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Half Yearly Examination 2015-2016 Class XI Mathematics Time 3 hrs M.M 100 Note: Attempt all the questions Section A (1X6 =6) Q 1 If U = {1,2,3,4,5,6,7,8,9} A= { 2,4,6,8} and B ={ 2,3,5,7} .Find Aββ© Bβ 11 Q2 Find the degree measure corresponding to the radian measure ( 16) Q 3 Find the multiplicative inverse of 3- 4i Q 4 Find solution of 3 ( π₯β2) 5 β€ 5(2βπ₯) 3 Q 5Find coefficient of π₯ 5 in ( π₯ + 3)8 Q6 If nth term of a series is n( n+1 ) .Find the sum of n terms of the series Section B (4 X 13= 52) Q 7Let A and B be sets: If A β©X = Bβ© π= β and A U X = B U X for some set X .Show that A =B . Q8 Find n , If the ratio of the fifth term from the beginning to fifth term from the end in the 1 expansion ( β2 + β3 ) is β6 : `1 Q 9 Between 1 and 31 , m numbers have been inserted in such way that the resulting sequence is an A.P and the ratio of 7th and (m-1) th number is 5:9 .Find value of m . Q 10 Find angles between the lines β3 x+ y = 1 and x+ β3 y =1 Q11The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and two different consonant s can be formed from the alphabet? Q 12 Find the real numbers x and y if ( x-iy ) ( 3+ 5i ) is the conjugate of ( -6 -24i) β4 Q13 If tan x = 3 π₯ π₯ π₯ where x is in second quadrant. Find Sin 2 , cos 2 and ten 2 π 9π 3π 5π Q 14Prove that 2 cos 13 cos 13+ cos 13 + cos 13 = 0 Q 15 Find domain and range of the following real function . f (x) =β9 β π₯ 2 Q16 which of the following relations are function? Give regions if it is a function, determine it s domain and range. a) { ( 2,1) ,(5,1) ,(8,1), (11,1) ,( 14,1),(17,1) } b) { (2,1),(4,2),(6,3),(8,4),(10,5),(12,6),(14,7) } c) {(1,3),(1,5),(2,5)} Q17 Find sets A ,B, and C such that Aβ©B ,Bβ© πΆ and Aβ© πΆare nonempty sets and ( Aβ©B)β© C= β Q18 Using principle of mathematical induction. Prove that 12 +32 +52 + β¦β¦β¦β¦β¦β¦β¦β¦β¦+(2π β 1)2 = π(2πβ1)(2π+1) 3 Q19 Find the direction in which a straight line must be drawn through the point (-1,2) so that its point of intersection with the line x+y=4 may be at a distance of three units from the point . Section C (6X7=42) Q 20 In a survey of 60 people , it was found that 25 people read newspaper H ,26 read newspaper T , 26 read newspaper I , 9 read both H and I , 11read both H and T , 8 read both T and I , 3 read all three newspapers . Find a) The number of people who read at least one of the newspapers. b) The number of people who read exactly one newspaper Q 21 a) Prove that Cot x Cot 2x-Cot 2x Cot 3x βCot 3x Cot x =1 b) Find general solution of Cos 3x+ Cos x βCos 2x =0 Q 22 Solve the following system of in equations graphically 2x+yβ₯4, x+y β€ 3, 2x-3y β€ 6 Q 23 In how many ways can the letters of the word PERMUTATIONS be arranged, if the a) Word start with P and end with S b) Vowels are all together c) There are always 4 letters b between P and S Q 24 The coefficients of (r-1)th , rth and ( r+1) th terms in the expansion of (π₯ + 1)π are in the ratio 1:3:5 .Find n and r . Q 25 Find the sum up to n terms of the series 3+7+13+21+31+β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ Q 26 Find the image of the point (3, 8) w.r.t the line x+3y =7 assuming the line to be plane mirror . End