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If x is the arithmetic mean of y and z and the two geometric means between y and z are $G_1\mathrm{and}{G}_2$, then $G_1^3+G_2^3=$.

is equal to

A. tan x
+ cot x + C

B. tan x
+ cosec x + C

C. − tan x
+ cot x + C

D. tan x
+ sec x + C

The value of $\underset{0}{\overset{\pi }{\int }}{\cos }^{101}xdx$ is equal to

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}\underset{x\to \infty }{lim}(\frac{x^2+1}{x+1}-ax-b)=0,\text{then}& \text{(P)}& a=\frac{3}{2},b\in R\\ \text{(B)}& \text{If}\underset{x\to 0}{lim}(1+ax+b{x}^2{)}^{2/x}=e^3,\text{then}& \text{(Q)}& a=1,b=-\frac{1}{2}\\ \text{(C)}& \text{If}\underset{x\to 0}{lim}\left(\frac{a{e}^x-b}{x}\right)=2,\text{then}& \text{(R)}& a=1,b=-1\\ \text{(D)}& \text{If}\underset{x\to \infty }{lim}\{\sqrt{(x^2-x+1)}-ax-b\}=0,\text{then}& \text{(S)}& a=2,b=2\end{array}$



Which of the following is the only CORRECT combination?

The coefficient of $x^{160}$ in the expansion of $(x^8+1{)}^{60}{(x^{12}+3x^4+\frac{3}{x^4}+\frac{1}{x^{12}})}^{-10}$ is

If $sec(\theta +\alpha )+sec(\theta -\alpha )=2sec\theta ,\text{prove that}cos\theta =\pm \sqrt{2}cos\frac{\alpha }{2}$

The minimum distance between the origin and the plane which is perpendicular bisector of the line joining the points $(1,3,5)$ and $(3,7,-1)$, is

If $f\left(x\right)=k{x}^3-9x^2+9x+3$ is monotonically increasing in $R$, then

If $x+iy=\sqrt{\frac{a+ib}{c+id}},$ then $x^4+y^4+2x^2{y}^2$ is equal to

The family of curves satisfying the differential equation $\frac{dy}{dx}+\frac{1}{x}\sin 2y=x^3{\cos }^{2}y,$ is

(where $C$ is an arbitrary constant)

The equation of the hyperbola whose centre is (5, 2), vertex is (9, 2) and the length of conjugate axis is 6 is

A diagonal matrix will always be

If $\sin x\cos y=\frac{1}{2}$, then $\frac{d^{2}y}{d{x}^2}$ at $x=\frac{\pi }{4}$ is

If $f\left(x\right)=\sin \left(\ln x\right)-\cos \left(\ln x\right)$ is strictly increasing, then $x\in$

If $n>1$, the values of the positive integer $m$ for which $n^m+1$ divides $a=1+n+n^2+\dots \dots +n^{63}$ is/are

Given,
find the values of x,
y, z
and

w.

The area bounded by $y=2-|2-x|;y=\frac{3}{|x|}is\frac{k-3ln3}{2},$ then k = ___

The parametric coordinates of the circle whose center coordinates are $(-4,3)$ and touches the $y$-axis, is

In a $\Delta ABC,$ the value of $a(r{r}_1+r_2{r}_3)=$

Let $A=\{x:-1\le x\le 1\}$ and $f:A\to A$ is a function defined by $f\left(x\right)=x|x|$, then f is

If $S_n=[\frac{1}{1+\sqrt{n}}+\frac{1}{2+\sqrt{2n}}+...+\frac{1}{n+\sqrt{n^2}}],$ then the value of $\underset{n\to \infty }{lim}{S}_n$ is equal to

There are 4 cards numbered 1,3,5 and 7.one number on one card.Two cards are drawn at random without replacement.Let X denote the sum of the numbers on the two drawn cards.Find the mean and variance of X.

The asymptotes of a hyperbola have center at the point $(1,2)$ and are parallel to the lines $2x+3y=0$ and $3x+2y=0$. If the hyperbola passes through the point $(5,3)$, then its equation is

Solve the following system of equations in R.

$\frac{7x-1}{2}<-3,\frac{3x+8}{5}+11<0$

Equations of the line(s) which makes an angle of ${45}^{\circ }$ with $y$ axis and passing through the point $(2,3)$ is