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If the expansion of $\frac{1}{(1-ax)(1-bx)}=a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n+\cdots$, then $a_n$ is (where $a\ne b,|ax|,|bx|<1$)

$\sqrt{(1+sinA)}-\sqrt{(1-sinA)}=-2cos\left(A/2\right)$

An electric dipole has fixed dipole moment $\overrightarrow{p}$, which makes angle $\theta$ with respect to x-axis. When subjected to an electric field $\overrightarrow{E_1}=E\hat{i}$, it experiences a torque $\overrightarrow{T_1}=\tau \hat{k}$. When subjected to another electric field $\overrightarrow{E_2}=\sqrt{3}{E}_1\hat{j}$, it experiences a torque $\overrightarrow{T_2}=-\overrightarrow{T_1}$. The angle $\theta$ is

If the focus of a parabola is (2, 3) and its latus rectum is 8, then the locus of the vertex of the parabola is

If $z=\frac{3}{2+\cos \theta +i\sin \theta }$, then locus of $z$ is

Tangent at the point $(2\sqrt{2},3)$ to the hyperbola$\frac{x^2}{4}-\frac{y^2}{9}$= 1 meet its asymptotes at A and B, then area of the triangle OAB, O being the origin is

Let $\frac{dy}{dx}+y=f\left(x\right)$ where $y$ is a continuous function of $x$ with $y\left(0\right)=1$ and $f\left(x\right)=\{\begin{array}{cc}{e}^{-x},& 0\le x\le 2\\ e^{-2},& x>2\end{array}$.

Which of the following hold(s) good?

The maximum value of a cosx+b sinx is

[MNR 1991; MP PET 1999; UPSEAT 2000]

Fundamental period of the function $f\left(x\right)={\sin }^{2}x$ is

The
integrating factor of the differential equation
is

A. e^–x

B. e^–y

C.

D. x

Prove that the straight line $(a+b)x+(a-b)y=2ab,(a-b)x+(a+b)y=2ab$ and $x+y=0$ form an isosceles triangle whose vertical angle is $2ta{n}^{-1}\left(\frac{a}{b}\right)$.

If one of the root of the qudratic polynomial $f\left(x\right)=a{x}^2+bx+c;a>0$ is greater than $k_1$ and other root is less than $k_2$. Then select the correct statement(s) for $k_1<k_2$.

If the scalar projection of vector $2\hat{i}-x\hat{j}+\hat{k}$ on vector $-\hat{i}+\hat{j}-2\hat{k}$ is $\frac{2}{\sqrt{6}},$ then the value of $x$ is

If
findin
terms of y alone.

The value of $\underset{0}{\overset{\pi /4}{\int }}\log (1+\tan \theta )d\theta$ is equal to

If $A$ and $B$ are two independent events such that $P(A^{\prime }\cap B)=\frac{2}{15}$ and $P(A\cap B^{\prime })=\frac{1}{6},$ then $P\left(A\right)$ is

Compare the demographic indicators of India with China and Pakistan.

If f(x) is a differentiable function and ${\int }_0^{t^2}xf\left(x\right)\mathrm{dx}=\frac{2}{5}{t}^5$, then $f\left(\frac{4}{25}\right)$ equals

16 A.Ms are inserted between 5 and 50. Find the sum of all the A.Ms.

The nearest point on the line $3x-4y=25$ from the origin is

For the curve $x=t^2-1,t^2-t$, the tangent line is perpendicular to x-axis when

The number of different words that can be formed using all the letters of the word 'APPLICATION' such that two vowels never come together, is

If $\alpha$ and $\beta$ are values of $\theta$ satisfying the equation $\frac{\sqrt{3}-1}{\sin \theta }+\frac{\sqrt{3}+1}{\cos \theta }=4\sqrt{2},$ where $\theta \in (0,\frac{\pi }{2}),$ then the value of $|\alpha -\beta |$ is

Let w = f(x, y), where x and y are functions of t. Then according to the chain rule, $\frac{dw}{dt}$ is equal to

If $f\left(x\right)=\{\begin{array}{cc}{x}^3+1,& x<0\\ x^2+1,& x\ge 0\end{array},g\left(x\right)=\{\begin{array}{cc}(x-1{)}^{\frac{1}{3}},& x<1\\ (x-1{)}^{\frac{1}{2}},& x\ge 1\end{array}$ then (gof) (x) is equal to

In $R^3$, Let $L$ be a straight line passing through the origin. Suppose that all the points on $L$ are at a constant distance from the two planes $P_1:x+2y-z+1=0$ and $P_2:2x-y+z-1=0.$ Let $M$ be the locus of the feet of the perpendiculars drawn from the points on $L$ on the plane $P_1.$ Which of the following points lie(s) on $M?$

A tangent $PT$ is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3},1).$ A straight line $L,$ perpendicular to $PT$ is a tangent to the circle $(x-3{)}^2+y^2=1.$

A possible equation of $L$ is

Find all points of discontinuity of f,
where f is
defined by