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If $\int \sqrt{1+\sin x}f\left(x\right)dx=\frac{2}{3}(1+\sin x{)}^{3/2}+c,$ then $f\left(x\right)$ equals

Find the value of x.

The total number of $6$ digit numbers that can be made using digits $1,2,3,4$, if all the digits should appear in the number at least once is

The position vector of mid-point of joining the points (2, – 1, 3) and (4, 3, –5) is:

Which of the following would be the final arrangement?

The least value of 'a' for which $\frac{4}{sinx}+\frac{1}{1-sinx}=a$ has at least one solution in the interval $(0,\pi /2)$ is

Let the normals at the four points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ and $(x_4,y_4)$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be concurrent at some point (called as conormal point). Then $(x_1+x_2+x_3+x_4)(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\frac{1}{x_4})$ is equal to

In the following case, find the coordinates of the foot of the perpendicular drawn from the origin:
3y +4z -6 =0

If the roots of the equation $b{x}^2+cx+a=0$be nonreal, then for all real values of x, the expression $3b^2{x}^2+6bcx+2c^2$ is

${\int }_0^{1}x{e}^{x^2}dx=\lambda {\int }_0^1{e}^{x^2}dx,$ then

The solution set of $\cos 5\theta =-\frac{1}{2}$ is

$\int \left[f\right(x\left)g^{\prime \prime }\right(x)-f"(x\left)g\right(x\left)\right]dx$ is equal to

If ∝ and β are the roots of the equation a$x^2$ + bx + c = 0, (a,b,c ​ R) , then (1+α+${\alpha }^2$) (1+β+${\beta }^2$) is :

The probability that the ${13}^{th}$ day of a randomly chosen month is a Friday, is

The complete set of values of 'a' such that $x^2+ax+a^2+6a$ < 0 $\forall$ x $ϵ$ [-1, 1] is:

Find the values of x and y by adding and subtracting following pair of lines:

x + y = 8

x - y = 5

The number of values of $x\in [0,\pi ]$, that satisfies the equation ${\log }_{|\sin x|}(1+\cos x)=2$, is

The area bounded by the curve $x=a{\cos }^{3}t,y=a{\sin }^{3}t$ is

Which of the following must be a sock pair?

The value of $\cos \frac{\pi }{4}\times (\cos \frac{\pi }{12}-\sin \frac{\pi }{12})$ is

The value of $\sin {75}^{\circ }+\cos {75}^{\circ }$ is

If $A=\{x:x$ is a letter in the word 'QUARANTINE'$\}$, then the cardinality of $A$ is

Find the values of $co{s}^{-1}x$ in terms of given options.

Verify Rolle's theorem for the function $f\left(x\right)=x^2+2x-8,xϵ[-4,2]$

A normal is drawn on the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ at a point given by parameter $\frac{\pi }{3}$.

What is the x-intercept of the normal.

If PM is the perpendicular from P (2,3) on to the line x+y=3 then the co-ordinates of M are

If $\alpha ,\beta$ and $\gamma$ are the roots of $x^3+8=0$, then the equation whose roots are ${\alpha }^2,{\beta }^{2}and{\gamma }^2$ is

Let $O$ be the origin, and $\overrightarrow{OX},\overrightarrow{OY},\overrightarrow{OZ}$ be three unit vectors in the directions of the sides $\overrightarrow{QR},\overrightarrow{RP},\overrightarrow{PQ},$ respectively, of a triangle $PQR.$

If the triangle $PQR$ varies, then the minimum value of
$\cos (P+Q)+\cos (Q+R)+\cos (R+P)$

If $|z-3|\le 2$, then maximum of $|z+1|$ is

The pair of lines represented by $3a{x}^2+5xy+(a^2-2)y^2=0$ are perpendicular to each other for