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Number of real solutions of the equation $|||x^2-1|-1|+3|=1$ is

$If{e}_{1}and{e}_{2}arerespectivelytheeccentricitiesoftheellipse\frac{x^2}{18}+\frac{y^2}{4}=1andthehyperbola\frac{x^2}{9}-\frac{y^2}{4}=1,thentherelationbetween{e}_{1}and{e}_{2}is$

If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0|\overrightarrow{a}|=3,|\overrightarrow{b}|=5,|\overrightarrow{c}|=7$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

If $S=(1+x^2{)}^{100}+2x^2(1+x^2{)}^{99}+3x^4(1+x^2{)}^{98}+\cdots +101x^{200}$, then the coefficient of $x^{100}$ in $S$ is

If the minimum value of $f\left(x\right)={\left({\sin }^{-1}x\right)}^2+2\pi {\cos }^{-1}x+{\pi }^2$ is $\frac{a{\pi }^2}{b}$ where $a,b$ are coprime, then the value of $a+b$ is equal to

A normal inclined at an angle of $\frac{\pi }{4}$ to the x-axis of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is drawn. It meets the major and minor axes in P and Q respectively. If C is the centre of the ellipse then the area of the triangle CPQ is

For the equation $3x^2+px+3=0,p>0$, if one of the roots is the square of the other, then p is equal to

The number of solutions of the equation $min\{|x|,|x-1|,|x+1|\}=\frac{1}{2}$ is

The sum of all the local minimum values of the twice differentiable function $f:R\to R$ defined by $f\left(x\right)=x^3-3x^2-\frac{3f^{\prime \prime }\left(2\right)}{2}x+f^{\prime \prime }\left(1\right)$ is:

If $0\le A\le \frac{\pi }{4},$ then ${\tan }^{-1}\left(\frac{1}{2}\tan 2A\right)+{\tan }^{-1}\left(\cot A\right)+{\tan }^{-1}\left({\cot }^{3}A\right)$ is equal to

Find $\frac{dy}{dx}$ when x and y are connected by the relation given.

$sin\left(xy\right)+\frac{x}{y}=x^2-y$

A line makes angles α, β and y with the positive directions of X-axis, Y-axis and Z-axis, respectively, then the directions cosines of the line are:

If ${\int }_0^b\frac{dx}{1+x^2}={\int }_b^{\infty }\frac{dx}{1+x^2}$,then b=

$\underset{x\to 0}{lim}\frac{\sin 3x}{\sin 5x}$ is equal to

Equation of the line drawn through the point $(1,0,2)$ and perpendicular to the line $\frac{x+1}{3}=\frac{y-2}{-2}=\frac{z+1}{-1},$ is

Let $PQ$ be a focal chord of $y^2=4ax$. The tangents to parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a(a>0)$. Length of $PQ$(in units) is :

Let $p,q,r$ denote the arbitary statements then the logical equivalance of the statement $p\Rightarrow (q\vee r)$ is

~[(- p)^q] is logically equivalent to

The proposition $p\to \sim (p\wedge \sim q)$ is equivalent to

The degree
of the differential equation

is

(A) 3 (B) 2 (C) 1 (D) not
defined

If $1^3+2^3+3^3+\cdots +{50}^3=m^2,$ then $m=....$

(use principle of mathematical induction)

If $p,q$ are the roots of the equation $a{x}^2+bx+c=0,$ then the value of $(ap+b{)}^{-2}+(aq+b{)}^{-2}$ is

The angle between the planes $2x+y-2z+3=0$ and $6x+3y+2z=5$ is

Prove that : $sec(\frac{3\pi }{2}-\theta )sec(\theta -\frac{5\pi }{2})+tan(\frac{5\pi }{2}+\theta )tan(\theta -\frac{3\pi }{2})=-1$

An array X of n distinct integers is interpreted as a complete binary tree. The index of the first element of the array is 0.



If the root node is at level 0, the level of element X[i], i $\ne$ 0, is