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The normal at $P\left(\theta \right)$ and D$(\theta +\frac{\pi }{2})$ meet the major axis of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at Q and R. Then $P{Q}^2+D{R}^2=$

Equation of director circle of the ellipse $\frac{x^2}{3}+\frac{y^2}{6}=1$ is

The range of $|x-2|+|x-5|$ is

The locus of a point whose chord of contact with respect to parabola

$y^2=8x$ passes through focus is

A bag contains 7 red and 2 white balls and another bag contains 5 red and 4 white balls. Two balls are drawn, one from each bag. The probability that both the balls are white, is

The expression $(\sqrt{2x^2+1}+\sqrt{2x^2-1}{)}^6+{\left(\frac{2}{\sqrt{2x^2+1}+\sqrt{2x^2-1}}\right)}^6$ is a polynomial of degree

The complete solution set of the inequality $[co{t}^{-1}x{]}^2-6[co{t}^{-1}x]+9\le 0,$ where $[.]$ denotes greatest integer function is

The value(s) of $x$ satisfying the equation $||x-3|-4|=3$, is/are

Let $\alpha ,\beta$ be real and z be a complex number. If $z^2+\alpha z+\beta =0$ has two distinct roots on the line $Re\left(z\right)=1$, then it is necessary that

If $P=2^4\cdot 6^3\cdot 5^2\cdot {15}^2$, then the number of proper even divisors of $P$ will be

Find the sum of the series ${}^n{C}_0{}^n{C}_2+{}^n{C}_1{}^n{C}_3+{}^n{C}_2{}^n{C}_4.....{}^n{C}_{n-2}{}^n{C}_n$

The value of the expression $({\tan }^{4}x+2{\tan }^{2}x+1){\cos }^{2}x$, when $x=\frac{\pi }{12}$ is equal to

If for the matrix, $A=\left[\begin{array}{cc}1& -\alpha \\ \alpha & \beta \end{array}\right],A{A}^T=I_2,$ then the value of ${\alpha }^4+{\beta }^4$ is:

If $x^2-ax+b=0$ and $x^2-px+q=0$ have one root common and the second equation has equal roots, then $b+q$ is equal to

Let $f\left(x\right)=5-|x-2|$ and $g\left(x\right)=|x+1|,x\in R.$ If $f\left(x\right)$ attains maximum value at $\alpha$ and $g\left(x\right)$ attains minimum value at $\beta ,$ then $\underset{x\to -\alpha \beta }{lim}\frac{(x-1)(x^2-5x+6)}{x^2-6x+8}$ is equal to:

The image of the point $(-8,12)$ with respect to the line mirror $4x+7y+13=0$ is

Box I contain three cards bearing numbers 1,2,3; box II contains five cards bearing numbers 1,2,3,4,5; and box III contains seven cards bearing numbers 1,2,3,4,5,6,7. A card is drawn from each of the boxes. Let $x_i$ be the number on the card drawn from the $i^{th}$ box $i=1,2,3.$



The probability that $x_1,x_2$ and $x_3$ are in arithmetic progression, is ?

The vertices of $△ABC$ lie on a rectangular hyperbola such that the orthocentre of the triangle is $(3,2)$ and the asymptotes of the rectangular hyperbola are parallel to the coordinate axes. If two perpendicular tangents of the hyperbola intersect at the point $(1,1)$,then combined equation of the asymptotes is

If $-3x>-15$ and $x\in N$, then $x$ is equal to

For the equation ${\left({\log }_{2}x\right)}^2-4{\log }_{2}x-m^2-2m-13=0,m\in R$.
If the real roots are $x_1,x_2$ such that $x_1<x_2$, then the sum of maximum value of $x_1$ and minimum value of $x_2$ is

If the pairs of lines $x^2+2xy+a{y}^2=0$ and $a{x}^2+2xy+y^2=0$ have exactly one line in common, then the combined equation of the other two lines is given by

If in $△ABC,\angle A=\frac{\pi }{4}$ and $\tan B\tan C=p$, then the possible set of value(s) of $p$ is/are

If $|z_1-1|<2,|z_2-2|<1,$ then the maximum possible integral value of $|z_1+z_2|$ is

Find the locus of mid-point of chord of

parabola $y^2=4x$ which touches the parabola $x^2=4y$.

Find the sum of the series $2^2$ + $4^2$ + $6^2$ + $8^2$ +................ 20 terms ___

Solve the following quadratics $17x^2+28x+12=0$

Given the vertices of triangle by position vectors $\hat{i}+\hat{j}+\hat{k},\hat{i}+\hat{k}and\hat{j}+\hat{k}$ the centroid and Incentre of the triangle will be given by

If the equation $x^n-1=0,n>1,n\in N,$ has roots $1,a_1,a_2,\dots ,a_{n-1},$ then

The point of intersection of tangents at the points on the parabola $y^2=4x$ whose ordinates are $4$ and $6$ is

Consider a circle with its centre lying on the focus of the parabola $y^2=2px$ such that it touches the directrix of the parabola. Then the point of intersection of the circle and parabola can be

${lim}_{x\to 0}\frac{\sqrt{1-cos2x}}{\sqrt{2x}}is$

If a, b, c are the sides of a triangle ABC such that $x^2-2(a+b+c)x+3\lambda (ab+bc+ca)=0$ has two distinct real roots, then

If $P(2,8)$ is an interior point of a circle $x^2+y^2-2x+4y-p=0$ which neither touches nor intersects the axes, then set for $p$ is -

The equation of the perpendicular bisector of the line segment joining $(2,1)$ and $(3,4)$ is

$\underset{x\to 3}{lim}\left(\right[x-3]+[3-x]-x),\text{where[.]denote the greatest integer function, is equal to:}$

$Ifsi{n}^{3}xsin3x={\sum }_{m=0}^n{c}_{m}cosmxwhere{c}_0,c_1,c_2.\cdots \cdots ,c_{n}areconstantsand{c}_n\ne 0,thenthevalueofnis$

Let $A(6,-1),B(1,3)$ and $C(x,8)$ be three points such that $AB=BC.$ Then the value of $x$ can be

If $A=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$ and $A^2=\left[\begin{array}{cc}\alpha & \beta \\ \beta & \alpha \end{array}\right]$, then

A variable name in certain computer language must be either an alphabet or an alphabet followed by a decimal digit. The total number of different variable names that can exist in that language is equal to

Let $y=y\left(x\right)$ be the solution of the differential equation $\frac{dy}{dx}+2y=f\left(x\right),$ where $f\left(x\right)=\{\begin{array}{c}1,x\in [0,1]\\ 0,\text{otherwise}\end{array}$

If $y\left(0\right)=0$, then $y\left(\frac{3}{2}\right)$ is

The value of $\frac{\tan {225}^{\circ }-\cot {81}^{\circ }\cot {69}^{\circ }}{\cot {261}^{\circ }+\tan {21}^{\circ }}$ is

The number of values of $x$ satisfying the equation $(x^2+7x+11{)}^{(x^2-4x-21)}=1$ is

The equation of line whose slope is $\frac{1}{3}$ and $x-$ intercept is $4$ will be

The general solution of $\frac{dy}{dx}-\frac{y}{x}=\frac{y^2}{x^2}$ is

The value of $\frac{\cos 15\pi +\tan \left(\frac{5\pi }{4}\right)}{\sin 16\pi +\sec 6\pi }$ is

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 number is