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The equation of line passing through the point $(2,1,-1)$ and parallel to the line $\frac{x-3}{1}=\frac{y-2}{-1}=\frac{z+1}{2}$ is



[2 marks]

If the foot of the perpendicular from point $(4,3,8)$ on the line $L_1:\frac{x-a}{l}=\frac{y-2}{3}=\frac{z-b}{4},$

$l\ne 0$ is $(3,5,7),$ then the shortest distance between the line $L_1$ and line $L_2:\frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}$ is equal to :

In the equation k = PZ_AB e^-E_a^/RT, P is known as:

Let $f\left(x\right)=x{(-1)}^{\left[\frac{1}{x}\right]}.x\ne 0$, where [x] denotes the greatest integer less than or equal to x. then $\underset{x\to 0}{lim}f\left(x\right)$

Identify Z

The four lines drawn from the vertices of any tetrahedron to the centroid of the opposite faces meet in a point whose distance from each vertex is k times the distance from each vertex to the opposite face, where k is

If $(x^2-9)\sqrt{x^2-1}<0$, then what are the possible values of $x$?

If two matrices $A$ and $B$ are such that they follow commutative property, then $A^4{B}^2$ is equal to

Let $A=\left[\begin{array}{cc}x+2& 3x\\ 3& x+2\end{array}\right],B=\left[\begin{array}{cc}x& 0\\ 5& x+2\end{array}\right]$. Then all solutions of the equation
$detAB=0$ is

A question paper consisting of $10$ questions which is divided into $3$ parts with $5,3,2$ questions respectively. A candidate is to answer $6$ questions without neglecting a question from any part. The number of ways in which he can answer the paper is

The normal to the curve $x=a(1+\cos \theta ),y=a\sin \theta$ at $\theta$ always passes through the fixed point

Find both
the maximum value and the minimum value of

3x⁴
− 8x³ + 12x² − 48x
+ 25 on the interval [0, 3]

If $y={\cot }^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right](0<x<\frac{\pi }{2})$, then $\frac{dy}{dx}$ is equal to

The general solution of $8\cos x\cdot \cos 2x\cdot \cos 4x=\frac{\sin 6x}{\sin x}$ is

Let $P=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \alpha \\ 3& -5& 0\end{array}\right]$, where $\alpha ϵR$. Suppose $Q=\left[q_{ij}\right]$ is a matrix such that PQ = kI, where $kϵR,k\ne 0$ and I is the identity matrix of order 3. If $q_{23}=-\frac{k}{8}anddet\left(Q\right)=\frac{k^2}{2}$ then

A four digit number is formed using the digits 0, 1, 2, 3, 4 without repetition. Find the probability that it is divisible by 4.

Sketch the graph of the following functions on the same scale.
(i) y=cos x and y =cos $(x-\frac{\pi }{4})$
(ii) y=cos 2 x and y = cos2 $(x-\frac{\pi }{4})$
(iii) y =cos x and y = cos $\left(\frac{x}{2}\right)$

Let a function $f\left(x\right)=\underset{-\frac{\pi }{2}}{\overset{x}{\int }}(2{\sin }^{2}t+3\cos t)dt$ is defined in $[-\frac{\pi }{2},\frac{\pi }{2}].$ Then which of the following is/are correct

Let $f:[0,2]\to R$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f\left(0\right)=1$. Let

$F\left(x\right)=\underset{0}{\overset{x^2}{\int }}f\left(\sqrt{t}\right)dt$

for $x\in [0,2]$. If $F^{\prime }\left(x\right)=f^{\prime }\left(x\right)$ for all $x\in (0,2),$ then $F\left(2\right)$ equals

Let $f\left(x\right)=max(x+\left|x\right|,x-\left[x\right])$ where [x] = the greatest integer in $x\le x$. Then ${\int }_{-2}^{2}f\left(x\right)dx$ is equal to

In a party there are $25$ men and $20$ women, then in how many ways a couple (one man with one woman) can be formed, when $2$ particular men and $5$ particular women refuse to be part of any couple?

Two students while solving a quadratic equation in $x$, one copied the constant term incorrectly and got the roots as $3$ and $2$. The other copied coefficient of $x$ incorrectly and got roots as $-6$ and $1$ respectively. The correct root(s) is/are (Assume the leading coefficient of the quadratic equation as $1$)

If $(1+2x+3x^2{)}^{10}=k_1+k_{2}x+\dots +k_{21}{x}^{20}$, then which of the following is/are correct?

The value of $cot\left\{{\sum }_{n=1}^{23}co{t}^{-1}(1+{\sum }_{k=1}^{n}2k)\right\}$ is

If the straight line $L_1$ touches the parabola $y^2=6x$ and is perpendicular to the straight line $L_2$ which touches the ellipse $x^2+4y^2=4$ at $(\sqrt{2},\frac{1}{\sqrt{2}}),$ then $L_1$ passes through

Area bounded by the curve y = sin x between x = 0 and x = 2$\pi$ is

Sir / Ma'am....... How to learn Inverse trigonometric formulas easily.... Are there any shortcuts for learning these identities ?