The probabilities that Mr. A and Mr. B will die within a year are respectively, then the probability that only one of them will be alive at the end of the year, is

• $\frac{1}{2}$

• $\frac{2}{3}$

• None of the above

The maximum resultant of two forces is 2P and the minimum resultant is 2Q, the two forces are at right angles, then the resultant is

• P + Q

• P - Q

• $\sqrt{\frac{{\mathrm{P}}^2 + {\mathrm{Q}}^2}{2}}$

• $\sqrt{2\left({\mathrm{P}}^2 + {\mathrm{Q}}^2\right)}$

In a right angle $∆\mathrm{ABC}$, $\angle$A = 90° and sides a, b, c are respectively 10 cm, 8 cm and 6 cm. If a force F has moments 0, 64 and 36 N-cm respectively about vertices A, B and C, then magnitude of F is

• 9

• 4

• 10

• 8

The maximum horizontal range of a ball projected with a velocity of 40 m/s is (take g = 9.8m/s²)

• 157 m

• 127 m

• 163 m

• 153 m

15.

$\int \frac{\mathrm{dx}}{\sqrt{{\mathrm{x}}^4 + {\mathrm{x}}^6}}$ is equal to

• $- \frac{\sqrt{1 + {\mathrm{x}}^2}}{\mathrm{x}} + \mathrm{C}$

• $\frac{\sqrt{1 + {\mathrm{x}}^2}}{\mathrm{x}} + \mathrm{C}$

• $- \frac{\sqrt{1 - {\mathrm{x}}^2}}{\mathrm{x}} + \mathrm{C}$

• $- \frac{\sqrt{{\mathrm{x}}^2 - 1}}{\mathrm{x}} + \mathrm{C}$

If $\int {\sin }^{-1}\left(\mathrm{x}\right){\cos }^{-1}\left(\mathrm{x}\right)\mathrm{dx} = {\mathrm{f}}^{-1}\left(\mathrm{x}\right)\left[\frac{\mathrm{\pi }}{2}\mathrm{x} - \mathrm{x}{\mathrm{f}}^{-1}\left(\mathrm{x}\right) - 2\sqrt{1 - {\mathrm{x}}^2}\right]$

$\frac{\mathrm{\pi }}{2}\sqrt{1 - {\mathrm{x}}^2} + 2\mathrm{x} +\hspace{0.17em}\mathrm{C}$, then

• f(x) = sin(x)

• f(x) = cos(x)

• f(x) = tan(x)

• None of these

If the vectors a and b are linearly independent satisfying $\left(\sqrt{3}\tan \left(\mathrm{\theta }\right) + 1\right)\mathrm{a} + \left(\sqrt{3}\sec \left(\mathrm{\theta }\right) - 2\right)\mathrm{b} = 0$, then the value of $\mathrm{\theta }$ is

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{5\mathrm{\pi }}{6}$

• $\frac{11\mathrm{\pi }}{6}$

If a and b are two unit vectors inclined at an angle $\frac{\mathrm{\pi }}{3}$, then $\left\{\mathrm{a} \times \left(\mathrm{b} + \mathrm{a} \times \mathrm{b}\right)\right\} . \mathrm{b}$ is equal to

• $\frac{1}{4}$

• $\frac{- 3}{4}$

• $\frac{3}{4}$

• $\frac{1}{2}$

Let u, v and w be such that $\left|\mathrm{u}\right| = 1, \left|\mathrm{v}\right| = 3 \mathrm{and} \left|\mathrm{w}\right| = 2$. If the projection of v along u is equal to that of w along u and vectors v and w are perpendicular to each other, then $\left|\mathrm{u} - \mathrm{v} + \mathrm{w}\right|$ equals

• 2

• $\sqrt{7}$

• $\sqrt{14}$

• 14

The solution ofthe differential equation

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{y}\left(\log \left(\mathrm{y}\right) - \log \left(\mathrm{x}\right) + 1\right)}{\mathrm{x}}$ is

• x = ye^cy

• y = xe^cy

• x = ye^cx

• None of these