6−x+y+z=6, then the average value of x,y,zx, y, zx,y,z is ……….. (a) 21 (b) 15 (c) 9 (d) 7

23.The area of a square field is 144 m2144\ \text{m}^2144 m2, then its perimeter is ……… (a) 12 m (b) 48 m (c) 144 m (d) 150 m

24.Ram does 35\frac{3}{5}53​ of a work in 12 days. He finishes the remaining work in 4 days with the help of Sanu. In how many days will Sanu do the work alone? (a) 18 days (b) 20 days (c) 24 days (d) 30 days

25.Which of the following is a rational number? (a) 2\sqrt{2}2​ (b) 14\sqrt{14}14​ (c) 24\sqrt{24}24​ (d) 36\sqrt{36}36​

26.If A⊂BA \subset BA⊂B then A∪BA \cup BA∪B is equal to ……. (a) AAA (b) BBB (c) both (a) and (b) (d) neither AAA nor BBB

27.If 2 American dollars (USD) = Rs. 156 and 2 Pounds (£) = Rs. 234, then 55 American dollars (USD) is equal to …….. (a) £ 36.67 (b) £ 63.67 (c) £ 67.36 (d) £ 67.60

28.There are 40% men and 30% women in a village. If there are 321000 children, then the total population of the village is ……… (a) 1050000 (b) 1060000 (c) 1070000 (d) 1080000

29.One side of an equilateral triangle is 4 cm, its area is ….. (a) 83 cm28\sqrt{3}\ \text{cm}^283​ cm2 (b) 163 cm216\sqrt{3}\ \text{cm}^2163​ cm2 (c) 43 cm24\sqrt{3}\ \text{cm}^243​ cm2 (d) 123 cm212\sqrt{3}\ \text{cm}^2123​ cm2

30.3245\sqrt[5]{\sqrt[4]{3^2}}5432​​ equals ……….. (a) 3103^{10}310 (b) 31103^{\frac{1}{10}}3101​ (c) 3403^{40}340 (d) 31403^{\frac{1}{40}}3401​

31.The polynomial xxx is ……… (a) a monomial (b) a binomial (c) a trinomial (d) none of these

32.In △ABC\triangle ABC△ABC, if ∠C>∠B\angle C > \angle B∠C>∠B, then ………. (a) BC>ACBC > ACBC>AC (b) AB>ACAB > ACAB>AC (c) AB<ACAB < ACAB<AC (d) BC<ACBC < ACBC<AC

33.Signs of the abscissa and ordinate of a point in the third quadrant are respectively ………. (a) +,0+, 0+,0 (b) −,−-, -−,− (c) −,+-, +−,+ (d) +,++, ++,+

34.The perpendicular distance of the point P(7,5)P(7, 5)P(7,5) from the y-axis is …… (a) 5 (b) 12 (c) 7 (d) 2

35.Which of the lines represents a line parallel to the x-axis? (a) 5x+3=45x + 3 = 45x+3=4 (b) 3y−5=y+23y - 5 = y + 23y−5=y+2 (c) x+y=7x + y = 7x+y=7 (d) x+3=0x + 3 = 0x+3=0

36.If the diagonals of a quadrilateral bisect each other at right angles, then the quadrilateral is a …… (a) rhombus (b) rectangle (c) square (d) trapezium

37.Two adjacent angles of a parallelogram are in the ratio of 4 : 5. The angles are …….. (a) 72∘,108∘72^\circ, 108^\circ72∘,108∘ (b) 84∘,96∘84^\circ, 96^\circ84∘,96∘ (c) 70∘,110∘70^\circ, 110^\circ70∘,110∘ (d) 80∘,100∘80^\circ, 100^\circ80∘,100∘

38.In figure, PQ is a diameter of circle. ∠SQP is equal to ………. (a) 40° (b) 60° (c) 50° (d) 30°

39.The mean of ten numbers is 55. If one number is excluded, their mean becomes 50. The excluded number is ……. (a) 100 (b) 80 (c) 70 (d) 60

40.The median of the data 32, 35, 39, 31, 35, 36, 33 is ……… (a) 33 (b) 35 (c) 39 (d) 36

41.From a normal pack of 52 cards, a card is drawn at random. The probability of getting a king is ……… (a) 14\frac{1}{4}41​ (b) 126\frac{1}{26}261​ (c) 113\frac{1}{13}131​ (d) 12\frac{1}{2}21​

42.What is the probability of getting a 2 or a 6 in a roll of a die? (a) 12\frac{1}{2}21​ (b) 13\frac{1}{3}31​ (c) 14\frac{1}{4}41​ (d) 16\frac{1}{6}61​

43.The value of x2x−2\sqrt{x^2\sqrt{x^{-2}}}x2x−2​​ is …….. (a) 1 (b) xxx (c) x\sqrt{x}x​ (d) x−4x^{-4}x−4

44.The value of xy−11−xy\frac{\frac{x}{y} - 1}{1 - \frac{x}{y}}1−yx​yx​−1​ is ………. (a) 0 (b) 1 (c) −1-1−1 (d) xy\frac{x}{y}yx​

45.The fourth proportional to 4, 6, 8 is …….. (a) 12 (b) 32 (c) 48 (d) 50

46.If a:b=3:4a : b = 3 : 4a:b=3:4 and b:c=1:3b : c = 1 : 3b:c=1:3, then a:c=a : c = a:c= ? (a) 1 : 4 (b) 3 : 4 (c) 4 : 1 (d) 1 : 1

47.A mobile of cost Rs. 25,000 is depreciated at 12% p.a., then what will be its value after 2 years? (a) Rs. 22,000 (b) Rs. 20,336 (c) Rs. 19,360 (d) Rs. 19,250

48.A can do a piece of work in 6 days and B does the same work in 3 days. How long will A and B take working together? (a) 4 days (b) 2 days (c) 1 day (d) 1121\frac{1}{2}121​ day

49.H.C.F. of a3−1a^3 - 1a3−1 and a4+a2+1a^4 + a^2 + 1a4+a2+1 is …….. (a) a−1a - 1a−1 (b) a2−a+1a^2 - a + 1a2−a+1 (c) a2+a+1a^2 + a + 1a2+a+1 (d) a2−a−1a^2 - a - 1a2−a−1

50.The value of 127×286217×166\frac{12^7 \times 28^6}{21^7 \times 16^6}217×166127×286​ is ………. (a) 47\frac{4}{7}74​ (b) 74\frac{7}{4}47​ (c) 34\frac{3}{4}43​ (d) 43\frac{4}{3}34​

51.If N=50N = 50N=50 and Xˉ=15\bar{X} = 15Xˉ=15 then ∑fx\sum fx∑fx is ……. (a) 600 (b) 750 (c) 500 (d) 450

52.The sum of the digits of a two digit number is 7 and their product is 12. Then the numbers are ………. (a) 52 and 25 (b) 34 and 43 (c) 61 and 16 (d) 62 and 26

53.Ananda spends 70% of his income and saves Rs. 540. What is his income? (a) Rs. 1500 (b) Rs. 1650 (c) Rs. 1750 (d) Rs. 1800

54.If the circumference of a great circle of a metallic sphere is 44 cm, then the surface area of the sphere is ……. (a) 308 cm2308\ \text{cm}^2308 cm2 (b) 416 cm2416\ \text{cm}^2416 cm2 (c) 616 cm2616\ \text{cm}^2616 cm2 (d) 710 cm2710\ \text{cm}^2710 cm2

56.One of the factors of r7+rs6r^7 + rs^6r7+rs6 is …….. (a) (r+s)(r + s)(r+s) (b) r2+s2r^2 + s^2r2+s2 (c) neither (a) nor (b) (d) both (a) and (b)

57.What is the area of a circle whose circumference is π\piπ? (a) π4\frac{\pi}{4}4π​ (b) π2\frac{\pi}{2}2π​ (c) π\piπ (d) 2π2\pi2π

58.In a race of 1 km, if A beats B by 100 m and B beats C by 50 m, then by how much does A beat C? (a) 250 m (b) 150 m (c) 147.5 m (d) 145 m

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

60.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.The surface area of a spherical ball is 616 cm2616\ \text{cm}^2616 cm2. What is its diameter? (a) 7 cm (b) 14 cm (c) 21 cm (d) 28 cm

62.In a race of 1 km, if A beats B by 50 m and B beats C by 50 m, then by how much does A beat C? (a) 90 m (b) 100 m (c) 97.5 m (d) none of these

63.The cotangents of π3,π4,π6\frac{\pi}{3}, \frac{\pi}{4}, \frac{\pi}{6}3π​,4π​,6π​ are in ……… (a) A. P. (b) G.P. (c) H.P. (d) None

64.Two vectors a⃗\vec{a}a and b⃗\vec{b}b are parallel if (a) a⃗=kb⃗\vec{a} = k\vec{b}a=kb (b) a⃗=2b⃗\vec{a} = 2\vec{b}a=2b (c) a⃗=b⃗\vec{a} = \vec{b}a=b (d) All of these

66.The equations 3x−4y=53x - 4y = 53x−4y=5 and 12x−16y=2012x - 16y = 2012x−16y=20 have …. (a) no solution (b) exactly one solution (c) infinitely many solutions (d) none of above

67.Which of the following statement is/are correct? i) every integer is a rational number. ii) every rational number is an integer. iii) a real number is either a rational or irrational number. iv) every whole number is a natural number. (a) (ii) (b) (iii) (c) (i) & (iii) (d) (i), (ii) & (iii)

68.The value of 2.999…2.999\ldots2.999… in the form of pq\frac{p}{q}qp​, where ppp and qqq are integers and q≠0q \ne 0q=0, is ……… (a) 29991000\frac{2999}{1000}10002999​ (b) 1910\frac{19}{10}1019​ (c) 3 (d) 269\frac{26}{9}926​

69.3245\sqrt[5]{\sqrt[4]{3^2}}5432​​ equals ………. (a) 3103^{10}310 (b) 13−10\frac{1}{3^{-10}}3−101​ (c) 31103^{\frac{1}{10}}3101​ (d) 3−103^{-10}3−10

70.If x11+101x^{11} + 101x11+101 is divided by x+1x + 1x+1, the remainder is ……… (a) 0 (b) −1-1−1 (c) 100 (d) 102

71.Which of the following polynomials has (x+1)(x + 1)(x+1) as a factor? (a) x4+x3+x2+x+1x^4 + x^3 + x^2 + x + 1x4+x3+x2+x+1 (b) x3+x2+x+1x^3 + x^2 + x + 1x3+x2+x+1 (c) x3+2x2−2x+1x^3 + 2x^2 - 2x + 1x3+2x2−2x+1 (d) x4+3x3+3x2+x+1x^4 + 3x^3 + 3x^2 + x + 1x4+3x3+3x2+x+1

72.A polynomial of degree 4 in yyy has at most ……… (a) 4 terms (b) 5 terms (c) 3 terms (d) 8 terms

73.If the sides of a triangle are doubled then its area ……… (a) is doubled (b) remains the same (c) becomes three times (d) becomes four times

74.In parallelogram ABCD, the bisectors of angles A and B intersect each other at O. The value of ∠AOB\angle AOB∠AOB is (a) 30∘30^\circ30∘ (b) 60∘60^\circ60∘ (c) 90∘90^\circ90∘ (d) cannot be determined

76.If sin⁡A2=45\sin\frac{A}{2} = \frac{4}{5}sin2A​=54​, then the value of cos⁡A\cos AcosA is ……. (a) 35\frac{3}{5}53​ (b) 725\frac{7}{25}257​ (c) −725\frac{-7}{25}25−7​ (d) −2125\frac{-21}{25}25−21​

77.The value of sin⁡18∘\sin 18^\circsin18∘ is ….. (a) 1+54\frac{1+\sqrt{5}}{4}41+5​​ (b) −1+54\frac{-1+\sqrt{5}}{4}4−1+5​​ (c) 1−54\frac{1-\sqrt{5}}{4}41−5​​ (d) −1+52\frac{-1+\sqrt{5}}{2}2−1+5​​

79.The value of xxx when cos⁡2x=sin⁡x\cos 2x = \sin xcos2x=sinx is …….. (a) 0∘0^\circ0∘ (b) 30∘30^\circ30∘ (c) 60∘60^\circ60∘ (d) 90∘90^\circ90∘

80.What is the value of θ\thetaθ when cos⁡2θ2−cos⁡θ2+14=0\cos^2\frac{\theta}{2} - \cos\frac{\theta}{2} + \frac{1}{4} = 0cos22θ​−cos2θ​+41​=0? (a) 60∘60^\circ60∘ (b) 90∘90^\circ90∘ (c) 120∘120^\circ120∘ (d) 150∘150^\circ150∘

82.A circle passes through (0,0)(0, 0)(0,0), (a,0)(a, 0)(a,0) and (0,b)(0, b)(0,b). The co-ordinate of the centre is ……. (a) (a,b)(a, b)(a,b) (b) (b,a)(b, a)(b,a) (c) (−a,−b)(-a, -b)(−a,−b) (d) (a2,b2)\left(\frac{a}{2}, \frac{b}{2}\right)(2a​,2b​)

83.The value of kkk such that the equation 2x2+2y2−6x+8y+k=02x^2 + 2y^2 - 6x + 8y + k = 02x2+2y2−6x+8y+k=0 represents a point circle is …… (a) 0 (b) 25 (c) 252\frac{25}{2}225​ (d) −252\frac{-25}{2}2−25​

84.The general equation ax2+2hxy+by2+2gx+2fy+c=0ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0ax2+2hxy+by2+2gx+2fy+c=0 represents a circle if ……. (a) a=b=ha = b = ha=b=h (b) a=ba = ba=b and h=0h = 0h=0 (c) g=f=cg = f = cg=f=c (d) a=ba = ba=b and g=fg = fg=f

If the surface area of a spherical ball is $616\ \text{cm}^2$ , then its diameter is ………… (a) 14 cm (b) 7 cm (c) 21 cm (d) 28 cm

The greatest number of 3 digits, which is a perfect square is ……… (a) 981 (b) 999 (c) 951 (d) 961

If the side of a rhombus is 10 cm and one diagonal is 16 cm, the area of the rhombus is ……… (a) $24\ \text{cm}^2$ (b) $48\ \text{cm}^2$ (c) $96\ \text{cm}^2$ (d) $90\ \text{cm}^2$

The surface area of a ball is $16\pi\ \text{cm}^2$ , then what is its diameter? (a) 8 cm (b) 4 cm (c) 2 cm (d) 1 cm

The side of a rhombus is 10 cm and one diagonal is 16 cm. The area of the rhombus is ………. (a) $24\ \text{cm}^2$ (b) $48\ \text{cm}^2$ (c) $96\ \text{cm}^2$ (d) $90\ \text{cm}^2$

The greatest number of 4 digits, which is a perfect square is …….. (a) 9996 (b) 9801 (c) 9900 (d) 9604

The quadratic equation whose roots are $\frac{1}{2}$ and $-\frac{1}{2}$ is …….. (a) $x^2 - 4 = 0$ (b) $4x^2 - 1 = 0$ (c) $4x^2 + 1 = 0$ (d) $x^2 - \frac{1}{2} = 0$

Let $A = \{a, b, c\}$ , the possible number of subsets for the set $A$ is …… (a) 3 (b) 5 (c) 6 (d) 8

The simple interest on Rs. 2500 for 3 years at $12\frac{1}{2}\%$ per annum is …….. (a) Rs. 900 (b) Rs. 937.50 (c) Rs. 1000 (d) Rs. 1050

Rama bears a loss of 12% while selling apples at Rs. 1,320. What is her cost price? (a) Rs. 1500 (b) Rs. 1450 (c) Rs. 1400 (d) Rs. 1350

A couple plans to have three children. What is the probability that at least two will be boys? (a) $\frac{3}{8}$ (b) $\frac{1}{2}$ (c) $\frac{2}{8}$ (d) $\frac{1}{8}$

How many cubes with 4 cm can be made from a cube with 16 cm in length? (a) 128 (b) 64 (c) 32 (d) 16

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) $\frac{1}{8}$

What is the probability that a leap year has 53 Sundays? (a) $\frac{1}{4}$ (b) $\frac{1}{3}$ (c) $\frac{1}{7}$ (d) $\frac{2}{7}$

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

What is the probability for 53 Sundays in a leap year? (a) $\frac{6}{7}$ (b) $\frac{5}{7}$ (c) $\frac{2}{7}$ (d) $\frac{1}{7}$

Let $A = \{1, 2, 3\}$ , the possible number of subsets for the set $A$ is ………. (a) 8 (b) 7 (c) 6 (d) 3

The simple interest on Rs. 20,000 for 3 years at 13% per annum is …… (a) Rs. 7500 (b) Rs. 7800 (c) Rs. 8100 (d) Rs. 8500

Rama bears a loss of 12% while selling a mobile at Rs. 1,320, then her cost price is ……… (a) Rs. 1,500 (b) Rs. 1,550 (c) Rs. 1,575 (d) Rs. 1,600

If $a + b + c = 0$ , then $a^3 + b^3 + c^3$ is equal to …… (a) $abc$ (b) 0 (c) $3abc$ (d) $9abc$

If $x + \frac{1}{x} = 4$ , then the value of $x^2 + \frac{1}{x^2}$ is …… (a) 18 (b) 14 (c) 16 (d) 20

If $x + y - z = 7$ , $x - y + z = 8$ and $-x + y + z = 6$ , then the average value of $x, y, z$ is ……….. (a) 21 (b) 15 (c) 9 (d) 7

The area of a square field is $144\ \text{m}^2$ , then its perimeter is ……… (a) 12 m (b) 48 m (c) 144 m (d) 150 m

Ram does $\frac{3}{5}$ of a work in 12 days. He finishes the remaining work in 4 days with the help of Sanu. In how many days will Sanu do the work alone? (a) 18 days (b) 20 days (c) 24 days (d) 30 days

Which of the following is a rational number? (a) $\sqrt{2}$ (b) $\sqrt{14}$ (c) $\sqrt{24}$ (d) $\sqrt{36}$

If $A \subset B$ then $A \cup B$ is equal to ……. (a) $A$ (b) $B$ (c) both (a) and (b) (d) neither $A$ nor $B$

If 2 American dollars (USD) = Rs. 156 and 2 Pounds (£) = Rs. 234, then 55 American dollars (USD) is equal to …….. (a) £ 36.67 (b) £ 63.67 (c) £ 67.36 (d) £ 67.60

There are 40% men and 30% women in a village. If there are 321000 children, then the total population of the village is ……… (a) 1050000 (b) 1060000 (c) 1070000 (d) 1080000

One side of an equilateral triangle is 4 cm, its area is ….. (a) $8\sqrt{3}\ \text{cm}^2$ (b) $16\sqrt{3}\ \text{cm}^2$ (c) $4\sqrt{3}\ \text{cm}^2$ (d) $12\sqrt{3}\ \text{cm}^2$

$\sqrt[5]{\sqrt[4]{3^2}}$ equals ……….. (a) $3^{10}$ (b) $3^{\frac{1}{10}}$ (c) $3^{40}$ (d) $3^{\frac{1}{40}}$

The polynomial $x$ is ……… (a) a monomial (b) a binomial (c) a trinomial (d) none of these

In $\triangle ABC$ , if $\angle C > \angle B$ , then ………. (a) $BC > AC$ (b) $AB > AC$ (c) $AB < AC$ (d) $BC < AC$

Signs of the abscissa and ordinate of a point in the third quadrant are respectively ………. (a) $+, 0$ (b) $-, -$ (c) $-, +$ (d) $+, +$

The perpendicular distance of the point $P(7, 5)$ from the y-axis is …… (a) 5 (b) 12 (c) 7 (d) 2

Which of the lines represents a line parallel to the x-axis? (a) $5x + 3 = 4$ (b) $3y - 5 = y + 2$ (c) $x + y = 7$ (d) $x + 3 = 0$

If the diagonals of a quadrilateral bisect each other at right angles, then the quadrilateral is a …… (a) rhombus (b) rectangle (c) square (d) trapezium

Two adjacent angles of a parallelogram are in the ratio of 4 : 5. The angles are …….. (a) $72^\circ, 108^\circ$ (b) $84^\circ, 96^\circ$ (c) $70^\circ, 110^\circ$ (d) $80^\circ, 100^\circ$

In figure, PQ is a diameter of circle. ∠SQP is equal to ………. (a) 40° (b) 60° (c) 50° (d) 30°

The mean of ten numbers is 55. If one number is excluded, their mean becomes 50. The excluded number is ……. (a) 100 (b) 80 (c) 70 (d) 60

The median of the data 32, 35, 39, 31, 35, 36, 33 is ……… (a) 33 (b) 35 (c) 39 (d) 36

From a normal pack of 52 cards, a card is drawn at random. The probability of getting a king is ……… (a) $\frac{1}{4}$ (b) $\frac{1}{26}$ (c) $\frac{1}{13}$ (d) $\frac{1}{2}$

What is the probability of getting a 2 or a 6 in a roll of a die? (a) $\frac{1}{2}$ (b) $\frac{1}{3}$ (c) $\frac{1}{4}$ (d) $\frac{1}{6}$

The value of $\sqrt{x^2\sqrt{x^{-2}}}$ is …….. (a) 1 (b) $x$ (c) $\sqrt{x}$ (d) $x^{-4}$

The value of $\frac{\frac{x}{y} - 1}{1 - \frac{x}{y}}$ is ………. (a) 0 (b) 1 (c) $-1$ (d) $\frac{x}{y}$

The fourth proportional to 4, 6, 8 is …….. (a) 12 (b) 32 (c) 48 (d) 50

If $a : b = 3 : 4$ and $b : c = 1 : 3$ , then $a : c =$ ? (a) 1 : 4 (b) 3 : 4 (c) 4 : 1 (d) 1 : 1

A mobile of cost Rs. 25,000 is depreciated at 12% p.a., then what will be its value after 2 years? (a) Rs. 22,000 (b) Rs. 20,336 (c) Rs. 19,360 (d) Rs. 19,250

A can do a piece of work in 6 days and B does the same work in 3 days. How long will A and B take working together? (a) 4 days (b) 2 days (c) 1 day (d) $1\frac{1}{2}$ day

H.C.F. of $a^3 - 1$ and $a^4 + a^2 + 1$ is …….. (a) $a - 1$ (b) $a^2 - a + 1$ (c) $a^2 + a + 1$ (d) $a^2 - a - 1$

The value of $\frac{12^7 \times 28^6}{21^7 \times 16^6}$ is ………. (a) $\frac{4}{7}$ (b) $\frac{7}{4}$ (c) $\frac{3}{4}$ (d) $\frac{4}{3}$

If $N = 50$ and $\bar{X} = 15$ then $\sum fx$ is ……. (a) 600 (b) 750 (c) 500 (d) 450

The sum of the digits of a two digit number is 7 and their product is 12. Then the numbers are ………. (a) 52 and 25 (b) 34 and 43 (c) 61 and 16 (d) 62 and 26

Ananda spends 70% of his income and saves Rs. 540. What is his income? (a) Rs. 1500 (b) Rs. 1650 (c) Rs. 1750 (d) Rs. 1800

If the circumference of a great circle of a metallic sphere is 44 cm, then the surface area of the sphere is ……. (a) $308\ \text{cm}^2$ (b) $416\ \text{cm}^2$ (c) $616\ \text{cm}^2$ (d) $710\ \text{cm}^2$

One of the factors of $r^7 + rs^6$ is …….. (a) $(r + s)$ (b) $r^2 + s^2$ (c) neither (a) nor (b) (d) both (a) and (b)

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

In a race of 1 km, if A beats B by 100 m and B beats C by 50 m, then by how much does A beat C? (a) 250 m (b) 150 m (c) 147.5 m (d) 145 m

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

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

The surface area of a spherical ball is $616\ \text{cm}^2$ . What is its diameter? (a) 7 cm (b) 14 cm (c) 21 cm (d) 28 cm

In a race of 1 km, if A beats B by 50 m and B beats C by 50 m, then by how much does A beat C? (a) 90 m (b) 100 m (c) 97.5 m (d) none of these

The cotangents of $\frac{\pi}{3}, \frac{\pi}{4}, \frac{\pi}{6}$ are in ……… (a) A. P. (b) G.P. (c) H.P. (d) None

Two vectors $\vec{a}$ and $\vec{b}$ are parallel if (a) $\vec{a} = k\vec{b}$ (b) $\vec{a} = 2\vec{b}$ (c) $\vec{a} = \vec{b}$ (d) All of these

The equations $3x - 4y = 5$ and $12x - 16y = 20$ have …. (a) no solution (b) exactly one solution (c) infinitely many solutions (d) none of above

Which of the following statement is/are correct? i) every integer is a rational number. ii) every rational number is an integer. iii) a real number is either a rational or irrational number. iv) every whole number is a natural number. (a) (ii) (b) (iii) (c) (i) & (iii) (d) (i), (ii) & (iii)

The value of $2.999\ldots$ in the form of $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \ne 0$ , is ……… (a) $\frac{2999}{1000}$ (b) $\frac{19}{10}$ (c) 3 (d) $\frac{26}{9}$

$\sqrt[5]{\sqrt[4]{3^2}}$ equals ………. (a) $3^{10}$ (b) $\frac{1}{3^{-10}}$ (c) $3^{\frac{1}{10}}$ (d) $3^{-10}$

If $x^{11} + 101$ is divided by $x + 1$ , the remainder is ……… (a) 0 (b) $-1$ (c) 100 (d) 102

Which of the following polynomials has $(x + 1)$ as a factor? (a) $x^4 + x^3 + x^2 + x + 1$ (b) $x^3 + x^2 + x + 1$ (c) $x^3 + 2x^2 - 2x + 1$ (d) $x^4 + 3x^3 + 3x^2 + x + 1$

A polynomial of degree 4 in $y$ has at most ……… (a) 4 terms (b) 5 terms (c) 3 terms (d) 8 terms

If the sides of a triangle are doubled then its area ……… (a) is doubled (b) remains the same (c) becomes three times (d) becomes four times

In parallelogram ABCD, the bisectors of angles A and B intersect each other at O. The value of $\angle AOB$ is (a) $30^\circ$ (b) $60^\circ$ (c) $90^\circ$ (d) cannot be determined

If $\sin\frac{A}{2} = \frac{4}{5}$ , then the value of $\cos A$ is ……. (a) $\frac{3}{5}$ (b) $\frac{7}{25}$ (c) $\frac{-7}{25}$ (d) $\frac{-21}{25}$

The value of $\sin 18^\circ$ is ….. (a) $\frac{1+\sqrt{5}}{4}$ (b) $\frac{-1+\sqrt{5}}{4}$ (c) $\frac{1-\sqrt{5}}{4}$ (d) $\frac{-1+\sqrt{5}}{2}$

The value of $x$ when $\cos 2x = \sin x$ is …….. (a) $0^\circ$ (b) $30^\circ$ (c) $60^\circ$ (d) $90^\circ$

What is the value of $\theta$ when $\cos^2\frac{\theta}{2} - \cos\frac{\theta}{2} + \frac{1}{4} = 0$ ? (a) $60^\circ$ (b) $90^\circ$ (c) $120^\circ$ (d) $150^\circ$

A circle passes through $(0, 0)$ , $(a, 0)$ and $(0, b)$ . The co-ordinate of the centre is ……. (a) $(a, b)$ (b) $(b, a)$ (c) $(-a, -b)$ (d) $\left(\frac{a}{2}, \frac{b}{2}\right)$

The value of $k$ such that the equation $2x^2 + 2y^2 - 6x + 8y + k = 0$ represents a point circle is …… (a) 0 (b) 25 (c) $\frac{25}{2}$ (d) $\frac{-25}{2}$

The general equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a circle if ……. (a) $a = b = h$ (b) $a = b$ and $h = 0$ (c) $g = f = c$ (d) $a = b$ and $g = f$

The value of $m$ such that $mx^2 - 8xy + 8y^2 = 0$ represents two coincident straight lines is ……. (a) $-1$ (b) $-3$ (c) 4 (d) 2