If _ x 2 x^n - 2^n x - 2 = 80, where n is a positive integer, then n =

$\sin ^{-1}\left(\sum_{i=1}^{\infty} x^{i+1}-x \sum_{i=1}^{\infty}\left(\frac{x}{2}\right)^i\right)=\frac{\pi}{2}-\cos ^{-1}\left(\sum_{i=1}^{\infty}\left(-\frac{x}{2}\right)^i-\sum_{i=1}^{\infty}(-x)^i\right)$

lying in the interval $\left(-\frac{1}{2}, \frac{1}{2}\right)$ is. . . . .

(Here, the inverse trigonometric functions $\sin ^{-1} x$ and $\cos ^{-1} x$ assume values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $[0, \pi]$, respectively.)

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A bag contains $3$ red, $4$ white and $5$ blue balls. All balls are different. Two balls are drawn at random. The probability that they are of different colour is

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Let R = (a, b, c, d) and S = (1, 2, 3), then the number of functions f, from R to S, which are onto is:

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The angle of intersection of the curves ${y^2} = 2x/\pi $ and $y = \sin x$, is

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If ${^\text{m}}\text{C}_{\text{1}}={^\text{n}}\text{C}_{\text{2}},$ is then:

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The value of $\lambda $ such that sum of the squares of the roots of quadratic equation, $x^2 + (3 - \lambda )x + 2 = \lambda $ has the lest value is

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Let $m$ (respectively, $n$ ) be the number of $5$ -digit integers obtained by using the digits $1,2,3,4,5$ with repetitions (respectively, without repetitions) such that the sum of any two adjacent digits is odd. Then $\frac{m}{n}$ is equal to