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Maths

In any reaction, catalyst
(A) retards the rate of reaction
(B) increases the rate of reaction
(C) both increases and decreases the rate of reaction
(D) none of these

Answer: Increases the rate of reaction Explanation: Most catalysts work by lowering the 'activation energy' of a reaction. This allows less energy to be used, thus speeding up the reaction. The...

Solve the following linear programming problem graphically: Maximise $\mathrm{Z}=4 \mathrm{x}+\mathrm{y} \ldots(1)$ subject to the constraints: $\mathrm{x}+\mathrm{y} \leq 50 \ldots(2)$
$\begin{array}{l} 3 \mathrm{x}+\mathrm{y} \leq 90 \ldots(3) \\ \mathrm{x} \geq 0, \mathrm{y} \geq 0 \ldots(4) \end{array}$

Maths, Previous Year Question Papers

The shaded region in fig. is the feasible region determined by the system of constraints (2) to (4). We observe that the feasible region $\mathrm{OABC}$ is bounded. So,we now use Corner Point Method...

$\text { The binomial distribution whose mean is } 9 \text { and the variance is } 2.25 \text { is }$

Maths, Previous Year Question Papers

$$ \begin{array}{l} \text { Mean }=n p \\ \text { Variance }=n p(1-p) \end{array} $$ Therefore according to question $$ \begin{array}{l} 1-\mathrm{p}=\mathrm{q}=\frac{\text { Variance }}{\text {...

$\int_{1}^{\alpha} \log (1+\tan x) d x$ is squal to

Maths, Previous Year Question Papers

$ \begin{array}{l} \operatorname{Lst} I=\int^{\infty} \log (1+\tan x) d x . \ldots .(\mathrm{i}) \\ \rightarrow 1=\int_{0}^{\infty} \log \left|1+\tan \left(\frac{\pi}{4}-x\right)\right| d x...

If $y=3 e^{2 x}+2 e^{3 x}$ . prove that $\frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+6 y=0$

Maths, Previous Year Question Papers

$$ \begin{array}{l} \text { Given : } y=3 e^{2 x}+2 e^{3 x} \\ \Longrightarrow \frac{d y}{d x}=3 e^{2 x} \times 2+2 e^{3 x} \times 3=6 e^{2 x}+6 e^{3 x} \\ \therefore \frac{d y}{d x}=6 e^{2 x}+6...

If $f: R \rightarrow R$ defined by $f(x)=x$ , if $x>2 ; f(x)=5 x-2$ if $x \leq 2$ then $f$ is

Maths, Previous Year Question Papers

$$ \begin{array}{l} f(x)=\left\{\begin{array}{cl} x & x>2 \\ 5 x-2 & x \leqslant 2 \end{array}\right. \\ \& f^{\prime}(x)=\left\{\begin{array}{ll} 1 & x>2 \\ 5 & x \leqslant 2 \end{array}\right....

$\text { If }\left(\tan ^{-1} x\right)^{2}+\left(\cot ^{-1} x\right)^{2}=\frac{5 \pi^{2}}{8}, \text { then } x \text { equals to }$

Maths, Previous Year Question Papers

$$ \begin{array}{l} \text { We have, }\left(\tan ^{-1} x\right)^{2}+\left(\cot ^{-1} x\right)^{2}=\frac{5 \pi^{2}}{8} \\ \Rightarrow\left(\tan ^{-1} x+\cot ^{-1} x\right)^{2}-2 \tan ^{-1} x \cdot...

If $A_{3 \times 3}$ and $\operatorname{det} A=6$ , then $\operatorname{det}(2 \operatorname{adj} A):$

Maths, Previous Year Question Papers

$$ \begin{array}{l} \operatorname{det}(K A)=K^{n} \operatorname{det}(A) \\ \operatorname{det}(\operatorname{adj} A)=|A|^{n-1} \end{array} $$ using the above 2 thus $|2 \operatorname{adj}...

Evaluate $\int \frac{3 x^{2}}{x^{6}+1} d x$

Maths, Previous Year Question Papers

$\int \frac{3 \mathrm{x}^{2} \mathrm{dx}}{\mathrm{x}^{6}+1}$ $=\int \frac{3 \mathrm{x}^{2} \mathrm{dx}}{\left(\mathrm{x}^{3}\right)^{2}+1}$ $\mathrm{t}=\mathrm{x}^{3} \Rightarrow \mathrm{dt}=3...

The vertices of a triangle are $A(-1,3), B(1,-1)$ and $C(5,1)$ . Find the length of the median through the vertex $\mathrm{C}$

Maths, Previous Year Question Papers

Coordinates of $\mathrm{A} \equiv(-1,3)$ Coordinates of $\mathrm{B} \equiv(1,-1)$ Coordinates of $\mathrm{C} \equiv(5,1)$ The median through the vertex $\mathrm{C}$ will meet at the mid point of...

Evaluate

Maths, Previous Year Question Papers

$$ \begin{array}{l} \int \frac{2 x^{3}-1}{x^{4}+x} d x \\ \Rightarrow \int \frac{\left(4 x^{3}+1\right)-\left(2 x^{3}+2\right)}{x^{4}+x} d x \\ \Rightarrow \int \frac{4 x^{3}+1}{x^{4}+x} d x-2 \int...

Find $\operatorname{gof}$ and $f o g$ if $f(x)=8 x^{3}$ and $g(x)=x^{\frac{1}{3}}$

Maths, Previous Year Question Papers

Given, $\mathrm{f}(\mathrm{x})=8 \mathrm{x}^{3}$ and $\mathrm{g}(\mathrm{x})=\mathrm{x}^{1 / 3}$ $($ gof $)(x)=g[f(x)]=g\left(8 x^{3}\right)=g(y)$, where $y=8 x^{3}$ $$ =\mathrm{y}^{1 / 3}=\left(8...

Evaluate $\int e^{c}\left(\log x+\frac{1}{x^{2}}\right) d x$ A $\mathrm{en}^{2} \log =-\mathrm{c}$ (B) $\quad e^{x}\left(\log x-\frac{1}{x}\right)+e$ c $\log ^{x}\left(\log x+\frac{1}{x}\right)+c$ D $\frac{e^{x^{2}}}{x^{2}}+c$

Maths, Previous Year Question Papers

Correct ogtion is $$ \text { a } e^{0}\left(\log x-\frac{1}{x}\right) \div e $$ $\int e^{c}\left(\log x+\frac{1}{x^{2}}\right) d x$ $\int e^{\prime} \log x d x+\int e^{\prime} \frac{1}{x^{2}} d x$...

Solve $\tan y d x+\tan x d y=0$

Maths, Previous Year Question Papers

$$ \begin{array}{l} \tan y d x+\tan x y=0 \\ \frac{d x}{\tan x}=-\frac{d y}{\tan y} \end{array} $$ Integrating both side, $$ \begin{array}{l} \int \frac{\mathrm{dx}}{\tan x}=-\int...

If $\tan ^{-1} y=4 \tan ^{-1} x$ , then $\frac{1}{y}$ is zero for

Maths, Previous Year Question Papers

If we put $x=\tan \theta$, the given equality becomes $\tan ^{-1} y=4 \theta$ $$ \begin{array}{l} \Rightarrow y=\tan 4 \theta=\frac{2 \tan 2 \theta}{1-\tan ^{2} 2 \theta}=\frac{2\left[\frac{2 \tan...

Find the equation of the plane which cuts intercepts 2,3 and 4 on the $x, y$ and z-axes respectively.

Maths, Previous Year Question Papers

The equation of plane with intercepts a, b, c is given as $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ $\Rightarrow$ The equation of plane with intercepts $2,3,4$ is...

Find the angle between the vectors $\overrightarrow{\mathrm{a}}=\mathrm{i}+\mathrm{j}-\mathrm{k}$ and $\overrightarrow{\mathrm{b}}=\mathrm{i}-\mathrm{j}+\overrightarrow{\mathrm{k}}$

Maths, Previous Year Question Papers

$$ \begin{array}{l} \overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}} \quad \text { and } \quad...

$\text { Prove that: } 2 \tan ^{-1}\left(\frac{1}{5}\right)+\sec ^{-1}\left(\frac{5 \sqrt{2}}{7}\right)+2 \tan ^{-1}\left(\frac{1}{8}\right)=\frac{\pi}{4}$

Maths, Previous Year Question Papers

$$ \begin{array}{l} 2 \tan ^{-1} \frac{1}{5}+\sec ^{-1} \frac{5 \sqrt{2}}{7}+2 \tan ^{-1} \frac{1}{8} \\ =2\left[\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{8}\right]+\sec ^{-1} \frac{5 \sqrt{2}}{7}...

Evaluate $P(A \cup B)$ if $2 P(A)=P(B)=\frac{5}{13}$ and $P(A / B)=\frac{2}{5}$

Maths, Previous Year Question Papers

$$ \begin{array}{l} \mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{\mathrm{P}(\mathrm{A} / \mathrm{B})}{\mathrm{P}(\mathrm{B})} \\ \begin{array}{l} \frac{2}{5}=\frac{\mathrm{P}(\mathrm{A} /...

Find direction ratio and the direction cosines of the vector $\vec{r}=i+j+\vec{k}$ .

Maths, Previous Year Question Papers

Let $\vec{r}=\hat{i}+\bar{j}+\hat{k}$ $=11 \hat{i}+1 \bar{j}+1 \hat{k}$ Direction ratios are $\mathrm{a}=1, \mathrm{~b}=1, \mathrm{c}=1$ Magnitude of $$...

Evaluate $\int \frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}} d x$ .

Maths, Previous Year Question Papers

$$ \begin{array}{l} \int \frac{\mathrm{x} \sin ^{-1} \mathrm{x}}{\sqrt{1-\mathrm{x}^{2}}} \mathrm{dx} \quad\left[\text { Here } 1^{\mathrm{st}} \mathrm{F} \text { unction }=\sin ^{-1} \mathrm{x}...

Find $\frac{d y}{d x}$ if $y=\sec (\tan \sqrt{x})$ .

Maths, Previous Year Question Papers

$$ y=\sec (\sqrt{\tan x}) $$ applying chain nule $$ \begin{aligned} \frac{d y}{d x} &=(\sec \sqrt{\tan n} \times \tan \sqrt{\tan n}) \times \frac{1}{2}(\tan n)^{1 / 2-1} \times \sec ^{2} n \\...

Solve $\int_{0}^{x / 4} \tan ^{2} x d x$

Maths, Previous Year Question Papers

$\tan ^{2} \mathrm{x}=\sec ^{2} \mathrm{x}-1$ $\Rightarrow \int_{0}^{\pi / 4}-1 \mathrm{dx}+\int_{0}^{\pi / 4}\left(\sec ^{2} \mathrm{x}\right) \mathrm{dx}$ $=-\pi / 4+[\tan \mathrm{x}\}_{0}^{\pi /...

Evaluate $\int \frac{\sin ^{2} x}{1-\cos x} d x$ .

Maths, Previous Year Question Papers

$\int \frac{\sin ^{2} x}{1-\cos x} \cdot d x$ $\sin ^{2} x=1-\cos ^{2} x$ $\int\left(\frac{1-\cos ^{2} x}{1-\cos x}\right) \cdot d x$ $=\int(1+\cos x) d x$ $=\int 1 d x+\int \cos x d x$ $=x+\sin...

Evaluate the determinant:-
$\left|\begin{array}{ccc} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{array}\right|$

Maths, Previous Year Question Papers

Solution:- $$ \left|\begin{array}{ccc} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{array}\right| $$ Expanding the above determinant along $\mathrm{C}_{1}$, we have $$ \begin{array}{l}...

Find the general solution of the differential equation

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\begin{array}{l} \frac{d y}{d x}+\sqrt{\frac{1-y^{2}}{1-x^{2}}}=0

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Maths, Previous Year Question Papers

$$ \begin{array}{l} \frac{d y}{d x}+\sqrt{\frac{1-y^{2}}{1-x^{2}}}=0 \\ \frac{d y}{d x}=\frac{-\sqrt{1-y^{2}}}{1+x^{2}} \\ \Rightarrow \int \frac{d y}{\sqrt{1-y^{2}}}=\int \frac{-d...

Find the rate of change or the area of a circle with respect to its radius $\mathrm{r}$ when $\mathrm{r}=5 \mathrm{~cm}$ .

Maths, Previous Year Question Papers

Area of circle $=\pi r^{2}$ $$ \mathrm{A}=\pi \mathrm{r}^{2} $$ differentiating with respect to $\mathrm{r}$ we get $$ \frac{\mathrm{dA}}{\mathrm{dr}}=2 \pi...

$\int \operatorname{cosec} x d x=?$

Maths, Previous Year Question Papers

$$ \begin{array}{l} \mathrm{I}=\int \csc x \mathrm{dx} \\ \Rightarrow \mathrm{I}=\int \frac{\csc x(\csc x-\cot x)}{(\csc x-\cot x)} \mathrm{dx} \\ \text { Put } \csc x-\cot x=t \\...

If $\mathrm{A}$ is an invertible matrix of order 2 then $\operatorname{det}\left(\mathrm{A}^{-1}\right)$ is equal to
(A) $\operatorname{det}(A)$
(B) $\frac{1}{\operatorname{det}(A)}$
(C) 1
(D) 0

Maths, Previous Year Question Papers

Since A is an invertible matrix, As matrix $A$ is of order 2, let $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ Then, $|A|=a d-b c$ and $a d y=\left[\begin{array}{cc}d & -b \\ -c &...

$\int \sec ^{2}(3 x+5) d x=$ ?
(A) $\frac{1}{3} \tan (3 x+5)+k$
(B) $-\frac{1}{3} \tan (3 x+5)+k$
(C) $\frac{1}{5} \tan (3 x+5)+k$
(D) $-\frac{1}{5} \tan (3 x+5)+k$

Maths, Previous Year Question Papers

Correct option is (B) $\frac{1}{3} \tan (3 x+5)+c$ $$ \begin{array}{l} \int \sec ^{2}(3 x+5) d x \\ \text { let } 3 x+5=t \\ 3 d x=d t \\ d x=1 / 3 d t \\ \int \sec ^{2} t \cdot \frac{1}{3} \cdot d...

The value of $\vec{i} \cdot(\vec{j} \times \vec{k})+\vec{j} \cdot(\vec{i} \times \vec{k})+\vec{k} \cdot(\vec{i} \times \vec{j})$ is
(A) 1
(B) $-1$
(C) 2
(D) 3

Maths, Previous Year Question Papers

The answer is (A) 1 $=i(\hat{\imath})+\hat{\jmath} \cdot(-\hat{\jmath})+\hat{f} \cdot(\hat{k})$ $=1-1+1$ $=1$

Find the direction cosines and direction ratios of the line joining the points $\mathrm{A}(1,3,5), \mathrm{B}(-1,0,-1)$

Maths, Previous Year Question Papers

Given the points A $(1,3,5) \& B(-1,0,-1)$ Direction ratios are $\mathrm{a}=-1-1=-2$ $$ \begin{array}{l} \mathrm{b}=0-3=-3 \\ \mathrm{c}=-1-5=-6 \end{array} $$ Now,...

Let $f(x)=x^{2}$ and $g(x)=2 x+1$ be two real functions. Find $(f+g)(x),(f-$ g)(x),(f g)(x),\left(\frac{f}{g}\right)(x)

Maths, Previous Year Question Papers

$$ (f+g)(x)=f(x)+g(x) $$ $$ =x^{2}+2 x+1 $$ $$ (f-g)(x)=f(x)-g(x) $$ $$ =x^{2}-(2 x+1) $$ $$ =x^{2}-2 x-1 $$ $$ (f g)(x)=f(x) \cdot g(x) $$ $$ =x^{2}(2 x+1) $$ $$ =2 x^{3}+x^{2} $$ $$...

$\int x^{2} \sin x^{3} d x=$ ?
(A) $-\frac{1}{3} \cos x^{3}+k$
(B) $\frac{1}{3} \cos x^{3}+k$
(C) $\frac{1}{3} \sin x^{3}+k$
(D) $-\frac{1}{3} \sin x^{3}+k$

Maths, Previous Year Question Papers

The answer is (A) $-\frac{1}{3} \cos x^{3}+k$. Let $u=x^{3}$ so that $d u=3 x^{2} d x$ Out integral becomes: $$ \begin{array}{c} \frac{1}{3} \int\left(\sin x^{3}\right) 3 x^{2} d x=\frac{1}{3} \int...

If the tangent to the curve, $y=x^{3}+a x-b$ at the point $(1,-5)$ is perpendicular to the line, $-x+y+4=0$ , then which one of the following points lies on the curve? A $\quad(-2,2)$ B $(2,-2)$ C $\quad(2,-1)$ D $\quad(-2,1)$

Maths, Previous Year Question Papers

Correct option is B $(2,-2)$ $y=x^{2}+a x-b$ $(1,-5)$ lies on the curve $\Rightarrow-5=1+\mathrm{a}-\mathrm{b} \Rightarrow \mathrm{a}-\mathrm{b}=-6$.(i) $\mathrm{AlsO}, y^{\prime}=3 x^{2}+a$...

If $P(A)=\frac{1}{2}, P(B)=0$ then $P(A / B)$ is
(A) 0
(B) $\frac{1}{2}$
(C) 1
(D) not defined

Maths, Previous Year Question Papers

Correct option is D Not defined It is given that, $P(A)=\frac{1}{2}$ and $P(B)=0$ $$ P(A \mid B)=\frac{P(A \cap B)}{P(B)}=\frac{P(A \cap B)}{0} $$ Therefore, $P(A \mid B)$ is not defined. Thus, the...

The number of all possible matrices of order $3 \times 3$ with each entry 0 or 1 is.
A. 27
B. 18
C. 81
D. 512

Maths, Previous Year Question Papers

A matrix of order $3 \times 3$ has 9 elements. Now each element can be 0 or $1 .$ $\therefore 9$ places can be filled up in $2^{9}$ ways required number of matrices $=2^{9}$ $$ \begin{array}{l} =2...

$\int \frac{\sin x+\cos x}{\sqrt{1+\sin 2 x}} d x=$ ?
(A) $x+k$
(B) $2 x+k$
(C) $2 x-k$
(D) $3 x+k$

Maths, Previous Year Question Papers

The correct option is (A) $x+k$ $$ \begin{aligned} & \frac{(\sin x+\cos x) \cdot d x}{\sqrt{1+\sin 2 x}} \\ =& \frac{(\sin x+\cos x) \cdot d x}{\sqrt{(\sin x+\cos x)^{2}}} \\ =& \frac{(\sin x+\cos...

The integrating factor of the differential equation $x \frac{\mathrm{d} y}{\mathrm{dx}}+2 y=\mathrm{x}^{2}$ is

Maths, Previous Year Question Papers

Given $x \cdot \frac{d y}{d x}+2 y=x^{2}$ $$ \frac{\mathrm{dy}}{\mathrm{dx}}+\frac{2}{\mathrm{x}} \cdot \mathrm{y}=\mathrm{x} $$ This is in the form of $\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{P}...

$\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})$ is equal to
(A) $-\frac{\pi}{2}$
(B) $\pi$
(C) 0
(D) $2 \sqrt{3}$

Maths, Previous Year Question Papers

The correct answer is (A) $-\frac{\pi}{2}$. $$ \tan ^{-1} \sqrt{3}-\cot (-\sqrt{3}) $$ $\because \tan \frac{\pi}{3}=\sqrt{3}$ and $\cot \left(\pi-\frac{\pi}{6}\right)=-\sqrt{3}$ $$ \tan ^{-1}...

Find the minimum value of $\mathrm{Z}=3 \mathrm{x}+5 \mathrm{y}$ , subject to the constraints $-2 \mathrm{x}+\mathrm{y} \leq$ $4, \mathrm{x}+\mathrm{y} \geq 3, \mathrm{x}-2 \mathrm{y} \leq 2, \mathrm{x} \geq 0$ and $\mathrm{y} \geq 0$ $Z=3 x+5 y$ , subject to the constraints $-2 x+y \leq 4, x+y \geq 3, x-2 y \leq 2, x \geq 0$ and $y \geq 0$

Maths, Previous Year Question Papers

$Z=3 x+5 y$, subject to the constraints $-2 x+y \leq 4, x+y \geq 3, x-2 y \leq 2, x \geq 0$ and $y \geq 0$ Draw the line $-2 x+y=4, x+y=3$ and $x-2 y=2$ and shaded region which is satisfied by above...

$\int x^{2} e^{x} d x=$ ?
(A) $\frac{e^{x^{3}}}{3}+k$
(B) $\frac{1}{3} e^{x^{2}}+k$
(C) $\frac{e^{x^{3}}}{2}+k$
(D) $\frac{1}{2} e^{x^{2}}+k$

Maths, Previous Year Question Papers

Integrating by parts, $$ \begin{array}{l} \mathrm{u}=\mathrm{x}^{2}, \mathrm{v}=\mathrm{e}^{\mathrm{x}} \\ \mathrm{u}^{\prime}=2 \mathrm{x}, \int \mathrm{vdx}=\mathrm{e}^{\mathrm{x}} \\...

$\frac{d}{d x}(\sec x)=$ ?
(A) $\sec x \cot x$
(B) $\sec x \tan x$
(C) $\tan x$
(D) $\cot x$

Maths, Previous Year Question Papers

Answer is (B) $\sec x \tan x$ $$ \begin{array}{l} \frac{\mathrm{d}}{\mathrm{dx}}(\sec \mathrm{x})=\frac{\mathrm{d}}{\mathrm{dx}}\left\{\frac{1}{\cos \mathrm{x}}\right\} \\ =\frac{\cos \mathrm{x}...

$\int d x=$ ?
(A) $x+k$
(B) $1+\mathrm{k}$
(C) $0+\mathrm{k}$
(D) $-x+k$

Maths, Previous Year Question Papers

Correct option is (A) $x+k$ Explanation: $$ \int d x=\int 1 d x=\int x^{0} d x $$ Using the standard power rule of integration, $$ \int x^{0} d x=\frac{x^{0+1}}{0+1}+C=x+k $$

$\int \frac{\sec ^{2}(\log x)}{x} d x=$ ?
(A) $\tan (\log x)+k$
(B) $-\tan (\log x)+k$
(C) $\cot (\log x)+k$
(D) $-\cot (\log x)+k$

Maths, Previous Year Question Papers

Correct option is (A) $\tan (\log x)+k$ Let $I=\int \frac{\sec ^{2}(\log x)}{x} d x$ Put $\log x=t \Rightarrow \frac{1}{x} d x=d t$ $$ \therefore \int \sec ^{2} t d t=\tan t+c=\tan (\log x)+c $$

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P, then the determinant $\left|\begin{array}{lll}\mathrm{x}+2 & \mathrm{x}+3 & \mathrm{x}+2 \mathrm{a} \\ \mathrm{x}+3 & \mathrm{x}+4 & \mathrm{x}+2 \mathrm{~b} \\ \mathrm{x}+4 & \mathrm{x}+5 & \mathrm{x}+2 \mathrm{c}\end{array}\right|$ is
A 0
B 1
C $\mathrm{x}$
D $2 \mathrm{x}$

Maths, Previous Year Question Papers

Correct option is (A)0 Given $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P. $$ \Rightarrow 2 \mathrm{~b}=\mathrm{a}+\mathrm{c} $$ $$ \left|\begin{array}{lll} \mathrm{x}+2 & \mathrm{x}+3 &...

$\int \tan x d x=$ ?
(A) $\log |\sec x|+k$
(B) $\log |\cos x|+k$
(C) $\log |\sin x|+k$
(D) $\log |\operatorname{cosec} x|+k$

Maths, Previous Year Question Papers

Correct option is A $\int \tan x=\int(\sin x / \cos x) \cdot d x$ This can be rewritten as $\int 1 / \cos x \cdot \sin x \cdot d x$ Let us find the indefinite integral of $\tan x$ using the...

$\cot \left(\tan ^{-1} x+\cot ^{-1} x\right)=$ ?
(A) 0
(B) 1
(C) $\frac{1}{2}$
(D) $\frac{\pi}{4}$

Maths, Previous Year Question Papers

Correct option is (A) 0 Sol:- The given relation should be $$ \begin{array}{l} \cot \left(\tan ^{-1} x+\cot ^{-1} x\right) \\ \text { we know } \tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2} \\ \therefore...

The height of $T V$ tower at a certain place is $245 \mathrm{~m}$ . The maximum distance up to which its programme can be received is (a) $245 \mathrm{~m}$ (b) 245km (c) $\mathbf{5 6} \mathbf{~ k m}$ (iv) $112 \mathrm{~km}$

Maths, Previous Year Question Papers

Answer: Option (c)$\mathbf{5 6} \mathbf{~ k m}$ Answer: Option (b)245km

The Boolean expression for NAND gate is (a) $Y=A+B$ (b) $Y=A \cdot B$ (c) $Y=\overrightarrow{A \cdot B}$

Maths, Previous Year Question Papers

Answer: Option (d)

The value of angle of dip at the earth’s magnetic pole is (a) $0^{\circ}$ (b) $45^{\circ}$ (c) $90^{\circ}$ (d) $180^{\circ}$

Maths, Previous Year Question Papers

Answer: Option (c)$90^{\circ}$

The stored energy, of a capacitor charged to $100 \mathrm{~V}$ , is 1 J. capacitor is (a) $2 \times 10^{4} \mathrm{~F}$ (b) $2 \times 10^{4} \mathrm{~F}$ (c) $2 \times 10^{2} \mathrm{~F}$ (d) $1.6 \times 10^{19} \mathrm{~F}$

Maths, Previous Year Question Papers

Answer: Option (b) $2 \times 10^{4} \mathrm{~F}$

The current flowing; in a wire is 1 A. if the charge of an electron is $1.6 \times 10^{-19} C$ , then the number of electrons flowing through the wire per second is (a) $0.625 \times 10^{13}$ (b) $6.25 \times 10^{18}$ (c) $1.6 \times 10^{-19}$ (d) $1.6 \times 10^{19}$

Maths, Previous Year Question Papers

Answer: Option (b)$6.25 \times 10^{18}$

The energy of a photon of wavelength $\lambda$ is (a) $h c \lambda$ (b) $\frac{h c}{\lambda}$ (c) $\frac{h c}{c}$ (d) $\frac{\lambda}{h c}$

Maths, Previous Year Question Papers

Answer: Option (b)$\frac{h c}{\lambda}$

The ratio of root mean (rms) value and peak value of alternative current is (a) $\sqrt{2}$ (b) $\frac{1}{\sqrt{2}}$ (c) $\frac{1}{2}$ (d) $2 \sqrt{2}$

Maths, Previous Year Question Papers

Answer: Option (b)$\frac{1}{\sqrt{2}}$

At constant potential difference, the resistance of any electric circuit is halved, the value of heat produced will be (a) Half (b) Double (c) four (d) Same

Maths, Previous Year Question Papers

Answer: Option (b) Double

In an alternating current circuit, the phase difference between curren I and voltage $\phi$ , then the Wattles component of current will be (a) $I \cos \phi$ (b) Double (c) four (d) Same

Maths, Previous Year Question Papers

Answer: Option (c)four

The working of dynamo is based on the principle of (a) Heating effect of current (b) Electro-magnetic induction (c) Induced magnetism (d) Induced electricity

Maths, Previous Year Question Papers

Answer: Option (b)Electro-magnetic induction

The magnifying power of an astronomical telescope for normal adjustment is (a) $-\frac{f_{0}}{f_{e}}$ (b) $-f_{0} \times f_{e}$ (c) $-\frac{f_{e}}{f_{0}}$ (d) $-f_{0}+f_{e}$

Maths, Previous Year Question Papers

Answer: Option (c)$-\frac{f_{e}}{f_{0}}$

The resistance of ideal ammeter is (a) Zero (b) Very small (c) Very large (d) Infinite

Maths, Previous Year Question Papers

Answer: Option (a)Zero

The direction of propagation of electromagnetic wave is (a) Parallel to $\vec{B}$ (b) Parallel to $\vec{E}$ (c) Parallel to $\vec{B} \times \vec{E}$ (d) parallel to $\vec{E} \times \vec{B}$

Maths, Previous Year Question Papers

Answer: Option (d) parallel to $\vec{E} \times \vec{B}$

A convex lens $(\mathrm{n}=1.5)$ is immersed in water $(\mathrm{n}=1.33)$ , then it will behave as a (a) Converging lens (b) Diverging lens (c) Prism (d) Concave mirror

Maths, Previous Year Question Papers

Answer: Option (b)Diverging lens

The power factor of $\mathrm{L}-\mathrm{R}$ circuit is (a) R+WL (b) $\frac{R}{\sqrt{R^{2}+W^{2} L^{2}}}$ (c) $R \sqrt{R^{2}+W^{2}+L^{2}}$ (d) $\frac{W L}{R}$

Maths, Previous Year Question Papers

Answer: Option (b) $\frac{R}{\sqrt{R^{2}+W^{2} L^{2}}}$

The electric potential in equatorial position of an electric dipole is (a) $\frac{1}{4 \pi \varepsilon_{0}} \frac{p \cos \theta}{r^{2}}$ (b) $\frac{1}{4 \pi \varepsilon_{0}} \frac{p}{r^{2}}$ (c) $\frac{1}{4 \pi \varepsilon_{0}} \frac{p}{r}$ (d) Zero

Maths, Previous Year Question Papers

Answer: Option (d)Zero

With the increase on temperature the resistivity of semiconductor (a) Increases (b) Decreases (c) Remains constant (d) Becomes zero

Maths, Previous Year Question Papers

Answer: Option (b) Decreases

In a step-up transformer, no, of turns in primary and secondary coils are $\mathrm{N}_{1}$ and $\mathrm{N}_{2}$ then (a) $N_{1}>N_{2}$ (b) $N_{2}>N_{1}$ (c) $N_{1}=N_{2}$ (d) $N_{1}=0$

Maths, Previous Year Question Papers

Answer: Option (b) $N_{2}>N_{1}$

When two bulbs of power $60 \mathrm{~W}$ and $40 \mathrm{~W}$ are connected in series, then the power of their combination will be (a) $100 \mathrm{~W}$ (b) $2400 \mathrm{~W}$ (c) $30 \mathbf{W}$ (d) $24 \mathrm{~W}$

Maths, Previous Year Question Papers

Answer: Option (a)$100 \mathrm{~W}$

The wave front due to a point source at a finite distance from the source is (a) Spherical (b) Cylindrical (c) Plane (d) Circular

Maths, Previous Year Question Papers

Answer: Option (a)Spherical

Light owes its colour to its (a) Frequency (b) Velocity (c) Phase (d) Amplitude

Maths, Previous Year Question Papers

Answer: Option (b)Velocity

In earth’s magnetic field $\mathrm{B}_{\mathrm{H}}$ , if the frequency of oscillation of a magnetic needle is $\mathrm{n}$ , then (a) $n \infty B_{H}$ (b) $n^{2} \propto B_{H}$ (c) $n \infty B_{H^{2}}$ (d) $n^{2} \infty \frac{1}{B_{H}}$

Maths, Previous Year Question Papers

Answer: Option (b)$n^{2} \propto B_{H}$

If $+\mathrm{q}$ charge is placed inside any spherical surface then total flux coming out from whole surface will be (a) $q+\varepsilon_{0}$ (b) $\frac{q}{\varepsilon_{0}}$ (c) $\frac{\varepsilon_{0}}{q}$ (d) $\frac{q^{2}}{\varepsilon_{0}}$

Maths, Previous Year Question Papers

Answer: Option (b) $\frac{q}{\varepsilon_{0}}$

Cylindrical lenses are used to correct the eye defect called (a) Myopia (b) Hypermetropia (c) Astigmatism (d) Presbyopia

Maths, Previous Year Question Papers

Answer: Option (c)Astigmatism

Equivalent focal length of two lenses in contact having power -15D and +5D will be (a) $-20 \mathrm{~cm}$ (b) $-10 \mathrm{~cm}$ (c) $+10 \mathrm{~cm}$ (d) $+20 \mathrm{~cm}$

Maths, Previous Year Question Papers

Answer: Option (b)$-10 \mathrm{~cm}$

Power of an electric $-$ circuit is (a) V.R (b) $V^{2} \cdot R$ (C) $\frac{V^{2}}{R}$ (d) $V^{2} R t$

Maths, Previous Year Question Papers

Answer: Option (c)$\frac{V^{2}}{R}$

The solar spectrum is (a) Continuous (b) Line spectrum (C) Spectrum of blank lines (d) Spectrum of blank bands

Maths, Previous Year Question Papers

Answer: Option (c)Spectrum of blank lines

Two point charges of $+10 \mu c$ and $-10 \mu c$ are placed at a distance $\mathbf{4 0} \mathbf{c m}$ in air. Potential energy of the system will be (a) $2.25 \mathrm{~J}$ (b) $2.35 \mathrm{~J}$ (c) $-2.25 \mathrm{~J}$ (d) $-2.35 \mathrm{~J}$

Maths, Previous Year Question Papers

Answer: Option (c) $-2.25 \mathrm{~J}$

The time in which radioactive substance becomes half of its initial amount is called (a) Average life (b) Half-life (c) Time-period (d) Decay constant

Maths, Previous Year Question Papers

Answer: Option (b)Half-life

Deviation of a thin prism of refractive index $\mathbf{n}$ and angle of $\mathbf{p r i s m} \mathbf{A}$ is (a) $(1-n) A$ (b) $(n-1) A$ (c) $(n+1) A$ (d) $(1+n) A^{2}$

Maths, Previous Year Question Papers

Answer: Option (b)$(n-1) A$

Unit of surface charge density is (a) Coulomb/metre $^{2}$ (b) newton/metre (c) Coulomb/volt (d) Coulmb/metre

Maths, Previous Year Question Papers

Answer: Option (a)Coulomb/metre $^{2}$

The susceptibility of paramagnetic substance is (a) Constant (b) Zero (b) Infinity (d) Depends on magnetic field

Maths, Previous Year Question Papers

Answer: Option (a) Constant

The torque $(\vec{\tau})$ experienced by a current-loop of magnetic moment $(\vec{M})$ placed in magnetic field $\vec{B}$ is (a) $\vec{\tau}=\vec{M} \times \vec{B}$ (b) $\vec{\tau}=\vec{B} \times \vec{M}$ (C) $\vec{\tau}=\frac{\vec{M}}{\vec{B}}$ (d) $\vec{\tau}=\vec{M} \vec{B}$

Maths, Previous Year Question Papers

Answer: Option (a)$\vec{\tau}=\vec{M} \times \vec{B}$

Let $\mathrm{A}$ be a non-singular matrix of the order $2 \times 2$ then $|a d j A|=\ldots \ldots \ldots \ldots$ (a) $2|A|$ (b) $|A|$ (c) $|A|^{2}$ (d) $|A|^{3}$

Maths, Previous Year Question Papers

SOL: Correct option is C.$|A|^{2}$

solution of $\lim _{n \rightarrow \infty}\left[\frac{e^{\frac{1}{n}}+e^{\frac{2}{n}}+e^{\frac{3}{n}}+\ldots \ldots . .+e^{\frac{n}{n}}}{n}\right]$ is (a) $1-e$ (b) $e-1$ (c) $e$ (d) 1

Maths, Previous Year Question Papers

SOL: Correct option is C.

$\int \frac{d x}{x+\sqrt{x}}$ (a) $\log x+\log (1+\sqrt{x})+C$ (b) $2 \log (1+\sqrt{x})+C$ (c) $\log (1+\sqrt{x})+C$ (d) $\log (\sqrt{x})+C$

Maths, Previous Year Question Papers

SOL: Correct option is B. $2 \log (1+\sqrt{x})+C$

If $\mathrm{A}$ and $\mathrm{B}$ are two events such that $P(A) \neq 0$ and $P\left(\frac{B}{A}\right)=1$ (a) $B \subset A$ (b) $A \subset B$ (c) $B=\varphi$ (d) $A \cap B=\varphi$

Maths, Previous Year Question Papers

SOL: Correct option is B. $A \subset B$

If $P(A)=\frac{3}{8}, P(B)=\frac{1}{2}$ and $P(A \cap B)=\frac{1}{4}$ then $P\left(\frac{A}{B}\right)=$ (a) 2 (b) $\frac{1}{2}$ (c) $\frac{2}{3}$ (d) $\frac{3}{2}$

Maths, Previous Year Question Papers

SOL: Correct option is (c) $\frac{2}{3}$

If events $\mathrm{A}$ and $\mathrm{B}$ are mutually exclusive then: (a) $P(A \cap B)=P(A) \cdot P(B)$ (b) $P(A \cap B)=0$ (c) $P(A \cap B)=1$ (d) $P(A \cup B)=0$

Maths, Previous Year Question Papers

SOL: Correct option is B. $P(A \cap B)=0$

If A’ and B’ are independent events then: (a) $P\left(A^{\prime} B^{\prime}\right)=P(A) \cdot P(B)$ (b) $P\left(A^{\prime} B^{\prime}\right)=P\left(A^{\prime}\right)+P\left(B^{\prime}\right)$ 12 (c) $P\left(A^{\prime} B^{\prime}\right)=P\left(A^{\prime}\right) \cdot P\left(B^{\prime}\right)$ (d) $P\left(A^{\prime} B^{\prime}\right)=P\left(A^{\prime}\right)-P\left(B^{\prime}\right)$

Maths, Previous Year Question Papers

SOL: Correct option is C. $P\left(A^{\prime} B^{\prime}\right)=P\left(A^{\prime}\right) \cdot P\left(B^{\prime}\right)$

The distance between $(4,3,7)$ and $(1,-1,-5)$ is: (a) 13 (b) 15 (c) 12 (d) 5

Maths, Previous Year Question Papers

SOL: Correct option is A. 13

If two planes $-2 x-4 y+3 z=5$ and $x+2 y+\lambda z=12$ are perpendicular to each other, then $\lambda=$ (a) $-2$ (b) 2 (c) 3 (d) None of these

Maths, Previous Year Question Papers

SOL: Correct option is B.2

The equation of the $x y$ -plane is: (a) $x=0, y=0$ (b) $z=0$ (c) $x=0, y \neq 0$ (d) None of these

Maths, Previous Year Question Papers

SOL: Correct option is B. $z=0$

The direction Cosines of $y$ axis are: (a) $(1,0,1)$ (b) $(0,1,0)$ (c) $\left(\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}\right)$ (d) None of these

Maths, Previous Year Question Papers

SOL: Correct option is B. $(0,1,0)$

\vec{i} \times(\vec{i} \times \vec{j})+\vec{j} \times(\vec{j} \times \vec{k})+\vec{k}(\vec{k} \times \vec{i}) $(a)$ \vec{i}+\vec{j}+\vec{k} $(b) 0 (c) 1 (d)$ -(\vec{i}+\vec{j}+\vec{k})$

Maths, Previous Year Question Papers

SOL: Correct option is C. 1

\vec{i} \times(\vec{i} \times \vec{j})+\vec{j} \times(\vec{j} \times \vec{k})+\vec{k}(\vec{k} \times \vec{i}) $(a)$ \vec{i}+\vec{j}+\vec{k} $(b) 0 (c) 1 (d)$ -(\vec{i}+\vec{j}+\vec{k})$

Maths, Previous Year Question Papers

SOL: Correct option is C. 1

$x \vec{i}-3 \vec{j}+5 \vec{k},-x \vec{i}+x \vec{j}+2 \vec{k}$ are perpendicular to each other then value of $\mathrm{x}=$ (a) $-2,5$ (b) 2,5 (c) $-2,-5$ (d) $2,-5$

Maths, Previous Year Question Papers

SOL: Correct option is D. $2,-5$

The direction cosines of the vector $3 \vec{i}-4 \vec{j}+12 \vec{k}$ is (a) $\frac{3}{13}, \frac{4}{13}, \frac{12}{13}$ (b) $\frac{3}{13}, \frac{-4}{13}, \frac{12}{13}$ (c) $\frac{3}{\sqrt{13}}, \frac{4}{\sqrt{13}}, \frac{12}{\sqrt{13}}$ (d) $\frac{3}{\sqrt{13}}, \frac{-4}{\sqrt{13}}, \frac{12}{\sqrt{13}}$

Maths, Previous Year Question Papers

SOL: Correct option is B. $\frac{3}{13}, \frac{-4}{13}, \frac{12}{13}$

The direction cosines of the vector $3 \vec{i}-4 \vec{j}+12 \vec{k}$ is (a) $\frac{3}{13}, \frac{4}{13}, \frac{12}{13}$ (b) $\frac{3}{13}, \frac{-4}{13}, \frac{12}{13}$ (c) $\frac{3}{\sqrt{13}}, \frac{4}{\sqrt{13}}, \frac{12}{\sqrt{13}}$ (d) $\frac{3}{\sqrt{13}}, \frac{-4}{\sqrt{13}}, \frac{12}{\sqrt{13}}$

Maths, Previous Year Question Papers

SOL: Correct option is B. $\frac{3}{13}, \frac{-4}{13}, \frac{12}{13}$

If $\vec{a}=\vec{i}+\vec{j}+2 \vec{k}$ , then the corresponding unit vector $\hat{a}$ in the direction of $\vec{a}=$ (a) $\frac{\vec{i}+\vec{j}+\vec{k}}{\sqrt{6}}$ (b) $\frac{\vec{i}+\vec{j}+2 \vec{k}}{\sqrt{6}}$ (c) $\frac{\vec{i}+\vec{j}+2 \vec{k}}{6}$ (d) None of these

Maths, Previous Year Question Papers

SOL: Correct option is B. $\frac{\vec{i}+\vec{j}+2 \vec{k}}{\sqrt{6}}$

solution of $\frac{x d y-y d x}{x^{2}+y^{2}}=0$ is: (a) $\frac{x^{2}}{2}+\tan ^{-1} \frac{x}{y}=k$ (b) $\frac{x^{2}}{2}+\tan ^{-1} \frac{y}{x}=k$ (c) $\frac{x^{2}}{2}-\tan ^{-1} \frac{x}{y}=k$ (d) $\frac{x^{2}}{2}-\tan ^{-1} \frac{y}{x}=k$

Maths, Previous Year Question Papers

SOL: 10 Correct option is B. $\frac{x^{2}}{2}+\tan ^{-1} \frac{y}{x}=k$

Integrating factor (I.F) of differential equation $\frac{d y}{d x}+\frac{y}{x}=\frac{y^{2}}{x^{2}}$ is (a) $\log x$ (b) $^{x}$ (c) $\frac{1}{x}$ (d) None of these

Maths, Previous Year Question Papers

SOL: Correct option is B. $^{x}$

solution of the differential equation $y d x-x d y=x y d x$ is (a) $\frac{y^{2}}{2}-\frac{x^{2}}{2}=x y+c$ (b) $x=k y e^{x}$ (c) $x=k y e^{y}$ (d) None of these

Maths, Previous Year Question Papers

SOL: Correct option is B. $x=k y e^{x}$

The differential equation $1+\left(\frac{d y}{d x}\right)^{2}=\left(\frac{d^{2} y}{d x^{2}}\right)^{3}$ is of order $=\ldots \ldots \ldots \ldots$ and and degree $=\ldots \ldots \ldots \ldots$ (a) Order $=2$ , degree $=3$ (b) Order $=$ , degree $=2$ (c) Order $=2$ , degree $=2$ (d) None of these

Maths, Previous Year Question Papers

SOL: Correct option is A. Order $=2$, degree $=3$

$\int_{0}^{1} x d x=\ldots \ldots \ldots$ (a) 0 (b) 1 (c) 2 (d) $\frac{1}{2}$

Maths, Previous Year Question Papers

SOL: Correct option is D. $\frac{1}{2}$

Area between the $\mathrm{x}$ -axis and the curve $y=\sin x$ from $x=0$ to $x=\frac{\pi}{2}$ : (a)2 (b) $-1$ (c) 1 (d) None of these

Maths, Previous Year Question Papers

SOL: Correct option is (c) 1

$\int_{\alpha}^{\beta} \varphi(x) d x+\int_{\beta}^{\alpha} \varphi(x) d x=$ (a) 1 (b) $2 \int_{\alpha}^{\beta} \varphi(x) d x$ (c) $-2 \int_{\beta}^{\alpha} \varphi(x) d x$ (d) 0

Maths, Previous Year Question Papers

SOL: Correct option is D. 0

If $f(-x)=-f(x)$ then $\int_{-a}^{a} f(x) d x=$ (a) $2 \int_{0}^{a} f(x) d x$ (b) 0 (c) 1 (d) $-1$

Maths, Previous Year Question Papers

SOL: Correct option is (b) 0

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

Maths, Previous Year Question Papers

SOL: Correct option is (c) $\frac{1}{a} \tan ^{-1} \frac{x}{a}+c$

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

Maths, Previous Year Question Papers

SOL: Correct option is (c) $\frac{e^{x}}{x+1}+c$

$\int x^{2} \cdot e^{x^{3}} d x=$ (a) $e^{x^{3}}+c$ (b) $\frac{1}{3} e^{x^{3}}+c$ (c) $e^{x^{2}}+c$ (d) $\frac{1}{3} e^{x^{2}}+c$

Maths, Previous Year Question Papers

SOL: Correct option is B. $\frac{1}{3} e^{x^{3}}+c$

\cdot \int \sqrt{1+\cos 2 x} d x= $(a)$ \sqrt{2} \cos x+c $(b)$ \sqrt{2} \sin x+c $(c)$ -\cos x-\sin x+c $(d)$ \sqrt{2} \sin \frac{x}{2}+c$

Maths, Previous Year Question Papers

SOL: Correct option is B. $\sqrt{2} \sin x+c$

$\frac{d}{d x}\left[\lim _{x \rightarrow a} \frac{x^{5}-a^{5}}{x-a}\right]=$ (a) $5 a^{4}$ (b) $5 x^{4}$ (c) 1 (d) 0

Maths, Previous Year Question Papers

SOL: Correct option is D. 0

Equation of the tangent to the curve $x^{2}+y^{2}=a^{2}$ at $\left(x_{1}, y_{1}\right)$ is (a) $x x_{1}-y y_{1}=0$ (b) $x x_{1}+y y_{1}=0$ (c) $x x_{1}-y y_{1}=a^{2}$ (d) $x x_{1}+y y_{1}=a^{2}$

Maths, Previous Year Question Papers

SOL: Correct option is D. $x x_{1}+y y_{1}=a^{2}$

If $y=\log \cos x^{2}$ then $\frac{d y}{d x}$ at $x=\sqrt{x}$ has the value (a) 1 (b) $\frac{\pi}{4}$ (c) 0 (d) $\sqrt{\pi}$

Maths, Previous Year Question Papers

SOL; Correct option is (c) 0

$f(x)=\sqrt{3} \sin x+\cos x$ is maximum then value of $\mathrm{x}=\ldots \ldots \ldots \ldots \ldots$ (a) $\frac{\pi}{6}$ (b) $\frac{\pi}{2}$ (c) $\frac{\pi}{3}$ (d) $\frac{\pi}{4}$

Maths, Previous Year Question Papers

SOL: Correct option is C. $\frac{\pi}{3}$

$f(x)=\sqrt{3} \sin x+\cos x$ is maximum then value of $\mathrm{x}=\ldots \ldots \ldots \ldots \ldots$ (a) $\frac{\pi}{6}$ (b) $\frac{\pi}{2}$ (c) $\frac{\pi}{3}$ (d) $\frac{\pi}{4}$

Maths, Previous Year Question Papers

SOL: Correct option is C. $\frac{\pi}{3}$

\cdot \frac{d\left(2^{x}\right)}{d\left(3^{x}\right)}= $(a)$ \left(\frac{2}{3}\right)^{x} $(b)$ \frac{2^{x-1}}{3^{x-1}} $(c)$ \left(\frac{2}{3}\right)^{x} \log _{3} 2 $(d)$ \left(\frac{2}{3}\right)^{x} \log _{2} 3$

Maths, Previous Year Question Papers

\cdot \frac{d\left(2^{x}\right)}{d\left(3^{x}\right)}=$ (a) $\left(\frac{2}{3}\right)^{x}$ (b) $\frac{2^{x-1}}{3^{x-1}}$ (c) $\left(\frac{2}{3}\right)^{x} \log _{3} 2$ (d)...

$\frac{d}{d x}\left[\tan ^{-1} \sqrt{1+x^{2}}-\cot ^{-1}\left(-\sqrt{1+x^{2}}\right)\right]=$ (a) $\pi$ (b) 1 (c) 0 (d) $\frac{2 x}{\sqrt{1+x^{2}}}$

Maths, Previous Year Question Papers

SOL: Correct option is (c) 0

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

Maths, Previous Year Question Papers

Correct option is $B.$\frac{y}{x}$

$\frac{d}{d x}[\log (\sec x+\tan x)]=$ (a) $\frac{1}{\sec x+\tan x}$ (b) $\sec x$ (c) $\tan x$ (d) $\sec x+\tan x$

Maths, Previous Year Question Papers

SOL: Correct option is $A. $\frac{1}{\sec x+\tan x}$

Let $\mathrm{A}$ be a non-singular matrix of the order $2 \times 2$ then $|a d j A|=\ldots \ldots \ldots \ldots$ (a) $2|A|$ (b) $|A|$ (c) $|A|^{2}$ (d) $|A|^{3}$

Maths, Previous Year Question Papers

SOL: Correct option is C.$|A|^{2}$

If $\mathrm{A}$ is a matrix of order $A+A^{\prime}$ such that $A+A^{\prime}$ then $A+A^{\prime}$ is equal to? (a) $I_{3}$ (b) $\mathrm{A}$ (c) $3 \mathrm{~A}$ (d) $I_{3}-A$

Maths, Previous Year Question Papers

SOL: Correct option is A. $I_{3}$

If $\mathrm{A}$ be a square matrix. Then $A+A$ ‘ will be………… (a)Symmetric matrix (b) Skew symmetric matrix (c) Null matrix (d) Unit matrix

Maths, Previous Year Question Papers

SOL: Correct option is (a)Symmetric matrix

If $\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$ and $A+A^{\prime}=I_{2}$ then $\alpha=$ (a) $\pi$ (b) $\frac{\pi}{3}$ (c) $\frac{3 \pi}{2}$ (d) $\frac{\pi}{6}$

Maths, Previous Year Question Papers

SOL: Correct option is B. $\frac{\pi}{3}$

If $\omega \neq 1, \omega^{3}=1$ and $\left|\begin{array}{ccc}x+1 & \omega & \omega^{2} \\ \omega & x+\omega^{2} & 1 \\ \omega^{2} & 1 & x+\omega\end{array}\right|=0$ then $\mathrm{x}=$ (a) 1 (b) $\omega$ (c) $\omega^{2}$ (d) 0

Maths, Previous Year Question Papers

SOL; Correct option is D. 0

If 7 and 2 are two roots of the equation $\left|\begin{array}{lll}x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x\end{array}\right|=0$ then the third root is: (a) $-9$ (b) 14 (c) $\frac{1}{2}$ (d) None of these

Maths, Previous Year Question Papers

SOL: Correct option is A. $-9$

If $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ aren in A.P then $\left|\begin{array}{ccc}x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c\end{array}\right|=$ a) 3 (b) $-3$ (c) 0 (d) 1

Maths, Previous Year Question Papers

Correct option is C.0

If $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ aren in A.P then $\left|\begin{array}{ccc}x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c\end{array}\right|=$ a) 3 (b) $-3$ (c) 0 (d) 1

Maths, Previous Year Question Papers

If $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ aren in A.P then $\left|\begin{array}{ccc}x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c\end{array}\right|=$ a) 3 (b) $-3$...

.If $\lambda \in R$ and $\Lambda=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|$ then $\lambda \Lambda=$ (a) $\left|\begin{array}{ll}\lambda a & \lambda b \\ \lambda c & \lambda d\end{array}\right|$ (b) $\left|\begin{array}{ll}\lambda a & b \\ c & d\end{array}\right|$ (c) $\left|\begin{array}{ll}\lambda a & b \\ \lambda c & d\end{array}\right|$ , (d) None of these

Maths, Previous Year Question Papers

SOL: Correct option is A. $\left|\begin{array}{ll}\lambda a & \lambda b \\ \lambda c & \lambda d\end{array}\right|$

$\tan ^{-1}(1)+\cos ^{-1}\left(\frac{-1}{2}\right)+\sin ^{-1}\left(\frac{-1}{2}\right)=$ (a) $\frac{\pi}{4}$ (b) $\frac{3 \pi}{4}$ (c) $-\frac{\pi}{4}$ (d) $\frac{\pi}{2}$

Maths, Previous Year Question Papers

SOL: Correct option is B. $\frac{3 \pi}{4}$

$\cos ^{-1}(2 x-1)=$ (a) $2 \cos ^{-1} x$ (b) $\cos ^{-1} \sqrt{x}$ (c) $2 \cos ^{-1} \sqrt{x}$ (d) None of these

Maths, Previous Year Question Papers

SOL: Correct option is D. None of these

$\cos ^{-1}\left(\cos \frac{8 \pi}{5}\right)=$ (a) $\frac{8 \pi}{5}$ (b) $\frac{12 \pi}{5}$ (c) $\frac{2 \pi}{5}$ (d) $\frac{4 \pi}{5}$

Maths, Previous Year Question Papers

SOL: Correct option is D. $\frac{4 \pi}{5}$

What type of a relation is “Less then” in the set of real numbers ? (a)Only symmetric (b) Only transitive (c)Only reflexive (d) equivalence relation

Maths, Previous Year Question Papers

SOL: Correct option is B. Only transitive

If $A=\{1,2,3\} B=\{6,7,8\}$ and $f: A \rightarrow B$ is a function such that $f(x)=x+5$ then what type of a function is $\mathrm{f}$ ?(a)into (b) one-one onto (c)many-one onto (d)Constant function

Maths, Previous Year Question Papers

SOL: Correct option is B. one-one onto

let $\{1,2,3\}$ which of the following function $f: A \rightarrow A$ does not an inverse function ? 1 (A) $\{(1,1),(2,2),(3,3)\}$ (b) $\{(1,2),(2,1),(3,1)\}$ (c) $\{(1,3),(3,2),(2,1)\}$ (d) $\{(1,2),(2,3),(3,1)\}$

Maths, Previous Year Question Papers

Sol: Correct option is B. $\{(1,2),(2,1),(3,1)\}$

let $\vec{a}=\vec{i}+\vec{j}+2 \vec{k}$ How many binary operation can be defined on this set? (a) 8 (b) 10 (c) $-16$ (d) 20

Maths, Previous Year Question Papers

Sol: Correct option is C$-16$

If $P(A)=\frac{3}{8}: P(B)=\frac{1}{2}$ and $P(A \cap B)=\frac{1}{4}$ then $P(A \cup B)=\ldots \ldots \ldots$ (a) 0 (b) $\frac{5}{8}$ (c) 1 (d) 4

Maths, Previous Year Question Papers

Sol: Correct option is B.$\frac{5}{8}$

If $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are three event independent of each other then $P(A \cap B \cap C)=$ (a) $P(A)+P(B)+P(C)$ (b) $P(A)-P(B)+P(C)$ (c) $P(A)+P(B)-P(A \cap B)$ (d) $P(A) P(B) P(C)$

Maths, Previous Year Question Papers

Sol: Correct option is D.$P(A) P(B) P(C)$

If S be the sample space and E be the event then $P(E)=.$ (a) $\frac{n(E)}{n(S)}$ (b) $\frac{n(S)}{n(E)}$ (c) $n(E)$ (d) $n(S)$

Maths, Previous Year Question Papers

Sol: Correct option is A.$\frac{n(E)}{n(S)}$

If $\mathrm{A}$ and $\mathrm{B}$ are two independent event then $P(A \cap B)=$ (a) $P(A) \cdot P(B)$ (b) $P\left(\frac{A}{B}\right)$ (c) $P(A)+P(B)$ (d) $P(A)+P(B)-P(A \cap B)$

Maths, Previous Year Question Papers

Sol: Correct option is A.$P(A) \cdot P(B)$

The direction ratio of the normal to the plane $7 x+4 y-2 z+5=0$ are (a) $7,4,-2$ (b) $7,4,5$ (c) $7,4,2$ (d) $4,-2,5$

Maths, Previous Year Question Papers

Sol: Correct option is A.$7,4,-2$

A line is passing through $(\alpha, \beta, \gamma)$ and its direction cosines are $l, m, n$ then the equations of the line are- (a) $\frac{x}{l}=\frac{y}{m}=\frac{z}{n}$ (b) $\frac{x-\alpha}{l}=\frac{y-\beta}{m}=\frac{z-\gamma}{n}$ (c) $\frac{x+\alpha}{l}=\frac{y+\beta}{m}=\frac{z+\gamma}{n}$ (d) $\frac{x-\alpha}{l}=\frac{y+\beta}{m}=\frac{z-\gamma}{n}$

Maths, Previous Year Question Papers

Correct option is D $\frac{x-\alpha}{l}=\frac{y+\beta}{m}=\frac{z-\gamma}{n}$

let a,b,c be the direction ratios of a line then direction cosines are (a) $\frac{a}{\sqrt{\sum a^{2}}}, \frac{b}{\sqrt{\sum a^{2}}}, \frac{c}{\sqrt{\sum a^{2}}}$ (b) $\frac{1}{\sqrt{\sum a^{2}}}, \frac{1}{\sqrt{\sum a^{2}}}, \frac{1}{\sqrt{\sum a^{2}}}$ (c) $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ (d) $\frac{a}{\sqrt{\sum a^{2}}}, \frac{b}{\sqrt{\sum b^{2}}}, \frac{c}{\sqrt{\sum c^{2}}}$

Maths, Previous Year Question Papers

Sol: Correct option is D.$\frac{a}{\sqrt{\sum a^{2}}}, \frac{b}{\sqrt{\sum b^{2}}}, \frac{c}{\sqrt{\sum c^{2}}}$

Let $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ be the direction cosines of two st-lines both the lines are perpendicular to each other, if- (a) $l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}$ (b) $l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}=1$ (c) $\frac{l_{1}}{l_{2}}=\frac{m_{1}}{m_{2}}=\frac{n_{1}}{n_{2}}$ (d) $\frac{l_{1}}{l_{2}}+\frac{m_{1}}{m_{2}}+\frac{n_{1}}{n_{2}}=0$

Maths, Previous Year Question Papers

Sol: Correct option is A.$l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}$

If $\vec{a}=\vec{i}+\vec{j}+3 \vec{k} ; \vec{b}=2 \vec{i}+3 \vec{j}-5 \vec{k}$ then $\vec{a} \vec{b}=$ (a) 10 (b) $-10$ (c) 20 (d) 5

Maths, Previous Year Question Papers

Sol: Correct option is B.$-10$

If $\overrightarrow{O A}=2 \vec{i}+5 \vec{j}-2 \vec{k}$ and $\overrightarrow{O B}=3 \vec{i}+6 \vec{j}+5 \vec{k}$ then $\overrightarrow{A B}=$ (a) $\vec{i}+\vec{j}+7 \vec{k}$ (b) $5 \vec{i}+2 \vec{j}+7 \vec{k}$ (c) $2 \vec{i}+\vec{j}+\vec{k}$ (d) $\vec{i}+\vec{j}+7 \vec{k}$

Maths, Previous Year Question Papers

Sol: Correct option is A.$\vec{i}+\vec{j}+7 \vec{k}$

$|2 \vec{i}-3 \vec{j}+\vec{k}|=$ (a) 14 (b) $\sqrt{14}$ (c) $\sqrt{3}$ (d) 2

Maths, Previous Year Question Papers

Sol: Correct option is A.14

The position vector of the point $(4,5,6)$ is (a) $4 \vec{i}+5 \vec{j}+6 \vec{k}$ (b) $4 \vec{i}-5 \vec{j}-6 \vec{k}$ (c) $2 \vec{i}+\vec{j}+\vec{k}$ (d) $\vec{i}+\vec{j}+\vec{k}$

Maths, Previous Year Question Papers

Sol: Correct option is A.$4 \vec{i}+5 \vec{j}+6 \vec{k}$

The position vector of the point $(4,5,6)$ is (a) $4 \vec{i}+5 \vec{j}+6 \vec{k}$ (b) $4 \vec{i}-5 \vec{j}-6 \vec{k}$ (c) $2 \vec{i}+\vec{j}+\vec{k}$ (d) $\vec{i}+\vec{j}+\vec{k}$

Maths, Previous Year Question Papers

Sol: Correct option is A.$4 \vec{i}+5 \vec{j}+6 \vec{k}$

The degree of the equation $\left(\frac{d_{2} y}{d x^{2}}\right)^{3}-4 \frac{d y}{d x}=2$ is (a) 0 (b) 1 (c) 2 (d) 3

Maths, Previous Year Question Papers

Sol: Correct option is D. 3

The order of the differential equation $\frac{d y}{d x}+4 y=2 x$ is- (a)0 (b) 1 (c) 2 (d) 3

Maths, Previous Year Question Papers

Sol: Correct option is B.1

The solution of the differential equation $\frac{d y}{d x}=e^{x-y}$ is (a) $e^{x}+e^{-y}+k=0$ (b) $e^{2 x}=k e^{y}$ (c) $e^{x}-e^{y}=k$ (d) $e^{x+y}=k$

Maths, Previous Year Question Papers

Sol: Correct option is C.$e^{x}-e^{y}=k$

.The solution of $\frac{d y}{d x}=\frac{x}{y}$ is- (a) $\frac{y^{2}}{2}-\frac{x^{2}}{2}=k$ (b) $\frac{x^{2}}{2}+\frac{y^{2}}{2}=k$ (c) $\frac{x-y}{2}=k$ (d) $\frac{x+y}{5}=k$

Maths, Previous Year Question Papers

Sol: Correct option is A. $\frac{y^{2}}{2}-\frac{x^{2}}{2}=k$

$\int_{b}^{a} x^{3} d x=\ldots \ldots .$ (a) $\frac{b^{3}-a^{3}}{3}$ (b) $\frac{b^{4}-a^{4}}{4}$ (c) $\frac{b^{2}-a^{2}}{2}$ (d) 0

Maths, Previous Year Question Papers

Sol: Correct option is B. $\frac{b^{4}-a^{4}}{4}$

$\int \frac{d x}{x}=\ldots \ldots \ldots \ldots$ (a) $x+k$ (b) $\frac{1}{x^{2}}+k$ (c) $-\frac{1}{x^{2}}+k$ (d) $\log x+k$

Maths, Previous Year Question Papers

Sol: Correct option is D. $\log x+k$

$\int 0 . d x=\ldots \ldots \ldots \ldots$ (a)k (b) 0 (c) 1 (d) $-1$

Maths, Previous Year Question Papers

Sol: Correct option is B.0

The solution of $\frac{d y}{d x}=\frac{x}{y}$ is- (a) $\frac{y^{2}}{2}-\frac{x^{2}}{2}=k$ (b) $\frac{x^{2}}{2}+\frac{y^{2}}{2}=k$ (c) $\frac{x-y}{2}=k$ (d) $\frac{x+y}{5}=k$

Maths, Previous Year Question Papers

Sol: Correct option is A. $\frac{y^{2}}{2}-\frac{x^{2}}{2}=k$

$\int_{b}^{a} x^{3} d x=\ldots \ldots .$ (a) $\frac{b^{3}-a^{3}}{3}$ (b) $\frac{b^{4}-a^{4}}{4}$ (c) $\frac{b^{2}-a^{2}}{2}$ (d) 0

Maths, Previous Year Question Papers

Sol: Correct option is B. $\frac{b^{4}-a^{4}}{4}$

$\int \frac{d x}{x}=\ldots \ldots \ldots \ldots$ (a) $x+k$ (b) $\frac{1}{x^{2}}+k$ (c) $-\frac{1}{x^{2}}+k$ (d) $\log x+k$

Maths, Previous Year Question Papers

Sol: Correct option is D. $\log x+k$

If $y=\sin (\log x)$ , then $\frac{d y}{d x}=$ (a) $\frac{1}{x} \cos (\log x)$ (b) $\frac{1}{x} \sin (\log x)$ (c) 0 (d) 1

Maths, Previous Year Question Papers

Sol: Correct option is (A)$\frac{1}{x} \cos (\log x)$

$\frac{d}{d x}\left(\sin ^{-1} x+\cos ^{-1} x\right)=$ (a) $\frac{2}{1+x^{2}}$ (b) 0 (c) 2 (d) 1

Maths, Previous Year Question Papers

Sol: Correct option is (A)$\frac{2}{1+x^{2}}$

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

Maths, Previous Year Question Papers

Sol: Correct option is (C)$\frac{1}{\sqrt{1-x^{2}}}$

$\frac{d}{d x}(\sec x)=$ (a) $\sec ^{2} x$ (b) $\tan ^{2} x$ (c) $\sec x \tan x$ (d) 0

Maths, Previous Year Question Papers

Sol: Correct option is (C)$\sec x \tan x$

If $A=\left[\begin{array}{ccc}9 & 10 & 11 \\ 12 & 13 & 14\end{array}\right]$ and $B=\left[\begin{array}{ccc}11 & 10 & 9 \\ 8 & 7 & 6\end{array}\right]$ then $A+B=.$ (a) $\left[\begin{array}{lll}20 & 20 & 20 \\ 20 & 20 & 20\end{array}\right]$ (b) $\left[\begin{array}{lll}10 & 10 & 10 \\ 10 & 10 & 10\end{array}\right]$ (c) $\left[\begin{array}{ccc}10 & 5 & 10 \\ 5 & 10 & 10\end{array}\right]$ (d) $\left[\begin{array}{ccc}25 & 10 & 15 \\ 15 & 10 & 25\end{array}\right]$

Maths, Previous Year Question Papers

Sol: Correct option is(A).$\left[\begin{array}{lll}20 & 20 & 20 \\ 20 & 20 & 20\end{array}\right]$

$\left|\begin{array}{ll}10 & 2 \\ 35 & 7\end{array}\right|=$ (a) 4 (b) 0 (c) 3 (d)6

Maths, Previous Year Question Papers

Sol: Correct option is B.0

If $\left|\begin{array}{ll}x & 5 \\ 5 & x\end{array}\right|=0$ then $x=0$ (a) $\pm 5$ (b) 6 (c) 0 (d) 4

Maths, Previous Year Question Papers

Sol: Correct option is A.$\pm 5$

$\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{4}=$ (a) $\tan ^{-1} \frac{3}{2}$ (b) $\tan ^{-1} \frac{6}{7}$ (c) $\tan ^{-1} \frac{5}{6}$ (d) $\tan ^{-1} \frac{1}{2}$

Maths, Previous Year Question Papers

Sol; Correct option is B.$\tan ^{-1} \frac{6}{7}$

$\tan ^{-1}(1)=\ldots \ldots \ldots$ ? (a) $\frac{\pi}{4}$ (b) $\frac{\pi}{2}$ (c) $\frac{\pi}{3}$ (d) $\frac{\pi}{8}$

Maths, Previous Year Question Papers

Sol: Correct option is A. $\frac{\pi}{4}$

$\left|\begin{array}{lll}1 & 1 & 2 \\ 2 & 2 & 4 \\ 3 & 5 & 6\end{array}\right|=\ldots \ldots \ldots ?$ (a) 5 (b) 7 (c) 0 (d) 9

Maths, Previous Year Question Papers

Sol: Correct option is (C)0

$\frac{d}{d x}(\tan a x)=$ (a) $a \tan a x$ (b) $a \sec ^{2} a x$ (c) $a \sec x$ (d) $a \cot a x$

Maths, Previous Year Question Papers

Sol: Correct option is (B)$a \sec ^{2} a x$

$\frac{d}{d x}(\tan a x)=$ (a) $\cos x$ (b) $-\sin x$ (c) $-\cos x$ (d) $\tan x$

Maths, Previous Year Question Papers

Sol: Correct option is A.$\cos x$

.If $f: R \rightarrow R$ such that $f(x)=3 x-4$ then which of the following $f^{-1}(x) ?$ (a) $\frac{x+4}{3}$ (b) $\frac{1}{3} x-4$ (c) $3 x-4$ (d) $3 x+5$

Maths, Previous Year Question Papers

Sol: Correct option is A. $\frac{x+4}{3}$

If $n(A)=3$ and $n(B)=2$ then $(A \times B)=\ldots \ldots$ 1 (a) 6 (b) 4 (c)2 (d) 0

Maths, Previous Year Question Papers

Sol: Correct option is A.

1.If $A\{1,2,3\}$ ,then how many equivalence relation can be defined on A containing $(1,2)$ (a)2 (b) 3 (c) 8 (d) 6

Maths, Previous Year Question Papers

Sol: Correct option is A.2

$\int \frac{-1}{1+x^{2}} d x=$ (A) $\tan ^{-1} x+k$ (B) $\sec ^{-1} x+k$ (C) $\operatorname{cosec}^{-1} x+k$ (D) $\cot ^{-1} x+k$

Maths, Previous Year Question Papers

Sol. Correct option is (D).$\cot ^{-1} x+k$

$\int \frac{-1}{1+x^{2}} d x=$ (A) $\tan ^{-1} x+k$ (B) $\sec ^{-1} x+k$ (C) $\operatorname{cosec}^{-1} x+k$ (D) $\cot ^{-1} x+k$

Maths, Previous Year Question Papers

Sol. Correct option is (D).$\cot ^{-1} x+k$

$\tan ^{-1} \frac{2 x}{1-x^{2}}$ (A) $2 \sin ^{-1} x$ (B) $\sin ^{-1} 2 x$ 11 (C) $\tan ^{-1} 2 x$ (D) $2 \tan ^{-1} x$

Maths, Previous Year Question Papers

Sol. Correct option is (D).$2 \tan ^{-1} x$

$\left|\begin{array}{lll}2 & 3 & 5 \\ 0 & 4 & 7 \\ 0 & 0 & 5\end{array}\right|$ (A) 40 (B) 0 (C) 3 (D) 25

Maths, Previous Year Question Papers

Correct option is (A).40

The matrix $\left[\begin{array}{ll}3 & 5 \\ 2 & \mathrm{k}\end{array}\right]$ has no inverse if the value of $\mathrm{k}$ is (A) 0 (B) 5 (C) $\frac{10}{3}$ (D) $\frac{4}{9}$

Maths, Previous Year Question Papers

Sol. Correct option is (A).0

If $\mathrm{A}$ and $\mathrm{B}$ are two independent events, then (A) $P\left(A B^{\prime}\right)=P(A) P(B)$ (B) $P\left(A B^{\prime}\right)=P(A) P\left(B^{\prime}\right)$ (C) $P\left(A B^{\prime}\right)=P\left(A^{\prime}\right)+P(B)$ (D)
$P\left(A B^{\prime}\right)=P(A)+P\left(B^{\prime}\right)$

Maths, Previous Year Question Papers

Sol. Correct option is (A). $P\left(A B^{\prime}\right)=P(A) P(B)$

If $P(A)=\frac{3}{8}, P(B)=\frac{1}{2}, P(A \cap B)=\frac{1}{4}$ , then $P(A / B)=$ (A) $\frac{1}{4}$ (B) $\frac{1}{2}$ (C) $\frac{2}{3}$ (D) $\frac{3}{8}$

Maths, Previous Year Question Papers

Correct option is (D).$\frac{3}{8}$

The length of the perpendicular from the point $(0,-1,3)$ to the plane $2 x+y-2 z+1=0$ his. (A) 0 (B) $2 \sqrt{3}$ (C) $\frac{2}{3}$ (D) 2

Maths, Previous Year Question Papers

Sol. Correct option is (D).2

The lines $\frac{x-1}{l}=\frac{y+2}{m}=\frac{z-4}{n}$ and $\frac{x+3}{2}=\frac{y-4}{3}=\frac{z}{6}$ are parallel is each other if (A) $2 l=3 m=n$ (B) $3 l=2 m=n$ (C) $2 l+3 m+6 n=0$ (D) $l m n=36$

Maths, Previous Year Question Papers

Sol. Correct option is (D). $l m n=36$

A line passing through $(2,-1,3)$ and its direction ratios are $3,-1,2$ . The equation of the line is (A) $\frac{x+2}{3}=\frac{y-1}{-1}=\frac{z+1}{2}$ (B) $\frac{x-2}{3}=\frac{y+1}{-1}=\frac{z-3}{2}$ (C) $\frac{x-3}{2}=\frac{y+1}{-1}=\frac{z-2}{3}$ (D) $\frac{x-3}{2}=\frac{y+1}{-1}=\frac{z-1}{3}$

Maths, Previous Year Question Papers

9 Correct option is (B). $\frac{x-2}{3}=\frac{y+1}{-1}=\frac{z-3}{2}$

The direction ratios of two straight lines are $\mathbf{l}, \mathbf{m}, \mathbf{n}$ and $l_{1}, m_{1}, n_{1}$ . The lines will be perpendicular to each other if. (A) $\frac{l}{l_{1}}=\frac{m}{m_{1}}=\frac{n}{n_{1}}$ (B) $\frac{l}{l_{1}}+\frac{m}{m_{1}}+\frac{n}{n_{1}}=0$ (C) $l l_{1}+m m_{1}+n n_{1}=0$ (D) $l l_{1}+m m_{1}+n n_{1}=1$

Maths, Previous Year Question Papers

Sol. Correct option is (C). $l l_{1}+m m_{1}+n n_{1}=0$

The direction between the points $(4,3,7)$ and $(1,-1,-5)$ is (A) $\frac{1}{\sqrt{35}}, \frac{3}{\sqrt{35}}, \frac{5}{\sqrt{35}}$ (B) $\frac{1}{9}, \frac{1}{3}, \frac{5}{9}$ (C) $\frac{3}{\sqrt{35}}, \frac{5}{\sqrt{35}}, \frac{1}{\sqrt{35}}$ (D) $\frac{5}{\sqrt{35}}, \frac{3}{\sqrt{35}}, \frac{1}{\sqrt{35}}$

Maths, Previous Year Question Papers

Sol. Correct option is (A). $\frac{1}{\sqrt{35}}, \frac{3}{\sqrt{35}}, \frac{5}{\sqrt{35}}$

The distance between the points $(4,3,7)$ and $(1,-1,-5)$ is 8 (A) 7 (B) 12 (C) 13 (D) 25

Maths, Previous Year Question Papers

Sol. Correct option is (C).13

$1, \mathrm{~m}, \mathrm{n}$ , are the direction cosines of a straight line them (A) $l^{2}+m^{2}-n^{2}=1$ (B) $l^{2}-m^{2}+n^{2}=1$ (C) $l^{2}-m^{2}-n^{2}-1$ (D) $l^{2}+m^{2}+n^{2}=1$

Maths, Previous Year Question Papers

Correct option is (D)$l^{2}+m^{2}+n^{2}=1$

If $\vec{a}$ and $\vec{b}$ are perpendicular to each other then (A) $\vec{a} \cdot \vec{b}=0$ (B) $\vec{a} \times \vec{b}=\overrightarrow{0}$ (C) $\vec{a}+\vec{b}=\overrightarrow{0}$ (D) $\vec{a}-\vec{b}=0$

Maths, Previous Year Question Papers

Sol. Correct option is (A).$\vec{a} \cdot \vec{b}=0$

.If $\vec{a}=3 \vec{i}+2 \vec{j}+\vec{k}, \vec{b}=4 \vec{i}-5 \vec{j}+3 \vec{k}$ , then $\vec{a} \cdot \vec{b}$ (A) 2 (B) 3 (C) 5 (D) 7

Maths, Previous Year Question Papers

Sol. Correct option is (C)5