Explore topic-wise InterviewSolutions in .

If $\int {\sin }^{-1}\left(\sqrt{\frac{x}{1+x}}\right)dx=A\left(x\right){\tan }^{-1}\left(\sqrt{x}\right)+B\left(x\right)+C,$ where $C$ is a constant of integration, then the ordered pair $\left(A\right(x),B(x\left)\right)$ can be:

If $z=x+iy,x,y\in R$ and $Im\left(\frac{2z+1}{i\overline{z}+1}\right)=-2$, then

If $S_1,S_2,S_3\dots \dots ,S_n,\dots$ are the sums of infinite geometric series whose first terms are $1,2,3,\dots \dots n,\dots \dots$ and whose common ratios are $\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots \dots \frac{1}{n+1}\dots \dots$ respectively, then the value of $\sum _{r=1}^{2n-1}{S}_r^2$ is

(ii) Write all the values of k for which the quadratic equation x² + kx + 16 = 0 has equal roots. Find the roots of the equation so obtained.

If for some positive integer $n$, the coefficients of three consecutive terms in the binomial expansion of $(1+x{)}^{n+5}$ are in the ratio $5:10:14,$ then the largest coefficient in this expansion is

If $\Delta \left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$ and $A_{ij}$ is cofactor of $a_{ij}$, then value of $\Delta$ is given by

a) $a_{11}{A}_{31}+a_{12}{A}_{32}+a_{13}{A}_{33}$
b) $a_{11}{A}_{11}+a_{12}{A}_{21}+a_{13}{A}_{31}$
c) $a_{21}{A}_{11}+a_{22}{A}_{12}+a_{23}{A}_{13}$
d) $a_{11}{A}_{11}+a_{21}{A}_{21}+a_{31}{A}_{31}$

Prove that: (1+cot+ tan A)(sin A- cos A)= sec A/ cosec²A-cosec A/ sec² A.

The locus of the foot of the perpendicular drawn from origin to a straight line which passes through a fixed point $P(h,k)$ is denoted by S. The tangent at $P(h,k)$ on the curve S is given by

Let $S$ be the set of all points where the function $f\left(x\right)=|x-\pi |(e^{|x|}-1)\sin |x|$ is not differentiable, then $S$ is

The general solution of the equation $\ln \left(\frac{dy}{dx}\right)=ax+by$ is:

Let $f:S\to S$ where $S=(0,\infty )$ be a twice differentiable function such that $f(x+1)=xf\left(x\right).$ If $g:S\to R$ be defined as $g\left(x\right)={\log }_{e}f\left(x\right),$ then the value of $|g^{\prime \prime }\left(5\right)-g^{\prime \prime }\left(1\right)|$ is equal to :

Let the
vectors
and
be
such that
and,
then
is a unit vector, if the angle between
and
is

(A) (B) (C) (D)

The value of the expression $1.(x-\omega )(2-{\omega }^2)+2.(3-\omega )(3-{\omega }^2)+.........+(n-1).(n-\omega )(n-{\omega }^2)$ where $\omega$ is an imaginary cube root of unity, is

Domain of the function f(x) = $si{n}^{-1}(2x^2+3x+1)$ is

If ${\Delta }_1=\left|\begin{array}{cc}x& b\\ a& x\end{array}\right|$ and ${\Delta }_2=\left|\begin{array}{ccc}x& b& b\\ a& x& b\\ a& a& x\end{array}\right|$ are given determinants, then

Let $\overrightarrow{u}$ be a vector coplanar with the vectors $\overrightarrow{a}=2\hat{i}+3\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{j}+\hat{k}$. If $\overrightarrow{u}$ is perpendicular to $\overrightarrow{a}$ and $\overrightarrow{u}\cdot \overrightarrow{b}=24,$ then $|\overrightarrow{u}{|}^2$ is equal to

If $\sec x+\tan x=p,$ then the value of $\frac{\sec x-\tan x}{2\sec x}$ is

Let S be the set of all non - zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2-x+\alpha =0$ has two distinct real roots $x_1$ and $x_2$ satisfying the inequality $|x_1-x_2|<1$. Which of the following interval(s) is/are a subset of S?

The number of ordered triples (a,b,c) of positive integers which satisfy the simultaneous equations ab + bc = 44 and ac + bc = 23 is

If $\sum _{n=1}^{20}\sum _{m=1}^{20}{\tan }^{-1}\left(\frac{m}{n}\right)=k\pi ,$ then the value of $k$ is

Mean of goals scored by Ronaldo and Messi for each year for the past 5 years is 48 and 50. Standard deviation for each of them is 6 and 4 respectively. Who is more consistent?

If two circles $x^2+y^2-2ax+c^2=0$ and $x^2+y^2-2by+c^2=0$ touch each other externally, then