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The value of $\underset{-\pi }{\overset{\pi }{\int }}\frac{{\cos }^{2}x}{1+a^x}dx,a>0$, is

Who, in today’s world, is our chief source of information about personalities?

If a differentiable function f(x) has a relative minimum at x = 0, then the function y = f(x) + ax + b has a relative minimum at x = 0 for

The value of $\underset{0}{\overset{100\pi }{\int }}\sqrt{1-\cos 2x}dx$ is

Let $f$ be any function continuous on $[a,b]$ and twice differentaible on $(a,b)$. If for all $x\in (a,b),f^{\prime }\left(x\right)>0$ and $f^{\prime \prime }\left(x\right)<0$, then for any $c\in (a,b),\frac{f\left(c\right)-f\left(a\right)}{f\left(b\right)-f\left(c\right)}$ is greater than :

The coefficient of the term independent of x in the expansion of ${(ax+\frac{b}{x})}^{14}$ is

If $\theta$ is the angle subtended at the focus by the chord which is normal at the point $(\lambda ,\lambda ),\lambda \ne 0$ to the parabola $y^2=4x,$ then the equation of line making angle with positive x-axis and passing through (1, 2) is

The number of different sums can be formed with the following coins a rupee a fifty paise a twenty five paise a ten paise a five paise?

The value of the summation $\sum _{i=1}^{\infty }\frac{2i-1}{2^{i+1}}$ is

If a class consists of $6$ periods, then the number of ways in which $5$ subjects can be taught such that each subject must be allotted at least one period and no period remains vacant is

Let P be the image of the point (3, 1, 7) with respect to the plane x - y + z = 3. Then, the equation of the plane passing through P and containing the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{1}$ is

A company X Ltd. is setting up a new plant in India for manufacturing auto components. India has a highly competitive and cost-effective production base in this sector. Many reputed car manufacturers source their auto components from here. X Ltd. is planning to capture about 40% of the market share in India and also export to the tune of at least $ 5 million in about 2 years of its planned operations. To achieve these targets it requires a highly trained and motivated workforce. You have been retained by the com pany to advise it in this matter. While giving answer keep in mind the sector, the company is operating.

Outline the process of selecting the company should follow with reasons.

verify that the area of the triangle with vertices (2,3) ,(5,7) and (-3-1) aremains invariant under the translation of axes when the origin is shifted to the point(-1,3).

The mirror image of the following figure, when mirror placed on line AB, will be



निम्नलिखित चित्र का दर्पण प्रतिबिम्ब कौनसा होगा जब दर्पण, रेखा AB पर स्थित हो?

If $\sin x+\cos x=\sqrt{y^2+\frac{1}{y^2}}$ for $x\in [0,\pi ]$ has a solution, then which of the following is/are correct?

If (x^2+ax+20)(x^2+17x+b)=0 find the roots by using quadratic equation's properties????

Suppose $p,q,r$ are in A.P. and $p^2,q^2,r^2$ are in G.P. If $p<q<r$ and $p+q+r=\frac{3}{2},$ then the value of $p$ is

If the squared difference of the quadratic polynomial f(X)=x^{​​​​​​2​​​​​}​​​​​+px+45 is equal to 144, find the value of p.

If $A$ is the set of people who travel only by bus, $B$ is the set of people who travel only by car, then which of the following will be the correct representation?

Value of the $\tan \left[\frac{1}{2}{\cos }^{-1}\left(\frac{2}{3}\right)\right]$ is

,
for some fixed
and

The minimum value of $\frac{(6+x)(11+x)}{(2+x)},x\ge 0$ is

If $\frac{{\sin }^{4}x}{3}+\frac{{\cos }^{4}x}{4}=\frac{1}{7},$ then the value of $\frac{{\sec }^{2}x}{3}+\frac{{\text{cosec}}^{2}x}{4}$ is

The value of $\underset{n\to \infty }{lim}(\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+....+\frac{1}{4n})$ is

The equation to the pair of tangents drawn from (–1, –2) to parabola $x^2=2y$ is

Consider the following statements:

$p:2$ is a prime number

$q:\cos {30}^{\circ }=\frac{1}{2}$

$r:{\sec }^{2}x+{\tan }^{2}x=1$

$s:\sqrt{7}$ is an irrational number

$u:{\pi }^2$ is greater than $10$



Then which among the following contain only valid statements?

Tangents are drawn to the ellipse $\frac{x^2}{36}+\frac{y^2}{9}=1$ from any point on the parabola $y^2=4x.$ The corresponding chord of contact will touch a parabola, whose equation is

If R be a relation $<$ from A={1,2,3,4} to B={1,3,5} i.e., (a,b) $\in R\iff a<b,thenRo{R}^{-1}$ is