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If $f$ and $g$ are continuous functions in $[0,a]$ satisfying $f\left(x\right)=f(a-x)$ and $g\left(x\right)+g(a-x)=a$, then $\underset{0}{\overset{a}{\int }}f\left(x\right)\cdot g\left(x\right)dx$ is equal to

The value of $\int \frac{{\tan }^{-1}\left(\ln x\right)+{\cot }^{-1}\left(\ln x\right)}{1+x^2}dx$ such that $x>1,$ is

(where $C$ of constant of integration)

If $f:R\to [\frac{-\pi }{4},\frac{\pi }{2})$ defined by $f\left(x\right)={\tan }^{-1}(x^4-x^2-\frac{7}{4}+{\tan }^{-1}\alpha )$ is surjective, then

Two vertical poles $AB=15$m and $CD=10$m are standing apart on a horizontal ground with points $A$ and $C$ on the ground. If $P$ is the point of intersection of $BC$ and $AD$, then the height of $P$(in m) above the line $AC$ is

In the expansion of ${(x+\frac{2}{x^2})}^{15}$, the term independent of x is

2 tangents are drawn from point P to the circle as shown.M is the midpoint of OP. Whats the ratio $\frac{\angle QMR}{\angle QPR{}^{\prime }}$

By using
properties of determinants, show that:

The middle term in the expansion of $(1+x{)}^{2n}$

Distinct prime numbers $p,q,r$ satisfy the equation $2pqr+50pq=7pqr+55pr=8pqr+12qr=A,$ for some positive integer $A.$ Then $A$ is

If the line $x=\alpha$ divides the area of region $R=\left\{\right(x,y)\in R^2:x^3\le y\le x,0\le x\le 1\}$ into two equal parts, then

If the points $P(-{\lambda }^2,\lambda ,0)$ and $Q({\lambda }^2,\frac{7\lambda }{3},7)$ lies on the same sides of the plane $x-3y+2z=4.$ Then the integral value(s) of $\lambda$ can be ?

The approximate value of $(1.0002{)}^{3000}$ is

1. 3

Let the equation of two concentric circles is $x^2+y^2-2x+4y+\lambda =0$. If a chord of first circle is tangent to second circle and normal to a circle passing through centre of these circles and having a centre at $(2,-1)$ is also touching first circle. Then length of chord with respect to first circle is:

Perpendicular are drawn from points on the line $\frac{x+2}{2}=\frac{y+1}{-1}=\frac{z}{3}$ to the plane x + y + z = 3. The feet of perpendiculars lie on the line

The minimum value of $9ta{n}^2\theta +4co{t}^2\theta$ is

If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2+(3{)}^{1/4}x+3^{1/2}=0,$ then the value of ${\alpha }^{96}({\alpha }^{12}-1)+{\beta }^{96}({\beta }^{12}-1)$ is equal to

The function $f\left(z\right)=\frac{z^2+1}{z^2+4}$ is singular at

The value of $(1+\cos \theta +i\sin \theta {)}^n+(1+\cos \theta -i\sin \theta {)}^n$ is

If $a=lo{g}_x\left(yz\right),b=lo{g}_y\left(zx\right),c=lo{g}_z\left(xy\right)$ where $x,y,z$ are positive reals not equal to unity then $abc-a-b-c$ is equal to -

If the coefficient of $3^{\text{rd}},4^{\text{th}}$ and $5^{\text{th}}$ terms in the expansion of $(1+x{)}^n,n\in N$ are in A.P., then the number of possible value(s) of $n$ is

If $cosx-sin\alpha cot\beta sinx=cos\alpha$ ,

then the value of $tan\left(\frac{x}{2}\right)$ is

The value of the definite integral $\underset{0}{\overset{\pi /3}{\int }}\ln (1+\sqrt{3}\tan x)dx$ is equal to

For any $\theta \in (\frac{\pi }{4},\frac{\pi }{2})$, the expression $3(\sin \theta -\cos \theta {)}^4+6(\sin \theta +\cos \theta {)}^2+4{\sin }^6\theta$ equals:

Solve the following system of equations in R.

x+3 >0,2x<14

Which of the following equations represents the given graph.

For the complex number $z=\frac{3+\sqrt{-1}}{2-\sqrt{-1}}$, the correct option(s) is/are

If $2{\tan }^{-1}\left(\cos x\right)={\tan }^{-1}\left(2\text{cosec}x\right)$, then the value of $x$ is:

Let $f\left(x\right)=(4+y^2)x^2+2xy+1$ and $g\left(y\right)$ be the minimum value of $f\left(x\right)$. If $y\in R,$ then the maximum value of $g\left(y\right)$ is

Three schools send $2,4$ and $6$ students, respectively, to a summer camp. The $12$ students must be accommodated in 6 rooms numbered $1,2,3,4,5,6$ in such a way that each room has exactly $2$ students and both from the same school. The number of ways, the students can be accommodated in the rooms is

If a > 0, then $\sqrt{\{a+\sqrt{(a+....)}\}}$=