If \(m: n\) is the ratio in which the point \(\left(\frac{8}{5},-\frac{1}{5}, \frac{8}{5}\right)\) divides the line segment joining the points \((2, \mathrm{p}, 2)\) and \((\mathrm{p},-2, \mathrm{p})\) where p is an integer then \(\frac{3 m+n}{3 n}=\)

The vectors \(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) and \(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}\)

If the coefficient of \(3^{\text {rd }}\) term from the beginning in the expansion of \(\left(a x^2-\frac{8}{b x}\right)^9\) is equal to the coefficient of \(3^{\text {rd }}\) term from the end in the expansion of \(\left(a x-\frac{2}{b x^2}\right)^9\) then the relation between \(a\) and \(b\) is

If \(\alpha, \beta\) are the roots of the equation \(x^2+5 x+2=0\), then \(\left(\frac{\alpha}{2+5 \alpha}\right)^2+\left(\frac{\beta}{2+5 \beta}\right)^2=\)

If \(f(x)=\cosh ^1\left(\frac{1-x}{1+x}\right)\) is well defined, then \(f^{\prime}(x)=\)

A target is to be destroyed in a bombing exercise and there is a \(75 \%\) chance that a bomb will hit the target. Assuming that two direct hits are required to destroy the target completely, the minimum number of bombs to be dropped in order that the probability of destroying the target is not less than \(99 \%\), is

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\) Assertion (A) : The identity \(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for \(\vec{a}\). Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\), \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\) Which of the following is correct?

If cis \(\alpha\) is the common value of \((-1)^{1 / 4}\) and \((-i)^{1 / 2}\) then \(\tan \alpha=\)

If the degree of dissociation of formic acid is \(11.0 \%\), the molar conductivity of 0.02 M solution of it is (Given, \(\lambda^{\circ}\left(\mathrm{H}^{+}\right)=349.6 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}, \lambda^{\circ}\left(\mathrm{HCOO}^{-}\right)=54.6\) \(\mathrm{S} \mathrm{cm}^2 \mathrm{~mol}^{-1}\) )

If \(\alpha, \beta, \gamma, \delta\) are the roots of the equation \(x^4-8 x^3+11 x^2+32 x-60=0\) and \(\alpha < \beta < \gamma < \delta\), then \(4 \alpha+3 \beta+2 \gamma+\delta=\)

In the balanced equation of the following reaction, the ratio of $\mathrm{a}/\mathrm{b}$ is   ${\mathrm{aCaCO}}_3+{\mathrm{bH}}_3{\mathrm{PO}}_4\to {\mathrm{pCa}}_3{\left({\mathrm{PO}}_4\right)}_2+{\mathrm{qCO}}_2+{\mathrm{rH}}_2\mathrm{O}$

Addition of \(\mathrm{HBr}\) to propene in presence of a peroxide takes place contrary to Markovnikov rule. This can be explained by the mechanism involving

Two fair dice are rolled. The probability of the sum of digits on their faces to be greater than or equal to 10 is