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If $\sum _{r=1}^{50}{\tan }^{-1}\frac{1}{2r^2}=p,$ then the value of $\tan p$ is

In $△ABC,b^{2}cos2A-a^{2}cos2B$ =

Using the method of
integration find the area bounded by the curve

[Hint: the
required region is bounded by lines x + y = 1, x
– y = 1, – x + y = 1 and – x
– y = 11]

One mapping (function) is selected at random from all the mappings of the set A={1,2,3,.....,n} into itself. The probability that the mapping selected is one to one is

An infinite G.P. has first term $x$ and sum $5.$ Then which of the following is true

A circle has the same centre as an ellipse and passes through the foci $F_1$ and $F_2$ of the ellipse such that the two curves intersect in $4$ points. Let '$P$' be any one of their point of intersection. If the major axis of the ellipse is $17$ and the area of the triangle $P{F}_1{F}_2$ is $30$, then the distance between the foci is

The maximum value of $f\left(x\right)=\left|\begin{array}{ccc}{\sin }^{2}x& 1+{\cos }^{2}x& \cos 2x\\ 1+{\sin }^{2}x& {\cos }^{2}x& \cos 2x\\ {\sin }^{2}x& {\cos }^{2}x& \sin 2x\end{array}\right|,x\in R$ is :

Find the value of (1 + Cos π/8)×.(1+cos3π/8)×(1+cos 5π/8 ) ×(1+cos7π/8) .

A die is thrown $6$ times. If 'getting an odd number' is a success, then the probability of at most $5$ successes is:

In the intergal $\underset{0}{\overset{10}{\int }}\frac{\left[\sin 2\pi x\right]}{e^{x-\left[x\right]}}dx=\alpha e^{-1}+\beta e^{-\frac{1}{2}}+\gamma ,$ where $\alpha ,\beta ,\gamma$ are integers and $\left[x\right]$ denotes the greatest integer less than or equal to $x,$ then the value of $\alpha +\beta +\gamma$ is equal to:

If C(n, 12) = C(n, 8), then C(22, n) is equal to

If ${\cot }^{-1}x+{\tan }^{-1}3=\frac{\pi }{2}$, then $x=$

If $\int \frac{xdx}{\sqrt{e^{2x}-(x+1{)}^2}}={\cos }^{-1}\left(f\right(x\left)\right)+C,$ where $C$ is arbitrary constant of integration, then

In the following cases,
find the distance of each of the given points from the corresponding
given plane.

Point Plane

(a) (0, 0, 0)

(b) (3, −2,
1)

(c) (2, 3, −5)

(d) (−6, 0,
0)

Let $I_n=\underset{-\pi }{\overset{\pi }{\int }}\frac{\sin \left(nx\right)}{(1+3^x)\sin x}dx,n=0,1,2,\cdots ,$ then which of the following is/are CORRECT?

Given that $E_{Z{n}^{+2}|Zn}^0=-0.764V\&E_{C{d}^{+2}|Cd}^0=-0.403V$, the Emf of the cell $Zn|Z{n}^{+2}||C{d}^{+2}|Cd$ will be given by

$[a_{Zn|Z{n}^{+2}}=0.004\&a_{C{d}^{+2}|Cd}=0.2]$

The cofficient of $x^9$ in the polynomial given by $\sum _{r=1}^{11}(x+r)(x+r+1)(x+r+2).......(x+r+9)$ is

Maximise Z
= 3x + 4y

Subject to
the constraints:

Evaluate $\int \frac{3x^2+7}{x^8+3x^3+5x}dx$

(where $C$ is constant of integration)

If $x^m\cdot y^n=(x+y{)}^{m+n}$, then which of the following is/are true :