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If a and d are two complex numbers, then the sum

to (n+1) terms of the following series

a$C_0$ - (a + d)$C_1$ + (a + 2d)$C_2$ - ........... is

Find the inverse of the matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 1& 1& 1\\ 2& 3& 4\end{array}\right]$

A parabola has the origin as its focus and the line $x=4$ as the directrix. Then the vertex of the parabola is at

The value of $\int \frac{1}{x^2+4}dx$ is equal to



$\int \frac{1}{x^2+4}dx$ का मान बराबर है

A bag contains some white and some black balls, all combinations of balls being equally likely. The total number of balls in the bag is 10. If three balls are drawn at random without replacement and all of them are found to be black, the probability that the bag contains 1 white and 9 black balls is

If a and b are two positive quantities whose sum is $\lambda$, then the minimum value of $\sqrt{(1+\frac{1}{a})(1+\frac{1}{b})}$ is

The values of $a$ for which the quadratic equation $3x^2+2(a^2+1)x+(a^2-3a+2)=0$ has roots of opposite sign, is

If $f\left(x\right)=\underset{n\to \infty }{lim}(2x+4x^3+\dots +2^n{x}^{2n-1})$, where $x\in (0,\frac{1}{\sqrt{2}}),$ then $\int f\left(x\right)dx$ is equal to

A geometric progression with common ratio $r,$ consists of an even number of terms. If the sum of all terms is $5$ times the sum of the terms occupying the odd places, then $\sum _{i=1}^4(ir{)}^2$ is

The equation of the circle through the points of intersection of $x^2+y^2-1=0,x^2+y^2-2x-4y+1=0$ and touching the line x + 2y = 0, is

The equation of circle passing through the points $(4,1),(6,5)$ and having centre on the line $4x+y=16$ is

If $A(-2,1),B(2,3)$ and $C(-2,-5)$ are the vertices of an acute angled $△ABC,$ then the value of $\tan \angle B$ is

The range of $k$, for which the inequality $k{x}^2-3kx+3<0$, $($where $x\in R)$ holds true is

Find the equation of lines passing through intersection of lines x+y+4 = 0 and 3x-y-8 = 0 and equally inclined to axis..

The sum of all real values of $x$ satisfying the equation $(x^2-5x+5{)}^{(x^2+4x-60)}=1$ is

The unit vector in the direction of sum of the vectors $\overrightarrow{A}=2\hat{i}+4\hat{j}-7\hat{k}$ and $\overrightarrow{B}=4\hat{i}-6\hat{j}+4\hat{k}$ will be

Let $f:R\to R$ be defined as

$f\left(x\right)=\{\begin{array}{cc}-\frac{4}{3}{x}^3+2x^2+3x,& x>0\\ 3x{e}^x,& x\le 0\end{array}$

Then $f$ is increasing function in the interval:

If $2a{x}^2+3bx+5c=0,a\in R-\left\{0\right\},c>0$ does not have any real roots, then which of the following is/are always true?

The ${17}^{th}$ term from the end of the A.P. $-36,-31,-26,...,79$ is

Graph of f(x) is given. Find the value of left hand limit as x approaches 3.

Let $f:R\to R$ be a function such that $f\left(x\right)=x^3+x^2{f}^{\prime }\left(1\right)+x{f}^{\prime \prime }\left(2\right)+f^{\prime \prime \prime }\left(3\right),x\in R$. Then $f\left(2\right)$ equals :

If ${1690}^{2608}+{2608}^{1690}$ is divided by $7$, then the remainder is

Let $f\left(x\right)=(\begin{array}{c}{x}^2-1,\mathrm{if}0<x<2\\ 2x+3,\mathrm{if}2\le x<3\end{array}$, a quadratic equation whose roots are $\underset{x\to 2^{-}}{\lim }f\left(x\right)\mathrm{and}\underset{x\to 2^{+}}{\lim }f\left(x\right)$ is

If $P=\sum _{r=1}^{\infty }{\tan }^{-1}\left(\frac{1}{r+3}\right)Q=\sum _{r=1}^{\infty }{\tan }^{-1}\left(\frac{1}{r+1}\right)R=\sum _{r=1}^{\infty }{\tan }^{-1}\left(\frac{1}{r}\right)$

, then $P-2Q+R$ is

Let $x_1,x_2(x_1\ne x_2)$ be the roots of the equation $x^2+2(m-3)x+9=0$. If $-6<x_1,x_2<1,$ then ${}^{\prime }{m}^{\prime }$ lies in the interval

Match the lines given on the left side with their corresponding slopes on the right..

$\begin{array}{cc}Linepassesthroughthepoints& Slopeoftheline\\ p.\left)\right(1,6\left)and\right(-4,2)& 1.)0\\ q.\left)\right(5,9\left)and\right(2,9)& 2.)-3\\ r.\left)\right(-2,-1\left)and\right(-3,2)& 3.)\frac{4}{5}\\ s.\left)\right(4,0\left)and\right(3,3)& 4.)\frac{5}{3}\end{array}$

${lim}_{x\to \frac{\pi }{2}}(1-sinx)tanx$ will be equal to _____

The distance of the point $(1,3)$ from the line $2x-3y+9=0$ measured along a line $x-y+1=0$

The equation of the line(s) which passes through the point $(3,4)$ and its sum of the intercepts on the axes is 14 is/are

The value of ${\sin }^2\frac{\pi }{6}+{\cos }^2\frac{\pi }{3}-{\tan }^2\frac{\pi }{4}+{\cot }^2\frac{\pi }{2}$ is equal to

A signal which can be green or red with probability $\frac{4}{5}and\frac{1}{5}$ respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is $\frac{3}{4}$. If the signal received at station B is green, then the probability that the original signal green is

If $\alpha ,\beta$ be the roots of the equation $3\cos 2\theta +4\sin 2\theta =5$, then match the following from $\text{List I}$ to $\text{List II}$.



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \tan \alpha +\tan \beta & \text{(P)}& 0\\ \text{(B)}& \tan (\alpha +\beta )& \text{(Q)}& \frac{4}{3}\\ \text{(C)}& \tan (\alpha -\beta )& \text{(R)}& \frac{1}{4}\\ \text{(D)}& \tan \alpha \tan \beta & \text{(S)}& 1\end{array}$

If $\tan A$ and $\tan B$ are the roots of $x^2-3x-7=0,$ then the value of $\sin 2(A+B)$ is

Let $f:N\to R$ be a function satisfying the following conditions:

$f\left(1\right)=1$ and $f\left(1\right)+2f\left(2\right)+\dots +nf\left(n\right)=n(n+1)f\left(n\right)$ for $n\ge 2$.

If $f\left(999\right)=\frac{1}{K}$, then $K$ equals

The order of the differential equation whose solution is $y=acosx+bsinx+c{e}^{-x}$ is

Three numbers are choosen at random without replacement from {1, 2, 3, ....8}. The probability that their minimum is 3, given that their maximum is 6, is

If the tangent at the point $P\left(\theta \right)$ to the ellipse $16x^2+11y^2=256$ is also a tangent to the circle $x^2+y^2-2x=15,$ then possible value(s) of $\theta$ is/are

Let $f\left(x\right)=5x^3+px+q$, where $p$ and $q$ are real numbers. When $f\left(x\right)$ is divided by $x^2+x+1$, the remainder is $0$. Then the value of $p-q$ is

Let $f:R\to R$ be a function such that $f(x+y)=f\left(x\right)+f\left(y\right)+x^{2}y+x{y}^2$ $\forall x,y\in R$. If $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x}=1$, then $f\left(x\right)$ is

A plane meets the coordinate axes at points A, B, C and $(\alpha ,\beta ,\gamma )$ is the centroid of the triangle ABC. Then the equation of the plane is

The value of the determinat $\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right|$ is .

If the coefficients of $p^{th}$, $(p+1{)}^{th}$ and $(p+2{)}^{th}$ terms in the expansion of $(1+x{)}^n$ are in A.P., then

A and B are events such that $P\left(A\right)=0.3,P(A\cup B)=0.8$. If A and B are independent then P(B)=

The differentiation of ${\tan }^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)w.r.t.{\tan }^{-1}x$ is

Find the equation whose roots are the cubes of the roots of $x^3+3x^2+2=0$

If $\sin A+\sin B+\sin C=0$ and $\cos A+\cos B+\cos C=0$, then the value of $\sin \left(\frac{A-B}{2}\right)$ is

$(\text{where}A,B,C\in [0,2\pi ])$