Explore topic-wise InterviewSolutions in .

Write each of the following intervals in the set-builder from:
(i) A=(2,5)

(ii) B=[-4,7]

(iii) C=[-8,0)

(iv) D= (5,9]

The vector equation of the line passing through $3\hat{i}-5\hat{j}+7\hat{k}$ and perpendicular to the plane $3x-4y+5z=8$ is

(where $\lambda$ is a parameter)

Coefficient of $t^{24}in(1+t^2{)}^{12}(1+t^{12}\left)\right(1+t^{24})$ is ?

If $2x-3\le 5$, What are the range of values for x?

If a,b,c are non-coplanar unit vectors such that $a\times (b\times c)=\frac{b+c}{\sqrt{2}}$, then the angle between a and b is

In triangle ABC, a =2, b =3 and sin $A=\frac{2}{3},$ then B is equal to

For each binary operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative.
(i) On Z, define a $\ast$ b =a-b

(ii) On Q, define $a\ast b=ab+1$

(iii) On Q, define $a\ast b=\frac{ab}{2}$

(iv) On $Z^{+}$, define $a\ast b=2^{ab}$

(v) On $Z^{+}$, define $a\ast b=a^b$

(vi) On (R-{-1} ,define $a\ast b=\frac{a}{b+1}$

The equation of the plane passing through the point (3, - 3, 1) and perpendicular to the line joining the points (3, 4, - 1) and (2, - 1, 5) is:

In a class of $140$ students numbered $1$ to $140$, all even numbered students opted Mathematics course, those whose number is divisible by $3$ opted Physics course and those whose number is divisible by $5$ opted Chemistry course. Then the number of students who did not opt for any of the three courses is :

Find
the inverse of each of the matrices, if it exists.

In $\Delta ABC,$ the ratio $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ is always equal to:

(where for $△ABC$ usual notations are used and $h_1,h_2,h_3$ are altitudes from respective vertices.)

If a straight line parallel to the line $y=\sqrt{3}x$ passes through $Q(2,3)$ and cuts the line $2x+4y-27=0$ at $P$, then the length of $PQ$ is (units)

The area of the region bounded by the parabola $(y-2{)}^2=(x-1),$ the tangent to it at the point whose ordinate is $3$ and the $x-$axis is

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five question. The number of choices available to him is

Find $\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})$, if $\overrightarrow{a}=2\hat{i}+\hat{j}+3\hat{k},\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}$ and $\overrightarrow{c}=3\hat{i}+\hat{j}+2\hat{k}$.

The number of solutions of the equation $2\cos x=2x^2+1$ in $x\in [-\frac{\pi }{2},\frac{\pi }{2}]$ is

The function $y=f\left(x\right)$ is the solution of the differential equation

$\frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^4+2x}{\sqrt{1-x^2}}$

in $(-1,1)$ satisfying $f\left(0\right)=0$. Then

$\underset{-\sqrt{3}/2}{\overset{\sqrt{3}/2}{\int }}f\left(x\right)dx$

is

Let $\ast$ be the binary operation of the set {1,2,3,4,5} defined by $a\ast b=HCF$ of a and b. Is the operation $\ast$ is same as the operation $\ast$ defined in Q.4 above? Justify your answer.

The angle between the lines represented by the equation $x^2-2pxy+y^2=0$, is

If $n$ is a positive integer for the function $f\left(x\right)=5x(x-2{)}^n,x\in [0,2].$ By Rolle's theorem value of $c$ is $\frac{1}{5},$ then $n$ is equal to

${lim}_{n\to \infty }{2}^{2-1}sin\left(\frac{a}{2^n}\right)$

Manas removed one number from the sequence of ten consecutive natural numbers. The sum of the remaining nine numbers in 2007. Which of the following numbers did Manas possibly remove?

If $A=\left[\begin{array}{cc}2& a\\ -1& -2\end{array}\right]$ is an involutory matrix, then which of the following is/are TRUE