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If a variable takes the values $0,1,2,...,n$ with corresponding frequencies as binomial coefficients ${}^n{C}_0,{}^n{C}_1,...,{}^n{C}_n$, then the mean of distribution is

Insert 5 geometric means between 16 and $\frac{1}{4}$.

The number of arrangements of the letters of the word BHARAT taking 3 at a times is

If $\cos (\alpha +\beta )=\frac{3}{5},\sin (\alpha -\beta )=\frac{5}{13}$ and $0<\alpha ,\beta <\frac{\pi }{4},$ then $\tan \left(2\alpha \right)$ is equal to:

The value of $\left|\begin{array}{ccc}cos(\theta +\propto )& -sin(\theta +\propto )& cos2\propto \\ sin\theta & cos\theta & sin\propto \\ -cos\theta & sin\theta & \lambda cos\propto \end{array}\right|$ is

A circular disc with 3 sectors marked as alphabet O, A, B has circumference 1. The alphabet coming infront of marker is noted down after each spin. Assume boundary of sectors do not come infront of marker. It is spun 6 times around its center and resulting alphabet which comes infront of marker are noted down in order. If the probability that the word BAOBAA to be formed is maximized, then (where x, y, z respectively denotes the probability of occurrence of alphabet O, A, B),

If $x={\sin }^{-1}\left(\sin 10\right)$ and $y={\cos }^{-1}\left(\cos 10\right)$, then $y-x$ is equal to :

Find the equation of the circle which passes through (3, - 2), (-2, 0) and has its centre on the line 2x -y = 3

If $A=\{a,b,2,3\},B=\{a,3,c\},$ $C=\{1,3,c\},$ then $n\left(\right(A\times B)\cap (A\times C\left)\right)$ is

Suppose ABCDEF is a hexagon such that AB = BC = CD = 1 and DE = EF = FA = 2. If the vertices A,B,C,D,E,F are concyclic, the radius of the circle passing through them is.

If mean and standard deviation of $5$ observations $x_1,x_2,x_3,x_4,x_5$ are $10$ and $3$, respectively, then the variance of $6$ observations $x_1,x_2,\cdots ,x_5$ and $-50$ is equal to :

The equation of the parabola whose focus is (1,-1) and the directrix is x+y+7=0 is

If A = BB’ + CC’, where B’ + C’ are respectively transpose of B and C respectively, $B=\left[\begin{array}{c}cos\theta \\ sin\theta \end{array}\right]$ and $C=\left[\begin{array}{c}sin\theta \\ -cos\theta \end{array}\right],\theta \in R$, then A =

A line passes through (2,2) and is perpendicular in the line 3x+y = 3 its y-intercepts is ------------

The domain of the function $f\left(x\right)=\sqrt{\frac{1-|x|}{|x|-2}}$ is

If $x\in [-4,-1]$, then $\frac{1}{x^2+4x+7}$ belongs to

$\underset{x\to 0}{Lt}=\frac{sin(\frac{\pi }{5}+x)-sin(\frac{\pi }{5}-x)}{x}$ is equal to

$2^3+4^3+6^3+8^3+....$

If $\frac{\pi }{4}<\theta <\frac{\pi }{2}$, then write the value of $\sqrt{1-sin2\theta }$

If one real root of the quadratic equation $81x^2+kx+256=0$ is cube of the other root, then a value of $k$ is :

The quadratic equation ${3ax}^2+2bx+c=0$ has at least one root between 0 and 1 if

Considering only the principal values of inverse functions, the set $A=\{x\ge 0:{\tan }^{-1}(2x)+{\tan }^{-1}(3x)=\frac{\pi }{4}\}$

$se{c}^6\theta +co{s}^6\theta =1-3si{n}^2\theta co{s}^2\theta$

A circle and an ellipse have centres at $(0,0)$ and the circle passes through foci $F_1$ and $F_2$ of the ellipse, such that the two curves intersect at four points. Let $P$ be any one of their points of intersection. If the length of major axis of the ellipse is $17$ and area of the triangle $P{F}_1{F}_2$ is $30$, then distance between foci is

For any two statements $pandq$, the negation of the expression $p\vee (\sim p\wedge q)$ is :

Let $y=f\left(x\right)$ be the solution of the differential equation $(1-x^2)\frac{dy}{dx}-xy=1x\in (-1,1)$.
If $y\left(0\right)=0$ then $y\left(\frac{1}{2}\right)=$

If sum of three prime numbers is $r$ where two numbers out of three are twin prime numbers and $r$ is an even number less than $20,$ then product of the numbers can be

The area of the triangle formed by the asymptotes and any tangent to the hyperbola $x^2-y^2=a^2$

If n is positive integer and $(3\sqrt{3}+5{)}^{2n+1}=\alpha +\beta$ where $\alpha$ is an integer and $0<\beta <1$, then

If $A$ is a skew symmetric matrix such that $A^{T}A=I$, then $A^{4n–1},\left(nϵN\right)$ is equal to

If the coefficient of x in ${(x^2+\frac{k}{x})}^5$ is 270, then k =

If $x=3\tan t$ and $y=3\sec t$, then the value of $\frac{d^{2}y}{d{x}^2}\text{at}t=\frac{\pi }{4}$, is :

Let $S$ be the set of points whose abscissas and ordinates are natural numbers. Let $P\in S$ such that the sum of the distance of $P$ from $(8,0)$ and $(0,12)$ is minimum among all elements in $S$. Then the number of such points $P$ in $S$ is

If $\sum _{i=1}^{20}{\left(\frac{{}^{20}{C}_{i-1}}{{}^{20}{C}_i+{}^{20}{C}_{i-1}}\right)}^3=\frac{k}{21},$ then $k$ equals :

If $x+y\sqrt{2}=2\sqrt{2}$ is a tangent to the ellipse $x^2+2y^2=4,$ then the eccentric angle of the point of contact is

If $y=sin\left(sinx\right)$, prove that $\frac{d^{2}y}{d{x}^2}+tanx\frac{dy}{dx}+yco{s}^{2}x=0.$

Which of the following functions are strictly decreasing on $(0,\frac{\pi }{2})$.

i) cosx

ii) cos 2x

iii) cos 3x

iv) tan x

Show that $f\left(x\right)=|x-5|$ is continuous but not differentiable at x=5