Explore topic-wise InterviewSolutions in .

The value of $\cos 1^{\circ }\cdot \cos 2^{\circ }\cdot \cos 3^{\circ }\cdots \cos {180}^{\circ }$ is

The equation of the circle passsing through $(1,0)$ and $(0,1)$ and having the smallest possible radius is

If the maximum and the minimum values of $\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& 1+{\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+4\sin 2x\end{array}\right|$ are $M$ and $m$ respectively, then $\frac{M}{m}$ is

If the derivative of the function $f\left(x\right)=\{\begin{array}{cc}b{x}^2+ax+4;x\ge -1& \\ a{x}^2+b;x<-1^{\prime }& \end{array}$ is continuous everywhere. Then

The inequality $\left|x^2\sin x+{\cos }^{2}x{e}^x+{\ln }^{2}x\right|<x^2\left|\sin x\right|+{\cos }^{2}x{e}^x+{\ln }^{2}x$ is true for $x\in$

Find the equation of a sphere, whose end points of a diameter are (0,0,0) and (1,2,3)

Let $P=(-1,0),Q=(0,0)\text{and}R=(3,3\sqrt{3})$ be three points. The equation of the bisector of the angle $PQR$ is

For the equation $x^2+bx+c=0,$ if $1+b+c=0$ for all $b,c\in R,$ then roots are

If $(a,a^2)$ falls inside the angle made by the lines $y=\frac{x}{2},x>0,$ and $y=3x,x>0,$ then a belongs to

l, m, n, are the pth, qth and rth terms of a GP and all positive, then

If $\underset{x\to 1}{lim}\frac{x^2-ax+b}{x-1}=5,$ then $a+b$ is equal to :

What is the least value of $\frac{25{\sec }^{4}x-50{\sec }^{2}x+74}{{\tan }^{2}x}$ ?

$\int \frac{x^2+1}{x^4+1}dx$ will be equal to which of the following

If $S_n=(\frac{n}{1})\sin a+(\frac{n}{2})\sin 2a+\cdots +(\frac{n}{n})\sin na$ and $T_n=(\frac{n}{1})\cos a+(\frac{n}{2})\cos 2a+\cdots +(\frac{n}{n})\cos na$

where $n\in N$ and $a$ be the non-zero real number such that $a\ne (2n-1)\frac{\pi }{2}$, then which of the following is/are CORRECT?

$(\text{Here},(\frac{n}{r})={}^n{C}_r)$

If the vertices of a triangle be (a, 1), (b, 3) and (4, c), then the centroid of the triangle will lie on x-axis, if

If $cos3\theta$ = $\alpha cos\theta +\beta co{s}^3\theta$, then $(\alpha ,\beta )$ =

The number of solutions of $5{\cos }^2\theta -3{\sin }^2\theta +6\sin \theta \cos \theta =7$ in the interval $[0,2\pi ]$ is

If atleast one of the root of the equation $x^2-(a-3)x+a=0$ is greater than $2$, then $a$ lies in the interval

The length and foot of the perpendicular from the point (7, 14, 5) to the plane 2x + 4y - z = 2, are

If $\alpha ,\beta$ are the roots of the equation $x^2-px+r=0$ and $\frac{\alpha }{2},2\beta$ are the roots of the equation $x^2-qx+r=0$, then the value of $r$ in terms of $p$ and $q$ is

The separate equations of the asymptotes of rectangular hyperbola $x^2+2xy\cot 2\alpha -y^2=a^2$ are :

If $\tan {70}^{\circ }=\tan {20}^{\circ }+\lambda \tan {50}^{\circ },$ then $\lambda$ is equal to

The domain of $f\left(x\right)=\ln \left[x\right]$ is

(where [.] denotes the greatest integer function)

If two functions f(x) and g(x) defined on R intersect each other at x = a & x =b then the area enclosed between these two functions will be -

The locus of the point which divides the double ordinate of the parabola $y^2=6ax$ in the ratio $5:6,$ is

If $b$ is the first term of an infinite $G.P.$ whose sum is five, then $b$ lies in the interval :

PSQ is a focal chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}$ = 1 then $\frac{1}{SP}+\frac{1}{SQ}$ =

The locus of intersection of the lines $\mathrm{xcos}\alpha +\mathrm{ysin}\alpha =a\mathrm{and}\mathrm{xsin}\alpha -\mathrm{ycos}\alpha =b$ is , where a and b are constants.

If $x^2-hx-21=0,x^2-3hx+35=0(h>0)$ has a common root, then the value of h is equal to

The solution of $\frac{dy}{dx}=\frac{ax+b}{cy+d}$ represents a parabola if

If the function $f$ given by

$f\left(x\right)=x^3-3(a-2)x^2+3ax+7$, for some $a\in R$ is increasing in $(0,1]$ and decreasing in $[1,5)$, then a root of the equation,

$\frac{f\left(x\right)-14}{(x-1{)}^2}=0(x\ne 1)$ is :

Two persons each makes a single throw with a pair of dice. Then the probability that the sum on both dice are unequal, is:

Find the equation of a sphere whose centre is (1,2,3) and touches the plane x+2y+3z = 0

The set of roots of the equations $3x^2-5x-2=0$ can be written in roster form as

The area (in sq. units) of the region $\{x\in R:x\ge 0,y\ge 0,y\ge x-2\text{and}y\le \sqrt{x}\},$ is :

An auto mobile dealer provides motor cycles and scooters in 3 body patterns and 4 different colours each. The number of choices open to customer is

$$\begin{array}{cc}coloumn1& coloumn2\\ a& p)x\\ b& q)-x\\ c& r)x^2\\ d& s)x^4\end{array}$$

The value of $\underset{x\to 2}{lim}\frac{2^x+2^{3-x}-6}{\sqrt{2^{-x}}-2^{1-x}}$ is

Equation(s) of the common tangent to the parabola $y^2=24x$ and the circle $x^2+y^2=18$ is/are

If $x+\frac{1}{x}=2\cos \theta ,$ then find the value of $x^3+\frac{1}{x^3}.$

A letter is known to have come from either $\text{TATANAGAR}$ or $\text{CALCUTTA}$. On the envelope, just two consecutive letter $\text{TA}$ are visible. The probability that the letter has come from $\text{CALCUTTA}$ is

If the straight lines $\frac{x-1}{2}=\frac{y+1}{K}=\frac{z}{2}$ and $\frac{x+1}{5}=\frac{y+1}{2}=\frac{z}{K}$ are coplanar, then the plane(s) containing these two lines is/are

A person writes a letter to six friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in wrong places?

Which of the following are equal matrixes.

$A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$

$B=\left[\begin{array}{ccc}1& 2& 4\\ 3& 5& 6\\ 7& 8& 9\end{array}\right]$

$C=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

$D=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$

$E=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]$

The number of positive integral solutions of ${\tan }^{-1}x+{\cos }^{-1}\frac{y}{\sqrt{1+y^2}}={\sin }^{-1}\frac{3}{\sqrt{10}}$ is