Explore topic-wise InterviewSolutions in Current Affairs.

If y = 3 cos (log x) + 4 sin (log x), show that $x^2{y}_2+x{y}_1+y=0$
where $y_1$ & $y_2$ are the successive derivatives of y.

x+y+z = π, tan x.tan z = 2 and tan y.tan z =18, then tan² z = ?

A letter lock consists of three rings each narked with 10 different letters. In how many nays it is possible to make an unsuccessful attempt to open the lock ?

Match the following :

$\begin{array}{cccc}& coloumn1& & coloumn2\\ \left(A\right)& Theshortestdistancebetweenthelines& \left(P\right)& 13\\ & \frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}and\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}& & \\ \left(B\right)& Thedistanceofthepointofintersectionoftheline& \left(Q\right)& \sqrt{\frac{2109}{110}}\\ & \frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}andtheplanex-y+z=5fromthepoint& & \\ & (-1,-5,-10)is& & \\ \left(C\right)& Thelengthoftheperpendiculardrawnfromthepoint(5,4,-1)& \left(R\right)& 2\\ & ontheline\frac{x-1}{2}=\frac{y}{9}=\frac{z}{5}is& & \\ \left(D\right)& ThedistancebetweenthepointsPandQisdandthelengthof& \left(S\right)& 3\sqrt{30}\\ & theprojectionofPQontheco-ordinateplanesare{d}_1,d_2,d_{3}then& & \\ & {d_1}^2+{d_2}^2+k{d}^{2}whenkis& & \end{array}$

${lim}_{x\to 0}\frac{tanmx}{tannx}$

If $z=cos\theta +isin\theta$, then

The degree of the differential equation ${\left(\frac{d^{2}y}{d{x}^2}\right)}^3+{\left(\frac{dy}{dx}\right)}^2+sin\frac{dy}{dx}+1=0$ is

If $\frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ are in A.P., prove that :
(i) $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P.
(ii) bc, ca, ab are in A.P.

Prove that:
(i) $sin\alpha +sin\beta +sin\gamma -sin(\alpha +\beta +\gamma )=4sin\left(\frac{\alpha +\beta }{2}\right)sin\left(\frac{\beta +\gamma }{2}\right)sin\left(\frac{\gamma +\alpha }{2}\right)$

(ii) $cos(A+B+C)+cos(A-B+C)+cos(A+B-C)+cos(-A+B+C)=4cosAcosBcosC$

If $A,B,C,D$ are four points in a space and $|AB\times CD+BC\times AD+CA\times BD|=\lambda$(area of the triangle ABC). Then the value of $\lambda$ is

A card is drawn at random from a pack of 100 cards numbered 1 to 100. The probability of drawing a number which is a square is

${lim}_{n\to \infty }\frac{n!}{(n+1)!+n!}$ is equal to

If the roots of the equation $x^2-(p+4)x+2p+5=0$ are equal, then the value(s) of $p$ is/are

Sigma(r=1 to n) of [r/{(r^4)+(r^2)+1}] is equal to

If $\alpha$ and $\beta$ are the roots of $x^2-p(x+1)-q=0$, then

Mark the correct alternative in each of the following :

$\text{In a triangle}ABC,a=4,b=3,\angle A={60}^{\circ }\text{then c is a root of the equation}$

If the matrices $A=\left[\begin{array}{ccc}1& 1& 2\\ 1& 3& 4\\ 1& -1& 3\end{array}\right],B=\text{adj}A$ and $C=3A$, then $\frac{|adjB|}{|C|}$ is equal to :

If $\tan A=\frac{x\sin B}{1-x\cos B}$ and $\tan B=\frac{y\sin A}{1-y\cos A}$, then the value of $\frac{\sin A}{\sin B}$ for all permissible values of $A,B$ is

If sum of the coefficients of first, second and third terms in the expansion of ${(x^2+\frac{1}{x})}^m$ is $46$, then coefficient of the term that is independent of $x$ is:

If $x$ and $y$ co-ordinates of a point $P$ in the $xy$ plane are given by $x=\left(u\cos \alpha \right)t,y=\left(u\sin \alpha \right)t-\frac{1}{2}g{t}^2$ where $t$ is a parameter and $g,u$ and $\alpha$ are given constants. Then the locus of the point $P$ is a parabola whose vertex is

If $\overline{a},\overline{b},\overline{c}$ form a left handed orthogonal system and $\overline{a}\cdot \overline{a}=4,\overline{b}\cdot \overline{b}=9,\overline{c}\cdot \overline{c}$=16 then $\left[\overline{a}\overline{b}\overline{c}\right]$ =

When M.V.T is verified for f(x) on [a, b] then there exists c such that

The sum of all the solutions of the equation $\cos 2x+{\sin }^{2}2x=1$ which lie in the interval$[0,2\pi ]$ is equal to

If A is any square matrix, then AA’ is a

The vertices of the ellipse $\frac{(x+1{)}^2}{25}+\frac{(y-3{)}^2}{16}$ = 1 is

A circle which touches the positive axes, and whose centre it at distance $2\sqrt{2}$ from the origin, then the equation is

If the circles $x^2+y^2=a^2$ and $x^2+y^2-2gx+g^2-b^2=0$ touch each other externally, then

${lim}_{x\to 0}\frac{8}{x^8}[1-cos\frac{x^2}{2}cos\frac{x^2}{4}+cos\frac{x^2}{2}cos\frac{x^2}{4}]$ is equal to

Find the probability that the 3N’s come consecutive in the arrangement of the letters of the word “CONSTANTINOPLE”.

For positive integers $n,n^3+2n$ is always divisible by

If x sin^3 A + y cos^3 A = sinAcosA and x sinA= y cosA, prove that x^2 + y^2 = 1

If a,b,c, are in A.P. and p, p' are respectively A.M. and G.M. between a and b while q, q' are respectively AM. And G.M. between b and c, then

How is phi a subset of cross product of any two sets?

Show by the Principle of Mathematical induction that the sum $S_n$ of the n terms of the series $1^2+2\times 2^2+3^2+2\times 4^2+5^2+2\times 6^2+7^2+.....$ is given by $S_n=\{\begin{array}{cc}\frac{n(n+1{)}^2}{2},\text{if n is even}& \\ \frac{n^2(n+1)}{2},\text{if n is odd}& \end{array}$

The mean deviation of the series a, a+d , a+2d,.... a+2n from its mean is

The area (in sq. units) of the smaller of the two circles that touch the parabola, $y^2=4x$ at the point $(1,2)$ and the $x$-axis is :

If the coefficient of the $(n+1{)}^{th}$ term and the $(n+3{)}^{th}$ termin the expansion of $(1+x{)}^{20}$ are equal, then the value of n is

If sum of perpendicular distances of a variable point $P(x,y)$ from the lines $x+y-5=0$ and $3x-2y+7=0$ is always 10. Show that P must move on a line.

What is the corresponding point of $(acos\theta ,bsin\theta )$ of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ on its auxiliary cirlcle?

${lim}_{n\to \infty }\sqrt{x}[\sqrt{x+1}-\sqrt{x}]$

$\begin{array}{cccccc}& ColumnI& & Column2& & Column3\\ \left(I\right)& f\left(x\right)=tan\left(2ta{n}^{-1}\sqrt{\frac{\sqrt{1+\sqrt{x}}-1}{\sqrt{1+\sqrt{x}}+1}}\right)& \left(i\right)& \int f\left(x\right)dx=\frac{4}{3}{x}^{\frac{3}{4}}+C& \left(P\right)& {\int }_0^{1}f\left(x\right)dx=\frac{4}{3}\\ \left(II\right)& f\left(x\right)=cot\left(2ta{n}^{-1}\sqrt{\frac{\sqrt{1+\sqrt{x}}-\sqrt[4]{4x}}{\sqrt{1+\sqrt{x}}+\sqrt[4]{4x}}}\right)& \left(ii\right)& \int f\left(x\right)dx=\frac{4}{5}{x}^{\frac{5}{4}}+C& \left(Q\right)& {\int }_0^{1}f\left(x\right)dx=\frac{4}{5}\\ \left(III\right)& f\left(x\right)\left(\frac{1-tan\left(\frac{1}{2}si{n}^{-1}\right(\frac{1-\sqrt{x}}{1+\sqrt{x}}\left)\right)}{1+tan\left(\frac{1}{2}si{n}^{-1}\right(\frac{1-\sqrt{x}}{1+\sqrt{x}}\left)\right)}\right)& \left(iii\right)& \int f\left(x\right)dx=\frac{2}{3}{x}^{\frac{3}{4}}+C& \left(R\right)& {\int }_0^{1}f\left(x\right)dx=\frac{2}{3}\\ \left(IV\right)& f\left(x\right)=\sqrt{x}tan\left(2ta{n}^{-1}\left(\frac{\sqrt{\sqrt{1+\sqrt{x}+1}}-\sqrt{\sqrt{1+\sqrt{x}-1}}}{\sqrt{\sqrt{1+\sqrt{x}+1}}+\sqrt{\sqrt{1+\sqrt{x}-1}}}\right)\right)& \left(iv\right)& \int f\left(x\right)dx=\frac{2}{5}{x}^{\frac{5}{4}}+C& \left(S\right)& {\int }_0^{1}f\left(x\right)dx=\frac{2}{5}\end{array}$

Which of the following options is only correct combination?

If $f\left(x\right)=|x^2+(k-1\left)\right|x|-k|$ is non differentiable at five real points, then k will lie in

${lim}_{x\to \pi }\frac{1-sin\frac{x}{2}}{cos\frac{x}{2}(cosx\frac{x}{4}-sin\frac{x}{4})}$

$1.25+\mathrm{2.3.6}+\mathrm{3.4.7}+....$