Explore topic-wise InterviewSolutions in .

If $\alpha ,\beta ,\gamma$ are non zero roots of $x^3+p{x}^2+qx+r=0$, then the equation whose roots are $\alpha (\beta +\gamma ),\beta (\gamma +\alpha ),\gamma (\alpha +\beta )$

If $x^2+2ax+a<0,\forall x\in [1,2]$ then the values of ${}^{\prime }{a}^{\prime }$ lies in the interval

A line makes angles $a,b,c,d$ with the four diagonals of a cube, then ${\cos }^{2}a+{\cos }^{2}b+{\cos }^{2}c+{\cos }^{2}d=$

The slope of a straight line passing through $A(-2,3)$ is $-\frac{4}{3}$. The point(s) on the line that are $10$ units away from $A$ is/are

If tangents are drawn to the parabola $(x-2{)}^2+(y-3{)}^2=\frac{(3x+4y-5{)}^2}{25}$ at the extremities of the chord $3x-y-3=0$, then angle between the tangents is

If $x\in R-\left\{3\right\},$ then the possible value(s) of the expression $\frac{\sqrt{x^2-6x+9}}{3-x}+2$ can be

If $z_1=24+7i$ and $|z_2|=6$ then $|z_1+z_2|$ lies in

If the equation $x^2+y^2=2x$ is written in polar coordinate system, then $r$ can be

If $z=\sqrt{\frac{1+i}{1-i}},$ then the modulus of $z$ is

If $f\left(x\right)$ is an invertible function and $g\left(x\right)=2f\left(x\right)+5$, then the value of $g^{-1}\left(x\right)$, is

Two lines $\frac{x-3}{1}=\frac{y+1}{3}=\frac{z-6}{-1}$ and $\frac{x+5}{7}=\frac{y-2}{-6}=\frac{z-3}{4}$ intersect at the point $R$. The reflection of $R$ in the $xy$- plane has coordinates :

If the straight line $x\cos \theta +y\sin \theta =2$ touches the circle $x^2+y^2-2x=0$ then

If the circles $x^2+y^2-2x-4y=0$ and $x^2+y^2-8y-k=0$ touches each other internally, then the possible value of $k$ is

Let `head` means 1 and `tail` means 2 and coefficients of the equation $a{x}^2+bx+c=0$ are chosen by tossing a fair coin. The probability that the roots of the equation are non-real, is equal to

The value of $\sqrt{-3}\cdot \sqrt{-75}$ is

The equation of the plane containing the line 2x - 5y + z =3, x + y + 4z = 5 and parallel to the plane x + 3y + 6z =1 is

If $\alpha +\beta +\gamma =0,$ ${\alpha }^3+{\beta }^3+{\gamma }^3=12$ and ${\alpha }^5+{\beta }^5+{\gamma }^5=40,$ then the value of ${\alpha }^4+{\beta }^4+{\gamma }^4$ is equal to

(correct answer + 1, wrong answer - 0.25)

If $2x^2+7xy+3y^2+8x+14y+k=0$ represents a pair of straight lines, then the value of $k$ is

​​​​​​A circle $C_1$ with centre at the origin meets $x$-axis at $A$ and $B$ (where $A\&B$ lies on negative and positive $x-$axis respectively). Two points $P\left(a\right)$ and $Q\left(b\right)$ are on the circle such that $b-a$ is a constant, where $a$ and $b$ are the parametric angles of the points. $BP$ and $AQ$ meets at $R$. Locus of $R$ is a circle $C_2$.

Let $c$ be the radius of $C_1$ and $d$ be the radius of $C_2$.



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(1)}& \text{For}b-a=\frac{\pi }{2}\text{, the value of}\frac{d^2}{c^2}\text{is}& \text{(P)}& 0\\ \text{(2)}& \text{For}b-a=\frac{\pi }{2}\text{and}c=\sqrt{2}\text{, circle}{C}_2\text{intersects}& \text{(Q)}& 1\\ & \text{the coordinate axes at four points}L,M,N,O.& & \\ & \text{Let the area of the quadrilateral LMNO is}2\sqrt{2}p.& & \\ & \text{Then the value of}p\text{is}& & \\ \text{(3)}& \text{Let}{m}_1,m_2\text{be the slopes of the line BQ,AP}& \left(R\right)& 2\\ & \text{respectively.}\text{If}{m}_1{m}_2=-1\text{, then}\frac{a}{b}\text{is}& & \\ \text{(3)}& \text{Let}{m}_1,m_2\text{be the slopes of the line BQ,AP}& \left(S\right)& 3\\ & \text{respectively.}\text{If}{m}_1=m_2\text{, then}\frac{3|b-a|}{\pi }\text{is}& & \\ & & \left(T\right)& 4\end{array}$



Then the INCORRECT option is:

Let the function, $f:[-7,0]\to R$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f^{\prime }\left(x\right)\le 2,$ for all $x\in (-7,0),$ then for all such functions $f,f(-1)+f\left(0\right)$ lies in the interval:

If $\alpha$ and $\beta$ are different complex numbers with $|\beta |=1,$ then the value of $\left|\frac{\beta -\alpha }{1-\overline{\alpha }\beta }\right|$ is

If two vertices of a triangle are $(5,-1)$ and $(-2,3)$ and if its orthocentre lies at the origin, then the coordinates of the third vertex are

The inverse of $\left[\begin{array}{ccc}3& 5& 7\\ 2& -3& 1\\ 1& 1& 2\end{array}\right]$ is

If $\cos x\frac{dy}{dx}+y\sin x=1$ and $y\left(\frac{\pi }{4}\right)=\sqrt{2}$, then $y\left(\frac{-\pi }{3}\right)$ is

Let ${\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2$ be the roots of $a{x}^2+bx+c=0andp{x}^2+qx+r=0$, respectively. If the system of equations ${\alpha }_{1}y+{\alpha }_{2}z=0and{\beta }_{1}y+{\beta }_{2}z=0$ has a non - trivial solution, then

The equation of the circle which touches the lines $3x+4y-5=0$ and $3x+4y+25=0$ and whose centre lies on the line $x+2y=0$, is

Find the value of $\underset{x\to 2}{lim}\frac{x^2-4}{x+2}$

The marks of some students were listed from total marks of $75.$ The standard deviation of marks was found to be $9.$ Subsequently, the marks were raised to a maximum of $100$ and variance of new marks was calculated. The new variance is

The value of $tan[si{n}^{-1}\left(\frac{3}{5}\right)+co{s}^{-1}\left(\frac{3}{\sqrt{13}}\right)]$ is

The difference between a natural number and twice its reciprocal is $\frac{47}{7}$, then the number is

In a given standrard hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.What is the length of transverse axis?

The equation $\frac{x^2}{2-\lambda }+\frac{y^2}{\lambda -5}+1=0$ represents an ellipse, if

Consider a right angled triangle $ABC$





If the sides of the triangle are $a=6,b=10,c=8$ units, then the distance between incentre and circumcentre will be

In the expansion of $(\frac{1+x}{1-x}{)}^2$, the coefficient of $x^n$

The locus of feet of perpendiculars drawn from the origin to the straight lines passing through $(2,1)$ is

If sum of the coefficients in the expansion of $(x+3y-2z{)}^n$ is 128, then greatest coefficient in the expansion $(1+x{)}^n$ is ____

If a normal drawn at one end of the latus rectum of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ meets the axes at points $A\&B$ respectively, then area of $△OAB$ (in sq.units) is

If ${}^{56}{P}_{r+6}:{}^{54}{P}_{r+3}$ = 30800:1, then r =

If the line $3x-4y=0$ is tangent in the first quadrant to the curve $y=x^3+k,$ then value of $k$ is

Given parabola $y^2=4ax$, find the equation of Normal which will interest the normal at (8, 8) and the parabola at the same point.

The direction ratios of normal to the plane through the points $(0,-1,0)$ and $(0,0,1)$ and making an angle $\frac{\pi }{4}$ with the plane $y-z+5=0$ are :

If the vectors AB=$-3\hat{i}+4\hat{k}$ and AC=$5\hat{i}-2\hat{j}+4\hat{k}$ are the sides of a triangle ABC,then the length of median through the point A is

The circle $C_1:x^2+y^2=3$ having centre at origin,O intersects the parabola $x^2=2y$ at the point P in the first quadrant. Let the tangent to the circle $C_1$ at P touches other two circles $C_2$ and $C_3$ at $R_2$ and $R_3$, respectively. Suppose $C_2$ and $C_3$ have equal radii $2\sqrt{3}$ and centres $Q_2$ and $Q_3$, respectively. If $Q_2$ and $Q_3$ lie on the Y-axis then