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If $A=\left[\begin{array}{cc}\cos 2\theta & -\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{array}\right]$ and $A+A^T=I,$ where $I$ is $2\times 2$ unit matrix and $A^T$ is the transpose of $A,$ then the value of $\theta$ is equal to

The range of $f\left(x\right)=\sec \left(\frac{\pi }{4}{\cos }^{2}x\right)$ is

All the values of $m$ for which both the roots of $x^2-2mx+m^2-1=0$ are greater than $-2$ but less than $4$, lies in the interval

Equation of one of the latus rectum of the hyperbola $(10x-5{)}^2+(10y-2{)}^2=9(3x+4y-7{)}^2$ is

If two unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ inclined at angles $\alpha$ and $2\alpha$ with another vector $\overrightarrow{c},$ such that their projection are same on $\overrightarrow{c}.$ Then $\alpha (\alpha \ne 0)$ can be?

If $y=f\left(x\right)$ and $y=g\left(x\right)$ are symmetrical about the line $x=\frac{\alpha +\beta }{2},$ then $\underset{\alpha }{\overset{\beta }{\int }}f\left(x\right)g^{\prime }\left(x\right)dx$ is equal to

Let $f\left(x\right)=Ax+B,A,B\in R$ and $y=f\left(x\right)$ passes through the points $(A,2A-B^2)$ and $(2B+3,(A+B{)}^2-1)$. If $B_1,B_2\cdots B_n,n\in N,$ are different possible value(s) of $B$, then the value of $\sum _{r=1}^n{B}_r$ is

A farmer buys a used tractor for Rs. 12000. He pays Rs. 6000 cash and agrees to pay the balancein annual instalments of Rs. 500 plus $12\%$ interest on the unpaid amount. How much the tractor cost him ?

Differentiate the following functions with respect to x :

$\frac{a+bsinx}{c+dcosx}$

The maximum value of $1+2\sin x+3{\cos }^{2}x$ is

If a class consits of $6$ periods, then the number of ways in which $5$ subjects can be taught such that each subject must be allotted at least one period and no period remains vacant is

Differential equation, having $y=(si{n}^{-1}x{)}^2+A(co{s}^{-1}x)+B$ where A and B are arbitary constants is $(p-x^2)\frac{d^{2}y}{d{x}^2}-\frac{xdy}{dx}=q$; then $p+q=$ ___

Solve the given inequality graphically in two-dimensional plane: y < –2

If $a\ne b\ne c$ such that

$\left|\begin{array}{ccc}{a}^3-1& b^3-1& c^3-1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right|=0$ then,

Let $x,y$ be real numbers such that $x{\cos }^2{39}^{\circ }=\tan {26}^{\circ }\cot {39}^{\circ }\tan {86}^{\circ }\cos {78}^{\circ }\tan {34}^{\circ }$ and $y\cos {81}^{\circ }=\cos {36}^{\circ }-\sin {36}^{\circ }.$ Then the value of $x+y^2$ is

Formation of the differential equation of equation of all the parabolas having their axes of symmetry coincident with the X-axis is:

Number of $2$ digit even numbers that can be formed using the digits $1,2,3,4,5,$ if the digits can be repeated is

The equation of the common tangent of the parabola $y^2=8x$ and $x^2+12y^2=48$ is

The region represented by $z=x+iy\in C:\left|\begin{array}{c}z\end{array}\right|-Re\left(z\right)\le 1$ is also given by the inequality:

Which among the following pairs of linear equations has unique solution?

if $f\left(x\right)=\sqrt{x}$, $g\left(x\right)=\frac{3x^2-x^6}{1-3x^4}$ and $h\left(x\right)=\tan x,$ then $(g\circ f\circ h)\left(\frac{\pi }{3}\right)$ is

If $f\left(x\right)$ and $g\left(x\right)$ are both continous and diffrentiable functions of '$x$' then,

$\int \left(3f\left(x\right)+4g\left(x\right)\right]dx$ can be simplified as

If $2x^2+4x+n>2\pi +{\sec }^{-1}\left(\sec 6\right)+{\cot }^{-1}\left(\cot 13\right)\forall x\in R,$ then possible integral value(s) of $n$ can be

In $△$ ABC, if 2(bc cos A + ca cos B + ab cos C) =

Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for x² = 6y

Let$P=(-1,0),Q=(0,0)$ and $R=(3,3\sqrt{3})$ be three points, Then the equation of the bisector of the angle PQR is -

What is the inverse fourier transform of $X\left(\omega \right)=4\sum _{n=-\infty }^{\infty }\delta (\omega -\frac{\pi }{2}n)?$