Explore topic-wise InterviewSolutions in .

If two circles of equal radii of $5$ unit each, touch externally at $(1,2)$ and the equation of one common tangent is $3x-4y+30=0,$ then the equations of the other two common tangents are

Let $y=y\left(x\right)$ be the solution of the differential equation, $\frac{dy}{dx}+y\tan x=2x+x^2\tan x,x\in (-\frac{\pi }{2},\frac{\pi }{2})$, such that $y\left(0\right)=1.$ Then :

The value of $\int (\sin 7x\cdot \sin 3x)dx$ is

(where $C$ is constant of integration)

The imaginary part of $(z-1)(\cos \alpha -i\sin \alpha )+(z-1{)}^{-1}\times (\cos \alpha +i\sin \alpha )$ is zero, if

Number of solutions of $8\cos x=x$ will be

If $\underset{n\to \infty }{lim}[an-\frac{1+n^2}{1+n}]=b$, where $b$ is a finite number, then

Which among the following points lies outside the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$.

If $\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k},\overrightarrow{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k},\overrightarrow{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$ and $\left[\begin{array}{ccc}3\overrightarrow{a}+\overrightarrow{b}& 3\overrightarrow{b}+\overrightarrow{c}& 3\overrightarrow{c}+\overrightarrow{a}\end{array}\right]=\lambda \left|\begin{array}{ccc}\overrightarrow{a}\cdot \hat{i}& \overrightarrow{a}\cdot \hat{j}& \overrightarrow{a}\cdot \hat{k}\\ \overrightarrow{b}\cdot \hat{i}& \overrightarrow{b}\cdot \hat{j}& \overrightarrow{b}\cdot \hat{k}\\ \overrightarrow{c}\cdot \hat{i}& \overrightarrow{c}\cdot \hat{j}& \overrightarrow{c}\cdot \hat{k}\end{array}\right|,$ then the value of $\frac{\lambda }{4}$ is :

Sort the following values of $\cos x$ in descending order.

In an acute angled triangle $ABC,\cos A+\cos B+\cos C\le \lambda ,$ then $\lambda$ is equal to

Find the tangent to the parabola $y^2=8x$ which makes an angle of ${45}^{\circ }$ to the line

$2x+y+3=0$

Choose the correct pair of equation and number of positive integral root

If a point moves such that the sum of the squares of its distance from the four sides of a unit square having $2$ sides along coordinate axis is $2$ , then the locus of the point is

In the cube of side ${}^{\prime }{a}^{\prime }$ shown in the figure, the vector from the central point of the face $ABOD$ to the central point of the face $BEFO$ will be

The normal at an end of a latus rectum of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through an end of the minor axis if

If $\underset{0}{\overset{y}{\int }}\cos t^{2}dt=\underset{0}{\overset{x^2}{\int }}\frac{\sin t}{t}dt$, then the value of $\frac{dy}{dx}$ is

(where $y=f\left(x\right)$ is a differentiable function)

If a student can select maximum $n$ books (and at least $1$ book) from $2n+1$ books in $63$ ways. Then value of $n$ is

Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time $t_1$. On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time $t_2$. The time taken by her walk up on the moving escalator will be

Integrate the following functions w.r.t. x.

$\int \frac{si{n}^{8}x-co{s}^{8}x}{1-2si{n}^{2}xco{s}^{2}x}dx.$

${lim}_{x\to 0}\frac{a^{mx}-b^{nx}}{x}$

The value of $\sum _{0\le i<}^{}\sum _{j\le n}^{}({}^n{C}_i+{}^n{C}_j)is:$

Let $a_1,a_2,a_3,...$ be an A.P. with $a_6=2.$Then the common difference of this A.P., which maximises the product $a_1{a}_4{a}_5,$ is :

Statements

All birds are crows.

All peacocks are birds.

Conclusions

I. All peacocks are crows.

II. All crows are peacocks.

If $3\text{and}4$ lies between the roots of the equation $x^2+2kx+9=0$ then $k$ lies in the interval

Let $\ast$ be the binary operation on N given by $a\ast b=LCM$ of a and b.
(i) Find $5\ast 7,20\ast 16$

(ii)Is $\ast$ commutative?

(iii)Is $\ast$ associative?

(iv) Find the identity of $\ast$ in N

(v)Which elements of N are invertible for the operation $\ast$ ?

If the normal at a point P to the hyperbola meets the transverse axis at G, and the value of $\frac{SG}{SP}$ is 2 then the eccentricity of the hyperbola is (where S is the focus of the hyperbola)

Integrate the following functions w.r.t. x.

$\int \frac{1}{\sqrt{x+a}+\sqrt{x+b}}dx.$

If $a$ and $b$ are rational numbers and $b$ is not a perfect square, then the quadratic equation with rational coefficients whose one root is $\frac{1}{a+\sqrt{b}}$, is

The solution of the equation $(x+2y^3)\frac{dy}{dx}-y=0$ is

[MP PET 1998; 2002]

For a square matrix $A,$ if $A^3=O,$ then $I+A+A^2$ is equal to

If the equations $x^2+ax+12=0,x^2+bx+15=0$ and $x^2+(a+b)x+36=0$ have a common positive root, then the value of $a+b$ is equal to

$Ifx\sqrt{1+y}+y\sqrt{1+x}=0,for-1<x<1,provethat\frac{dy}{dx}=-\frac{1}{(1+x{)}^2}$

A dice is thrown two times and the sum of the scores appearing on the dice is observed to be a multiple of $4$. Then the conditional probability that the score $4$ has appeared atleast once is

If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.