Explore topic-wise InterviewSolutions in .

The values of $\lambda$ for which the circle $x^2+y^2+6x+5+\lambda (x^2+y^2-8x+7)=0$ dwindles into a point are

If $f\left(x\right)+2f\left(\frac{1}{x}\right)=3x,x\ne 0andS=xϵR:f\left(x\right)=f(-x)$; then S

The coefficient of $x^{11}$ in the expansion of $(1+2x+2x^2{)}^6$ is

In what direction a line be drawn through the point (1, 2) so that its points of intersection with the line x+y=4 is at a distance $\frac{\sqrt{6}}{3}$ from the given point

$IfA=\left[\begin{array}{ccc}2& 1& 3\\ 1& 4& 5\\ 3& 5& 6\end{array}\right]B=\left[\begin{array}{ccc}1& 4& 5\\ 4& 2& 7\\ 5& 8& 3\end{array}\right]$ then which of the following is correct?

The total number of irrartional terms in the binomial expansion of ${(7^{1/5}-3^{1/10})}^{60}$ is :

The value of $2^n\left\{\mathrm{1.3.5.....}\right(2n-3\left)\right(2n-1\left)\right\}$ is

If $\underset{x\to 0}{lim}\varphi \left(x\right)=a^3,a\ne 0$, then $\underset{x\to 0}{lim}\varphi \left(\frac{x}{a}\right)$ is

Define $x\%a$ as the remainder obtained when $x$ is divided by $a$.

Let $f:Z\to R$ be defined as $f\left(x\right)=x\%k$, where $kϵN$. If Y is the range of $f\left(x\right)$, then

If $a{x}^2+bx+c=0$, then x =

The equation of the conjugate hyperbola of the hyperbola $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y-3=0$ is

If the normals at $(x_i,y_i),\text{where,}i=1,2,3,4$ on the rectangular hyperbola $xy=c^2$ meet at $(\alpha ,\beta ).$ and $x_1{x}_2{x}_3{x}_4=a$ and $y_1{y}_2{y}_3{y}_4=b$, then $a+b$ is

Which of the following are true statements.

Consider a convex polygon which has $35$ diagonals. Then match the following columns:

​​​​​​$\begin{array}{cc}Column1& Column2\\ \text{a. Number of triangles joining the vertices of the}& \\ \text{polygon}& \text{p.}210\\ \text{b. Number of points of intersections of diagonal}& \\ \text{which lies inside the polygon}& \text{q.}120\\ \text{c. Number of triangle in which exactly one side}& \\ \text{is common with that of polygon}& \text{r.}10\\ \text{d. Number of triangles in which exactly two sides}& \\ \text{are common with that of polygon}& \text{s.}60\end{array}$

The values of $x$ for which ${\log }_3(x+2)(x+4)+{\log }_{1/3}(x+2)<\frac{1}{2}{\log }_{\sqrt{3}}7$ holds, is

The sum of coefficients of integral powers of $x$ in the binomial expansion $(1-2\sqrt{x}{)}^{50}$ is

The number of real roots of the equation $(x^2+2{)}^2+8x^2=6x(x^2+2)$ is

Complete set of values of a such that $\frac{x^2-x}{1-ax}$ attains all real values is

The characteristic of $27.321$ is

Find the $6^{\text{th}}$ term of the expansion $(y^{1/2}+x^{1/3}{)}^n,$ if the value of coefficient of $3^{\text{rd}}$ term from the end is $45$

If $\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]A\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then $A=$

The sum of an infinite geometric series with positive terms is $3$ and the sum of the cubes of its terms is $\frac{27}{19}$. Then the common ratio of this series is :

If for x≠0, y≠0, then D is

The radius of a right circular cylinder increases at the rate of $0.1$ cm/min, and the height decreases at the rate of $0.2$ cm/min. The rate of change of the volume of the cylinder, in ${\text{cm}}^3/\text{min},$ when the radius is $2$ cm and the height is $3$ cm, is

A line which passes through $P(4,5)$ and making an angle of ${30}^{\circ }$ with positive direction of $x-$axis. Then coordinates of point which is at a distance $4$ unit's from the line on either side of $P$, is

If $a_1,a_2,a_3,a_4$ are the coefficients of any four consecutive terms in the expansion of $(1+x{)}^n$, then $\frac{a_1}{a_1+a_2}+\frac{a_3}{a_3+a_4}=$

The distance of the point (4, 3, 5) from the y-axis is

[MP PET 2003]

The value of the integral $\int (x^2+x)(x^{-8}+2x^{-9}{)}^{1/10}dx$ is

In a triangle $ABC,a:b:c=4:5:6$. The ratio of the radius of the circumcircle to that to the incircle is

The total number of times, the digit ${}^{\prime }{3}^{\prime }$ will be written, when the integers having less than $4$ digits are listed, is

Let the coordinates of the two vertices of a triangle are $(4,-5),$ $(-6,5)$ respectively. If the coordinates of the centroid are $(3,-1),$ then coordinates of the third vertex are

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

If $(3+x^{2020}+x^{2021}{)}^{2022}=a_0+a_{1}x+a_2{x}^2+\cdots +a_{10}{x}^n$, then the value of $a_0-\frac{1}{2}{a}_1-\frac{1}{2}{a}_2+a_3-\frac{1}{2}{a}_4-\frac{1}{2}{a}_5+a_6\cdots$ is

(correct answer + 2, wrong answer - 0.50)

Which of the following options is the correct graph of $y={\left(\frac{1}{2}\right)}^x-2$ ?

Let $\Delta \left(x\right)=\left|\begin{array}{ccc}x+a& x+b& x+a-c\\ x+b& x+c& x-1\\ x+c& x+d& x-b+d\end{array}\right|$ and $\underset{0}{\overset{2}{\int }}\Delta \left(x\right)dx=-16,$ where $a,b,c,d$ are in A.P. Then the common difference of the A.P. can be

The solution of $\frac{dy}{dx}+1=e^{x+y}$ is

e and $e^1$ are the eccentricities of the hyperbolas $16x^2-9y^2=144$ and $9x^2-16y^2$= - 144 then e - $e^1$ =

Equation of the line perpendicular to the line $6x+2y+7=0$ and which dIvides the line segment joining the points $(5,-3)$ and $(0,2)$ internally in the ratio $7:4$ is

A standard hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is drawn along with its auxiliary circle. A point P $(asec\theta ,btan\theta )$ is taken. A perpendicular is dropped from P to the x axis which meets at the axis at R. A tangent is drawn from R to auxiliary circle. Which angle is equal to θ

Find the value of cos$\frac{\pi }{6}$.cos$\frac{\pi }{3}$.cos$\frac{4\pi }{6}$.cos$\frac{8\pi }{6}$.cos$\frac{16\pi }{6}$.cos$\frac{32\pi }{6}$

If

${\Delta }_1$, $\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right|$, ${\Delta }_2$ = $\left|\begin{array}{ccc}1& bc& a\\ 1& ca& b\\ 1& ab& c\end{array}\right|$ then

PSP’ is a focal chord of the ellipse $16x^2+25y^2=400.$ If SP = 8 then SP’ =

Let P(6, 3) be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If the normal at the point P intersects the X-axis at (9, 0), then the eccentricity of the hyperbola is

All real values of $x$ which satisfy $x^2-3x+2>0$ and $x^2-3x-4\le 0$ lie in the interval