If a plane meets the co-ordinate axes at A, B and C such that the centroid of the triangle is (1, 2, 4), then the equation of the plane is:

• x + 2y + 4z = 12

• 4x + 2y + z = 12

• x + 2y + 4z = 3

• 4x + 2y + z = 3

The position vector of the point where the line $\overrightarrow{\mathrm{r}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} + \mathrm{t}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$ meets the plane $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$ = 5 is :

• $5\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$

• $5\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

• $5\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$

If the distance of the point (1, 1, 1) from the origin is half its distance from the plane x + y + z+ k = 0, then k is equal to :

• ± 3

• ± 6

• - 3, 9

• 3, - 9

Two persons A and B throw a die alternately till one of them gets a 3 and wins the game, the respective probabilities of winning, if A begins, are :

• $\frac{7}{11}, \frac{4}{11}$

• $\frac{6}{11}, \frac{5}{11}$

• $\frac{5}{6}, \frac{1}{6}$

• $\frac{4}{7}, \frac{3}{7}$

If three natural numbers from 1 to 100 are selected randomly, then probability that all are divisible by both 2 and 3, is :

• $\frac{4}{105}$

• $\frac{4}{33}$

• $\frac{4}{35}$

• $\frac{4}{1155}$

56.

$\int \frac{\mathrm{dx}}{\sin \left(\mathrm{x} - \mathrm{a}\right)\sin \left(\mathrm{x} - \mathrm{b}\right)}$ is :

• $\frac{1}{\sin \left(\mathrm{a} - \mathrm{b}\right)}\log \left|\frac{\sin \left(\mathrm{x} - \mathrm{a}\right)}{\sin \left(\mathrm{x} - \mathrm{b}\right)}\right|$ + c

• $\frac{- 1}{\sin \left(\mathrm{a} - \mathrm{b}\right)}\log \left|\frac{\sin \left(\mathrm{x} - \mathrm{a}\right)}{\sin \left(\mathrm{x} - \mathrm{b}\right)}\right|$ + c

• $\log \left(\sin \left(\mathrm{x} - \mathrm{a}\right)\sin \left(\mathrm{x} - \mathrm{b}\right)\right)$ + c

• $\log \left|\frac{\sin \left(\mathrm{x} - \mathrm{a}\right)}{\sin \left(\mathrm{x} - \mathrm{b}\right)}\right|$

If an antiderivative of f(x) is e^x and that of g(x) is cos(x), then $\int \mathrm{f}\left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)\mathrm{dx}$ + $\int \mathrm{g}\left(\mathrm{x}\right){\mathrm{e}}^{\mathrm{x}}\mathrm{dx}$ is equal to :

• f(x)g(x) + c

• f(x) + g(x) + c

• e^xcos(x) + c

• f(x) - g(x) + c

$\int {\mathrm{e}}^{\mathrm{xlog}\left(\mathrm{a}\right)}{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}$ is equal to :

• $\frac{{\mathrm{a}}^{\mathrm{x}}}{\log \left(\mathrm{ae}\right)} + \mathrm{c}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{1 + {\log }_{\mathrm{e}}\left(\mathrm{a}\right)} + \mathrm{c}$

• ${\left(\mathrm{ae}\right)}^{\mathrm{x}} + \mathrm{c}$

• $\frac{{\displaystyle {\left(\mathrm{ae}\right)}^{\mathrm{x}}}}{{\displaystyle {\log }_e\left(\mathrm{ae}\right)}} + \mathrm{c}$

$\int \sqrt{{\mathrm{e}}^{\mathrm{x}} - 1}\mathrm{dx}$ is equal to :

• $2\left[\sqrt{{\mathrm{e}}^{\mathrm{x}} - 1} - {\tan }^{-1}\left(\sqrt{{\mathrm{e}}^{\mathrm{x}} - 1}\right)\right] + \mathrm{c}$

• $\sqrt{{\mathrm{e}}^{\mathrm{x}} - 1} - {\tan }^{-1}\left(\sqrt{{\mathrm{e}}^{\mathrm{x}} - 1}\right) + \mathrm{c}$

• $\sqrt{{\mathrm{e}}^{\mathrm{x}} - 1} + {\tan }^{-1}\left(\sqrt{{\mathrm{e}}^{\mathrm{x}} - 1}\right) + \mathrm{c}$

• $2\left[\sqrt{{\mathrm{e}}^{\mathrm{x}} - 1} + {\tan }^{-1}\left(\sqrt{{\mathrm{e}}^{\mathrm{x}} - 1}\right)\right] + \mathrm{c}$

If I₁ = $\int {\sin }^{-1}\left(\mathrm{x}\right)\mathrm{dx}$ and I₂ = $\int {\sin }^{-1}\left(\sqrt{1 - {\mathrm{x}}^2}\right)\mathrm{dx}$ then : 

• I₁ = I₂

• ${\mathrm{I}}_{2 }= \frac{\mathrm{\pi }}{2}{\mathrm{I}}_1$

• I₁ + I₂ = $\frac{\mathrm{\pi }}{2}\mathrm{x}$

• I₁ + I₂ = $\frac{\mathrm{\pi }}{2}$