$\mathrm{If} \int \left(1 - \cos \left(\mathrm{x}\right)\right){\csc }^2\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{f}\left(\mathrm{x}\right) + \mathrm{c}, \mathrm{then} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\tan \left(\frac{\mathrm{x}}{2}\right)$

• $\cot \left(\frac{\mathrm{x}}{2}\right)$

• $2\tan \left(\frac{\mathrm{x}}{2}\right)$

• $\frac{1}{2}\tan \left(\frac{\mathrm{x}}{2}\right)$

$$\begin{gathered} \mathrm{If} {\mathrm{I}}_{\mathrm{n}} = {\int }_0^{\raisebox{1ex}{${\mathrm{n}}^{}\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\tan \left(\mathrm{x}\right)\mathrm{dx}, \mathrm{then} \\ {\mathrm{I}}_2 + {\mathrm{I}}_4, {\mathrm{I}}_3 + {\mathrm{I}}_5, {\mathrm{I}}_4 + {\mathrm{I}}_6, . . . , \mathrm{are} \mathrm{in} \end{gathered}$$

• arithmatic progression

• geometric progression

• harmonic progression

• arithmetic-geometric progression

The area (in square units) of the region enclosed by the two circles x² + y² = 1 and (x - 1)² + y² = 1 is

• $\frac{2\mathrm{\pi }}{3} + \frac{\sqrt{3}}{2}$

• $\frac{\mathrm{\pi }}{3} + \frac{\sqrt{3}}{2}$

• $\frac{\mathrm{\pi }}{3} - \frac{\sqrt{3}}{2}$

• $\frac{2\mathrm{\pi }}{3} - \frac{\sqrt{3}}{2}$

The values of a function f(x) at different values of x are as follows

x	0	1	2	3	4	5
f(x)	2	3	6	11	18	27

Then, the approximate area (in square units) bounded by the curve y = f(x) and x-axis between x = 0 and 5, using the Trapezoidal rule, is

• 50

• 75

• 52.5

• 62.5

$\mathrm{The} \mathrm{solution} \mathrm{of} \tan \left(\mathrm{y}\right)\frac{\mathrm{dy}}{\mathrm{dx}} = \sin \left(\mathrm{x} + \mathrm{y}\right) + \sin \left(\mathrm{x} - \mathrm{y}\right) \mathrm{is}$

• sec(y) = 2cos(x) + c

• sec(y) = - 2cos(x) + c

• tan(y) =  - 2cos(x) + c

• sec²(y) = - 2cos(x) + c

A family of curves has the differential equation $\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 2{\mathrm{y}}^2 - {\mathrm{x}}^2$. Then, the family of curves is

• ${\mathrm{y}}^2 = {\mathrm{cx}}^2 + {\mathrm{x}}^3$

• ${\mathrm{y}}^2 = {\mathrm{cx}}^4 + {\mathrm{x}}^3$

• ${\mathrm{y}}^2 = \mathrm{x} + {\mathrm{cx}}^4$

• ${\mathrm{y}}^2 = {\mathrm{x}}^2 + {\mathrm{cx}}^4$

The length of the common chord of the circles of radii 15 and 20, whose centres are 25 unit of distance apart, is

• 12

• 16

• 24

• 25

Let M be the foot of the perpendicular from a point P on the parabola y = 8(x - 3) onto its directrix and let S be the focus ofthe parabola. If  $∆$SPM is an equilateral triangle, then P is equal to

• $(4\sqrt{3}, 8)$

• $(8, 4\sqrt{3})$

• $(9, 4\sqrt{3})$

• $(4\sqrt{3}, 9)$

79.

The point dividing the join of (3, - 2, 1) and( - 2, 3 11) in the ratio 2 : 3 is

• (1, 1, 4)

• (1, 0, 5)

• (2, 3, 5)

• (0, 6, - 1)

The longest distance of the point (a, 0) from the curve 2x² + y² = 2x is

• 1 + a

• $\left|1 - \mathrm{a}\right|$

• $\sqrt{1 - 2\mathrm{a} +2{\mathrm{a}}^2}$

• $\sqrt{1 - 2\mathrm{a} + 3{\mathrm{a}}^2}$