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$si{n}^{-1}\frac{\sqrt{x}}{\sqrt{x+a}}$ is equal to

If $A$ and $B$ are two independent events, such that $P\left(A\right)=\frac{1}{2}$ and $P\left(B\right)=\frac{1}{5},$ then which of the following is/are correct :

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three non-zero vectors, no two of which are collinear, $\overrightarrow{a}+\overrightarrow{b}$ is collinear with $\overrightarrow{c}$ and $\overrightarrow{b}+3\overrightarrow{c}$ is collinear with $\overrightarrow{a}$, then $|\overrightarrow{a}+2\overrightarrow{b}+6\overrightarrow{c}|$ will be equal to

A hyperbola having the transverse axis of length $2\sin \theta$ unit, is confocal with the ellipse $3x^2+4y^2=12$, then its equation is

The angles of a right - angled triangle are in AP. The ratio of the inradius and the perimeter is

Determine the direction cosines of the normal to plane and the distance from the origin:

x + y + z = 1

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

The range of the function $f\left(x\right)={\tan }^{-1}(x^2+4|x|+1)$ is equal to

There is a rectangle drawn such that their diagonals meet at the origin and their sides are parallel to the coordinate axes. The length of the sides parallel to y-axis is equal to the length of the conjugate axis of the hyperbola and the length of other two sides is equal to the distance between focii of the hyperbola. Then the area bounded by the hyperbola and the rectangle, is

The domain of the function $f\left(x\right)=\log |\log x|,$ is

Choose
the correct answer in the following questions:

If
is
such that
then

If the roots of the equation $x^2-(p+4)x+2p+5=0$ are equal, then the value(s) of $p$ is/are

Solve
system of linear equations, using matrix method.

Find the number of solutions of $e^x=x^4$.

Prove that the function f given
by f(x) = x² − x + 1 is
neither strictly increasing nor strictly decreasing on (−1, 1).

Pair the addition statements that give the same answer.

If $\sin x+{\sin }^{2}x=1$, then

${\cos }^{12}x+3{\cos }^{10}x+3{\cos }^{8}x+{\cos }^{6}x+2{\cos }^{4}x+{\cos }^{2}x-2=$

The value of $5.63\overline{74}-2.094\overline{26}$ is

Suppose the limit
$L=\underset{n\to \infty }{lim}\sqrt{n}\underset{0}{\overset{1}{\int }}\frac{1}{(1+x^2{)}^n}dx$
exists and is larger than $\frac{1}{2}$. Then

If four points P (1,2,3) , Q (2,3,4), R(3,4,5), S(4,5, k) are coplanar then the value of k is / are -

In a bag, there are 2 blue balls, 5 green balls and 6 red balls. If a person took two balls without replacement, what is the probability that the second ball will be red given that the first ball is red?

Find
X and Y,
if

(i) and

(ii) and

The value of (3 + 4i) (6 - 7i) is

If equation of a plane is given $4x+2y+12z=7$ then $x,y\&z$ intercepts will be

If $z_1$ and $z_2$ are two non zero complex numbers satisfying the equation $|z_1|=|z_2|+|z_1-z_2|,$ then which of the following is/are true?

In a quadrilateral ABCD, which is not a trapezium, it is known that $\angle$ DAB = $\angle$ABC = ${60}^0$. Moreover, $\angle$CAB = $\angle$CBD. Then,

The area between $x=y^2$ and x = 4 is divided into two equal parts by the line x = a, find the value of a.