31.

The differential equation of the curve for which the initial ordinate of any tangent is equal to the corresponding subnormal

• is linear

• is homogeneous of second degree

• has separable variables

• is of second order

The solution of the equation $\frac{d\mathrm{y}}{d\mathrm{x}} = \cos \left(\mathrm{x} - \mathrm{y}\right)$ is

• $\mathrm{x} + \cot \left(\frac{\mathrm{x} - \mathrm{y}}{2}\right) = \mathrm{C}$

• $\mathrm{y} + \cot \left(\frac{\mathrm{x} - \mathrm{y}}{2}\right) = \mathrm{C}$

• $\mathrm{x} + \tan \left(\frac{\mathrm{x} - \mathrm{y}}{2}\right) = \mathrm{C}$

• None of these

The differential equation of all parabolas each of which has a latusrectum 4a and whose axes are parallel to the Y-axis is

• of order 1 and degree 2

• of order 2 and degree 3

• of order 2 and degree 1

• of order 2 and degree 2

The value of the parameter a such that the area bounded by y = a²x² + ax + 1, coordinate axes and the line x = 1 attains its least value is equal to

• - 1

• $- \frac{1}{4}$

• $- \frac{3}{4}$

• $- \frac{1}{2}$

${\int }_0^{\mathrm{x}}\left|\sin \left(\mathrm{t}\right)\right|\mathrm{dt}, \mathrm{where} \mathrm{x} \in \left[2\mathrm{n\pi }, \left(2\mathrm{n} + 1\right)\mathrm{\pi }\right], \mathrm{n} \in \mathrm{N}$, is equal to

• 4n - 1 - cos(x)

• 4n - sin(x)

• 4n - cos(x)

• 4n + 1 - cos(x)

Let I₁ = ${\int }_0^1\frac{{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}}{1 + \mathrm{x}}$ and I₂ = ${\int }_0^1\frac{{\mathrm{x}}^2\mathrm{dx}}{{\mathrm{e}}^{{\mathrm{x}}^3}\left(2 - {\mathrm{x}}^3\right)}$ Then, $\frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}$ is equal to

• $\frac{1}{3\mathrm{e}}$

• 3e

• $\frac{\mathrm{e}}{3}$

• $\frac{3}{\mathrm{e}}$

${\int }_{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^5\frac{\sqrt{{\left(25 - {\mathrm{x}}^2\right)}^3}}{{\mathrm{x}}^4}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{3}$

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{5\mathrm{\pi }}{6}$

If n integers taken at random are multiplied together, then the probability that the last digit of the product is 1, 3, 7 or 9 is

• $\frac{4^{\mathrm{n}}}{5^{\mathrm{n}}}$

• $\frac{2^{\mathrm{n}}}{5^{\mathrm{n}}}$

• 4 - $\frac{2^{\mathrm{n}}}{5^{\mathrm{n}}}$

• None

A bag contains (n + 1) coins. It is known that one of these coins shows heads on both sides, whereas the other coins are fair. One coin is selected at random and tossed. If the probability that toss results in heads is $\frac{7}{12}$, then the value of n is

• 5

• 4

• 3

• None of these

If A and B are two given events, then P(A ∩ B) is

• equal to P(A) + P(B)

• equal to P(A) + P(B) + P(A $\cup$ B)

• not less than P(A) + P(B) - 1

• not greater than P(A) + P(B) - P(A $\cup$ B)