Explore topic-wise InterviewSolutions in .

1. 1.8

If $A=\{x:x\in R,0<x<2\}$, $B=\{x:x\in R,1<x\le 3\}$, then $A-B=$

(3x² + 8x + 6 )ⁿ - (2 + x )ⁿ is divisible by

Show that the points (0, 7, 10), (-1, 6, 6) and (-4, 9, 6) are the vertices of an isosceles right-angled triangle.

If the vertices of a triangle are A (-2, -3), B (3, 2) and C (-1, -8), then the area of triangle ADC, where D is the mid-point of side BC, is:

If $f\left(x\right)$ is a polynomial of degree $4$ such that $\underset{x\to -1}{lim}\frac{f\left(x\right)}{(x+1{)}^3}=1$ and $f^{\prime \prime \prime }\left(0\right)=-12$, then the maximum value of $f\left(x\right)$ is

​​​​​​​(correct answer + 1, wrong answer - 0.25)

In a school $\frac{4}{9}$ of students are boys and the number of girls is $775.$ find the number of boys in the school.

Differentiate
the following w.r.t. x:

Find the probability distribution of the number of successes in two tosses of a die, where a succeess is defined as

number greater than 4

six appears on the die.

A ship is fitted with three engines $E_1,E_{2}and{E}_3$. The engines function independently of each other with respective probabilities $\frac{1}{2},\frac{1}{4}and\frac{1}{4}$. For the ship to be operational, at least two of its engines must function. Let X denote the event that the ship is operational and let $X_1,X_{2}and{X}_3$ denote, respectively the events that the engines $E_1,E_{2}and{E}_3$ are functioning.

Which of the following is/are true?

Find the Cartesian
equation of the line which passes through the point

(−2, 4, −5)
and parallel to the line given by

$bsinB-csinC=asin(B-C)$

If $I_n=\int {\cot }^{n}xdx,$ then $I_0+I_1+2(I_2+I_3)+I_4+I_5=$

(where $C$ is integration constant)

The direction cosines of the line equally inclined with the axes are:

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.

Define a relation R on the set N of natural numbers by R = {(x, y) : y = x + 5, x} is a natural number less than $4,x,yϵN$}

Depict this relationship using roster form

The points $D,E,F$ divide $\overrightarrow{BC},\overrightarrow{CA}$ and $\overrightarrow{AB}$ of the triangle $ABC$ in the ratio $1:4,3:2$ and $3:7$ respectively and the point $K$ divides $\overrightarrow{AB}$ in the ratio $1:3$, then $(\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}):\overrightarrow{CK}$ is equal to

There are $n$ urns, each of these contain $n+1$ balls. The $i^{th}$ urn contains $i$ white balls and $n+1-i$ red balls. Let $u_i$ be the event of selecting $i^{th}$ urn, $i=1,2,3,\cdots n$ and $w$ be the event of getting a white ball.

If $P\left(u_i\right)\propto i$, where $i=1,2,3\cdots n,$ then $\underset{n\to \infty }{lim}P\left(w\right)$ equals to

${lim}_{x\to 0}\frac{a^x+a^{-x}-2}{x^2}$

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

Differentiate the
function with respect to x.

Which of the following will be the coordinates of point $A$, if you translates point $A$ $4$ units to the right and $4$ units down?

Match the elements from $\text{Column-I}$ to $\text{Column-II}$.
$\begin{array}{cccc}& \text{Column-I}& & \text{Column-II}\\ \text{(A)}& \text{Let}f\left(x\right)\text{be a continuous function, where}f\left(1\right)=3& \text{(P)}& 1\\ & \text{and}F\left(x\right)\text{is defined as}F\left(x\right)=\underset{0}{\overset{x}{\int }}(t^2\cdot \underset{1}{\overset{t}{\int }}f\left(u\right)du)dt\text{.}& & \\ & \text{Then the value of}{F}^{\prime \prime }\left(1\right)\text{is}& & \\ \text{(B)}& f_a,f_b\text{and}{f}_c\text{denote the lengths of the interior angle}& \text{(Q)}& 10\\ & \text{bisector in a triangle of side lengths}a,b,c\text{and area}T.& & \\ & \text{If}\frac{f_a\cdot f_b\cdot f_c}{abc}=\frac{\lambda T(a+b+c)}{(a+b)(b+c)(c+a)}\text{, then the value}& & \\ & \text{of}\lambda \text{is}& & \\ \text{(C)}& \text{Let}{a}_n\text{be the}{n}^{\text{th}}\text{term of an A.P. Let}{S}_n\text{be the sum}& \text{(R)}& 3\\ & \text{of the first}n\text{terms of the A.P. where}{a}_1=1\text{and}& & \\ & a_3=3a_8.\text{If}{S}_n\text{is maximum, then the value of}n\text{is}& & \\ \text{(D)}& \text{If}x={\tan }^{-1}\left(t\right)\text{is substituted in the differential}& \text{(S)}& 4\\ & \text{equation}\frac{d^{2}y}{d{x}^2}+xy\frac{dy}{dx}+{\sec }^{2}x=0,\text{it becomes}& & \\ & (1+t^2)\frac{d^{2}y}{d{t}^2}+(2t+y{\tan }^{-1}(t\left)\right)\frac{dy}{dt}=k.\text{Then}& & \\ & \text{the value of}k\text{is}& & \\ & & \text{(T)}& -1\end{array}$

Which of the following is correct combination?

The Boolean expression $\sim (p\Rightarrow (\sim q\left)\right)$ is equivalent to :

Find the
principal and general solutions of the equation

If $I_1=\underset{0}{\overset{1}{\int }}{e}^{-x}{\cos }^{2}xdx$,

$I_2=\underset{0}{\overset{1}{\int }}{e}^{-x^2}{\cos }^{2}xdx$ and

$I_3=\underset{0}{\overset{1}{\int }}{e}^{-x^3}dx$ ; then :

Let $\alpha ,\beta \in R$. If $\alpha ,{\beta }^2$ be the roots of quadratic equation $x^2-px+1=0$ and ${\alpha }^2,\beta$ be the roots of quadratic equation $x^2-qx+8=0$, then the value of ${}^{\prime }{r}^{\prime }$ if $\frac{r}{8}$ be arithmetic mean of $p$ and $q$, is

If P(x) be a polynomial of degree 4, with P(2)=-1, P'(2)=0, P”(2)=2, P”'(2)=-12 and P^ir(2) =24, then P”(1) is equal to

After striking the floor ball rebounds ${\frac{4}{5}}^{\text{th}}$ of its height from which it has fallen. If it is released from a height of $120$m, then the total distance travelled by the ball (in $\text{m}$) before it comes to rest is

If $a,b,c,d$ and $p$ are distinct non-zero real numbers such that $(a^2+b^2+c^2)p^2-2(ab+bc+dc)p+(b^2+c^2+d^2)\le 0$, then $ac$ is equal to