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211.
ln $∆$ ABC, if the sides a, b, c are in geometric progression and the largest angle exceeds the smallest angle by 60°, then cos(B) is equal to

$\frac{\sqrt{13} + 1}{4}$

$\frac{1 - \sqrt{13}}{4}$

1

$\frac{\sqrt{13} - 1}{4}$

$\frac{\sqrt{13} - 1}{4}$

$Given, a, b, c are GP ∴ b^{2} = ac . . . (i) and C - A = 60 ° . . . (ii) \Rightarrow \cos (C - A) = \cos (60 °) \Rightarrow \cos (C) \cos (A) + \sin (C) \sin (A) = \frac{1}{2} \Rightarrow 2 \cos (C) \cos (A) + 2 \sin (C) \sin (A) = 1$

$\Rightarrow \cos (C + A) + \cos (C - A) + 2 k^{2} ac = 1 [∴ \frac{\sin (A)}{a} = \frac{\sin (B)}{b} = \frac{\sin (C)}{c} = k] \Rightarrow \cos (π - B) + \cos (60 °) + 2 k^{2} b^{2} = 1 [∵ ac = b^{2}] \Rightarrow - \cos (B) + \frac{1}{2} + 2 \sin^{2} (B) = 1 [∵ bk = \sin (B)] \Rightarrow - \cos (B) + 1 + 4 \sin^{2} (B) = 2 \Rightarrow - \cos (B) + 1 + 4 (1 - \cos (B)) = 2 \Rightarrow 4 \cos^{2} (B) + 2 \cos (B) - 3 = 0 ∴ \cos (B) = \frac{- 2 \pm \sqrt{4 + 48}}{2 \times 4} = \frac{- 2 \pm 2 \sqrt{13}}{8} = \frac{\sqrt{13} - 1}{4} or \frac{- \sqrt{13} - 1}{4} ∴ Option (d) is correct$

212.

$The coefficient of x^{5} in the expansion of (1 + x)^{21} + (1 + x)^{22} + . . . + (1 + x)^{30} is$

${}_{31}C_{6} - {}_{21}C_{6}$
${}_{51}C_{5}$
${}_{9}C_{5}$
${}_{30}C_{5} + {}_{20}C_{5}$

213.

If the roots of the equation x² -7x² + 14x - 8 = 0 are in geometric progression, then the difference between the largest and the smallest roots is

4
2
$\frac{1}{2}$
3

214.

$p, x_{1}, x_{2}, . . ., x_{n} and q, y_{1}, y_{2}, . . ., y_{n}$ are two arithmetic progressions with common differences a and b respectively. If $α and β$ are the arithmetic means of $x_{1}, x_{2}, . . ., x_{n} and y_{1}, y_{2}, . . ., y_{n}$ respectively. Then the locus of P $(α, β)$ is

$a (x - p) = b (y - q)$
$b (x - p) = a (y - q)$
$α (x - p) = β (y - q)$
$p (x - α) = q (y - β)$

215.

$The sum of the n terms of \frac{1}{2 . 5} + \frac{1}{5 . 8} + \frac{1}{8 . 11} + . . . is$

$\frac{3 n}{2 (3 n + 2)}$
$\frac{3 n}{3 n + 2}$
$\frac{n}{2 (3 n + 2)}$
$\frac{n}{3 n + 2}$

216.

$If x = \frac{1 . 3}{3 . 6} + \frac{1 . 3 . 5}{3 . 6 . 9} + \frac{1 . 3 . 5 . 7}{3 . 6 . 9 . 12} + . . . to infinite terms, then 9 x^{2} + 24 x = ?$

217.

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in

$(- \infty, - 3) \cup (9, \infty)$
$(- \infty, - 9) \cup (3, \infty)$
$(- 3, \infty)$
$(- \infty, 9)$

218.

If |x| < 1, |y| < 1 and x $\neq$ y, then the sum to infinity of the following series (x + y) + (x² + xy + y²) + (x³+ x²y + xy² + y³) + ..... is :

$\frac{x + y + xy}{(1 - x) (1 - y)}$
$\frac{x + y - xy}{(1 - x) (1 - y)}$
$\frac{x + y - xy}{(1 - x) (1 + y)}$
$\frac{x + y + xy}{(1 + x) (1 + y)}$

219.

Let S be the sum of the first 9 term of the series :(x + ka} + (x² + (k + 2)a} + {x³ + (k + 4)a} + {x⁴ + (k + 6)a} + ........ where a $\neq$ 0 and x $\neq$ 1. If $S = \frac{x^{10} - x + 45 a (x - 1)}{x - 1}$ , then k is equal to

220.

$If the sum of first 11 terms of an A . P ., a_{1}, a_{2}, a_{3} . . . . . is 0 (a_{1} \neq 0), then the sum of the A . P ., a_{1}, a_{3}, a_{5}, . . . . . a_{23} is {ka}_{1}, where k is equal to :$