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If f(x) = sgn (x) and $g\left(x\right)=x(1-x^2)$ then (fog)(x) is discontinuous at

The number of points at which the function $f\left(x\right)=\{\begin{array}{cc}\left[\cos \pi x\right],& 0\le x\le 1\\ |x-1|[x-2],& 1<x\le 2\end{array}$

is discontinuous, is

$\left(\right[.]$ denotes the greatest integral function $)$

Each of the persons $A$ and $B$ independently tosses three fair coins. The probability that both of them get the same number of heads is :

Following bar graph shows the number of books sold by a bookstore during five consecutive years.

a) In which year, 200 books were sold?

b) How many books were sold in the year 1989?

c) What is the difference between the number of books sold in 1993 and 1992?

Given 3 points given by position vectors $\overline{a},\overline{b}$ and $\overline{c}$. The plane which passes through these 3 points can be given by

Copper crystallises in a face-centred cubic lattice with a unit cell length of $361\text{pm}$. What is the radius of copper atom in $\text{pm}$?

If θ
is the angle between any two vectors
and,
then
when
θisequal
to

(A) 0 (B) (C) (D) π

The value of ${\int }_0^{si{n}^{2}x}si{n}^{-1}\left(\sqrt{t}\right)dt+{\int }_0^{co{s}^{2}x}co{s}^{-1}\left(\sqrt{t}\right)dt$ is

Let $S=\{1,2,...,100\}.$ Suppose $b$ and $c$ are chosen at random from the set $S.$ The probability that $4x^2+bx+c$ has equal roots is

Let $Q=\left(\begin{array}{cc}\cos \frac{\pi }{4}& -\sin \frac{\pi }{4}\\ \sin \frac{\pi }{4}& \cos \frac{\pi }{4}\end{array}\right)$ and
$x=\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right)$ then$Q^{3}x$ is equal to

Area of the region in which point $p(x,y),x>0$ lies: such that $y\le \sqrt{16-x^2}and\left|ta{n}^{-1}\left(\frac{y}{x}\right)\right|\le \frac{\pi }{3}$ is

If $15{\sin }^4\alpha +10{\cos }^4\alpha =6,$ for some $\alpha \in R,$ then the value of $27{\sec }^6\alpha +8{\text{cosec}}^6\alpha$ is equal to

If ${\cos }^{-1}{x}_1+{\cos }^{-1}{x}_2+\cdots +{\cos }^{-1}{x}_{10}=10\pi ,$ then the value of $\frac{\sum x_1+\sum x_1{x}_2{x}_3}{\sum x_1{x}_2+\sum x_1{x}_2{x}_3{x}_4}$ is equal to

What is the value of $\frac{1+i}{1-i}$ (give answer in 'i')

Let $A=\left[a_{ij}\right]$ be a $3\times 3$ matrix, where

$a_{ij}=\{\begin{array}{ccc}1& ,& \text{if}i=j\\ -x& ,& \text{if}|i-j|=1\\ 2x+1& ,& \text{otherwise}\end{array}$

Let a function $f:R\to R$ be defined as $f\left(x\right)=det\left(A\right).$ Then the sum of maximum and minimum values of $f$ on $R$ is equal to

Two numbers $a$ and $b$ are chosen at random from the first $30$ natural numbers. The probability that $a^2-b^2$ is divisible by $3$ is

$\underset{x\to \frac{\pi }{4}}{lim}\frac{16\sqrt{2}-(\sin x+\cos x{)}^9}{1-\sin 2x}$ equals

10 men and 6 women are to be seated in a row so that no two women sit together. the number of ways they can be seated is

Find the equation of the normals to the
curve y = x³ + 2x + 6 which are
parallel to the line x + 14y + 4 = 0.

The correct graph of $y=x^3-1$ is

Which of the following represents the condition for a matrix A to be skew hermitian Matrix. Given that the general element of the matrix is aij.

If the chord through the points whose eccentric angles are $\theta$ and $\varphi$ on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ passes through a focus, then the value of $\tan \left(\frac{\theta }{2}\right)\tan \left(\frac{\varphi }{2}\right)$is

The vector equation of the plane which passes through the point (5, 2,-4) and perpendicular to the line with direction ratios 2, 3, -1 is

Let $T_n=(n^2+1)n!$ and $S_n=T_1+T_2+T_3+\cdots +T_n.$ If $\frac{T_{10}}{S_{10}}=\frac{a}{b},$ where $a$ and $b$ are relatively prime natural numbers, then the value of $(b-a)$ is

How many ways are there to arrange the letters of the word EDUCATION so that all the following three conditions hold?

– the vowels occur in the same order (EUAIO);

– the consonants occur in the same order(DCTN);

– no two consonants are next to each other.

A logic circuit implements the Boolean function, $F=\overline{X}Y+X\overline{Y}\overline{Z}.$ It is found that the input combination $X=Y=1$ can never occur. Taking this into account, a simplified expression for F is given by

The roots of the quadratic equation $2(3x+5)(2x-4)=0$ are

Let $f\left(x\right)=\{\begin{array}{cc}{\tan }^{-1}\alpha -5x^2,& 0<x<1\\ -6x,& x\ge 1\end{array}.$

If $f\left(x\right)$ has a maximum at $x=1,$ then