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If $C_r$ stands for ${}^n{C}_r$ , the sum of the given series $\frac{2\left(\frac{n}{2}\right)!\left(\frac{n}{2}\right)!}{n!}[C_0^2-2C_1^2+3C_2^2-\cdots +(-1{)}^n(n+1)C_n^2]$

If $a(p+q{)}^2+2bpq+c=0$ and $a(p+r{)}^2+2bpr+c=0$, then $p^2+\frac{c}{a}$ is

The G.M of the number $3,3^2,3^3,..........,3^n$ is

$-\frac{2\pi }{5}$ is the principal value of

$\text{Find the equation of the hyperbola whose}\left(i\right)\text{foci are (6,4)and (-4,4) and eccentricity is 2.}\left(ii\right)\text{vertices are (-8,-1) and (16,-1) and focus is(17,-1)}\left(iii\right)\text{foci are (4,2) and (8,2) and eccentricity is 2.}\left(iv\right)\text{vertices are at}(0\pm 7)andfociat(0,\pm \frac{28}{4}).$

If A = {1, 3, 4} and B = {1, 4, 3, 2} then which of the following is true?

Differentiate sin x³ with respect to x³.

Two lines whose equation are $L_1:\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{\lambda }$ and $L_2:\frac{x-2}{3}=\frac{y-3}{2}=\frac{z-2}{3}$ lie in the same plane. If $L_1$ intersects a plane $x+y+z=15$ at P, then distance of P from (3, 4, 3) is

If $\alpha ,\beta$ are roots of the equation $4x^2+3x+7=0,$ then~,

$\frac{1}{\alpha }+\frac{1}{\beta }$ is equal to

If x=A/2 at t=0, find phase constant α in x=Asin(wt+α) . At t=0 , a particle executing shm is going along x axis.

The domain of the function$f\left(x\right)=\sqrt{1-4x}$ is

If $y=b[{\cot }^{-1}(-x)+{\tan }^{-1}(-x)],x\in R$ then $\frac{dy}{dx}=$

Two distinct polynomial f(x) and g(x) are defined as follows:

$f\left(x\right)=x^2+ax+2;g\left(x\right)=x^2+2x+a$

If the equation f (x) = 0 and g(x) = 0 have a common root, then the sum of the roots of the equation f (x) + g(x) = 0 is

The sum of all solutions of the equation $\cos 3\theta =\sin 2\theta$ in the interval $[-\frac{\pi }{2},\frac{\pi }{2}]$ is

If the term independent of $x$ in the expansion of ${(\sqrt{x}-\frac{m}{x^2})}^{10}$ is $405$, the value of $m$ is :

The value of $20!+\frac{21!}{1!}+\frac{22!}{2!}+.....+\frac{60!}{40!}$ is

The interval in which the function $y=x-2\sin x;0\le x\le 2\pi$ increases throughout is :

The point (4,1) undergoes the following three transformations successively:-

i) Reflection about the line y=x

ii)Translation through a distance of 2units along the positive direction of X axis.The final position of the point is

For the differential equation in given question find a particular solution satisfying the given condition.

$(x^3+x^2+x+1)\frac{dy}{dx}=2x^2+x,y=1whenx=0.$

Find the vector equation of the line passing through the point (1, 2, -4) and perpendicular to the two lines
$\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}and\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}$

${\int }_0^{\pi /4}cosx{e}^{sinx}dx$ is equal to
(a) e + 1 (b) e - 1 (c) e (d) -e

Let $f:R\to R$ be the function defined by $f\left(x\right)=\frac{1}{2-cos{x}^{\prime }}\forall x\in R$.Then, find the range of f.

The probability that a bulb produced by a factory will fuse after 150 days of used is 0.05. Find the probability that out of 5 such bulbs
more than one will fuse after 150 days

A water jet coming out of the small opening O of a fountain reaches its maximum height of 4 metres at a distance of 0.5 metre from the vertical. Find the height of the jet above the horizontal OX at a distance of 0.75 metre from the point O.

The complex number having least positive argument and satisfying the inequality $|z-5i|\le 3$ is

Let f(x) = x + 1; where $xϵ[0,\infty ]$. Choose the right option.

If sin x+ sin y + sin z = 3 than what is the value of cos x + cos y + cos z

Which of the following expression(s) represent a polynomial

Let $A$ and $B$ be two finite sets. The set $A$ has $480$ more subsets than $B.$ If $A\cap B$ has $2$ elements, then the number of elements in $A\cup B$ is

The normal to curve $xy=4$ at the point $(1,4)$ meets the curve again at point

If $\sin \theta =\frac{3}{5},\cos \varphi =\frac{12}{13}$ (where $\theta$ and $\varphi$ both belongs to $1^{\text{st}}$ quadrant), then which of the following is/are true?

If $n$ is an odd integer, then $(1+i{)}^{6n}+(1-i{)}^{6n}=$

If A={x:x=4 power n minus 3n minus 1 and n belongs to mode of N } and B = {y:y = 9(n -1) and n belongs to mode of N } Prove that A subset of B

The arithmetic mean of nC0,nC1,nC2....nCn is
(A)2^n/n
(B)2^n-1/n
(C)2^n/n+1
(D)2^n-1/n+1

If ${\int }_{ln2}^{x^2(1+x)}f\left(t\right)dt=x,$ then value of $f\left(2\right)$ is