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The product of all roots of the equation $(x^2-5x+7{)}^2-(x-2\left)\right(x-3)=1$ is

If $\theta$ is an acute angle between the lines y=2x+3, y=x+1 then the value of tan$\theta$ =

If $\left(\begin{array}{cc}2& 4\\ -1& k\end{array}\right)$ is a nilpotent matrix of index $2,$ then $k$ equals to

The value of $\tan ({\tan }^{-1}\frac{1}{2}-{\tan }^{-1}\frac{1}{3})$is

A pair of tangents are drawn to the parabola $y^2=4ax$ which are equally inclined to a straight line $y=mx+c$, whose inclination to the axis is $\alpha$ then locus of their point of intersection is

Let $f\left(x\right)=7{\tan }^{8}x+7{\tan }^{6}x-3{\tan }^{4}x-3{\tan }^{2}x$ for all $x\in (-\frac{\pi }{2},\frac{\pi }{2}).$ Then the correct expression(s) is(are)

Let $C_n=\underset{\frac{1}{n+1}}{\overset{\frac{1}{n}}{\int }}\frac{{\tan }^{-1}\left(nx\right)}{{\sin }^{-1}\left(nx\right)}dx.$ Then $\underset{n\to \infty }{lim}{n}^2.C_n$ equals

$\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0and\gamma ,\delta aretherootsofp{x}^2+qx+r=0and{D}_1,D_2$ be the respective discriminants of these equations. If $\alpha ,\beta ,\gamma ,and\delta$ are in A.P. then $D_1:D_2=(where\alpha ,\beta ,\gamma \delta ,ϵR\&a,b,c,p,q,rϵR)$

For $0<\alpha <\beta ,$ which of the following is true

The number of distinct real roots of the cubic polynomial equation $x^3-3x^2+3x-1=0$ is

Let $A=\{1,2,4\},B=\{2,4,5\},C=\{2,5\}$, then $(A-B)\times (B-C)$ is

If mean and variance for the following series of numbers:
$15,30,a,25,27,b,13,20$ is $20\text{and}39.5$ then the value of $\sqrt[3]{3a}b$ is

Show that the line $a^{2}x+ay+1=0$ is perpendicular to the line $x-ay=1$ for all non-zero real values of a.

The set of value(s) of $x$ for which $\underset{n\to \infty }{lim}\frac{n^3\cdot 7^n}{n^3(2x-3{)}^n+3n^3\cdot 7^{n+1}+7}=\frac{1}{21}$ is

If the roots of equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ be equal. then $a,b,c$ are in

Angle between line
$\frac{x-5}{1}=\frac{y-2}{2}=\frac{z-8}{2}$ and plane $2x+y+2z+5=0$ is

The domain of the function $\sqrt{\sin 2x}$ is

$A=\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& -2& 4\end{array}\right|,I=\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|and{A}^{-1}=\frac{1}{6}(A^2+cA+dI),thenthevalueofcanddare$

The value of $\underset{x\to 0}{lim}\frac{(\sqrt{2+x\sin x}-\sqrt{2\cos x})(1+\cos x)}{1-\cos x}$ is

Differential equation of the family of parabolas whose vertex lie on the $x-$ axis and focus as origin is

The asymptotes of a hyperbola have center at the point $(1,2)$ and are parallel to the lines $2x+3y=0$ and $3x+2y=0$. If the hyperbola passes through the point $(5,3)$, then its equation is

Two sides of a parallelogram are along the lines $4x+5y=0$ and $7x+2y=0$. If the equation of one of the diagonals of the parallelogram is $11x+7y=9$, then other diagonal passes through the point

Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 2& 1\end{array}\right]$. If $u_1$ and $u_2$ are column matrices such that $A{u}_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$ and $A{u}_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$, then $u_1+u_2$ is equal to :

The solution curve of the differential equation $(1+e^{-x})(1+y^2)\frac{dy}{dx}=y^2$, which passes through the point $(0,1)$ is

The number of real roots of the equation ${\tan }^{-1}\sqrt{x(x+1)}+{\sin }^{-1}\sqrt{x^2+x+1}=\frac{\pi }{4}$ is