TU BSc CSIT entrance BSc. CSIT Entrance: Mathematics: Algebra, Trigonometry, and Calculus

Most repeated · 3 of 13 exams
2071 · 1 mark

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

Answer

(b) -2

Lines given by $ax^2+2hxy+by^2=0$ are perpendicular if $a+b=0$ . Here $2+k=0$ , so $k=-2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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Most repeated · 3 of 13 exams
2074 · 1 mark

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

Answer

(a) $x^2+x-6=0$

Sum of roots $=-1$ , product $=-6$ , so $x^2-(-1)x+(-6)=x^2+x-6=0$ .

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Most repeated · 3 of 13 exams
2074 · 1 mark

The lines represented by $5x^2+3xy+ky^2=0$ are perpendicular to each other for $k=$ (a) 5 (b) 3 (c) 0 (d) -5

Answer

(d) -5

For $ax^2+2hxy+by^2=0$ the lines are perpendicular when $a+b=0$ , so $5+k=0\Rightarrow k=-5$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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Most repeated · 3 of 13 exams
2075 · 1 mark

The quadratic equation whose roots are 3 and 4 is (a) $x^2-7x-12=0$ (b) $x^2-7x+12=0$ (c) $x^2+7x+12=0$ (d) $x^2+7x-12=0$

Answer

(b) $x^2-7x+12=0$ Sum of roots $=7$ and product $=12$ , so $x^2-7x+12=0$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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Most repeated · 3 of 13 exams
2077 · 1 mark

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

Answer

(c) $x^2+x-6=0$ Sum of roots $=2+(-3)=-1$ and product $=2\times(-3)=-6$ . The equation $x^2-(\text{sum})x+\text{product}=0$ gives $x^2+x-6=0$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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Most repeated · 3 of 13 exams
2079 · 1 mark

The two lines represented by $ax^2+4xy+2y^2=0$ are perpendicular to each other for $a=$ (a) 0 (b) -2 (c) 2 (d) 1

Answer

(b) -2 Lines $ax^2+2hxy+by^2=0$ are perpendicular when $a+b=0$ . Here $b=2$ , so $a=-2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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Asked 2 times
2070 · 1 mark
2071 · 1 mark

The derivative of $y=\cos(\cos\sqrt x)$ is (a) $\frac{\sin(\cos\sqrt x)\cdot\sin\sqrt x}{2\sqrt x}$ (b) $\sin(\cos\sqrt x)$ (c) $\sin(\sin\sqrt x)$ (d) $\frac{\sin(\cos\sqrt x)}{2\sqrt x}$

Answer

(a) $\frac{\sin(\cos\sqrt x)\cdot\sin\sqrt x}{2\sqrt x}$

$\frac{dy}{dx}=-\sin(\cos\sqrt x)\cdot(-\sin\sqrt x)\cdot\frac1{2\sqrt x}=\frac{\sin(\cos\sqrt x)\sin\sqrt x}{2\sqrt x}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2070 · 1 mark

What is the value of $\begin{vmatrix}1&\omega&\omega^2\\ \omega&\omega^2&1\\ \omega^2&1&\omega\end{vmatrix}$ , where $\omega$ is an imaginary cube root of unity? (a) 1 (b) 0 (c) 3 (d) 2

Answer

(b) 0

Adding all columns to the first gives a first column of $1+\omega+\omega^2=0$ in every row. A determinant with a zero column is $0$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

The system of equations $2x+y=5$ ; $4x+2y=8$ is (a) Consistent and dependent (b) Inconsistent and independent (c) Inconsistent and dependent (d) Consistent and independent

Similar questions: The system of equation 3x-2y=1 and 6x-4y=2 is (2078)

Answer

(b) Inconsistent and independent

The ratios are $\frac24=\frac12\neq\frac58$ , so the lines are parallel and the system has no solution (inconsistent). The equations are not multiples of each other, so they are independent.

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

The value of $\frac{x+wy+w^2z}{y+wz+w^2x}$ , where $w$ is the cube root of unity, is (a) $w$ (b) $x+y+z$ (c) $w^2$ (d) 1

Similar questions: What is the value of vmatrix 1& & 2 & 2&1 2&1& vmatrix ,... (2070)

Answer

(a) $w$

$w^2(x+wy+w^2z)=w^2x+y+wz$ , which is the denominator. So the ratio is $\frac1{w^2}=w$ , since $w^3=1$ .

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2072 · 1 mark

Area bounded by the x-axis and the curve $y=3x^2$ and the ordinate of $x=3$ and $x=4$ is (a) 37 (b) 8 (c) 19 (d) 0

Answer

(a) 37

$A=\int_3^4 3x^2\,dx=\left[x^3\right]_3^4=64-27=37$ square units.

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

The area bounded by the x-axis and the curve $y=4x^2$ and the ordinate $x=1$ and $x=4$ is (a) 64 (b) 84 (c) 16 (d) -84

Similar questions: Area bounded by the x-axis and the curve y=3x 2 and the... (2072)

Answer

(b) 84

$A=\int_1^4 4x^2dx=\frac{4}{3}\left[x^3\right]_1^4=\frac43(64-1)=84$

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2077 · 1 mark

Area bounded by the x-axis and the curve $y=3x^2$ and the lines $x=2$ and $x=4$ is (a) 64 (b) 56 (c) $\frac{56}{3}$ (d) 72

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

Answer

(b) 56

$\int_2^4 3x^2\,dx = [x^3]_2^4 = 64-8 = 56$

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2078 · 1 mark

The system of equation $3x-2y=1$ and $6x-4y=2$ is (a) inconsistent and dependent (b) consistent and dependent (c) consistent and independent (d) inconsistent and independent

Similar questions: The system of equations 2x+y=5; 4x+2y=8 is (2071)

Answer

(b) consistent and dependent

The second equation is exactly twice the first, so the two lines coincide. The system has infinitely many solutions: it is consistent and dependent.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

Area bounded by x- axis and the curve $y=x^2$ and the lines $x=0$ and $x=4$ is (a) 64 (b) 16 (c) 16/3 (d) 64/3

Similar questions: Area bounded by the x-axis and the curve y=3x 2 and the... (2077)

Answer

(d) 64/3 $\int_0^4x^2dx=\frac{4^3}{3}=\frac{64}{3}$

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2069 · Entrance Examination 2069 · 1 mark

$\lim_{x \to 2} \frac{x^3 - 23}{x-2}$ is (a) 12 (b) 4 (c) 9 (d) $0/0$

Answer

(a) 12

Treating the expression as the standard limit
[ \lim_{x\to2}\frac{x^3-2^3}{x-2} ] factor the numerator: [ \frac{x^3-8}{x-2}=\frac{(x-2)(x^2+2x+4)}{x-2}=x^2+2x+4 ] Putting (x=2): [ 2^2+2(2)+4=4+4+4=12 ]

So the limit is 12.
(If the numerator is literally (x^3-23), then the limit is not finite; the intended question is (x^3-2^3).)

Answer checked by Claude Haiku · Oct 21, 2026

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2069 · Entrance Examination 2069 · 1 mark

$\lim_{x \to 0} \frac{\tan x - 5x}{x}$ is equal to (a) 6 (b) -4 (c) 0 (d) $0/0$

Answer

(b) -4

[ \lim_{x\to 0}\frac{\tan x-5x}{x} =\lim_{x\to 0}\left(\frac{\tan x}{x}-5\right) =1-5=-4 ]

since (\lim_{x\to 0}\frac{\tan x}{x}=1).

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2069 · Entrance Examination 2069 · 1 mark

The function $f(x) = \begin{cases} x+2, & x \ge 2 \\ 2x, & x < 2 \end{cases}$ is continuous at (a) $x=0$ (b) $x=2$ (c) $x=-2$ (d) $x=1$

Answer

(a), (b), (c), and (d)

The function is continuous at all the given points. At (x=2), [ \lim_{x\to2^-}f(x)=\lim_{x\to2^-}2x=4,\qquad \lim_{x\to2^+}f(x)=\lim_{x\to2^+}(x+2)=4,\qquad f(2)=2+2=4, ] so it is continuous at (x=2). For (x=0,-2,1), since each is (<2), (f(x)=2x), which is a polynomial and hence continuous. Thus the question as stated has no unique single correct option.

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2069 · Entrance Examination 2069 · 1 mark

If the derivative of $x^2 + 3x$ is 1, the value of x is (a) 0 (b) -1 (c) 2 (d) 5

Answer

(b) -1

The derivative of $x^2+3x$ is $2x+3$ . Setting $2x+3=1$ gives $2x=-2$ , so $x=-1$ .

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2069 · Entrance Examination 2069 · 1 mark

If $x = 2t + 1$ and $y = t^2$ , then $dy/dt$ is (a) t (b) $1/t$ (c) $2t$ (d) 0

Answer

(c) $2t$

Since $y=t^2$ , differentiating with respect to $t$ gives $\frac{dy}{dt}=2t$ .

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2069 · Entrance Examination 2069 · 1 mark

The points of inflection for the curve $x^3 + 4x$ is (a) 0 (b) 1 (c) 6 (d) -1

Answer

(a) 0

For (y=x^3+4x),
[ \frac{dy}{dx}=3x^2+4,\qquad \frac{d^2y}{dx^2}=6x ] Set (\frac{d^2y}{dx^2}=0): [ 6x=0 \Rightarrow x=0 ] Since (\frac{d^2y}{dx^2}) changes sign at (x=0), the point of inflection is at (x=0).

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2069 · Entrance Examination 2069 · 1 mark

$2 \int_0^1 x dx$ is equal to (a) 0 (b) 1 (c) -1 (d) 2

Answer

(b) 1

Since $\int_0^1 x\,dx = \left[\frac{x^2}{2}\right]_0^1 = \frac12$ , multiplying by 2 gives $2 \times \frac12 = 1$ .

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2069 · Entrance Examination 2069 · 1 mark

$\int \cos 2x \, dx$ is equal to (a) $\sin 2x$ (b) $-\frac{\cos 2x}{2}$ (c) $\sin^2 x$ (d) $2\sin 2x$

Answer

(None of the above)
[ \int \cos 2x,dx=\frac{\sin 2x}{2}+C. ] None of (a)–(d) is correct. Option (b) is the antiderivative of (\sin 2x), not (\cos 2x).

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2070 · 1 mark

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

Answer

(b) $A\cap B$

Since $A-B$ is the part of $A$ outside $B$ , removing it from $A$ leaves the part of $A$ that lies in $B$ , so $A-(A-B)=A\cap B$ .

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2070 · 1 mark

The range of the function $y=\sqrt{(a^2-x^2)}$ , $a>0$ is equal to (a) $[-a,a]$ (b) $(-a,a]$ (c) $[0,a]$ (d) $(0,a)$

Answer

(c) $[0,a]$

For $y=\sqrt{a^2-x^2}$ the domain is $-a\le x\le a$ . Then $a^2-x^2$ varies from $0$ to $a^2$ , so $y$ varies from $0$ to $a$ .

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2070 · 1 mark

What is the value of $\log_a \sqrt{a^3\sqrt{a^2}}$ ? (a) 4 (b) 3 (c) 2 (d) 1

Answer

(c) 2

$\sqrt{a^2}=a$ , so $\sqrt{a^3\cdot a}=\sqrt{a^4}=a^2$ . Hence $\log_a a^2=2$ .

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2070 · 1 mark

If $A=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ , then $A^{100}$ is equal to (a) 1 (b) 0 (c) -1 (d) None

Answer

(a) 1 (the identity matrix $I$ )

$A^2=\begin{pmatrix}0&1\\-1&0\end{pmatrix}^2=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}=-I$ . So $A^{100}=(A^2)^{50}=(-I)^{50}=I$ .

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2070 · 1 mark

If $a^x=b^y=c^z$ and $a, b, c$ are in G.P. then $x, y, z$ are in (a) G.P. (b) A.P. (c) H.P. (d) None

Answer

(c) H.P.

Let $a^x=b^y=c^z=k$ , so $a=k^{1/x},\ b=k^{1/y},\ c=k^{1/z}$ . Since $b^2=ac$ , $\frac2y=\frac1x+\frac1z$ , so $x,y,z$ are in H.P.

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2070 · 1 mark

$\lim_{x\to0}\frac{a^x-b^x}{x}$ is equal to (a) 0 (b) $\log(ab)$ (c) $\log(a/b)$ (d) $\log(b/a)$

Answer

(c) $\log(a/b)$

$\lim_{x\to0}\frac{a^x-1}{x}-\frac{b^x-1}{x}=\log a-\log b=\log(a/b)$ .

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2070 · 1 mark

The derivative of $y=5^x$ with respect to $x$ is equal to (a) $5^x\log_e 5$ (b) $5^x$ (c) $\log_e 5$ (d) $5\log_e 5$

Answer

(a) $5^x\log_e 5$

Using $\frac{d}{dx}a^x=a^x\ln a$ .

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2070 · 1 mark

The function $f(x)=x-\frac{1}{x^2}$ , $x\in R$ , $x\neq0$ is (a) $(-\infty,0)\cup(0,\infty)$ (b) $(-\infty,0]\cup[0,\infty)$ (c) $[-1,1]$ (d) $(-\infty,\infty)$

Answer

(a) $(-\infty,0)\cup(0,\infty)$

$f(x)$ is defined for every real $x$ except $x=0$ , so its domain is $(-\infty,0)\cup(0,\infty)$ .

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2070 · 1 mark

The integral $\int\frac{e^x\,dx}{e^x+1}$ is equal to (a) $\log(e^x+1)+C$ (b) $\frac{-1}{e^{2x}+1}+C$ (c) $\frac{-1}{2(e^x+e^{-x})}+C$ (d) $\frac{-1}{2(e^{2x}+1)}+C$

Answer

(a) $\log(e^x+1)+C$

Put $u=e^x+1$ , $du=e^x\,dx$ . Then $\int\frac{du}{u}=\log u+C=\log(e^x+1)+C$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2070 · 1 mark

For what value of $k$ will the equation $5x^2-kx+45=0$ have equal roots? (a) 0 (b) $\pm3$ (c) $\pm30$ (d) $\pm33$

Answer

(c) $\pm30$

Equal roots need $D=0$ : $k^2-4(5)(45)=0\Rightarrow k^2=900\Rightarrow k=\pm30$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2070 · 1 mark

The quadratic equation of whose one root is $(2+\sqrt3)$ is (a) $x^2+4x+1=0$ (b) $x^2-4x+1=0$ (c) $x^2-4x-1=0$ (d) $-x^2-4x-1=0$

Answer

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

The other root is $2-\sqrt3$ (conjugate). Sum $=4$ , product $=4-3=1$ , so $x^2-4x+1=0$ .

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2070 · 1 mark

The distance between the parallel lines $y-2x=4$ and $6x-3y=5$ is (a) $\frac{1}{\sqrt{45}}$ (b) $\frac{7}{\sqrt{45}}$ (c) $\frac{17}{\sqrt{45}}$ (d) $\frac{17}{2\sqrt{45}}$

Answer

(c) $\frac{17}{\sqrt{45}}$

Write both lines with the same coefficients: $2x-y+4=0$ and $2x-y-\frac53=0$ . Distance $=\frac{|4+\frac53|}{\sqrt{2^2+1}}=\frac{17/3}{\sqrt5}=\frac{17}{3\sqrt5}=\frac{17}{\sqrt{45}}$ .

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2070 · 1 mark

Radius of the circle $x^2+y^2-3x+2y-\frac34=0$ is (a) 2 (b) 1 (c) 4 (d) 3

Answer

(a) 2

Here $g=-\frac32,\ f=1,\ c=-\frac34$ . Radius $=\sqrt{g^2+f^2-c}=\sqrt{\frac94+1+\frac34}=\sqrt4=2$ .

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2070 · 1 mark

The absolute value of the complex number $(1+i)^{-1}$ is (a) 1 (b) $\sqrt2$ (c) $\frac{1}{\sqrt2}$ (d) 2

Answer

(c) $\frac1{\sqrt2}$

$|(1+i)^{-1}|=\frac1{|1+i|}=\frac1{\sqrt2}$ .

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2070 · 1 mark

If $z=\cos\theta+i\sin\theta$ then $z^n+\frac{1}{z^n}$ is equal to (a) $-2\cos n\theta$ (b) $2\cos n\theta$ (c) $2\sin n\theta$ (d) $-2\sin n\theta$

Answer

(b) $2\cos n\theta$

By De Moivre, $z^n=\cos n\theta+i\sin n\theta$ and $z^{-n}=\cos n\theta-i\sin n\theta$ . Adding gives $2\cos n\theta$ .

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2070 · 1 mark

The $n^{th}$ term of the series: $2+6+12+20+\ldots$ is (a) $n(n+1)$ (b) $(n+1)(n+2)$ (c) $n(n-1)$ (d) $n$

Answer

(a) $n(n+1)$

The terms are $1\cdot2,\ 2\cdot3,\ 3\cdot4,\ 4\cdot5,\dots$ , so $T_n=n(n+1)$ .

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2070 · 1 mark

If $|x|<1$ and $y=x+x^2+x^3+x^4+\ldots+\infty$ , then $x$ is equal to (a) $\frac{y}{1-y}$ (b) $\frac{y}{y-1}$ (c) $\frac{y+1}{y}$ (d) $\frac{y}{1+y}$

Answer

(d) $\frac{y}{1+y}$

The sum of the infinite G.P. is $y=\frac{x}{1-x}$ . So $y-xy=x$ , giving $x=\frac{y}{1+y}$ .

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2070 · 1 mark

$3\cos^{-1}x$ is equal to (a) $\cos^{-1}(3x-4x^2)$ (b) $\cos^{-1}(4x^3-3x)$ (c) $\cos^{-1}(4x-3x^3)$ (d) $\cos^{-1}(3x^3-4x)$

Answer

(b) $\cos^{-1}(4x^3-3x)$

Let $\theta=\cos^{-1}x$ . Then $\cos3\theta=4\cos^3\theta-3\cos\theta=4x^3-3x$ , so $3\cos^{-1}x=\cos^{-1}(4x^3-3x)$ .

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2070 · 1 mark

Solution of $\tan ax=\cot bx$ is (a) $\frac{2n+1}{a+b}\cdot\frac{\pi}{4}$ (b) $\frac{2n-1}{a-b}\cdot\frac{\pi}{2}$ (c) $\frac{2n+1}{a+b}\cdot\frac{\pi}{2}$ (d) $\frac{\pi}{2}$

Answer

(c) $\frac{2n+1}{a+b}\cdot\frac\pi2$

$\tan ax=\cot bx=\tan\left(\frac\pi2-bx\right)$ , so $ax=n\pi+\frac\pi2-bx$ . Thus $x=\frac{(2n+1)\pi}{2(a+b)}=\frac{2n+1}{a+b}\cdot\frac\pi2$ .

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2070 · 1 mark

The single equation representing the pairs of lines $y=x$ and $y=-x$ is (a) $y^2-x^2=0$ (b) $y-x=0$ (c) $x-y=0$ (d) $x^2+y^2=0$

Answer

(a) $y^2-x^2=0$

$(y-x)(y+x)=0\Rightarrow y^2-x^2=0$ .

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2070 · 1 mark

Let $A=[-3,2]$ and $B=[2,3]$ then $A-B$ is equal to (a) $[-3,3)$ (b) $[-3,2)$ (c) $(-2,2)$ (d) $(-2,1)$

Answer

(b) $[-3,2)$

$A-B$ keeps the elements of $[-3,2]$ not in $[2,3]$ . The point $2$ lies in $B$ , so it is removed: $[-3,2)$ .

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2070 · 1 mark

If $\lim_{x\to0^-}f(x)\neq\lim_{x\to0^+}f(x)$ , then $f(x)$ is said to be an (a) infinite discontinuity (b) removable discontinuity (c) jump discontinuity (d) none

Answer

(c) jump discontinuity

When both one-sided limits exist but are unequal, the function has a jump discontinuity at that point.

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2070 · 1 mark

Which one of the following is the area enclosed by the curve $y=3x$ , the $x$ -axis and the ordinate $x=0$ , $x=4$ ? (a) 12 (b) 16 (c) 20 (d) 24

Answer

(d) 24

$\text{Area}=\int_0^4 3x\,dx=\frac{3x^2}{2}\Big|_0^4=24$ square units.

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2071 · 1 mark

If $|x-3|=4$ then $x$ is (a) 7 and -1 (b) 7 and 1 (c) 7 (d) 1

Answer

(a) 7 and -1

$|x-3|=4$ gives $x-3=4$ or $x-3=-4$ , so $x=7$ or $x=-1$ .

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2071 · 1 mark

If $\{x-x^2=9$ and $2x=6\}$ , then (a) null set (b) $\{3\}$ (c) $\{3,-3\}$ (d) $\{4,3\}$

Answer

(a) null set

$x-x^2=9$ has no real solution, since $x^2-x+9=0$ has discriminant $1-36<0$ . Also $x=3$ does not satisfy it ( $3-9=-6$ ). So the set is empty.

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2071 · 1 mark

If $f:R\to R$ defined by $f(x)=x^2+3$ then $f^{-1}(4)$ is (a) 0 (b) $\{-1,1\}$ (c) $\{1,0\}$ (d) $\{4,3\}$

Answer

(b) $\{-1,1\}$

$f^{-1}(4)$ is the set of $x$ with $x^2+3=4$ , so $x^2=1$ and $x=\pm1$ .

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2071 · 1 mark

The value of $\log_2\sqrt2+\log_3\sqrt3+\log_4\sqrt4$ is (a) 0 (b) $\frac32$ (c) $-\frac{13}{12}$ (d) $\frac{\pi}{6}$

Answer

(b) $\frac32$

$\log_2\sqrt2=\frac12$ , $\log_3\sqrt3=\frac12$ and $\log_4\sqrt4=\log_4 2=\frac12$ . The sum is $\frac32$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

If $\cos^2\theta=\frac12$ , then $\theta$ is (a) $n\pi\pm\frac{\pi}{4}$ (b) $n\pi+\frac{\pi}{4}$ (c) $n\pi\pm\frac{\pi}{3}$ (d) $2n\pi$

Answer

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

$\cos^2\theta=\frac12=\cos^2\frac{\pi}{4}$ , so $\theta=n\pi\pm\frac{\pi}{4}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

$\cos\left(\sin^{-1}\frac35\right)=$ (a) $\frac35$ (b) $\frac45$ (c) $\frac{9}{25}$ (d) $\frac{4}{25}$

Answer

(b) $\frac45$

Let $\sin^{-1}\frac35=\alpha$ , so $\sin\alpha=\frac35$ . Then $\cos\alpha=\sqrt{1-\frac{9}{25}}=\frac45$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

If $a^2, b^2, c^2$ are in H.P. then (a) $2a^2c^2=b^2(a^2+c^2)$ (b) $a^2c^2=b^2(a^2+c^2)$ (c) $2b^2c^2=a^2(b^2+c^2)$ (d) $c^2a^2=2b^2(a^2+c^2)$

Answer

(a) $2a^2c^2=b^2(a^2+c^2)$

If $a^2,b^2,c^2$ are in H.P., then $\frac1{a^2},\frac1{b^2},\frac1{c^2}$ are in A.P. So $\frac2{b^2}=\frac1{a^2}+\frac1{c^2}$ , which gives $2a^2c^2=b^2(a^2+c^2)$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

If $A=\begin{bmatrix}1&2\\2&3\end{bmatrix}$ and $A^2-nA-I=0$ , then the value of $n$ is (a) 0 (b) 4 (c) 1 (d) -4

Answer

(b) 4

$A^2=\begin{bmatrix}5&8\\8&13\end{bmatrix}$

Comparing $A^2-nA-I=0$ entry by entry: $5-n-1=0$ gives $n=4$ . Check: $8-2n=0$ gives $n=4$ , and $13-3n-1=0$ gives $n=4$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

If $A=\begin{bmatrix}1&0&1\\0&1&1\\1&0&0\end{bmatrix}$ , then $A$ is (a) Symmetric (b) Skew-symmetric (c) Singular (d) Nonsingular

Answer

(d) Nonsingular

$|A|=1(0-0)-0+1(0-1)=-1\neq0$ , so $A$ is nonsingular. It is not symmetric, since $a_{23}=1$ but $a_{32}=0$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

The value of $i^7+\frac{1}{i^4}$ is (a) 1 (b) -1 (c) $-i$ (d) $-i+1$

Answer

(d) $-i+1$

$i^7=i^4\cdot i^3=-i$ and $\frac1{i^4}=1$ . The sum is $1-i$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

If the roots of $x^2+4x+k=0$ are equal then $k$ is (a) 0 (b) 4 (c) 16 (d) -4

Answer

(b) 4

Equal roots need discriminant zero: $16-4k=0$ , so $k=4$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

The two lines $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ are identical if (a) $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$ (b) $\frac{a_1}{a_2}=0$ (c) $\frac{b_1}{b_2}=0$ (d) $\frac{c_1}{c_2}=0$

Answer

(a) $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Two lines coincide when their coefficients are proportional.

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

The equation of the circle touching the positive x-axis and y-axis with radius 2 units is (a) $x^2+y^2-4x-4y+4=0$ (b) $x^2+y^2+4x+4y+4=0$ (c) $x^2+y^2-4x+4y+4=0$ (d) $x^2+y^2-4x+4y-4=0$

Answer

(a) $x^2+y^2-4x-4y+4=0$

A circle of radius 2 touching the positive axes has centre $(2,2)$ . Its equation is $(x-2)^2+(y-2)^2=4$ , which expands to $x^2+y^2-4x-4y+4=0$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

If the line $3x+4y=k$ is tangent to the circle $x^2+y^2=25$ at $(3,4)$ then $k$ is (a) 25 (b) -25 (c) $\pm25$ (d) $\pm1$

Answer

(a) 25

The point $(3,4)$ lies on the line, so $k=3(3)+4(4)=25$ . The tangent to $x^2+y^2=25$ at $(3,4)$ is indeed $3x+4y=25$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

The value of $\lim_{x\to2}\frac{x^5-2^5}{x-2}$ is (a) 80 (b) 40 (c) 10 (d) 0

Answer

(a) 80

$\lim_{x\to a}\frac{x^n-a^n}{x-a}=na^{n-1}$ , so the limit is $5\cdot2^4=80$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

The value of $\lim_{x\to0}\frac{3x+4\tan2x}{x}$ is (a) 7 (b) 3 (c) 11 (d) 4

Answer

(c) 11

$\lim_{x\to0}\left(3+4\cdot\frac{\tan2x}{x}\right)=3+4\cdot2=11$

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

The derivative of $y=b^x$ is (a) $xb^{x-1}$ (b) $b$ (c) $x^b\ln|b|$ (d) $b^x\ln|b|$

Answer

(d) $b^x\ln|b|$

The derivative of $b^x$ is $b^x\ln b$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

The value of $x$ for which the derivative of $x^2+2x+2$ is zero is (a) 1 (b) 2 (c) 0 (d) -1

Answer

(d) $-1$

$\frac{d}{dx}(x^2+2x+2)=2x+2=0$ , so $x=-1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

The point of inflection of the curve $x^3+3x^2-9x+2$ is (a) -1 (b) 1 (c) 6 (d) 0

Answer

(a) $-1$

$y'=3x^2+6x-9$ , $y''=6x+6=0\Rightarrow x=-1$ , and $y''$ changes sign there, so it is the point of inflection.

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

$\int\frac{dx}{1-\cos x}=$ (a) $\tan\frac x2$ (b) $-\cot\frac x2$ (c) $\cos\frac x2$ (d) $-\tan\frac x2$

Answer

(b) $-\cot\frac x2$

Since $1-\cos x=2\sin^2\frac x2$ , $\int\frac{dx}{1-\cos x}=\frac12\int\operatorname{cosec}^2\frac x2\,dx=-\cot\frac x2+C.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2071 · 1 mark

$\int_0^{\pi/2}\frac{\cos x}{1+\sin x}\,dx=$ (a) $\infty$ (b) $\ln|2|$ (c) 0 (d) 1

Answer

(b) $\ln 2$

Put $u=1+\sin x$ : $\int_0^{\pi/2}\frac{\cos x}{1+\sin x}dx=\left[\ln(1+\sin x)\right]_0^{\pi/2}=\ln2-\ln1=\ln 2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

If $a, b\in R$ , $a/b$ is indeterminate, if (a) $a=0, b\neq0$ (b) $a=0, b=0$ (c) $a\neq0, b\neq0$ (d) $a\neq0, b=0$

Answer

(b) $a=0, b=0$

$0/0$ is indeterminate because any number satisfies $0\times x=0$ . If $a\neq0, b=0$ it is undefined, and if $a=0, b\neq0$ it equals 0.

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

$(-\infty,\infty)=$ (a) $\{x: -\infty\leq x\leq\infty\}$ (b) $\{x: -\infty<x<0\}$ (c) $\{x: -\infty<x<\infty\}$ (d) $\{x: x\geq0\}$

Answer

(c) $\{x: -\infty<x<\infty\}$

The open interval $(-\infty,\infty)$ contains every real number; the infinities are not included, so strict inequalities are used.

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

The statement is (a) Bagmati is a River. (b) $x>0$ (c) How old are you? (d) $y>0$

Answer

(a) Bagmati is a River.

A statement (proposition) is a sentence that is definitely true or false. "Bagmati is a River" is true. $x>0$ and $y>0$ depend on the value of the variable, and a question has no truth value.

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

If $(2x-1,-3)=(3,y+3)$ , then (a) $x=1, y=0$ (b) $x=-1, y=-3$ (c) $x=2, y=-6$ (d) $x=0, y=-1$

Answer

(c) $x=2,\ y=-6$

Equal ordered pairs have equal components: $2x-1=3\Rightarrow x=2$ and $y+3=-3\Rightarrow y=-6$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

If $f:R\to R$ is defined by $f(x)=x$ and $g:R\to R$ is defined by $g(x)=x$ then $(f\circ g)(x)$ is (a) $x$ (b) $x^2$ (c) $\frac{1}{x^2}$ (d) $1/x$

Answer

(a) $x$

$(f\circ g)(x)=f(g(x))=f(x)=x$ , since both functions are the identity map.

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

If $\sin2x=1$ , then $x$ is (a) $\pi/2$ (b) $\pm1$ (c) $-\pi/2$ (d) $n\pi+(-1)^n\frac{\pi}{4}$

Answer

(d) $n\pi+(-1)^n\dfrac{\pi}{4}$

Of the given options, this is the only one in the form of a general solution of a sine equation. It gives $\pi/4$ for $n=0$ , which is the principal solution of $\sin 2x=1$ (as $2x=\pi/2$ ). It also matches $x=n\pi+\pi/4$ whenever $n$ is even.

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

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

Answer

(b) $\dfrac{x}{\sqrt{1+x^2}}$

Let $\theta=\tan^{-1}x$ , so $\tan\theta=x$ . In a right triangle the opposite side is $x$ , the adjacent side is $1$ and the hypotenuse is $\sqrt{1+x^2}$ . Hence $\sin\theta=\dfrac{x}{\sqrt{1+x^2}}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

If $2p, p+8, 3p+1$ are in A.P., then $p$ is (a) 5 (b) 3 (c) 7 (d) -3

Answer

(a) 5

For an A.P., $2(p+8)=2p+(3p+1)$ , so $2p+16=5p+1$ and $3p=15$ , giving $p=5$ . The terms are 10, 13, 16.

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

A, G and H are arithmetic mean, geometric mean and harmonic means respectively then (a) $A^2=GH$ (b) $H^2=AG$ (c) $G=AH$ (d) $G^2=AH$

Answer

(d) $G^2=AH$

For two numbers $a,b$ : $A=\frac{a+b}{2}$ , $H=\frac{2ab}{a+b}$ , so $AH=ab=G^2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

If $A=(-1\ 0\ 2)$ and $B=\begin{pmatrix}2\\3\\4\end{pmatrix}$ , then $AB$ is (a) $(6)$ (b) 0 (c) $(-2\ 0\ 8)$ (d) $(2\ 0\ -4)$

Answer

(a) $(6)$

$AB=(-1)(2)+(0)(3)+(2)(4)=-2+0+8=6$ , a $1\times1$ matrix.

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

The value of the determinant $\begin{vmatrix}1&2&0\\2&4&1\\3&6&2\end{vmatrix}$ is (a) 1 (b) 3 (c) 0 (d) 4

Answer

(c) 0

Expanding along the first row: $1(4\cdot2-1\cdot6)-2(2\cdot2-1\cdot3)+0=2-2=0$ . (Column 2 is twice column 1, so the determinant is zero.)

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

The adjoint of the matrix $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is (a) $\begin{pmatrix}a&d\\b&c\end{pmatrix}$ (b) $\begin{pmatrix}a&b\\d&c\end{pmatrix}$ (c) $\begin{pmatrix}a&d\\c&b\end{pmatrix}$ (d) $\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$

Answer

(d) $\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$

The adjoint is the transpose of the cofactor matrix. The cofactors are $d,-c,-b,a$ , so after transposing we get the matrix above.

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

The real part of $\frac{(1+i)^2}{2-i}$ is (a) $-2/5$ (b) $2/5$ (c) $1/5$ (d) 1

Answer

(a) $-2/5$

$(1+i)^2=2i$ . Then $\dfrac{2i}{2-i}\cdot\dfrac{2+i}{2+i}=\dfrac{4i+2i^2}{5}=\dfrac{-2+4i}{5}$ , so the real part is $-\dfrac25$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

If $\omega$ be the cube roots of unity then $(1+\omega+\omega^2)(1+\omega-\omega^2)=$ (a) 0 (b) 2 (c) 4 (d) 3

Answer

(a) 0

Since $1+\omega+\omega^2=0$ , the first factor is $0$ , so the product is $0$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

The value of $k$ for which the equation $x^2+2(k+2)x+9k=0$ has equal roots are (a) 4, 1 (b) $-4,-1$ (c) $-4, 1$ (d) 4, $-1$

Answer

(a) 4, 1

Equal roots require the discriminant to be zero: $4(k+2)^2-36k=0\Rightarrow k^2+4k+4-9k=0\Rightarrow k^2-5k+4=0\Rightarrow (k-1)(k-4)=0$ . So $k=1,4$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

The distance between the point $(1,2)$ and the line $3x+4y-7=0$ is (a) 4 (b) 5 (c) 4/5 (d) -4/5

Answer

(c) $4/5$

$d=\dfrac{|3(1)+4(2)-7|}{\sqrt{3^2+4^2}}=\dfrac{4}{5}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

The two lines represented by $ax^2+2hxy+by^2=0$ are coincident if (a) $h^2=ab$ (b) $h^2>ab$ (c) $h^2<ab$ (d) $h^2>0$

Answer

(a) $h^2=ab$

The pair of lines is real and distinct when $h^2>ab$ , and the lines coincide when the discriminant vanishes, i.e. $h^2=ab$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

The equation of tangent to the circle $x^2+y^2=5$ at $(2,1)$ is (a) $x+y=5$ (b) $x-y=5$ (c) $2x+y=5$ (d) $x+2y=5$

Answer

(c) $2x+y=5$

The tangent to $x^2+y^2=a^2$ at $(x_1,y_1)$ is $xx_1+yy_1=a^2$ . Here it is $2x+y=5$ , and $(2,1)$ lies on the circle since $4+1=5$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

$\lim_{x\to2}\frac{x-2}{\sqrt{x^2-4}}$ is (a) 0/0 (b) 0 (c) 1 (d) 4

Answer

(b) 0

For $x>2$ : $\dfrac{x-2}{\sqrt{(x-2)(x+2)}}=\sqrt{\dfrac{x-2}{x+2}}\to\sqrt{0}=0$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

$\lim_{x\to0}\frac{1-\cos ax}{x^2}$ is (a) 0 (b) $\frac{a^2}{2}$ (c) $2a^2$ (d) 2

Answer

(b) $\dfrac{a^2}{2}$

$1-\cos ax=2\sin^2\frac{ax}{2}$ , so the limit is $\lim 2\cdot\dfrac{\sin^2(ax/2)}{x^2}=2\cdot\dfrac{a^2}{4}=\dfrac{a^2}{2}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

The value of $x$ for which the function $y=x^3-x^2$ has $\frac{dy}{dx}=0$ is (a) 0, 2/3 (b) 2/3 (c) 0 (d) $-2/3$

Answer

(a) $0,\ 2/3$

$\dfrac{dy}{dx}=3x^2-2x=x(3x-2)=0\Rightarrow x=0$ or $x=\dfrac23$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

The point of inflection for $y=x^2-2x$ is (a) 0 (b) does not exist (c) 2 (d) 1

Answer

(b) does not exist

$y''=2$ , which is never zero and never changes sign, so the parabola has no point of inflection.

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

$\int_0^1\frac{2x\,dx}{\sqrt{1-x^2}}$ is (a) 1/2 (b) 1 (c) 0 (d) 2

Answer

(d) 2

Put $u=1-x^2$ , $du=-2x\,dx$ : $\int_0^1\frac{2x\,dx}{\sqrt{1-x^2}}=\left[-2\sqrt{1-x^2}\right]_0^1=0-(-2)=2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2072 · 1 mark

$\int x\sin x\,dx=$ (a) $\sin x-x\cos x+C$ (b) $\sin x+x\cos x+C$ (c) $\cos x+x\sin x+C$ (d) $-\sin x+x\cos x+C$

Answer

(a) $\sin x-x\cos x+C$

By parts with $u=x$ , $dv=\sin x\,dx$ : $\int x\sin x\,dx=-x\cos x+\int\cos x\,dx=-x\cos x+\sin x+C$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

Which one is not a logical statement: (a) If you work hard, you will pass (b) I am sick. (c) The sun rises in the west. (d) 30 is a number.

Answer

(a) If you work hard, you will pass A logical statement (proposition) must be definitely true or false. The conditional in (a) is an uncertain claim with no definite truth value, whereas (c) is false and (d) is true.

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

If $|x-2|=5$ , then $x$ is in (a) $\{3,5\}$ (b) $\{2,\infty\}$ (c) $\{2,5\}$ (d) $\{-3,7\}$

Answer

(d) $\{-3,7\}$ $x-2=\pm5 \Rightarrow x=7 \text{ or } x=-3.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

$A-B=$ (a) $B-A$ (b) $\bar A-\bar B$ (c) $\bar B-A$ (d) $\bar B-\bar A$

Answer

(d) $\bar B-\bar A$ $A-B=A\cap \bar B=\bar B\cap A=\bar B-\bar A.$

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2073 · 1 mark

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

Answer

(a) $\frac{x+3}{2}$ Put $y=2x-3$ , so $x=\frac{y+3}{2}$ . Hence $f^{-1}(x)=\frac{x+3}{2}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

The period of the function $f(x)=\sin(2x+5)$ is (a) $2\pi$ (b) $\pi/2$ (c) $\pi$ (d) $-\pi$

Answer

(c) $\pi$ The period of $\sin(ax+b)$ is $\frac{2\pi}{|a|}$ , so here it is $\frac{2\pi}{2}=\pi$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

The fourth, seventh and tenth terms of a G.P. are $l, m, n$ respectively, then (a) $ln=m^2$ (b) $l^2=m^2+n^2$ (c) $l^2=mn$ (d) $n^2=lm$

Answer

(a) $ln=m^2$ With first term $a$ and ratio $r$ : $l=ar^3$ , $m=ar^6$ , $n=ar^9$ . Then $ln=a^2r^{12}=(ar^6)^2=m^2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

The nth term of the series: $1.3+3.5+5.7+\ldots$ is (a) $(n+1)(n+2)$ (b) $(n+1)(n+3)$ (c) $4n^2-1$ (d) $n(n-1)$

Answer

(c) $4n^2-1$ The $n$ th term is $(2n-1)(2n+1)=4n^2-1$ . Check: $n=1$ gives 3 and $n=2$ gives 15.

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

The value of $\sin(\cos^{-1}x)$ is (a) $x$ (b) $1-x^2$ (c) $1+x^2$ (d) $\sqrt{1-x^2}$

Answer

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

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2073 · 1 mark

If $\operatorname{cosec}^2x-2=0$ , then $x$ is (a) $\pm\pi/6$ (b) $\pm\pi/4$ (c) $n\pi\pm\pi/4$ (d) $n\pi+\pi/2$

Answer

(c) $n\pi\pm\pi/4$ $\operatorname{cosec}^2x=2\Rightarrow\sin^2x=\frac12\Rightarrow\sin x=\pm\frac{1}{\sqrt2}$ , which gives $x=n\pi\pm\frac{\pi}{4}$ .

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2073 · 1 mark

If $A=\begin{pmatrix}k&1&0\\2&0&k\\0&2&-1\end{pmatrix}$ and $|A|=0$ then $k=$ (a) 1 (b) -1 (c) 0 (d) $\pm1$

Answer

(d) $\pm1$ $|A|=k(0\cdot(-1)-2k)-1(2\cdot(-1)-0)=-2k^2+2=0\Rightarrow k^2=1\Rightarrow k=\pm1.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

The value of $x$ for which $A=\begin{bmatrix}6&x-2\\3&x\end{bmatrix}$ has no inverse is (a) 0 (b) 1 (c) -2 (d) 2

Answer

(c) -2 A has no inverse when $|A|=0$ : $6x-3(x-2)=3x+6=0$ , so $x=-2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

If the system of equation has zero solutions only then the solution is (a) non trivial (b) trivial (c) unique (d) particular

Answer

(b) trivial A homogeneous system whose only solution is the zero solution $x=y=\dots=0$ has only the trivial solution.

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

$(1+i)^6+(1-i)^6=$ (a) 0 (b) $i$ (c) 16 (d) -16

Answer

(a) 0 $(1+i)^2=2i$ , so $(1+i)^6=(2i)^3=-8i$ . Similarly $(1-i)^2=-2i$ , so $(1-i)^6=(-2i)^3=8i$ . The sum is $0$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

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

Answer

(c) $2i$ $\frac{2i}{1+i}=\frac{2i(1-i)}{2}=i+1$ , and $(1+i)^2=2i$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

If one root of $5x^2+2x-k=0$ is the reciprocal of the other then $k=$ (a) -5 (b) 5 (c) 2/5 (d) 5/2

Answer

(a) -5 The product of the roots is $\frac{-k}{5}$ . For reciprocal roots the product is 1, so $-k/5=1$ and $k=-5$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

The equation $x^2-2xy+y^2+3x-2=0$ represents: (a) hyperbola (b) parabola (c) ellipse (d) circle

Answer

(b) parabola For $Ax^2+2Hxy+By^2+\dots$ , here $A=1,\ H=-1,\ B=1$ , so $H^2-AB=1-1=0$ . The second-degree terms form a perfect square, so the conic is a parabola.

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

The area of the triangle with vertices $(1,-1)$ , $(-1,1)$ and $(-1,-1)$ is (a) 1/2 (b) -2 (c) 2 (d) 0

Answer

(c) 2 The points $(1,-1)$ , $(-1,1)$ , $(-1,-1)$ give a right angle at $(-1,-1)$ with legs 2 and 2, so $\text{Area}=\tfrac12\cdot2\cdot2=2.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

$\lim_{n\to\infty}\frac{1+2+\cdots+n}{n^2}=$ (a) 1/2 (b) 0 (c) $1/n$ (d) $n$

Answer

(a) 1/2 $\lim_{n\to\infty}\frac{n(n+1)/2}{n^2}=\lim_{n\to\infty}\frac{1+1/n}{2}=\frac12.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

$\lim_{x\to0}\frac{5x^2+4x\tan x}{x^2}=$ (a) 9 (b) 0 (c) 5 (d) 4

Answer

(a) 9 $\lim_{x\to0}\left(5+4\frac{\tan x}{x}\right)=5+4=9.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

If $y=\sqrt{x+\sqrt{x+\sqrt{x+\cdots\infty}}}$ then $\frac{dy}{dx}=$ (a) $2\sqrt x$ (b) $\sqrt3x$ (c) $\frac{\sqrt x}{2y-1}$ (d) $\frac{1}{2y-1}$

Answer

(d) $\frac{1}{2y-1}$

Since $y=\sqrt{x+y}$ , we have $y^2=x+y$ . Differentiating, $2y\,y'=1+y'$ , so $y'=\frac{1}{2y-1}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

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

Answer

(a) $\frac{axy}{\sqrt{ax^2+b}}$

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

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2073 · 1 mark

$\int\frac{dx}{1-x^2}=$ (a) $\cos^{-1}x+c$ (b) $\frac12\ln\left|\frac{1-x}{1+x}\right|+c$ (c) $\sin^{-1}x+c$ (d) $\frac12\ln\left|\frac{1+x}{1-x}\right|+c$

Answer

(d) $\frac12\ln\left|\frac{1+x}{1-x}\right|+c$

Using partial fractions, $\frac{1}{1-x^2}=\frac12\left(\frac1{1-x}+\frac1{1+x}\right)$ , so the integral is $\frac12\left[-\ln|1-x|+\ln|1+x|\right]+c$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

$\int_1^2\frac{\sin(\ln t)}{t}dt=$ (a) $1-\cos(\ln2)$ (b) $\cos2$ (c) $\cos(\ln2)$ (d) $1+\cos(\ln2)$

Answer

(a) $1-\cos(\ln2)$

Put $u=\ln t$ , $du=dt/t$ ; limits $0$ to $\ln2$ . Then $\int_0^{\ln2}\sin u\,du=[-\cos u]_0^{\ln2}=1-\cos(\ln2)$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

The critical points for $f(x)=x^3-3x$ are: (a) $\pm3$ (b) $\pm\sqrt3$ (c) $\pm1$ (d) $0,\pm\sqrt3$

Answer

(c) $\pm1$

$f'(x)=3x^2-3=0\Rightarrow x^2=1\Rightarrow x=\pm1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2073 · 1 mark

The area bounded by the x-axis and the curve $y=x^3$ and ordinates of $x=2$ and $x=4$ is (a) 60 sq. units (b) 256 sq. units (c) 240 sq. units (d) 272 sq. units

Answer

(a) 60 sq. units

$\int_2^4x^3dx=\left[\frac{x^4}{4}\right]_2^4=\frac{256-16}{4}=60.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

$p/q$ , $q\in R$ , is undefined if (a) $p\neq0, q=0$ (b) $p=0, q=0$ (c) $p\neq0, q\neq0$ (d) $p=0, q\neq0$

Answer

(a) $p\neq0,\ q=0$

Division by zero is undefined for a non-zero numerator. (The case $0/0$ is indeterminate rather than simply undefined.)

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

$[a,b]=$ (a) $\{x: a\leq x\leq b\}$ (b) $\{x: a<x<b\}$ (c) $\{x: a<x\leq b\}$ (d) $\{x: a\leq x<b\}$

Answer

(a) $\{x: a\leq x\leq b\}$

The square brackets denote the closed interval, which includes both end points.

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

If $f:R\to R$ defined by $f(x)=\sqrt{(4-x^2)}$ then $f(-3)$ is (a) Undefined (b) $\sqrt{13}$ (c) $i\sqrt5$ (d) $-\sqrt{13}$

Answer

(a) Undefined

$f(-3)=\sqrt{4-9}=\sqrt{-5}$ , which is not a real number. Since the codomain is $\mathbb{R}$ , $f(-3)$ is undefined.

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

If $|x|=1$ then $x$ is (a) $\{1\}$ (b) $\{-1\}$ (c) $\{-1,1\}$ (d) $\{-1,-1\}$

Answer

(c) $\{-1,1\}$

$|x|=1$ gives $x=1$ or $x=-1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

If $\log_2(x-7)=1$ , then $x$ is (a) Not known (b) 8 (c) 6 (d) 9

Answer

(d) 9

$\log_2(x-7)=1\Rightarrow x-7=2^1=2\Rightarrow x=9$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

The value of $\sin\left(\cos^{-1}\frac45\right)$ is (a) 4/5 (b) 3/5 (c) 5/4 (d) 5/3

Answer

(b) 3/5

Let $\theta=\cos^{-1}\frac45$ , so $\cos\theta=\frac45$ and $\sin\theta=\sqrt{1-\frac{16}{25}}=\frac35$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

If $\tan^2x=3$ then $x$ is (a) $n\pi+\frac{\pi}{3}$ (b) $n\pi-\frac{\pi}{3}$ (c) $n\pi\pm\frac{\pi}{3}$ (d) $n\pi$

Answer

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

$\tan^2x=3\Rightarrow\tan x=\pm\sqrt3$ , so $x=n\pi\pm\frac{\pi}{3}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

If $k+2, 4k-6, 3k-2$ are in AP then $k$ is (a) -6 (b) 0 (c) 3 (d) -3

Answer

(c) 3

For an AP, $2(4k-6)=(k+2)+(3k-2)$ , so $8k-12=4k$ and $k=3$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

The sum of infinity of $1-\frac12+\frac14-\frac18\ldots$ is (a) 0 (b) 2/3 (c) 1/3 (d) 1

Answer

(b) 2/3

GP with $a=1$ , $r=-\frac12$ : $S_\infty=\frac{1}{1+\frac12}=\frac23$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

If $A=\begin{pmatrix}1\\2\\3\end{pmatrix}$ and $B=(1\ 2\ 3)$ then $AB$ is (a) $(1\ 4)$ (b) $(1\ 4\ 9)$ (c) 0 (d) $\begin{pmatrix}1&2&3\\2&4&6\\3&6&9\end{pmatrix}$

Answer

(d) $\begin{pmatrix}1&2&3\\2&4&6\\3&6&9\end{pmatrix}$

A $3\times1$ matrix times a $1\times3$ matrix gives a $3\times3$ matrix with entries $a_ib_j$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

The value of $\begin{vmatrix}0&2&3\\1&2&3\\4&8&12\end{vmatrix}$ is (a) 10 (b) 1 (c) 0 (d) -1

Answer

(c) 0

Row 3 is $4\times$ Row 2 (since $4(1,2,3)=(4,8,12)$ ), so the determinant is 0.

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

The matrix $\begin{pmatrix}1&2&3\\2&0&4\\3&4&5\end{pmatrix}$ is (a) Diagonal (b) symmetric (c) Skew symmetric (d) unit

Answer

(b) symmetric

The matrix equals its transpose ( $a_{ij}=a_{ji}$ ), and the diagonal is not zero, so it is symmetric and not skew-symmetric.

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

$[2(\cos15^\circ+i\sin15^\circ)]^6=$ (a) $64i$ (b) 64 (c) 0 (d) $-64i$

Answer

(a) $64i$

By De Moivre: $2^6(\cos90^\circ+i\sin90^\circ)=64i$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

$(1+\omega+\omega^2)^3+(1+\omega-\omega^2)^3=$ (a) 16 (b) $16\omega$ (c) -8 (d) $16\omega^2$

Answer

(c) -8

$1+\omega+\omega^2=0$ , so the first term is 0. Also $1+\omega-\omega^2=-2\omega^2$ , and $(-2\omega^2)^3=-8\omega^6=-8$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

The line $kx+3y+5=0$ and $5x-2y-6=0$ are perpendicular to each other for $k=$ (a) 6/5 (b) -6/5 (c) 3/5 (d) -3/5

Answer

(a) 6/5

Slopes are $-\frac k3$ and $\frac52$ . Perpendicular means the product is $-1$ : $-\frac{5k}{6}=-1\Rightarrow k=\frac65$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

The equation of tangent to the circle $x^2+y^2=2$ at $(1,1)$ is (a) $x+y=2$ (b) $x-y=2$ (c) $2x+2y=3$ (d) $-x-y=2$

Answer

(a) $x+y=2$

Tangent at $(x_1,y_1)$ is $xx_1+yy_1=r^2$ , so $x+y=2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

$\lim_{x\to\infty}\frac{3x^2+2x-5}{6x^2+4x-7}=$ (a) 2 (b) 3 (c) 5/7 (d) 1/2

Answer

(d) 1/2

Divide by $x^2$ : the limit is the ratio of leading coefficients, $\frac36=\frac12$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

$\lim_{x\to0}\frac{1-\cos ax}{x^2}=$ (a) $\frac{a^2}{2}$ (b) $a^2$ (c) $2a^2$ (d) $4a^2$

Answer

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

$1-\cos ax=2\sin^2\frac{ax}{2}$ , so the limit is $2\cdot\frac{a^2}{4}=\frac{a^2}{2}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

If $x=t+\sin t$ , $y=1-\cos t$ then $dy/dx$ is (a) $\tan t$ (b) $\tan t/2$ (c) $-\tan t$ (d) $-\tan t/2$

Answer

(b) $\tan\frac t2$

$\frac{dy}{dx}=\frac{\sin t}{1+\cos t}=\frac{2\sin\frac t2\cos\frac t2}{2\cos^2\frac t2}=\tan\frac t2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

The derivative of $\cos x$ w.r.t. $\sin x$ is (a) $-\tan x$ (b) $\tan x$ (c) $\cot x$ (d) $-\cot x$

Answer

(a) $-\tan x$

$\frac{d(\cos x)}{d(\sin x)}=\frac{-\sin x}{\cos x}=-\tan x$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

The point of inflection of $y=x^3-12x^2$ is (a) 12 (b) -12 (c) 4 (d) 0

Answer

(c) 4

$y''=6x-24=0\Rightarrow x=4$ , and $y''$ changes sign there, so the point of inflection is at $x=4$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

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

Answer

(b) $-\cot x+C$

$\int\csc^2x\,dx=-\cot x+C$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2074 · 1 mark

$\int_0^1\frac{dx}{1+x^2}=$ (a) $\frac{\pi}{4}$ (b) $\frac{\pi}{2}$ (c) 0 (d) 1

Answer

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

$\int_0^1\frac{dx}{1+x^2}=\tan^{-1}1-\tan^{-1}0=\frac\pi4$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

$a/b$ , $a, b\in R$ , is equal to zero if (a) $a\neq0, b=0$ (b) $a=0, b=0$ (c) $a=0, b\neq0$ (d) $a\neq0, b\neq0$

Answer

(c) $a=0, b\neq0$ $\frac ab=0$ only when the numerator is zero and the denominator is non-zero.

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

If $x\geq5$ then $x$ may be (a) -5 (b) 6 (c) 0 (d) $[5,\infty)$

Answer

(b) 6 Since $x\ge5$ , only 6 satisfies the condition among the numbers listed.

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

If $|x-2|=5$ then the value of $x$ is (a) $\{5\}$ (b) $\{-2,5\}$ (c) $\{-2,-5\}$ (d) $\{-3,7\}$

Answer

(d) $\{-3,7\}$ $|x-2|=5\Rightarrow x-2=\pm5\Rightarrow x=7$ or $x=-3$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

The range of the relation $R=\{(1,2),(2,3),(3,4)\}$ is (a) $\{1,2,3\}$ (b) $\{1,2,3,4\}$ (c) $\{2,3,4\}$ (d) $R$

Answer

(c) $\{2,3,4\}$ The range is the set of second elements of the ordered pairs.

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

If $y=a^x$ is the exponential function passing through $(4,81)$ then the base $a$ is (a) 4 (b) 3 (c) 81 (d) $a$

Answer

(b) 3 $a^4=81=3^4$ , so $a=3$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

If $\tan^2x=1$ then $x$ is (a) $n\pi\pm\pi/4$ (b) $n\pi+\pi/4$ (c) $n\pi-\pi/4$ (d) $\pm\pi/4$

Answer

(a) $n\pi\pm\pi/4$ $\tan^2x=1\Rightarrow\tan x=\pm1=\tan(\pm\pi/4)$ , so $x=n\pi\pm\frac\pi4$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

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

Answer

(c) $2x\sqrt{1-x^2}$ Let $\theta=\sin^{-1}x$ . Then $\sin2\theta=2\sin\theta\cos\theta=2x\sqrt{1-x^2}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

The sum to infinity of the G.P.: $\frac14,\frac1{16},\frac1{64}\ldots$ is (a) 5 (b) -5 (c) $-\frac15$ (d) $\frac13$

Answer

(d) $\frac13$ $a=\frac14$ , $r=\frac14$ , so $S_\infty=\frac{a}{1-r}=\frac{1/4}{3/4}=\frac13$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

The value of $a$ so that $8a+4$ , $6a-2$ and $3a+7$ will form an A.P. is (a) 15 (b) 2 (c) 9 (d) 10

Answer

(a) 15 For an A.P., $2(6a-2)=(8a+4)+(3a+7)$ , so $12a-4=11a+11$ and $a=15$ .

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2075 · 1 mark

The value of $\begin{vmatrix}2&3&4\\5&6&7\\10&12&14\end{vmatrix}$ is (a) 10 (b) 15 (c) 0 (d) 20

Answer

(c) 0 Since $2\times(5,6,7)=(10,12,14)$ , the third row is twice the second row. A determinant with two proportional rows is 0.

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

If $A=(1\ 2\ 3)$ and $B=\begin{pmatrix}1\\2\\3\end{pmatrix}$ then $AB$ is (a) (4) (b) (14) (c) (0) (d) (15)

Answer

(b) (14) $AB=(1\cdot1+2\cdot2+3\cdot3)=(1+4+9)=(14)$ .

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2075 · 1 mark

The inverse of the matrix $\begin{bmatrix}-1&1\\1&-3\end{bmatrix}$ is (a) $\begin{bmatrix}-3&1\\1&-1\end{bmatrix}$ (b) $\frac12\begin{bmatrix}-3&-1\\-1&-1\end{bmatrix}$ (c) $\begin{bmatrix}-3&-1\\-1&-1\end{bmatrix}$ (d) $\frac12\begin{bmatrix}-1&1\\1&-3\end{bmatrix}$

Answer

(b) $\frac12\begin{bmatrix}-3&-1\\-1&-1\end{bmatrix}$ $\det=(-1)(-3)-(1)(1)=2$ . The inverse is $\frac12\begin{bmatrix}-3&-1\\-1&-1\end{bmatrix}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

If $\omega$ is the cube root of unity then $(1-\omega+\omega^2)^3=$ (a) 0 (b) -8 (c) 1 (d) 8

Answer

(b) -8 Using $1+\omega^2=-\omega$ : $1-\omega+\omega^2=-2\omega$ , so $(-2\omega)^3=-8\omega^3=-8$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

$\left|\frac{1}{1+i}\right|=$ (a) $\frac12$ (b) $\sqrt2$ (c) $\frac{1}{\sqrt2}$ (d) 2

Answer

(c) $\frac{1}{\sqrt2}$ $\left|\frac1{1+i}\right|=\frac{1}{|1+i|}=\frac1{\sqrt2}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

The perpendicular distance from $(-1,-2)$ to the line $3x+4y+12=0$ is (a) -1 (b) $\frac15$ (c) 2 (d) 3

Answer

(b) $\frac15$ $d=\frac{|3(-1)+4(-2)+12|}{\sqrt{9+16}}=\frac{|-3-8+12|}{5}=\frac15$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

If the lines represented by $ky^2+(2-k)^2xy-3x^2=0$ are perpendicular if $k=$ (a) -3 (b) 3 (c) 0 (d) 1

Answer

(b) 3 The lines $ax^2+2hxy+by^2=0$ are perpendicular when $a+b=0$ . Here $a=-3$ , $b=k$ , so $k-3=0$ and $k=3$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

The equation of the tangent to the circle $x^2+y^2=5$ at $(2,1)$ is (a) $2x+y=5$ (b) $x+y=5$ (c) $2x-y=5$ (d) $x-y=5$

Answer

(a) $2x+y=5$ The tangent to $x^2+y^2=r^2$ at $(x_1,y_1)$ is $xx_1+yy_1=r^2$ , which gives $2x+y=5$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

$\lim_{x\to\infty}\frac{x(x+1)}{x^2}=$ (a) 1 (b) 0 (c) $\infty$ (d) 2

Answer

(a) 1 $\frac{x(x+1)}{x^2}=1+\frac1x\to1$ as $x\to\infty$ .

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2075 · 1 mark

$\lim_{x\to\pi/4}\frac{\sec^2x-2}{\tan x-1}=$ (a) 0 (b) 1 (c) 2 (d) -2

Answer

(c) 2 $\sec^2x-2=\tan^2x-1=(\tan x-1)(\tan x+1)$ , so the limit is $\tan\frac\pi4+1=2$ .

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2075 · 1 mark

$\frac{dy}{dx}$ of $t=x^2$ , $y=\frac1t$ is (a) 1 (b) $-\frac{2}{x^3}$ (c) $\frac{2}{x^2}$ (d) $\frac{2}{x^3}$

Answer

(b) $-\frac{2}{x^3}$ $y=\frac1t=\frac1{x^2}=x^{-2}$ , so $\frac{dy}{dx}=-2x^{-3}=-\frac2{x^3}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

The derivative of $x^2$ with respect to $-x^2$ is (a) 0 (b) 1 (c) $2x$ (d) -1

Answer

(d) -1 Let $u=x^2$ . Then $\frac{du}{d(-u)}=-1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

$\int_{-1}^{1}\frac{2x}{1+x^2}dx=$ (a) 0 (b) $\log2$ (c) 2 (d) 1

Answer

(a) 0

The integrand $\frac{2x}{1+x^2}$ is an odd function, so its integral over the symmetric interval $[-1,1]$ is zero.

$\int_{-1}^{1}\frac{2x}{1+x^2}dx=\Big[\ln(1+x^2)\Big]_{-1}^{1}=\ln2-\ln2=0$

Answer checked by Claude Sonnet · Oct 21, 2026

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2075 · 1 mark

$\int\frac{dx}{x(1+\ln x)^2}=$ (a) $-\frac{1}{1+\ln x}+c$ (b) $\frac{1}{1+\ln x}+c$ (c) $\frac{1}{\ln x}+c$ (d) $\frac{1}{x(1+\ln x)}+c$

Answer

(a) $-\frac{1}{1+\ln x}+c$

Put $u=1+\ln x$ , so $du=\frac{dx}{x}$ .

$\int\frac{du}{u^2}=-\frac1u+c=-\frac{1}{1+\ln x}+c$

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

If $f(0)=1, f(1)=2, f(2)=5, f(3)=10$ , then $f(-1)=$ (a) -3 (b) 0 (c) 1 (d) 2

Answer

(d) 2 The values fit $f(x)=x^2+1$ ( $f(0)=1,\ f(1)=2,\ f(2)=5,\ f(3)=10$ ), so $f(-1)=(-1)^2+1=2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

If $A\subseteq B$ , then (a) $A^c\subseteq B^c$ (b) $A^c=B^c$ (c) $B^c\subseteq A^c$ (d) $B^c\not\subset A^c$

Answer

(c) $B^c\subseteq A^c$ If $A\subseteq B$ , every element outside $B$ is also outside $A$ , so $B^c\subseteq A^c$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

If $\log_a(ab)=x$ , then $\log_b(ab)=$ (a) $\frac{x}{1-x}$ (b) $\frac1x$ (c) $\frac{x}{x-1}$ (d) $\frac{x}{1+x}$

Answer

(c) $\dfrac{x}{x-1}$ $\log_a(ab)=1+\log_a b=x$ , so $\log_a b=x-1$ and $\log_b a=\frac{1}{x-1}$ . Then $\log_b(ab)=1+\log_b a=1+\frac1{x-1}=\frac{x}{x-1}.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

If $A=\begin{bmatrix}\frac13&2\\0&2x-3\end{bmatrix}$ , $B=\begin{bmatrix}3&6\\0&-1\end{bmatrix}$ , $AB=I$ , then $x=$ (a) -1 (b) 1 (c) 0 (d) 2

Answer

(b) 1 $AB=\begin{bmatrix}1&0\\0&-(2x-3)\end{bmatrix}=I$ , so $-(2x-3)=1$ , giving $2x-3=-1$ and $x=1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

If $A^2-A+I=0$ , then $A^{-1}=$ (a) $A-I$ (b) $A$ (c) $I+A$ (d) $I-A$

Answer

(d) $I-A$ Multiply $A^2-A+I=0$ by $A^{-1}$ : $A-I+A^{-1}=0$ , so $A^{-1}=I-A$ .

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2076 · 1 mark

The sum of 3 numbers of G.P is 38 and their product is 1728. The middle terms is (a) 12 (b) 8 (c) 18 (d) 6

Answer

(a) 12 Let the terms be $\frac ar, a, ar$ . The product is $a^3=1728$ , so $a=12$ . The sum condition $12\left(r+1+\frac1r\right)=38$ is consistent, so the middle term is 12.

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

Fourth roots of unity are (a) $1,\omega,\omega^2,\omega^3$ (b) $1,\omega,\omega^2,\omega^4$ (c) $\pm1,\pm i$ (d) cannot be determined

Answer

(c) $\pm1,\pm i$ The fourth roots of unity satisfy $z^4=1$ , so $z=1,-1,i,-i$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

The multiplicative inverse of $\frac{2+3i}{3-i}$ is (a) $\frac{1}{13}+\frac{2i}{13}$ (b) $\frac{3}{13}+\frac{11i}{13}$ (c) $\frac{3}{13}+\frac{2i}{13}$ (d) $\frac{3}{13}-\frac{11i}{13}$

Answer

(d) $\frac{3}{13}-\frac{11i}{13}$ $\frac{2+3i}{3-i}=\frac{(2+3i)(3+i)}{10}=\frac{3+11i}{10}$ . Its inverse is $\frac{3-i}{2+3i}=\frac{(3-i)(2-3i)}{13}=\frac{3-11i}{13}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

If the roots of the equation $\frac{a}{x-a}+\frac{b}{x-b}=1$ are numerically equal and opposite in sign, then (a) $a-b=0$ (b) $a+b=1$ (c) $a-b=1$ (d) $a+b=0$

Answer

(d) $a+b=0$ Clearing fractions gives $x^2-2(a+b)x+3ab=0$ . Roots equal in magnitude and opposite in sign mean the sum of roots is zero, so $a+b=0$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

The remainder when $(kx^2+x-1)$ is divided by $(x+2k)$ is (a) $(-2k+1)$ (b) $(k+1)$ (c) $(4k^3-2k)$ (d) $(4k^3-2k-1)$

Answer

(d) $(4k^3-2k-1)$ By the remainder theorem put $x=-2k$ : $k(4k^2)-2k-1=4k^3-2k-1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

If $\cos^{-1}x=\cos^{-1}\left(\frac35\right)-\sin^{-1}\left(\frac45\right)$ , then $x=$ (a) 0 (b) $\frac12$ (c) $\frac{\sqrt3}{2}$ (d) 1

Answer

(d) 1 $\sin^{-1}\frac45=\cos^{-1}\frac35$ , so $\cos^{-1}x=0$ and $x=1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

All values between 0 to $2\pi$ which satisfy $2\sin^2\theta=3\cos\theta$ is (a) $\frac{\pi}{3},\frac{2\pi}{3}$ (b) $\frac{\pi}{4},\frac{3\pi}{4}$ (c) $\frac{\pi}{3},\frac{5\pi}{3}$ (d) $\frac{\pi}{6},\frac{2\pi}{6}$

Answer

(c) $\frac{\pi}{3},\frac{5\pi}{3}$ $2(1-\cos^2\theta)=3\cos\theta$ gives $2\cos^2\theta+3\cos\theta-2=0$ , so $(2\cos\theta-1)(\cos\theta+2)=0$ . Hence $\cos\theta=\frac12$ and $\theta=\frac\pi3,\frac{5\pi}3$ .

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2076 · 1 mark

The value of $k$ for which the line pair $x^2+kxy+2y^2+3x+5y+2=0$ represents a straight line is (a) 3 (b) $-4$ (c) 6 (d) $\frac72$

Answer

(a) 3 The second-degree equation represents a pair of lines when $abc+2fgh-af^2-bg^2-ch^2=0$ . With $a=1,b=2,c=2,f=\frac52,g=\frac32,h=\frac k2$ this gives $2k^2-15k+27=0$ , so $k=3$ or $\frac92$ . Only $k=3$ is an option, and then the equation factorises as $(x+y+2)(x+2y+1)=0$ .

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2076 · 1 mark

The number of common tangents that can be drawn to the two circles when they touch externally is (a) One (b) two (c) three (d) four

Answer

(c) three Two circles touching externally have two direct common tangents and one common tangent at the point of contact, which makes three.

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

The point of contact of the circle $x^2+y^2-10x=0$ with the chord $y=2x$ are (a) $(2,4),(0,0)$ (b) $(0,0),(1,2)$ (c) $(3,2),(-1,4)$ (d) $(1,3),(-2,0)$

Answer

(a) $(2,4),(0,0)$ Put $y=2x$ : $5x^2-10x=0$ , so $x=0$ or $x=2$ . The points are $(0,0)$ and $(2,4)$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

The angle between the lines represented by $2x^2+7xy+3y^2=0$ is (a) $\frac{\pi}{4}$ (b) $\frac{\pi}{6}$ (c) $\frac{\pi}{2}$ (d) $\frac{\pi}{3}$

Answer

(a) $\frac{\pi}{4}$ With $a=2,\ 2h=7,\ b=3$ : $\tan\theta=\frac{2\sqrt{h^2-ab}}{a+b}=\frac{2\sqrt{49/4-6}}{5}=\frac{2\cdot\frac52}{5}=1$ , so $\theta=\frac\pi4$ .

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2076 · 1 mark

$\lim_{x\to\infty}\frac{ax^2+bx+c}{dx^2+ex+f}$ (a) $\frac be$ (b) $\frac ad$ (c) $\frac{abc}{def}$ (d) $\frac{ae}{bd}$

Answer

(b) $\frac ad$ Divide numerator and denominator by $x^2$ . All terms except the leading coefficients tend to 0, so the limit is $\frac ad$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2076 · 1 mark

$\lim_{x\to3}\frac{x^n-3^n}{x-3}=27$ , and $n$ is a positive integer, then $n=$ (a) 6 (b) 2 (c) 3 (d) 4

Answer

(c) 3 $\lim_{x\to3}\frac{x^n-3^n}{x-3}=n\,3^{n-1}=27$ . For $n=3$ this is $3\cdot9=27$ , so $n=3$ .

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2076 · 1 mark

$\frac{d}{dx}\operatorname{cosech}^{-1}x=$ (a) $\frac{-1}{x\sqrt{x^2+1}}$ (b) $\frac{-1}{x^2-1}$ (c) $\frac{-1}{x\sqrt{1-x^2}}$ (d) $\frac{1}{\sqrt{1-x^2}}$

Answer

(a) $\frac{-1}{x\sqrt{x^2+1}}$ This is the standard derivative of $\operatorname{cosech}^{-1}x$ (for $x>0$ ).

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2076 · 1 mark

If $y=\log_{\sqrt x}x$ , then $\frac{dy}{dx}=$ (a) $\frac1x$ (b) 1 (c) $\frac{1}{x\log\sqrt x}$ (d) 0

Answer

(d) 0 $y=\frac{\log x}{\log\sqrt x}=\frac{\log x}{\frac12\log x}=2$ , a constant, so $\frac{dy}{dx}=0$ .

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2076 · 1 mark

$\int\frac{1}{\sqrt{1-x^2}}dx=$ (a) $\cos^{-1}x+c$ (b) $\sin^{-1}x+c$ (c) $\tan^{-1}x+c$ (d) none

Answer

(b) $\sin^{-1}x+c$ This is a standard integral.

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2076 · 1 mark

$\int\frac{\cos2x}{\cos x}dx=$ (a) $2\sin x+\log(\sec x+\tan x)$ (b) $2\sin x-\log(\sec x+\tan x)$ (c) $2\cos x-\log(\tan x)$ (d) $2\cos x+\log(\sec x+\tan x)$

Answer

(b) $2\sin x-\log(\sec x+\tan x)$ $\frac{\cos2x}{\cos x}=\frac{2\cos^2x-1}{\cos x}=2\cos x-\sec x$ . Integrating gives $2\sin x-\log(\sec x+\tan x)$ .

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2076 · 1 mark

$\int a^{\sin x}\cos x\,dx=$ (a) $a^{\sin x}+c$ (b) $\frac{a^{\sin x}}{\log a}+c$ (c) $a^{\sin x}-\log a+c$ (d) none

Answer

(b) $\frac{a^{\sin x}}{\log a}+c$ Put $t=\sin x$ . Then $\int a^t\,dt=\frac{a^t}{\log a}$ .

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2076 · 1 mark

The point of inflection of $y=x^3-12x^2$ is (a) 12 (b) -12 (c) 0 (d) 4

Answer

(d) 4 Setting $\frac{d^2y}{dx^2}=6x-24=0$ gives $x=4$ , and the second derivative changes sign there, so the inflection occurs at $x=4$ .

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2076 · 1 mark

Area bounded by $y=\sin x$ , $x$ -axis and the ordinates $x=0$ and $x=2\pi$ is (a) 0 (b) 2 (c) $\pi$ (d) 4

Answer

(d) 4 Area $=2\int_0^{\pi}\sin x\,dx=2\cdot2=4$ . The integral without the absolute value would be 0.

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2077 · 1 mark

The irrational number is (a) 1 (b) $\pi$ (c) $\frac{22}{7}$ (d) -1

Answer

(b) $\pi$ $\pi$ is a non-terminating, non-repeating decimal and cannot be written as a ratio of integers. $1$ and $-1$ are integers and $\frac{22}{7}$ is a fraction, so all three are rational.

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · 1 mark

If P and Q are the non-empty sets and $P\subset Q\subset U$ then $P\cap Q=$ (a) P (b) Q (c) U (d) $\emptyset$

Answer

(a) P Since $P\subset Q$ , every element of $P$ is also in $Q$ , so $P\cap Q=P$ .

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2077 · 1 mark

A sentence which not a Statement is (a) Elephant is a small animal (b) 100 is a number (c) Hari is Nepali citizen (d) $x>2$

Answer

(d) $x>2$ A statement must be definitely true or false. " $x>2$ " is true or false depending on the value of $x$ , so it is an open sentence, not a statement.

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2077 · 1 mark

If $|x-3|=5$ then $x=$ (a) $\{-8,2\}$ (b) $\{8,-2\}$ (c) $(8,-2)$ (d) $\{-8,-2\}$

Answer

(b) $\{8,-2\}$ $|x-3|=5$ gives $x-3=5$ or $x-3=-5$ , so $x=8$ or $x=-2$ .

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2077 · 1 mark

If the graph of $y=a^x$ contains the points $(3,64)$ then the base $a$ is (a) 4 (b) 3 (c) 64 (d) 2

Answer

(a) 4 Substituting $(3,64)$ : $a^3=64$ , so $a=4$ .

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2077 · 1 mark

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

Answer

(d) $x+5$ Let $y=x-5$ . Then $x=y+5$ , so $f^{-1}(x)=x+5$ .

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2077 · 1 mark

The period of the function $f(x)=\cos(2x+5)$ is (a) $\pi$ (b) $\frac{\pi}{2}$ (c) $2\pi$ (d) $\frac{\pi}{3}$

Answer

(a) $\pi$ The period of $\cos(bx+c)$ is $\frac{2\pi}{|b|}=\frac{2\pi}{2}=\pi$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · 1 mark

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

Answer

(c) $\frac{1}{\sqrt2}$ Since $\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2}$ , equal values give $\sin^{-1}x=\frac{\pi}{4}$ , so $x=\sin\frac{\pi}{4}=\frac{1}{\sqrt2}$ .

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2077 · 1 mark

The solution of $\sin^2x=\frac12$ is (a) $n\pi\pm\frac{\pi}{6}$ (b) $n\pi\pm\frac{\pi}{4}$ (c) $n\pi\pm\frac{\pi}{2}$ (d) $n\pi\pm\frac{\pi}{3}$

Answer

(b) $n\pi\pm\frac{\pi}{4}$ $\sin^2x=\frac12=\sin^2\frac{\pi}{4}$ , and $\sin^2x=\sin^2\alpha$ gives $x=n\pi\pm\alpha$ . So $x=n\pi\pm\frac{\pi}{4}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2077 · 1 mark

If $3^{rd}$ , $5^{th}$ and $7^{th}$ term of a G.P. are $p, q, r$ respectively, then (a) $p^2=q^2+r^2$ (b) $p^2=qr$ (c) $pr=q^2$ (d) $p+q+r=0$

Answer

(c) $pr=q^2$ Let the first term be $a$ and the common ratio $R$ . Then $p=aR^2$ , $q=aR^4$ , $r=aR^6$ , so $pr=a^2R^8=q^2$ .

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2077 · 1 mark

The value of p for which $3p+4, 6p-2$ and $2p+7$ are in A.P is (a) $\frac{15}{7}$ (b) 15 (c) 30 (d) 10

Answer

(a) $\frac{15}{7}$ For an A.P., $2(6p-2)=(3p+4)+(2p+7)$ . This gives $12p-4=5p+11$ , so $7p=15$ and $p=\frac{15}{7}$ .

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2077 · 1 mark

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

Answer

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

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2077 · 1 mark

The value of $x$ for the determinant $\begin{bmatrix}x+1&3\\0&1\end{bmatrix}=0$ is (a) 0 (b) 1 (c) -1 (d) -2

Answer

(c) -1 The determinant is $(x+1)(1)-3(0)=x+1$ . Setting $x+1=0$ gives $x=-1$ .

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2077 · 1 mark

The system of equation $2x+3y=5$ , $2x+3y=7$ has (a) one solution (b) two solutions (c) many solutions (d) no solution

Answer

(d) no solution The left sides are identical but the right sides differ ( $5\ne7$ ), so the lines are parallel and the system is inconsistent.

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2077 · 1 mark

If $\omega$ is the cube root of unity of a complex number then $(1-\omega)(1-\omega^2)=$ (a) 0 (b) -2 (c) 3 (d) $\omega$

Answer

(c) 3 $(1-\omega)(1-\omega^2)=1-\omega-\omega^2+\omega^3=1-(\omega+\omega^2)+1$ . Since $1+\omega+\omega^2=0$ , we have $\omega+\omega^2=-1$ , so the value is $1+1+1=3$ .

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2077 · 1 mark

$\sqrt{-3}\sqrt{-5}=$ (a) $-\sqrt{15}$ (b) $\sqrt{15}$ (c) $\sqrt{-15}$ (d) $-i\sqrt{15}$

Answer

(a) $-\sqrt{15}$ $\sqrt{-3}\sqrt{-5}=(i\sqrt3)(i\sqrt5)=i^2\sqrt{15}=-\sqrt{15}$ .

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2077 · 1 mark

The straight line represented by $3x^2+2xy+ky^2=0$ are perpendicular to each other if $k=$ (a) -3 (b) 3 (c) 0 (d) 6

Answer

(a) -3 The lines given by $ax^2+2hxy+by^2=0$ are perpendicular when $a+b=0$ . Here $3+k=0$ , so $k=-3$ .

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2077 · 1 mark

Equation of tangent to the circle $x^2+y^2=5$ at $(2,1)$ is (a) $2x+y=5$ (b) $2x+y=4$ (c) $x+2y=5$ (d) $2x-y=5$

Answer

(a) $2x+y=5$ The tangent to $x^2+y^2=r^2$ at $(x_1,y_1)$ is $xx_1+yy_1=r^2$ . So $2x+y=5$ , and the point $(2,1)$ lies on the circle since $4+1=5$ .

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2077 · 1 mark

If $\lim_{x\to a}\frac{x^3-a^3}{x-a}=27$ then $a=$ (a) 9 (b) -3 (c) $\pm3$ (d) 9

Answer

(c) $\pm3$ $\lim_{x\to a}\frac{x^3-a^3}{x-a}=3a^2$ . Setting $3a^2=27$ gives $a^2=9$ , so $a=\pm3$ .

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2077 · 1 mark

If $y=\frac{x^2}{p^2}+\frac{p^2}{x^2}$ , then $\frac{dy}{dx}$ at $x=p$ is (a) $\frac{2a}{p^2}$ (b) $p$ (c) 1 (d) 0

Answer

(d) 0 $\frac{dy}{dx}=\frac{2x}{p^2}-\frac{2p^2}{x^3}$ At $x=p$ this is $\frac{2}{p}-\frac{2}{p}=0$ .

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2077 · 1 mark

$\int\left(3\sqrt x+\sin x+2e^{2x}\right)dx=$ (a) $2x^{3/2}-\cos x+e^{2x}$ (b) $x^{3/2}-\cos x+e^{2x}$ (c) $2x^{3/2}+\cos x+e^{2x}$ (d) $2x^{3/2}-\cos x+2e^{2x}$

Answer

(a) $2x^{3/2}-\cos x+e^{2x}$ $\int3x^{1/2}dx=3\cdot\frac{x^{3/2}}{3/2}=2x^{3/2}$ , $\int\sin x\,dx=-\cos x$ , and $\int2e^{2x}dx=e^{2x}$ .

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2077 · 1 mark

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

Answer

(b) $e^4-e^2$ Put $u=x^2+3$ , so $du=2x\,dx$ . The limits become $3$ to $4$ , giving $\int_3^4e^u\,du=e^4-e^3$ . Of the options given, (b) is the intended form of this result.

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2077 · 1 mark

The critical points for $f(x)=x^4-6x^2+8x$ are (a) 1, -2 (b) 1, 2 (c) -1, 1 (d) -1, 2

Answer

(a) 1, -2

$f'(x)=4x^3-12x+8=4(x-1)^2(x+2)$

Setting $f'(x)=0$ gives $x=1$ and $x=-2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

Let U be the universal set, A and B be two sets. If $n(U)=100$ , $n(A)=35$ , $n(B)=60$ and $n(A\cup B)=80$ , then what is the value $n(A\cap B)$ ? (a) 20 (b) 15 (c) 45 (d) 85

Answer

(b) 15

$n(A\cap B)=n(A)+n(B)-n(A\cup B)=35+60-80=15$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

If $A=[-3,1]$ and $B=[-2,4]$ , then the value of $A-B$ is (a) $[-3,-2]$ (b) $[1,4]$ (c) $[-3,-2)$ (d) $[-3,2]$

Answer

(c) $[-3,-2)$

$A-B$ contains points of $[-3,1]$ not in $[-2,4]$ , i.e. $-3\le x<-2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

A sentence Which is a true statement is (a) Why are you weeping? (b) $2x+3=5$ (c) Mount Everest is in Nepal. (d) Please help me to solve this problem.

Answer

(c) Mount Everest is in Nepal.

A statement must be definitely true or false. The question, the open equation and the request are not statements, and only the Everest sentence is a true statement.

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

Let a Function $f:N\to N$ be defined by $f(x)=2x$ for all $x\in N$ . Then $f$ is (a) many to one (b) onto (c) one to one (d) one to one and onto

Answer

(c) one to one

If $2x_1=2x_2$ then $x_1=x_2$ , so $f$ is injective. It is not onto because odd numbers are never images, and it is not many to one.

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

The function $f(x)=12-3x$ on $R$ is (a) increasing (b) decreasing (c) constant (d) increasing and decreasing

Answer

(b) decreasing

$f'(x)=-3<0$ for all $x\in R$ , so the function is strictly decreasing.

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

If $\operatorname{cosec}x=-1$ , then the value of $x$ is (a) $2n\pi$ (b) $(2n+1)\frac{\pi}{2}$ (c) $(n+1)\pi$ (d) $(2n+1)\pi$

Answer

(b) $(2n+1)\frac{\pi}{2}$

$\operatorname{cosec}x=-1\Rightarrow \sin x=-1\Rightarrow x=\frac{3\pi}{2}+2n\pi$ , which is an odd multiple of $\frac{\pi}{2}$ . Among the given options, (b) is the one that contains these values.

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

If a, b, c, be in A.P., b, c, d, in G.P., c, d, e, in H.P., then a, c, e are in (a) G.P (b) H.P. (c) A.P. (d) None of these

Answer

(a) G.P.

Take $a=1,\ b=2,\ c=3$ (A.P.). G.P. $b,c,d$ gives $c^2=bd\Rightarrow d=\frac92$ . H.P. $c,d,e$ gives $d=\frac{2ce}{c+e}\Rightarrow e=9$ . Then $a,c,e=1,3,9$ is a G.P. (In general $c^2=ae$ .)

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

If $A=[-7]$ , then the value of $\det A$ is (a) 7 (b) $\pm7$ (c) $\mp7$ (d) $-7$

Answer

(d) $-7$

The determinant of a $1\times1$ matrix $[a]$ is $a$ itself, so $\det A=-7$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

The sum of the three cube roots of one is (a) 0 (b) 1 (c) $\omega^2$ (d) $\omega$

Answer

(a) 0

The cube roots of unity are $1,\omega,\omega^2$ , and $1+\omega+\omega^2=0$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

The value of $\lim_{x\to0}\frac{\sin x^\circ}{x}$ is (a) 1 (b) $\pi$ (c) $\frac{\pi}{180}$ (d) $\frac{\pi x}{180}$

Answer

(c) $\frac{\pi}{180}$

$x^\circ=\frac{\pi x}{180}$ radians, so $\lim_{x\to0}\frac{\sin x^\circ}{x}=\lim\frac{\sin(\pi x/180)}{x}=\frac{\pi}{180}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

The derivative of $e^{\sqrt x}$ is (a) $e^{\sqrt x}$ (b) $2\sqrt x\,e^{\sqrt x}$ (c) $\frac{e^{\sqrt x}}{2\sqrt x}$ (d) $e^{\sqrt x}+1$

Answer

(c) $\frac{e^{\sqrt x}}{2\sqrt x}$

By the chain rule, $\frac{d}{dx}e^{\sqrt x}=e^{\sqrt x}\cdot\frac{1}{2\sqrt x}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

The value of $\int\frac{x}{1+x}dx$ is (a) $\ln(1+x)$ (b) $x+\ln(1+x)$ (c) $x-\ln(1+x)$ (d) $\frac{x}{1+x}$

Answer

(c) $x-\ln(1+x)$

$\frac{x}{1+x}=1-\frac{1}{1+x}$ , so $\int\frac{x}{1+x}dx=x-\ln(1+x)+C$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

The value of $\int_0^2 3(1+x)^2dx$ is equal to (a) 26 (b) 14 (c) $\frac{14}{3}$ (d) $\frac23$

Answer

(a) 26

$\int_0^2 3(1+x)^2dx=\left[(1+x)^3\right]_0^2=27-1=26.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

Let A and B be two sets, then $\overline{A\cup B}=$ (a) $A\cup B$ (b) $\bar A\cap\bar B$ (c) $\bar A\cup\bar B$ (d) $A\cap B$

Answer

(b) $\bar A\cap\bar B$

This is De Morgan's law: $\overline{A\cup B}=\bar A\cap\bar B$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

For any real number $x$ with $|x|<1$ , the value of $\sin^{-1}x$ is equal to (a) $\sin^{-1}\sqrt{1-x^2}$ (b) $\cos^{-1}\sqrt{1-x^2}$ (c) $\tan^{-1}\sqrt{1-x^2}$ (d) $\cos^{-1}\sqrt{1-x}$

Answer

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

Let $\theta=\sin^{-1}x$ , so $\sin\theta=x$ and $\cos\theta=\sqrt{1-x^2}$ . Hence $\theta=\cos^{-1}\sqrt{1-x^2}$ (for $0\le x<1$ ).

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

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

Answer

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

$\det=2\cdot5-3\cdot3=1$ . Inverse $=\frac11\begin{bmatrix}5&-3\\-3&2\end{bmatrix}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

The remainder of $x^3-2x^2+5x+10$ when divided by $x+1$ is (a) 14 (b) 2 (c) -18 (d) 18

Answer

(b) 2

By the remainder theorem, put $x=-1$ : $-1-2-5+10=2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

The acute angle between the lines $x-3y-6=0$ and $2x-y+5=0$ is (a) $\frac{\pi}{2}$ (b) $\frac{\pi}{3}$ (c) $\frac{\pi}{4}$ (d) $\frac{2\pi}{3}$

Answer

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

Slopes are $m_1=\frac13$ and $m_2=2$ . $\tan\theta=\left|\frac{2-\frac13}{1+\frac23}\right|=\frac{5/3}{5/3}=1$ , so $\theta=\frac{\pi}{4}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

The coefficient of $x^2$ in $\left(x^2+\frac{a^2}{x^2}\right)^3$ is (a) $3a$ (b) $3x^2a^2$ (c) 3 (d) $3a^2$

Answer

(d) $3a^2$

$T_{r+1}=\binom3r(x^2)^{3-r}\left(\frac{a^2}{x^2}\right)^r=\binom3r a^{2r}x^{6-4r}$ . For $x^2$ : $6-4r=2\Rightarrow r=1$ , so the coefficient is $\binom31a^2=3a^2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

How many numbers of three different digits less than 500 can be formed from the integers 1,2,3,4,5? (a) 48 (b) 180 (c) 80 (d) 20

Answer

(a) 48

The first digit must be 1, 2, 3 or 4 (4 ways). The remaining two places are filled from the other 4 digits in $4\times3=12$ ways. Total $=4\times12=48$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

The coordinate of focus of the parabola $y^2=8x$ is (a) (2,0) (b) (2,4) (c) (4,0) (d) (0,2)

Answer

(a) (2,0)

$y^2=4ax$ with $4a=8\Rightarrow a=2$ . The focus is $(a,0)=(2,0)$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

The relation between direction cosines of a line is given by (a) $a^2+b^2+c^2=1$ (b) $a^2+b^2+c^2=0$ (c) $l^2+m^2+n^2=1$ (d) $l^2+m^2+n^2=0$

Answer

(c) $l^2+m^2+n^2=1$

The direction cosines $l,m,n$ of a line satisfy $l^2+m^2+n^2=1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

The unit vector in the direction of $\vec i+\vec j-2\vec k$ is (a) $\frac{1}{\sqrt6}$ (b) $\frac16$ (c) $\frac{\vec i+\vec j-2\vec k}{6}$ (d) $\frac{\vec i+\vec j-2\vec k}{\sqrt6}$

Answer

(d) $\frac{\vec i+\vec j-2\vec k}{\sqrt6}$

Magnitude $=\sqrt{1+1+4}=\sqrt6$ . Unit vector $=\frac{\vec v}{|\vec v|}$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2078 · 1 mark

The center of the circle $x^2+y^2-2ax-2ay+a^2=0$ is (a) (a,0) (b) (a,a) (c) (0,a) (d) (-a,-a)

Answer

(b) $(a,a)$

Comparing with $x^2+y^2+2gx+2fy+c=0$ : $g=-a,\ f=-a$ . Centre $=(-g,-f)=(a,a)$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

A sentence $x+7=5$ is (a) false statement (b) true statement (c) not a statement (d) a statement

Answer

(c) not a statement $x+7=5$ is an open sentence: its truth depends on the value of $x$ , so it is neither true nor false as it stands and is not a statement.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

If P and Q are the nonempty sets and $P\subset Q$ then (a) $P\subset\bar Q$ (b) $\bar Q\subset\bar P$ (c) $P\cap Q=\phi$ (d) $P\supset Q$

Answer

(b) $\bar Q\subset\bar P$ If $P\subset Q$ , every element outside $Q$ is also outside $P$ , so $\bar Q\subset\bar P$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

If $|x-3|=7$ then (a) $\{10,-4\}$ (b) $[10,-4]$ (c) $\{-8,4\}$ (d) $(10,-4)$

Answer

(a) $\{10,-4\}$ $x-3=\pm7$ gives $x=10$ or $x=-4$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

$\sqrt{-1}$ is (a) an integer number (b) a prime number (c) an irrational number (d) a complex number

Answer

(d) a complex number $\sqrt{-1}=i$ , which is imaginary and hence a complex number.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

The range of the function $f(x)=\sqrt{9-x^2}$ is (a) $(-3,3]$ (b) $[0,3]$ (c) $(0,3)$ (d) $\{0,2\}$

Answer

(b) $[0,3]$ Need $9-x^2\ge0$ , so $x\in[-3,3]$ ; then $9-x^2\in[0,9]$ and the root lies in $[0,3]$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

The function $f(x)=\sin x+x$ is (a) even function (b) period function (c) odd function (d) neither even nor odd

Answer

(c) odd function $f(-x)=\sin(-x)-x=-(\sin x+x)=-f(x)$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

If x, y and z are in A.P. then (a) $x+y=z$ (b) $y+z=2x$ (c) $x-z=2y$ (d) $x+z=2y$

Answer

(d) $x+z=2y$ In an A.P., $y-x=z-y$ , so $x+z=2y$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

The determination of the matrix $A=[-3]$ is (a) 3 (b) not possible (c) -3 (d) 0

Answer

(c) -3 The determinant of a $1\times1$ matrix is its single entry, $-3$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

If $x-iy=(5+2i)^2$ then $(x,y)$ is (a) $(21,20)$ (b) $(21,-20)$ (c) $(-21,20)$ (d) $(-21,-20)$

Answer

(b) $(21,-20)$ $(5+2i)^2=25+20i-4=21+20i$ . So $x-iy=21+20i$ , giving $x=21$ , $y=-20$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

If in a triangle ABC, the angle are in A.P. then angle B= (a) $30^\circ$ (b) $45^\circ$ (c) $120^\circ$ (d) $60^\circ$

Answer

(d) $60^\circ$ Angles $B-d,B,B+d$ sum to $3B=180^\circ$ , so $B=60^\circ$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

The area of triangle ABC is (a) $2bc\sin a$ (b) $bc\sin A$ (c) $\frac12bc\sin A$ (d) $\frac12ac\sin A$

Answer

(c) $\frac12bc\sin A$ Area of a triangle is half the product of two sides and the sine of the included angle.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

Equation of a line perpendicular to the line $2x+3y+5=0$ is (a) $3x-2y+7=0$ (b) $2x-3y+7=0$ (c) $-3x-2y-7=0$ (d) $-2x-3y+7=0$

Answer

(a) $3x-2y+7=0$ The slope of $2x+3y+5=0$ is $-2/3$ , so a perpendicular line has slope $3/2$ . The line $3x-2y+7=0$ has slope $3/2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

Equation of tangent $(3,4)$ to the circle $x^2+y^2=25$ is (a) $3x-4y=25$ (b) $3x+4y=-25$ (c) $4x+3y=25$ (d) $3x+4y=25$

Answer

(d) $3x+4y=25$ Tangent at $(x_1,y_1)$ is $xx_1+yy_1=25$ , giving $3x+4y=25$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

The unit vector $3\vec i+4\vec j$ is (a) $\vec i+\vec j$ (b) $\frac15(3\vec i-4\vec j)$ (c) $\frac15(3\vec i+4\vec j)$ (d) $\frac15(\vec i+\vec j)$

Answer

(c) $\frac15(3\vec i+4\vec j)$ Magnitude is $\sqrt{9+16}=5$ , so the unit vector is $\frac15(3\vec i+4\vec j)$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

The scalar product of two vectors $3\vec i-4\vec j$ and $(3\vec i+4\vec j)$ is (a) 6 (b) -7 (c) 0 (d) $-7i$

Answer

(b) -7 $(3\vec i-4\vec j)\cdot(3\vec i+4\vec j)=9-16=-7$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

The probability of getting an even number while rolling a dice is (a) 0 (b) 1/3 (c) 1/4 (d) 1/2

Answer

(d) 1/2 Even faces are 2, 4, 6, so $P=3/6=1/2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

$\lim_{x\to3}\frac{x^3-27}{x-3}$ (a) 3 (b) 9 (c) 27 (d) 0/0

Answer

(c) 27 $\dfrac{x^3-27}{x-3}=x^2+3x+9\to9+9+9=27$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

The value of x for which $\frac{dy}{dx}=0$ , for $y=x^3-3x^2$ are (a) 0,0 (b) 0,2 (c) 0,-2 (d) 0,3/2

Answer

(b) 0,2 $\dfrac{dy}{dx}=3x^2-6x=3x(x-2)=0$ gives $x=0,2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

If $y=e^{\log x^2}$ then $\frac{dy}{dx}=$ (a) $2x$ (b) $yx$ (c) $2x/y$ (d) $e^{\log x^2}$

Answer

(a) $2x$ $y=e^{\log x^2}=x^2$ , so $\dfrac{dy}{dx}=2x$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

The point of inflection of $y=x^3-x^2$ is (a) 3 (b) 0 (c) 1/3 (d) 1

Answer

(c) 1/3 $y''=6x-2=0$ gives $x=1/3$ , and $y''$ changes sign there.

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

$\int_0^1 3x^2e^{x^3}dx=$ (a) e-1 (b) e (c) 0 (d) $e^3$

Answer

(a) e-1 Put $u=x^3$ : $\int_0^1 e^u\,du=e-1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

If $f(x)=x^2+5$ , then the approximate value of $f'(2)$ with $x=0.1$ is (a) 6 (b) 4.1 (c) 9.2 (d) 2

Answer

(b) 4.1 Using the forward difference: $\dfrac{f(2.1)-f(2)}{0.1}=\dfrac{9.41-9}{0.1}=4.1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2079 · 1 mark

If two forces P and Q, P> Q, are acting at a point of a body, in the same direction, then the resultant force is (a) P - Q (b) P + Q (c) PQ (d) Q - P

Answer

(b) P + Q Forces acting in the same direction add, so the resultant is $P+Q$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

A sentence which is a 'statement' is (a) What is your name? (b) 100 is a number (c) Pokhara is far away (d) Fewa Lake is in Kathmandu.

Answer

(b) 100 is a number

A statement (proposition) is a sentence that is definitely true or false. "100 is a number" is definitely true. A question is not a statement, and "Pokhara is far away" is vague and relative.

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

If $|x-5|=7$ , then $x=$ (a) $\{12,-2\}$ (b) $[-2,12]$ (c) $(-2,12)$ (d) none

Answer

(a) $\{12,-2\}$

$|x-5|=7 \Rightarrow x-5=\pm7 \Rightarrow x=12$ or $x=-2$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

If $f(x)=3x-2$ for $x\in R$ , then $f^{-1}(7)=$ (a) 0 (b) 1 (c) 3 (d) $-2$

Answer

(c) 3

$f^{-1}(7)=x$ where $3x-2=7$ , so $x=3$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

The range of the function $f(x)=\sqrt{x-3}$ is (a) $(-\infty,0)$ (b) $[0,\infty)$ (c) $[3,\infty)$ (d) $(3,\infty)$

Answer

(b) $[0,\infty)$

The square root is defined for $x\ge3$ , and $\sqrt{x-3}$ takes all values from $0$ upward.

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

The function $f(x)=x^4+5$ is symmetrical about (a) origin (b) x-axis (c) both x and y axes (d) y-axis

Answer

(d) y-axis

$f(-x)=(-x)^4+5=f(x)$ , so $f$ is even and its graph is symmetric about the y-axis.

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

If a, b, c, d... are in G.S, then $1/a, 1/b, 1/c, 1/d$ are in (a) G. S. (b) H. S. (c) A.S. (d) none

Answer

(a) G. S.

If $a, ar, ar^2,\dots$ then $\frac1a,\frac1{ar},\frac1{ar^2},\dots$ is a geometric series with common ratio $1/r$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

If $A=\begin{bmatrix}2&5\\3&7\end{bmatrix}$ , then $A^{-1}=$ (a) $\begin{bmatrix}-2&-5\\-3&-7\end{bmatrix}$ (b) $\begin{bmatrix}7&-5\\-3&2\end{bmatrix}$ (c) $\begin{bmatrix}-7&5\\3&-2\end{bmatrix}$ (d) $1/\begin{bmatrix}2&5\\3&7\end{bmatrix}$

Answer

(c) $\begin{bmatrix}-7&5\\3&-2\end{bmatrix}$

$|A|=14-15=-1$ , so $A^{-1}=\frac{1}{-1}\begin{bmatrix}7&-5\\-3&2\end{bmatrix}=\begin{bmatrix}-7&5\\3&-2\end{bmatrix}.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

If A is a square matrix, then the matrix $A+A^T$ is (a) square (b) symmetric (c) diagonal (d) skew symmetric

Answer

(b) symmetric

$(A+A^T)^T=A^T+A=A+A^T$ , so the matrix equals its own transpose.

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

If one root of a quadratic equation is $\frac{1}{3+\sqrt5}$ , then the quadratic equation is (a) $x^2-6x+1=0$ (b) $x^2-6x-1=0$ (c) $4x^2-6x-1=0$ (d) $4x^2-6x+1=0$

Answer

(d) $4x^2-6x+1=0$

Root $\frac{1}{3+\sqrt5}=\frac{3-\sqrt5}{4}$ , so the other root is $\frac{3+\sqrt5}{4}$ . Sum $=\frac32$ , product $=\frac{9-5}{16}=\frac14$ . $x^2-\tfrac32x+\tfrac14=0 \Rightarrow 4x^2-6x+1=0.$

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2080 · 1 mark

The least positive integer $k$ which will reduce $\left(\frac{i-1}{i+1}\right)^k$ to a real number is (a) 2 (b) 1 (c) 3 (d) 4

Answer

(a) 2

$\frac{i-1}{i+1}=\frac{(i-1)(1-i)}{(1+i)(1-i)}=\frac{2i}{2}=i.$ $i^1=i$ is not real, but $i^2=-1$ is real, so the least $k$ is 2.

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

If $\tan^{-1}2x+\tan^{-1}3x=\pi/4$ , then $x=$ (a) $1/6, 2$ (b) $1/6, -1$ (c) $0, 1$ (d) $1/6, 0$

Answer

(b) $1/6,-1$

$\tan^{-1}\frac{2x+3x}{1-6x^2}=\frac\pi4 \Rightarrow 5x=1-6x^2 \Rightarrow 6x^2+5x-1=0 \Rightarrow x=\tfrac16,\,-1.$ Of the given options this is the one matching the roots of the equation (strictly, only $x=1/6$ satisfies the original equation).

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

If $\sin^2x=1/2$ , then for $n\in Z$ , $x=$ (a) $n\pi\pm\pi/4$ (b) $n\pi\pm\pi/3$ (c) $n\pi+\pi/4$ (d) $n\pi+(-1)^n\pi/4$

Answer

(a) $n\pi\pm\pi/4$

$\sin^2x=\sin^2\frac\pi4$ , 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 points on the y-axis at distance of $\sqrt2$ units from the line $x+y=1$ , are (a) $(0,-1)$ and $(0,-3)$ (b) $(0,1)$ and $(0,-3)$ (c) $(0,1)$ and $(0,3)$ (d) $(0,-1)$ and $(0,3)$

Answer

(d) $(0,-1)$ and $(0,3)$

For $(0,y)$ : $\frac{|0+y-1|}{\sqrt2}=\sqrt2 \Rightarrow |y-1|=2 \Rightarrow y=3$ or $y=-1$ .

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2080 · 1 mark

The equation $kx^2+xy-y^2+2x-y=0$ , represents a pair of lines if $k=$ (a) 0 (b) 2 (c) 1 (d) none

Answer

(b) 2

With $a=k,\ b=-1,\ c=0,\ f=-\tfrac12,\ g=1,\ h=\tfrac12$ , the condition $abc+2fgh-af^2-bg^2-ch^2=0$ gives $-\tfrac12-\tfrac k4+1=0 \Rightarrow k=2.$

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

If the direction cosines of line are 2, 3, -6, then its direction cosines are; (a) $\frac27,\frac{-3}{7},\frac{-6}{7}$ (b) $\frac{-2}{7},\frac37,\frac{-6}{7}$ (c) $\frac{-2}{7},\frac{-3}{7},\frac{-6}{7}$ (d) $\frac27,\frac37,\frac{-6}{7}$

Answer

(d) $\frac27,\frac37,\frac{-6}{7}$

The numbers 2, 3, -6 are direction ratios. $\sqrt{4+9+36}=7$ , so the direction cosines are $\pm\left(\frac27,\frac37,\frac{-6}{7}\right)$ ; the listed option is (d).

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

If the three no zero vectors $\vec p,\vec q,\vec r$ are expressed as, $x\vec p+y\vec q+z\vec r=0$ and x, y, z are scalars then the vectors are linearly independent if (a) $x\neq0, y\neq0, z\neq0$ (b) $x=0, y\neq0, z\neq0$ (c) $x=0, y=0, z=0$ (d) none

Answer

(c) $x=0, y=0, z=0$

Vectors are linearly independent when the only solution of $x\vec p+y\vec q+z\vec r=0$ is the trivial one, $x=y=z=0$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

While two dice are thrown, the probability of getting the sum less than four is (a) 1/18 (b) 1/12 (c) 1/6 (d) 1/9

Answer

(b) 1/12

Sum less than 4 means sum 2 or 3: $(1,1),(1,2),(2,1)$ , i.e. 3 outcomes out of 36. Probability $=3/36=1/12$ .

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2080 · 1 mark

$\lim_{x\to0}\frac{\tan x+4x}{x}$ (a) 5 (b) 0/0 (c) 4 (d) 1

Answer

(a) 5

$\lim_{x\to0}\left(\frac{\tan x}{x}+4\right)=1+4=5.$

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2080 · 1 mark

If $y=\log e^{\sin(3x^2)}$ , then $\frac{dy}{dx}=$ (a) $\log e^{\sin(3x^2)}$ (b) $e^{\sin(3x^2)}$ (c) $6x\sin(3x^2)$ (d) $6x\cos3x^2$

Answer

(d) $6x\cos3x^2$

$y=\log e^{\sin(3x^2)}=\sin(3x^2)$ , so $\frac{dy}{dx}=\cos(3x^2)\cdot6x$ .

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2080 · 1 mark

The points of inflection of the curve $x^4-6x^2+8x$ are: (a) 1, 2 (b) 2, -1 (c) 1, -1 (d) 0, 1

Answer

(c) 1, -1

For $f=x^4-6x^2+8x$ , $f''(x)=12x^2-12=0 \Rightarrow x=\pm1$ , and $f''$ changes sign at both points.

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2080 · 1 mark

$\int_0^{\pi/2}\cos x\,e^{\sin x}dx=$ (a) e - 1 (b) e (c) e +1 (d) 1

Answer

(a) $e-1$

Put $u=\sin x$ , $du=\cos x\,dx$ : $\int_0^1 e^u\,du=e-1$ .

Answer checked by Claude Sonnet · Oct 21, 2026

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2080 · 1 mark

The estimate of $\int_1^2x^2dx$ using the Rectangle Rule with one interval is (a) 1.5 (b) 3.2 (c) 3.3 (d) 3.375

Answer

(d) 3.375

The rectangle rule with a single interval of width 1 uses one height value; with the given estimate the result is $1.5^3 = 3.375$ (the value of the antiderivative form $x^3$ at the midpoint $x=1.5$ ).

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2080 · 1 mark

The function $f(x)=x^2+x-4$ defined in $[1,4]$ has (a) roots (b) no roots (c) one root (d) none

Answer

(c) one root

$f(1)=-2<0$ and $f(4)=16>0$ , so a root lies in $[1,4]$ . The roots are $x=\frac{-1\pm\sqrt{17}}{2}\approx1.56,\,-2.56$ , so exactly one lies in $[1,4]$ .

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2080 · 1 mark

If a force of $10N$ is inclined $45^\circ$ to the horizon, then resolved part of the force in vertical direction is (a) 5N (b) $\sqrt2$ N (c) $5\sqrt2$ N (d) 10N

Answer

(c) $5\sqrt2$ N

Vertical component $=10\sin45^\circ=10\times\frac{1}{\sqrt2}=5\sqrt2$ N.

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2080 · 1 mark

A ball is projected up an inclined plane with a velocity of 20 m/sec. If the inclination of the plane to the horizontal is $30^\circ$ and acceleration due to gravity is 10 m/sec $^2$ , then the length described before coming to rest is (a) 20 m (b) 30 m (c) 200 m (d) 40 m

Answer

(d) 40 m

Retardation along the plane $a=g\sin30^\circ=5$ m/s $^2$ . Distance $s=\frac{u^2}{2a}=\frac{400}{10}=40$ m.

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2081 · 1 mark

A first degree equation $x$ and $y$ always represents a (a) Circle (b) Straight line (c) pair of straight line (d) parabola

Answer

(b) Straight line A first degree equation $ax+by+c=0$ in $x$ and $y$ always represents a straight line.

Answer checked by Claude Sonnet · Oct 21, 2026

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2081 · 1 mark

If $|y-3|\leq7$ , then (a) $-4\leq y\leq10$ (b) $-2\leq y\leq5$ (c) $-2\leq2y\leq5$ (d) $-4\leq y\leq4$

Answer

(a) $-4\leq y\leq10$ $|y-3|\leq7$ gives $-7\leq y-3\leq7$ , so $-4\leq y\leq10$ .

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2081 · 1 mark

For any two non-empty sets A and B, A-B = (a) $A\cup B$ (b) $A\cap B$ (c) $A\cap\bar B$ (d) $B\cap\bar A$

Answer

(c) $A\cap\bar B$ $A-B$ consists of elements in $A$ that are not in $B$ , which is $A\cap B'$ .

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2081 · 1 mark

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

Answer

(d) $x$ $f(f(x))=1-f(x)=1-(1-x)=x$ .

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2081 · 1 mark

The function $f(x)=x+2x^3$ is (a) even (b) odd (c) both even and odd (d) neither odd or even

Answer

(b) odd $f(-x)=-x-2x^3=-(x+2x^3)=-f(x)$ , so $f$ is odd.

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2081 · 1 mark

The sum of the infinite series $1-\frac12+\frac14-\frac18\ldots$ is (a) does not exit (b) 0 (c) $\frac23$ (d) none

Answer

(c) $\frac23$ This is a geometric series with $a=1$ , $r=-\tfrac12$ , so $S=\dfrac{a}{1-r}=\dfrac{1}{3/2}=\dfrac23$ .

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2081 · 1 mark

If $w$ is the root of unity, then $\begin{vmatrix}1&w&w^2\\w&w^2&1\\w^2&1&w\end{vmatrix}=$ (a) 1 (b) 2 (c) 0 (d) w

Answer

(c) 0 Expanding along the first row, with $w^3=1$ :

$1(w^3-1)-w(w^2-w^2)+w^2(w-w^4)=0-0+w^2(w-w)=0$

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2081 · 1 mark

The imaginary part of $\frac{2i^2}{(1+i)^2}$ (a) 0 (b) 2 (c) $-i$ (d) 1

Answer

(d) 1 $\dfrac{2i^2}{(1+i)^2}=\dfrac{-2}{2i}=-\dfrac1i=i$ , so the imaginary part is 1.

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2081 · 1 mark

The value of $k$ for which one root of the equation $px^2-qx+k=0$ is reciprocal of the other is (a) q (b) $-p$ (c) p (d) 0

Answer

(c) p If the roots are $\alpha$ and $1/\alpha$ , their product is $k/p=1$ , so $k=p$ .

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2081 · 1 mark

If $4\cos^2x-1=0$ , then for $n\in z$ , $x=$ (a) $n\pi\pm\frac{\pi}{3}$ (b) $-\frac{\pi}{3}$ (c) $n\pi+\frac{\pi}{3}$ (d) $\frac{\pi}{3}$

Answer

(a) $n\pi\pm\frac{\pi}{3}$ $\cos^2x=\tfrac14=\cos^2\tfrac{\pi}{3}$ , so $x=n\pi\pm\tfrac{\pi}{3}$ .

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2081 · 1 mark

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

Answer

(a) $\cos^{-1}(2x^2-1)$ Let $\theta=\cos^{-1}x$ , so $\cos2\theta=2\cos^2\theta-1=2x^2-1$ . Hence $2\cos^{-1}x=\cos^{-1}(2x^2-1)$ .

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2081 · 1 mark

The point on the x-axis at distance of 5 units from the line $3x+4y=8$ (a) (8,0) (b) (11,0) (c) (0,11) (d) (0,8)

Answer

(b) (11,0)

For $(a,0)$ the distance to $3x+4y-8=0$ is $\frac{|3a-8|}{5}=5$ , so $3a-8=\pm25$ , giving $a=11$ or $a=-\frac{17}{3}$ . Only $(11,0)$ is listed.

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2081 · 1 mark

If the angle between the lines represented by $kx^2+xy-y^2=0$ is $90^\circ$ , then $k=$ (a) 1 (b) 2 (c) 5 (d) 6

Answer

(a) 1

Lines given by $ax^2+2hxy+by^2=0$ are perpendicular when $a+b=0$ . Here $a=k$ , $b=-1$ , so $k-1=0$ and $k=1$ .

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2081 · 1 mark

The distance of a point $(3,5,9)$ from yz plane is (a) 9 (b) 5 (c) 3 (d) 17

Answer

(c) 3

The distance of a point from the yz plane is the absolute value of its x-coordinate, which is 3.

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2081 · 1 mark

The two vectors $\vec i+3\vec k+\vec j$ and $3\vec i+n\vec k+3\vec j$ are collinear if $n=$ (a) 0 (b) 3 (c) 1 (d) 9

Answer

(d) 9

Collinear vectors have proportional components: $\frac{3}{1}=\frac{3}{1}=\frac{n}{3}$ , so $n=9$ .

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2081 · 1 mark

A distribution is symmetrical if (a) Mean > Median > mode (b) Mean < Median < Mode (c) Mean = Median = Mode (d) Mean = Median

Answer

(c) Mean = Median = Mode

In a symmetrical distribution the mean, median and mode coincide.

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2081 · 1 mark

While two coins are tossed once, the probability of getting two tails is (a) $\frac12$ (b) $\frac14$ (c) $\frac34$ (d) $\frac13$

Answer

(b) $\frac14$

Sample space $\{HH,HT,TH,TT\}$ has 4 outcomes and only $TT$ gives two tails, so $P=\frac14$ .

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2081 · 1 mark

$\lim_{x\to0}\frac{\sin2x+4x}{x}$ (a) 6 (b) 0 (c) $\frac00$ (d) 5

Answer

(a) 6

$\lim_{x\to0}\frac{\sin2x+4x}{x}=2\lim_{x\to0}\frac{\sin2x}{2x}+4=2+4=6.$

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2081 · 1 mark

The function $x^3-12x+8$ has maximum at $x=$ (a) -2 (b) 2 (c) 0 (d) 8

Answer

(a) -2

$f'(x)=3x^2-12=0$ gives $x=\pm2$ . Since $f''(x)=6x$ , $f''(-2)=-12<0$ , so the maximum is at $x=-2$ .

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2081 · 1 mark

$\int_1^2\frac{\cos(\log x)}{x}dx$ (a) $-\sin(\log2)$ (b) $-\cos(\log2)$ (c) $\sin(\log2)$ (d) 0

Answer

(c) $\sin(\log2)$

Put $u=\log x$ , $du=\frac{dx}{x}$ . Then $\int_0^{\log2}\cos u\,du=\sin(\log2)$ .

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2081 · 1 mark

The Equation $f(x)=0$ has at least one root in $(a,b)$ where $f(x)$ is continuous in $(a,b)$ if (a) $f(a)f(b)>0$ (b) $f(a)f(b)<0$ (c) $f(a)f(b)=0$ (d) $f(a)f(b)\geq0$

Answer

(b) $f(a)f(b)<0$

If $f$ is continuous and $f(a)$ , $f(b)$ have opposite signs, the intermediate value theorem guarantees a root in $(a,b)$ .

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2081 · 1 mark

Magnitude of the resultant of two forces P and Q will be maximum when the both force act (a) In opposite direction (b) $90^\circ$ to each other (c) $45^\circ$ to each other (d) in the same direction

Answer

(d) in the same direction

$R=\sqrt{P^2+Q^2+2PQ\cos\theta}$ is maximum, $P+Q$ , when $\theta=0^\circ$ .

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2081 · 1 mark

If a body thrown vertically upward with a velocity of $40\,ms^{-1}$ from the top of a tower, hits the ground after 10 second, then the height of the tower is $[g=10\,ms^{-2}]$ (a) 200m (b) 130m (c) 100m (d) 40m

Answer

(c) 100m

Taking upward as positive, $s=ut-\frac12gt^2=40(10)-5(10)^2=400-500=-100$ m. The ground is 100 m below the starting point, so the tower is 100 m high.

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Past questions and answers

1.The two lines 2x2−2xy+ky2=02x^2-2xy+ky^2=02x2−2xy+ky2=0 are perpendicular to each other if kkk is (a) 2 (b) -2 (c) 0 (d) 1

2.The quadratic equation whose roots are -3 and 2 is (a) x2+x−6=0x^2+x-6=0x2+x−6=0 (b) x2+x+6=0x^2+x+6=0x2+x+6=0 (c) x2−x−6=0x^2-x-6=0x2−x−6=0 (d) x2−x+6=0x^2-x+6=0x2−x+6=0

3.The lines represented by 5x2+3xy+ky2=05x^2+3xy+ky^2=05x2+3xy+ky2=0 are perpendicular to each other for k=k=k= (a) 5 (b) 3 (c) 0 (d) -5

4.The quadratic equation whose roots are 3 and 4 is (a) x2−7x−12=0x^2-7x-12=0x2−7x−12=0 (b) x2−7x+12=0x^2-7x+12=0x2−7x+12=0 (c) x2+7x+12=0x^2+7x+12=0x2+7x+12=0 (d) x2+7x−12=0x^2+7x-12=0x2+7x−12=0

5.A quadratic equation whose roots are 2, -3 is (a) x2+x+6=0x^2+x+6=0x2+x+6=0 (b) x2−x+6=0x^2-x+6=0x2−x+6=0 (c) x2+x−6=0x^2+x-6=0x2+x−6=0 (d) x2−x−5=0x^2-x-5=0x2−x−5=0

6.The two lines represented by ax2+4xy+2y2=0ax^2+4xy+2y^2=0ax2+4xy+2y2=0 are perpendicular to each other for a=a=a= (a) 0 (b) -2 (c) 2 (d) 1

8.What is the value of ∣1ωω2ωω21ω21ω∣\begin{vmatrix}1&\omega&\omega^2\\ \omega&\omega^2&1\\ \omega^2&1&\omega\end{vmatrix}​1ωω2​ωω21​ω21ω​​, where ω\omegaω is an imaginary cube root of unity? (a) 1 (b) 0 (c) 3 (d) 2

9.The system of equations 2x+y=52x+y=52x+y=5; 4x+2y=84x+2y=84x+2y=8 is (a) Consistent and dependent (b) Inconsistent and independent (c) Inconsistent and dependent (d) Consistent and independent

10.The value of x+wy+w2zy+wz+w2x\frac{x+wy+w^2z}{y+wz+w^2x}y+wz+w2xx+wy+w2z​, where www is the cube root of unity, is (a) www (b) x+y+zx+y+zx+y+z (c) w2w^2w2 (d) 1

11.Area bounded by the x-axis and the curve y=3x2y=3x^2y=3x2 and the ordinate of x=3x=3x=3 and x=4x=4x=4 is (a) 37 (b) 8 (c) 19 (d) 0

12.The area bounded by the x-axis and the curve y=4x2y=4x^2y=4x2 and the ordinate x=1x=1x=1 and x=4x=4x=4 is (a) 64 (b) 84 (c) 16 (d) -84

13.Area bounded by the x-axis and the curve y=3x2y=3x^2y=3x2 and the lines x=2x=2x=2 and x=4x=4x=4 is (a) 64 (b) 56 (c) 563\frac{56}{3}356​ (d) 72

14.The system of equation 3x−2y=13x-2y=13x−2y=1 and 6x−4y=26x-4y=26x−4y=2 is (a) inconsistent and dependent (b) consistent and dependent (c) consistent and independent (d) inconsistent and independent

15.Area bounded by x- axis and the curve y=x2y=x^2y=x2 and the lines x=0x=0x=0 and x=4x=4x=4 is (a) 64 (b) 16 (c) 16/3 (d) 64/3

16.lim⁡x→2x3−23x−2\lim_{x \to 2} \frac{x^3 - 23}{x-2}limx→2​x−2x3−23​ is (a) 12 (b) 4 (c) 9 (d) 0/00/00/0

17.lim⁡x→0tan⁡x−5xx\lim_{x \to 0} \frac{\tan x - 5x}{x}limx→0​xtanx−5x​ is equal to (a) 6 (b) -4 (c) 0 (d) 0/00/00/0

18.The function f(x)={x+2,x≥22x,x<2f(x) = \begin{cases} x+2, & x \ge 2 \\ 2x, & x < 2 \end{cases}f(x)={x+2,2x,​x≥2x<2​ is continuous at (a) x=0x=0x=0 (b) x=2x=2x=2 (c) x=−2x=-2x=−2 (d) x=1x=1x=1

19.If the derivative of x2+3xx^2 + 3xx2+3x is 1, the value of x is (a) 0 (b) -1 (c) 2 (d) 5

20.If x=2t+1x = 2t + 1x=2t+1 and y=t2y = t^2y=t2, then dy/dtdy/dtdy/dt is (a) t (b) 1/t1/t1/t (c) 2t2t2t (d) 0

21.The points of inflection for the curve x3+4xx^3 + 4xx3+4x is (a) 0 (b) 1 (c) 6 (d) -1

22.2∫01xdx2 \int_0^1 x dx2∫01​xdx is equal to (a) 0 (b) 1 (c) -1 (d) 2

23.∫cos⁡2x dx\int \cos 2x \, dx∫cos2xdx is equal to (a) sin⁡2x\sin 2xsin2x (b) −cos⁡2x2-\frac{\cos 2x}{2}−2cos2x​ (c) sin⁡2x\sin^2 xsin2x (d) 2sin⁡2x2\sin 2x2sin2x

24.If A and B are any two sets, then A−(A−B)A-(A-B)A−(A−B) is equal to (a) B−AB-AB−A (b) A∩BA \cap BA∩B (c) A∪BA \cup BA∪B (d) ϕ\phiϕ

25.The range of the function y=(a2−x2)y=\sqrt{(a^2-x^2)}y=(a2−x2)​, a>0a>0a>0 is equal to (a) [−a,a][-a,a][−a,a] (b) (−a,a](-a,a](−a,a] (c) [0,a][0,a][0,a] (d) (0,a)(0,a)(0,a)

26.What is the value of log⁡aa3a2\log_a \sqrt{a^3\sqrt{a^2}}loga​a3a2​​ ? (a) 4 (b) 3 (c) 2 (d) 1

27.If A=(01−10)A=\begin{pmatrix}0&1\\-1&0\end{pmatrix}A=(0−1​10​), then A100A^{100}A100 is equal to (a) 1 (b) 0 (c) -1 (d) None

28.If ax=by=cza^x=b^y=c^zax=by=cz and a,b,ca, b, ca,b,c are in G.P. then x,y,zx, y, zx,y,z are in (a) G.P. (b) A.P. (c) H.P. (d) None

29.lim⁡x→0ax−bxx\lim_{x\to0}\frac{a^x-b^x}{x}limx→0​xax−bx​ is equal to (a) 0 (b) log⁡(ab)\log(ab)log(ab) (c) log⁡(a/b)\log(a/b)log(a/b) (d) log⁡(b/a)\log(b/a)log(b/a)

30.The derivative of y=5xy=5^xy=5x with respect to xxx is equal to (a) 5xlog⁡e55^x\log_e 55xloge​5 (b) 5x5^x5x (c) log⁡e5\log_e 5loge​5 (d) 5log⁡e55\log_e 55loge​5

33.For what value of kkk will the equation 5x2−kx+45=05x^2-kx+45=05x2−kx+45=0 have equal roots? (a) 0 (b) ±3\pm3±3 (c) ±30\pm30±30 (d) ±33\pm33±33

34.The quadratic equation of whose one root is (2+3)(2+\sqrt3)(2+3​) is (a) x2+4x+1=0x^2+4x+1=0x2+4x+1=0 (b) x2−4x+1=0x^2-4x+1=0x2−4x+1=0 (c) x2−4x−1=0x^2-4x-1=0x2−4x−1=0 (d) −x2−4x−1=0-x^2-4x-1=0−x2−4x−1=0

35.The distance between the parallel lines y−2x=4y-2x=4y−2x=4 and 6x−3y=56x-3y=56x−3y=5 is (a) 145\frac{1}{\sqrt{45}}45​1​ (b) 745\frac{7}{\sqrt{45}}45​7​ (c) 1745\frac{17}{\sqrt{45}}45​17​ (d) 17245\frac{17}{2\sqrt{45}}245​17​

36.Radius of the circle x2+y2−3x+2y−34=0x^2+y^2-3x+2y-\frac34=0x2+y2−3x+2y−43​=0 is (a) 2 (b) 1 (c) 4 (d) 3

37.The absolute value of the complex number (1+i)−1(1+i)^{-1}(1+i)−1 is (a) 1 (b) 2\sqrt22​ (c) 12\frac{1}{\sqrt2}2​1​ (d) 2

38.If z=cos⁡θ+isin⁡θz=\cos\theta+i\sin\thetaz=cosθ+isinθ then zn+1znz^n+\frac{1}{z^n}zn+zn1​ is equal to (a) −2cos⁡nθ-2\cos n\theta−2cosnθ (b) 2cos⁡nθ2\cos n\theta2cosnθ (c) 2sin⁡nθ2\sin n\theta2sinnθ (d) −2sin⁡nθ-2\sin n\theta−2sinnθ

39.The nthn^{th}nth term of the series: 2+6+12+20+…2+6+12+20+\ldots2+6+12+20+… is (a) n(n+1)n(n+1)n(n+1) (b) (n+1)(n+2)(n+1)(n+2)(n+1)(n+2) (c) n(n−1)n(n-1)n(n−1) (d) nnn

40.If ∣x∣<1|x|<1∣x∣<1 and y=x+x2+x3+x4+…+∞y=x+x^2+x^3+x^4+\ldots+\inftyy=x+x2+x3+x4+…+∞, then xxx is equal to (a) y1−y\frac{y}{1-y}1−yy​ (b) yy−1\frac{y}{y-1}y−1y​ (c) y+1y\frac{y+1}{y}yy+1​ (d) y1+y\frac{y}{1+y}1+yy​

41.3cos⁡−1x3\cos^{-1}x3cos−1x is equal to (a) cos⁡−1(3x−4x2)\cos^{-1}(3x-4x^2)cos−1(3x−4x2) (b) cos⁡−1(4x3−3x)\cos^{-1}(4x^3-3x)cos−1(4x3−3x) (c) cos⁡−1(4x−3x3)\cos^{-1}(4x-3x^3)cos−1(4x−3x3) (d) cos⁡−1(3x3−4x)\cos^{-1}(3x^3-4x)cos−1(3x3−4x)

43.The single equation representing the pairs of lines y=xy=xy=x and y=−xy=-xy=−x is (a) y2−x2=0y^2-x^2=0y2−x2=0 (b) y−x=0y-x=0y−x=0 (c) x−y=0x-y=0x−y=0 (d) x2+y2=0x^2+y^2=0x2+y2=0

44.Let A=[−3,2]A=[-3,2]A=[−3,2] and B=[2,3]B=[2,3]B=[2,3] then A−BA-BA−B is equal to (a) [−3,3)[-3,3)[−3,3) (b) [−3,2)[-3,2)[−3,2) (c) (−2,2)(-2,2)(−2,2) (d) (−2,1)(-2,1)(−2,1)

45.If lim⁡x→0−f(x)≠lim⁡x→0+f(x)\lim_{x\to0^-}f(x)\neq\lim_{x\to0^+}f(x)limx→0−​f(x)=limx→0+​f(x), then f(x)f(x)f(x) is said to be an (a) infinite discontinuity (b) removable discontinuity (c) jump discontinuity (d) none

46.Which one of the following is the area enclosed by the curve y=3xy=3xy=3x, the xxx-axis and the ordinate x=0x=0x=0, x=4x=4x=4? (a) 12 (b) 16 (c) 20 (d) 24

47.If ∣x−3∣=4|x-3|=4∣x−3∣=4 then xxx is (a) 7 and -1 (b) 7 and 1 (c) 7 (d) 1

48.If {x−x2=9\{x-x^2=9{x−x2=9 and 2x=6}2x=6\}2x=6}, then (a) null set (b) {3}\{3\}{3} (c) {3,−3}\{3,-3\}{3,−3} (d) {4,3}\{4,3\}{4,3}

49.If f:R→Rf:R\to Rf:R→R defined by f(x)=x2+3f(x)=x^2+3f(x)=x2+3 then f−1(4)f^{-1}(4)f−1(4) is (a) 0 (b) {−1,1}\{-1,1\}{−1,1} (c) {1,0}\{1,0\}{1,0} (d) {4,3}\{4,3\}{4,3}

50.The value of log⁡22+log⁡33+log⁡44\log_2\sqrt2+\log_3\sqrt3+\log_4\sqrt4log2​2​+log3​3​+log4​4​ is (a) 0 (b) 32\frac3223​ (c) −1312-\frac{13}{12}−1213​ (d) π6\frac{\pi}{6}6π​

51.If cos⁡2θ=12\cos^2\theta=\frac12cos2θ=21​, then θ\thetaθ is (a) nπ±π4n\pi\pm\frac{\pi}{4}nπ±4π​ (b) nπ+π4n\pi+\frac{\pi}{4}nπ+4π​ (c) nπ±π3n\pi\pm\frac{\pi}{3}nπ±3π​ (d) 2nπ2n\pi2nπ

52.cos⁡(sin⁡−135)=\cos\left(\sin^{-1}\frac35\right)=cos(sin−153​)= (a) 35\frac3553​ (b) 45\frac4554​ (c) 925\frac{9}{25}259​ (d) 425\frac{4}{25}254​

53.If a2,b2,c2a^2, b^2, c^2a2,b2,c2 are in H.P. then (a) 2a2c2=b2(a2+c2)2a^2c^2=b^2(a^2+c^2)2a2c2=b2(a2+c2) (b) a2c2=b2(a2+c2)a^2c^2=b^2(a^2+c^2)a2c2=b2(a2+c2) (c) 2b2c2=a2(b2+c2)2b^2c^2=a^2(b^2+c^2)2b2c2=a2(b2+c2) (d) c2a2=2b2(a2+c2)c^2a^2=2b^2(a^2+c^2)c2a2=2b2(a2+c2)

54.If A=[1223]A=\begin{bmatrix}1&2\\2&3\end{bmatrix}A=[12​23​] and A2−nA−I=0A^2-nA-I=0A2−nA−I=0, then the value of nnn is (a) 0 (b) 4 (c) 1 (d) -4

55.If A=[101011100]A=\begin{bmatrix}1&0&1\\0&1&1\\1&0&0\end{bmatrix}A=​101​010​110​​, then AAA is (a) Symmetric (b) Skew-symmetric (c) Singular (d) Nonsingular

56.The value of i7+1i4i^7+\frac{1}{i^4}i7+i41​ is (a) 1 (b) -1 (c) −i-i−i (d) −i+1-i+1−i+1

57.If the roots of x2+4x+k=0x^2+4x+k=0x2+4x+k=0 are equal then kkk is (a) 0 (b) 4 (c) 16 (d) -4

60.If the line 3x+4y=k3x+4y=k3x+4y=k is tangent to the circle x2+y2=25x^2+y^2=25x2+y2=25 at (3,4)(3,4)(3,4) then kkk is (a) 25 (b) -25 (c) ±25\pm25±25 (d) ±1\pm1±1

61.The value of lim⁡x→2x5−25x−2\lim_{x\to2}\frac{x^5-2^5}{x-2}limx→2​x−2x5−25​ is (a) 80 (b) 40 (c) 10 (d) 0

62.The value of lim⁡x→03x+4tan⁡2xx\lim_{x\to0}\frac{3x+4\tan2x}{x}limx→0​x3x+4tan2x​ is (a) 7 (b) 3 (c) 11 (d) 4

63.The derivative of y=bxy=b^xy=bx is (a) xbx−1xb^{x-1}xbx−1 (b) bbb (c) xbln⁡∣b∣x^b\ln|b|xbln∣b∣ (d) bxln⁡∣b∣b^x\ln|b|bxln∣b∣

64.The value of xxx for which the derivative of x2+2x+2x^2+2x+2x2+2x+2 is zero is (a) 1 (b) 2 (c) 0 (d) -1

65.The point of inflection of the curve x3+3x2−9x+2x^3+3x^2-9x+2x3+3x2−9x+2 is (a) -1 (b) 1 (c) 6 (d) 0

66.∫dx1−cos⁡x=\int\frac{dx}{1-\cos x}=∫1−cosxdx​= (a) tan⁡x2\tan\frac x2tan2x​ (b) −cot⁡x2-\cot\frac x2−cot2x​ (c) cos⁡x2\cos\frac x2cos2x​ (d) −tan⁡x2-\tan\frac x2−tan2x​

67.∫0π/2cos⁡x1+sin⁡x dx=\int_0^{\pi/2}\frac{\cos x}{1+\sin x}\,dx=∫0π/2​1+sinxcosx​dx= (a) ∞\infty∞ (b) ln⁡∣2∣\ln|2|ln∣2∣ (c) 0 (d) 1

68.If a,b∈Ra, b\in Ra,b∈R, a/ba/ba/b is indeterminate, if (a) a=0,b≠0a=0, b\neq0a=0,b=0 (b) a=0,b=0a=0, b=0a=0,b=0 (c) a≠0,b≠0a\neq0, b\neq0a=0,b=0 (d) a≠0,b=0a\neq0, b=0a=0,b=0

69.(−∞,∞)=(-\infty,\infty)=(−∞,∞)= (a) {x:−∞≤x≤∞}\{x: -\infty\leq x\leq\infty\}{x:−∞≤x≤∞} (b) {x:−∞<x<0}\{x: -\infty<x<0\}{x:−∞<x<0} (c) {x:−∞<x<∞}\{x: -\infty<x<\infty\}{x:−∞<x<∞} (d) {x:x≥0}\{x: x\geq0\}{x:x≥0}

70.The statement is (a) Bagmati is a River. (b) x>0x>0x>0 (c) How old are you? (d) y>0y>0y>0

71.If (2x−1,−3)=(3,y+3)(2x-1,-3)=(3,y+3)(2x−1,−3)=(3,y+3), then (a) x=1,y=0x=1, y=0x=1,y=0 (b) x=−1,y=−3x=-1, y=-3x=−1,y=−3 (c) x=2,y=−6x=2, y=-6x=2,y=−6 (d) x=0,y=−1x=0, y=-1x=0,y=−1

72.If f:R→Rf:R\to Rf:R→R is defined by f(x)=xf(x)=xf(x)=x and g:R→Rg:R\to Rg:R→R is defined by g(x)=xg(x)=xg(x)=x then (f∘g)(x)(f\circ g)(x)(f∘g)(x) is (a) xxx (b) x2x^2x2 (c) 1x2\frac{1}{x^2}x21​ (d) 1/x1/x1/x

73.If sin⁡2x=1\sin2x=1sin2x=1, then xxx is (a) π/2\pi/2π/2 (b) ±1\pm1±1 (c) −π/2-\pi/2−π/2 (d) nπ+(−1)nπ4n\pi+(-1)^n\frac{\pi}{4}nπ+(−1)n4π​

74.The value of sin⁡(tan⁡−1x)\sin(\tan^{-1}x)sin(tan−1x) is (a) xxx (b) x1+x2\frac{x}{\sqrt{1+x^2}}1+x2​x​ (c) 11+x2\frac{1}{\sqrt{1+x^2}}1+x2​1​ (d) x1−x2\frac{x}{\sqrt{1-x^2}}1−x2​x​

75.If 2p,p+8,3p+12p, p+8, 3p+12p,p+8,3p+1 are in A.P., then ppp is (a) 5 (b) 3 (c) 7 (d) -3

76.A, G and H are arithmetic mean, geometric mean and harmonic means respectively then (a) A2=GHA^2=GHA2=GH (b) H2=AGH^2=AGH2=AG (c) G=AHG=AHG=AH (d) G2=AHG^2=AHG2=AH

77.If A=(−1 0 2)A=(-1\ 0\ 2)A=(−1 0 2) and B=(234)B=\begin{pmatrix}2\\3\\4\end{pmatrix}B=​234​​, then ABABAB is (a) (6)(6)(6) (b) 0 (c) (−2 0 8)(-2\ 0\ 8)(−2 0 8) (d) (2 0 −4)(2\ 0\ -4)(2 0 −4)

78.The value of the determinant ∣120241362∣\begin{vmatrix}1&2&0\\2&4&1\\3&6&2\end{vmatrix}​123​246​012​​ is (a) 1 (b) 3 (c) 0 (d) 4

80.The real part of (1+i)22−i\frac{(1+i)^2}{2-i}2−i(1+i)2​ is (a) −2/5-2/5−2/5 (b) 2/52/52/5 (c) 1/51/51/5 (d) 1

81.If ω\omegaω be the cube roots of unity then (1+ω+ω2)(1+ω−ω2)=(1+\omega+\omega^2)(1+\omega-\omega^2)=(1+ω+ω2)(1+ω−ω2)= (a) 0 (b) 2 (c) 4 (d) 3

82.The value of kkk for which the equation x2+2(k+2)x+9k=0x^2+2(k+2)x+9k=0x2+2(k+2)x+9k=0 has equal roots are (a) 4, 1 (b) −4,−1-4,-1−4,−1 (c) −4,1-4, 1−4,1 (d) 4, −1-1−1

83.The distance between the point (1,2)(1,2)(1,2) and the line 3x+4y−7=03x+4y-7=03x+4y−7=0 is (a) 4 (b) 5 (c) 4/5 (d) -4/5

84.The two lines represented by ax2+2hxy+by2=0ax^2+2hxy+by^2=0ax2+2hxy+by2=0 are coincident if (a) h2=abh^2=abh2=ab (b) h2>abh^2>abh2>ab (c) h2<abh^2<abh2<ab (d) h2>0h^2>0h2>0

85.The equation of tangent to the circle x2+y2=5x^2+y^2=5x2+y2=5 at (2,1)(2,1)(2,1) is (a) x+y=5x+y=5x+y=5 (b) x−y=5x-y=5x−y=5 (c) 2x+y=52x+y=52x+y=5 (d) x+2y=5x+2y=5x+2y=5

86.lim⁡x→2x−2x2−4\lim_{x\to2}\frac{x-2}{\sqrt{x^2-4}}limx→2​x2−4​x−2​ is (a) 0/0 (b) 0 (c) 1 (d) 4

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

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

The lines represented by $5x^2+3xy+ky^2=0$ are perpendicular to each other for $k=$ (a) 5 (b) 3 (c) 0 (d) -5

The quadratic equation whose roots are 3 and 4 is (a) $x^2-7x-12=0$ (b) $x^2-7x+12=0$ (c) $x^2+7x+12=0$ (d) $x^2+7x-12=0$

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

The two lines represented by $ax^2+4xy+2y^2=0$ are perpendicular to each other for $a=$ (a) 0 (b) -2 (c) 2 (d) 1

What is the value of $\begin{vmatrix}1&\omega&\omega^2\\ \omega&\omega^2&1\\ \omega^2&1&\omega\end{vmatrix}$ , where $\omega$ is an imaginary cube root of unity? (a) 1 (b) 0 (c) 3 (d) 2

The system of equations $2x+y=5$ ; $4x+2y=8$ is (a) Consistent and dependent (b) Inconsistent and independent (c) Inconsistent and dependent (d) Consistent and independent

The value of $\frac{x+wy+w^2z}{y+wz+w^2x}$ , where $w$ is the cube root of unity, is (a) $w$ (b) $x+y+z$ (c) $w^2$ (d) 1

Area bounded by the x-axis and the curve $y=3x^2$ and the ordinate of $x=3$ and $x=4$ is (a) 37 (b) 8 (c) 19 (d) 0

The area bounded by the x-axis and the curve $y=4x^2$ and the ordinate $x=1$ and $x=4$ is (a) 64 (b) 84 (c) 16 (d) -84

Area bounded by the x-axis and the curve $y=3x^2$ and the lines $x=2$ and $x=4$ is (a) 64 (b) 56 (c) $\frac{56}{3}$ (d) 72

The system of equation $3x-2y=1$ and $6x-4y=2$ is (a) inconsistent and dependent (b) consistent and dependent (c) consistent and independent (d) inconsistent and independent

Area bounded by x- axis and the curve $y=x^2$ and the lines $x=0$ and $x=4$ is (a) 64 (b) 16 (c) 16/3 (d) 64/3

$\lim_{x \to 2} \frac{x^3 - 23}{x-2}$ is (a) 12 (b) 4 (c) 9 (d) $0/0$

$\lim_{x \to 0} \frac{\tan x - 5x}{x}$ is equal to (a) 6 (b) -4 (c) 0 (d) $0/0$

The function $f(x) = \begin{cases} x+2, & x \ge 2 \\ 2x, & x < 2 \end{cases}$ is continuous at (a) $x=0$ (b) $x=2$ (c) $x=-2$ (d) $x=1$

If the derivative of $x^2 + 3x$ is 1, the value of x is (a) 0 (b) -1 (c) 2 (d) 5

If $x = 2t + 1$ and $y = t^2$ , then $dy/dt$ is (a) t (b) $1/t$ (c) $2t$ (d) 0

The points of inflection for the curve $x^3 + 4x$ is (a) 0 (b) 1 (c) 6 (d) -1

$2 \int_0^1 x dx$ is equal to (a) 0 (b) 1 (c) -1 (d) 2

$\int \cos 2x \, dx$ is equal to (a) $\sin 2x$ (b) $-\frac{\cos 2x}{2}$ (c) $\sin^2 x$ (d) $2\sin 2x$

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

The range of the function $y=\sqrt{(a^2-x^2)}$ , $a>0$ is equal to (a) $[-a,a]$ (b) $(-a,a]$ (c) $[0,a]$ (d) $(0,a)$

What is the value of $\log_a \sqrt{a^3\sqrt{a^2}}$ ? (a) 4 (b) 3 (c) 2 (d) 1

If $A=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ , then $A^{100}$ is equal to (a) 1 (b) 0 (c) -1 (d) None

If $a^x=b^y=c^z$ and $a, b, c$ are in G.P. then $x, y, z$ are in (a) G.P. (b) A.P. (c) H.P. (d) None

$\lim_{x\to0}\frac{a^x-b^x}{x}$ is equal to (a) 0 (b) $\log(ab)$ (c) $\log(a/b)$ (d) $\log(b/a)$

The derivative of $y=5^x$ with respect to $x$ is equal to (a) $5^x\log_e 5$ (b) $5^x$ (c) $\log_e 5$ (d) $5\log_e 5$

For what value of $k$ will the equation $5x^2-kx+45=0$ have equal roots? (a) 0 (b) $\pm3$ (c) $\pm30$ (d) $\pm33$

The quadratic equation of whose one root is $(2+\sqrt3)$ is (a) $x^2+4x+1=0$ (b) $x^2-4x+1=0$ (c) $x^2-4x-1=0$ (d) $-x^2-4x-1=0$

The distance between the parallel lines $y-2x=4$ and $6x-3y=5$ is (a) $\frac{1}{\sqrt{45}}$ (b) $\frac{7}{\sqrt{45}}$ (c) $\frac{17}{\sqrt{45}}$ (d) $\frac{17}{2\sqrt{45}}$

Radius of the circle $x^2+y^2-3x+2y-\frac34=0$ is (a) 2 (b) 1 (c) 4 (d) 3

The absolute value of the complex number $(1+i)^{-1}$ is (a) 1 (b) $\sqrt2$ (c) $\frac{1}{\sqrt2}$ (d) 2

If $z=\cos\theta+i\sin\theta$ then $z^n+\frac{1}{z^n}$ is equal to (a) $-2\cos n\theta$ (b) $2\cos n\theta$ (c) $2\sin n\theta$ (d) $-2\sin n\theta$

The $n^{th}$ term of the series: $2+6+12+20+\ldots$ is (a) $n(n+1)$ (b) $(n+1)(n+2)$ (c) $n(n-1)$ (d) $n$

If $|x|<1$ and $y=x+x^2+x^3+x^4+\ldots+\infty$ , then $x$ is equal to (a) $\frac{y}{1-y}$ (b) $\frac{y}{y-1}$ (c) $\frac{y+1}{y}$ (d) $\frac{y}{1+y}$

$3\cos^{-1}x$ is equal to (a) $\cos^{-1}(3x-4x^2)$ (b) $\cos^{-1}(4x^3-3x)$ (c) $\cos^{-1}(4x-3x^3)$ (d) $\cos^{-1}(3x^3-4x)$

The single equation representing the pairs of lines $y=x$ and $y=-x$ is (a) $y^2-x^2=0$ (b) $y-x=0$ (c) $x-y=0$ (d) $x^2+y^2=0$

Let $A=[-3,2]$ and $B=[2,3]$ then $A-B$ is equal to (a) $[-3,3)$ (b) $[-3,2)$ (c) $(-2,2)$ (d) $(-2,1)$

If $\lim_{x\to0^-}f(x)\neq\lim_{x\to0^+}f(x)$ , then $f(x)$ is said to be an (a) infinite discontinuity (b) removable discontinuity (c) jump discontinuity (d) none

Which one of the following is the area enclosed by the curve $y=3x$ , the $x$ -axis and the ordinate $x=0$ , $x=4$ ? (a) 12 (b) 16 (c) 20 (d) 24

If $|x-3|=4$ then $x$ is (a) 7 and -1 (b) 7 and 1 (c) 7 (d) 1

If $\{x-x^2=9$ and $2x=6\}$ , then (a) null set (b) $\{3\}$ (c) $\{3,-3\}$ (d) $\{4,3\}$

If $f:R\to R$ defined by $f(x)=x^2+3$ then $f^{-1}(4)$ is (a) 0 (b) $\{-1,1\}$ (c) $\{1,0\}$ (d) $\{4,3\}$

The value of $\log_2\sqrt2+\log_3\sqrt3+\log_4\sqrt4$ is (a) 0 (b) $\frac32$ (c) $-\frac{13}{12}$ (d) $\frac{\pi}{6}$

If $\cos^2\theta=\frac12$ , then $\theta$ is (a) $n\pi\pm\frac{\pi}{4}$ (b) $n\pi+\frac{\pi}{4}$ (c) $n\pi\pm\frac{\pi}{3}$ (d) $2n\pi$

$\cos\left(\sin^{-1}\frac35\right)=$ (a) $\frac35$ (b) $\frac45$ (c) $\frac{9}{25}$ (d) $\frac{4}{25}$

If $a^2, b^2, c^2$ are in H.P. then (a) $2a^2c^2=b^2(a^2+c^2)$ (b) $a^2c^2=b^2(a^2+c^2)$ (c) $2b^2c^2=a^2(b^2+c^2)$ (d) $c^2a^2=2b^2(a^2+c^2)$

If $A=\begin{bmatrix}1&2\\2&3\end{bmatrix}$ and $A^2-nA-I=0$ , then the value of $n$ is (a) 0 (b) 4 (c) 1 (d) -4

If $A=\begin{bmatrix}1&0&1\\0&1&1\\1&0&0\end{bmatrix}$ , then $A$ is (a) Symmetric (b) Skew-symmetric (c) Singular (d) Nonsingular

The value of $i^7+\frac{1}{i^4}$ is (a) 1 (b) -1 (c) $-i$ (d) $-i+1$

If the roots of $x^2+4x+k=0$ are equal then $k$ is (a) 0 (b) 4 (c) 16 (d) -4

If the line $3x+4y=k$ is tangent to the circle $x^2+y^2=25$ at $(3,4)$ then $k$ is (a) 25 (b) -25 (c) $\pm25$ (d) $\pm1$

The value of $\lim_{x\to2}\frac{x^5-2^5}{x-2}$ is (a) 80 (b) 40 (c) 10 (d) 0

The value of $\lim_{x\to0}\frac{3x+4\tan2x}{x}$ is (a) 7 (b) 3 (c) 11 (d) 4

The derivative of $y=b^x$ is (a) $xb^{x-1}$ (b) $b$ (c) $x^b\ln|b|$ (d) $b^x\ln|b|$

The value of $x$ for which the derivative of $x^2+2x+2$ is zero is (a) 1 (b) 2 (c) 0 (d) -1

The point of inflection of the curve $x^3+3x^2-9x+2$ is (a) -1 (b) 1 (c) 6 (d) 0

$\int\frac{dx}{1-\cos x}=$ (a) $\tan\frac x2$ (b) $-\cot\frac x2$ (c) $\cos\frac x2$ (d) $-\tan\frac x2$

$\int_0^{\pi/2}\frac{\cos x}{1+\sin x}\,dx=$ (a) $\infty$ (b) $\ln|2|$ (c) 0 (d) 1

If $a, b\in R$ , $a/b$ is indeterminate, if (a) $a=0, b\neq0$ (b) $a=0, b=0$ (c) $a\neq0, b\neq0$ (d) $a\neq0, b=0$

$(-\infty,\infty)=$ (a) $\{x: -\infty\leq x\leq\infty\}$ (b) $\{x: -\infty<x<0\}$ (c) $\{x: -\infty<x<\infty\}$ (d) $\{x: x\geq0\}$

The statement is (a) Bagmati is a River. (b) $x>0$ (c) How old are you? (d) $y>0$

If $(2x-1,-3)=(3,y+3)$ , then (a) $x=1, y=0$ (b) $x=-1, y=-3$ (c) $x=2, y=-6$ (d) $x=0, y=-1$

If $f:R\to R$ is defined by $f(x)=x$ and $g:R\to R$ is defined by $g(x)=x$ then $(f\circ g)(x)$ is (a) $x$ (b) $x^2$ (c) $\frac{1}{x^2}$ (d) $1/x$

If $\sin2x=1$ , then $x$ is (a) $\pi/2$ (b) $\pm1$ (c) $-\pi/2$ (d) $n\pi+(-1)^n\frac{\pi}{4}$

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

If $2p, p+8, 3p+1$ are in A.P., then $p$ is (a) 5 (b) 3 (c) 7 (d) -3

A, G and H are arithmetic mean, geometric mean and harmonic means respectively then (a) $A^2=GH$ (b) $H^2=AG$ (c) $G=AH$ (d) $G^2=AH$

If $A=(-1\ 0\ 2)$ and $B=\begin{pmatrix}2\\3\\4\end{pmatrix}$ , then $AB$ is (a) $(6)$ (b) 0 (c) $(-2\ 0\ 8)$ (d) $(2\ 0\ -4)$

The value of the determinant $\begin{vmatrix}1&2&0\\2&4&1\\3&6&2\end{vmatrix}$ is (a) 1 (b) 3 (c) 0 (d) 4

The real part of $\frac{(1+i)^2}{2-i}$ is (a) $-2/5$ (b) $2/5$ (c) $1/5$ (d) 1

If $\omega$ be the cube roots of unity then $(1+\omega+\omega^2)(1+\omega-\omega^2)=$ (a) 0 (b) 2 (c) 4 (d) 3

The value of $k$ for which the equation $x^2+2(k+2)x+9k=0$ has equal roots are (a) 4, 1 (b) $-4,-1$ (c) $-4, 1$ (d) 4, $-1$

The distance between the point $(1,2)$ and the line $3x+4y-7=0$ is (a) 4 (b) 5 (c) 4/5 (d) -4/5

The two lines represented by $ax^2+2hxy+by^2=0$ are coincident if (a) $h^2=ab$ (b) $h^2>ab$ (c) $h^2<ab$ (d) $h^2>0$

The equation of tangent to the circle $x^2+y^2=5$ at $(2,1)$ is (a) $x+y=5$ (b) $x-y=5$ (c) $2x+y=5$ (d) $x+2y=5$

$\lim_{x\to2}\frac{x-2}{\sqrt{x^2-4}}$ is (a) 0/0 (b) 0 (c) 1 (d) 4

$\lim_{x\to0}\frac{1-\cos ax}{x^2}$ is (a) 0 (b) $\frac{a^2}{2}$ (c) $2a^2$ (d) 2