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If ${\sin }^{-1}(x-\frac{x^2}{2}+\frac{x^3}{4}-\cdots )+{\cos }^{-1}(x^2-\frac{x^4}{2}+\frac{x^6}{4}-\cdots )=\frac{\pi }{2}$ for $0<x<\sqrt{2},$ then the value of $x$ is

The domain of $y=si{n}^{-1}(1+3x+2x^2)$ is:

For two sets $A\&B$ such that:

$A=\{12,14,15,16,19,20\}$

$B=\{18,21,20,15,17,13\}$

Choose the correct options.

One of the factors of the determinant

$\left|\begin{array}{ccc}x& 5& 3\\ x^2& 9& 9\\ x^3& 16& 27\end{array}\right|$ is

The value of $\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}dx$ is

(where $C$ is constant of integration)

A chord attached about an end to a vibrating fork divides it into $6$ loops, when its tension is $36\text{N}$. The tension at which it will vibrate in $4$ loops is
(correct answer + 1, wrong answer - 0.25)

Prove that (sinA+cosecA)²+(cosA+secA)²=7+tan^{​​​​2}​​​A+cot^{​​​​2}​​​A

For the
function

Prove that

If,
then find the value of x.

The number of solution(s) of the equation ${\sin }^{4}x+{\cos }^{4}x=\sin x\cos x$ in $[\pi ,5\pi ]$ is

If the three normals drawn to the parabola, $y^2=2x$ pass through the point $(a,0),a\ne 0$ then ${}^{\prime }{a}^{\prime }$ must be greater than:

If $z=\sqrt{-7-24i}$, then $z$ is equal to

The equation of a plane which passes through the point of intersection of lines $\frac{x-1}{3}=\frac{y-2}{1}=$ $\frac{z-3}{2}$, and $\frac{x-3}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ and at greatest distance from point $(0,0,0)$ is

Let $a,b\in R^{+}$ such that ${\log }_{27}a+{\log }_{9}b=\frac{7}{2}$ and ${\log }_{27}b+{\log }_{9}a=\frac{2}{3}.$ Then the value of $ab$ is

If p
and q are the lengths of perpendiculars from the origin to
the lines x cos θ – y
sin θ = k
cos 2θ and x sec θ+
y cosec θ =
k, respectively, prove that p²
+ 4q² =
k²

If AB is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that $\Delta OAB$ is an equilateral triangle O being the origin, then the eccentricity of the hyperbola satisfies

Let $X$ and $Y$ be two events such that $P\left(X\right)=\frac{1}{3},P\left(X|Y\right)=\frac{1}{2}$ and $P\left(Y|X\right)=\frac{2}{5}$. Then

2/3 + 5/3 is same as ___ .

If Q(0, 1) is equidistant from P(5, –3) and R(x, 6), then the value of x is .

If $\alpha$ is the $n^{th}$ root of unity, then $1+2\alpha +3{\alpha }^2+\dots$ to $n$ terms is equal to

Find the area of the region bounded by
the parabola y = x² and

The number of relations defined from the set of letters of the word 'DIGITAL' to the set of letters of the word 'OFFLINE' is

The slope of normal to curve $y=f\left(x\right)$, where $x=t^2-5t+4$ and $y=t^2-7t+10$ at $(-2,-2),$ is

If $a,b,c$ are non-zero real numbers and $a{x}^2+bx+c=0,$ $b{x}^2+cx+a=0$ have one root in common, then $\frac{a^3+b^3+c^3}{abc}=$

If $\int \frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}dx=ax+b\ln (4e^x+5e^{-x})+c$, then

If $x^2-3x+2$ is a factor of $x^4-a{x}^2+b$ then the equation whose roots are $a,b$ is

Find the equation of the ellipse whose foci are (4,0) and (-4,0), eccentricity = 1/3.

If $I={\int }_0^{1}x\sqrt{\frac{1-x}{1+x}}dx$, the I equals

The system of linear equations $x+y+z=6,x+2y+3z=10,$
$x+2y+2z=4$ will have

What is the condition for a function y = f(x) to be a monotonically increasing function

If the sum of n terms of an A.P. is cn(n-1), where c$\ne$0, then sum of the squares of these terms is

Number of five digit numbers that contain digit $7$ exactly once (repetition of digits is allowed) is

Find the sum to n
terms of the series 3 × 1²
+ 5 × 2² + 7 ×
3² + …

Calculate mean deviation about mean and median for given observations {1, 1, 4, 2, 6, 100, 150, 200, 400}

1. 6

Vertex of the parabola $x^2+4x+2y-7=0$ is