\(\int\left(\frac{\tan \left(\frac{1}{x}\right)}{x}\right)^2 \mathrm{~d} x=\)

The differential equation of \(y=\mathrm{e}^x\left(\mathrm{a}+\mathrm{b} x+x^2\right)\) is

Let \(\quad \overline{\mathrm{a}}=\alpha \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \quad \overline{\mathrm{b}}=3 \hat{\mathrm{i}}-\beta \hat{\mathrm{j}}+4 \hat{\mathrm{k}} \quad\) and \(\overline{\mathrm{c}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}\), where \(\alpha, \beta \in \mathbb{R}\), be three vectors. If the projection at \(\overline{\mathrm{a}}\) on \(\overline{\mathrm{c}}\) is \(\frac{10}{3}\) and \(\overline{\mathrm{b}} \times \overline{\mathrm{c}}=-6 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}\), then the value of \(\alpha^2+\beta^2-\alpha \beta\) is equal to

If \(\int \frac{5 \tan x}{\tan x-2} d x=x+a \log |\sin x-2 \cos x|+c\), then \(a\) (Where \(\mathrm{c}\) is constant of integration)

If \(\int_0^{\mathrm{k}} \frac{d x}{2+8 x^2}=\frac{\pi}{16}\), then value of \(\mathrm{k}\) is

The equation of the line, through \(A(1,2,3)\) and perpendicular to the vector \(2 \hat{i}+\hat{j}-\hat{k}\) and \(\hat{i}+3 \hat{j}+2 \hat{k}\), is

\(\int \frac{\tan ^4 \sqrt{x} \cdot \sec ^2 \sqrt{x}}{\sqrt{x}} d x=\)

For the following combination of logic gates, when all the three inputs \(A\), \(B\) and \(C\) are first 'high' and then 'low', the output \(Y\) will be respectively.

In a thermodynamic system, working substance is an ideal gas. Its internal energy is in the form of

Which of the following is NOT dihydric alcohol?

If the apertaure of a telescope is decreased the resolving power will

A coil having effective area \(A\), is held with its plane normal to magnetic field of induction \(\mathrm{B}\). The magnetic induction is quickly reduced by \(25 \%\) of its initial value in 2 second. Then the e.m.f. induced across the coil will be

In a normal healthy person, if total number of WBCs is about \(9000 / mm ^3\), then the number of Agranulocytes is about __________ \(/ mm ^3\) of blood.