\(\int_0^1(\sqrt{10})^{2 x} d x=\)

\(\int \frac{d x}{(\sin x)(\cos x)}\) is equal to

\(\int \frac{3 x^9+7 x^8}{\left(x^2+2 x+5 x^8\right)^2} d x=\)

If \(\bar{a}, \bar{b}\) are two non-parallel unit vectors and the vector \(\alpha \bar{a}+\bar{b}\) bisects the internal angle between \(\bar{a}\) and \(\bar{b}\), then \(\alpha\) is equal to

The distance from a point \((1,1,1)\) to a variable plane \(\pi\) is 12 units and the points of intersections of the plane \(\pi\) and \(\mathrm{X}, \mathrm{Y}, \mathrm{Z}\)-axes are \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) respectively. If the point of intersection of the planes through the points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and parallel to the coordinate planes is P , then the equation of the locus of P 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 \(A=\left[a_{i j}\right], 1 \leq i, j \leq n\) with \(\mathrm{n} \geq 2\) and \(a_{i j}=i+j\) is a matrix, then the rank of \(A\) is

Two bodies A and B of masses 20 kg and 5 kg respectively are at rest. Due to the action of a force of 40 N separately, if the two bodies acquire equal kinetic energies in times \(t_A\) and \(t_B\) respectively, then \(t_A: t_B=\)

\(\mathrm{G}(1,0,1)\) is the centroid of the triangle ABC. If \(\mathrm{A}=(1,-4,2)\) and \(\mathrm{B}=(3,1,0)\) then \(\mathrm{AG}^2+\mathrm{CG}^2=\)

A toroid has a non ferromagnetic core of inner radius $24 \mathrm{cm}$ and outer radius $25 \mathrm{cm}$, around which $4900$ turns of a wire are wound. If the current in the wire is $12 \mathrm{A}$, the magnetic field inside the core of the toroid is

When phenol is treated with bromine water, the white precipitate formed is

Two statements are given below.
Statement-I: In dry cleaning, the solvent \(\mathrm{Cl}_2 \mathrm{C}=\mathrm{CCl}_2\) was earlier used and now it is replaced by liquified \(\mathrm{CO}_2\)
Statement-II: In bleaching of paper, \(\mathrm{H}_2 \mathrm{O}_2\) was used earlier and now it is replaced by chlorine gas
Correct answer is