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You are given $cosx=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}......;$

$sinx=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}......$

$tanx=x+\frac{x^3}{3}+\frac{2.x^5}{15}......$

Find the value of $\underset{x\to 0}{lim}\frac{xcosx+sinx}{x^2+tanx}$

The values of $cos\alpha ,tan\alpha ,cot\alpha ,$ respectively if $sin\alpha =\frac{5}{13}$ and $\frac{\pi }{2}<\alpha <\pi$

The general solution of $tanx=tan\alpha ,\alpha ϵ(-\frac{\pi }{2},\frac{\pi }{2})$ is

A uniform heavy rod of weight W, cross-sectional area A and length L is hanging from a fixed support. Young modulus of the material of the rod is Y. Neglect the lateral contraction. Find the elongation of the rod.

Give the formula for calculating the median of ungrouped data.

If $(1+x+2x^2{)}^{20}=a_0+a_{1}x+a_2{x}^2.....a_{40}{x}^{40}$, then $a_0+a_2+a_4....a_{38}$ =

The relation R defined on the set of natural numbers as {(a, b) : a differs from b by 3}, is given by

Let $f:[-1,1]\to [0,2]$ be a function such that the range of $f=$ co-domain of $f$, then the number of distinct linear function(s) $f$ is

In a random survey 250 people participated. Out of 250 people who took part in the survey, 40 people have listened to Pink Floyd. 30 people have listened to metallica and 20 people have listened to John Denver. If 10 people have listened to all three then find the no. of people who have listened only Pink Floyd.

$Themagnitudeoftheprojectionof2\hat{i}+3\hat{j}+4\hat{k}onthevector\hat{i}+\hat{j}+\hat{k}willbe\underset{–}{}$

If n is any integer, then ${\int }_0^{\pi }{e}^{co{s}^{2}x}co{s}^3(2n+1)xdx=$ [IIT 1985; RPET 1995; UPSEAT 2001]

If $y={[x+\sqrt{x^2+a^2}]}^n,then\frac{dy}{dx}$ is equal to

The roots of the equation $i{x}^2-4x-4i$ = 0 are

Solving integrals of the form $\int \frac{\mathrm{dx}}{\sqrt{x^2+a^2}}$ requires substitution of x =

Three vertices of a parallelogram ABCD are A$(3,-1,2),B(1,2,-4)andC(-1,1,2)$. Which of the follwing could be the fourth vertex ?

Number of ways of selection of 8 letters from 24 letters of which 8 are a, 8 are b and the rest unlike, is given by

Find the sum of he first n terms of the series $3+7+12+21+31+\cdots$

Let $\theta ,\varphi \in [0,2\pi ]$ be such that $2cos\theta (1-sin\varphi )=si{n}^2\theta (tan\frac{\theta }{2}+cot\frac{\theta }{2})cos\theta -1,tan(2\pi -\theta )>0$ and -1 < $sin\theta <-\frac{\sqrt{3}}{2}$. Then $\varphi$ cannot satisfy

If $x\in (0,\frac{\pi }{6})$, then the number of solutions of the equation $\left\{2x\right\}+\left\{\tan x\right\}=0$ is



(Here, $\left\{x\right\}$ denotes the fractional part of $x$.)

The term independent of x in the expansion of $(1+x{)}^n{(1+\frac{1}{x})}^n$ is

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 number of points at which $f\left(x\right)=\{\begin{array}{cc}min(x,x^2),& \text{if}-\infty <x<1\\ min(2x-1,x^2),& \text{if}x\ge 1\end{array}$ is not differentiable is

If the graph of $y=x^2$ is given by





then graph of $y-2=3(x-5{)}^2$ will be given by:

The intercept on the line y = x by the circle $x^2+y^2-2x=0$ is AB. The equation of the circle on AB as a diameter is

$\text{The value of}\underset{x\to \infty }{lim}\frac{x+cosx}{x+sinx}is$

The points on the parabola $y^2=36x$ whose ordinate is three times the abscissa are

A single card is chosen at random from a standard deck of 52 playing cards. What is the probability that the card will be a club or a king?

The most general solutions of the equation $secx-1=(\sqrt{2}-1)tanx$ are given by

The range of $f\left(x\right)=\frac{x}{x^2+4}$ is

Differentiate $x{e}^x$ using first principle.

$\underset{x\to \frac{\pi }{2}}{lim}\frac{sinx-(sinx{)}^{sinx}}{1-sinx+1nsinx}$ is equal to

If $A+B+C=\pi$ then,$\tan \frac{A}{2}\cdot \tan \frac{B}{2}+\tan \frac{B}{2}\cdot \tan \frac{C}{2}+\tan \frac{C}{2}\cdot \tan \frac{A}{2}$ =

The Boolean expression $(p\wedge \sim q)\vee q\vee (\sim p\wedge q)$ is equivalent to

The slope of a line is double of the slope of another line. If tangent of the angle between them is $\frac{1}{3}$, find the slopes of the lines.

Find the equation of a sphere, whose centre is $(1,1,1)$ and radius is $5$ units

Which of the following is the average rate of change of f(x) with respect to x over the interval

[a, a+h]?

The value of $(\cot {18}^{\circ }\cot {12}^{\circ }-\cot {45}^{\circ })({\sin }^2{15}^{\circ }-{\sin }^2{3}^{\circ })$ is

If y=$co{s}^2{x}^2,find\frac{dy}{dx}.$

What is the length of focal chord of a parabola $y^2=8xmakingangle{30}^{\circ }$ with x axis?

If the foci and vertices of an ellipse be (±1,0) and ( ±2,0) , then the minor axis of the ellipse os

For what value of n, $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ is the arithmetic mean of a and b?

The locus of the point p(x,y) satisfying the relation $\sqrt{(x-3{)}^2+(y-1{)}^2}+\sqrt{(x+3{)}^2+(y-1{)}^2}=6$ is

Select all the number(s) that would change the direction of inequality when multiplied with $2x>3$.

The distance between the foci of the hyperbola $9x^2-16y^2+18x+32y-151$ = 0 is

The coefficients of three successive terms in the

expansion of $(1+x{)}^n$ are 165, 330 and 462

respectively, then the value of n will be