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If $△=\left|\begin{array}{ccc}{e}^x& sinx& 1\\ cosx& ln(1+x^2)& 1\\ x& x^2& 1\end{array}\right|=a+bx+c{x}^2$ then the value of b is

If the sum of the ordinate and the abscissa of a point P (x, y) is 2n, where x and y are natural numbers, then probability that the point does not lie on y = x is:

A= $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$which of the following is true?

Equation of pair of tangents drawn from $(4,3)$ to the circle $x^2+y^2=4$ is

The coefficient of $x^9$ in the expansion of $1+x\left)\right((1+x^2)$ $(1+x^3)$ $\cdots$ $(1+x^{100})$ is

If $\overrightarrow{a},\overrightarrow{b}$ are unit vectors such that the vector $\overrightarrow{a}+3\overrightarrow{b}$ is perpendicular to $7\overrightarrow{a}-5\overrightarrow{b}$ and $\overrightarrow{a}-4\overrightarrow{b}$ is perpendicular to $7\overrightarrow{a}-2\overrightarrow{b}$,then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

Let $x+y=0$ and $2x-y+3=0$ be the major and minor axis of an ellipse respectively. If the foot of perpendicular drawn from vertex of the parabola $x^2-4x+4y+16=0$ to these lines is focus and one endpoint of minor axis respectively, then the eccentricity of the ellipse is

The sum of the infinite series

${\sin }^{-1}\frac{1}{\sqrt{2}}+{\sin }^{-1}\left(\frac{\sqrt{2}-1}{\sqrt{6}}\right)+{\sin }^{-1}\left(\frac{\sqrt{3}-\sqrt{2}}{\sqrt{12}}\right)+\cdots +{\sin }^{-1}\left(\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n(n+1)}}\right)+\cdots$ is

If $A$ and $B$ are two independent events such that $P\left(B\right)=\frac{2}{7},P(A\cup B^c)=0.8,$ then $P\left(A\right)=$

If ${\log }_3(x^3-x^2-x+1)-{\log }_3(x-1)-{\log }_3(x+1)=2$, then $x=$

If the greatest and the least values of $f\left(x\right)={\sin }^{-1}\left(\frac{x}{\sqrt{x^2+1}}\right)-\ln x$ in $[\frac{1}{\sqrt{3}},\sqrt{3}]$ are $M$ and $m$ respectively, then

If $\omega (\ne 1)$ is a cube root of unity and



$A=\left[\begin{array}{ccc}1+2{\omega }^{100}+{\omega }^{200}& {\omega }^2& 1\\ 1& 1+2{\omega }^{100}+{\omega }^{200}& \omega \\ \omega & {\omega }^2& 2+{\omega }^{100}+2{\omega }^{200}\end{array}\right]$ then

The value of determinant using Bagula rule $\left|\begin{array}{ccc}4& -2& 1\\ 9& 2& -3\\ 2& 1& 0\end{array}\right|$ is

Let f:$[0,1]\to R$ (the set of all real numbers) be a function. Suppose the function f is twice differentiable,$f\left(0\right)=f\left(1\right)=0$ and satisfies $f"\left(x\right)-2f^{\prime }\left(x\right)+f\left(x\right)\ge e^x,x\in [0,1]$

Which of the following is true for $0<x<1$?

In a meeting, $70\%$ of the members favour a certain proposal, $30\%$ being opposite. A member is selceted at random and let X=0, if he opposed and X=1, if he is in favour. Find E(X) and V(X)

Ten pair of shoes are in a closet. Four shoes are selected at random. The probability that there will be at least one pair among the four selected shoes is

If x = a $co{s}^3\theta$,y=a $si{n}^3\theta$, then $1+{\left(\frac{dy}{dx}\right)}^2$ is

A box contains $12$ white and $12$ black balls. The balls are drawn at random from the box, one at a time with replacement. The probability that a white ball is drawn for the fourth time on the seventh draw, is

The graph of the function $f\left(x\right)=x^2+\frac{1}{x}$ is:

Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three vectors such that $\overrightarrow{a}=\overrightarrow{b}\times (\overrightarrow{b}\times \overrightarrow{c})$. If magnitudes of the vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are $\sqrt{2},1$ and $2$ respectively and the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\theta (0<\theta <\frac{\pi }{2})$, then the value of $1+\tan \theta$ is equal to

The equation $a{x}^2+bx+c$ = 0 does not have real roots and c < 0. Which of these is true?

The area (in sq. units) of the region $A=\left\{\right(x,y):\frac{y^2}{2}\le x\le y+4\}$ is :

For an n-variable Boolean function, the maximum number of prime implicants are

Which of the following is a monotonically decreasing function?

Find the sum 1 + 3 + 5 + … + 51 (the sum of all odd numbers from 1 to 51) without actually adding them.

The number of real solutions of the equation $2\sin 3x+\sin 7x-3=0$ which lie in the interval $[-2\pi ,2\pi ]$ is

The equation (cos p - 1)$x^2$ + (cos p)x + sin p = 0 in variable x has real roots, if p belongs to the interval

Solve the following system of equations in R.

$10\le -5(x-2)<20$