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The principal value of ${\sec }^{-1}\left(\frac{-2}{\sqrt{3}}\right)$ is

Let the function $f\left(x\right)=\sqrt{x^2-|x+2|+x}$ be a real valued function, then the domain of $f\left(x\right)$ is

If the line 2x - y + I = 0 touches the circle at the point (2, 5) and the centre of the circle lies on the line x + y - 9 = 0. Find the equation of the circle.

Find the value of ${\sum }_{r=0}^{n}r{}^n{C}_r$

The coordinates of an endpoint of the latusrectum of the parabola $(y-1{)}^2=4(x+1)$ are

The range of numbers $x_1,x_2,x_3,x_4,x_5$ is $r$ where $x_i<x_{i+1}$. If each number is multiplied by $p(p>0)$ and then $k$ is subtracted from each number, then the range of new numbers is

Four persons are selected at random out of 3 men, 2 women and 4 children. The probability that there are exactly 2 children in the selection is

Study the following carefully to answer these questions.
${}^{\prime }P\times Q^{\prime }$ means 'P is the wife of Q'.
${}^{\prime }P÷Q^{\prime }$ means 'P is the father of Q'.
${}^{\prime }P+Q^{\prime }$ means 'P is the son of Q'.
${}^{\prime }P-Q^{\prime }$ means 'P is the sister of Q'.
Which of the following represents 'S is the mother of T'?

If $(h,k)$ is the centre of the circle touches $y-\text{axis}$ at a distance of $12\text{units}$ from the origin and makes an intercept of $10\text{units}$ on $x-\text{axis}$, then the equation of circle for which $(h+k)$ is minimum, is

Match the images with their symmetric part.

Let A = {1, 2, 3, 4} and let R = {(2, 2), (3, 3), (4, 4), (1, 2)} be a relation on A. Then, R is

If $A=\text{diag}(2,4,5),$ then the trace of $A^{-1}$ is

The set of all possible values of $\theta$ in the interval $(0,\pi )$ for which the points $(1,2)$ and $(\sin \theta ,\cos \theta )$ lie on the same side of the line $x+y=1$ is:

Solve the equation

If $\alpha ,\beta$ are the roots of the quadratic equation $x^2-7x+12=0$. Then the equation with roots $-\alpha ,-\beta$ is

From a pack of 52 cards two are drawn with replacement. The probability, that the first is a diamond and the second is a king, is

If $(a\sec \theta ,b\tan \theta )$ and $(a\sec \varphi ,b\tan \varphi )$ are the ends of a focal chord of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, the value of $\tan \frac{\theta }{2}\cdot \tan \frac{\varphi }{2}$ can be equal to :

The value of $\underset{x\to 0}{lim}\frac{1-8^x-9^x+(72{)}^x}{(9^x-8^x)(\tan 9x-\sin 8x)}$ is

The solution of $x^2\frac{dy}{dx}-xy=1+\cos \frac{y}{x}$ is

If
X = {a,
b,
c,
d}
and Y = {f,
b,
d,
g},
find

(i) X
– Y

(ii) Y
– X

(iii) X
∩
Y

Let $\alpha =\underset{x\in R}{max}\{8^{2\sin 3x}\cdot 4^{4\cos 3x}\}$ and $\beta =\underset{x\in R}{min}\{8^{2\sin 3x}\cdot 4^{4\cos 3x}\}.$

If $8x^2+bx+c=0$ is a quadratic equation whose roots are ${\alpha }^{1/5}$ and ${\beta }^{1/5}$, then the value of $c-b$ is equal to

Evaluate $\int (\frac{\cos x}{x}-\log x^{\sin x})dx$

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are the position vectors of points A and B respectively, find the position vector of point C on AB produced such that $\overrightarrow{AC}=3\overrightarrow{AB}$

If the line joining the points (6, 6) and (1, – 3) is divided by the line x + λy = 2, internally in the ration 2 : 3 , then the value of $\lambda$ is

If ${lim}_{x\to a}\frac{x^3-a^3}{x-a}={lim}_{x\to 1}\frac{x^4-1}{x-1}$, find all possible values of a.

In a compact single dimensional array representation for lower triangular matrices (i.e. all the elements above the diagonal are zero) of size n $\times$ n, non-zero elements (i.e. elements of the lower triangle) of each row are stored one after another, starting from the first row, the index of the $(i,j{)}^{th}$ element of the lower triangular matrix in this new representation is

In the parabola $y^2+4=4x$, a chord passing through point $(2,0)$ cuts the parabola at $P$ and $Q$. If coordinates of $P$ are $(5,4)$ and the tangents at $P$ and $Q$ meet at $R$, then



$\begin{array}{cccc}& \text{List-I}& & \text{List-II}\\ \text{(I)}& \text{The focus of the parabola is}& \text{(P)}& (0,\frac{3}{2})\\ \text{(II)}& \text{The centroid of}△PQR\text{is}& \text{(Q)}& (4,0)\\ \text{(III)}& \text{The circumcentre of}△PQR\text{is}& \text{(R)}& (\frac{25}{12},\frac{3}{2})\\ \text{(IV)}& \text{The orthocentre of}△PQR\text{is}& \text{(S)}& (\frac{25}{8},\frac{3}{2})\\ & & \text{(T)}& (\frac{25}{4},\frac{3}{2})\\ & & \text{(U)}& (2,0)\end{array}$



Which of the following is the only INCORRECT combination?

If $\frac{2}{3},H_1,H_2,H_3,H_4,\frac{2}{13}$ is an H.P., then the value of $\frac{H_1{H}_4}{H_2{H}_3}$ is

Solve the differential equation $\frac{dy}{dx}=1+x+y^2+x{y}^2,$ when y=0 and x=0.

If p and q are order and degree of differential equation $y^2(\frac{d^{2}y}{d{x}^2}{)}^2+3x(\frac{dy}{dx}{)}^{\frac{1}{3}}+x^2{y}^2=sinx$, then

If the parabolas $y^2=4b(x-c)$ and $y^2=8ax$ have a common normal except the axis of symmetry of the parabolas, then the range of $\frac{c}{2a-b}$ is