Half Yearly Examination – 2013

Half Yearly Examination – 2013
Class XI (Subject : Mathematics)
M.M. :100
TIME : 3 Hrs.
General Instructions:
(i)
(ii)
(iii)
All questions are compulsory. Question paper is divided in to three sections.
Section-A carries 10 questions, Section-B carries 12 questions and Section-C carries 07 questions.
In some questions internal choices have been provided, you have to attempt one of these two
SECTION – A (Each question of ONE mark)
Q-1
If A&B are two disjoint sets, then find the value of n( AUB).
Q-2
If A= {5,6,7,8,9,10} & B = {5,7,9,11}. Find A – B.
Q-3
Find the number of relations that are possible from a set A of m elements to another set B of n elements.
Q-4
Express one radian in to degrees and minutes.
Q-5
Find the multiplicative inverse of
Q-6
Write the complex number in the form of x + iy ,
Q-7
List the solution set of 30 - 4(2x – 1) < 30, given that x is a positive integer.
Q-8
There are 10 buses running between two towns X & Y. In how many ways can a man go from X to Y and
return by a different bus.
Q-9
Find the number of diagonals that can be drawn by joining the angular points of a heptagon.
- 7 +0i .
------ 1+i² + i⁴ + i⁶ .
Q-10 If 0° <θ< 90° and cosθ = 4/5, find the value of cosec(180° + θ ).
SECTION – B (each of 4 marks)
Q-11 Hundred of 120 students at a college take at least one of the languages French, German and Russian.
Suppose 65 study French, 45 study German, 42 study Russian, 20 study French and German, 25 study
French and Russian and 15 study German and Russian. Find the number of students who study all the three
languages.
Q-12 If A = {1,2,3} , B = { 4,5 } , C = {1,2,3,4,5} , Find (i) A X B (ii) C X B (iii) B X B .
Hence prove that (C X B) - ( A X B) = B X B .
Q- 13 If A + B = 45°, prove that ( 1 + tanA) ( 1 + tanB) = 2, Hence show that tan22.5° = √2 – 1.
OR
If A B C D is a cyclic quadrilateral, prove that; SinA + sinB - sinC – sinD = 0
3
3
3
Q-14 In any triangle ABC, prove that : a cos ( B – C ) + b cos( C – A ) + c cos ( A – B ) = 3abc
OR
In Δ ABC prove that : a ( cos C – cos B ) = 2 ( b – c ) cos² A/2
Q-15 Find the general solution of the equation :
Cos 3x + cos x – cos 2x = 0
Q- 16 Using principle of mathematical induction, prove that : 13 + 23 + 33 + ---------- + n3 = [ n( n+ 1)]²/4.
Q-17 For any two complex numbers z₁ and z₂ , prove that : Re( z₁ z₂) = Re z₁ Re z₂
-
Im z₁ Im z₂
Q- 18 Find the real numbers x and y if ( x – iy) ( 3 + 5i) is the conjugate of - 6 – 24i.
Q-19 Solve the equation :: 21 x² - 28x + 10 = 0.
Q-20 Solve the system of inequalities graphically :: 2x + y ≥ 6 ; 3x + 4y ≤ 12.
s
Q-21 In how many of the distinct permutations of the letters in MISSISSIPPI do the four i not come together?
OR
Find the number of different 8- letter arrangements that can be made from the letters of the word
DAUGHTER so that (i) all vowels occur together (ii) all vowels do not occur together.
Q-22 In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can
bowl if each cricket team of 11 must include exactly 4 bowlers?
OR
In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific
courses are compulsory for every student?
SECTION – C (Each of 6 marks )
Q-23 In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C . If
14 people liked products A & B , 12 people liked products C & A , 14 people liked products B & C and 8
liked all the three products . Find how many liked product C only.
Q- 24 Define and draw the graphs of the following functions.
(i)
(ii)
(iii)
Identity function.
Modulus function.
Greatest- integer function.
Q- 25 Prove that : sin3x + sin2x – sinx = 4 sinx cos 1/₂ x cos 3/₂ x .
Q-26 Prove that by using principle of induction : ( 2n + 3 ) < ( n + 3 )² OR 1 + 2 + 3 +……………. + n < 1/8 ( 2n + 1 )²
Q- 27 If ( x+ iy)3 = u + iv , then show that : ( u/x) + ( v/y) = 4( x² - y² ) .
Q -28 A manufacturer has 600 litres of a 12% solution of acid. How many litres of a 30% acid solution must be
added to it so that acid content in the resulting mixture will be more than 15% but less than 18%?
OR
How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the
resulting mixture will contain more than 25% but less than 30% acid content?
Q-29 What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these :
i.
iii.
4 cards are of the same suit.
Are face cards.
ii.
Iv.
4 cards belong to 4 different suits.
Two are red cards and two are black cards.

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