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$2ta{n}^{-1}\left[tan\frac{\alpha }{2}tan(\frac{\pi }{4}-\frac{\beta }{2})\right]$ is equal to

The radius of the circle which touches the line $x+y=0$ at $M(-1,1)$ and cuts the circle $x^2+y^2+6x-4y+18=0$ orthogonally, is

If $\Delta =\left|\begin{array}{ccc}abc& b^{2}c& c^{2}b\\ abc& c^{2}a& c{a}^2\\ abc& a^{2}b& b^{2}a\end{array}\right|=0$ $(a,b,c\in R$

and are all different and non-zero), then the value of $a+b+c$ is:

$\int \frac{x^2}{x^4-x^2-12}dx$

The corner points of the feasible region determined by a system of linear constraints are $(0,10),(5,5),(15,15),(0,20).$ Let $z=px+qy$ where $p,q>0.$ If the maximum of $z$ occurs at both the points $(15,15)$ and $(0,20),$ then which of the following is true:

If the variance of a random variable $X$ is $9,$ the $S.D.$ of the random variables $-2X+5,3X-6$ and $-4X-3$ is $a,b$ and $c$ respectively. Then which of the following statement(s) is/are correct $?$

The area of the loop of the curve $y^2=x^4(x+2)$ is [in square units]

Let $f:R\to R$ be defined as $f\left(x\right)=\{\begin{array}{ccc}\frac{x^3}{(1-\cos 2x{)}^2}\cdot {\log }_e\left(\frac{1+2x{e}^{-2x}}{(1-x{e}^{-x}{)}^2}\right)& ,& x\ne 0\\ \alpha & ,& x=0\end{array}$

If $f$ is continuous at $x=0$, then $\alpha$ is equal to:

If ${\sin }^4\theta +\frac{1}{{\sin }^4\theta }=194,\theta \ne n\pi ,n\in Z$, then the value(s) of $(2\text{cosec}\theta -\cot \theta \cos \theta )$ can be

The coordinates of the foot of perpendicular drawn from the point $(1,-2)$ on the line $y=2x+1$,is

Which of the following is true for $\alpha ,\beta \in (0,\pi )$ and $\alpha \ne \beta$

Which among the following can be categorized as a mathematical statement?

The area of a square whose two sides lie on lines x+y=1 and x+y+2=0 will be .

$ta{n}^{-1}\left[\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right]=$

If the function $e^{-x}f\left(x\right)$ assumes its minimum in the interval $[0,1]\text{at}x=\frac{1}{4},$ which of the following is true?

If $f\left(x\right)=3x-2$ and $(gof{)}^{-1}=x-2$, then $\int g\left(x\right)dx$ is (where $C$ is constant of integration)

The number of different n$\times$n symmetric matrices with each element being either 0 or 1 is: (Note: power (2, x) is same as $2^x$)

In some of the cases we can split the integrand into the sum of the two functions such that the integration of one of them by parts produces an integral which cancels the other integral. Suppose we have an integral of the type
$\int \left[f\right(x\left)h\right(x)+g(x\left)\right]dx$
Let $\int f\left(x\right)h\left(x\right)dx=I_1$ and $\int g\left(x\right)dx=I_2$
Integrating $I_1$ by parts, we get
$I_1=f\left(x\right)\int h\left(x\right)dx-\int \left\{f^{\prime }\right(x)\int h(x\left)dx\right\}dx$
Now find $\int (\frac{1}{logx}-\frac{1}{(logx{)}^2})dx$ (x > 0)

The sum of the series $ta{n}^{-1}\frac{1}{2}+ta{n}^{-1}\frac{1}{8}+ta{n}^{-1}\frac{1}{18}+ta{n}^{-1}\frac{1}{32}+\cdots +\infty$ is

If the three equations $x^2+ax+12=0,$ $x^2+bx+15=0,$ $x^2+(a+b)x+36=0$ have a common possible root. Then, the sum of roots is

The latus rectum of parabola $y^2=5x+4y+1$ is

Which of the following is/are valid PSD?

If f : R $\to$ R defined by f(x) = $\frac{2x+3}{5}$, then $f^{-1}\left(x\right)=$

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse

If $\frac{dy}{dx}=\frac{xy}{x^2+y^2};y\left(1\right)=1;$ then a value of $x$ satisfying $y\left(x\right)=e$ is:

Find the equation of circle circumscribing the quadrilateral formed by four lines $3x+4y-5=0$, $4x-3y-5=0$, $3x+4y+5=0$ and $4x-3y+5=0$

If $f(x+2y,x-2y)=xy,$ then $f(x,y)$ equals

The mean ans dtandard deviation of 20 observations are found to be 10 and 2 respectively. On rechecking it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases :

(i) If worng item is omitted

(ii) If it is replaced by 12.

There exists a positive real number $x$ satisfying $\cos \left({\tan }^{-1}x\right)=x.$ Then the value of ${\cos }^{–1}\left(\frac{x^2}{2}\right)$ is

If $z=se{c}^{-1}(x+\frac{1}{x})+se{c}^{-1}(y+\frac{1}{y})$ , where xy > 0, then the value of z (among the given values) which is not possible is

Let ${\Delta }_r=\left|\begin{array}{ccc}r-1& n& 6\\ (r-1{)}^2& 2n^2& 4n-2\\ (r-1{)}^3& 3n^3& 3n^2-3n\end{array}\right|$. Then the value of $\sum _{r=1}^n{\Delta }_r$ is:

The angle between diagonals of a cube is?

Find the equations of direct common tangents for two circles

$x^2+y^2+6x-2y+1=0,$ $x^2+y^2-2x-6y+9=0$