41.

The differential equation of the family of circles touching the y-axis at the origin is

• xy' - 2y = 0

• y'' - 4y' + 4y = 0

• 2xyy' + x² = y²

• 2yy' + y² = x²

$\int \frac{\mathrm{dx}}{{\mathrm{x}}^4 - 1}$ is equal to

• $\frac{1}{4}\log \left|\frac{\mathrm{x} - 1}{\mathrm{x} + 1}\right| - \frac{1}{2}{\tan }^{-1}\left(\mathrm{x}\right) + \mathrm{C}$

• $\log \left|\frac{\mathrm{x} - 1}{\mathrm{x} + 1}\right| + \mathrm{C}$

• $\frac{1}{4}\log \left|\frac{\mathrm{x} - 1}{\mathrm{x} + 1}\right| + \frac{1}{2}{\tan }^{-1}\left(\mathrm{x}\right) + \mathrm{C}$

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

The equation of a straight line parallel to the x-axis is given by

• $\frac{\mathrm{x} - \mathrm{a}}{1} = \frac{\mathrm{y} - \mathrm{b}}{1} = \frac{\mathrm{z} - \mathrm{c}}{1}$

• $\frac{\mathrm{x} - \mathrm{a}}{0} = \frac{\mathrm{y} - \mathrm{b}}{0} = \frac{\mathrm{z} - \mathrm{c}}{1}$

• $\frac{\mathrm{x} - \mathrm{a}}{0} = \frac{\mathrm{y} - \mathrm{b}}{1} = \frac{\mathrm{z} - \mathrm{c}}{1}$

• $\frac{\mathrm{x} - \mathrm{a}}{1} = \frac{\mathrm{y} - \mathrm{b}}{0} = \frac{\mathrm{z} - \mathrm{c}}{0}$

The shortest distance between the lines

$\frac{\mathrm{x} - 3}{1} = \frac{\mathrm{y} - 5}{- 2} = \frac{\mathrm{z} - 7}{1}$ and $\frac{\mathrm{x} + 1}{1} = \frac{\mathrm{y} + 1}{- 6} = \frac{\mathrm{z} + 1}{1}$ is

• $\frac{1}{2}\sqrt{29} \mathrm{units}$

• $2\sqrt{29} \mathrm{units}$

• $\sqrt{29} \mathrm{units}$

• $\frac{1}{4}\sqrt{29} \mathrm{units}$

Area of the region bounded by the parabola y = x² and the curve y = $\left|\mathrm{x}\right|$ is

• 3

• $\frac{1}{3}$

• 2

• $\frac{1}{2}$

The angle between the lines $\frac{\mathrm{x} - 2}{3} = \frac{\mathrm{y} + 1}{- 2}; \mathrm{z}= 2$ and $\frac{\mathrm{x} - 1}{1} = \frac{2\mathrm{y} + 3}{3}; \frac{\mathrm{z} +\hspace{0.17em}5}{2}$ is

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

If a, b and c are three non-zero, non-coplanar vectors, then the value of a x a' + b x b'+ c x c' is

• 1

• 0

• - 1

• None of the above

If e⁰ = 1, e¹ = 2.72, e² = 7.39, e³ = 20.09, e⁴ = 54.60, then the value of ${\int }_0^4{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}$ using Simpson's rule, will be

• 5.387

• 53.87

• 52.78

• 53.17

A and B are two independent events. Probability of happening of both A and B is 1/6 and probability of happening of neither of them is 1/3, then the probability of events A and B are respectively

• $\frac{1}{2} \mathrm{and} \frac{1}{3}$

• $\frac{1}{5} \mathrm{and} \frac{1}{6}$

• $\frac{1}{2} \mathrm{and} \frac{1}{6}$

• $\frac{2}{3} \mathrm{and} \frac{1}{4}$

${\int }_{- 1}^2\left|{\mathrm{x}}^3 - \mathrm{x}\right|\mathrm{dx}$ is equal to

• 11

• 4

• $\frac{11}{4}$

• $\frac{{\displaystyle 4}}{{\displaystyle 11}}$