Explore topic-wise InterviewSolutions in Current Affairs.

Integrate the function.
$\int \frac{x{e}^x}{(1+x{)}^2}dx.$

Shaded region in the following figure illustrates

If the following system of linear equations

$2x+y+z=5x-y+z=3x+y+az=b$

has no solution, then :

If the function $f\left(x\right)=\frac{ax+b}{(x-1)(x-4)}$is monotonic decreasing at $x=2$, then the possible values of $a$ and $b$ are:

A determinant of second order is made with the elements $0$ and $1.$ The number of determinants with non-negative values is

The area (in sq. units) of the largest rectangle $ABCD$ whose vertices $A$ and $B$ lie on the $x-$axis and vertices $C$ and $D$ lie on the parabola, $y=x^2-1$ below the $x-$axis, is

If the multiplicative inverse of $\frac{x^2-i{y}^2}{x+iy}$ is purely imaginary, where $x,y\ne 0$, then the value of $\frac{x}{y}$ is

If the roots of the quadratic equation $a(x-1{)}^2+2b(x-2)+c(x-1)+4=0$ are imaginary, where $a,b,c\in R$ and $b>2$, then

Let $\left[x\right]$ be the greatest integer less than or equal to $x$. Then, at which of the following point(s) the function $f\left(x\right)=x\cos \left(\pi \right(x+\left[x\right]\left)\right)$ is discontinuous?

For the reaction $H_2+B{r}_2\to 2HBr$ overall order is found to be 3/2. The rate of reaction can be expressed as

If $I_{(n,m)}=\int \frac{{\sin }^{n}x}{{\cos }^{m}x}dx,$ then the value of $I_{(3,4)}=$

(where $C$ is integration constant)

If $y={\tan }^{-1}\left(\frac{2^x}{1+2^{2x+1}}\right)$, then $\frac{dy}{dx}$ at $x=0$ is

Let $f$ and $g$ be differentiable funcitons on $R$, such that $fog$ is the identity funciton. If for some $a,b\in R,g^{\prime }\left(a\right)=5$ and $g\left(a\right)=b$, then $f^{\prime }\left(b\right)$ is equal to :

If $|\overrightarrow{x}|=|\overrightarrow{y}|=|\overrightarrow{x}+\overrightarrow{y}|=1$, then $|\overrightarrow{x}-\overrightarrow{y}|=$_______

The point of intersection of lines $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{1}$ and $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ is
[AISSE 1986; AMU 2005]

If $y=2^{x^3},$ then $\frac{dy}{dx}=$

If $\omega (\ne 1)$ is cube root of unity satisfying $\frac{1}{a+\omega }+\frac{1}{b+\omega }+\frac{1}{c+\omega }=2{\omega }^2$ and $\frac{1}{a+{\omega }^2}+\frac{1}{b+{\omega }^2}+\frac{1}{c+{\omega }^2}=2\omega$, then the value of $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}$ is :

​​​​​​$\begin{array}{cc}Column-I& Column-II\\ \left(I\right)\text{A circle}{x}^2+y^2-6x-10y+k=0& \\ \text{doesn't touch or intersect the coordinate axes}& \\ \text{and the point}(2,4)\text{lies inside the circle}& \\ \text{then the number of integral value of}k\text{is}& \left(P\right)5\\ \left(II\right)\text{If}\overrightarrow{V_1}=\hat{i}-2\hat{j}+3\hat{k},\overrightarrow{V_2}=a\hat{i}+b\hat{j}+c\hat{k}& \\ V_2\text{is non-zero vectors}\&a,b,c\in \{-1,0,1,2,3\},& \\ a,b,c\text{are choosen such that}\overrightarrow{V_1}\cdot \overrightarrow{V_2}=0& \\ \text{then the maximum of}(a+b+c)\text{is}& \left(Q\right)2\\ \left(III\right)\text{If a circle having radius}r\text{described on}& \\ \text{a normal chord of}{y}^2=4x\text{as diameter and}& \\ \text{passes through the vertex of the parabola}& \\ \text{, then}\left[r\right]\text{is}\left(\right[\left]\text{denotes G.I.F.}\right)& \left(R\right)6\\ \left(IV\right)\text{Consider}f\left(x\right)={\sin }^{-1}(\frac{x+3}{2x+5})\text{and}& \\ g\left(x\right)={\sin }^{-1}(\frac{a{x}^2+b}{x^2+5})\&\underset{x\to \infty }{lim}\left(f\right(x)-g(x\left)\right)=0,& \\ \underset{x\to 0}{lim}\left(f\right(x)+g(x\left)\right)=\frac{\pi }{4}\text{, then}10b^2\text{is}& \left(S\right)7\end{array}$

Which of the following is only "CORRECT" combination?

The set of values of $a$ for which the point$(a-1,a+1)$ lies outside the circle $x^2+y^2=8$ and inside the circle $x^2+y^2-12x+12y-62=0$ is

If $y={\tan }^{-1}\left(\frac{\cos x}{1+\sin x}\right),x\in (-\frac{\pi }{2},\frac{\pi }{2})$, then $\frac{dy}{dx}$ is equal to

The three vectors $7i-11j+k,5i+3j-2kand12i-8j-k$ form the sides of

If $x,y,z$ are in arithmetic progression with common difference $d,x\ne 3d,$ and the determinant of the matrix $\left[\begin{array}{ccc}3& 4\sqrt{2}& x\\ 4& 5\sqrt{2}& y\\ 5& k& z\end{array}\right]$ is zero, then the value of $k^2$ is:

The general solution of the differential equation $\frac{dy}{dx}+x(x+y)=x^3(x+y{)}^3-1$ is

(where ${}^{\prime }{C}^{\prime }$ is the constant of integration)

f(x) = $\{\begin{array}{c}-2,ifx\le -1\\ 2x,if-1<x\le 1\\ 2,ifx>1\end{array}$

Find the area of the circle $4x^2+4y^2=9$ which is interior to the parabola $x^2=4y$

The principal argument of the complex number $\frac{2+i}{4i+(1+i{)}^2},$ ( where $i=\sqrt{-1}$) is

The odds in favour of A solving a problem are 3 to 4 and the odds against B solving the same problem are 5 to 7. If they both try the problem, the probability that the problem is solved is:

The solution of the differential equation $\cos xdy=y\left(\sin \right(x)-y)dx,$ $0<x<\frac{\pi }{2}$ is

$($Here, $C$ is a constant of integration$)$

Reduce $\frac{2a^2-2ac+3ab-3bc}{3a^2-3ac-2ab+2bc}$

Let $y=y\left(x\right)$ be a curve passing through the point $(1,1)$ and satisfying $\frac{dy}{dx}+\frac{\sqrt{(x^2-1)(y^2-1)}}{xy}=0.$ If the curve passes through the point $(\sqrt{2},k),$ then the largest value of $|\left[k\right]|$ is

(Here, $[.]$ represents the greatest integer function)

Let P be the point on the parabola, $y^2=8x$ which is at a minimum distance from the centre C of the circle, $x^2+(y+6{)}^2=1.$ Then the equation of the circle, passing through C and having its centre at P is: