If \(\frac{\sin ^{-1} x}{a}=\frac{\cos ^{-1} x}{b}=\frac{\tan ^{-1} y}{c} ; 0< x< 1,\) then the value of \(\cos \left(\frac{\pi c }{ a + b }\right)\) is

For \(k \in R\), let the solutions of the equation \(\cos \left(\sin ^{-1}\left(x \cot \left(\tan ^{-1}\left(\cos \left(\sin ^{-1} x\right)\right)\right)\right)\right)=k, 0\,<\,|x|<\,\frac{1}{\sqrt{2}}\) be \(\alpha\) and \(\beta\), where the inverse trigonometric functions take only principal values. If the solutions of the equation \(x ^{2}- bx -5=0\) are \(\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}\) and \(\frac{\alpha}{\beta}\), then \(\frac{b}{k^{2}}\) is equal to\(......\)

If \(\overrightarrow x  = 3\hat i - 6\hat j - \hat k\) , \(\overrightarrow y  = \hat i + 4\hat j - 3\hat k\) and \(\,\,\overrightarrow z  = 3\hat i - 4\hat j - 12\hat k\) , then the magnitude of the projection of \(\overrightarrow x  \times \overrightarrow y \) on \(\overrightarrow z\) is

The coefficient of \(x^{50}\) in the binomial expansion of \({\left( {1 + x} \right)^{1000}} + x{\left( {1 + x} \right)^{999}} + {x^2}{\left( {1 + x} \right)^{998}} + ..... + {x^{1000}}\) is

A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are \(\dfrac{2}{5}\), \(\dfrac{1}{5}\) and \(\dfrac{2}{5}\). The probabilities that the candidate reaches late at the examination centre are \(\dfrac{1}{5}\), \(\dfrac{1}{3}\) and \(\dfrac{1}{4}\) if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is:

If \(y=y(x)\) is the solution of the differential equation,
\(\sqrt{4-x^2} \frac{\mathrm{~d} y}{\mathrm{~d} x}=\left(\left(\sin ^{-1}\left(\frac{x}{2}\right)\right)^2-y\right) \sin ^{-1}\left(\frac{x}{2}\right),-2 \leq x \leq 2, y(2)=\frac{\pi^2-8}{4}\), then \(y^2(0)\) is equal to

Let \(\vec{a}=3 \hat{i}+\hat{j}-\hat{k}\) and \(\overrightarrow{ c }=2 \hat{ i }-3 \hat{ j }+3 \hat{k}\). If \(\vec{b}\) is \(a\) vector such that \(\vec{a}=\vec{b} \times \vec{c}\) and \(|\vec{b}|^2=50\), then \(|72-| \vec{b}+\left.\vec{c}\right|^2 \mid\) is equal to \(..........\).

The coefficient of \(x^{1012}\) in the expansion of \({\left( {1 + {x^n} + {x^{253}}} \right)^{10}}\) , (where \(n \leq 22\) is any positive integer), is

The inverse of \(y=5^{\log x}\) is

If \(x^{2}+9 y^{2}-4 x+3=0, x, y \in R\), then \(x\) and \(y\) respectively lie in the intervals:

Let \(f\) be a polynomial function such that \(\log_2(f(x)) = \left(\log_2\left(2+\dfrac{2}{3}+\dfrac{2}{9}+\ldots\infty\right)\right)\cdot\log_3\left(1+\dfrac{f(x)}{f(1/x)}\right)\), \(x>0\) and \(f(6)=37\). Then \(\displaystyle\sum_{n=1}^{10}f(n)\) is equal to ________.

The mean of five observations is \(5\) and their variance is \(9.20\). If three of the given five observations are \(1, 3\) and \(8\), then a ratio of other two observations is

The value of \({1^2} + {3^2} + {5^2} + ....... + {25^2}\) is