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Let $f\left(x\right)=\{\begin{array}{cc}\frac{e^{\frac{-x^2}{2}}-\cos x}{x\ln (1+x)\sin x(e^x-1)},& x\ne 0\\ k,& x=0\end{array}.$

If $f\left(x\right)$ is continuous at $x=0$, then $k$ equals

If $x=3,y=\omega +2{\omega }^2$ and $z={\omega }^2+2\omega ,$ then $xyz$ is equal to

A function f(x) is called as a strictly increasing function about a point ‘a’ If -

Where 'h' is a positive real number tending to zero.

The equation of circle whose two diameters are $2x-3y=5,$ $3x-2y=10$ and having perimeter $6\pi \text{cm},$ is

If a² +b² + c² =-2 and

then f(x) is a polynomial of degree

The line $4x-3y+2=0$ is rotated through an angle of $\frac{\pi }{4}$ in clockwise direction about the point $(1,2).$ The equation of the line in its new position is

If f(x) is a differentiable function and ${\int }_0^{x^3}{t}^{2}f\left(t\right)dt=\frac{3}{13}{x}^{13}+5$ then $f\left(\frac{8}{27}\right)$

If A = {3, 6, 9, 12, 15, 18, 21}

B = {2, 4, 6, 8, 10, 12, 14, 16}

C = {5, 10, 15, 20}

$$\begin{array}{cc}A& B\\ 1.B-C& A.\{5,10,20\}\\ 2.A-C& B.\{3,9,15,21\}\\ 3.C-(A-B)& C.\{3,6,9,12,18,21\}\\ 4.C\cap (A-B)& D.\{15\}\\ & E.\{2,4,6,8,12,14,16\}\end{array}$$

If the value of ${lim}_{n\to \infty }\left(n^{-3/2}\right){\sum }_{j=1}^{6n}\sqrt{j}$ is equal to $\sqrt{N}$, then value of N/12 is___

Length of the straight line $x-3y=1$ intercepted by the hyperbola $x^2-4y^2=1$ is

Let R be the realtion on the set R of all real

numbers defined by a R b if |a-b| $\le$ 1. then R is

If $x{\log }_e\left({\log }_{e}x\right)-x^2+y^2=4(y>0)$, then $\frac{dy}{dx}$ at $x=e$ is equal to :

If $\alpha ,\beta$ and $\gamma$ are real numbers such that ${\alpha }^2+{\beta }^2+{\gamma }^2=1$ and $\alpha +\beta +\gamma =\sqrt{3},$ then $\beta =$
(correct answer + 1, wrong answer - 0.25)

If roots of the equation $x^3+3p{x}^2+3qx+r=0,p,q,r\ne 0$ are in H.P., then which of the following is correct?

Let $f\left(x\right)=[x^3-3],$ where $[.]$ denotes the greatest integer function. Then the number of points in the interval $(1,2)$ where the function is discontinuous, is

and f '(1) = 15, then the value of n is

If $S$ is the solution set of the inequality ${\log }_5(x^2-2)<{\log }_5(\frac{3}{2}|x|-1),$ then which of the following intervals lie(s) in $S$?

If $f\left(x\right)=\underset{0}{\overset{x}{\int }}{e}^t\sin (x-t)dt,$ then $f^{\prime \prime }x-f\left(x\right)$ is equal to

The tangents at the points $(a{t}_1^2,2a{t}_1)$,$\left(a{t}_2^2\right),2a{t}_2)$ on the parabola $y^2=4ax$ are at right angles if

Least value of the function $f\left(x\right)=|x-a|+|x-b|+|x-c|+|x-d|$, where $a<b<c<d$ are fixed real numbers & $x$ takes arbitary values, is

The area of the triangle formed by the intersection of a line parallel to x-axis and passing through P(h,k) with the lines y=x and x+y=2 is $4h^2$. Find the locus of the point P.

The value of $\sum _{n=1}^5\tan \left(\frac{\theta }{2^{n+1}}\right)\sec \left(\frac{\theta }{2^n}\right)$ is

A plane passes through (1, -2, 1) and is perpendicular to two planes $2x-2y+z=0$ and $x-y+2z=4,$ then the distance of the plane form the point (1, 2, 2) is

Between any two real roots of the equation $e^x$ sin x = 1, the equation $e^x$ cos x = - 1 has

If f(x) = sin log $\left(\frac{\sqrt{4-x^2}}{1-x}\right)$, then the domain of f(x) is ___

If $\cot \theta -pq\tan \theta =p-q;p,q\ne 0$, then the general value of $\theta$ is

If the tangents drawn to the parabola at the extremities of a common chord $AB$ of the circle $x^2+y^2=5$ and the parabola $y=b{x}^2$ intersect at the point $T$ which lies on the directrix of the parabola, then $\frac{1}{b}=$

If the roots of the equation $8x^3-14x^2+7x-1=0$ are in G.P., then the roots are

If the system of equations $2x+3y-z=0,x+ky-2z=0$ and $2x-y+z=0$ has a non-trivial solution $(x,y,z)$, then $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+k$ is equal to :

If one vertex of a chord of parabola $y^2=8ax$ is at $(0,0),$ then the locus of a point which divides the chord in the ratio $1:2$ is

If the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular to each other, then a vector $\overrightarrow{v}$ in terms of $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying the equations

$\overrightarrow{v}.\overrightarrow{a}=0,\overrightarrow{v}.\overrightarrow{b}=1and\left[\overrightarrow{v}\overrightarrow{a}\overrightarrow{b}\right]=1$ is

Let $y=e^{\left\{\right({\sin }^{2}x+{\sin }^{4}x+{\sin }^{6}x+\dots \left){\log }_{e}2\right\}}$ satisfy the equation $x^2-17x+16=0,$ where $0<x<\frac{\pi }{2}.$ Then match the correct value of $\text{List I}$ from $\text{List II}.$



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(a)}& \frac{2\sin 2x}{1+{\cos }^{2}x}& \left(p\right)& 1\\ \text{(b)}& \frac{2\sin x}{\sin x+\cos x}& \left(q\right)& \frac{4}{9}\\ \text{(c)}& \sum _{n=1}^{\infty }(\cot x{)}^n& \left(r\right)& \frac{2}{3}\\ \text{(d)}& \sum _{n=1}^{\infty }n(\cot x{)}^{2n}& \left(s\right)& \frac{4}{3}\end{array}$

The coordinates of focus of parabola $9y^2-9x-12y-57=0$ is

Let $S_1,S_2,S_3$ and $S_4$ be four sets defined as

$S_1=\{x:x\in Z\text{and}{\log }_2|4-3x|\le 2\}$

$S_2=\{x:x\in Z\text{and}|1-\frac{|x|}{1+|x|}|\ge \frac{1}{3}\}$

$S_3=\left\{x:x^2-3x+2\text{sgn}\left(x\right)=0\right\},$ where $\text{sgn}\left(x\right)$ represents the signum function.

$S_4=\left\{\right(x,y):x,y\in Z,x^2+y^2\le 4\}$.



$\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be correctly matched with a unique entry of $\text{List II}.$



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& n\left(S_1\Delta S_2\right)& \text{(P)}& 9\\ \text{(B)}& n((S_1\times S_2)\cap (S_2\times S_1))& \text{(Q)}& 12\\ \text{(C)}& n(S_1\cap S_2\cap S_3^{\prime })& \text{(R)}& 36\\ \text{(D)}& n(S_4\times S_3)& \text{(S)}& 2\\ & & \text{(T)}& 0\end{array}$



Which of the following is the only CORRECT combination?

If the normal at $P$ to the hyperbola $x^2-y^2=4$ meets the axes in $G$ and $g$ and $C$ is centre of the hyperbola, then

If $Li{m}_{x\to 0}\frac{4+sin2x+Asinx+Bcosx}{x^2}$ exists, then the values A and B are

The solution set of $\frac{3}{x+1}-\frac{2}{x-1}<0$ is

If $co{s}^6\alpha +si{n}^6\alpha +Ksi{n}^{2}2\alpha =1,$ then K=

If $2\sec \theta \text{cosec}\theta -\cot \theta =3$, then the value of $\tan \theta$ is/are

The roots of the equation $x^2+|x|-2=0$ is/are

The value of $\frac{\sin 8\theta \cos \theta -\sin 6\theta \cos 3\theta }{\cos 2\theta \cos \theta -\sin 3\theta \sin 4\theta }$ is

Let $z$ be an imaginary complex number satisfying $|z-1|=1.$ If $\alpha =2z,$ $\beta =2\alpha$ and $\gamma =2\beta ,$ then the value of $|z{|}^2+|\alpha {|}^2+|\beta {|}^2+|\gamma {|}^2+|z-2{|}^2+|\alpha -4{|}^2+|\beta -8{|}^2+|\gamma -16{|}^2$ is

Find the equation of the sphere having extremities of one of its diameters as the

points (2,3, 5) and (-4, 7,11).

The Equation of chord of contact of $\frac{x^2}{25}+\frac{y^2}{16}=1is4x+5y-20=0$ Find the point from which the tangents are drawn

If $A$ is the area and $2s$ the sum of 3 sides of triangle then