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If the matrix $A=\left(\begin{array}{cc}0& 2\\ K& -1\end{array}\right)$ satisfies $A(A^3+3I)=2I$, then the value of $K$ is

If $tan{1}^{\circ }tan{2}^{\circ }.............tan{89}^{\circ }=x^2-8$,then the value of x can be

Evaluate $\int x^2{\tan }^{-1}xdx$

(where $C$ is constant of integration)

$\int e^x\left(\frac{1+sinx}{1+cosx}\right)$ dx is......................

If $y=\sin \left(x^2\right),$ then $\frac{dy}{dx}=$

Write the expression ${}^n{C}_{r+1}+{}^n{C}_{r-1}+2\times {}^n{C}_r$ in the simplest form.

$Ina\Delta ABC,ifsi{n}^{2}A+si{n}^{2}B=si{n}^2,C\text{show that the triangle is right angled.}$

If $\alpha ,\beta \ne 0,$ and $f\left(n\right)={\alpha }^n+{\beta }^n$ and

$\left|\begin{array}{ccc}3& 1+f\left(1\right)& 1+f\left(2\right)\\ 1+f\left(1\right)& 1+f\left(2\right)& 1+f\left(3\right)\\ 1+f\left(2\right)& 1+f\left(3\right)& 1+f\left(4\right)\end{array}\right|=K(1-\alpha {)}^2(1-\beta {)}^2(\alpha -\beta {)}^2$, then $K$ is equal to:

Let $A=\left[\begin{array}{cc}2& 3\\ a& 0\end{array}\right],$ $a\in R$ be written as $P+Q$ where $P$ is a symmetric matrix and $Q$ is skew symmetric matrix. If $det\left(Q\right)=9,$ then the modulus of the sum of all possible values of determinant of $P$ is equal to

If $p_1$ and $p_2$ are the lengths of the perpendiculars from the origin to the straight lines $x\sec \theta +y\text{cosec}\theta =a$ and $x\cos \theta -y\sin \theta =a\cos 2\theta$ respectively, then the value of $4p_1^2+p_2^2$ is

If $A_r=\cos \frac{\pi }{3^r}+i\sin \frac{\pi }{3^r},$ then the value of $A_1\cdot A_2\cdot A_3\cdot \cdots \cdot A_{\infty }$ is (where $i=\sqrt{-1}$)

Two or more vectors having the same initial point are called

Solution of $|x+1|+|2x+3|=5$ is

If $\overrightarrow{A}=2\hat{i}+3\hat{j}+4\hat{k}$ and $\overrightarrow{B}=4\hat{i}+3\hat{j}+2\hat{k}$, find $\overrightarrow{A}\times \overrightarrow{B}$.

The area (in sq. units) of the region $A=\left\{\right(x,y):|x|+|y|\le 1,2y^2\ge |x|\}$

Let $f:R-\left(\frac{3}{5}\right\}\to R-\left(\frac{3}{5}\right\}bedefinedbyf\left(x\right)=\frac{3x+2}{5x-3}.$

Then

${lim}_{x\to -2}\frac{x^3+x^2+4x+12}{x^2-3x+2}$

If $\alpha and\beta$ ($\alpha <\beta$ ) are the roots of equation $x^2+2x-5=0$, then the value of $\frac{1}{\alpha }-\frac{1}{\beta }$ is:

Using integration find the area of region bounded by the triangle whose vertices are (-1, 0),(1, 3) and (3, 2).

If A lies in the third quadrant and 3 tan A - 4 = 0, then 5 sin 2A + 3 sin A + 4 cos A =

One in nine ships is likely to be wrecked, when they are set on sail. When $6$ ships set on sail, the probability for exactly $3$ will not arrive safely is:

How many intersecting lines are required to form a rectangle?

If $\overrightarrow{A}.\overrightarrow{B}=|\overrightarrow{A}\times \overrightarrow{B}|$ then the angle between $\overrightarrow{A}and\overrightarrow{B}$ is

${\int }_0^{\pi }\frac{co{s}^{4}x}{co{s}^{4}x+si{n}^{4}x}dx=$

Show that in an infinite G.P. with common ratio $r(|r|<1)$, each term bears a constant ratio to the sum of all terms that follow it.