CTEVT diploma and PCL entrance Diploma and PCL Entrance (all programmes): Mathematics

2079 · 1 mark

What is the total surface area of a cone when the base radius of the cone is 7 cm and the sum of the radius and slant height is 28 cm? (a) 231 cm² (b) 400 cm² (c) 462 cm² (d) 616 cm²

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Answer

(d) 616 cm²

Given $r = 7$ cm and $r + l = 28$ cm.

$\text{TSA} = \pi r(r + l) = \frac{22}{7} \times 7 \times 28 = 616 \text{ cm}^2$

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2075 · 1 mark

If the total surface area of the cone is $704\ cm^2$ and its base radius is 7 cm, the slant height of the cone is (a) 24 cm (b) 22 cm (c) 25 cm (d) 44 cm

Similar questions: What is the total surface area (2079)

Answer

(c) 25 cm Total surface area $=\pi r(r+l)$ : $\frac{22}{7}\times 7\,(7+l)=704 \Rightarrow 22(7+l)=704 \Rightarrow 7+l=32 \Rightarrow l=25\text{ cm}$

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2079 · 1 mark

The area of triangle XYZ is 45 cm². If $XY = 12$ cm and $YZ = 15$ cm, what will be the value of $\angle XYZ$ ? (a) 30° (b) 45° (c) 60° (d) 90°

Answer

(a) 30°

$\text{Area} = \tfrac{1}{2}\, XY \cdot YZ \sin\angle XYZ$

$45 = \tfrac{1}{2} \times 12 \times 15 \times \sin\angle XYZ = 90\sin\angle XYZ$

So $\sin\angle XYZ = \frac{1}{2}$ and $\angle XYZ = 30°$ .

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2079 · 1 mark

If $x = 10^a$ , $y = 10^b$ and $x^b y^a = 100$ , what is the value of $ab$ ? (a) 1 (b) $\sqrt{2}$ (c) 2 (d) 10

Answer

(a) 1

$x^b y^a = (10^a)^b (10^b)^a = 10^{ab} \cdot 10^{ab} = 10^{2ab}$

Since this equals $100 = 10^2$ , $2ab = 2$ , so $ab = 1$ .

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2079 · 1 mark

If $\sqrt{3m+1} - 2 = 0$ , what is the value of $m$ ? (a) 0 (b) -1 (c) 1 (d) 4

Answer

(c) 1

$\sqrt{3m+1} = 2 \Rightarrow 3m + 1 = 4 \Rightarrow m = 1$

Check: $\sqrt{4} - 2 = 0$ .

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2079 · 1 mark

If $m + n = 7$ and $2m - 3n = 4$ , what will be the value of $m - n$ ? (a) 2.5 (b) 5 (c) 3 (d) 5.2

Answer

(c) 3

From $m + n = 7$ , $m = 7 - n$ . Then $2(7 - n) - 3n = 4$ gives $14 - 5n = 4$ , so $n = 2$ and $m = 5$ .

$m - n = 5 - 2 = 3$

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2079 · 1 mark

6 black and 7 red marbles of the same size and shape are kept in a bag. Two marbles are drawn one at a time without replacement. What is the probability of getting a red marble at first and then a black marble? (a) $\frac{7}{26}$ (b) $\frac{27}{26}$ (c) $\frac{78}{84}$ (d) $\frac{9}{26}$

Answer

(a) $\frac{7}{26}$

Total marbles = 13. Red first: $\frac{7}{13}$ ; then black from the remaining 12: $\frac{6}{12}$ .

$P = \frac{7}{13} \times \frac{6}{12} = \frac{42}{156} = \frac{7}{26}$

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2079 · 1 mark

If 10 is added to the number that is seven times the square of a natural number, the result is 353. What will be the number? (a) $\pm 7$ (b) -7 (c) 7 (d) 9

Answer

(c) 7

Let the natural number be $x$ . Then $7x^2 + 10 = 353$ , so $7x^2 = 343$ and $x^2 = 49$ , giving $x = \pm 7$ . Since the number is natural, $x = 7$ .

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2079 · 1 mark

The area of an equilateral triangle is $9\sqrt{3}$ cm². What is the perimeter of the triangle? (a) 6 cm (b) 9 cm (c) 18 cm (d) $27\sqrt{3}$ cm

Answer

(c) 18 cm

$\frac{\sqrt{3}}{4}a^2 = 9\sqrt{3} \Rightarrow a^2 = 36 \Rightarrow a = 6 \text{ cm}$

$\text{Perimeter} = 3a = 18 \text{ cm}$

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2079 · 1 mark

What is the result when $\frac{x^2 - xy + y^2}{x - y} - \frac{x^2 + xy + y^2}{x + y}$ is simplified? (a) $2(x^3 + x^3)$ (b) $\frac{2x^3}{x^2 - y^2}$ (c) $\frac{2y^3}{x^2 - y^2}$ (d) $\frac{(2y)^3}{x^2 - y^2}$

Answer

(c) $\dfrac{2y^3}{x^2 - y^2}$

Take the common denominator $(x-y)(x+y) = x^2 - y^2$ :

$\frac{(x^2 - xy + y^2)(x+y) - (x^2 + xy + y^2)(x-y)}{x^2 - y^2} = \frac{(x^3 + y^3) - (x^3 - y^3)}{x^2 - y^2} = \frac{2y^3}{x^2 - y^2}$

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2079 · 1 mark

What is the highest common factor (H.C.F.) of two algebraic expressions $(x+2)$ and $(x-2)$ ? (a) $x + 2$ (b) $x - 2$ (c) $x^2 - 4$ (d) 1

Answer

(d) 1

$(x+2)$ and $(x-2)$ are different linear factors with no common factor other than 1, so their H.C.F. is 1.

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2079 · 1 mark

What is the order of the surd $3\sqrt[4]{7}$ ? (a) 1 (b) 2 (c) 3 (d) 4

Answer

(d) 4

The order of a surd $\sqrt[n]{a}$ is $n$ . In $3\sqrt[4]{7}$ the root index is 4, so the order is 4.

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2079 · 1 mark

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

Answer

(a) $x - 1$

$x^2-1=(x-1)(x+1),\qquad x^2-3x+2=(x-1)(x-2)$

The only common factor is $x-1$ , so it is the H.C.F.

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2079 · 1 mark

If $A = \{2, 3, 5\}$ and $B = \{1, 4\}$ , what is the value of $n(A \cap B)$ ? (a) $\theta$ (b) $\{\ \}$ (c) 0 (d) 5

Answer

(c) 0

$A=\{2,3,5\}$ and $B=\{1,4\}$ have no common element, so $A\cap B=\varnothing$ and $n(A\cap B)=0$ .

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2079 · 1 mark

How many different values of the variable of unit place digit in a two digit number? (a) 1 (b) 9 (c) 10 (d) $y$

Answer

(c) 10

The unit place digit of a two-digit number can be any of $0,1,2,\dots,9$ , which gives 10 different values.

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2079 · 1 mark

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

Answer

(c) 1

$2^{x+1}=4=2^2 \Rightarrow x+1=2 \Rightarrow x=1$

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2079 · 1 mark

Which one is the formula to calculate the curved surface area of a cylinder? (a) $\pi r h$ (b) $2\pi r h$ (c) $2\pi r^2 h$ (d) $2\pi(r + h)$

Answer

(b) $2\pi r h$

Unrolling the curved surface of a cylinder gives a rectangle of length $2\pi r$ (the circumference) and width $h$ , so the curved surface area is $2\pi r h$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

What is the relation between the area of a square and a triangle standing on the same base and between the same parallel lines? (a) Area of square = area of triangle (b) Area of square = $\frac{1}{2}$ area of triangle (c) Area of square = 2 area of triangle (d) Area of triangle = 2 area of square

Answer

(c) Area of square = 2 area of triangle

Let the common base be $b$ and the height between the parallels be $h$ . The area of the triangle is $\frac{1}{2}bh$ and the area of the parallelogram (square or rectangle) is $bh$ , which is twice the triangle. So area of square = 2 area of triangle.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

The marked price of an article is Rs. $x$ . If the price of the article after discount is Rs. $y$ , what percent discount is provided? (a) $\frac{y}{x} \times 100\%$ (b) $\frac{x}{y} \times 100\%$ (c) $\frac{x-y}{x}\%$ (d) $\frac{x-y}{x} \times 100\%$

Answer

(d) $\frac{x-y}{x} \times 100\%$

Discount $= x-y$ , and discount percent is taken on the marked price, so it is $\frac{x-y}{x}\times 100\%$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Paying · 1 mark

If $X = \{1, 4, 6\}$ and $Y = \{2, 5\}$ , what is the value of $n(X \cup Y)$ ? (a) 0 (b) 2 (c) 3 (d) 5

Answer

(d) 5 $X \cup Y = \{1, 2, 4, 5, 6\}$ , and the sets have no common element, so $n(X \cup Y) = 3 + 2 = 5$ .

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2079 Paying · 1 mark

The present population of a town is $X$ . If the population of the town increases every year by $R\%$ , what will be the population of the town after $Y$ years? (a) $X\left(1 + \frac{R}{100}\right)^T$ (b) $X\left(1 - \frac{R}{100}\right)^T$ (c) $X\left(1 + \frac{R}{100}\right)^Y$ (d) $X\left(1 + \frac{R}{100}\right)^R$

Answer

(c) $X\left(1 + \frac{R}{100}\right)^Y$ For growth at $R\%$ per year, population after $Y$ years is $P = X\left(1 + \frac{R}{100}\right)^Y$ . Option (a) uses $T$ and (d) uses $R$ , neither of which is the number of years.

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2079 Paying · 1 mark

How much more is Rs. 6,000 including 13% VAT on it than Rs. 6,500? (a) Rs. 200 (b) Rs. 280 (c) Rs. 300 (d) Rs. 380

Answer

(b) Rs. 280

Price with VAT: $6000 \times 1.13 = 6780$ .

Difference: $6780 - 6500 = \text{Rs. } 280$ .

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2079 Paying · 1 mark

The present price of a television is Rs. 35,000. After how many years will it be Rs. 25,515 if its price reduces by 10% per annum? (a) 2 years (b) $2\frac{1}{2}$ years (c) 3 years (d) 4 years

Answer

(c) 3 years

Using $A = P\left(1 - \frac{r}{100}\right)^n$ :

$25515 = 35000(0.9)^n \Rightarrow (0.9)^n = 0.729 = (0.9)^3$

So $n = 3$ years.

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2079 Paying · 1 mark

What is a triangle whose three sides are 4 cm, 6 cm and 12 cm respectively? (a) Acute angled triangle (b) Equilateral triangle (c) Obtuse angle triangle (d) No triangle

Answer

(d) No triangle

A triangle exists only if the sum of any two sides exceeds the third. Here $4 + 6 = 10 < 12$ , so these lengths cannot form a triangle.

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2079 Paying · 1 mark

What is the vertical height of a square based pyramid having its base area $64\ cm^2$ and volume $640\ cm^3$ ? (a) 5 cm (b) 8 cm (c) 10 cm (d) 30 cm

Answer

(d) 30 cm

$V = \frac{1}{3} \times \text{base area} \times h \Rightarrow 640 = \frac{1}{3} \times 64 \times h$

$h = \frac{1920}{64} = 30 \text{ cm}$

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2079 Paying · 1 mark

The total surface area of a hemisphere is $243\pi\ cm^2$ . What will be its diameter? (a) 9 cm (b) 18 cm (c) 27 cm (d) 30 cm

Answer

(b) 18 cm

Total surface area of a hemisphere $= 3\pi r^2$ , so $3\pi r^2 = 243\pi \Rightarrow r^2 = 81 \Rightarrow r = 9$ cm.

Diameter $= 2r = 18$ cm.

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2079 Paying · 1 mark

What is the L.C.M. of two algebraic expressions $(x+1)$ and $(x^2 - 1)$ ? (a) $x + 1$ (b) $x^2 - 1$ (c) $(x^2 - 1)(x + 1)$ (d) $x^3 - 1$

Answer

(b) $x^2 - 1$

Since $x^2 - 1 = (x+1)(x-1)$ , it already contains the factor $(x+1)$ . So the L.C.M. of $(x+1)$ and $(x+1)(x-1)$ is $(x+1)(x-1) = x^2 - 1$ .

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2079 Paying · 1 mark

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

Answer

(a) $x + 1$

$x^2 - 1 = (x+1)(x-1)$ and $x^2 + 3x + 2 = (x+1)(x+2)$ .

The only common factor is $x + 1$ .

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2079 Paying · 1 mark

Which one is a quadratic equation in which the values of $x$ are 2 and $-1$ ? (a) $x^2 + x - 2 = 0$ (b) $x^2 - x + 2 = 0$ (c) $x^2 - x - 2 = 0$ (d) $2x^2 - x - 1 = 0$

Answer

(c) $x^2 - x - 2 = 0$

Sum of roots $= 2 + (-1) = 1$ , product $= 2 \times (-1) = -2$ .

Equation: $x^2 - (\text{sum})x + \text{product} = x^2 - x - 2 = 0$ . Check: $(x-2)(x+1) = x^2 - x - 2$ .

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2079 Paying · 1 mark

If the sum of a positive number and its square is 30, what is the number? (a) 3 (b) 5 (c) $\pm 5$ (d) 6

Answer

(b) 5

Let the number be $x$ : $x + x^2 = 30 \Rightarrow x^2 + x - 30 = 0 \Rightarrow (x+6)(x-5) = 0$ .

So $x = 5$ or $-6$ ; the number is positive, so $x = 5$ .

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2079 Paying · 1 mark

What is the relation between the areas of the triangles standing on the same base and between the same parallel lines? (a) double (b) unequal (c) equal (d) half one

Answer

(c) equal

Triangles on the same base and between the same parallel lines have the same base and the same height, so their areas are equal.

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2079 Paying · 1 mark

In $\triangle PQR$ , $PQ = 7$ cm, $QR = 16$ cm and the area of $\triangle PQR$ is $28\sqrt{3}\ cm^2$ . What is the measure of $\angle PQR$ ? (a) 30° (b) 45° (c) 60° (d) 75°

Answer

(c) 60°

$\text{Area} = \frac{1}{2} \cdot PQ \cdot QR \sin Q \Rightarrow 28\sqrt{3} = \frac{1}{2}(7)(16)\sin Q = 56 \sin Q$

$\sin Q = \frac{\sqrt{3}}{2} \Rightarrow \angle PQR = 60°$

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2079 Paying · 1 mark

What is the angle of elevation when a man observes the top of a tower of height $80\sqrt{3}$ m from 240 m from the foot of the tower? (a) 30° (b) 45° (c) 60° (d) 75°

Answer

(a) 30°

$\tan\theta = \frac{\text{height}}{\text{distance}} = \frac{80\sqrt{3}}{240} = \frac{\sqrt{3}}{3} = \frac{1}{\sqrt{3}}$

So $\theta = 30°$ .

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2079 Paying · 1 mark

The third quartile is calculated by the formula ........ (a) $L + \frac{\frac{N}{3} - c.f}{f} \times i$ (b) $L + \frac{\frac{N}{4} - c.f}{f} \times i$ (c) $L + \frac{\frac{3N}{4} - c.f}{f} \times i$ (d) $L + \frac{\frac{3N}{4} + c.f}{f} \times i$

Answer

(c) $L + \frac{\frac{3N}{4} - c.f}{f} \times i$

The third quartile $Q_3$ is the value at the $\frac{3N}{4}$ th item; $L$ is the lower limit of the $Q_3$ class, $c.f$ the cumulative frequency before it, $f$ its frequency and $i$ its class width.

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2079 Paying · 1 mark

Two players, A and B play a tennis match. If the probability of winning the match by A is $\frac{1}{3}$ , what is the probability of winning the match by B? (a) $\frac{2}{3}$ (b) $\frac{1}{3}$ (c) $\frac{5}{6}$ (d) $\frac{1}{6}$

Answer

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

In a match one player wins and the other loses (no draw), so $P(B) = 1 - P(A) = 1 - \frac{1}{3} = \frac{2}{3}$ .

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2076 · 1 mark

If $2\sec 2\alpha = \tan\beta + \cot\beta$ , then the value of $(\alpha + \beta)$ = (a) $\frac{\pi}{2}$ (b) $\frac{\pi}{3}$ (c) $\frac{\pi}{4}$ (d) $\frac{\pi}{6}$

Answer

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

$\tan\beta+\cot\beta=\dfrac{1}{\sin\beta\cos\beta}=\dfrac{2}{\sin 2\beta}$ , so $\sec 2\alpha=\dfrac{1}{\sin 2\beta}$ , i.e. $\cos 2\alpha=\sin 2\beta$ .

Hence $2\alpha+2\beta=\dfrac{\pi}{2}$ , so $\alpha+\beta=\dfrac{\pi}{4}$ .

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2076 · 1 mark

If $(A + B) = \frac{\pi}{4}$ , then $(\cot A - 1)(\cot B - 1)$ = (a) 1 (b) 12 (c) 2 (d) $-1$

Answer

(c) 2

Since $\cot(A+B)=1$ , we have $\dfrac{\cot A\cot B-1}{\cot A+\cot B}=1$ , so $\cot A\cot B-1=\cot A+\cot B$ .

Then $(\cot A-1)(\cot B-1)=\cot A\cot B-(\cot A+\cot B)+1=1+1=2$ .

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2076 · 1 mark

In $\triangle ABC$ , if $2\cos A = \frac{\sin B}{\sin C}$ , then the $\triangle$ is: (a) Right angled triangle (b) Equilateral triangle (c) Isosceles triangle (d) Scalene triangle

Answer

(c) Isosceles triangle

$2\cos A\sin C=\sin B=\sin(A+C)=\sin A\cos C+\cos A\sin C$ .

So $\cos A\sin C-\sin A\cos C=0$ , i.e. $\sin(C-A)=0$ , giving $C=A$ . Two angles are equal, so the triangle is isosceles.

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2076 · 1 mark

The number of non-empty proper subsets of set $A = \{1, 2, 3, 4\}$ is: (a) 16 (b) 14 (c) 15 (d) 13

Answer

(b) 14

A set with 4 elements has $2^4=16$ subsets. Removing the empty set and the set itself gives $16-2=14$ non-empty proper subsets.

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2076 · 1 mark

If $f(x + 1) = x + 2$ , then $f(x - 1)$ = (a) 2 (b) $x$ (c) $x + 1$ (d) $x - 1$

Answer

(b) $x$

Put $t=x+1$ : $f(t)=t-1+2=t+1$ . So $f(x-1)=(x-1)+1=x$ .

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2076 · 1 mark

For any two square matrices A and B, which of the following is not true? (a) $(A + B)^T = A^T + B^T$ (b) $(AB)^T = B^T A^T$ (c) $A(B + C) = AB + AC$ (d) $(A - B)^T = B^T - A^T$

Answer

(d) $(A-B)^T = B^T - A^T$

The correct rule is $(A-B)^T=A^T-B^T$ , so the statement in (d) is not true. The other three are standard properties.

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2076 · 1 mark

If $AB = C$ , then the order of matrices A, B and C are: (a) $A_{2\times3}$ , $B_{2\times3}$ , $C_{2\times3}$ (b) $A_{3\times2}$ , $B_{2\times3}$ , $C_{3\times2}$ (c) $A_{3\times3}$ , $B_{3\times2}$ , $C_{3\times2}$ (d) $A_{3\times2}$ , $B_{2\times3}$ , $C_{3\times3}$

Answer

(d) $A_{3\times2}$ , $B_{2\times3}$ , $C_{3\times3}$

For $AB$ to exist, the columns of $A$ must equal the rows of $B$ , and the product has the rows of $A$ and the columns of $B$ . Here $(3\times2)(2\times3)=3\times3$ .

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2076 · 1 mark

If $\frac{1}{q+r}, \frac{1}{r+p}, \frac{1}{p+q}$ are in AP, then: (a) $p^2, q^2, r^2$ are in AP (b) $p, q, r$ are in GP (c) $p^2, q^2, r^2$ are in GP (d) $p^2, q^2, r^2$ are in HP

Answer

(a) $p^2, q^2, r^2$ are in AP

Since the terms are in AP, $\dfrac{1}{r+p}-\dfrac{1}{q+r}=\dfrac{1}{p+q}-\dfrac{1}{r+p}$ , so $\dfrac{q-p}{(r+p)(q+r)}=\dfrac{r-q}{(p+q)(r+p)}$ .

Cancelling $(r+p)$ gives $(q-p)(p+q)=(r-q)(q+r)$ , i.e. $q^2-p^2=r^2-q^2$ . Hence $p^2,q^2,r^2$ are in AP.

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2076 · 1 mark

If $m^{th}$ term of an AP is $n$ and $n^{th}$ term is $m$ , then $r^{th}$ term is: (a) $m - n - r$ (b) $m + n - r$ (c) $m - n + r$ (d) $n - r - m$

Answer

(b) $m+n-r$

$a+(m-1)d=n$ and $a+(n-1)d=m$ . Subtracting: $(m-n)d=n-m$ , so $d=-1$ and $a=m+n-1$ .

$T_r=a+(r-1)d=m+n-1-(r-1)=m+n-r$ .

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2076 · 1 mark

If $\alpha$ and $\beta$ are roots of the equation $x^2 - cp(x + 1) - c = 0$ then $(1 + \alpha)(1 + \beta)$ = (a) $1 + p$ (b) $1 - p$ (c) $1 + c$ (d) $1 - c$

Answer

(d) $1-c$

The equation is $x^2-cpx-(cp+c)=0$ , so $\alpha+\beta=cp$ and $\alpha\beta=-(cp+c)$ .

$(1+\alpha)(1+\beta)=1+(\alpha+\beta)+\alpha\beta=1+cp-cp-c=1-c$ .

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2076 · 1 mark

If 7 is added to the square of a natural number then the result is 43. What is the number? (a) $\sqrt{50}$ (b) $\pm 7$ (c) $\pm 6$ (d) 6

Answer

(d) 6

Let the number be $n$ : $n^2+7=43$ , so $n^2=36$ and $n=\pm6$ . Since the number is natural, $n=6$ .

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2076 · 1 mark

The length of three sides of a triangle are $x$ cm, $y$ cm, and $z$ cm respectively. What is the area of the triangle? (a) $xyz\ cm^2$ (b) $\frac{1}{2}xyz\ cm^2$ (c) $\sqrt{s(s-x)(s-y)(s-z)}\ cm^2$ (d) $\frac{1}{2}xy\ cm^2$

Answer

(c) $\sqrt{s(s-x)(s-y)(s-z)}\ cm^2$

This is Heron's formula, where $s=\dfrac{x+y+z}{2}$ is the semi-perimeter.

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

If set $A = \{a, b, e, f\}$ and set $B = \{b, e, g, a\}$ , what is $n(B - A)$ ? (a) 1 (b) 2 (c) 4 (d) 6

Answer

(a) 1

$B-A$ contains the elements of $B$ not in $A$ : $B-A=\{g\}$ , so $n(B-A)=1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

If $x + y + z = 0$ , then $x^3 + y^3 + z^3$ is equal to (a) $xyz$ (b) $2xyz$ (c) $3xyz$ (d) $4xyz$

Answer

(c) $3xyz$

Using $x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$ , the right side is 0 when $x+y+z=0$ . So $x^3+y^3+z^3=3xyz$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

In a two digits' number, the sum of the digits is 7 and difference is 3. What will be the digit of ten's place? (a) 2 (b) 4 (c) 5 (d) 7

Answer

(c) 5

Let the tens digit be $x$ and the units digit $y$ : $x+y=7$ , $x-y=3$ . Adding: $2x=10$ , so $x=5$ and $y=2$ . The number is 52.

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

The values of the variable $x$ in a quadratic equation are 2 and 3. What will be the quadratic equation? (a) $x^2 - x - 6$ (b) $x^2 - 2x + 6$ (c) $x^2 - 5x + 6$ (d) $x^2 - 5x - 6$

Answer

(c) $x^2-5x+6$

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

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

The mean weight of 150 students is 60 kg. The mean weight of boys is 70 kg and that of girls is 55 kg. The number of girls is: (a) 50 (b) 150 (c) 25 (d) 100

Answer

(d) 100

Let girls $=g$ , boys $=150-g$ . Total weight $=150\times60=9000$ .

$70(150-g)+55g=9000 \Rightarrow 10500-15g=9000 \Rightarrow g=100$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

What is the probability of getting a number divisible by 2 or a cube number when a fair die is thrown? (a) 1/3 (b) 1/2 (c) 2/3 (d) 1/6

Answer

(c) 2/3

Numbers divisible by 2: $\{2,4,6\}$ . Cube numbers on a die: $\{1\}$ (since $8>6$ ). The union is $\{1,2,4,6\}$ , which has 4 outcomes.

$P=\dfrac{4}{6}=\dfrac{2}{3}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

What is the H.C.F. of relative prime numbers a, b and c? (a) bc (b) abc (c) ac (d) 1

Answer

(d) 1

Relatively prime (co-prime) numbers have no common factor other than 1, so their H.C.F. is 1.

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

What is the order of the surd $\sqrt{3x}$ ? (a) 1 (b) 2 (c) 3 (d) 0

Answer

(b) 2

$\sqrt{3x}=(3x)^{1/2}$ is a square root, so the order (index) of the surd is 2.

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

For what value of $x$ , $\sqrt[x]{a}$ equals $a^3$ ? (a) 1 (b) $1/3$ (c) $2/3$ (d) 3

Answer

(b) $1/3$ $\sqrt[x]{a}=a^{1/x}$ , so $a^{1/x}=a^3$ gives $1/x=3$ , hence $x=\dfrac{1}{3}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

If the length of the rectangle is twice the breadth and its perimeter is 60 cm. Find its length. (a) 10 cm (b) 20 cm (c) 30 cm (d) 40 cm

Answer

(b) 20 cm Let breadth $=b$ , length $=2b$ . Perimeter $=2(2b+b)=6b=60$ , so $b=10$ cm and length $=2b=20$ cm.

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

If in a certain figure, AB is a tangent, OB is the radius and angle $OAB = 30°$ , what is the value of angle AOB? (a) 90° (b) 60° (c) 30° (d) 15°

Answer

(b) 60° A tangent is perpendicular to the radius at the point of contact, so $\angle OBA=90^\circ$ . In $\triangle OAB$ : $\angle AOB=180^\circ-90^\circ-30^\circ=60^\circ$

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

What is the area of a triangle whose sides are 16 cm, 30 cm and 34 cm? (a) $37.9\ cm^2$ (b) $75.8\ cm^2$ (c) $120\ cm^2$ (d) $240\ cm^2$

Answer

(d) $240\ cm^2$ Since $16^2+30^2=256+900=1156=34^2$ , it is a right triangle with legs 16 and 30: $A=\tfrac12\times16\times30=240\ cm^2$

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

What is the probability of getting 6 on a dice and head on a coin when a dice is rolled and a coin is tossed simultaneously? (a) $1/8$ (b) $2/3$ (c) $1/12$ (d) $3/2$

Answer

(c) $1/12$ The events are independent: $P(6)=\tfrac16$ and $P(\text{head})=\tfrac12$ , so $P=\frac16\times\frac12=\frac1{12}$

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

If U.S. $1 = Rs. 105.52, how much dollar can be exchanged with Rs. 90312? (a) $855.87 (b) $90206.48 (c) $90417.52 (d) $9529722.2

Answer

(a) $855.87 $\text{Dollars}=\frac{90312}{105.52}\approx 855.9$ which matches option (a) $855.87.

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

If A and B can do a piece of work in 10 days and 20 days respectively, how many part of the work do they work together in 3 days? (a) $1/10$ (b) $1/20$ (c) $3/20$ (d) $9/20$

Answer

(d) $9/20$ Work done together in one day $=\frac1{10}+\frac1{20}=\frac3{20}$ , so in 3 days it is $3\times\frac3{20}=\frac9{20}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

What percent of discount should be given on a doll costing Rs. 1800 so that a customer can purchase it at Rs. 1500? (a) 13% (b) 16.66% (c) 20% (d) 83.33%

Answer

(b) 16.66% Discount $=1800-1500=$ Rs 300. $\text{Discount \%}=\frac{300}{1800}\times100=16.66\%$

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

What does the formula $P\left(1 + \frac{R}{100}\right)^n - 1$ represent? (a) compound amount (b) compound interest (c) simple interest (d) simple amount

Answer

(b) compound interest The compound amount is $A=P\left(1+\frac{R}{100}\right)^n$ , and compound interest is $CI=A-P=P\left[\left(1+\frac{R}{100}\right)^n-1\right]$ , which is what the formula represents.

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

The area of an isosceles triangle is given by: (a) $\frac{1}{4}b\sqrt{4a^2 - b^2}$ (b) $\frac{1}{4}a\sqrt{4a^2 - a^2}$ (c) $\frac{1}{4}b\sqrt{4a^2 + b^2}$ (d) $\frac{1}{4}a\sqrt{4a^2 + a^2}$

Answer

(a) $\frac{1}{4}b\sqrt{4a^2-b^2}$ For an isosceles triangle with equal sides $a$ and base $b$ , the height is $\sqrt{a^2-b^2/4}$ , so $A=\frac12 b\sqrt{a^2-\frac{b^2}{4}}=\frac14 b\sqrt{4a^2-b^2}$

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

The compound interest of Rs. 1500 at 5% p.a. for 2 years will be: (a) Rs. 150.75 (b) Rs. 151.375 (c) Rs. 152.75 (d) Rs. 153.75

Answer

(d) Rs. 153.75

Amount after 2 years: $A = 1500\left(1+\frac{5}{100}\right)^2 = 1500 \times 1.1025 = \text{Rs } 1653.75$ .

Compound interest $= 1653.75 - 1500 = \text{Rs } 153.75$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

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

Answer

(d) $(x^2-2x+2)(x^2+2x+2)$

Add and subtract $4x^2$ to complete a square:

$x^4+4 = (x^2+2)^2 - 4x^2 = (x^2+2-2x)(x^2+2+2x).$

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

If the ratio of two digit numbers is 5:6 and their sum is 33, then the numbers will be (a) 15 and 18 (b) 21 and 12 (c) 9 and 24 (d) 45 and 54

Answer

(a) 15 and 18

Let the numbers be $5k$ and $6k$ . Then $11k = 33$ , so $k = 3$ , giving $15$ and $18$ . Check: $15:18 = 5:6$ and $15+18=33$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

The values of $x$ in the equation $x^2 - 5x + 6 = 0$ are: (a) 2 and 3 (b) -2 and 3 (c) 2 and -3 (d) -2 and -3

Answer

(a) 2 and 3

$x^2-5x+6 = (x-2)(x-3) = 0 \Rightarrow x = 2 \text{ or } 3.$

Check: sum of roots $=5$ , product $=6$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

The value of $\sin 50° - \sin 70° + \sin 10°$ = (a) 1 (b) -1 (c) 0 (d) 2

Answer

(c) 0

$\sin 50° + \sin 10° = 2\sin 30°\cos 20° = \cos 20° = \sin 70°.$

So $\sin 50° - \sin 70° + \sin 10° = \sin 70° - \sin 70° = 0$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

The sum of two positive numbers is 60 and their difference is 20. Find the numbers. (a) 40 and 20 (b) 45 and 15 (c) 33 and 13 (d) 25 and 35

Answer

(a) 40 and 20

Let the numbers be $x$ and $y$ with $x+y=60$ and $x-y=20$ . Adding gives $2x=80$ , so $x=40$ and $y=20$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

What type of sets are $A = \{1, 2, 3, 4, 0\}$ and $B = \{10, 5, 6, 7, 8\}$ ? (a) Equal (b) Disjoint (c) Connected (d) Infinite

Answer

(b) Disjoint

$A$ and $B$ have no element in common ( $A\cap B=\varnothing$ ), so they are disjoint sets. They are finite and not equal.

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

The item of a statistical data that divides it into two equal parts (halves) is called: (a) Mode (b) Quartile (c) Median (d) Mean

Answer

(c) Median

The median is the middle value that divides ordered data into two equal halves. Quartiles divide it into four parts.

Answer checked by Claude Sonnet · Oct 21, 2026

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Past questions and answers

1.What is the total surface area of a cone when the base radius of the cone is 7 cm and the sum of the radius and slant height is 28 cm? (a) 231 cm² (b) 400 cm² (c) 462 cm² (d) 616 cm²

2.If the total surface area of the cone is 704 cm2704\ cm^2704 cm2 and its base radius is 7 cm, the slant height of the cone is (a) 24 cm (b) 22 cm (c) 25 cm (d) 44 cm

3.The area of triangle XYZ is 45 cm². If XY=12XY = 12XY=12 cm and YZ=15YZ = 15YZ=15 cm, what will be the value of ∠XYZ\angle XYZ∠XYZ? (a) 30° (b) 45° (c) 60° (d) 90°

4.If x=10ax = 10^ax=10a, y=10by = 10^by=10b and xbya=100x^b y^a = 100xbya=100, what is the value of ababab? (a) 1 (b) 2\sqrt{2}2​ (c) 2 (d) 10

5.If 3m+1−2=0\sqrt{3m+1} - 2 = 03m+1​−2=0, what is the value of mmm? (a) 0 (b) -1 (c) 1 (d) 4

6.If m+n=7m + n = 7m+n=7 and 2m−3n=42m - 3n = 42m−3n=4, what will be the value of m−nm - nm−n? (a) 2.5 (b) 5 (c) 3 (d) 5.2

8.If 10 is added to the number that is seven times the square of a natural number, the result is 353. What will be the number? (a) ±7\pm 7±7 (b) -7 (c) 7 (d) 9

9.The area of an equilateral triangle is 939\sqrt{3}93​ cm². What is the perimeter of the triangle? (a) 6 cm (b) 9 cm (c) 18 cm (d) 27327\sqrt{3}273​ cm

11.What is the highest common factor (H.C.F.) of two algebraic expressions (x+2)(x+2)(x+2) and (x−2)(x-2)(x−2)? (a) x+2x + 2x+2 (b) x−2x - 2x−2 (c) x2−4x^2 - 4x2−4 (d) 1

12.What is the order of the surd 3743\sqrt[4]{7}347​? (a) 1 (b) 2 (c) 3 (d) 4

13.What is the H.C.F. of x2−1x^2 - 1x2−1 and x2−3x+2x^2 - 3x + 2x2−3x+2? (a) x−1x - 1x−1 (b) x+1x + 1x+1 (c) (x−1)(x−2)(x-1)(x-2)(x−1)(x−2) (d) (x+1)(x−1)(x+1)(x-1)(x+1)(x−1)

14.If A={2,3,5}A = \{2, 3, 5\}A={2,3,5} and B={1,4}B = \{1, 4\}B={1,4}, what is the value of n(A∩B)n(A \cap B)n(A∩B)? (a) θ\thetaθ (b) { }\{\ \}{ } (c) 0 (d) 5

15.How many different values of the variable of unit place digit in a two digit number? (a) 1 (b) 9 (c) 10 (d) yyy

16.If 2x+1−4=02^{x+1} - 4 = 02x+1−4=0, what is the value of xxx? (a) 0 (b) -1 (c) 1 (d) 2

17.Which one is the formula to calculate the curved surface area of a cylinder? (a) πrh\pi r hπrh (b) 2πrh2\pi r h2πrh (c) 2πr2h2\pi r^2 h2πr2h (d) 2π(r+h)2\pi(r + h)2π(r+h)

20.If X={1,4,6}X = \{1, 4, 6\}X={1,4,6} and Y={2,5}Y = \{2, 5\}Y={2,5}, what is the value of n(X∪Y)n(X \cup Y)n(X∪Y)? (a) 0 (b) 2 (c) 3 (d) 5

22.How much more is Rs. 6,000 including 13% VAT on it than Rs. 6,500? (a) Rs. 200 (b) Rs. 280 (c) Rs. 300 (d) Rs. 380

23.The present price of a television is Rs. 35,000. After how many years will it be Rs. 25,515 if its price reduces by 10% per annum? (a) 2 years (b) 2122\frac{1}{2}221​ years (c) 3 years (d) 4 years

24.What is a triangle whose three sides are 4 cm, 6 cm and 12 cm respectively? (a) Acute angled triangle (b) Equilateral triangle (c) Obtuse angle triangle (d) No triangle

25.What is the vertical height of a square based pyramid having its base area 64 cm264\ cm^264 cm2 and volume 640 cm3640\ cm^3640 cm3? (a) 5 cm (b) 8 cm (c) 10 cm (d) 30 cm

26.The total surface area of a hemisphere is 243π cm2243\pi\ cm^2243π cm2. What will be its diameter? (a) 9 cm (b) 18 cm (c) 27 cm (d) 30 cm

27.What is the L.C.M. of two algebraic expressions (x+1)(x+1)(x+1) and (x2−1)(x^2 - 1)(x2−1)? (a) x+1x + 1x+1 (b) x2−1x^2 - 1x2−1 (c) (x2−1)(x+1)(x^2 - 1)(x + 1)(x2−1)(x+1) (d) x3−1x^3 - 1x3−1

28.What is the H.C.F. of x2−1x^2 - 1x2−1 and x2+3x+2x^2 + 3x + 2x2+3x+2? (a) x+1x + 1x+1 (b) x−1x - 1x−1 (c) x2−1x^2 - 1x2−1 (d) x−2x - 2x−2

29.Which one is a quadratic equation in which the values of xxx are 2 and −1-1−1? (a) x2+x−2=0x^2 + x - 2 = 0x2+x−2=0 (b) x2−x+2=0x^2 - x + 2 = 0x2−x+2=0 (c) x2−x−2=0x^2 - x - 2 = 0x2−x−2=0 (d) 2x2−x−1=02x^2 - x - 1 = 02x2−x−1=0

30.If the sum of a positive number and its square is 30, what is the number? (a) 3 (b) 5 (c) ±5\pm 5±5 (d) 6

31.What is the relation between the areas of the triangles standing on the same base and between the same parallel lines? (a) double (b) unequal (c) equal (d) half one

32.In △PQR\triangle PQR△PQR, PQ=7PQ = 7PQ=7 cm, QR=16QR = 16QR=16 cm and the area of △PQR\triangle PQR△PQR is 283 cm228\sqrt{3}\ cm^2283​ cm2. What is the measure of ∠PQR\angle PQR∠PQR? (a) 30° (b) 45° (c) 60° (d) 75°

33.What is the angle of elevation when a man observes the top of a tower of height 80380\sqrt{3}803​ m from 240 m from the foot of the tower? (a) 30° (b) 45° (c) 60° (d) 75°

35.Two players, A and B play a tennis match. If the probability of winning the match by A is 13\frac{1}{3}31​, what is the probability of winning the match by B? (a) 23\frac{2}{3}32​ (b) 13\frac{1}{3}31​ (c) 56\frac{5}{6}65​ (d) 16\frac{1}{6}61​

36.If 2sec⁡2α=tan⁡β+cot⁡β2\sec 2\alpha = \tan\beta + \cot\beta2sec2α=tanβ+cotβ, then the value of (α+β)(\alpha + \beta)(α+β) = (a) π2\frac{\pi}{2}2π​ (b) π3\frac{\pi}{3}3π​ (c) π4\frac{\pi}{4}4π​ (d) π6\frac{\pi}{6}6π​

37.If (A+B)=π4(A + B) = \frac{\pi}{4}(A+B)=4π​, then (cot⁡A−1)(cot⁡B−1)(\cot A - 1)(\cot B - 1)(cotA−1)(cotB−1) = (a) 1 (b) 12 (c) 2 (d) −1-1−1

38.In △ABC\triangle ABC△ABC, if 2cos⁡A=sin⁡Bsin⁡C2\cos A = \frac{\sin B}{\sin C}2cosA=sinCsinB​, then the △\triangle△ is: (a) Right angled triangle (b) Equilateral triangle (c) Isosceles triangle (d) Scalene triangle

39.The number of non-empty proper subsets of set A={1,2,3,4}A = \{1, 2, 3, 4\}A={1,2,3,4} is: (a) 16 (b) 14 (c) 15 (d) 13

40.If f(x+1)=x+2f(x + 1) = x + 2f(x+1)=x+2, then f(x−1)f(x - 1)f(x−1) = (a) 2 (b) xxx (c) x+1x + 1x+1 (d) x−1x - 1x−1

41.For any two square matrices A and B, which of the following is not true? (a) (A+B)T=AT+BT(A + B)^T = A^T + B^T(A+B)T=AT+BT (b) (AB)T=BTAT(AB)^T = B^T A^T(AB)T=BTAT (c) A(B+C)=AB+ACA(B + C) = AB + ACA(B+C)=AB+AC (d) (A−B)T=BT−AT(A - B)^T = B^T - A^T(A−B)T=BT−AT

43.If 1q+r,1r+p,1p+q\frac{1}{q+r}, \frac{1}{r+p}, \frac{1}{p+q}q+r1​,r+p1​,p+q1​ are in AP, then: (a) p2,q2,r2p^2, q^2, r^2p2,q2,r2 are in AP (b) p,q,rp, q, rp,q,r are in GP (c) p2,q2,r2p^2, q^2, r^2p2,q2,r2 are in GP (d) p2,q2,r2p^2, q^2, r^2p2,q2,r2 are in HP

44.If mthm^{th}mth term of an AP is nnn and nthn^{th}nth term is mmm, then rthr^{th}rth term is: (a) m−n−rm - n - rm−n−r (b) m+n−rm + n - rm+n−r (c) m−n+rm - n + rm−n+r (d) n−r−mn - r - mn−r−m

45.If α\alphaα and β\betaβ are roots of the equation x2−cp(x+1)−c=0x^2 - cp(x + 1) - c = 0x2−cp(x+1)−c=0 then (1+α)(1+β)(1 + \alpha)(1 + \beta)(1+α)(1+β) = (a) 1+p1 + p1+p (b) 1−p1 - p1−p (c) 1+c1 + c1+c (d) 1−c1 - c1−c

46.If 7 is added to the square of a natural number then the result is 43. What is the number? (a) 50\sqrt{50}50​ (b) ±7\pm 7±7 (c) ±6\pm 6±6 (d) 6

48.If set A={a,b,e,f}A = \{a, b, e, f\}A={a,b,e,f} and set B={b,e,g,a}B = \{b, e, g, a\}B={b,e,g,a}, what is n(B−A)n(B - A)n(B−A)? (a) 1 (b) 2 (c) 4 (d) 6

49.If x+y+z=0x + y + z = 0x+y+z=0, then x3+y3+z3x^3 + y^3 + z^3x3+y3+z3 is equal to (a) xyzxyzxyz (b) 2xyz2xyz2xyz (c) 3xyz3xyz3xyz (d) 4xyz4xyz4xyz

50.In a two digits' number, the sum of the digits is 7 and difference is 3. What will be the digit of ten's place? (a) 2 (b) 4 (c) 5 (d) 7

51.The values of the variable xxx in a quadratic equation are 2 and 3. What will be the quadratic equation? (a) x2−x−6x^2 - x - 6x2−x−6 (b) x2−2x+6x^2 - 2x + 6x2−2x+6 (c) x2−5x+6x^2 - 5x + 6x2−5x+6 (d) x2−5x−6x^2 - 5x - 6x2−5x−6

52.The mean weight of 150 students is 60 kg. The mean weight of boys is 70 kg and that of girls is 55 kg. The number of girls is: (a) 50 (b) 150 (c) 25 (d) 100

53.What is the probability of getting a number divisible by 2 or a cube number when a fair die is thrown? (a) 1/3 (b) 1/2 (c) 2/3 (d) 1/6

54.What is the H.C.F. of relative prime numbers a, b and c? (a) bc (b) abc (c) ac (d) 1

55.What is the order of the surd 3x\sqrt{3x}3x​? (a) 1 (b) 2 (c) 3 (d) 0

56.For what value of xxx, ax\sqrt[x]{a}xa​ equals a3a^3a3? (a) 1 (b) 1/31/31/3 (c) 2/32/32/3 (d) 3

57.If the length of the rectangle is twice the breadth and its perimeter is 60 cm. Find its length. (a) 10 cm (b) 20 cm (c) 30 cm (d) 40 cm

58.If in a certain figure, AB is a tangent, OB is the radius and angle OAB=30°OAB = 30°OAB=30°, what is the value of angle AOB? (a) 90° (b) 60° (c) 30° (d) 15°

59.What is the area of a triangle whose sides are 16 cm, 30 cm and 34 cm? (a) 37.9 cm237.9\ cm^237.9 cm2 (b) 75.8 cm275.8\ cm^275.8 cm2 (c) 120 cm2120\ cm^2120 cm2 (d) 240 cm2240\ cm^2240 cm2

60.What is the probability of getting 6 on a dice and head on a coin when a dice is rolled and a coin is tossed simultaneously? (a) 1/81/81/8 (b) 2/32/32/3 (c) 1/121/121/12 (d) 3/23/23/2

61.If U.S. $1 = Rs. 105.52, how much dollar can be exchanged with Rs. 90312? (a) $855.87 (b) $90206.48 (c) $90417.52 (d) $9529722.2

62.If A and B can do a piece of work in 10 days and 20 days respectively, how many part of the work do they work together in 3 days? (a) 1/101/101/10 (b) 1/201/201/20 (c) 3/203/203/20 (d) 9/209/209/20

63.What percent of discount should be given on a doll costing Rs. 1800 so that a customer can purchase it at Rs. 1500? (a) 13% (b) 16.66% (c) 20% (d) 83.33%

64.What does the formula P(1+R100)n−1P\left(1 + \frac{R}{100}\right)^n - 1P(1+100R​)n−1 represent? (a) compound amount (b) compound interest (c) simple interest (d) simple amount

65.The area of an isosceles triangle is given by: (a) 14b4a2−b2\frac{1}{4}b\sqrt{4a^2 - b^2}41​b4a2−b2​ (b) 14a4a2−a2\frac{1}{4}a\sqrt{4a^2 - a^2}41​a4a2−a2​ (c) 14b4a2+b2\frac{1}{4}b\sqrt{4a^2 + b^2}41​b4a2+b2​ (d) 14a4a2+a2\frac{1}{4}a\sqrt{4a^2 + a^2}41​a4a2+a2​

66.The compound interest of Rs. 1500 at 5% p.a. for 2 years will be: (a) Rs. 150.75 (b) Rs. 151.375 (c) Rs. 152.75 (d) Rs. 153.75

67.The factors of x4+4x^4 + 4x4+4 are: (a) (x+2)(x2+2x+4)(x+2)(x^2+2x+4)(x+2)(x2+2x+4) (b) (x+2)(x2−2x+2)(x+2)(x^2-2x+2)(x+2)(x2−2x+2) (c) (x−2x+2)(x+2x+2)(x-2x+2)(x+2x+2)(x−2x+2)(x+2x+2) (d) (x2−2x+2)(x2+2x+2)(x^2-2x+2)(x^2+2x+2)(x2−2x+2)(x2+2x+2)

68.If the ratio of two digit numbers is 5:6 and their sum is 33, then the numbers will be (a) 15 and 18 (b) 21 and 12 (c) 9 and 24 (d) 45 and 54

69.The values of xxx in the equation x2−5x+6=0x^2 - 5x + 6 = 0x2−5x+6=0 are: (a) 2 and 3 (b) -2 and 3 (c) 2 and -3 (d) -2 and -3

70.The value of sin⁡50°−sin⁡70°+sin⁡10°\sin 50° - \sin 70° + \sin 10°sin50°−sin70°+sin10° = (a) 1 (b) -1 (c) 0 (d) 2

71.The sum of two positive numbers is 60 and their difference is 20. Find the numbers. (a) 40 and 20 (b) 45 and 15 (c) 33 and 13 (d) 25 and 35

72.What type of sets are A={1,2,3,4,0}A = \{1, 2, 3, 4, 0\}A={1,2,3,4,0} and B={10,5,6,7,8}B = \{10, 5, 6, 7, 8\}B={10,5,6,7,8}? (a) Equal (b) Disjoint (c) Connected (d) Infinite

73.The item of a statistical data that divides it into two equal parts (halves) is called: (a) Mode (b) Quartile (c) Median (d) Mean

What is the total surface area of a cone when the base radius of the cone is 7 cm and the sum of the radius and slant height is 28 cm? (a) 231 cm² (b) 400 cm² (c) 462 cm² (d) 616 cm²

If the total surface area of the cone is $704\ cm^2$ and its base radius is 7 cm, the slant height of the cone is (a) 24 cm (b) 22 cm (c) 25 cm (d) 44 cm

The area of triangle XYZ is 45 cm². If $XY = 12$ cm and $YZ = 15$ cm, what will be the value of $\angle XYZ$ ? (a) 30° (b) 45° (c) 60° (d) 90°

If $x = 10^a$ , $y = 10^b$ and $x^b y^a = 100$ , what is the value of $ab$ ? (a) 1 (b) $\sqrt{2}$ (c) 2 (d) 10

If $\sqrt{3m+1} - 2 = 0$ , what is the value of $m$ ? (a) 0 (b) -1 (c) 1 (d) 4

If $m + n = 7$ and $2m - 3n = 4$ , what will be the value of $m - n$ ? (a) 2.5 (b) 5 (c) 3 (d) 5.2

If 10 is added to the number that is seven times the square of a natural number, the result is 353. What will be the number? (a) $\pm 7$ (b) -7 (c) 7 (d) 9

The area of an equilateral triangle is $9\sqrt{3}$ cm². What is the perimeter of the triangle? (a) 6 cm (b) 9 cm (c) 18 cm (d) $27\sqrt{3}$ cm

What is the highest common factor (H.C.F.) of two algebraic expressions $(x+2)$ and $(x-2)$ ? (a) $x + 2$ (b) $x - 2$ (c) $x^2 - 4$ (d) 1

What is the order of the surd $3\sqrt[4]{7}$ ? (a) 1 (b) 2 (c) 3 (d) 4

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

If $A = \{2, 3, 5\}$ and $B = \{1, 4\}$ , what is the value of $n(A \cap B)$ ? (a) $\theta$ (b) $\{\ \}$ (c) 0 (d) 5

How many different values of the variable of unit place digit in a two digit number? (a) 1 (b) 9 (c) 10 (d) $y$

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

Which one is the formula to calculate the curved surface area of a cylinder? (a) $\pi r h$ (b) $2\pi r h$ (c) $2\pi r^2 h$ (d) $2\pi(r + h)$

If $X = \{1, 4, 6\}$ and $Y = \{2, 5\}$ , what is the value of $n(X \cup Y)$ ? (a) 0 (b) 2 (c) 3 (d) 5

How much more is Rs. 6,000 including 13% VAT on it than Rs. 6,500? (a) Rs. 200 (b) Rs. 280 (c) Rs. 300 (d) Rs. 380

The present price of a television is Rs. 35,000. After how many years will it be Rs. 25,515 if its price reduces by 10% per annum? (a) 2 years (b) $2\frac{1}{2}$ years (c) 3 years (d) 4 years

What is a triangle whose three sides are 4 cm, 6 cm and 12 cm respectively? (a) Acute angled triangle (b) Equilateral triangle (c) Obtuse angle triangle (d) No triangle

What is the vertical height of a square based pyramid having its base area $64\ cm^2$ and volume $640\ cm^3$ ? (a) 5 cm (b) 8 cm (c) 10 cm (d) 30 cm

The total surface area of a hemisphere is $243\pi\ cm^2$ . What will be its diameter? (a) 9 cm (b) 18 cm (c) 27 cm (d) 30 cm

What is the L.C.M. of two algebraic expressions $(x+1)$ and $(x^2 - 1)$ ? (a) $x + 1$ (b) $x^2 - 1$ (c) $(x^2 - 1)(x + 1)$ (d) $x^3 - 1$

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

Which one is a quadratic equation in which the values of $x$ are 2 and $-1$ ? (a) $x^2 + x - 2 = 0$ (b) $x^2 - x + 2 = 0$ (c) $x^2 - x - 2 = 0$ (d) $2x^2 - x - 1 = 0$

If the sum of a positive number and its square is 30, what is the number? (a) 3 (b) 5 (c) $\pm 5$ (d) 6

What is the relation between the areas of the triangles standing on the same base and between the same parallel lines? (a) double (b) unequal (c) equal (d) half one

In $\triangle PQR$ , $PQ = 7$ cm, $QR = 16$ cm and the area of $\triangle PQR$ is $28\sqrt{3}\ cm^2$ . What is the measure of $\angle PQR$ ? (a) 30° (b) 45° (c) 60° (d) 75°

What is the angle of elevation when a man observes the top of a tower of height $80\sqrt{3}$ m from 240 m from the foot of the tower? (a) 30° (b) 45° (c) 60° (d) 75°

Two players, A and B play a tennis match. If the probability of winning the match by A is $\frac{1}{3}$ , what is the probability of winning the match by B? (a) $\frac{2}{3}$ (b) $\frac{1}{3}$ (c) $\frac{5}{6}$ (d) $\frac{1}{6}$

If $2\sec 2\alpha = \tan\beta + \cot\beta$ , then the value of $(\alpha + \beta)$ = (a) $\frac{\pi}{2}$ (b) $\frac{\pi}{3}$ (c) $\frac{\pi}{4}$ (d) $\frac{\pi}{6}$

If $(A + B) = \frac{\pi}{4}$ , then $(\cot A - 1)(\cot B - 1)$ = (a) 1 (b) 12 (c) 2 (d) $-1$

In $\triangle ABC$ , if $2\cos A = \frac{\sin B}{\sin C}$ , then the $\triangle$ is: (a) Right angled triangle (b) Equilateral triangle (c) Isosceles triangle (d) Scalene triangle

The number of non-empty proper subsets of set $A = \{1, 2, 3, 4\}$ is: (a) 16 (b) 14 (c) 15 (d) 13

If $f(x + 1) = x + 2$ , then $f(x - 1)$ = (a) 2 (b) $x$ (c) $x + 1$ (d) $x - 1$

For any two square matrices A and B, which of the following is not true? (a) $(A + B)^T = A^T + B^T$ (b) $(AB)^T = B^T A^T$ (c) $A(B + C) = AB + AC$ (d) $(A - B)^T = B^T - A^T$

If $\frac{1}{q+r}, \frac{1}{r+p}, \frac{1}{p+q}$ are in AP, then: (a) $p^2, q^2, r^2$ are in AP (b) $p, q, r$ are in GP (c) $p^2, q^2, r^2$ are in GP (d) $p^2, q^2, r^2$ are in HP

If $m^{th}$ term of an AP is $n$ and $n^{th}$ term is $m$ , then $r^{th}$ term is: (a) $m - n - r$ (b) $m + n - r$ (c) $m - n + r$ (d) $n - r - m$

If $\alpha$ and $\beta$ are roots of the equation $x^2 - cp(x + 1) - c = 0$ then $(1 + \alpha)(1 + \beta)$ = (a) $1 + p$ (b) $1 - p$ (c) $1 + c$ (d) $1 - c$

If 7 is added to the square of a natural number then the result is 43. What is the number? (a) $\sqrt{50}$ (b) $\pm 7$ (c) $\pm 6$ (d) 6

If set $A = \{a, b, e, f\}$ and set $B = \{b, e, g, a\}$ , what is $n(B - A)$ ? (a) 1 (b) 2 (c) 4 (d) 6

If $x + y + z = 0$ , then $x^3 + y^3 + z^3$ is equal to (a) $xyz$ (b) $2xyz$ (c) $3xyz$ (d) $4xyz$

In a two digits' number, the sum of the digits is 7 and difference is 3. What will be the digit of ten's place? (a) 2 (b) 4 (c) 5 (d) 7

The values of the variable $x$ in a quadratic equation are 2 and 3. What will be the quadratic equation? (a) $x^2 - x - 6$ (b) $x^2 - 2x + 6$ (c) $x^2 - 5x + 6$ (d) $x^2 - 5x - 6$

The mean weight of 150 students is 60 kg. The mean weight of boys is 70 kg and that of girls is 55 kg. The number of girls is: (a) 50 (b) 150 (c) 25 (d) 100

What is the probability of getting a number divisible by 2 or a cube number when a fair die is thrown? (a) 1/3 (b) 1/2 (c) 2/3 (d) 1/6

What is the H.C.F. of relative prime numbers a, b and c? (a) bc (b) abc (c) ac (d) 1

What is the order of the surd $\sqrt{3x}$ ? (a) 1 (b) 2 (c) 3 (d) 0

For what value of $x$ , $\sqrt[x]{a}$ equals $a^3$ ? (a) 1 (b) $1/3$ (c) $2/3$ (d) 3

If the length of the rectangle is twice the breadth and its perimeter is 60 cm. Find its length. (a) 10 cm (b) 20 cm (c) 30 cm (d) 40 cm

If in a certain figure, AB is a tangent, OB is the radius and angle $OAB = 30°$ , what is the value of angle AOB? (a) 90° (b) 60° (c) 30° (d) 15°

What is the area of a triangle whose sides are 16 cm, 30 cm and 34 cm? (a) $37.9\ cm^2$ (b) $75.8\ cm^2$ (c) $120\ cm^2$ (d) $240\ cm^2$

What is the probability of getting 6 on a dice and head on a coin when a dice is rolled and a coin is tossed simultaneously? (a) $1/8$ (b) $2/3$ (c) $1/12$ (d) $3/2$

If U.S. $1 = Rs. 105.52, how much dollar can be exchanged with Rs. 90312? (a) $855.87 (b) $90206.48 (c) $90417.52 (d) $9529722.2

If A and B can do a piece of work in 10 days and 20 days respectively, how many part of the work do they work together in 3 days? (a) $1/10$ (b) $1/20$ (c) $3/20$ (d) $9/20$

What percent of discount should be given on a doll costing Rs. 1800 so that a customer can purchase it at Rs. 1500? (a) 13% (b) 16.66% (c) 20% (d) 83.33%

What does the formula $P\left(1 + \frac{R}{100}\right)^n - 1$ represent? (a) compound amount (b) compound interest (c) simple interest (d) simple amount

The area of an isosceles triangle is given by: (a) $\frac{1}{4}b\sqrt{4a^2 - b^2}$ (b) $\frac{1}{4}a\sqrt{4a^2 - a^2}$ (c) $\frac{1}{4}b\sqrt{4a^2 + b^2}$ (d) $\frac{1}{4}a\sqrt{4a^2 + a^2}$

The compound interest of Rs. 1500 at 5% p.a. for 2 years will be: (a) Rs. 150.75 (b) Rs. 151.375 (c) Rs. 152.75 (d) Rs. 153.75

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

If the ratio of two digit numbers is 5:6 and their sum is 33, then the numbers will be (a) 15 and 18 (b) 21 and 12 (c) 9 and 24 (d) 45 and 54

The values of $x$ in the equation $x^2 - 5x + 6 = 0$ are: (a) 2 and 3 (b) -2 and 3 (c) 2 and -3 (d) -2 and -3

The value of $\sin 50° - \sin 70° + \sin 10°$ = (a) 1 (b) -1 (c) 0 (d) 2

The sum of two positive numbers is 60 and their difference is 20. Find the numbers. (a) 40 and 20 (b) 45 and 15 (c) 33 and 13 (d) 25 and 35

What type of sets are $A = \{1, 2, 3, 4, 0\}$ and $B = \{10, 5, 6, 7, 8\}$ ? (a) Equal (b) Disjoint (c) Connected (d) Infinite

The item of a statistical data that divides it into two equal parts (halves) is called: (a) Mode (b) Quartile (c) Median (d) Mean