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Let $A=\{a,b,c,d\},$ $B=\{b,c,e,f\},$ then $n\left(\right(A-B)\times (B-A\left)\right)=$

For two sets $X\&Y,X-Y=$, $\text{and}Y-X=$

If $1\le |x|<4$, then $x$ belongs to

If cos x= -1/2 and x lies in quadrant II. What is the value of tan x?

Evaluate the following limits in
$\underset{x\to \frac{\pi }{2}}{\text{lim}}\frac{\text{tan 2x}}{x-\frac{\pi }{2}}$

If $\tan A,\tan B$ are the roots of quadratic equation $a{x}^2+3x+4=0,a\ne 0$ and $\frac{\tan A}{\cot B-\tan A}=\frac{1}{3}$, then the value of $a$ is

Let $n\left(U\right)=700$, $n\left(A\right)=200$, $n\left(B\right)=300$ and $n(A\cap B)=100,\text{then}n(A^c\cap B^c)$ is -

Sum to infinite terms of the series $co{t}^{-1}(1^2+\frac{3}{4})+co{t}^{-1}(2^2+\frac{3}{4})+co{t}^{-1}(3^2+\frac{3}{4})+.......$ is

Find the mean deviation about the median for the following data:

$\begin{array}{ccccccc}{x}_i& 3& 5& 7& 9& 11& 13\\ f_i& 6& 8& 15& 3& 8& 4\end{array}$

For two events A and B, if

P(A)=P(A | B) =1/4 and P(B | A) =1/2, then

$n(n+1)(n+5)$ is a multiple of 3

If $x_1,x_2$ are the roots of $a{x}^2$ + bx + c = 0 and $x_1+d,x_2+d$ are the roots of $p{x}^2$ + qx + r = 0, d $\ne$ 0 then

The digits of a three-digit positive integer are in A.P. and their sum is $15$. The number obtained by reversing the digits is $594$ less than the original number. Then the unit place of the number is

How many 6 digit telephone numbers can be constructed with the digits 0 to 9 ,if each number starts with 23 and if repetition of digits is not allowed.

Find the value of $\frac{cosec{60}^o+sec{60}^o+cosec{45}^o}{sec{45}^o+cosec{60}^o+sec{30}^o}$

If $cos\left(2si{n}^{-1}x\right)=\frac{1}{9}$, then x=

In a triangle ABC, right angled at C, the value of cot A + cot B is

A card is selected from a pack of 52 cards.
(a) How many points are there in the sample space?
(b) Calculate the probability that the card is an ace of spades.
(c) Calculate the probability that the card is (i) an ace (ii) black card.

If the length of latus rectum of a hyperbola whose eccentricity is $\frac{3}{\sqrt{5}}$, centre $(4,3)$ and axis is parallel to coordinate axis is $8$ units, then the equation(s) of the hyperbola is/are

A function f(x) is such that it is not differentiable at two points h, k in its domain and f’(x) becomes zero at 3 points a, b, c in the domain. The critical points and stationary points will be -

The probability that a patient visiting a dentist will have a tooth extracted is 0.06, the probability that he will have a cavity filled is 0.2, and the probability that he will have a tooth extracted or a cavity filled is 0.23. What is the probability that he will have a tooth extracted as well as a cavity filled ?

If T = $(5+2\sqrt{6}{)}^n$ = M + f , n \in N , 0 $\le$ f < 1 , Then M =

The vertex of a parabola is the point $(a,b)$ and latus rectum is of the length $l.$ If the parabola is upward opening and its axis is parallel to the $y-$axis, then its equation is

The statement $A\to (B\to A)$ is equivalent to:

Which of the following graphs correctly represents the variation of $\beta$ = - $(\frac{\frac{dV}{dP}}{V}{)}_{\gamma }$ with P for an ideal gas at constant temperature.

For 0< $\varphi$ < $\frac{\pi }{2},ifx={\sum }_{n=0}^{\infty }co{s}^{2n}\varphi ,y={\sum }_{n=0}^{\infty }si{n}^{2n}\varphi ,z={\sum }_{n=0}^{\infty }co{s}^{2n}\varphi ,then$

If $4\hat{i}+7\hat{j}+8\hat{k},2\hat{i}+3\hat{j}+4\hat{k}$ and $2\hat{i}+5\hat{j}+7\hat{k}$ are the position vectors of the vertices A, B and C respectively of triangle ABC. The position vector of the point where the bisector of angle A meets BC is

If A and B be the points $(3,4,5)$ and $(-1,3,-7)$ respectively, find the equation of the set of points P such that $P{A}^2+P{B}^2=k^2$ where k is a constant.

Coefficient of x in the expansion of $(x^2+\frac{a}{x}{)}^5$ is

What is the vertex of the quadratic function y = $x^2-6x+12$?

If the sides of a right-angled triangle form an A.P. Then the sines of the acute angle are

The number of proper subsets of the set $\{1,2,3\}$ is ___.

Let $p,p_1$ be A.M. and G.M. between $a$ and $b$ respectively and $q,q_1$ be the A.M. and G.M. between $b$ and $c$ respectively where $a,b,c>0$. If $a,b,c$ are in A.P., then which of the following is CORRECT?

If ${(x^2+2x+4)}^n={\sum }_{r=0}^{2n}{a}_r{x}^r,then{\sum }_{r=n}^{2r}\left(\frac{{}^{a}2n-r}{a_r}\right)$ is

Find the equation of the tangents drawn form the point (5,3) to the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1.$

$r{\cdot }^n{C}_r=$

If $-5<p<-2$ and $7<q<9$, then what is the range of p + q?

The coefficient of $t^4$ in the expansion of ${\left(\frac{1-t^6}{1-t}\right)}^3$ is :

The integral $\int \cos \left({\log }_{e}x\right)dx$ is equal to: (where $C$ is a constant of integration)

$K$ for the synthesis of $HI$ is $50$. $K$ for dissociation of $HI$ is

An element has a body centered cubic (bcc) structure with a cell edge length of $288\text{pm}$. The shortest interatomic distance is

Let $\overrightarrow{a}=\hat{i}+2\hat{j}+4\hat{k},\overrightarrow{b}=\hat{i}+\lambda \hat{j}+4\hat{k}$ and $\overrightarrow{c}=2\hat{i}+4\hat{j}+({\lambda }^2-1)\hat{k}$ be coplanar vectors. Then the non-zero vector $\overrightarrow{a}\times \overrightarrow{c}$ is :

Consider the following system of linear equations

$\left[\begin{array}{ccc}2& 1& -4\\ 4& 3& -12\\ 1& 2& -8\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}a\\ 5\\ 7\end{array}\right]$

Number of values of $a$ for which system has infinitely many solutions.

The sum of the roots of the equation $x+1-2{\log }_2(2^x+3)+2{\log }_4(10-2^{-x})=0$ is

1. -5

Solve the inequalities $x+y\le 9,y>x$ and $x\ge 1$ graphically.

If a, b,and c are in A.P., p, q, and r are in H.P. and ap, bq, and cr are in G.P., then $\frac{p}{r}+\frac{r}{p}$ is equal to