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If \(\alpha\) and \(\beta\) are roots of \(a x^{2}+b x+c=0\) then the equation whose roots are \(\alpha^{2}\) and \(\beta^{2}\) is
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The sum of \(n\) terms of the following series; \(1^{3}+3^{3}+5^{3}+7^{3}+\ldots .\) is
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If the matrix \(A=\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 2 & 0 \\ 2 & 0 & 2\end{array}\right)\), then \(A^{n}=\left(\begin{array}{lll}a & 0 & 0 \\ 0 & a & 0 \\ b & 0 & a\end{array}\right), n \in N\) where
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The letters of the word COCHIN are permuted and all permutation are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is
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The points of the ellipse \(16 x^{2}+9 y^{2}=400\) at which the ordinate decreases at the same rate at which the abscissa increases is/are given by
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General solution of \(y \frac{d y}{d x}+b y^{2}=a \cos x, 0<x<1\) is
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For non-zero vectors \(\vec{a}\) and \(\vec{b}\) if \(|\vec{a}+\vec{b}|<|\vec{a}-\vec{b}|\), then \(\vec{a}\) and \(\vec{b}\) are
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If \([\mathrm{x}]\) denotes the greatest integer less than or equal to \(\mathrm{x}\), then the value of the integral \(\int_{0}^{2} \mathrm{x}^{2}[\mathrm{x}] \mathrm{dx}\) equals
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Standard Deviation of n observations \(\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3} \ldots . \mathrm{a}_{\mathrm{n}}\) is \(\sigma .\) Then the standard deviation of the observations \(\lambda \mathrm{a}_{1}, \lambda \mathrm{a}_{2}, \ldots \lambda \mathrm{a}_{n}\)
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If the function \(f: \mathbb{R} \rightarrow R\) is defined by \(f(x)=\left(x^{2}+1\right)^{35} \forall \in \mathbb{R}\), then \(f\) is
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If \(A=\left\{5^{n}-4 n-1: n \in N\right\}\) and \(B=\{16(n-1): n \in N\}\), then
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Let \(R\) be a relation defined on the set \(Z\) of all integers and \(x R y\) when \(x+2 y\) is divisible by 3 . Then
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Let \(Q=\left(\begin{array}{cc}\cos \frac{\pi}{4}-\sin \frac{\pi}{4} \\ \sin \frac{\pi}{4} & \cos \frac{\pi}{4}\end{array}\right)\) and \(x=\left(\begin{array}{c}\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}\end{array}\right)\) then \(Q^{3} x\) is equal to
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Let \(A\) is a \(3 \times 3\) matrix and \(B\) is its adjoint matrix. If \(|B|=64\), then \(|A|=\)
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If \(x, y\) and \(z\) be greater than 1 , then the value of \(\left|\begin{array}{ccc}1 & \log _{x} y & \log _{x} z \\ \log _{y} x & 1 & \log _{y} z \\ \log _{z} x & \log _{z} y & 1\end{array}\right|\) is
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\(1+{ }^{n} C_{1} \cos \theta+{ }^{n} C_{2} \cos 2 \theta+\ldots \ldots+{ }^{n} C_{n} \cos n \theta\) equals
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In the expansion of \((x-1)(x-2) \ldots .(x-18)\), the coefficient of \(x^{17}\) is
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For \(+\) ve integer \(n, n^{3}+2 n\) is always divisible by
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If, \(\frac{1}{{ }^{5} C_{t}}+\frac{1}{{ }^{6} C_{r}}=\frac{1}{{ }^{4} C_{t}}\), then the value of r equals to
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The number of ways in which the letters of the word ARRANGE can be permuted such that the R's occur together is
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