The acute angle between the line r = $\left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ and the plane $\left(2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ = 5

• ${\cos }^{-1}\left(\frac{\sqrt{2}}{3}\right)$

• ${\sin }^{-1}\left(\frac{\sqrt{2}}{3}\right)$

• ${\tan }^{-1}\left(\frac{\sqrt{2}}{3}\right)$

• ${\sin }^{-1}\left(\frac{\sqrt{2}}{\sqrt{3}}\right)$

The area of the region bounded by the curve y = 2x - x² and X - axis is

• $\frac{2}{3} \mathrm{sq} \mathrm{units}$

• $\frac{4}{3} \mathrm{sq} \mathrm{units}$

• $\frac{5}{3} \mathrm{sq} \mathrm{units}$

• $\frac{8}{3} \mathrm{sq} \mathrm{units}$

23.

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

• cot(x)

• tan(x)

• sec(x)

• csc(x)

If A and B are foot of perpendicular drawn from point Q(a, b, c) to the planes yz and zx, then equation of plane through the points A, B and O is

• $\frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} - \frac{\mathrm{z}}{\mathrm{c}} = 0$

• $\frac{\mathrm{x}}{\mathrm{a}} - \frac{\mathrm{y}}{\mathrm{b}} + \frac{\mathrm{z}}{\mathrm{c}} = 0$

• $\frac{\mathrm{x}}{\mathrm{a}} - \frac{\mathrm{y}}{\mathrm{b}} - \frac{\mathrm{z}}{\mathrm{c}} = 0$

• $\frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} + \frac{\mathrm{z}}{\mathrm{c}} = 0$

If a = $\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}, \mathrm{b} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and c = $3\hat{\mathrm{i}} - \hat{\mathrm{k}}$ and c = ma + nb, then m + n is equal to

• 0

• 1

• 2

• - 1

${\int }_0^{\frac{\mathrm{\pi }}{2}}\left(\frac{\sqrt[\mathrm{n}]{\sec \left(\mathrm{x}\right)}}{\sqrt[\mathrm{n}]{\sec \left(\mathrm{x}\right)} +\sqrt[\mathrm{n}]{\csc \left(\mathrm{x}\right)}}\right)\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

If the probability density function of a random variable X is given as

xi	- 2	- 1	0	1	2
P(X = xi)	0.2	0.3	0.15	0.25	0.1

then F(0) is equal to

• P(X < 0)

• P(X > 0)

• 1 - P(X > 0)

• 1 - P(X < 0)

The particular solution of the differential equation

$\mathrm{y}\left(1 + \log \left(\mathrm{x}\right)\right)\frac{d\mathrm{x}}{d\mathrm{y}} - \mathrm{xlog}\left(\mathrm{x}\right) = 0$, when, x = e, y = e² is

• y = exlog(x)

• ey = xlog(x)

• xy = elog(x)

• ylog(x) = ex

M and N are the mid-points of the diagonals AC and BO respectively of quadrilateral ABCD, then AB + AD + CB + CD is equal to

• 2MN

• 2NM

• 4MN

• 4NM

If $\sin \left(\mathrm{x}\right)$  is the integrating factor (IF) of the linear differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{Py} = \mathrm{Q}$, then P is

• $\log \left(\sin \left(\mathrm{x}\right)\right)$

• $\cos \left(\mathrm{x}\right)$

• $\tan \left(\mathrm{x}\right)$

• $\cot \left(\mathrm{x}\right)$