61.

Let f: $\mathrm{X} \to \mathrm{X} \mathrm{be} \mathrm{such} \mathrm{that} \mathrm{f}\left[\mathrm{f}\right(\mathrm{x}\left)\right] = \mathrm{x}, \mathrm{for} \mathrm{all} \mathrm{x} \in \mathrm{X} \mathrm{and} \mathrm{X} \subseteq \mathrm{R}$ then

• f is one - to - one

• f is onto

• f is one - to - one but not onto

• f is onto but not one - to - one

If f(x) is a function such that f'(x) = (x - 1)²(4 - x), then

• f(0) = 0

• f(x) is increasing in (0, 3)

• x = 4 is a critical point of f(x)

• f(x) is decreasing in (3, 5)

If the solution of the differential equation $\mathrm{x}\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = {\mathrm{xe}}^{\mathrm{x}} \mathrm{be} \mathrm{xy} = {\mathrm{e}}^{\mathrm{x}}\mathrm{\varphi }\left(\mathrm{x}\right) + \mathrm{C}$, then $\mathrm{\varphi }\left(\mathrm{x}\right)$ is equal to

• x + 1

• x - 1

• 1 - x

• x

The order of the differential equation of all parabolas whose axis of symmetry along X-axis is

• 2

• 3

• 1

• None of these

The area enclosed by y = $\sqrt{5 - {\mathrm{x}}^2}$ and y = $\left|\mathrm{x} - 1\right|$ is

• $\left(\frac{5\mathrm{\pi }}{4} - 2\right)$ sq. units

• $\left(\frac{5\mathrm{\pi } - 2}{2}\right)$ sq. units

• $\left(\frac{5\mathrm{\pi }}{4} - \frac{1}{2}\right)$ sq. units

• $\left(\frac{\mathrm{\pi }}{2} - 5\right)$ sq. units

A straight line joining the points (1, 1, 1) and (0, 0, 0) intersects the plane 2x + 2y + z = 10 at

• (1, 2, 5)

• (2, 2, 2)

• (2, 1, 5)

• (1, 1, 6)

Angle between the planes x + y + 2z = 6 and 2x - y + z = 9 is

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

$\int \frac{\log \left(\sqrt{\mathrm{x}}\right)}{3\mathrm{x}}\mathrm{dx}$ is equal to

• $\frac{1}{3}{\left(\log \left(\sqrt{\mathrm{x}}\right)\right)}^2 + \mathrm{C}$

• $\frac{2}{3}{\left(\log \left(\sqrt{\mathrm{x}}\right)\right)}^2 + \mathrm{C}$

• $\frac{2}{3}{\left(\log \left(\mathrm{x}\right)\right)}^2 + \mathrm{C}$

• $\frac{1}{3}{\left(\log \left(\mathrm{x}\right)\right)}^2 + \mathrm{C}$

$\int 2^{\mathrm{x}}\left[\mathrm{f}\text{'}\left(\mathrm{x}\right) + \mathrm{f}\left(\mathrm{x}\right)\log \left(2\right)\right]\mathrm{dx}$

• 2^xf'(x) + C

• 2^xlog(2) + C

• 2^xf(x) + C

• 2^x + C

${\int }_0^1\log \left(\frac{1}{\mathrm{x}} - 1\right)d\mathrm{x}$

• 1

• 0

• 2

• None of the above