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If A and B are 2 sets such that A U B has 40 elements, A has 18 elements and B has 29 elements, how many elements does A ∩ B have? __

Find the general solution of (2 sin x - cos x) (1 + cos x) = $si{n}^2$ x

Foci of the Ellipse $25x^2+9y^2-150x-90y+225$ = 0 are

Let $a_1,a_2,a_3,.............$ be in harmonic progression with $a_1=5and{a}_{20}=25.$ The least positive integer n for which $a_n<0$ is

Equation of the parabola, if coordinates of vertex and focus are $(0,0)$ and $(2,3)$ respectively, is

If $A_r,B_r,C_r$ denotes the coefficients of $x^r$ in the expansion of $(1+x{)}^{10},(x+1{)}^{20},(1+x{)}^{30}$ respectively, then the value of $\sum _{r=1}^{10}{A}_r(B_{10}{B}_r-C_{10}{A}_r)$ is

If the angle between two lines is $\frac{\pi }{6}$ and slope of one of the lines is $\frac{1}{2}$, then the slope of the other line can be

In $\Delta ABC,$ the median AD divides $\angle BAC$ such that $\angle BAD:\angle CAD=2:1.$ then $cos\frac{A}{3}$ is equal to

Which of the following is/are irrational numbers ?

If the difference between two complementary angles is ${52}^{\circ }$, then the smaller angle is

Let $a,b,c$ are in AP and $a^2,b^2,c^2$ are in G.P. If $a<b<c$ and $a+b+c=\frac{3}{2}$ then $a=$

Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be the reciprocal to that of the ellipse $x^2+4y^2=4$. If the hyperbola passes through the focus of the ellipse, then the focus of the hyperbola is at :

An urn contains $5$ red and $2$ green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is :

The value of the expression $1(2-\omega )(2-{\omega }^2)+2(3-\omega )(3-{\omega }^2)+\cdots \cdots +(n-1)(n-\omega )(n-{\omega }^2)$ is where $\omega$ is an imaginary cube root of unity is

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 possible value(s) of $\tan \left(\frac{\theta }{2}\right)\tan \left(\frac{\varphi }{2}\right)$ is/are

The domain of f(x) is (0, 1). Then the domain of $f\left(e^x\right)+f\left(ln|x|\right)$ is

What is the angle between two planes having normal vectors as

$\overrightarrow{n_1}=\hat{i}+2\hat{j}+2\hat{k}$ and $\overrightarrow{n_2}=4\hat{i}-4\hat{j}+2\hat{k}$ ?

In a survey of 200 students from 7 different schools, 50 people do not play NFS, 40 people do not play Dota and 10 people play no online game. Then find the no. of people out of 200 people who do not play both the games provided these are the only two games on offer.

The differential equation of all circles which pass through the origin and whose centers lie on the y-axis is

If the $x$-intercept of the focal chord of the parabola $y^2-2y-4x+5=0$ is $\frac{11}{4}$, then find the length of chord.

$li{m}_{x\to 1}$$\frac{(2x-3)(\sqrt{x}-1)}{({2x}^2+x-3)}$ =

[IIT 1977]

If $[x+[x+[x+[x+\left[x\right]]]]]=10$, then $x$ lies in

If $y=y\left(x\right)$ is the solution of the differential equation, $e^y(\frac{dy}{dx}-1)=e^x$ such that $y\left(0\right)=0$, then $y\left(1\right)$ is equal to

Common tangents are drawn to the parabola $y^2=4x$ and the ellipse $3x^2+8y^2=48$ touching the parabola $A$ and $B$ and the ellipse at $C$ and $D$. Area of quadrilateral $ABCD$ (in sq.units) is:

If $y=ln{\left(\frac{x}{a+bx}\right)}^x$,then $x^3\frac{d^{2}y}{d{x}^2}$ is equal to

If a vertex of a triangle is (1, 1) and the mid points of two sides through this vertex are (-1, 2) and (3, 2), then what type of triangle is this?

If equations $x^2+bx+c=0$ and $b{x}^2+cx+1=0$ have a common root then

If $\theta \in (0,\frac{\pi }{4})$ and $t_1=(\tan \theta {)}^{\tan \theta },$ $t_2=(\tan \theta {)}^{\cot \theta },$ $t_3=(\cot \theta {)}^{\tan \theta }$ and $t_4=(\cot \theta {)}^{\cot \theta }$ then

Let $r,s,t$ and $u$ be the roots of the equation $x^4+A{x}^3+B{x}^2+Cx+D=0;$

$A,B,C,D\in R.$ If $rs=tu$, then $A^{2}D$ is equal to

The parallelism condition for two straight lines one of which is specified by the equation ax+by+c=0 the other being represented parametrically by $x=\alpha t+\beta ,y=\gamma t+\delta$ is given by

The general solution of $\tan \alpha +2\tan 2\alpha +4\tan 4\alpha +8\cot 8\alpha =\sqrt{3}$ is

$(\text{where}n\in Z)$

Let $f\left(x\right)=\frac{x^2-1}{x},g\left(x\right)=\frac{x+2}{x-3}$ then domain of $\frac{f\left(x\right)}{g\left(x\right)}$ is

The range of the function $2\sin x+7$ is

What is the inverse of the matrix $\left[\begin{array}{cc}-3& 2\\ 5& -1\end{array}\right]$

A unit vector $\overrightarrow{a}$ makes an angle of $45°$ with positive $z-$axis and if $\overrightarrow{a}+\hat{i}+\hat{j}$ is a unit vector, then $\overrightarrow{a}=$

If $P$ and $Q$ be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, whose centre is $C$ such that $CP$ is perpnediuclar to $CQ,a<b$, then the value of $\frac{1}{C{P}^2}+\frac{1}{C{Q}^2}$ is

The area (in sq. units) of the region $\left\{\right(x,y):x\ge 0,x+y\le 3,x^2\le 4y\text{and}y\le 1+\sqrt{x}\}$

is:

The complete set of values of $a$ for which the inequality $(a-1)x^2-(a+1)x+(a-1)\ge 0$ is true for all $x\ge 2$ is

The letters of the word $\text{PROBABILITY}$ are written down at random in a row. Let $E_1$ denote the event that two $I^{\prime }s$ are together and $E_2$ denote the event that two $B^{\prime }s$ are together. Then which of the following is (are) CORRECT?

If the system of equations ax + y = 3, x + 2y =3, 3x + 4y =7 is consistent, then value of a is equal to

Set of values of $a$ for which both the roots of the quadratic polynomial $f\left(x\right)=a{x}^2+(a-3)x+1$ lie on one side of the $y-$axis is

Interval in which {x} is monotonically increasing function, where { } is fractional part function -

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

If the coordinates of two points A and B are (3,4) and (5,-2) respectively. Then, the coordinates of any point P, if PA = PB and area of $\Delta$ PAB = 10, are

The area of the region bounded by the curve y = x + 1 and the lines x = 2 and x = 3 is