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If $(1+x{)}^n=C_0+C_{1}x+\cdots +C_n{x}^n,$ then the value of $\sum _{r=0}^n\sum _{s=0}^n{C}_r{C}_s$ is equal to

f(x) = $x^2$+2x+5. Find the average rate of change of f(x) in the interval [0,2]

From given figure, $\sin x+\sec x+\tan x$ is $\alpha$, find $156\alpha$.

Number of words that can be formed by taking $4$ letters at a time out of the letters of the word $MATHEMATICS$ is

Let $\alpha ,\beta$ be the roots of the equation $x^2-px+r=0$ and $\frac{\alpha }{2},2\beta$ be the roots of the equation $x^2-qx+r=0$. Then the value of r is:

The number of solution(s) of $y=x^2+10x+22$ and $y=\log x,$ is

If $\sin 2A+\sin 2B=\frac{1}{2}$ and $\cos 2A+\cos 2B=\frac{3}{2}$, then the value of $|\cos (A-B)|$ is

Equation of plane passing through points given by position vectors $\overline{a}and\overline{b}$ and origin will be

Suppose that the three quadratic equations $a{x}^2-2bx+c=0,b{x}^2-2cx+a=0$ and $c{x}^2-2ax+b=0$ all have only positive roots. Then,

The foci of the hyperbola are $S(5,6),S^{\prime }(-3,-2)$. If its eccentricity is $2$, then the equation of its directrix corresponding to focus $S$ is

Solve the irrational inequality:

$\frac{3}{\sqrt{2-x}}-\sqrt{2-x}\le 2$

If the distance between the points $(5,-2)$ and $(1,a)$ is $5$ units, then the sum of all possible values(s) of $a$ is

If $2\tan A+\cot A=\tan B$, then the value of $\cot A+2\tan (A-B)$ is

If a,b,c be in A.P. and b,c,d be in H.P., then

Let $S$ and $S^{\prime }$ be foci of an ellipse and $B$ be any one of the extremities of its minor axis. If $\Delta S^{\prime }BS$ is a right angled triangle with right angle at $B$ and area of $△S^{\prime }BS=8\text{sq. units}$, then the length of a latus rectum of the ellipse (in units) is :

Two poles standing on a horizontal ground are of heights $5$ m and $10$ m respectively. The line joining their tops makes an angle of ${15}^{\circ }$ with ground. Then the distance (in m) between the poles, is :

Let $y=y\left(x\right)$ be the solution of the differential equation, $\frac{dy}{dx}+y\tan x=2x+x^2\tan x,x\in (-\frac{\pi }{2},\frac{\pi }{2})$, such that $y\left(0\right)=1.$ Then :

The locus of the moving point $P$ such that $2PA=3PB$, where $A$ is $(0,0)$ and $B$ is $(4,-3)$, is

If $\alpha ,\beta$ and $\gamma$ are the roots of $x^3+8=0$, then the equation whose roots are ${\alpha }^2,{\beta }^{2}and{\gamma }^2$ is

If we convert the denominator of the integral into a perfect square, $\int \frac{1}{x^2-x+1}dx$ then the correct integral will be

For $3\times 3$ matrices M and N, which of the following sttements (s) is/are not correct?

$\frac{5+9+13+.......upton.terms}{If7+9+11+.....upto(n+1)terms}=\frac{17}{16}$ then n is equal to

If a < 0 and D < 0, what can be inferred about $f\left(x\right)$ = $a{x}^2+bx+c$?

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

If the product of $3$ consecutive numbers in G.P. is $216$ and the sum of their products in pairs is $156$, then the smallest term in the three is

In some of the cases we can split the integrand into the sum of the two functions such that the integration of one of them by parts produces an integral which cancels the other integral. Suppose we have an integral of the type

$\int \left[f\right(x\left)h\right(x)+g(x\left)\right]dx$

Let $\int f\left(x\right)h\left(x\right)dx=I_1$ and $\int g\left(x\right)dx=I_2$

Integrating $I_1$ by parts, we get

$I_1=f\left(x\right)\int h\left(x\right)dx-\int \left\{f^{\prime }\right(x)\int h(x\left)dx\right\}dx$



$\int \frac{x{e}^x}{(1+x{)}^2}dx$ is equal to

$\frac{\sin 2A}{1+\cos 2A}$ =

Let $OPQR$ be a square and $M$ and $N$ be the midpoints of the sides $PQ$ and $QR$ respectively. The ratio of the area of square to the triangle $OMN$ is

If $f\left(x\right)=x^3$, then the graph of $g\left(x\right)=f(x+2)$ is

The number of $5$-digit numbers that can be formed using the digits $0,2,3,6,8,9$ (without repetition) such that it is divisible by $4$ is

Let $z_1$ and $z_2$ be roots of the equation $z^2+pz+q=0$ where the coefficients $p$ and $q$ may be complex numbers. Let $A$ and $B$ represent $z_1$ and $z_2$ in the complex plane, If $\angle AOB=\theta \ne 0$ and $OA=OB$ where $O$ is the origin, then $p^2$ is

There are two die $A$ and $B$ both having six faces. Die $A$ has three faces marked with $1$, two faces marked with $2$, and one face marked with $3$. Die $B$ has one face marked with $1$, two faces marked with $2$, and three faces marked with $3$. Both dices are thrown randomly once. If $E$ be the event of getting sum of the numbers appearing on top faces equal to $x$, let $P\left(E\right)$ be the probability of event $E$, then

$P\left(E\right)$ is minimum when $x$ equals to

The point(s) on $y-$ axis which is equidistant from the points $(12,3)$ and $(-5,10)$ is/are

If the coefficients of $r^{\text{th}},(r+1{)}^{\text{th}}\text{and}(r+2{)}^{\text{th}}$ terms in the expansion of $(1+x{)}^{14}$ are in A.P., then value $r$ can be

The value of ${\sin }^2\frac{\pi }{8}+{\sin }^2\frac{3\pi }{8}+{\sin }^2\frac{5\pi }{8}+{\sin }^2\frac{7\pi }{8}$ is

If the length of the perpendicular from the point $(\beta ,0,\beta )(\beta \ne 0)$ to the line, $\frac{x}{1}=\frac{y-1}{0}=\frac{z+1}{-1}$ is $\sqrt{\frac{3}{2}}$, then $\beta$ is equal to :

If the line's $(1+t)x-2ty+3t^2=0$ and $t^{2}x-(3-t)y+6=0,t\ne 0$ are perpendicular to each other, then the number of possible value(s) of $t$ is

$si{n}^{-1}[x\sqrt{1-x}-\sqrt{x}\sqrt{1-x^2}]=$

In a three dimensional co - ordinate system P, Q and R are images of a point A(a, b, c) in the xy the yz and the zx planes respectively. If G is the centroid of triangle PQR then area of triangle AOG is (O is the origin)

The coordinates of the foci of the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$ is/are

The number of triangles whose vertices are at the vertices of an octagon but none of whose sides happen to come from the octagon, is

The number of words that can be formed by the letters of $SOURCE$ is

If the ratio of the $5^{\text{th}}$ term from the beginning to the $5^{\text{th}}$ term from the end in the expansion of ${(2^{1/4}+\frac{1}{3^{1/4}})}^n$ is $\sqrt{6}:1$, then the value of $n$ is

If $x^2-(a-3)x+a=0$ has atleast one positive root, then $a\in$