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The equation of the circle of area $4\pi$ lying in the first quadrant and touching both the coordinate axes is

Let $M$ be a $3\times 3$ invertible matrix with real entries and let $I$ denote the $3\times 3$ identity matrix. If $M^{-1}=\text{adj}\left(\text{adj}M\right),$ then which of the following statements is/are ALWAYS TRUE?

The position of the point (2, 5) relative to the hyperbola $9x^2-y^2=1$

Evaluate $\int \frac{x^4(5+4x)}{(x^5+x+1{)}^2}dx$

Let $(2+i)z+(2-i)\overline{z}=\lambda ,\lambda ϵR$, be a straight line in the complex plane. If $A\left(z_1\right)$ and $B\left(z_2\right)$ are 2 points in the plane such that AB is perpendicular to the given line and also the midpoint of AB lies on the given line, then $\lambda$ is equal to

Prove the following by using the principle of mathematical induction for all n ∈ N:

The number of possible tangent(s) drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$, which is/are perpendicular to $5x+2y=10$, is

If $\int x\ln (1+\frac{1}{x})dx=f\left(x\right)\ln (1+\frac{1}{x})+A\ln |x+1|+Bx+C$, then which of the following is(are) correct

(where $C$ is constant of integration)

If $f\left(x\right)=x^2+1$ and $g\left(x\right)=2x$, then find the domain of the function

$h\left(x\right)=\frac{(f+g)x}{(f-g)x}$

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Which of the following curve has no asymptote:

If $y=\ln \left(\tan x\right),$ then $\frac{dy}{dx}=$

If,
find values of x
and y.

If the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are inclined at an angle $2\theta$ such that $0\le \theta \le \pi$ and $|\overrightarrow{a}-\overrightarrow{b}|<1,$ then $\theta$ lies in the interval

The set of values of $a$ for which $a{x}^2-(4-2a)x-8<0$ for exactly three integral values of $x$ is -

A straight line given by the equation  $\left|\begin{array}{ccc}x+3& y-1& 1\\ 7& -1& 1\\ -3& 9& 1\end{array}\right|=0$ passes through the point. 

If a+b+c=10 and a²+b²=58,find the value of a³+b³

tan6*tan42*tan66*tan78=1

The missing term in the third figure is

Let $f\left(x\right)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|.$ The value of $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x}$ is equal to

Choose the
correct answer.

If x, y, z are nonzero real numbers, then the
inverse of matrix
is

A. B.

C. D.

$Ify=(ta{n}^{-1}x{)}^2,showthat(x^2+1{)}^2{y}_2+2x(x^2+1)y_1=2.$

Given $A=\left[\begin{array}{cccc}1& 3& -2& 1\\ 5& 1& 0& -1\\ 0& 1& 0& -2\\ 2& -1& 0& 3\end{array}\right]$ . Find det(A) wrt to row 1 and column 3.

Which of the followings set of intervals of $x$ satisfying the inequality $[{\tan }^{-1}x{]}^2-\frac{3\left[{\tan }^{-1}x\right]}{2}+\frac{1}{2}>0$

(where $[.]$ denotes greatest integer function)

Find the lengths of the major and minor axes, coordinates of the vertices and the foci, the eccentricity and length of the latus rectum of the ellipse $4x^2+9y^2=144.$

Integrate the function.
$\int (si{n}^{-1}x{)}^{2}dx.$

The equations of the tangents to the hyperbola $3x^2-4y^2=12$ which are parallel to the line 2x + y + 7 = 0 are

The circle passing through the points $(1,0),(2,-7)$ and $(8,1)$ also passes through

If 2$ta{n}^{-1}\left(cos\theta \right)=ta{n}^{-1}\left(2cosec\theta \right),\text{then show that}\theta =\frac{\pi }{4},$ where $\theta$ is any integer.

If $y=mx-b\sqrt{1+m^2}$ is a common tangent to $x^2+y^2=b^2$ and $(x-a{)}^2+y^2=b^2$, where $a>2b>0$, then the positive value of $m$ is

Prove that:

$co{s}^{3}2\theta +3cos2\theta =4(co{s}^6\theta -si{n}^6\theta )$

A person standing at the junction of two straight paths represented by the equations $2x-3y+4=0$ and $3x+4y-5=0$. If he wants to reach the path whose equation is $6x-7y+8=0$ in the least time, then the equation of the path he should follow is