1.

For the circle C with the equation x² + y² - 16 - 12y + 64 = 0 match the List I with the List II given below.

	List I		List II
(i)	The equation of the polar of (- 5, 1) with respect to	(A)	y = 0
(ii)	The equation of the tangent at (8, 0) to C	(B)	y = 6
(iii)	The equation of the normal at (2, 6) to C	(C)	13x + 5y = 98
(iv)	The equation of the diameter of C through (8, 12)	(D)	13x + 5y = 98
		(E)	x = 6

The correct match is

A. (i) (ii) (iii) (iv)	(i) (D) (B) (A) (E)
B. (i) (ii) (iii) (iv)	(ii) (D) (A) (B) (E)
C. (i) (ii) (iii) (iv)	(iii) (C) (D) (A) (B)
D. (i) (ii) (iii) (iv)	(iv) (C) (E) (B) (A)

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = {\left(\mathrm{p} - {\mathrm{x}}^{\mathrm{n}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{n}$}\right.}, \mathrm{p} > 0 \mathrm{and} \mathrm{n} \mathrm{is} \mathrm{a} \mathrm{positive} \mathrm{integer}, \mathrm{then} \mathrm{f}\left[\mathrm{f}\left(\mathrm{x}\right)\right] = ?$

• x

• xⁿ

• p^1/n

• p - xⁿ

$\mathrm{The} \mathrm{value} \mathrm{of} \left\{\mathrm{x} \in \mathrm{R} ⋮ \left[\log \left(\log {\left(1.6\right)}^{1 - {\mathrm{x}}^2} - {\left(0.625\right)}^{6\left(1 + \mathrm{x}\right)}\right)\right] \in \mathrm{R}\right\} \mathrm{is}$

• $\left(- \infty , - 1\right) \cup \left(7, 0\right)$

• (- 1, 5)

• (1, 7)

• (- 1, 7)

$\mathrm{If} { }^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r} - 1} = 330, {}^{\mathrm{n}}\mathrm{C}_{\mathrm{r}} = 462, \mathrm{and}{ }^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r} + 1} = 462, \mathrm{then} \mathrm{r} = ?$

• 3

• 4

• 5

• 6

10 men and 6 women are to be seated in a row so that no two women sit together. The number of ways they can be seated, is

• 11! 10!

• $\frac{11!}{6! 5!}$

• $\frac{10! 9!}{5!}$

• $\frac{11! 10!}{5!}$

If t_n denotes the number of triangles formed with n points in a plane, no three of which are collinear and if t_{n + 1} - t_n= 36_, then n is equal to

• 7

• 8

• 9

• 10

$$\begin{gathered} \mathrm{The} \mathrm{term} \mathrm{independent} \mathrm{of} \mathrm{x}\left(\mathrm{x} > 0, \mathrm{x} \ne 1\right) \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} \\ {\left[\frac{\left(\mathrm{x} + 1\right)}{{\mathrm{x}}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}} - {\mathrm{x}}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}} + 1} - \frac{\left(\mathrm{x} - 1\right)}{\left(\mathrm{x} - \sqrt{\mathrm{x}}\right)}\right]}^{10} \mathrm{is} \end{gathered}$$

• 105

• 210

• 315

• 420

If x is small, so that x² and higher powers can be neglected, then the approximate value for $\frac{{\left(1 - 2\mathrm{x}\right)}^{ - 1}{\left(1 - 3\mathrm{x}\right)}^{ - 2}}{{\left(1 - 4\mathrm{x}\right)}^{ - 3}} \mathrm{is}$

• 1 - 2x

• 1 - 3x

• 1 - 4x

• 1 - 5x

$\mathrm{If} \frac{1}{{\mathrm{x}}^4 + {\mathrm{x}}^2 + 1} = \frac{\mathrm{Ax} +\hspace{0.17em}\mathrm{B}}{{\mathrm{x}}^2 + \mathrm{x} +\hspace{0.17em}1 } + \frac{\mathrm{Cx} + \mathrm{D}}{{\mathrm{x}}^2 - \mathrm{x} +\hspace{0.17em}1}, \mathrm{then} \mathrm{C} + \mathrm{D} = ?$

• - 1

• 1

• 2

• 0

$\frac{1}{2 . 3} + \frac{1}{4 . 5} + \frac{1}{6 . 7} + \frac{1}{8 . 9} + ... \infty = ?$

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

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

• $\log \left(2\mathrm{e}\right)$

• e - 1