The tangents to curve y = x³ - 2x² + x - 2 which are parallel to straight line y = x, are

• $\mathrm{x} + \mathrm{y} = 2 \mathrm{and} \mathrm{x} - \mathrm{y} = \frac{86}{27}$

• $\mathrm{x} - \mathrm{y} = 2 \mathrm{and} \mathrm{x} - \mathrm{y} = \frac{86}{27}$

• $\mathrm{x} - \mathrm{y} = 2 \mathrm{and} \mathrm{x} + \mathrm{y} = \frac{86}{27}$

• $\mathrm{x} + \mathrm{y} = 2 \mathrm{and} \mathrm{x} + \mathrm{y} = \frac{86}{27}$

The value of ${\int }_0^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.}\frac{{\sin }^{-1}\left(\mathrm{x}\right)}{{\left(1 - {\mathrm{x}}^2\right)}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\mathrm{dx}$ is

• $\frac{\mathrm{\pi }}{2} - \log \left(2\right)$

• $\frac{\mathrm{\pi }}{4} - \frac{1}{2}\log \left(2\right)$

   

• $\frac{\mathrm{\pi }}{4} + \frac{1}{2}\log \left(2\right)$

• $\mathrm{\pi } - \frac{1}{2}\log \left(2\right)$

Integral of $\frac{1}{2 + \cos \left(\mathrm{x}\right)}$

• $\frac{1}{\sqrt{3}}{\tan }^{-1}\left(\frac{1}{2}\tan \left(\mathrm{x}\right)\right) +\hspace{0.17em}\mathrm{C}$

• $\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{1}{\sqrt{3}}\tan \left(\frac{\mathrm{x}}{2}\right)\right) +\hspace{0.17em}\mathrm{C}$

• $- \sin \left(\mathrm{x}\right)\log \left(2 + \cos \left(\mathrm{x}\right)\right) + \mathrm{C}$

• $\sin \left(\mathrm{x}\right)\log \left(2 + \cos \left(\mathrm{x}\right)\right) + \mathrm{C}$

14.

A plane is flying horizontally at a height of 1 km from ground. Angle of elevation of the plane at a certain instant is 60°. After 20 s, angle of elevation is found 30°. The speed of plane is

• $100\sqrt{3} \mathrm{m}/\mathrm{s}$

• $200\sqrt{3} \mathrm{m}/\mathrm{s}$

• $\frac{100}{\sqrt{3}} \mathrm{m}/\mathrm{s}$

• $\frac{200}{\sqrt{3}} \mathrm{m}/\mathrm{s}$

Probability of solving a particular question by person A is 1/3 and probability ofsolving that question by person B is 2/5. what is the probability of solving that question by atleast one of them ?

• $\frac{3}{5}$

• $\frac{7}{9}$

• $\frac{2}{5}$

• $\frac{2}{3}$

The centre of gravity (centre of mass) of a rod (of length L), whose linear mass density varies as the square of the distance from one end is at

• $\frac{3\mathrm{L}}{5}$

• $\frac{2\mathrm{L}}{5}$

• $\frac{\mathrm{L}}{3}$

• $\frac{3\mathrm{L}}{4}$

Three forces each of magnitude F are applied along the edges of a regular hexagon as shown in the figure. Each side of hexagon is a. What is the resultant moment (torque) of these three forces about centre O ?

• $\frac{3\sqrt{3}}{2}\mathrm{aF}$

• $\frac{1}{2}\mathrm{aF}$

• 3aF

• $\frac{\sqrt{3}}{2}\mathrm{aF}$

The coordinates of a moving point particle in a plane at time t is given by x = a(t + sin(t)), y = a(1 - cos(t)). The magnitude of acceleration of the particle is acceleration ofthe particle is

• 2a

• $\frac{\sqrt{3}}{2}\mathrm{a}$

• a

• $\sqrt{3}\mathrm{a}$

Two vectors A = 3 and B = 4 are perpendicular. Resultant of both these vectors is R. The projection of the vector B on the vector R is

• 5

• 1.25

• 3.2

• 2.4

A vector R is given by R = A x (B x C), which of the following is true ?

• R must be perpendicular to 8

• R is parallel to A

• R must be parallel to B

• None of these