61.

If the vector $\mathrm{a} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ and b are collinear and $\left|\mathrm{b}\right| = 21$, then b is equal to

• $\pm \left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}\right)$

• $\pm 3\left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}\right)$

• $\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\pm 21\left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}\right)$

If a and b are unit vectors, then the vector (a + b) x (a x b) is parallel to the vector

• a - b

• a + b

• 2a - b

• 2a + b

I. Two non-zero, non-collinear vectors arelinearly independent.

II. Any three coplanar vectors are linearlydependent.Which ofthe above statements is/are true?

• Only I

• Only II

• Both I and II

• Neither I nor II

Observe the following statements

A. Three vectors are coplanar if one of them is expressible as a linear combination of the other two.

R. Any three coplanar vectors are linearly dependent.Then, which of the following is true ?

• Both A and R are true and R is the correct explainaton of A

• Both A and R are true but R is not the correct explainaton of A

• A is true, but R is false

• A is false, but R is true

If the range of a random variable X is {0, 1, 2, 3, 4.....} with P(X = k) = $\frac{\left(\mathrm{k} + 1\right)\mathrm{a}}{3^{\mathrm{k}}}$, fork  > 0, then a is equal to

• $\frac{2}{3}$

• $\frac{4}{9}$

• $\frac{8}{27}$

• $\frac{16}{81}$

For a binomial variate X with n = 6, if P(X = 2) = 9 P(X = 4), then its variance is

• $\frac{8}{9}$

• $\frac{1}{4}$

• $\frac{9}{8}$

• 4

The direction cosines of the line passing through P(2, 3, - 1) and the origin are

• $\frac{2}{\sqrt{14}}, \frac{{\displaystyle 3}}{{\displaystyle \sqrt{14}}}, \frac{{\displaystyle 1}}{{\displaystyle \sqrt{14}}}$

• $\frac{2}{\sqrt{14}}, \frac{{\displaystyle - 3}}{{\displaystyle \sqrt{14}}}, \frac{{\displaystyle 1}}{{\displaystyle \sqrt{14}}}$

• $\frac{- 2}{\sqrt{14}}, \frac{{\displaystyle - 3}}{{\displaystyle \sqrt{14}}}, \frac{{\displaystyle 1}}{{\displaystyle \sqrt{14}}}$

• $\frac{ 2}{\sqrt{14}}, \frac{{\displaystyle - 3}}{{\displaystyle \sqrt{14}}}, \frac{{\displaystyle - 1}}{{\displaystyle \sqrt{14}}}$

If $\int \frac{\sin \left(\mathrm{x}\right)}{\cos \left(\mathrm{x}\right)\left(1 + \cos \left(\mathrm{x}\right)\right)}\mathrm{dx}$ = f(x) + c, then f(x) is equal to

• $\log \left|\frac{1 + \cos \left(\mathrm{x}\right)}{\cos \left(\mathrm{x}\right)}\right|$

• $\log \left|\frac{\cos \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}\right|$

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

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

$\int \frac{{\mathrm{x}}^{49}{\tan }^{-1}\left({\mathrm{x}}^{50}\right)}{\left(1 + {\mathrm{x}}^{100}\right)}\mathrm{dx} = \mathrm{k}{\left({\tan }^{-1}\left({\mathrm{x}}^{50}\right)\right)}^2 +\hspace{0.17em}\mathrm{c}$, then k is equal to

• $\frac{1}{50}$

• $- \frac{1}{50}$

• $\frac{1}{100}$

• $- \frac{1}{100}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{200\sin \left(\mathrm{x}\right) + 100\cos \left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $50\mathrm{\pi }$

• $25\mathrm{\pi }$

• $75\mathrm{\pi }$

• $150\mathrm{\pi }$