${\int }_{\frac{\pi }{18}}^{\frac{4\pi }{9}}\frac{2\sqrt{\mathrm{s}\mathrm{i}\mathrm{n}x}}{\sqrt{\mathrm{s}\mathrm{i}\mathrm{n}x}+\sqrt{\mathrm{c}\mathrm{o}\mathrm{s}x}}dx=\dots$

If \(\mathrm{f}(x)=3 x^{10}-7 x^8+5 x^6-21 x^3+3 x^2-7\), then \(\lim _{\alpha \rightarrow 0} \frac{f(1-\alpha)-f(1)}{\alpha^3+3 \alpha}=\)

Let \(\vec{v}=2 \hat{i}+2 \hat{j}-\hat{k}\) and \(\bar{w}=\hat{i}+3 \hat{k}\). If \(\bar{u}\) is a unit vector, then the maximum value of the scalar triple product \([\overline{\mathrm{u}} \overline{\mathrm{v}} \overline{\mathrm{W}}]\) is

The derivative of \(\sin ^{-1}\left(2 x \sqrt{1-x^2}\right)\) w.r.t. \(\sin ^{-1}\left(3 x-4 x^3\right)\) is

The equation of the tangent to the curve \(y=4 x e^{x}\) at \(\left(-1, \frac{-4}{e}\right)\) is

Which one of the following is the pair of equivalent circuits?
i.
ii.
iii.
iv.
v.

The particular solution of the differential equation \(\frac{d y}{d x}-e^x=y e^x\), when \(x=0\) and \(y=1\) is

Three charges \(Q,+q\) and \(+q\) are placed at the vertices of right-angle triangle as shown. The net electrostatic energy of the configuration is zero. The value of \(Q\) is

An electrical circuit consists of the resistances \(R, r_1, R_2\) and voltage sources \(E_1\) and \(E_2\) as shown in the figure The correct equation for the loop \(E B C D E\) is

Aldol condensation reaction is

If \(\bar{x}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \bar{y}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}\) and \(\overline{\mathrm{z}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}\) where \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non-coplanar vectors, then value of \(\bar{x} \cdot(\overline{\mathrm{a}}+\overline{\mathrm{b}})+\bar{y} \cdot(\overline{\mathrm{~b}}+\overline{\mathrm{c}})+\overline{\mathrm{z}} \cdot(\overline{\mathrm{c}}+\overline{\mathrm{a}})\) is

Identify the angle \(\mathrm{O}-\mathrm{S}-\mathrm{O}\) in \(\mathrm{SO}_2\) molecule.

For a reaction \(\mathrm{A} \rightarrow\) product, rate constant is \(2 \times 10^{-2} \mathrm{~s}^{-1}\). The initial concentration of \(\mathrm{A}\) is \(1.0 \mathrm{~mol} \mathrm{dm}^{-3}\). What is the value of \(\log \frac{1}{[\mathrm{~A}]_{\mathrm{t}}}\) after 100 seconds?