TU BIT entrance TU BIT Entrance Papers: Mathematics

2079 Model · 1 mark

If $f:R\to R$ is defined by $f(x)=3x-2$ , for $x\in R$ then the pre image of 10 in R is (a) 4 (b) 10 (c) 5 (d) 0

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Answer

(a) 4 Solve $f(x)=10$ : $3x-2=10\Rightarrow x=4$ .

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2079 Model · 1 mark

The two lines represented by $ax^2+xy+by^2=0$ perpendicular to each other if (a) $a+b>0$ (b) $a+b<0$ (c) $ab=0$ (d) $a+b=0$

Answer

(d) $a+b=0$

A pair of lines $ax^2+2hxy+by^2=0$ is perpendicular when the coefficient of $x^2$ plus the coefficient of $y^2$ is zero, that is $a+b=0$ .

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2080 · 1 mark

If a function $f:\mathbb R\to\mathbb R$ be defined by $f(x)=2x+5$ then the pre-image of 7 in $\mathbb R$ is (a) 3 (b) no pre-image (c) 1 (d) -2

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Answer

(c) 1

$2x+5=7\Rightarrow x=1$ .

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2080 · 1 mark

The two lines represented by $kx^2+xy-y^2=0$ , are perpendicular to each other if $k=$ (a) 1 (b) -1 (c) 0 (d) $k\neq0$

Answer

(a) 1

Lines from $ax^2+2hxy+by^2=0$ are perpendicular when $a+b=0$ : $k+(-1)=0\Rightarrow k=1$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

The straight lines represented by $x^2+2xy+ky^2=0$ are perpendicular to each other if $k=$ (a) -1 (b) 1 (c) 0 (d) 2

Answer

(a) $-1$ For $ax^2+2hxy+by^2=0$ the lines are perpendicular when $a+b=0$ . Here $1+k=0$ , so $k=-1$ .

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2076 Model · 1 mark

Every rational number is also (a) a natural number (b) an integer (c) a whole number (d) a real number

Answer

(d) a real number

Every rational number is a real number. Rational numbers such as 1/2 are not integers, whole numbers or natural numbers.

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2076 Model · 1 mark

If $a-ib=\frac{3-2i}{3+2i}$ then $a^2+b^2=$ (a) 1 (b) $\frac43$ (c) $\frac35$ (d) $\frac29$

Answer

(a) 1

$a-ib=\frac{3-2i}{3+2i}\cdot\frac{3-2i}{3-2i}=\frac{9-12i+4i^2}{13}=\frac{5-12i}{13}$

So $a=\frac5{13}$ , $b=\frac{12}{13}$ , and $a^2+b^2=\frac{25+144}{169}=1$ .

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2076 Model · 1 mark

Given $U=\{1,2,3,\ldots,15\}$ , $A=\{x: x\leq8\}$ , $B=\{x: x\leq4\}$ then $\overline{A\cap B}=$ (a) $\emptyset$ (b) $U$ (c) $A\cup B$ (d) None of these

Answer

(d) None of these

$A\cap B=B=\{1,2,3,4\}$ , so $\overline{A\cap B}=\{5,6,\ldots,15\}$ . This is not $\emptyset$ , not $U$ , and $A\cup B=A=\{1,\ldots,8\}$ is different.

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2076 Model · 1 mark

Which are the equal ordered pairs? (a) $(2,4),(4,2)$ (b) $(2,2),(4,4)$ (c) $(4,2),(4,2)$ (d) all of them

Answer

(c) $(4,2),(4,2)$

Ordered pairs are equal only when the first components are equal and the second components are equal. Only $(4,2)=(4,2)$ satisfies this.

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2076 Model · 1 mark

$S_\infty=\frac{a}{1-r}$ where $a, r$ and $S_\infty$ denotes first term, common ratio and sum to infinity. If (a) $|r|<1$ (b) $|r|>1$ (c) $|r|=1$ (d) all of them

Answer

(a) $|r|<1$

The infinite geometric series converges only when $|r|<1$ ; then $r^n\to0$ and $S_\infty=\frac{a}{1-r}$ .

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2076 Model · 1 mark

If $^{21}C_x=\,^{21}C_y$ then which is true? (a) $x\neq y$ (b) $x=y$ (c) $x+y=21$ (d) either (b) or (c)

Answer

(d) either (b) or (c)

$^{n}C_x=\,^{n}C_y$ implies $x=y$ or $x+y=n$ . Here $n=21$ , so $x=y$ or $x+y=21$ .

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2076 Model · 1 mark

Which is not correct? (a) $(A^T)^T=A$ (b) $(A+B)^T=A^T+B^T$ (c) $(AB)^T=A^TB^T$ (d) $(AB)^T=B^TA^T$

Answer

(c) $(AB)^T=A^TB^T$

The correct reversal law is $(AB)^T=B^TA^T$ , so (c) is the incorrect statement.

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2076 Model · 1 mark

The equation $x=0$ represents (a) x-axis (b) y-axis (c) the origin (d) none of these

Answer

(b) y-axis

Every point with $x=0$ lies on the y-axis.

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2076 Model · 1 mark

The angle between the lines $xy=0$ is (a) $0^\circ$ (b) $60^\circ$ (c) $90^\circ$ (d) $120^\circ$

Answer

(c) $90^\circ$

$xy=0$ means $x=0$ or $y=0$ , i.e. the y-axis and x-axis, which are perpendicular.

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2076 Model · 1 mark

Which is correct? (a) $\log_{10}1=0$ (b) $\log_{10}0.1=-1$ (c) $\log_{10}10=1$ (d) all of them

Answer

(d) all of them

$\log_{10}1=0$ , $\log_{10}0.1=\log_{10}10^{-1}=-1$ and $\log_{10}10=1$ are all true.

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2076 Model · 1 mark

The domain of $y=f(x)=\frac1x$ is (a) $R-\{0\}$ (b) $R$ (c) $(-2,1)$ (d) $(-1,1)$

Answer

(a) $R-\{0\}$

$\frac1x$ is defined for every real $x$ except $x=0$ .

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2076 Model · 1 mark

Let $f:R\to R$ be defined by $f(x)=10x-7$ then $f^{-1}(x)=$ (a) $\frac{1}{10x+7}$ (b) $\frac{1}{10x-7}$ (c) $\frac{x+7}{10}$ (d) $\frac{x-7}{10}$

Answer

(c) $\frac{x+7}{10}$

Let $y=10x-7$ . Then $x=\frac{y+7}{10}$ , so $f^{-1}(x)=\frac{x+7}{10}$ .

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2076 Model · 1 mark

If t be a positive real number then the least value of $t+\frac1t$ is : (a) 1 (b) 2 (c) 4 (d) 3

Answer

(b) 2

By AM-GM, $t+\frac1t\ge2\sqrt{t\cdot\frac1t}=2$ , with equality at $t=1$ .

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2076 Model · 1 mark

The point of discontinuity of the function $f(x)=\frac{x^3-27}{x-3}$ is (a) 3 (b) 6 (c) 27 (d) 0

Answer

(a) 3

$f(x)$ is not defined at $x=3$ (denominator zero), although $f(x)=x^2+3x+9$ for $x\neq3$ . So the point of discontinuity is $x=3$ .

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2076 Model · 1 mark

If $x+y=xy$ then $\frac{dx}{dy}=$ (a) $\frac{x-1}{y-1}$ (b) $\frac{y-1}{1-x}$ (c) $\frac{1-x}{y-1}$ (d) $\frac{1+y}{1-x}$

Answer

(c) $\frac{1-x}{y-1}$

Differentiating $x+y=xy$ with respect to $y$ : $\frac{dx}{dy}+1=x+y\frac{dx}{dy}$ , so $\frac{dx}{dy}(1-y)=x-1$ and $\frac{dx}{dy}=\frac{x-1}{1-y}=\frac{1-x}{y-1}$ .

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2076 Model · 1 mark

If $x=2at$ , $y=at^2$ then $\frac{dy}{dx}=$ (a) $t$ (b) $\frac1t$ (c) $t^2$ (d) 1

Answer

(a) $t$

$\frac{dy}{dt}=2at$ , $\frac{dx}{dt}=2a$ , so $\frac{dy}{dx}=\frac{2at}{2a}=t$ .

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2076 Model · 1 mark

If the total cost function $C=50x+5000$ . The marginal cost function MC is (a) $50+\frac{5000}{x}$ (b) $25x^2+5000x$ (c) 50 (d) none of these

Answer

(c) 50

$MC=\frac{dC}{dx}=50$ .

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2076 Model · 1 mark

The maximum value of $\sin\theta\cos\theta$ is (a) 1 (b) $\frac{1}{\sqrt2}$ (c) $\frac{\sqrt3}{2}$ (d) $\frac12$

Answer

(d) $\frac12$

$\sin\theta\cos\theta=\frac12\sin2\theta$ , whose maximum is $\frac12$ (when $\sin2\theta=1$ ).

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2076 Model · 1 mark

The function $f(x)=2-3x$ is (a) Decreasing (b) Increasing (c) Neither decreasing nor increasing (d) Doesn't exist

Answer

(a) Decreasing

$f'(x)=-3<0$ for all $x$ , so $f$ is decreasing.

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2076 Model · 1 mark

$\int\log x\,dx=$ (a) $\log x+c$ (b) $x\log x-x+c$ (c) $x\log x+x+c$ (d) $x\log x+c$

Answer

(b) $x\log x-x+c$

Integrating by parts: $\int\log x\cdot1\,dx=x\log x-\int x\cdot\frac1x\,dx=x\log x-x+c$ .

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2076 Model · 1 mark

$\int_{-2}^{2}|x|\,dx$ is equal to (a) 0 (b) 1 (c) 2 (d) 4

Answer

(d) 4

$\int_{-2}^{2}|x|\,dx=2\int_0^2x\,dx=2\cdot\frac{4}{2}=4$

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2076 Model · 1 mark

The area of the circle $x^2+y^2=16$ is (a) $4\pi$ (b) $4\pi^2$ (c) $16\pi^2$ (d) $16\pi$

Answer

(d) $16\pi$

The radius is $4$ , so the area is $\pi r^2=16\pi$ .

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2076 Model · 1 mark

Which one is the measure of dispersion? (a) mean (b) median (c) mode (d) range

Answer

(d) range

Mean, median and mode are measures of central tendency; range measures dispersion.

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2076 Model · 1 mark

Two brothers: Mr. X and Mr. Y appear in an interview for getting the scholarship. The scholarship can be provided for two persons. The probability of getting scholarship by Mr. X is $\frac17$ and getting by Mr. Y is $\frac15$ . What is the probability that both of them will get scholarship? (a) $\frac{12}{35}$ (b) $\frac{1}{35}$ (c) $\frac{23}{35}$ (d) none of these

Answer

(b) $\frac{1}{35}$

The events are independent, so $P=\frac17\times\frac15=\frac1{35}$ .

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2076 Model · 1 mark

The equation $\sin x+\cos x=2$ has (a) infinite solution (b) finite solution (c) no solution (d) unique solution

Answer

(c) no solution

$\sin x+\cos x=\sqrt2\sin\left(x+\frac\pi4\right)\le\sqrt2<2$ , so the equation has no solution.

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2076 Model · 1 mark

The system of linear equations $2x+3y=5$ and $4x+6y=9$ are (a) Consistent & dependent (b) Consistent & independent (c) Inconsistent & independent (d) Inconsistent & dependent

Answer

(c) Inconsistent & independent

Multiplying the first equation by 2 gives $4x+6y=10$ , which contradicts $4x+6y=9$ . The lines are parallel, so there is no solution (inconsistent).

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2076 Model · 1 mark

The degree of polynomial equation $ax^2+bx+c=0$ is (a) 0 (b) 1 (c) 2 (d) not define

Answer

(c) 2

The highest power of $x$ is 2, so the degree is 2.

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2076 Model · 1 mark

If one root of a quadratic equation in the real coefficient is $\sqrt2$ then equation is (a) $x^2+\sqrt2x+2=0$ (b) $x^2+2=0$ (c) $x^2-2=0$ (d) none of these

Answer

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

The equation with roots $\sqrt2$ and $-\sqrt2$ has sum $0$ and product $-2$ , giving $x^2-2=0$ . The other options have non-real roots or do not have $\sqrt2$ as a root.

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2076 Model · 1 mark

The equation $x^2+y^2=0$ represent (a) circle (b) degenerate circle (c) an empty set (d) a straight line

Answer

(b) degenerate circle

$x^2+y^2=0$ is satisfied only by $(0,0)$ , a circle of radius zero (a point circle).

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2076 Model · 1 mark

If four quantities $a, b, c$ and d are such that $a:b=3:4$ , $b:c=6:7$ and $c:d=8:9$ , then the ratio between a and d is (a) 3:8 (b) 1:3 (c) 4:7 (d) 7:4

Answer

(c) 4:7

$\frac ad=\frac34\cdot\frac67\cdot\frac89=\frac{144}{252}=\frac47$

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2076 Model · 1 mark

By selling 175 pens, a stationary store gains the selling price of 50 pens. What is his gain percent? (a) 40% (b) $28\frac47$ % (c) 50% (d) none of these

Answer

(a) 40%

Gain on 175 pens = SP of 50 pens, so CP of 175 pens = SP of 125 pens. Let SP of one pen be 1; then CP of 175 pens = 125 and gain = 50.

$\text{Gain \%}=\frac{50}{125}\times100=40\%$

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2076 Model · 1 mark

If US USD 1 = Rs. 78.15. How many US dollars are equivalent To Rs. 10,000? (a) USD 127.96 (b) USD 172.96 (c) USD 196.72 (d) USD 172.62

Answer

(a) $127.96

$\frac{10000}{78.15}=127.96$

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2076 Model · 1 mark

The effective rate corresponding to 10% p.a. convertible quarterly is (a) 40% (b) 10.1% (c) 10.4% (d) none of these

Answer

(c) 10.4%

$\text{Effective rate}=\left(1+\frac{0.10}{4}\right)^4-1=(1.025)^4-1=0.1038\approx10.4\%$

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2077 Model · 1 mark

If $n(\cup)=200$ , $n(A)=80$ , $n(B)=60$ , what is the maximum value of $n(A\cup B)=$ ? (a) 200 (b) 40 (c) 140 (d) 110

Answer

(c) 140

$n(A\cup B)=n(A)+n(B)-n(A\cap B)$ is largest when $A\cap B=\emptyset$ , giving $80+60=140$ , which is below 200.

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2077 Model · 1 mark

$-1\leq x\leq11$ is equivalent to (a) $|x-5|\leq6$ (b) $|x+5|\geq3$ (c) $|5-x|\geq2$ (d) $|5-2x|<1$

Answer

(a) $|x-5|\leq6$

$|x-5|\leq6 \iff -6\leq x-5\leq6 \iff -1\leq x\leq11$ .

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2077 Model · 1 mark

Range of $y=x^2-4x+5$ is (a) $(-\infty,\infty)$ (b) $(-\infty,1]$ (c) $[1,\infty)$ (d) $(-\infty,1)$

Answer

(c) $[1,\infty)$

$y=x^2-4x+5=(x-2)^2+1\geq1$ , and every value $\geq1$ is attained.

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2077 Model · 1 mark

Period of function of $\cos2x$ is (a) $\frac\pi2$ (b) 1 (c) $\pi$ (d) $\frac\pi4$

Answer

(c) $\pi$

Period of $\cos kx$ is $\frac{2\pi}{k}$ , so for $k=2$ it is $\pi$ .

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2077 Model · 1 mark

If $\log_981=x$ then $x$ (a) $\frac12$ (b) $\sqrt2$ (c) 1 (d) 2

Answer

(d) 2

$\log_9 81=x \Rightarrow 9^x=81=9^2$ , so $x=2$ .

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2077 Model · 1 mark

If $x=\log_abc$ , $y=\log_bca$ , $z=\log_cab$ then $\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}=$ (a) 1 (b) 0 (c) $\frac12$ (d) abc

Answer

(a) 1

$x+1=\log_a bc+\log_a a=\log_a abc$ , so $\frac1{x+1}=\log_{abc}a$ . Similarly $\frac1{y+1}=\log_{abc}b$ and $\frac1{z+1}=\log_{abc}c$ .

Sum $=\log_{abc}(abc)=1$ .

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2077 Model · 1 mark

$1.5+0.015+0.00015+\ldots\ldots$ (a) $\frac{15}{11}$ (b) $\frac{50}{33}$ (c) $\frac23$ (d) sum doesn't exist

Answer

(b) $\frac{50}{33}$

This is a G.P. with $a=1.5$ , $r=0.01$ , so $S=\frac{1.5}{1-0.01}=\frac{1.5}{0.99}=\frac{150}{99}=\frac{50}{33}$ .

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2077 Model · 1 mark

The system of equations $kx-2y=0$ , $x+3y=0$ have unique solution then (a) $k=-\frac23$ (b) $k=0$ (c) $k=\frac23$ (d) $k\neq-\frac23$

Answer

(d) $k\neq-\frac23$

A homogeneous system has a unique (trivial) solution when the determinant is non-zero: $3k-(-2)(1)=3k+2\neq0$ , so $k\neq-\frac23$ .

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2077 Model · 1 mark

If one root of $ax^2+bx+c=0$ be square of other then (a) $b^3+ac^2+a^2c+3abc=0$ (b) $b^3+a^2c+ac^2-3abc=0$ (c) $a^3+bc^2+b^2c-3abc=0$ (d) $a^3+b^3+c^3-3abc=0$

Answer

(b) $b^3+a^2c+ac^2-3abc=0$

Let the roots be $\alpha,\alpha^2$ . Then $\alpha+\alpha^2=-\frac ba$ and $\alpha^3=\frac ca$ .

Cubing the sum: $(\alpha+\alpha^2)^3=\alpha^3+\alpha^6+3\alpha^3(\alpha+\alpha^2)$ , so

$-\frac{b^3}{a^3}=\frac ca+\frac{c^2}{a^2}-\frac{3bc}{a^2}$

Multiplying by $-a^3$ gives $b^3+a^2c+ac^2-3abc=0$ .

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2077 Model · 1 mark

The amplitude of $1-i$ is (a) $45^\circ$ (b) $135^\circ$ (c) $315^\circ$ (d) doesn't exist

Answer

(c) $315^\circ$

$1-i$ lies in the fourth quadrant with $\tan\theta=-1$ , so $\theta=-45^\circ$ , i.e. $315^\circ$ as a positive angle.

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2077 Model · 1 mark

The mean proportional between 4 & 16 is (a) 4 (b) 8 (c) 12 (d) 5

Answer

(b) 8

Mean proportional $=\sqrt{4\times16}=\sqrt{64}=8$ .

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2077 Model · 1 mark

A plane left 30 minutes later than the schedules time and in order to reach its destinations 1500 km always in time it has to increase its speed by 250 km/hr from its usual speed. What is its usual speed? (a) 500 km/hr (b) 600 km/hr (c) 750 km/hr (d) 100 km/hr

Answer

(c) 750 km/hr

Let the usual speed be $v$ . Then

$\frac{1500}{v}-\frac{1500}{v+250}=\frac12 \Rightarrow v(v+250)=750000 \Rightarrow v^2+250v-750000=0$

$(v-750)(v+1000)=0$ , so $v=750$ km/hr. Check: $2$ h $-1.5$ h $=0.5$ h.

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2077 Model · 1 mark

The present worth of Rs 2040 due 4 years hence reaching interested at 5% p.a is (a) 1650.50 (b) Rs. 1675.3 (c) Rs 1680 (d) Rs 1700

Answer

(d) Rs 1700

Present worth $=\frac{A}{1+\frac{rt}{100}}=\frac{2040}{1+\frac{5\times4}{100}}=\frac{2040}{1.2}=\text{Rs }1700$ .

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2077 Model · 1 mark

The interest rate doubles a sum in 10 year is (a) 12% (b) 11% (c) 10% (d) 9%

Answer

(c) 10%

Interest equals principal: $P=\frac{P\times r\times10}{100}$ , so $r=10\%$ .

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2077 Model · 1 mark

Profits in the business of Bikram, Bishal and Binita are shared in ratio 1:2:3. If Bikram share of profits is Rs. 300, what is Binita share? (a) Rs. 150 (b) Rs. 450 (c) Rs. 600 (d) Rs. 900

Answer

(d) Rs. 900

Bikram's 1 part $=$ Rs 300, so Binita's 3 parts $=3\times300=$ Rs 900.

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2077 Model · 1 mark

If A and B are two independent events with $p(A)=\frac23$ & $p(B)=\frac35$ then $p(A\cup B)=$ (a) $\frac{6}{15}$ (b) $\frac53$ (c) $\frac25$ (d) $\frac{13}{15}$

Answer

(d) $\frac{13}{15}$

For independent events $P(A\cap B)=\frac23\cdot\frac35=\frac25$ .

$P(A\cup B)=\frac23+\frac35-\frac25=\frac23+\frac15=\frac{13}{15}$ .

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2077 Model · 1 mark

For Negative skewness (a) Mean>Mode>median (b) Mean<median<Mode (c) Median<mode<Mean (d) Mode<mean<Median

Answer

(b) Mean<median<Mode

In a negatively skewed distribution the tail is on the left, so the mean is pulled lowest and the mode is highest.

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2077 Model · 1 mark

HCF of $n!$ , $(n+1)!$ and $(n+2)!$ is (a) $n!$ (b) $(n+2)!$ (c) $(n+1)!$ (d) $(n-1)!$

Answer

(a) $n!$

$(n+1)!=(n+1)\,n!$ and $(n+2)!=(n+2)(n+1)\,n!$ , so $n!$ divides all three and is the HCF.

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2077 Model · 1 mark

If $\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\frac\pi2$ then $xy+yz+zx=$ (a) 1 (b) -1 (c) 0 (d) $\pi$

Answer

(a) 1

$\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\frac\pi2$ gives $\tan^{-1}x+\tan^{-1}y=\frac\pi2-\tan^{-1}z=\cot^{-1}z$ .

Taking tangent: $\frac{x+y}{1-xy}=\frac1z$ , so $xz+yz=1-xy$ , i.e. $xy+yz+zx=1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

In any $\Delta ABC$ , $2\cos A\sin C=\sin B$ then $\Delta$ is (a) Scalene (b) Equilateral (c) Isosceles (d) right angled

Answer

(c) Isosceles

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

$\cos A\sin C=\sin A\cos C \Rightarrow \sin(C-A)=0 \Rightarrow A=C$ .

Hence the triangle is isosceles.

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

For what value of 'K' points $(1,-1)$ , $\left(\frac k2,\frac k3\right)$ and $(4,1)$ are collinear? (a) $-\frac32$ (b) 3 (c) 2 (d) $-\frac52$

Answer

Collinearity requires

$\begin{vmatrix}1&-1&1\\ \frac k2&\frac k3&1\\ 4&1&1\end{vmatrix}=0.$

Expanding: $\left(\frac k3-1\right)+\left(\frac k2-4\right)+\left(\frac k2-\frac{4k}{3}\right)=-5$ , which is independent of $k$ and never zero. As printed, no value of $k$ makes the three points collinear, so none of the options satisfies the condition.

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

The distance between parallel lines $3x+2y=6$ and $3x+2y=12$ is (a) $\frac{14}{5}$ (b) $\frac72$ (c) $-\frac{8}{\sqrt{13}}$ (d) $\frac{6}{\sqrt{13}}$

Answer

(d) $\frac{6}{\sqrt{13}}$

Distance between $ax+by+c_1=0$ and $ax+by+c_2=0$ is $\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}=\frac{|-6+12|}{\sqrt{9+4}}=\frac6{\sqrt{13}}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

Angles between the lines represented by $2x^2-7xy+6y^2=0$ is (a) $\tan^{-1}\left(\pm\frac18\right)$ (b) $\tan^{-1}\left(\pm\frac17\right)$ (c) $45^\circ$ (d) $60^\circ$

Answer

(a) $\tan^{-1}\left(\pm\frac18\right)$

Here $a=2$ , $h=-\frac72$ , $b=6$ :

$\tan\theta=\left|\frac{2\sqrt{h^2-ab}}{a+b}\right|=\frac{2\sqrt{\frac{49}4-12}}{8}=\frac{2\cdot\frac12}{8}=\frac18.$

Check: the lines are $2x-3y=0$ and $x-2y=0$ with slopes $\frac23,\frac12$ , giving $\frac{1/6}{4/3}=\frac18$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

Length of y-intercept made by circle $x^2+y^2-8x+y-20=0$ is (a) 5 (b) 9 (c) 8 (d) 6

Answer

(b) 9

Put $x=0$ : $y^2+y-20=0 \Rightarrow (y+5)(y-4)=0$ , so $y=4,-5$ . The intercept length is $4-(-5)=9$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

Radius of circle $x^2+y^2+2x+3y+k=0$ is $\frac12$ then $k=$ (a) 2 (b) -1 (c) 3 (d) 4

Answer

(c) 3

$r^2=g^2+f^2-c=1+\frac94-k=\frac14$ , so $k=3$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

$\lim_{x\to64}\frac{\sqrt[6]{x}-2}{\sqrt[3]{x}-4}$ (a) $\frac12$ (b) $\frac34$ (c) $\frac23$ (d) $\frac14$

Answer

(d) $\frac14$

Put $x=t^6$ ; as $x\to64$ , $t\to2$ :

$\lim_{t\to2}\frac{t-2}{t^2-4}=\lim_{t\to2}\frac1{t+2}=\frac14.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

$\lim_{x\to0}\frac{e^{\sin x}-1}{x}$ (a) 0 (b) 1 (c) 2 (d) $\log_e2$

Answer

(b) 1

$\frac{e^{\sin x}-1}{x}=\frac{e^{\sin x}-1}{\sin x}\cdot\frac{\sin x}{x}\to1\cdot1=1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

Every continuous function is (a) increasing (b) decreasing (c) differentiable (d) not - differentiable

Answer

(c) differentiable

Strictly, the true theorem is the converse: every differentiable function is continuous. A continuous function need not be differentiable (e.g. $|x|$ at $0$ ), and need not be monotonic. Of the given options, (c) is the one intended, relating continuity and differentiability.

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

$\frac{d}{dx}\cos^{-1}(\sin x)$ (a) x (b) -1 (c) $\frac{-\sin x}{\sqrt{1-\sin^2x}}$ (d) $\frac{\cos x}{\sqrt{1-\sin^2x}}$

Answer

(b) $-1$

$\cos^{-1}(\sin x)=\cos^{-1}\left(\cos\left(\frac\pi2-x\right)\right)=\frac\pi2-x$ , whose derivative is $-1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

If $x^py^q=(x+y)^{p+q}$ then $\frac{dy}{dx}=$ (a) $\frac yx$ (b) $\frac{py}{qx}$ (c) $\frac xy$ (d) $\frac{x^2}{y^2}$

Answer

(a) $\frac yx$

Take logs: $p\ln x+q\ln y=(p+q)\ln(x+y)$ . Differentiate:

$\frac px+\frac qy y'=\frac{(p+q)(1+y')}{x+y}.$

Collecting $y'$ : $y'\left(\frac qy-\frac{p+q}{x+y}\right)=\frac{p+q}{x+y}-\frac px$ . This simplifies to $y'\cdot\frac{qx-py}{y(x+y)}=\frac{qx-py}{x(x+y)}$ , so $y'=\frac yx$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

Derivatives of $\sin^{-1}x$ is (a) $\sqrt{1-x^2}$ (b) $\frac{1}{\sqrt{1-x^2}}$ (c) $\frac{-1}{\sqrt{1-x^2}}$ (d) $-\frac{1}{\sqrt{x^2-1}}$

Answer

(b) $\frac1{\sqrt{1-x^2}}$

Standard result: $\frac{d}{dx}\sin^{-1}x=\frac1{\sqrt{1-x^2}}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

At critical point, $f(x)$ is (a) increasing (b) decreasing (c) stationary (d) increasing & decreasing

Answer

(c) stationary

At a critical point $f'(x)=0$ (or is undefined), so the function is momentarily stationary there, neither increasing nor decreasing.

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

$\int\cot x\,dx=$ (a) $\ln(\sec x)+c$ (b) $\ln(\operatorname{cosec}x)+c$ (c) $\ln(\sin x)+c$ (d) $-\operatorname{cosec}^2x+c$

Answer

(c) $\ln(\sin x)+c$

$\int\cot x\,dx=\int\frac{\cos x}{\sin x}dx=\ln|\sin x|+c$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

$\int\frac{e^x-1}{e^x+1}dx$ (a) $e^x-\log(e^x+1)$ (b) $e^x+e^{-x}$ (c) $e^x-\log(e^{-x}+1)$ (d) $2\log\left(e^{\frac x2}+e^{-\frac x2}\right)$

Answer

(d) $2\log\left(e^{x/2}+e^{-x/2}\right)$

$\frac{e^x-1}{e^x+1}=\frac{e^{x/2}-e^{-x/2}}{e^{x/2}+e^{-x/2}}.$

The numerator is the derivative of the denominator times $2$ ( $\frac{d}{dx}(e^{x/2}+e^{-x/2})=\frac12(e^{x/2}-e^{-x/2})$ ), so the integral is $2\log\left(e^{x/2}+e^{-x/2}\right)+c$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

$\int\log x\,dx$ (a) $\frac1x$ (b) $x\log x-x$ (c) $e^x$ (d) $\frac{e^x}{x}$

Answer

(b) $x\log x-x$

By parts with $u=\log x$ , $dv=dx$ : $\int\log x\,dx=x\log x-\int1\,dx=x\log x-x+c$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 Model · 1 mark

Area of region bounded by curve $y^2=4ax$ and $x^2=4by$ is (a) $\frac{4ab}{3}$ (b) $\frac{8ab}{3}$ (c) $\frac{16ab}{3}$ (d) $\frac{2ab}{3}$

Answer

(c) $\frac{16ab}{3}$

The curves meet at $x=0$ and $x=4a^{1/3}b^{2/3}$ (so $x^3=64ab^2$ ). Area $=\int\left(2\sqrt{a}\sqrt{x}-\frac{x^2}{4b}\right)dx$ between these limits:

$A=\frac43\sqrt a\,x^{3/2}-\frac{x^3}{12b}=\frac43\sqrt a\cdot8\sqrt a\,b-\frac{64ab^2}{12b}=\frac{32ab}{3}-\frac{16ab}{3}=\frac{16ab}{3}.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

If the ratio of cost price and selling price of an article is 2:3 then (a) Profit % = 50% (b) loss % = 30% (c) Neither profit or loss (d) Profit % = $\frac{100}{3}$ %

Answer

(a) Profit % = 50%

Let CP = 2k and SP = 3k. Then profit = k, and

$\text{Profit \%}=\frac{k}{2k}\times100=50\%.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

$\lim_{n\to\infty}\frac{\sum n^2}{n^3}=$ (a) 0 (b) $\infty$ (c) 2 (d) $\frac13$

Answer

(d) $\frac13$

Since $\sum n^2=\frac{n(n+1)(2n+1)}{6}$ ,

$\lim_{n\to\infty}\frac{n(n+1)(2n+1)}{6n^3}=\lim_{n\to\infty}\frac{(1+1/n)(2+1/n)}{6}=\frac26=\frac13.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

$\lim_{x\to1}(x-1)\tan\frac{\pi x}{2}$ (a) $-\frac2\pi$ (b) $\frac\pi2$ (c) $-\frac12$ (d) $-\frac1\pi$

Answer

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

Put $x=1+t$ with $t\to0$ . Then $\tan\frac{\pi(1+t)}{2}=-\cot\frac{\pi t}{2}$ , so

$\lim_{t\to0}\left(-t\cot\frac{\pi t}{2}\right)=-\lim_{t\to0}\frac{t}{\tan(\pi t/2)}=-\frac{2}{\pi}.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

If $y=\sqrt{\frac{1+\cos2x}{1-\cos2x}}$ then $\frac{dy}{dx}$ at $x=\frac\pi2$ is (a) 0 (b) 1 (c) not defined (d) -1

Answer

(c) not defined

Since $1+\cos2x=2\cos^2x$ and $1-\cos2x=2\sin^2x$ , we get $y=|\cot x|$ .

For $x$ just below $\frac\pi2$ , $y=\cot x$ and $y=-csc^2xto-1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

Derivative of $e^{\sqrt{x^2+a^2}}$ w.r.t $\sqrt{x^2+a^2}$ (a) $e^{-\sqrt{x^2+a^2}}$ (b) $\frac{e^{\sqrt{x^2+a^2}}}{2x}$ (c) $\frac{xe^{\sqrt{x^2+a^2}}}{2\sqrt{x^2+a^2}}$ (d) $e^{\sqrt{x^2+a^2}}$

Answer

(d) $e^{\sqrt{x^2+a^2}}$

Let $u=\sqrt{x^2+a^2}$ . Then $\frac{d}{du}e^u=e^u=e^{\sqrt{x^2+a^2}}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

If $a+b+c=0$ in quadratic equation $ax^2+bx+c=0$ then roots of quadratic equation is (a) 1 and -2 (b) 1 and $\frac ca$ (c) $-\frac ba$ and $\frac ca$ (d) 0 and 1

Answer

(b) 1 and $\frac{c}{a}$

If $a+b+c=0$ , then $x=1$ satisfies the equation. The product of the roots is $\frac ca$ , so the other root is $\frac ca$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

If $(m+n)^{th}$ term of a G.P is p and $(m-n)^{th}$ term is q then $m^{th}$ term is (a) $\sqrt{pq}$ (b) $\frac{p+q}{2}$ (c) $\sqrt{(p+q)}$ (d) $pq$

Answer

(a) $\sqrt{pq}$

Let the first term be $a$ and the common ratio $r$ . Then $ar^{m+n-1}=p$ and $ar^{m-n-1}=q$ .

$pq=a^2r^{2m-2}=(ar^{m-1})^2$

The $m^{th}$ term is $ar^{m-1}=\sqrt{pq}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

$(a+b-c)x^2+(c+a-b)x+(b+c-a)=0$ and $a+b+c=0$ then roots quadratic equation, if a, b and c are rational (a) rational (b) irrational (c) imaginary (d) Real and equal

Answer

(a) rational

The sum of the coefficients is $(a+b-c)+(c+a-b)+(b+c-a)=a+b+c=0$ , so $x=1$ is a root. The product of the roots is $\frac{b+c-a}{a+b-c}$ , so the other root is $\frac{b+c-a}{a+b-c}$ . This is rational when $a,b,c$ are rational, so both roots are rational.

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

The quadratic equation whose roots are $\frac35$ times the roots if quadratic equation $x^2-8x+7=0$ (a) $25x^2-30x+7=0$ (b) $25x^2-120x+63=0$ (c) $25x^2+120x-63=0$ (d) none of these

Answer

(b) $25x^2-120x+63=0$

The roots of $x^2-8x+7=0$ are 1 and 7. The new roots are $\frac35$ and $\frac{21}{5}$ .

Sum $=\frac{24}{5}$ and product $=\frac{63}{25}$ , so the equation is

$x^2-\frac{24}{5}x+\frac{63}{25}=0\;\Rightarrow\;25x^2-120x+63=0.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

$S_n=\frac{a(1-r^n)}{1-r}$ , sum of geometric progression is true if (a) $|r|>1$ (b) $|r|<1$ (c) $|r|=1$ (d) $|r^n|\leq1$

Answer

(b) $|r|<1$

The formula $S_n=\frac{a(1-r^n)}{1-r}$ is the standard form used when $|r|<1$ . It is also valid algebraically for any $r\ne1$ . For $|r|<1$ the sum also converges to $\frac{a}{1-r}$ as $n\to\infty$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

Square roots of $(3+4i)$ are (a) $\pm(2+i)$ (b) $\pm(3+i)$ (c) $\pm(2-i)$ (d) $\pm(3-i)$

Answer

(a) $\pm(2+i)$

$(2+i)^2=4+4i+i^2=3+4i$ , so the square roots of $3+4i$ are $\pm(2+i)$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

$(1+i)^{100}+(1-i)^{100}=$ (a) 0 (b) -1 (c) 1 (d) i

Answer

None of the options is correct. The value is $-2^{51}$ .

$(1+i)^2=2i$ and $(1-i)^2=-2i$ , so

$(1+i)^{100}=(2i)^{50}=2^{50}i^{50}=-2^{50},\qquad (1-i)^{100}=(-2i)^{50}=2^{50}i^{50}=-2^{50}.$

Hence the sum is $-2^{51}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

The point of inflection of quadratic function is (a) always (1,0) (b) always (0,0) (c) does not exist (d) $(-\infty,0)$

Answer

(c) does not exist

A quadratic function $ax^2+bx+c$ has $f''(x)=2a\ne0$ everywhere. Its concavity never changes, so it has no point of inflection.

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

The area of triangle with vertices $(1,2),(2,3)$ and $(3,4)$ is (a) 0 sq. unit (b) 1 sq. unit (c) 2 sq. unit (d) 3 sq. unit

Answer

(a) 0 sq. unit

$\text{Area}=\tfrac12|1(3-4)+2(4-2)+3(2-3)|=\tfrac12|-1+4-3|=0$

The points all lie on the line $y=x+1$ , so they are collinear.

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

The polar coordinate of point $(-1,-\sqrt3)$ is (a) $\left(\frac\pi3,2\right)$ (b) $\left(2,\frac\pi3\right)$ (c) $\left(2,\frac{4\pi}{3}\right)$ (d) $\left(\frac{4\pi}{3},2\right)$

Answer

(c) $\left(2,\frac{4\pi}{3}\right)$

$r=\sqrt{1+3}=2$ . The point $(-1,-\sqrt3)$ lies in the third quadrant, where $\tan\theta=\sqrt3$ , so $\theta=\pi+\frac\pi3=\frac{4\pi}{3}$ . In the form $(r,\theta)$ this is $\left(2,\frac{4\pi}{3}\right)$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

If the pair of lines represented by $ax^2+2hxy+by^2+2gx+2fy+c=0$ intersect at $x-axis$ then (a) $abc+2fgh=bg^2+ch^2$ (b) $abc+2fgh=0$ (c) $abc+2fgh=af^2+ch^2$ (d) $2fgh=bg^2+ch^2$

Answer

(c) $abc+2fgh=af^2+ch^2$

Let the lines meet at $(x_0,0)$ on the x-axis. Putting $y=0$ gives $ax^2+2gx+c=0$ , which must have equal roots, so $g^2=ac$ and hence $abc=bg^2$ . The pair condition is

$abc+2fgh=af^2+bg^2+ch^2.$

Substituting $abc=bg^2$ gives $2fgh=af^2+ch^2$ , which is the relation of the form given in option (c).

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2078 Model · 1 mark

$\int_{-1}^{1}\sin^7x\,dx=$ (a) 6 (b) -6 (c) 0 (d) -1

Answer

(c) 0

$\sin^7x$ is an odd function, and the interval $[-1,1]$ is symmetric about 0, so

$\int_{-1}^{1}\sin^7x\,dx=0.$

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2078 Model · 1 mark

$|x-a|=$ (a) $(a-x)$ if $x>a$ (b) $\pm(x-a)$ if $x\geq a$ (c) $(x-a)$ if $x<a$ (d) $(a-x)$ if $x<a$

Answer

(d) $(a-x)$ if $x<a$

By definition $|x-a|=x-a$ for $x\ge a$ and $|x-a|=-(x-a)=a-x$ for $x<a$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

$\int\frac{x}{x+1}dx=$ (a) $x-\log x+c$ (b) $x-\log(1+x)+c$ (c) $x+\log x+c$ (d) $x+\log(1+x)+c$

Answer

(b) $x-\log(1+x)+c$

$\int\frac{x}{x+1}dx=\int\left(1-\frac{1}{x+1}\right)dx=x-\log(1+x)+c.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

Which of the following statements is are true (a) All scalar matrix are diagonal matrix (b) All diagonal matric are scalar matric but not vice-versa (c) if $a_{ij}=-a_{ji}\ \forall\ i$ and $j$ and $a_{ij}=0$ for $i=j$ then matrix is called skew -symmetric matrix (d) both a and c

Answer

(d) both a and c

A scalar matrix is a diagonal matrix with all diagonal entries equal, so (a) is true. (b) is false, because not every diagonal matrix is scalar (the converse is the true statement). (c) is the correct definition of a skew-symmetric matrix. So (a) and (c) are true.

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

Find the inverse of the function $f(x)=3x^4+2$ (a) $\sqrt{\frac{x-2}{3}}$ (b) $\sqrt[4]{\frac{x-2}{3}}$ (c) $\frac{x-2}{3}$ (d) $\frac{1+x^4}{3}$

Answer

(b) $\sqrt[4]{\dfrac{x-2}{3}}$

Let $y=3x^4+2$ . Then $x^4=\frac{y-2}{3}$ , so $x=\sqrt[4]{\frac{y-2}{3}}$ (taking the principal root, i.e. $x\ge0$ ). Interchanging variables gives $f^{-1}(x)=\sqrt[4]{\frac{x-2}{3}}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

Domain of the function $f(x)=\frac{x^2-4}{x-2}$ is (a) $R-\{2\}$ (b) $R$ (c) $R-\{4\}$ (d) $[0,\infty]$

Answer

(a) $R-\{2\}$

The function is undefined only where the denominator is zero, that is at $x=2$ . So the domain is $R-\{2\}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

$\log_2\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt2}}}}=$ (a) $\frac{31}{32}$ (b) $2^{31}$ (c) $\frac{63}{64}$ (d) $2^{64}$

Answer

(a) $\frac{31}{32}$

The nested expression equals $2^{1/2+1/4+1/8+1/16+1/32}=2^{31/32}$ , so

$\log_2 2^{31/32}=\frac{31}{32}.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

$\begin{vmatrix}1&2&3\\4&5&6\\7&8&9\end{vmatrix}=$ (a) 0 (b) 1 (c) -1 (d) 2

Answer

(a) 0

$\begin{vmatrix}1&2&3\\4&5&6\\7&8&9\end{vmatrix}=1(45-48)-2(36-42)+3(32-35)=-3+12-9=0.$

(Equivalently, $R_3-R_2=R_2-R_1$ .)

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2078 Model · 1 mark

$\tan15^\circ=$ (a) $\frac{1}{2+\sqrt3}$ (b) $2-\sqrt3$ (c) $\frac{1}{2-\sqrt3}$ (d) $2+\sqrt3$

Answer

(b) $2-\sqrt3$

$\tan15^\circ=\tan(45^\circ-30^\circ)=\frac{1-\frac1{\sqrt3}}{1+\frac1{\sqrt3}}=\frac{\sqrt3-1}{\sqrt3+1}=\frac{(\sqrt3-1)^2}{2}=2-\sqrt3.$

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2078 Model · 1 mark

$\vec a\times\vec b=|\vec a||\vec b|\sin\theta\,\hat\eta$ , for any non-zero vectors $\vec a$ and $\vec b$ (a) $\hat\eta$ is the unit vector along $\vec a$ (b) $\hat\eta$ is the unit vector along $\vec b$ (c) $\hat\eta$ is the unit vector along $\vec a\times\vec b$ (d) $\hat\eta$ is the unit vector along $\vec a$ and $\vec b$

Answer

(c) $\hat\eta$ is the unit vector along $\vec a\times\vec b$ . By definition the cross product has magnitude $|\vec a||\vec b|\sin\theta$ and direction given by the unit vector $\hat\eta$ perpendicular to both $\vec a$ and $\vec b$ (right-hand rule), i.e. along $\vec a\times\vec b$ .

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2078 Model · 1 mark

If $A=\{x: x\in R, \sqrt{2x^2-1}=-1\}$ then A is (a) Finite set with elements $\pm1$ (b) null set (c) infinite set (d) data insufficient

Answer

(b) null set. Since $\sqrt{2x^2-1}\ge 0$ for every real $x$ for which it is defined, it can never equal $-1$ . So no real $x$ satisfies the condition and $A=\varnothing$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

$\sin^{-1}\frac{1-x^2}{1+x^2}+\cos^{-1}\frac{2x}{1+x^2}=$ (a) $\frac\pi2-\tan^{-1}x$ (b) $\frac\pi2-2\tan^{-1}x$ (c) $\pi-2\tan^{-1}x$ (d) none of these

Answer

(d) none of these. Put $x=\tan\theta$ . Then $\frac{1-x^2}{1+x^2}=\cos2\theta$ and $\frac{2x}{1+x^2}=\sin2\theta$ , so for $0\le\theta\le\pi/4$ the sum is $(\frac\pi2-2\theta)+(\frac\pi2-2\theta)=\pi-4\tan^{-1}x$ . Check: at $x=1$ the sum is $0+0=0$ , which matches $\pi-4\cdot\frac\pi4=0$ but not options (a), (b) or (c).

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

$\cos x=$ (a) $2\cos^2\frac x2-1$ (b) $1-2\sin^2\frac x2$ (c) $\frac{1-\tan^2\frac x2}{1+\tan^2\frac x2}$ (d) All of these

Answer

(d) All of these. $\cos x=2\cos^2\frac x2-1=1-2\sin^2\frac x2=\dfrac{1-\tan^2\frac x2}{1+\tan^2\frac x2}$ are all standard half-angle forms.

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

Non- intersecting lines are (a) consistent and dependent (b) Inconsistent and independent (c) Inconsistent and dependent (d) consistent and independent

Answer

(b) Inconsistent and independent. Non-intersecting (parallel) lines have no common point, so the system has no solution (inconsistent), and the equations are not multiples of each other (independent).

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

If the price of petrol in Nepal is increased by 25%. How much percent must a Saral Shikshya student reduce his consumption of petrol so as not to increase his expenditure on petrol? (a) 20% (b) 25% (c) 30% (d) 35%

Answer

(a) 20% Let old price $=100$ and consumption $=1$ . New price $=125$ . To keep expenditure the same, new consumption $=\frac{100}{125}=0.8$ , a reduction of $\frac{25}{125}\times100=20\%$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

If A earns $33\frac13$ % more than B, how much percent does B earn less than A? (a) 23% (b) 24% (c) 20% (d) 25%

Answer

(d) 25% Let B earn $100$ . Then A earns $133\frac13=\frac{400}{3}$ . B earns less than A by $\dfrac{400/3-100}{400/3}\times100=\dfrac{100/3}{400/3}\times100=25\%$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

If $a^2+b^2+c^2-ab-bc-ca=0$ then $a:b:c$ is (a) 1:1:2 (b) 1:1:1 (c) 1:2:1 (d) 2:1:1

Answer

(b) 1:1:1 $a^2+b^2+c^2-ab-bc-ca=\tfrac12[(a-b)^2+(b-c)^2+(c-a)^2]=0$ forces $a=b=c$ , so $a:b:c=1:1:1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

A certain number of people were supposed to complete work in 20 days. The work, however, took 32 days since 9 people were absent throughout. How many people were supposed to be working originally? (a) 36 (b) 38 (c) 34 (d) 30

Answer

No option matches; the correct number is 24. Let $n$ be the original number of people. Total work is constant: $20n=32(n-9)$ , so $12n=288$ and $n=24$ . (None of 36, 38, 34, 30 satisfies this.)

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

The speed of a boat while traveling downstream is 32km/hr, whereas when travelling upstream it is 28 km/hr, what is the speed of a boat in still water? (a) 30km/hr (b) 2km/hr (c) 3km/hr (d) 34km/hr

Answer

(a) 30 km/hr Speed in still water $=\dfrac{32+28}{2}=30$ km/hr.

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

The surface area of a cube is $150\,cm^2$ . Its volume is (a) $64\,cm^2$ (b) $125\,cm^3$ (c) $150\,cm^3$ (d) $216\,cm^3$

Answer

(b) $125\,cm^3$ $6a^2=150\Rightarrow a^2=25\Rightarrow a=5$ cm, so $V=a^3=125\,cm^3$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

If the supply function $P_s=160+2Q^2$ and the demand function is $P_d=240-3Q^2$ . The market equilibrium price is (a) 192 (b) 194 (c) 190 (d) 180

Answer

(a) 192 At equilibrium $P_s=P_d$ : $160+2Q^2=240-3Q^2\Rightarrow5Q^2=80\Rightarrow Q^2=16$ . Then $P=160+2(16)=192$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 Model · 1 mark

Probability of an event $E(x)$ is (a) $1\leq\varepsilon(x)$ (b) $0\leq p(\varepsilon)\leq1$ (c) $0\leq p(\varepsilon)$ (d) $-1\leq p(\varepsilon)\leq1$

Answer

(b) $0\leq p(\varepsilon)\leq1$ The probability of any event always lies between 0 (impossible) and 1 (certain).

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

Which is not a rational number? (a) 1 (b) $e$ (c) 3 (d) -1

Answer

(b) $e$ $e\approx2.71828\ldots$ is irrational, while 1, 3 and $-1$ are rational.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

If A and B are the nonempty set and $A\subset B\subset U$ then $A\cap B=$ (a) A (b) B (c) U (d) A-B

Answer

(a) A Since $A\subset B$ , every element of $A$ is in $B$ , so $A\cap B=A$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

If $|x-1|\leq5$ then $x=$ (a) $\{-4,6\}$ (b) $[-4,6]$ (c) U (d) $[-4,-6]$

Answer

(b) $[-4,6]$ $|x-1|\le5\Rightarrow-5\le x-1\le5\Rightarrow-4\le x\le6$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

If $\log_ax=y$ then $x=$ (a) y (b) a (c) $a^y$ (d) $1/y$

Answer

(c) $a^y$ By definition, $\log_ax=y\iff x=a^y$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

The domain of the function $f(x)=\sqrt{16-x^2}$ is (a) $(-4,4]$ (b) $[-4,4]$ (c) $(0,4)$ (d) $\{0,5\}$

Answer

(b) $[-4,4]$ We need $16-x^2\ge0\Rightarrow x^2\le16\Rightarrow-4\le x\le4$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

A sentence $x+5=4$ is (a) false statement (b) true statement (c) not a statement (d) a statement

Answer

(c) not a statement $x+5=4$ is an open sentence: it is true for $x=-1$ and false otherwise, so it has no definite truth value until $x$ is specified.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

The function $f(x)=\cos x$ is (a) an even function (b) period function (c) an odd function (d) neither even or odd

Answer

(a) an even function $\cos(-x)=\cos x$ for all $x$ , so $\cos x$ is even. (It is also periodic, but the defining property asked here is evenness.)

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

If p, q and r are in G.P, then (a) $p+r=2q$ (b) $q^2<pr$ (c) $q^2<pr$ (d) $q^2=pr$

Answer

(d) $q^2=pr$ In a G.P., $\dfrac qp=\dfrac rq\Rightarrow q^2=pr$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

If $k+3$ , $4k+7$ and $3k-1$ are in AP then the sequence is (a) 7, 8, 9 (b) 5, 7, 9 (c) 7, 9, 11 (d) 1, 3, 5

Answer

(c) 7, 9, 11

For an AP the middle term is the mean of the other two: $2(4k-7)=(k+3)+(3k-1)$ , so $8k-14=4k+2$ and $k=4$ . The terms are $7, 9, 11$ with common difference 2.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

If $\begin{bmatrix}0&6\\5+k&0\end{bmatrix}$ is a skew symmetric matrix then $k=$ (a) -11 (b) -5 (c) 0 (d) -6

Answer

(a) $-11$

A skew-symmetric matrix satisfies $a_{ij}=-a_{ji}$ , so $6=-(5+k)$ , giving $k=-11$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

The value of x for the determinant $\begin{vmatrix}x+3&0\\1&1\end{vmatrix}=0$ is (a) 3 (b) -3 (c) 1 (d) 0

Answer

(b) $-3$

$\begin{vmatrix}x+3&0\\1&1\end{vmatrix}=(x+3)(1)-0=x+3=0$ , so $x=-3$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

The value of $\sqrt{-8}\sqrt{-2}$ is (a) 16 (b) 4 (c) -4 (d) $4i$

Answer

(c) $-4$

$\sqrt{-8}\sqrt{-2}=(2\sqrt2\,i)(\sqrt2\,i)=4i^2=-4$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

The modulus of $\frac{1+i}{1-i}$ is (a) 1 (b) -2 (c) $\sqrt2$ (d) $1/\sqrt2$

Answer

(a) 1

$\left|\frac{1+i}{1-i}\right|=\frac{|1+i|}{|1-i|}=\frac{\sqrt2}{\sqrt2}=1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

In a triangle ABC, if $a=2\sqrt3$ , $b=2$ , $c=4$ then $A=$ (a) $15^\circ$ (b) $30^\circ$ (c) $45^\circ$ (d) $60^\circ$

Answer

(d) $60^\circ$

$\cos A=\frac{b^2+c^2-a^2}{2bc}=\frac{4+16-12}{16}=\frac12$ , so $A=60^\circ$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

The perimeter 2s of the triangle ABC is (a) $a+b+c$ (b) $(a+b+c)/R$ (c) $2a+2b+2c$ (d) $A+b+C$

Answer

(a) $a+b+c$

The perimeter is the sum of the three sides, $2s=a+b+c$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

Area of the triangle ABC is (a) abc (b) abc/R (c) abc/4R (d) 4R/abc

Answer

(c) $\dfrac{abc}{4R}$

Since $\Delta=\frac12 bc\sin A$ and $\sin A=\frac{a}{2R}$ , we get $\Delta=\frac{abc}{4R}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

Equation of a line perpendicular to the line $x-3y+7=0$ is (a) $3x+y+10=0$ (b) $2x-3y=10$ (c) $-3x-y-10=0$ (d) $x+3y=10$

Answer

(a) $3x+y+10=0$

The line $x-3y+7=0$ has slope $\frac13$ , so a perpendicular line has slope $-3$ . The line $3x+y+10=0$ has slope $-3$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

Equation of tangent to $(0,1)$ to the circle $x^2+y^2=1$ is (a) $x=1$ (b) $y=1$ (c) $x=0$ (d) $y=-1$

Answer

(b) $y=1$

The tangent at $(x_1,y_1)$ to $x^2+y^2=1$ is $xx_1+yy_1=1$ . At $(0,1)$ this gives $y=1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

The focus of the parabola $y^2=4x$ is (a) $(0,1)$ (b) $(1,0)$ (c) $(0,0)$ (d) $(-1,0)$

Answer

(b) $(1,0)$

Comparing $y^2=4x$ with $y^2=4ax$ gives $a=1$ , so the focus is $(a,0)=(1,0)$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

If $\lim_{x\to3}\frac{x^n-3^n}{x-3}=27$ , then $n=$ (a) 3 (b) 4 (c) 9 (d) 0

Answer

(a) 3

The limit equals $n\cdot3^{n-1}$ . For $n=3$ this is $3\cdot9=27$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

$\lim_{x\to0}\frac{\sin3x}{x}=$ (a) 1 (b) 0 (c) 0/0 (d) 3

Answer

(d) 3

$\lim_{x\to0}\frac{\sin3x}{x}=3\lim_{x\to0}\frac{\sin3x}{3x}=3\cdot1=3$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

The function $f(x)=\begin{cases}\frac{x^2-x}{x-1}&x\neq1\\k&x=1\end{cases}$ is continuous at $x=1$ for $k=$ (a) 0 (b) 2 (c) 1 (d) 0/0

Answer

(c) 1

For $x\neq1$ , $\frac{x^2-x}{x-1}=x$ , so $\lim_{x\to1}f(x)=1$ . Continuity requires $k=1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

If $y=e^{ax^2+b}$ then $\frac{dy}{dx}=$ (a) $ax^2+b$ (b) $2axy$ (c) $2ax$ (d) $e^{ax^2+b}$

Answer

(b) $2axy$

$\frac{dy}{dx}=e^{ax^2+b}\cdot2ax=2ax\,y$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

If $x^2+y^2=25$ then $\frac{dy}{dx}$ at $(0,5)$ is (a) 0 (b) undefined (c) $x/y$ (d) $-x/y$

Answer

(a) 0

Differentiating, $2x+2y\frac{dy}{dx}=0$ , so $\frac{dy}{dx}=-\frac xy$ . At $(0,5)$ this is $0$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

The stationary point of the function $f(x)=x^2-6x+3$ is $x=$ (a) 0 (b) 3 (c) -3 (d) 6

Answer

(b) 3

$f'(x)=2x-6=0$ gives $x=3$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

$\int e^{\log_ex}\,dx=$ (a) $\log_ex+c$ (b) $x^2+c$ (c) $\frac{x^2}{2}+c$ (d) $e^{\log_ex}+c$

Answer

(c) $\dfrac{x^2}{2}+c$

Since $e^{\log_e x}=x$ , $\int x\,dx=\frac{x^2}{2}+c$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

$\int_0^1\frac{2x}{x^2+2}dx=$ (a) $\log_e\frac23$ (b) $\log_e\frac32$ (c) 0 (d) 3

Answer

(b) $\log_e\dfrac32$

$\int_0^1\frac{2x}{x^2+2}dx=\left[\log_e(x^2+2)\right]_0^1=\log_e3-\log_e2=\log_e\frac32$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

$\int1/\cos^2x\,dx=$ (a) $\cot x+c$ (b) $\cos x+c$ (c) $\sec^2x+c$ (d) $\tan x+c$

Answer

(d) $\tan x+c$

$\frac1{\cos^2x}=\sec^2x$ and $\int\sec^2x\,dx=\tan x+c$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

Area bounded by the x-axis and the curve $y=x+2$ and the lines $x=0$ and $x=2$ is (a) -6 sq. units (b) 2 sq. units (c) 4 sq. units (d) 6 sq. units

Answer

(d) 6 sq. units

$A=\int_0^2(x+2)dx=\left[\frac{x^2}{2}+2x\right]_0^2=2+4=6$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

The unit vector of $\vec i+\vec j$ is (a) $\frac{\vec i}{\sqrt2}+\frac{\vec i}{\sqrt2}$ (b) $\frac{\vec i}{\sqrt2}-\frac{\vec i}{\sqrt2}$ (c) $\vec i-\vec j$ (d) 1

Answer

(a) $\dfrac{\vec i}{\sqrt2}+\dfrac{\vec j}{\sqrt2}$ (the option as printed, with the second term written as $\vec i$ )

$|\vec i+\vec j|=\sqrt2$ , so the unit vector is $\frac{\vec i+\vec j}{\sqrt2}$ . Option (a) is the only one of this form.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

The direction of vector $\vec v=a\vec i+b\vec j$ is (a) $\tan^{-1}\frac ab$ (b) $\sin^{-1}\frac ab$ (c) $\tan^{-1}\frac ba$ (d) $\cos^{-1}\frac ab$

Answer

(c) $\tan^{-1}\dfrac ba$

The direction angle $\theta$ of $a\vec i+b\vec j$ satisfies $\tan\theta=\frac ba$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

The probability of selecting a vowel from the English alphabet is (a) 1/26 (b) 0 (c) 1/5 (d) 5/26

Answer

(d) $\dfrac5{26}$

There are 5 vowels among the 26 letters.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

The standard deviation of the data: 5, 10, 15, 20 and 25 is (a) $5\sqrt2$ (b) 5 (c) $\sqrt2$ (d) 10

Answer

(a) $5\sqrt2$

The mean is $15$ . The squared deviations are $100, 25, 0, 25, 100$ , which sum to $250$ . The variance is $\frac{250}{5}=50$ , so the standard deviation is $\sqrt{50}=5\sqrt2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

The solution set of $3x+5>2x+1$ is (a) $x>4$ (b) $x>-4$ (c) $x<-4$ (d) $x\geq-4$

Answer

(b) $x>-4$

$3x+5>2x+1$ gives $x>-4$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

If $f(x)=2x+1$ then the approximate value of $f'(2)$ with $h=0.1$ is (a) 5.1 (b) 0 (c) 2.0 (d) 2.1

Answer

(c) 2.0

$f'(2)\approx\frac{f(2.1)-f(2)}{0.1}=\frac{5.2-5}{0.1}=2.0$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

In a manufacturing company the production cost of 50 items is Rs 5000 in one day and Rs. 12000 to produce 150 items in one day. If x represents the number of items and y represents the cost, the model equation is (a) $y=70x-1500$ (b) $y=70x-5000$ (c) $y=-70x-1500$ (d) $y=70x+1500$

Answer

(d) $y=70x+1500$

Slope $=\frac{12000-5000}{150-50}=70$ . Using $(50, 5000)$ : $5000=70(50)+b$ gives $b=1500$ , so $y=70x+1500$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

A company finds the variable cost of Rs .5 per unit output and fixed cost of Rs 25, where Q is the number of outputs, then the total cost function, T is (a) $T=5Q+25$ (b) $T=25Q+5$ (c) $T=5Q-25$ (d) $T=25Q-5$

Answer

(a) $T=5Q+25$

Total cost = variable cost + fixed cost = $5Q+25$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 Model · 1 mark

If a ball is projected vertically upwards with a velocity of 24.9 m/s, then the maximum height described by the ball is (a) 44 m (b) 14.7 m (c) 78.4 m (d) 44.1 m

Answer

(d) 44.1 m

Maximum height $H=\frac{u^2}{2g}$ with $g=9.8$ m/s $^2$ . The option 44.1 m corresponds to $u=29.4$ m/s, since $\frac{29.4^2}{19.6}=44.1$ . It is the intended answer.

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

In the followings, the statement is (a) give me a book (b) $x-2=10$ (c) Rara lake is in Pokhara (d) $y>0$

Answer

(c) Rara lake is in Pokhara

A statement (proposition) is a sentence that is definitely true or false. This one is a (false) statement. "Give me a book" is a command, and $x-2=10$ and $y>0$ are open sentences whose truth depends on the variable.

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

If $S=\{x: x\in\mathbb R\text{ and }x^2+4=0\}$ then the set S is (a) singleton set (b) $x-2=10$ (c) $S=\{-2i,2i\}$ (d) $S=\emptyset$

Answer

(d) $S=\emptyset$

$x^2+4=0$ gives $x=\pm2i$ , which are not real. So no real $x$ satisfies it and $S$ is the empty set.

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

The set is a collection of (a) long rivers of Nepal (b) longest rivers of Nepal (c) tall plants in a garden (d) nice books

Answer

(b) longest rivers of Nepal

A set must be well defined. "Longest rivers" can be decided exactly, whereas "long", "tall" and "nice" are vague.

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

$S=\{x: x\geq-2, x\in\mathbb R\}$ is represented by (a) $[-2,\infty)$ (b) $(-2,\infty)$ (c) $(\infty,-2]$ (d) $[-2,\infty]$

Answer

(a) $[-2,\infty)$

$x\ge-2$ includes $-2$ (closed bracket), and $\infty$ is never included.

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

$|x-5|\leq3$ has values of $x=$ (a) $(2,8)$ (b) $[2,8]$ (c) $\{-2,8\}$ (d) $\{2,8\}$

Answer

(b) $[2,8]$

$|x-5|\le3\Rightarrow -3\le x-5\le3\Rightarrow 2\le x\le8$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

The ordered pair $(a,b)$ for $(2a-1,-3)=(1,b+3)$ is (a) $(1,-6)$ (b) $[1,-6]$ (c) $\{1,-6\}$ (d) $[1,6]$

Answer

(a) $(1,-6)$

$2a-1=1\Rightarrow a=1$ and $b+3=-3\Rightarrow b=-6$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

The domain of the function $f:\mathbb R\to\mathbb R$ defined by $f(x)=\sqrt{1-x^2}$ is (a) $[0,1]$ (b) $\{1,-1\}$ (c) $[0,\infty)$ (d) $[-1,1]$

Answer

(d) $[-1,1]$

We need $1-x^2\ge0\Rightarrow x^2\le1\Rightarrow -1\le x\le1$ .

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2080 · 1 mark

The base of an exponential function $y=a^x$ passing through $(2,4)$ is (a) 4 (b) 2 (c) 8 (d) 1

Answer

(b) 2

$a^2=4\Rightarrow a=2$ (the base must be positive).

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2080 · 1 mark

$3^{2\log_3x}=$ (a) $2x$ (b) $x$ (c) $x^2$ (d) $x/2$

Answer

(c) $x^2$

$3^{2\log_3x}=3^{\log_3x^2}=x^2$ .

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2080 · 1 mark

The function $f(x)=x^2-2$ is (a) an even function (b) an odd number (c) not a function (d) neither odd nor even

Answer

(a) an even function

$f(-x)=(-x)^2-2=x^2-2=f(x)$ .

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2080 · 1 mark

A number which can be expressed in the form $p/q$ ; $p,q$ are integers, $q\neq0$ is called (a) a rational number (b) an integer number (c) an irrational number (d) a prime number

Answer

(a) a rational number

This is the definition of a rational number.

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2080 · 1 mark

The sum of an infinite geometric sequence $a, ak, ak^2,\ldots$ exists if (a) $k>1$ (b) $k>-1$ (c) $-1\leq k\leq1$ (d) $-1<k<1$

Answer

(d) $-1<k<1$

The sum $\dfrac{a}{1-k}$ exists only when the common ratio satisfies $|k|<1$ .

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2080 · 1 mark

If $2k+1$ , $3k+2$ and $5k+1$ are in A.P., then $k=$ (a) 2 (b) 6 (c) -8 (d) 0

Answer

(a) 2

For an A.P., $2(3k+2)=(2k+1)+(5k+1)$ , so $6k+4=7k+2$ and $k=2$ .

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2080 · 1 mark

The matrix $\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ is (a) symmetric (b) skew-symmetric (c) unit (d) none

Answer

(b) skew-symmetric

The transpose is $\begin{bmatrix}0&-1\\1&0\end{bmatrix}=-A$ , so $A^T=-A$ .

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2080 · 1 mark

If $A=\begin{bmatrix}-5&3\\2&-1\end{bmatrix}$ then $A^{-1}=$ (a) $\begin{bmatrix}5&3\\-2&-1\end{bmatrix}$ (b) $\begin{bmatrix}-5&-3\\-2&1\end{bmatrix}$ (c) $\begin{bmatrix}5&3\\2&1\end{bmatrix}$ (d) $\begin{bmatrix}1&3\\-2&-5\end{bmatrix}$

Answer

(d) $\begin{bmatrix}1&3\\-2&-5\end{bmatrix}$ (nearest option)

$|A|=(-5)(-1)-(3)(2)=-1$ , so $A^{-1}=\dfrac{1}{-1}\begin{bmatrix}-1&-3\\-2&-5\end{bmatrix}=\begin{bmatrix}1&3\\2&5\end{bmatrix}$ . Option (d) is the closest of the given options (only the sign of the second-row entries differs).

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2080 · 1 mark

If $\begin{vmatrix}2a+1&1\\3a&2\end{vmatrix}=0$ , then $a=$ (a) 0 (b) 2 (c) -2 (d) 1

Answer

(c) $-2$

$2(2a+1)-3a=a+2=0\Rightarrow a=-2$ .

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2080 · 1 mark

A square matrix A is said to be singular if (a) $|A|>0$ (b) $|A|<0$ (c) $|A|\neq0$ (d) $|A|=0$

Answer

(d) $|A|=0$

A square matrix is singular when its determinant is zero (no inverse exists).

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2080 · 1 mark

If the roots of the equation $px^2+qx+r=0$ are equal in magnitude but opposite in sign, then (a) $p=0$ (b) $q=0$ (c) $r=0$ (d) $p=q$

Answer

(b) $q=0$

Sum of roots $=-q/p=0$ , so $q=0$ .

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2080 · 1 mark

If the discriminant of the equation $x^2+kx+4=0$ is zero then $k=$ (a) $\pm4$ (b) $-4$ (c) 4 (d) 0

Answer

(a) $\pm4$

$D=k^2-16=0\Rightarrow k=\pm4$ .

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2080 · 1 mark

If $i$ is the imaginary unit, then $\frac{1-i}{1+i}=$ (a) 1 (b) $-i$ (c) $i$ (d) $-1$

Answer

(b) $-i$

$\dfrac{1-i}{1+i}=\dfrac{(1-i)^2}{2}=\dfrac{-2i}{2}=-i$ .

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2080 · 1 mark

The square roots of a complex number $5+12i$ is (a) $\pm(3+2i)$ (b) $\pm(2+3i)$ (c) $\pm(3-2i)$ (d) $\pm(2-3i)$

Answer

(a) $\pm(3+2i)$

$(3+2i)^2=9-4+12i=5+12i$ .

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2080 · 1 mark

$\sin(\cos^{-1}x)=$ (a) $\sqrt{1+x^2}$ (b) $\sin\theta$ (c) $\sqrt{1-x^2}$ (d) $x\sqrt{1-x^2}$

Answer

(c) $\sqrt{1-x^2}$

Let $\theta=\cos^{-1}x$ , so $\cos\theta=x$ and $\sin\theta=\sqrt{1-x^2}$ (as $\theta\in[0,\pi]$ ).

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

The general solution of $4\sin^2x=1$ , for $n\in Z$ is (a) $x=n\pi\pm\frac\pi6$ (b) $x=n\pi\pm\frac\pi4$ (c) $x=n\pi\pm\frac\pi3$ (d) $x=2n\pi\pm\frac\pi6$

Answer

(a) $x=n\pi\pm\dfrac{\pi}{6}$

$\sin^2x=\tfrac14=\sin^2\dfrac{\pi}{6}$ , and $\sin^2x=\sin^2\alpha$ gives $x=n\pi\pm\alpha$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

The slope of the line $px+qy+c=0$ , is (a) p (b) q (c) p/q (d) -p/q

Answer

(d) $-p/q$

$y=-\dfrac{p}{q}x-\dfrac{c}{q}$ , so the slope is $-p/q$ .

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2080 · 1 mark

If the length of the perpendicular from the point $(k,2)$ to $3x-4y+6=0$ is 2, then $k=$ (a) 8 (b) 6 (c) 4 (d) 5

Answer

(c) 4

$\dfrac{|3k-8+6|}{\sqrt{9+16}}=2\Rightarrow|3k-2|=10$ , so $k=4$ or $k=-\tfrac83$ . Only $k=4$ is among the options.

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

The homogeneous equation is (a) $ax^2+bxy+c=0$ (b) $ax^2+xy+by^2=0$ (c) $ax^2+bxy+by^2+c=0$ (d) $x^2+by^2+y^3=0$

Answer

(b) $ax^2+xy+by^2=0$

A homogeneous equation has every term of the same degree (here 2); the others contain a constant or a cubic term.

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

The distance of the point $(1,2,3)$ from origin is (a) 1 (b) 3 (c) $\sqrt{14}$ (d) 2

Answer

(c) $\sqrt{14}$

$\sqrt{1^2+2^2+3^2}=\sqrt{14}$ .

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2080 · 1 mark

The direction ratios of the line AB joining the points $A(4,5,0)$ and $B(2,3,4)$ are (a) $-2,2,-4$ (b) $2,2,-4$ (c) $2,2,4$ (d) $-2,-2,-4$

Answer

(b) $2,2,-4$

Direction ratios are the differences of coordinates: $A-B=(4-2,\,5-3,\,0-4)=(2,2,-4)$ . This is the only option proportional to the vector joining the two points ( $B-A=(-2,-2,4)$ is its negative, and is not offered).

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

The direction of the vector $a\mathbf i+b\mathbf j$ is (a) $\tan^{-1}\left(-\frac ab\right)$ (b) $\tan^{-1}\left(\frac ba\right)$ (c) $\tan^{-1}\left(-\frac ba\right)$ (d) none

Answer

(b) $\tan^{-1}\left(\frac ba\right)$

The vector $a\mathbf i+b\mathbf j$ makes angle $\theta$ with the x-axis where $\tan\theta=\frac ba$ .

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2080 · 1 mark

The unit vector along $x\mathbf i-y\mathbf j$ is (a) $\frac{x\mathbf i-y\mathbf j}{\sqrt{x^2-y^2}}$ (b) $\frac{x\mathbf i-y\mathbf j}{\sqrt{x^2y^2}}$ (c) $\frac{x\mathbf i-y\mathbf j}{x^2+y^2}$ (d) $\frac{x\mathbf i-y\mathbf j}{\sqrt{x^2+y^2}}$

Answer

(d) $\frac{x\mathbf i-y\mathbf j}{\sqrt{x^2+y^2}}$

A unit vector is the vector divided by its magnitude, and $|x\mathbf i-y\mathbf j|=\sqrt{x^2+y^2}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

The standard deviation of 3, 6, 9, 12, 15 is (a) $3\sqrt2$ (b) 12 (c) $2\sqrt3$ (d) 2

Answer

(a) $3\sqrt2$

Mean $=\frac{45}{5}=9$ . Squared deviations: $36+9+0+9+36=90$ .

$\sigma=\sqrt{\frac{90}{5}}=\sqrt{18}=3\sqrt2$

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2080 · 1 mark

The probability of getting an ace of club or a king of spade from a pack of 52 cards is (a) 1/26 (b) 1/13 (c) 1/52 (d) 1/2

Answer

(a) $\frac1{26}$

The events are mutually exclusive, so $P=\frac1{52}+\frac1{52}=\frac2{52}=\frac1{26}$ .

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2080 · 1 mark

If $\lim_{x\to a}\frac{x^4-a^4}{x-a}=256$ , then $a=$ (a) 5 (b) 3 (c) 4 (d) 6

Answer

(c) 4

$\lim_{x\to a}\frac{x^4-a^4}{x-a}=4a^3=256$ , so $a^3=64$ and $a=4$ .

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2080 · 1 mark

$\lim_{x\to0}\frac{\tan3x}{x}=$ (a) 3 (b) 0/0 (c) 0 (d) $x$

Answer

(a) 3

$\lim_{x\to0}\frac{\tan3x}{x}=3\lim_{x\to0}\frac{\tan3x}{3x}=3\cdot1=3$ .

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2080 · 1 mark

A function $f(x)=2x+1$ is continuous at $x=1$ , if (a) $\lim_{x\to1}f(x)=3$ (b) $\lim_{x\to1}f(x)$ exists (c) $\lim_{x\to1}f(x)=f(1)$ (d) none

Answer

(c) $\lim_{x\to1}f(x)=f(1)$

A function is continuous at a point when the limit exists and equals the function value there. Here both equal 3, but the defining condition is (c).

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2080 · 1 mark

If $y=e^{\sin x^2}$ , then $\frac{dy}{dx}=$ (a) $e^{\sin x^2}$ (b) $2xy\cos x^2$ (c) $2xe^{\sin x^2}$ (d) $y\cos x^2$

Answer

(b) $2xy\cos x^2$

By the chain rule, $\frac{dy}{dx}=e^{\sin x^2}\cdot\cos x^2\cdot2x=2xy\cos x^2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

If $y=x^3-x^2$ , then $\frac{dy}{dx}=0$ , $x=$ (a) $0,-2/3$ (b) $0,3/2$ (c) $0,3$ (d) $0,2/3$

Answer

(d) $0,\frac23$

$\frac{dy}{dx}=3x^2-2x=x(3x-2)=0$ , so $x=0$ or $x=\frac23$ .

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2080 · 1 mark

The function $f(x)=2x+5$ , on $R$ is (a) decreasing (b) stationary (c) increasing (d) none

Answer

(c) increasing

$f'(x)=2>0$ for all real $x$ , so $f$ is increasing on $R$ .

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2080 · 1 mark

$\int\frac{dx}{ax+b}=$ (a) $\frac1a\log|ax+b|+c$ (b) $a\log|ax+b|+c$ (c) $|\log(ax+b)|+c$ (d) $\log|ax+b|+c$

Answer

(a) $\frac1a\log|ax+b|+c$

Put $u=ax+b$ , so $du=a\,dx$ and the integral becomes $\frac1a\int\frac{du}{u}$ .

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2080 · 1 mark

$\int\frac{dx}{2\cos^2x\sin^2x}=$ (a) $2\cot2x+c$ (b) $-2\cot2x+c$ (c) $\cot2x+c$ (d) $-\cot2x+c$

Answer

(d) $-\cot2x+c$

Since $\sin^22x=4\sin^2x\cos^2x$ , we have $\frac1{2\sin^2x\cos^2x}=\frac{2}{\sin^22x}=2\csc^22x$ .

$\int2\csc^22x\,dx=2\cdot\left(-\frac{\cot2x}{2}\right)+c=-\cot2x+c$

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2080 · 1 mark

$\int_1^3\frac{\sin(\ln t)}{t}dt=$ (a) $\cos(\ln3)$ (b) $-\cos(\ln3)$ (c) $1-\cos(\ln3)$ (d) $1-\sin(\ln3)$

Answer

(c) $1-\cos(\ln3)$

Put $u=\ln t$ , so $du=\frac{dt}{t}$ . The limits become $0$ to $\ln3$ .

$\int_0^{\ln3}\sin u\,du=\left[-\cos u\right]_0^{\ln3}=1-\cos(\ln3)$

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2080 · 1 mark

The area under the curve $y=x^2$ bounded by the x axis and the ordinates $x=1$ and $x=4$ is (a) 21 (b) 64/3 (c) 64 (d) 7

Answer

(a) 21

$\int_1^4x^2dx=\left[\frac{x^3}{3}\right]_1^4=\frac{64-1}{3}=21$

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2080 · 1 mark

The general formula for Newton-Raphson method is (a) $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$ (b) $x_{n+1}=x_n-\frac{f'(x_n)}{f(x_n)}$ (c) $x_{n+1}=x_n+\frac{f'(x_n)}{f(x_n)}$ (d) $x_{n+1}=x_{n+1}-\frac{f'(x_n)}{f(x_n)}$

Answer

(a) $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$

This is the Newton-Raphson iteration, obtained by taking the x-intercept of the tangent at $x_n$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

The approximate value of $\int_0^2x^2dx$ using midpoint rule with one interval is (a) 4 (b) 8 (c) 2 (d) 0

Answer

(c) 2

With one interval $[0,2]$ , the midpoint is $x=1$ . The approximation is $h\,f(1)=2\times1^2=2$ .

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2080 · 1 mark

If three equal forces P, Q and R are acting at a point of rigid body are in equilibrium, then the angle between any two forces is (a) $120^\circ$ (b) $30^\circ$ (c) $60^\circ$ (d) none

Answer

(a) $120^\circ$

Three equal forces in equilibrium form an equilateral triangle when drawn head to tail. The exterior angle of that triangle, which is the angle between the forces, is $120^\circ$ . Check: $P^2+P^2+2P^2\cos\theta=P^2$ gives $\cos\theta=-\frac12$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

A force equal to $4N$ acting upwards is resolved into forces, one being horizontal and equal to $3N$ . The magnitude of the other remaining force is (a) $7N$ (b) $1N$ (c) $2N$ (d) $5N$

Answer

(d) $5N$

The 4 N force is the resultant, so the vertical component of the other force is 4 N and its horizontal component is 3 N (equal and opposite to the given one, which has no vertical part). Magnitude $=\sqrt{3^2+4^2}=5\,N$ .

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2080 · 1 mark

If the acceleration of a particle, starting from rest is $5m/sec^2$ , then the velocity at the end of 3 seconds is (a) $8m/sec^2$ (b) $15m/sec^2$ (c) $5m/sec^2$ (d) none

Answer

(b) 15 (m/sec)

$v=u+at=0+5\times3=15$ m/sec. The unit of velocity is m/sec, not m/sec $^2$ , but 15 is the value.

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

A stone is thrown vertically upwards with a velocity of 30m/sec from the top of a tower and hits the ground after 10 secs, then the height of the tower is, (a) $100\,m$ (b) $300\,m$ (c) $200\,m$ (d) $50\,m$

Answer

(c) $200\,m$

Take upward as positive, $u=30$ m/s, $g=10$ m/s $^2$ and displacement $-h$ after $t=10$ s:

$-h=30(10)-\frac12(10)(10)^2=300-500=-200$

So $h=200$ m.

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If one root of $ax^2+bx+c=0$ is $2+3i$ then the other root is (a) $4+6i$ (b) $2-3i$ (c) $2i-3$ (d) $2i+3$

Similar questions: If one root of ax 2+bx+c=0 be square of other then (2077 Model)

Answer

(b) $2-3i$ For a quadratic with real coefficients, complex roots occur in conjugate pairs, so the other root is $2-3i$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

Area bounded by the x axis and the curve $y=3x^2$ and the lines $x=0$ and $x=2$ is (a) 8 (b) -8 (c) 3 (d) 0

Similar questions: Area bounded by the x-axis and the curve y=x+2 and the... (2079 Model)

Answer

(a) 8 $\int_0^23x^2\,dx=\left[x^3\right]_0^2=8$

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

The number which is not rational is (a) 1 (b) $\pi$ (c) $22/7$ (d) $-1$

Answer

(b) $\pi$ $\pi$ is irrational (non-terminating, non-recurring decimal); $1$ , $22/7$ and $-1$ are all ratios of integers.

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If A and B are the nonempty sets $A\subset B\subset U$ then $A\cup B=$ (a) B (b) A (c) U (d) $A+B$

Answer

(a) B Since $A\subset B$ , every element of $A$ is already in $B$ , so $A\cup B=B$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If $|x-1|=5$ then $x=$ (a) $\{-6,4\}$ (b) $[6,-4]$ (c) $\{6,-4\}$ (d) $\{-6,-4\}$

Answer

(c) $\{6,-4\}$ $|x-1|=5\Rightarrow x-1=\pm5\Rightarrow x=6$ or $x=-4$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If $\log_3x=2$ then $x=$ (a) $2x$ (b) 2 (c) 9 (d) 8

Answer

(c) 9 $\log_3x=2\Rightarrow x=3^2=9$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If $f(x)=2x+5$ for $x\in R$ , then $f^{-1}(x)=$ (a) $\frac{1}{2x-5}$ (b) $\frac{1}{f(x)+5}$ (c) $\frac{5}{y-2x}$ (d) $(x-5)/2$

Answer

(d) $(x-5)/2$ Let $y=2x+5$ . Solving for $x$ gives $x=(y-5)/2$ , so $f^{-1}(x)=(x-5)/2$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If $\sin^{-1}x=\cos^{-1}x$ then $x=$ (a) $-\sqrt2$ (b) $-1/\sqrt2$ (c) $1/\sqrt2$ (d) $\sqrt2$

Answer

(c) $1/\sqrt2$ $\sin^{-1}x=\cos^{-1}x$ and $\sin^{-1}x+\cos^{-1}x=\pi/2$ , so $\sin^{-1}x=\pi/4$ and $x=\sin(\pi/4)=1/\sqrt2$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

The solution of $\cos x=1/2$ is (a) $2n\pi\pm\pi/3$ (b) $n\pi+\pi/3$ (c) $2n\pi\pm\pi/2$ (d) $n\pi-\pi/3$

Answer

(a) $2n\pi\pm\pi/3$ $\cos x=1/2=\cos(\pi/3)$ , and $\cos x=\cos\alpha$ gives $x=2n\pi\pm\alpha$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If p, q, r are in A.P. then (a) $q=p+r$ (b) $p+r=2q$ (c) $pr=q^2$ (d) $p+q+r=0$

Answer

(b) $p+r=2q$ In an A.P. the middle term is the mean of its neighbours: $q-p=r-q\Rightarrow p+r=2q$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

The fifth term of the sequence: 6, 12, 24, 48, ... is (a) 56 (b) 70 (c) 192 (d) 96

Answer

(d) 96 The sequence is a G.P. with ratio 2, so the fifth term is $48\times2=96$ (also $6\cdot2^4=96$ ).

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If $A=[a_{ij}]$ , where $a_{ij}=i+2j$ , then $2\times2$ matrix A is (a) $\begin{bmatrix}3&6\\5&4\end{bmatrix}$ (b) $\begin{bmatrix}3&5\\4&6\end{bmatrix}$ (c) $\begin{bmatrix}3&6\\5&6\end{bmatrix}$ (d) $\begin{bmatrix}3&4\\6&5\end{bmatrix}$

Answer

(b) $\begin{bmatrix}3&5\\4&6\end{bmatrix}$ With $a_{ij}=i+2j$ : $a_{11}=3,\ a_{12}=5,\ a_{21}=4,\ a_{22}=6$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

The value of x for the determinant $\begin{vmatrix}x+5&3\\0&1\end{vmatrix}=0$ is (a) 0 (b) 5 (c) -5 (d) 3

Answer

(c) $-5$ The determinant is $(x+5)(1)-3(0)=x+5=0$ , so $x=-5$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

The system of equation $x+y=5$ , $x+y=7$ has (a) one solution (b) two solutions (c) many solutions (d) no solution.

Answer

(d) no solution. Both equations have the same left side $x+y$ but different right sides ( $5\ne7$ ), so they are inconsistent (parallel lines).

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If $\omega$ is the cube root of unity of a complex number then $\omega(1+\omega)(1+\omega^2)=$ (a) $\omega$ (b) -2 (c) 1 (d) $-\omega$

Answer

(a) $\omega$ Using $1+\omega+\omega^2=0$ : $1+\omega=-\omega^2$ and $1+\omega^2=-\omega$ , so the product is $\omega(-\omega^2)(-\omega)=\omega\cdot\omega^3=\omega$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

A quadratic equation whose roots are 2, 3 is (a) $x^2-5x+6=0$ (b) $x^2+5x+6=0$ (c) $x^2+x-6=0$ (d) $x^2-5x-6=0$

Answer

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

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

Distance between the lines $2x+2y=5$ and $x+y=10$ is (a) $15/2$ (b) $15/(2\sqrt2)$ (c) $-15/(2\sqrt2)$ (d) $2\sqrt2$

Answer

(b) $15/(2\sqrt2)$ Write the first line as $x+y=5/2$ , i.e. $x+y-5/2=0$ , and the second as $x+y-10=0$ . Distance $=\dfrac{|10-5/2|}{\sqrt2}=\dfrac{15/2}{\sqrt2}=\dfrac{15}{2\sqrt2}$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

$\lim_{x\to3}\frac{x^5-3^5}{x-3}=$ (a) 0 (b) 504 (c) 405 (d) 81

Answer

(c) 405 $\lim_{x\to a}\dfrac{x^n-a^n}{x-a}=na^{n-1}$ , so the limit is $5\cdot3^4=405$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

$\lim_{x\to0}\frac{x}{\tan x}=$ (a) undefined (b) 0 (c) 1 (d) 0/0

Answer

(c) 1 $\dfrac{x}{\tan x}=\dfrac{x}{\sin x}\cos x\to1\cdot1=1$ as $x\to0$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

The function $f(x)=\begin{cases}\frac{x^2-5x}{x-5}&x\neq5\\p&x=5\end{cases}$ is continuous at $x=5$ , for $p=$ (a) 0 (b) 5 (c) -5 (d) 1

Answer

(b) 5 For $x\ne5$ , $f(x)=\dfrac{x(x-5)}{x-5}=x$ , so $\lim_{x\to5}f(x)=5$ . Continuity requires $p=5$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If $y=\sin(ax^2+b)$ then $dy/dx=$ (a) $\cos(ax^2+b)$ (b) $-\cos(ax^2+b)$ (c) $-2ax\cos(ax^2+b)$ (d) $2ax\cos(ax^2+b)$

Answer

(d) $2ax\cos(ax^2+b)$ By the chain rule, $\dfrac{dy}{dx}=\cos(ax^2+b)\cdot2ax$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If $x^2+y^2=25$ then $dy/dx=$ (a) $\frac{dy}{dx}=-\frac xy$ (b) $\frac{dy}{dx}=\frac yx$ (c) $\frac{dy}{dx}=\frac xy$ (d) $\frac{dy}{dx}=-\frac yx$

Answer

(a) $\dfrac{dy}{dx}=-\dfrac xy$ Differentiating, $2x+2y\dfrac{dy}{dx}=0$ , so $\dfrac{dy}{dx}=-\dfrac xy$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

$\int_0^12xe^{x^2+1}dx=$ (a) $e^3$ (b) $e^2-e$ (c) $e^4-1$ (d) 1

Answer

(b) $e^2-e$ Put $u=x^2+1$ , $du=2x\,dx$ ; limits $x=0\to u=1$ , $x=1\to u=2$ . $\int_1^2e^u\,du=e^2-e$

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

The critical point for $f(x)=x^3-6x^2+9x-2$ are (a) $1,-3$ (b) $-1,3$ (c) $1,3$ (d) $-1,-3$

Answer

(c) $1,3$ $f'(x)=3x^2-12x+9=3(x-1)(x-3)=0\Rightarrow x=1,3$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

The letter of the word 'union' can be arranged in the number of ways: (a) 60 (b) 120 (c) 5 (d) 4

Answer

(b) 120 "Union" has 5 distinct letters, so the arrangements number $5!=120$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If $^{18}C_r=\,^{18}C_{r+2}$ then $r=$ (a) 18 (b) 9 (c) 8 (d) 16

Answer

(c) 8 $^{n}C_a={}^{n}C_b$ with $a\ne b$ implies $a+b=n$ . So $r+(r+2)=18\Rightarrow r=8$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If the revenue function $R=x-3x^2$ and cost function $C=x^2-x$ , then the maximum profit is at $x=$ (a) 1/4 (b) 4 (c) 2 (d) 3

Answer

(a) $1/4$ Profit $\pi=R-C=(x-3x^2)-(x^2-x)=2x-4x^2$ . Then $\pi'=2-8x=0\Rightarrow x=1/4$ , and $\pi''=-8<0$ , so this is a maximum.

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

The probability that a card drawn from a pack of 52 cards will be a diamond or a king is (a) 1/52 (b) 1/13 (c) 2/13 (d) 4/13

Answer

(d) $4/13$ Diamonds $=13$ , kings $=4$ , king of diamonds counted twice, so favourable cases $=13+4-1=16$ . Probability $=16/52=4/13$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

The median of the distribution: 5, 3, 2, 4, 10, 7, 12 is (a) 4 (b) 6 (c) 7 (d) 5

Answer

(d) 5 Arranged in order: 2, 3, 4, 5, 7, 10, 12. There are 7 values, so the median is the 4th value, 5.

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

The ratio of the prices of the first and second items is 3:5. If the price of the second item is Rs125, then the price of the first item is (a) 375 (b) 75 (c) 625 (d) 15

Answer

(b) 75 $\dfrac{P_1}{125}=\dfrac35\Rightarrow P_1=75$ , i.e. Rs 75.

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If a profit of 10 percent is made by selling an article for Rs 1320, then the cost price in Rupees (a) 1200 (b) 1100 (c) 1250 (d) 1150

Answer

(a) 1200 $SP=1.10\,CP\Rightarrow CP=1320/1.10=$ Rs 1200.

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

In an examination 25 percent of the examinee have been failed and 240 are passed, then the total number of examinees are (a) 300 (b) 320 (c) 265 (d) 350

Answer

(b) 320 Passed $=75\%$ of total, so total $=240/0.75=320$ .

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

A, B and C are in partnership. Towards capitals A contributed Rs 3600 and B Rs 3000. The profit amounted is Rs 3800 out of which C receives Rs 1600. The capital of C is Rs (a) 4800 (b) 4000 (c) 3000 (d) 2000

Answer

(a) 4800 A and B together receive $3800-1600=$ Rs 2200 on capitals $3600+3000=$ Rs 6600, i.e. Rs 1 profit per Rs 3 of capital. So C's capital $=3\times1600=$ Rs 4800.

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If 1US dollar = Rs 110 then Rs 5000 will give US dollar (a) 46.5 (b) 44.5 (c) 45.45 (d) 50

Answer

(c) 45.45 $5000/110=45.45$ US dollars.

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If P, r, A are the principal, rate and amount of interest per annum respectively, then after n years the amount A will be (a) $P(1+r/100)^n$ (b) $P(1-r/100)^n$ (c) $(P+r100)^n$ (d) $P(r+1/100)^n$

Answer

(a) $P(1+r/100)^n$ This is the compound amount formula, with $r$ the annual rate in percent.

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

The amount of annuity of Rs 25 a year for 5 years at 4 percent per year is (a) 140.0 (b) 140.4 (c) 125.0 (d) 150

Answer

(a) 140.0 Taking each instalment of Rs 25 at the start of the year with simple interest at 4%, the amounts for 5, 4, 3, 2, 1 years are $25(1+0.04t)$ , so the total is $125+25(0.04)(15)=125+15=$ Rs 140.

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TU BIT Entrance Exam Model Questions Set 2 · 1 mark

If $x+iy=(2+i3)^2$ then $(x,y)=$ (a) $(5,-12)$ (b) $(-5,-12)$ (c) $(5,12)$ (d) $(-5,12)$

Answer

(d) $(-5,12)$ $(2+3i)^2=4+12i+9i^2=-5+12i$ , so $(x,y)=(-5,12)$ .

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Past questions and answers

1.If f:R→Rf:R\to Rf:R→R is defined by f(x)=3x−2f(x)=3x-2f(x)=3x−2, for x∈Rx\in Rx∈R then the pre image of 10 in R is (a) 4 (b) 10 (c) 5 (d) 0

2.The two lines represented by ax2+xy+by2=0ax^2+xy+by^2=0ax2+xy+by2=0 perpendicular to each other if (a) a+b>0a+b>0a+b>0 (b) a+b<0a+b<0a+b<0 (c) ab=0ab=0ab=0 (d) a+b=0a+b=0a+b=0

3.If a function f:R→Rf:\mathbb R\to\mathbb Rf:R→R be defined by f(x)=2x+5f(x)=2x+5f(x)=2x+5 then the pre-image of 7 in R\mathbb RR is (a) 3 (b) no pre-image (c) 1 (d) -2

4.The two lines represented by kx2+xy−y2=0kx^2+xy-y^2=0kx2+xy−y2=0, are perpendicular to each other if k=k=k= (a) 1 (b) -1 (c) 0 (d) k≠0k\neq0k=0

5.The straight lines represented by x2+2xy+ky2=0x^2+2xy+ky^2=0x2+2xy+ky2=0 are perpendicular to each other if k=k=k= (a) -1 (b) 1 (c) 0 (d) 2

6.Every rational number is also (a) a natural number (b) an integer (c) a whole number (d) a real number

7.If a−ib=3−2i3+2ia-ib=\frac{3-2i}{3+2i}a−ib=3+2i3−2i​ then a2+b2=a^2+b^2=a2+b2= (a) 1 (b) 43\frac4334​ (c) 35\frac3553​ (d) 29\frac2992​

8.Given U={1,2,3,…,15}U=\{1,2,3,\ldots,15\}U={1,2,3,…,15}, A={x:x≤8}A=\{x: x\leq8\}A={x:x≤8}, B={x:x≤4}B=\{x: x\leq4\}B={x:x≤4} then A∩B‾=\overline{A\cap B}=A∩B= (a) ∅\emptyset∅ (b) UUU (c) A∪BA\cup BA∪B (d) None of these

9.Which are the equal ordered pairs? (a) (2,4),(4,2)(2,4),(4,2)(2,4),(4,2) (b) (2,2),(4,4)(2,2),(4,4)(2,2),(4,4) (c) (4,2),(4,2)(4,2),(4,2)(4,2),(4,2) (d) all of them

10.S∞=a1−rS_\infty=\frac{a}{1-r}S∞​=1−ra​ where a,ra, ra,r and S∞S_\inftyS∞​ denotes first term, common ratio and sum to infinity. If (a) ∣r∣<1|r|<1∣r∣<1 (b) ∣r∣>1|r|>1∣r∣>1 (c) ∣r∣=1|r|=1∣r∣=1 (d) all of them

11.If 21Cx= 21Cy^{21}C_x=\,^{21}C_y21Cx​=21Cy​ then which is true? (a) x≠yx\neq yx=y (b) x=yx=yx=y (c) x+y=21x+y=21x+y=21 (d) either (b) or (c)

12.Which is not correct? (a) (AT)T=A(A^T)^T=A(AT)T=A (b) (A+B)T=AT+BT(A+B)^T=A^T+B^T(A+B)T=AT+BT (c) (AB)T=ATBT(AB)^T=A^TB^T(AB)T=ATBT (d) (AB)T=BTAT(AB)^T=B^TA^T(AB)T=BTAT

13.The equation x=0x=0x=0 represents (a) x-axis (b) y-axis (c) the origin (d) none of these

14.The angle between the lines xy=0xy=0xy=0 is (a) 0∘0^\circ0∘ (b) 60∘60^\circ60∘ (c) 90∘90^\circ90∘ (d) 120∘120^\circ120∘

15.Which is correct? (a) log⁡101=0\log_{10}1=0log10​1=0 (b) log⁡100.1=−1\log_{10}0.1=-1log10​0.1=−1 (c) log⁡1010=1\log_{10}10=1log10​10=1 (d) all of them

16.The domain of y=f(x)=1xy=f(x)=\frac1xy=f(x)=x1​ is (a) R−{0}R-\{0\}R−{0} (b) RRR (c) (−2,1)(-2,1)(−2,1) (d) (−1,1)(-1,1)(−1,1)

17.Let f:R→Rf:R\to Rf:R→R be defined by f(x)=10x−7f(x)=10x-7f(x)=10x−7 then f−1(x)=f^{-1}(x)=f−1(x)= (a) 110x+7\frac{1}{10x+7}10x+71​ (b) 110x−7\frac{1}{10x-7}10x−71​ (c) x+710\frac{x+7}{10}10x+7​ (d) x−710\frac{x-7}{10}10x−7​

18.If t be a positive real number then the least value of t+1tt+\frac1tt+t1​ is : (a) 1 (b) 2 (c) 4 (d) 3

19.The point of discontinuity of the function f(x)=x3−27x−3f(x)=\frac{x^3-27}{x-3}f(x)=x−3x3−27​ is (a) 3 (b) 6 (c) 27 (d) 0

20.If x+y=xyx+y=xyx+y=xy then dxdy=\frac{dx}{dy}=dydx​= (a) x−1y−1\frac{x-1}{y-1}y−1x−1​ (b) y−11−x\frac{y-1}{1-x}1−xy−1​ (c) 1−xy−1\frac{1-x}{y-1}y−11−x​ (d) 1+y1−x\frac{1+y}{1-x}1−x1+y​

21.If x=2atx=2atx=2at, y=at2y=at^2y=at2 then dydx=\frac{dy}{dx}=dxdy​= (a) ttt (b) 1t\frac1tt1​ (c) t2t^2t2 (d) 1

22.If the total cost function C=50x+5000C=50x+5000C=50x+5000. The marginal cost function MC is (a) 50+5000x50+\frac{5000}{x}50+x5000​ (b) 25x2+5000x25x^2+5000x25x2+5000x (c) 50 (d) none of these

23.The maximum value of sin⁡θcos⁡θ\sin\theta\cos\thetasinθcosθ is (a) 1 (b) 12\frac{1}{\sqrt2}2​1​ (c) 32\frac{\sqrt3}{2}23​​ (d) 12\frac1221​

24.The function f(x)=2−3xf(x)=2-3xf(x)=2−3x is (a) Decreasing (b) Increasing (c) Neither decreasing nor increasing (d) Doesn't exist

25.∫log⁡x dx=\int\log x\,dx=∫logxdx= (a) log⁡x+c\log x+clogx+c (b) xlog⁡x−x+cx\log x-x+cxlogx−x+c (c) xlog⁡x+x+cx\log x+x+cxlogx+x+c (d) xlog⁡x+cx\log x+cxlogx+c

26.∫−22∣x∣ dx\int_{-2}^{2}|x|\,dx∫−22​∣x∣dx is equal to (a) 0 (b) 1 (c) 2 (d) 4

27.The area of the circle x2+y2=16x^2+y^2=16x2+y2=16 is (a) 4π4\pi4π (b) 4π24\pi^24π2 (c) 16π216\pi^216π2 (d) 16π16\pi16π

28.Which one is the measure of dispersion? (a) mean (b) median (c) mode (d) range

30.The equation sin⁡x+cos⁡x=2\sin x+\cos x=2sinx+cosx=2 has (a) infinite solution (b) finite solution (c) no solution (d) unique solution

31.The system of linear equations 2x+3y=52x+3y=52x+3y=5 and 4x+6y=94x+6y=94x+6y=9 are (a) Consistent & dependent (b) Consistent & independent (c) Inconsistent & independent (d) Inconsistent & dependent

32.The degree of polynomial equation ax2+bx+c=0ax^2+bx+c=0ax2+bx+c=0 is (a) 0 (b) 1 (c) 2 (d) not define

33.If one root of a quadratic equation in the real coefficient is 2\sqrt22​ then equation is (a) x2+2x+2=0x^2+\sqrt2x+2=0x2+2​x+2=0 (b) x2+2=0x^2+2=0x2+2=0 (c) x2−2=0x^2-2=0x2−2=0 (d) none of these

34.The equation x2+y2=0x^2+y^2=0x2+y2=0 represent (a) circle (b) degenerate circle (c) an empty set (d) a straight line

35.If four quantities a,b,ca, b, ca,b,c and d are such that a:b=3:4a:b=3:4a:b=3:4, b:c=6:7b:c=6:7b:c=6:7 and c:d=8:9c:d=8:9c:d=8:9, then the ratio between a and d is (a) 3:8 (b) 1:3 (c) 4:7 (d) 7:4

36.By selling 175 pens, a stationary store gains the selling price of 50 pens. What is his gain percent? (a) 40% (b) 284728\frac472874​% (c) 50% (d) none of these

37.If US USD 1 = Rs. 78.15. How many US dollars are equivalent To Rs. 10,000? (a) USD 127.96 (b) USD 172.96 (c) USD 196.72 (d) USD 172.62

38.The effective rate corresponding to 10% p.a. convertible quarterly is (a) 40% (b) 10.1% (c) 10.4% (d) none of these

39.If n(∪)=200n(\cup)=200n(∪)=200, n(A)=80n(A)=80n(A)=80, n(B)=60n(B)=60n(B)=60, what is the maximum value of n(A∪B)=n(A\cup B)=n(A∪B)=? (a) 200 (b) 40 (c) 140 (d) 110

40.−1≤x≤11-1\leq x\leq11−1≤x≤11 is equivalent to (a) ∣x−5∣≤6|x-5|\leq6∣x−5∣≤6 (b) ∣x+5∣≥3|x+5|\geq3∣x+5∣≥3 (c) ∣5−x∣≥2|5-x|\geq2∣5−x∣≥2 (d) ∣5−2x∣<1|5-2x|<1∣5−2x∣<1

41.Range of y=x2−4x+5y=x^2-4x+5y=x2−4x+5 is (a) (−∞,∞)(-\infty,\infty)(−∞,∞) (b) (−∞,1](-\infty,1](−∞,1] (c) [1,∞)[1,\infty)[1,∞) (d) (−∞,1)(-\infty,1)(−∞,1)

42.Period of function of cos⁡2x\cos2xcos2x is (a) π2\frac\pi22π​ (b) 1 (c) π\piπ (d) π4\frac\pi44π​

43.If log⁡981=x\log_981=xlog9​81=x then xxx (a) 12\frac1221​ (b) 2\sqrt22​ (c) 1 (d) 2

44.If x=log⁡abcx=\log_abcx=loga​bc, y=log⁡bcay=\log_bcay=logb​ca, z=log⁡cabz=\log_cabz=logc​ab then 1x+1+1y+1+1z+1=\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}=x+11​+y+11​+z+11​= (a) 1 (b) 0 (c) 12\frac1221​ (d) abc

45.1.5+0.015+0.00015+……1.5+0.015+0.00015+\ldots\ldots1.5+0.015+0.00015+…… (a) 1511\frac{15}{11}1115​ (b) 5033\frac{50}{33}3350​ (c) 23\frac2332​ (d) sum doesn't exist

46.The system of equations kx−2y=0kx-2y=0kx−2y=0, x+3y=0x+3y=0x+3y=0 have unique solution then (a) k=−23k=-\frac23k=−32​ (b) k=0k=0k=0 (c) k=23k=\frac23k=32​ (d) k≠−23k\neq-\frac23k=−32​

48.The amplitude of 1−i1-i1−i is (a) 45∘45^\circ45∘ (b) 135∘135^\circ135∘ (c) 315∘315^\circ315∘ (d) doesn't exist

49.The mean proportional between 4 & 16 is (a) 4 (b) 8 (c) 12 (d) 5

50.A plane left 30 minutes later than the schedules time and in order to reach its destinations 1500 km always in time it has to increase its speed by 250 km/hr from its usual speed. What is its usual speed? (a) 500 km/hr (b) 600 km/hr (c) 750 km/hr (d) 100 km/hr

51.The present worth of Rs 2040 due 4 years hence reaching interested at 5% p.a is (a) 1650.50 (b) Rs. 1675.3 (c) Rs 1680 (d) Rs 1700

52.The interest rate doubles a sum in 10 year is (a) 12% (b) 11% (c) 10% (d) 9%

53.Profits in the business of Bikram, Bishal and Binita are shared in ratio 1:2:3. If Bikram share of profits is Rs. 300, what is Binita share? (a) Rs. 150 (b) Rs. 450 (c) Rs. 600 (d) Rs. 900

54.If A and B are two independent events with p(A)=23p(A)=\frac23p(A)=32​ & p(B)=35p(B)=\frac35p(B)=53​ then p(A∪B)=p(A\cup B)=p(A∪B)= (a) 615\frac{6}{15}156​ (b) 53\frac5335​ (c) 25\frac2552​ (d) 1315\frac{13}{15}1513​

55.For Negative skewness (a) Mean>Mode>median (b) Mean<median<Mode (c) Median<mode<Mean (d) Mode<mean<Median

56.HCF of n!n!n!, (n+1)!(n+1)!(n+1)! and (n+2)!(n+2)!(n+2)! is (a) n!n!n! (b) (n+2)!(n+2)!(n+2)! (c) (n+1)!(n+1)!(n+1)! (d) (n−1)!(n-1)!(n−1)!

57.If tan⁡−1x+tan⁡−1y+tan⁡−1z=π2\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\frac\pi2tan−1x+tan−1y+tan−1z=2π​ then xy+yz+zx=xy+yz+zx=xy+yz+zx= (a) 1 (b) -1 (c) 0 (d) π\piπ

58.In any ΔABC\Delta ABCΔABC, 2cos⁡Asin⁡C=sin⁡B2\cos A\sin C=\sin B2cosAsinC=sinB then Δ\DeltaΔ is (a) Scalene (b) Equilateral (c) Isosceles (d) right angled

59.For what value of 'K' points (1,−1)(1,-1)(1,−1), (k2,k3)\left(\frac k2,\frac k3\right)(2k​,3k​) and (4,1)(4,1)(4,1) are collinear? (a) −32-\frac32−23​ (b) 3 (c) 2 (d) −52-\frac52−25​

60.The distance between parallel lines 3x+2y=63x+2y=63x+2y=6 and 3x+2y=123x+2y=123x+2y=12 is (a) 145\frac{14}{5}514​ (b) 72\frac7227​ (c) −813-\frac{8}{\sqrt{13}}−13​8​ (d) 613\frac{6}{\sqrt{13}}13​6​

61.Angles between the lines represented by 2x2−7xy+6y2=02x^2-7xy+6y^2=02x2−7xy+6y2=0 is (a) tan⁡−1(±18)\tan^{-1}\left(\pm\frac18\right)tan−1(±81​) (b) tan⁡−1(±17)\tan^{-1}\left(\pm\frac17\right)tan−1(±71​) (c) 45∘45^\circ45∘ (d) 60∘60^\circ60∘

62.Length of y-intercept made by circle x2+y2−8x+y−20=0x^2+y^2-8x+y-20=0x2+y2−8x+y−20=0 is (a) 5 (b) 9 (c) 8 (d) 6

63.Radius of circle x2+y2+2x+3y+k=0x^2+y^2+2x+3y+k=0x2+y2+2x+3y+k=0 is 12\frac1221​ then k=k=k= (a) 2 (b) -1 (c) 3 (d) 4

64.lim⁡x→64x6−2x3−4\lim_{x\to64}\frac{\sqrt[6]{x}-2}{\sqrt[3]{x}-4}limx→64​3x​−46x​−2​ (a) 12\frac1221​ (b) 34\frac3443​ (c) 23\frac2332​ (d) 14\frac1441​

65.lim⁡x→0esin⁡x−1x\lim_{x\to0}\frac{e^{\sin x}-1}{x}limx→0​xesinx−1​ (a) 0 (b) 1 (c) 2 (d) log⁡e2\log_e2loge​2

66.Every continuous function is (a) increasing (b) decreasing (c) differentiable (d) not - differentiable

67.ddxcos⁡−1(sin⁡x)\frac{d}{dx}\cos^{-1}(\sin x)dxd​cos−1(sinx) (a) x (b) -1 (c) −sin⁡x1−sin⁡2x\frac{-\sin x}{\sqrt{1-\sin^2x}}1−sin2x​−sinx​ (d) cos⁡x1−sin⁡2x\frac{\cos x}{\sqrt{1-\sin^2x}}1−sin2x​cosx​

68.If xpyq=(x+y)p+qx^py^q=(x+y)^{p+q}xpyq=(x+y)p+q then dydx=\frac{dy}{dx}=dxdy​= (a) yx\frac yxxy​ (b) pyqx\frac{py}{qx}qxpy​ (c) xy\frac xyyx​ (d) x2y2\frac{x^2}{y^2}y2x2​

69.Derivatives of sin⁡−1x\sin^{-1}xsin−1x is (a) 1−x2\sqrt{1-x^2}1−x2​ (b) 11−x2\frac{1}{\sqrt{1-x^2}}1−x2​1​ (c) −11−x2\frac{-1}{\sqrt{1-x^2}}1−x2​−1​ (d) −1x2−1-\frac{1}{\sqrt{x^2-1}}−x2−1​1​

70.At critical point, f(x)f(x)f(x) is (a) increasing (b) decreasing (c) stationary (d) increasing & decreasing

71.∫cot⁡x dx=\int\cot x\,dx=∫cotxdx= (a) ln⁡(sec⁡x)+c\ln(\sec x)+cln(secx)+c (b) ln⁡(cosec⁡x)+c\ln(\operatorname{cosec}x)+cln(cosecx)+c (c) ln⁡(sin⁡x)+c\ln(\sin x)+cln(sinx)+c (d) −cosec⁡2x+c-\operatorname{cosec}^2x+c−cosec2x+c

72.∫ex−1ex+1dx\int\frac{e^x-1}{e^x+1}dx∫ex+1ex−1​dx (a) ex−log⁡(ex+1)e^x-\log(e^x+1)ex−log(ex+1) (b) ex+e−xe^x+e^{-x}ex+e−x (c) ex−log⁡(e−x+1)e^x-\log(e^{-x}+1)ex−log(e−x+1) (d) 2log⁡(ex2+e−x2)2\log\left(e^{\frac x2}+e^{-\frac x2}\right)2log(e2x​+e−2x​)

73.∫log⁡x dx\int\log x\,dx∫logxdx (a) 1x\frac1xx1​ (b) xlog⁡x−xx\log x-xxlogx−x (c) exe^xex (d) exx\frac{e^x}{x}xex​

74.Area of region bounded by curve y2=4axy^2=4axy2=4ax and x2=4byx^2=4byx2=4by is (a) 4ab3\frac{4ab}{3}34ab​ (b) 8ab3\frac{8ab}{3}38ab​ (c) 16ab3\frac{16ab}{3}316ab​ (d) 2ab3\frac{2ab}{3}32ab​

75.If the ratio of cost price and selling price of an article is 2:3 then (a) Profit % = 50% (b) loss % = 30% (c) Neither profit or loss (d) Profit % = 1003\frac{100}{3}3100​%

76.lim⁡n→∞∑n2n3=\lim_{n\to\infty}\frac{\sum n^2}{n^3}=limn→∞​n3∑n2​= (a) 0 (b) ∞\infty∞ (c) 2 (d) 13\frac1331​

77.lim⁡x→1(x−1)tan⁡πx2\lim_{x\to1}(x-1)\tan\frac{\pi x}{2}limx→1​(x−1)tan2πx​ (a) −2π-\frac2\pi−π2​ (b) π2\frac\pi22π​ (c) −12-\frac12−21​ (d) −1π-\frac1\pi−π1​

78.If y=1+cos⁡2x1−cos⁡2xy=\sqrt{\frac{1+\cos2x}{1-\cos2x}}y=1−cos2x1+cos2x​​ then dydx\frac{dy}{dx}dxdy​ at x=π2x=\frac\pi2x=2π​ is (a) 0 (b) 1 (c) not defined (d) -1

80.If a+b+c=0a+b+c=0a+b+c=0 in quadratic equation ax2+bx+c=0ax^2+bx+c=0ax2+bx+c=0 then roots of quadratic equation is (a) 1 and -2 (b) 1 and ca\frac caac​ (c) −ba-\frac ba−ab​ and ca\frac caac​ (d) 0 and 1

81.If (m+n)th(m+n)^{th}(m+n)th term of a G.P is p and (m−n)th(m-n)^{th}(m−n)th term is q then mthm^{th}mth term is (a) pq\sqrt{pq}pq​ (b) p+q2\frac{p+q}{2}2p+q​ (c) (p+q)\sqrt{(p+q)}(p+q)​ (d) pqpqpq

82.(a+b−c)x2+(c+a−b)x+(b+c−a)=0(a+b-c)x^2+(c+a-b)x+(b+c-a)=0(a+b−c)x2+(c+a−b)x+(b+c−a)=0 and a+b+c=0a+b+c=0a+b+c=0 then roots quadratic equation, if a, b and c are rational (a) rational (b) irrational (c) imaginary (d) Real and equal

If $f:R\to R$ is defined by $f(x)=3x-2$ , for $x\in R$ then the pre image of 10 in R is (a) 4 (b) 10 (c) 5 (d) 0

The two lines represented by $ax^2+xy+by^2=0$ perpendicular to each other if (a) $a+b>0$ (b) $a+b<0$ (c) $ab=0$ (d) $a+b=0$

If a function $f:\mathbb R\to\mathbb R$ be defined by $f(x)=2x+5$ then the pre-image of 7 in $\mathbb R$ is (a) 3 (b) no pre-image (c) 1 (d) -2

The two lines represented by $kx^2+xy-y^2=0$ , are perpendicular to each other if $k=$ (a) 1 (b) -1 (c) 0 (d) $k\neq0$

The straight lines represented by $x^2+2xy+ky^2=0$ are perpendicular to each other if $k=$ (a) -1 (b) 1 (c) 0 (d) 2

Every rational number is also (a) a natural number (b) an integer (c) a whole number (d) a real number

If $a-ib=\frac{3-2i}{3+2i}$ then $a^2+b^2=$ (a) 1 (b) $\frac43$ (c) $\frac35$ (d) $\frac29$

Given $U=\{1,2,3,\ldots,15\}$ , $A=\{x: x\leq8\}$ , $B=\{x: x\leq4\}$ then $\overline{A\cap B}=$ (a) $\emptyset$ (b) $U$ (c) $A\cup B$ (d) None of these

Which are the equal ordered pairs? (a) $(2,4),(4,2)$ (b) $(2,2),(4,4)$ (c) $(4,2),(4,2)$ (d) all of them

$S_\infty=\frac{a}{1-r}$ where $a, r$ and $S_\infty$ denotes first term, common ratio and sum to infinity. If (a) $|r|<1$ (b) $|r|>1$ (c) $|r|=1$ (d) all of them

If $^{21}C_x=\,^{21}C_y$ then which is true? (a) $x\neq y$ (b) $x=y$ (c) $x+y=21$ (d) either (b) or (c)

Which is not correct? (a) $(A^T)^T=A$ (b) $(A+B)^T=A^T+B^T$ (c) $(AB)^T=A^TB^T$ (d) $(AB)^T=B^TA^T$

The equation $x=0$ represents (a) x-axis (b) y-axis (c) the origin (d) none of these

The angle between the lines $xy=0$ is (a) $0^\circ$ (b) $60^\circ$ (c) $90^\circ$ (d) $120^\circ$

Which is correct? (a) $\log_{10}1=0$ (b) $\log_{10}0.1=-1$ (c) $\log_{10}10=1$ (d) all of them

The domain of $y=f(x)=\frac1x$ is (a) $R-\{0\}$ (b) $R$ (c) $(-2,1)$ (d) $(-1,1)$

Let $f:R\to R$ be defined by $f(x)=10x-7$ then $f^{-1}(x)=$ (a) $\frac{1}{10x+7}$ (b) $\frac{1}{10x-7}$ (c) $\frac{x+7}{10}$ (d) $\frac{x-7}{10}$

If t be a positive real number then the least value of $t+\frac1t$ is : (a) 1 (b) 2 (c) 4 (d) 3

The point of discontinuity of the function $f(x)=\frac{x^3-27}{x-3}$ is (a) 3 (b) 6 (c) 27 (d) 0

If $x+y=xy$ then $\frac{dx}{dy}=$ (a) $\frac{x-1}{y-1}$ (b) $\frac{y-1}{1-x}$ (c) $\frac{1-x}{y-1}$ (d) $\frac{1+y}{1-x}$

If $x=2at$ , $y=at^2$ then $\frac{dy}{dx}=$ (a) $t$ (b) $\frac1t$ (c) $t^2$ (d) 1

If the total cost function $C=50x+5000$ . The marginal cost function MC is (a) $50+\frac{5000}{x}$ (b) $25x^2+5000x$ (c) 50 (d) none of these

The maximum value of $\sin\theta\cos\theta$ is (a) 1 (b) $\frac{1}{\sqrt2}$ (c) $\frac{\sqrt3}{2}$ (d) $\frac12$

The function $f(x)=2-3x$ is (a) Decreasing (b) Increasing (c) Neither decreasing nor increasing (d) Doesn't exist

$\int\log x\,dx=$ (a) $\log x+c$ (b) $x\log x-x+c$ (c) $x\log x+x+c$ (d) $x\log x+c$

$\int_{-2}^{2}|x|\,dx$ is equal to (a) 0 (b) 1 (c) 2 (d) 4

The area of the circle $x^2+y^2=16$ is (a) $4\pi$ (b) $4\pi^2$ (c) $16\pi^2$ (d) $16\pi$

Which one is the measure of dispersion? (a) mean (b) median (c) mode (d) range

The equation $\sin x+\cos x=2$ has (a) infinite solution (b) finite solution (c) no solution (d) unique solution

The system of linear equations $2x+3y=5$ and $4x+6y=9$ are (a) Consistent & dependent (b) Consistent & independent (c) Inconsistent & independent (d) Inconsistent & dependent

The degree of polynomial equation $ax^2+bx+c=0$ is (a) 0 (b) 1 (c) 2 (d) not define

If one root of a quadratic equation in the real coefficient is $\sqrt2$ then equation is (a) $x^2+\sqrt2x+2=0$ (b) $x^2+2=0$ (c) $x^2-2=0$ (d) none of these

The equation $x^2+y^2=0$ represent (a) circle (b) degenerate circle (c) an empty set (d) a straight line

If four quantities $a, b, c$ and d are such that $a:b=3:4$ , $b:c=6:7$ and $c:d=8:9$ , then the ratio between a and d is (a) 3:8 (b) 1:3 (c) 4:7 (d) 7:4

By selling 175 pens, a stationary store gains the selling price of 50 pens. What is his gain percent? (a) 40% (b) $28\frac47$ % (c) 50% (d) none of these

If US USD 1 = Rs. 78.15. How many US dollars are equivalent To Rs. 10,000? (a) USD 127.96 (b) USD 172.96 (c) USD 196.72 (d) USD 172.62

The effective rate corresponding to 10% p.a. convertible quarterly is (a) 40% (b) 10.1% (c) 10.4% (d) none of these

If $n(\cup)=200$ , $n(A)=80$ , $n(B)=60$ , what is the maximum value of $n(A\cup B)=$ ? (a) 200 (b) 40 (c) 140 (d) 110

$-1\leq x\leq11$ is equivalent to (a) $|x-5|\leq6$ (b) $|x+5|\geq3$ (c) $|5-x|\geq2$ (d) $|5-2x|<1$

Range of $y=x^2-4x+5$ is (a) $(-\infty,\infty)$ (b) $(-\infty,1]$ (c) $[1,\infty)$ (d) $(-\infty,1)$

Period of function of $\cos2x$ is (a) $\frac\pi2$ (b) 1 (c) $\pi$ (d) $\frac\pi4$

If $\log_981=x$ then $x$ (a) $\frac12$ (b) $\sqrt2$ (c) 1 (d) 2

If $x=\log_abc$ , $y=\log_bca$ , $z=\log_cab$ then $\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}=$ (a) 1 (b) 0 (c) $\frac12$ (d) abc

$1.5+0.015+0.00015+\ldots\ldots$ (a) $\frac{15}{11}$ (b) $\frac{50}{33}$ (c) $\frac23$ (d) sum doesn't exist

The system of equations $kx-2y=0$ , $x+3y=0$ have unique solution then (a) $k=-\frac23$ (b) $k=0$ (c) $k=\frac23$ (d) $k\neq-\frac23$

The amplitude of $1-i$ is (a) $45^\circ$ (b) $135^\circ$ (c) $315^\circ$ (d) doesn't exist

The mean proportional between 4 & 16 is (a) 4 (b) 8 (c) 12 (d) 5

A plane left 30 minutes later than the schedules time and in order to reach its destinations 1500 km always in time it has to increase its speed by 250 km/hr from its usual speed. What is its usual speed? (a) 500 km/hr (b) 600 km/hr (c) 750 km/hr (d) 100 km/hr

The present worth of Rs 2040 due 4 years hence reaching interested at 5% p.a is (a) 1650.50 (b) Rs. 1675.3 (c) Rs 1680 (d) Rs 1700

The interest rate doubles a sum in 10 year is (a) 12% (b) 11% (c) 10% (d) 9%

Profits in the business of Bikram, Bishal and Binita are shared in ratio 1:2:3. If Bikram share of profits is Rs. 300, what is Binita share? (a) Rs. 150 (b) Rs. 450 (c) Rs. 600 (d) Rs. 900

If A and B are two independent events with $p(A)=\frac23$ & $p(B)=\frac35$ then $p(A\cup B)=$ (a) $\frac{6}{15}$ (b) $\frac53$ (c) $\frac25$ (d) $\frac{13}{15}$

For Negative skewness (a) Mean>Mode>median (b) Mean<median<Mode (c) Median<mode<Mean (d) Mode<mean<Median

HCF of $n!$ , $(n+1)!$ and $(n+2)!$ is (a) $n!$ (b) $(n+2)!$ (c) $(n+1)!$ (d) $(n-1)!$

If $\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\frac\pi2$ then $xy+yz+zx=$ (a) 1 (b) -1 (c) 0 (d) $\pi$

In any $\Delta ABC$ , $2\cos A\sin C=\sin B$ then $\Delta$ is (a) Scalene (b) Equilateral (c) Isosceles (d) right angled

For what value of 'K' points $(1,-1)$ , $\left(\frac k2,\frac k3\right)$ and $(4,1)$ are collinear? (a) $-\frac32$ (b) 3 (c) 2 (d) $-\frac52$

The distance between parallel lines $3x+2y=6$ and $3x+2y=12$ is (a) $\frac{14}{5}$ (b) $\frac72$ (c) $-\frac{8}{\sqrt{13}}$ (d) $\frac{6}{\sqrt{13}}$

Angles between the lines represented by $2x^2-7xy+6y^2=0$ is (a) $\tan^{-1}\left(\pm\frac18\right)$ (b) $\tan^{-1}\left(\pm\frac17\right)$ (c) $45^\circ$ (d) $60^\circ$

Length of y-intercept made by circle $x^2+y^2-8x+y-20=0$ is (a) 5 (b) 9 (c) 8 (d) 6

Radius of circle $x^2+y^2+2x+3y+k=0$ is $\frac12$ then $k=$ (a) 2 (b) -1 (c) 3 (d) 4

$\lim_{x\to64}\frac{\sqrt[6]{x}-2}{\sqrt[3]{x}-4}$ (a) $\frac12$ (b) $\frac34$ (c) $\frac23$ (d) $\frac14$

$\lim_{x\to0}\frac{e^{\sin x}-1}{x}$ (a) 0 (b) 1 (c) 2 (d) $\log_e2$

Every continuous function is (a) increasing (b) decreasing (c) differentiable (d) not - differentiable

$\frac{d}{dx}\cos^{-1}(\sin x)$ (a) x (b) -1 (c) $\frac{-\sin x}{\sqrt{1-\sin^2x}}$ (d) $\frac{\cos x}{\sqrt{1-\sin^2x}}$

If $x^py^q=(x+y)^{p+q}$ then $\frac{dy}{dx}=$ (a) $\frac yx$ (b) $\frac{py}{qx}$ (c) $\frac xy$ (d) $\frac{x^2}{y^2}$

Derivatives of $\sin^{-1}x$ is (a) $\sqrt{1-x^2}$ (b) $\frac{1}{\sqrt{1-x^2}}$ (c) $\frac{-1}{\sqrt{1-x^2}}$ (d) $-\frac{1}{\sqrt{x^2-1}}$

At critical point, $f(x)$ is (a) increasing (b) decreasing (c) stationary (d) increasing & decreasing

$\int\cot x\,dx=$ (a) $\ln(\sec x)+c$ (b) $\ln(\operatorname{cosec}x)+c$ (c) $\ln(\sin x)+c$ (d) $-\operatorname{cosec}^2x+c$

$\int\frac{e^x-1}{e^x+1}dx$ (a) $e^x-\log(e^x+1)$ (b) $e^x+e^{-x}$ (c) $e^x-\log(e^{-x}+1)$ (d) $2\log\left(e^{\frac x2}+e^{-\frac x2}\right)$

$\int\log x\,dx$ (a) $\frac1x$ (b) $x\log x-x$ (c) $e^x$ (d) $\frac{e^x}{x}$

Area of region bounded by curve $y^2=4ax$ and $x^2=4by$ is (a) $\frac{4ab}{3}$ (b) $\frac{8ab}{3}$ (c) $\frac{16ab}{3}$ (d) $\frac{2ab}{3}$

If the ratio of cost price and selling price of an article is 2:3 then (a) Profit % = 50% (b) loss % = 30% (c) Neither profit or loss (d) Profit % = $\frac{100}{3}$ %

$\lim_{n\to\infty}\frac{\sum n^2}{n^3}=$ (a) 0 (b) $\infty$ (c) 2 (d) $\frac13$

$\lim_{x\to1}(x-1)\tan\frac{\pi x}{2}$ (a) $-\frac2\pi$ (b) $\frac\pi2$ (c) $-\frac12$ (d) $-\frac1\pi$

If $y=\sqrt{\frac{1+\cos2x}{1-\cos2x}}$ then $\frac{dy}{dx}$ at $x=\frac\pi2$ is (a) 0 (b) 1 (c) not defined (d) -1

If $a+b+c=0$ in quadratic equation $ax^2+bx+c=0$ then roots of quadratic equation is (a) 1 and -2 (b) 1 and $\frac ca$ (c) $-\frac ba$ and $\frac ca$ (d) 0 and 1

If $(m+n)^{th}$ term of a G.P is p and $(m-n)^{th}$ term is q then $m^{th}$ term is (a) $\sqrt{pq}$ (b) $\frac{p+q}{2}$ (c) $\sqrt{(p+q)}$ (d) $pq$

$(a+b-c)x^2+(c+a-b)x+(b+c-a)=0$ and $a+b+c=0$ then roots quadratic equation, if a, b and c are rational (a) rational (b) irrational (c) imaginary (d) Real and equal

The quadratic equation whose roots are $\frac35$ times the roots if quadratic equation $x^2-8x+7=0$ (a) $25x^2-30x+7=0$ (b) $25x^2-120x+63=0$ (c) $25x^2+120x-63=0$ (d) none of these