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1.) 6 + 6 + 6 + .... n terms = 6 * n 2) (xb - c) /bc + (xc -a )/ca + (xa - b) / ab = [ ax b - ac + x bc - ab + xac - bc ] / abc = (x - 1) ( ab + bc + ca) / abc 3) case 1) perimeter of the sector exceeds the diameter by 16.8 cm (2 R + arc length) - 2 R = 16.8 cm => arc length = 16.8 cm case 2) arc length exceeds R Ф - 2 R = 16.8 cm. Here there are two parameters R & Ф. we need more information to solve this.

4) T3 = 5 = a + 2 d T7 = a + 8 d = 9 subtract equations, 6 d = 4 => d = 2/3 => a = 11/3

5) In a cyclic quadrilateral, sum of opposite angles is 180 deg. Cos A + Cos C = 2 Cos (A+C)/2 Cos (A-C)/2 = 0 As (A+C)/2 = 90 deg similarly Cos B + Cos D = 0 as (B+D)/2 = 90 deg. hence the answer is 0.

6) tan 1 tan2 tan 3 ...... tan 88 tan 89 = tan1 tan2 tan 3 .... tan 43 tan 44 tan 45 cot 44 cot 43 .... cot 3 cot 2 cot 1 = tan 1 cot 1 tan 2 cot2 tan3 cot3 .... tan43 cot43 tan44 cot44 * 1 = 1 Log 1 = 0

7) x / 5 , x/4 x/3 , x/2 median is the average of x/4 and x/3 so x/4 + x/3 = 2 * 8 = 16 7 x / 12 = 16 x = 12 * 16 / 7

8) D = discriminant = b² (c-a)² - 4 ac (b-c) (a-b) = b² (c² + a² - 2 a c) - 4 a c ( a b - b² - a c + b c) = b² c² + a² b² - 2 a c b² - 4 a² b c + 4 a c b² + 4 a² c² - 4 a b c² = b² c² + a² b² + 2 a c b² - 4 a² b c + 4 a² c² - 4 a b c² = b² (a +c)² - 4 a c (a b - a c + b c) = (a+c)² [ b² - 4 abc ] + 4 a² c²