Mathematics Tes...

An integer is chosen at random from the integers \(1,2,3, \ldots, 50\). The probability that the chosen integer is a multiple of atleast one of 4, 6 and 7 is:

Let \(A=\left[\begin{array}{llc}2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2\end{array}\right]\) and \(P=\left[\begin{array}{lll}1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5\end{array}\right]\). The sum of the prime factors of \(\left|\mathrm{P}^{-1} \mathrm{AP}-2 \mathrm{I}\right|\) is equal to:

Let \(\mathrm{f}:[-1,3] \rightarrow R\) be defined as

\(f(x)=\left\{\begin{array}{cc}|x|+[x], & -1 \leq x<1 \\ x+|x|, & 1 \leq x<2 \\ x+[x], & 2 \leq x \leq 3\end{array}\right.\)

where \([t]\) denotes the greatest integer less than or equal to \(t\). Then, \(f\) is discontinuous at:

Let \(\lambda \in \mathbb{R}, \vec{a}=\lambda \hat{i}+2 \hat{j}-3 \hat{k}, \vec{b}=\hat{i}-\lambda \hat{j}+2 \hat{k}\).

If \(\vec{a}+\vec{b}) \times(\vec{a} \times \vec{b} \times(\vec{a}-\vec{b})=8 \hat{i}-40 \hat{j}-24 \hat{k}\), then \(|\lambda(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})|^2\) is equal to:

If \(\vec{a}, \vec{b}, \vec{c}\) are three non-zero vectors and \(\hat{n}\) is a unit vector perpendicular to \(\overrightarrow{\mathrm{c}}\) such that \(\overrightarrow{\mathrm{a}}=\alpha \overrightarrow{\mathrm{b}}-\hat{\mathrm{n}},(\alpha \neq 0)\) and \(\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=12\), then \(|\overrightarrow{\mathrm{c}} \times(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})|\) is equal to:

Using the principle of mathematical induction, prove that \(1 \times 3+2 \times 3^{2}+3 \times 3^{3}+\ldots+n \times 3^{n}=\frac{(2 n-1) 3^{n+1}+3}{4}\) for all:

If \(4 x+3<6 x+7\), then \(x\) belongs to the interval

Evaluate the expression: \(\sqrt{\frac{1+\cos \theta}{1-\cos \theta}}=?\)

\(\text { If } A=\{x \in R:|x|<2\} \text { and } B=\{x \in R:|x-2| \geq 3\} \text {; then : }\)