Suppose \(E\) and \(F\) are two events of a random experiment. If the probability of occurrence of \(E\) is \(1 / 5\) and the probability of occurrence of \(F\) given \(E\) is \(1 / 10\), then the probability of non-occurrence of atleast one of the events \(E\) and \(F\) is

If \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) are three vectors such that \(|\mathbf{a}|=1,|\mathbf{b}|=2\), \(|\mathbf{c}|=3\) and \(\mathbf{a} \cdot \mathbf{b}=\mathbf{b} \cdot \mathbf{c}=\mathbf{c} \cdot \mathbf{a}=0\), then \(\|[\mathbf{a} \mathbf{b} \mathbf{c}] \mid\) is equal to

Match the items of List - I with those of the entires of List - II
\(List - I\)
\(\begin{aligned} & \text { (I) } \sin ^2 5^{\circ}+\sin ^2 10^{\circ}+ \\ & \sin ^2 15^{\circ}+\ldots+\sin ^2 90^{\circ}=\end{aligned}\)
\(\begin{aligned} & \text { (II) } \tan ^2 5^{\circ} \cdot \tan ^2 10^{\circ} \text {. } \\ & \tan ^2 15^{\circ} \ldots \tan ^2 85^{\circ}= \\ & \end{aligned}\)
\(\begin{aligned} & \text { (III) } \cos ^2 5^{\circ}+\cos ^2 10^{\circ} \\ & +\cos ^2 15^{\circ}+\ldots+\cos ^2 180^{\circ}=\end{aligned}\)
\(\begin{gathered}\text { (IV) } \cot 5^{\circ}+\cot 10^{\circ}+\cot 15^{\circ} \\ +\ldots .+\cot 175^{\circ}=\end{gathered}\)
\(List - II\)
(A) \(0\)
(B) \(\frac{19}{2}\)
(C) \(18\)
(D) \(1\)
(E) \(-1\)

If \(\alpha, \beta\) are the roots of \(x^2-2 x+4=0\), for \(n \in \mathbf{N}\), what is the value of \(\alpha^n+\beta^n=\)

Let \(A\) and \(B\) be two events with \(P(A)=\frac{1}{7}\), \(P\left(\frac{A}{B}\right)=\frac{2}{3}\) and \(P(B)=\frac{2}{7}\). Then, the value \(P\left(\frac{B}{A}\right)\) is

The product and sum of the roots of the equation \(\left|x^2\right|-5|x|-24=0\) are respectively

If a committee of 10 members is to be formed from 8 men and 6 women, then the number of different possible committees in which the men are in majority is

If \(\mathrm{C}\) is a point on the straight line joining the points \(\mathrm{A}(-2+\mathrm{i})\) and \(\mathrm{B}(3-4 \mathrm{i})\) in the Argand plane and \(\frac{\mathrm{AC}}{\mathrm{CB}}=\frac{1}{2}\), then the argument of \(\mathrm{C}\) is

A \(6 \mathrm{~V}\) cell with \(0.5 \Omega\) internal resistance, a \(10 \mathrm{~V}\) cell with \(1 \Omega\) internal resistance and a \(12 \Omega\) external resistance are connected in parallel. The current (in ampere) through the \(10 \mathrm{~V}\) cell is

If \(\rho\) is a complex cube root of unity, then
\(\cos \left(\sum_{k=1}^7(k-\omega)\left(k-\omega^2\right) \frac{\pi}{175}\right)=\)

Which one of the following ions has same number of unpaired electrons as those present in \(\mathrm{V}^{3+}\) ion?

The \(\mathrm{pH}\) of \(0.01 \mathrm{M} \mathrm{BOH}\) solution is 10 . What is its degree of dissociation?
(Given \(\mathrm{K}_{\mathrm{b}}\) of \(\mathrm{BOH}\) is \(1 \times 10^{-6}\) )