FOUNDATION & OLYMPIAD

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1 Concept maps provided for every chapter l Set of objective and subjective questions at the end of each chapter l Previous contest questions at the end of each chapter l Designed to fulfill the preparation needs for international/national talent eams, olympiads and all competitive eams UNIQUE ATTRACTIONS CLASS X Cross word Puzzles Graded Eercise Basic Practice Further Practice Brain Works Multiple Answer Questions Paragraph Questions Rs. 85 Detailed solutions for all problems of IIT Foundation & Olympiad Eplorer are available in this book FOUNDATION & OLYMPIAD CLASS - IX w ` 50 w w (F.bm re a e ta Sa le n m t. pl co e) m Simple, clear and systematic presentation l IIT Foundation & Olympiad Eplorer - Mathematics Class - IX l Integrated Syllabus YOUR COACH India s FIRST scientifically designed portal for Olympiad preparation Olympiad & Talent Eams preparation packages Analysis Reports Previous question papers Free Demo Packages Free Android Mobile App Get 5% discount on all packages by using the discount coupon code: KR57N A unique opportunity to take about 50 tests per subject.

2 FOUNDATION & OLYMPIAD MATHEMATICS CLASS - IX

3 Published by: #6 6//B, First Floor, Farhat Hospital Road, Saleem Nagar, Malakpet, Hyderabad Andhra Pradesh, India , E mail: info@bmatalent.com Website: ALL RIGHTS RESERVED No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. Publication Team Authors: Y.S. Srinivasu Design & Typing: P.S.Chakravarthi ISBN: Disclaimer Every care has been taken by the compilers and publishers to give correct, complete and updated information. In case there is any omission, printing mistake or any other error which might have crept in inadvertently, neither the compiler / publisher nor any of the distributors take any legal responsibility. In case of any dispute, all matters are subject to the eclusive jurisdiction of the courts in Hyderabad only.

4 Preface Speed and accuracy play an important role in climbing the competitive ladder. Students have to integrate the habit of being able to calculate and function quickly as well as efficiently in order to ecel in the learning culture. They need to think on their feet, understand basic requirements, identify appropriate information sources and use that to their best advantage. The preparation required for the tough competitive eaminations is fundamentally different from that of qualifying ones like the board eaminations. A student can emerge successful in a qualifying eamination by merely scoring the minimum percentage of marks, whereas in a competitive eamination, he has to score high and perform better than the others taking the eamination. This book provides all types of questions that a student would be required to tackle at the foundation level. It will also help the student in identifying the pattern of questions set for various competitive eaminations. Constant practice and familiarity with these questions will not only make him/her conceptually sound, but will also give the student the confidence to face any entrance eamination with ease. Students are advised to go through every question carefully and try to solve it on their own. They should also attempt different methods and alternate processes in reaching the desired solution and seek their teacher s help if required. Valuable suggestions as well as criticism from the teacher and student community are most welcome and will be incorporated in the ensuing edition. Publisher

5 CONTENTS. Surds Logarithms Relations Mensuration II Polynomials II Quadratic Equations - I Plane Geometry - II Permutations and Combinations Inequalities II Coordinate Geometry II Number Theory Trigonometry - I Matrices Answers

6 Chapter 6 SYNOPSIS QUADRATIC EQUATIONS An equation of the form a + b + c = 0 Where a, b, c C and a 0 is called a quadratic equation. The numbers a, b, c are called the coefficients of this equaiton. A root of the quadratic equation is a comple number α such that aα + bα+ c = 0. Discriminant (D) = b 4ac The roots of the above quadratic equation are given by the formula b± D = or a QUADRATIC Mathematical EQUATIONS Force and Pressure Induction - I b b 4ac ± = a Properties of Quadratic Equations. A quadratic equation has two and only two roots.. A quadratic equation cannot have more than two different roots. 3. If α be a root of the quadratic equation a + b + c = 0, then ( α ) is a factor of a + b + c = 0 Note: The possible values of which satisfy the quadratic equation are called the roots of the quadratic eqaution. Sum and Product of the roots of a Quadratic Equation Let α, β be the roots of a quadratic equation a + b + c = 0 ; a 0, then b coefficient of α+β= = a coefficient of and c constant term α. β = = a coefficient of 6. Quadratic Equation I

7 Therefore, v If the two roots α and β be reciprocal to each other, then a = c. v If the two roots α and β be equal in magnitude and opposite in sign, then b = 0. Sign of the Roots. The roots α, β are both negative, if ( ) b (i.e., if and c are both positive) a a α+β and α β are both positive. The roots α, β are both positive, if α + β is negative and αβ is positive (i.e, if b a is negative and c a is positive) 3. The roots α, β are of opposite signs, if α β is negative (i.e., c a Sign of ( α+β) Sign of ( ) αβ Sign of the αβ, is negative) + ve + ve α and β are positive ve + ve α and β are negative + ve ve α is positive and β is negative if α > β (numerically) ve ve α is negative and β is positive if α > β (numerically) Nature of Roots For a quadratic equation a + b + c = 0 where a, b, c R, a 0 and D = b 4ac. (i) If D < 0, roots are imaginary (ii) If D 0 roots are real. D< 0 (roots are comple with non - zero imaginary part) D= 0 (roots are rational and equal) D > 0 6. Quadratic Equation I D is a perfect square root are rational and unequal 3 D is not a perfect square root are irrational and conjugate pairs

8 Graph of a Quadratic Function a + b + c = 0, a 0 Characteristics of b 4 ac < 0 b 4 ac = 0 b 4ac > 0 the Function When a is positive i.e., a > 0 When a is negative i.e., a < 0 v v v v v Y O (minima) O Y (Miima) X X Y O (minima) O Y (Miima) X X O O Y Y (minima) (Miima) The graph of a quadratic function (epression) is called a parabola. The point at which its direction changes is called its turning point, commonly called the verte of the parabola. The graph of the function is concave upwards when a > 0 and concave downwards when a < 0. If the graph has no points in common with the -ais, the roots of the equation are imaginary and cannot be determined from the graph. If the graph is tangent to the ais, the roots are real and equal. If the graph cuts the -ais, the roots of the equation will be real and unequal. Their values will be given by the abscissae of the points of interesection of the graph and the -ais. Solutions of Equations Reducible to Quadratic Form Equation which are not quadratic at a glance but can be reduced to quadratic equations by suitable transformations. Some of the common types are : Type : a 4 + b + c = 0 Eample: Sol.: This can be reduced to a quadratic equation by substituting = y. i.e., ay + by + c = 0. Solve for y : 9y 4 9y + 0 = 0 X X 6. Quadratic Equation I 5

9 Type : Eample: Sol.: 9y 4 9y + 0 = 0 Put y = = = 0 ( ) (9 0) = 0 Solve : = or 0 = 9 y 0 = or y = 9 y ± and y = ± 5 3 q p + = r, q p a + = r, ( ) ( b) Multiply both sides by the LCD of LHS to get a quadratic equation : q p + = r p r + q = 0 and ( ) q p a + = r ( b) p ( a) ( b) r ( b) + q = 0 3 = 5 3 = 5 3 = 5 ( + ) ( 3) = 0 = or = Quadratic Equation I 6

10 Type 3: a = b+ c Eample: Sol.: Squaring on both sides and simplify. i.e., a = b + bc + c ( + b ) + bc + (c a) = 0 Solve = = = 3 squaring on both sides + 9 = (3 ) = 0 ( 8) ( 0) = 0 = 8, = 0 Type 4: a + b + c + d = e Eample: Sol.: Transform one of the radicals to RHS and square a + b = e c + d, such equations may require squaring and your solution must satisfy a + b 0 and c + d 0 Solve for : 6 7 0, + + = + R = + ( + 3)( ) ( ) = ( 5)( ) ( ) ( ) ( ) ( ) = 0 6. Quadratic Equation I Either = 0 = 7

11 or ( + 3) ( ) 5 = 0 Type 5: + 3 = 5 Squaring on both sides = 4 ( + 6) = 6, 0 = 3 0 Since the equation involves radical therefore substituting =, 6 and in the original 3 equation, we find that =, 6 a + + b + + c = 0 0 = does not satisfy the equation. 3 For this type of equations we use the following identity Thus Now put Eample: Sol.: a + + b + + c = 0 + = + + = y to get a quadratic equation i.e., ay + by + (c a) = 0. Solve = 0 Put + = = 0 6. Quadratic Equation I 8

12 Type 6: Substitute + = y (y ) 9y + 4 = 0 y 4 9y + 4 = 0 (y ) (y 5) = 0 y = 0 or y = 5 Since + = + = 0 ( ) = 0 = Also + = = 0 =,,. = or = a + + b + c = 0 Use the following identity. + = + Thus, a + + b + c = 0, put = y to get a quadratic equation in y. i.e., ay + by + (c a) = 0 Eample: Solve = 6. Quadratic Equation I 9

13 SOLVED EXAMPLES Eample 6. Solve: = 0 Solution: Let y = The given equation reduces to a quadratic equation in y as 9y 48y + 64 = 0 ( ) ( )( ) 48 ± y =.9 48 ± 9600 = 8 48 ± 40 = 8 = 6 or 4 9 = 6 or = 4 9 = + 4 or = + 3 Solve: Solution: Eample 6. 3 y + = 7 y This can be rewritten as y 7y + 3 = 0 y 6y y + 3 = 0 y (y 3) (y 3) = 0 (y ) (y 3) = 0 y = or y = Quadratic Equation I 3

14 Eample 6.3 Solve: ( + ) ( + 4) ( + 6) ( + 8) = 05 Solution: The given equation can be written as ( )( ) ( )( ) = 05 [ ] [ ] = 05 Substituting + 0 = y, we get (y + 6) (y + 4) = 05 y + 40y + 79 = 0 (y + 3) (y + 9) = 0 y = 3 or y = 9 Taking y = 3, we have + 0 = = 0 0 ± 00 4 = = 5± -6 Taking y = 9, we have + 0 = = 0 ( + 9) ( + ) = 0 = 9 or = Hence the roots are, 9, 5± 6 Solve: Solution: Let Eample = + = y. Then y + + = + = y 6. Quadratic Equation I 33

15 Substituting in the given equation, we have, 3 (y ) 0 (y) 94 = 0 3y 0y 00 = 0 Factorising, we get (3y + 0) (y 0) = 0 0 y = or 0 3 When we have 0 y=, = = 0 solving for, we get = 3 or When y = 0, we have 3 + = = 0 Solve: Solving for, we get = 5± 4 Hence the roots are 3, Eample 6.5 Solution: Let y = =. Then y =.. +, 3 5± 4 y + = + 6. Quadratic Equation I 34

16 The equation becomes y + 3y = 0 y 3y + = 0 (y ) (y ) = 0 y = or When y =, we have = 0 6. Quadratic Equation I = = + or = When y =, we have = 0 On solving, we get = + 5 = or 5 Hence the roots of the quadratic equation are +,, + 5, 5 Eample 6.6 Solve: = 0 Solution: Dividing the equation by. We get = 0 Grouping equidistant terms we have, = Let Then + = y. + = y The equation becomes 3(y ) 0 y 94 = 0 3y 0y 00 = 0 (3y + 0) (y 0) = 0 0 y = or

17 Solution: ( + 3)( 3) ( )( 7) ( )( ) + 3 = ( ) = = = = = 0 3 ( 4) 4 ( 4) = 0 (3 4) ( 4) = 0 = 4 3 or = 4 Hence the roots are 4 3, 4 Eample 6.9 Solve for : + 36 a = 43a Solution: Rearranging the terms, we get 43a + 36a = 0 6a 7a + 36a = 0 4 (3 4a) 9a (3 4a) = 0 (3 4a) (4 9a) = 0 Solution: 4a = or = 9a 3 4 Eample 6.0 Form the quadratic equation whose roots are 3 7 and 4. 5 We know that if αβ, are the roots, then the quadratic equation is ( α+β ) +αβ = 0. Hence, the required quadratic equation is -3 = = Quadratic Equation I 37

18 3 + = = 0 is the required quadratic equation. Eample 6. Solve for : = 0 Solution: Let = y. Then the equation is transformed to 9y 35y + 36 = 0 ( ) 35 ± y =.9 = 35 ± 33 8 = 648 or 8 8 = 36 or 9 If y = 36, then = 36 = 6 If y = 9, then = = 9 3 Eample 6. Solve for : = 0 Solution: = 0 is written as = 0 Let 3 = y 9y + y = 0 9y 0y + = 0 (9y ) (y ) = 0 y= 9 or If y= 9, then If y =, then 6. Quadratic Equation I 38

19 Solution: 3 = 9 = 3 3 = = 3 0 = = 0 Eample 6.3 Solve for : = 3 3 Let 3 y = y + = y = 3 3 (5y + 7)3 = 68y 5y 68y + = 0 5y 63y 5y + = 0 3y(5y ) (5y ) = 0 (3y ) (5y ) = 0 y = 3 or y = 5 If y =, then If y = 3 5, then 3 = or 3 3 = 3 = or 3 = = 7 or = = Quadratic Equation I 39

20 Quadratic Equation: Sum of the roots: Let An equation of the form a + b + c = 0, where a, b, c C and a 0 is called a quadratic equation. The root of the equation a + b + c = 0 are given by the formula = α& β be the roots of the quadratic equation a + b + c = 0; a 0, b coeff. of then α+β= = a coeff. of b ± b 4ac a Product of the roots: Let α& β be the roots of the quadratic equation a + b + c = 0, c a 0, then αβ = a = constant term coeff. of Formation of Quadratic equation: Let α& β be the roots then the quadratic equation is given by ( α β ) + ( αβ ) = 0 Properties:. A quadratic equation has two and only two roots.. A quadratic equation cannot have more than two different roots. 3. If α is a root of the quadratic equation a + b + c = 0, then ( α) is a factor of a + b + c = If the roots α & β be reciprocals to each other, then a = c. 5. If the two roots α & β be equal in magnitude and op posite in sign, then b = 0 6. CONCEPT MAP + α β = b c D < 0 roots are comple with nonzero imaginary part D is a perfect square Nature of roots D > 0 D = 0 Roots are rationals equal Condition for common roots: Consider: a + b + c = 0 a 0 a' + b + c' = 0 a 0 (a) if one root is common then D is not a perfect square (ab' a'b) (bc' b'c) = (ca' c'a) (b) If two roots are common, 7. α β = b 4ac a Roots are rational and unequal Roots are rational and conjugate pairs a b c = = a' b' c' 6. Quadratic Equation I 4

21 BASIC PRACTICE. For what value of k, (4 k) + (k + 4) + (8k +) = 0, is a perfect square.. Find the least positive value of k for which the equation + k + 4 = 0 has real roots. 3. If the roots of the equation (b c) + (c a) + (a b) = 0 are equal, then prove that b = a + c. 4. If the roots of the equation (a + b ) (ac + bd) + (c + d ) = 0 are equal, then prove that a c =. b d 5. If the roots of the equations a + b + c = 0 and b ac + b = 0 are simultaneously real, then prove that b = ac. 6. If the roots of the equation (c ab) (a bc) + b ac = 0 are equal, prove that either a = 0 or a 3 + b 3 + c 3 = 3 abc. 7. Show that the equation (a + b ) + (a + b) + = 0 has no real roots, when a b. 8. Prove that both the roots of the equation ( a) ( b) + ( b) ( c) + ( c) ( a) = 0 are real but they are equal only when a = b = c. 9. If the equation ( + m ) + mc + (c a ) = 0 has equal roots, then prove that c = a ( + m ). 0. If αβ, are the roots of ( k + ) + (k + k + ) = 0, then show that α +β = k.. For what values of k does the equation (k ) + (k 3) + (5k 6) = 0 have equal roots? Find the roots of the equations corresponding to those values of k?. If a root of p + q + r = 0 is thrice the other root, then show that 3q = 6pr. 3. If one root of 5 + k = 0 is, then find the value of k and the other root. FURTHER PRACTICE. Common root of + 6 = 0, = 0 is: (A) (B) (C) 3 (D) 5. Ratio of the sum of the roots of = 0 to the product of the roots is: (A) : (B) : (C) : (D) : 3. Quadratic equation whose one of the roots is 4+ 5 is: (A) + 8 = 0 (B) = 0 (C) 8 + = 0 (D) 8 + = 0 4. If the discriminant of k = 0 is 00, then k = (A) 8 (B) 3 (C) 6 (D) 4 5. If a root of k + 8 = 0 is 4, then k = (A) 7 (B) 3 (C) 6 (D) 8 6. Equation whose roots are 3± is: (A) = 0 (B) = 0 (C) = 0 (D) 6 7 = 0 7. If ( + ) is a factor of 3 + k + 4, then k = (A) 3 (B) 6 (C) 7 (D) 7 8. If p + 8p 5 = 0 has equal roots, then p = (A) 3 or 5 (B) 3 or 5 (C) 3 or 5 (D) 3 or 5 6. Quadratic Equation I 4

22 4. The roots of /3 + /3 = 0 are: (A) or 8 (B) or (C) or (D) or For what value of m, the equation (3m + ) + (m + ) + m = 0 have equal root? (A), / (B) or 4 (C) 4 (D) 3 6. The value of is: (A) 4 (B) 3 (C) 3.5 (D).5 7. The value of is: (A) 7 (B) 6 (C) 5 (D)3 8. If the roots of a quadratic equation are p, q, then the equaiton is: q p (A) q (q + p ) pq= 0 (B) pq (p q ) pq = 0 (C) p (p + ) + p = 0 (D) p (p q ) pq =0 9. Form a quadratic equations, whose roots are + and. (A) 4 + = 0 (B) + = 0 (C) + 4 = 0 (D) + 4 = If αβ, are the roots of the equation a + b + c = 0, find the value of α +β. b + 4ac 4ac b b ac b 4ac (A) (B) (C) (D) 4a a a a 3. If, y and z are real numbers such that + y + z = 5 and y + yz + z = 3, what is the largest value that can have? (A) 5 3 (B) 9 (C) 3 3 (D) 7 3. If + 5y + z = y ( + z) then which of the following statement (s) is/are necessarily true? (i) = y (ii) = z (iii) = z (A) only (i) (B) only (ii) and (iii) (C) only (i) and (ii) (D) only (ii) 6. Quadratic Equation I 44 A B 33. The number of real roots of the equation + =, where A and B are real numbers not equal to zero simultaneouly is: (A) none (B) (C) (D) or BRAIN WORKS. If both a and b belong to the set {,, 3, 4}, then the number of equations of the form a + b + = 0 having real roots is: A) 0 B) 7 C) 6 D). If a, b are the two roots of a quadratic equation such that a + b = 4 and a b = 8, then the quadratic equation having a and b as its roots is: A) = 0 B) = 0 C) = 0 D) = 0

23 3. One fourth of a herd of cows is in the forest. Twice the square root of the herd has gone to mountains and on the remaining 5 are on the banks of a river. The total number of cows is: A) 6 B) 00 C) 63 D) Which of the following equations has real roots? A) = 0 B) = 0 C) ( ) ( 5) = 0 D) = 0 5. Which of the following is a quadratic equation? A) / = 0 B) ( ) ( + 4) = + C) = 0 D) ( + ) (3 4) = If a and b are the roots of the equation = 0, then the value of a + b is: A) 36 B) 4 C) D) 6 7. The roots of = 0 are: A) Real, unequal and rational B) Real, unequal and irrational B) Real and equal D) Imaginary 8. A and B solved a quadratic equation. In solving it, A made a mistake in the constant term and obtained the roots as 5, 3, while B made a mistake in the coefficient of and obtained the roots as, 3. The correct roots of the equation are: A), 3 B), 3 C), 3 D), 9. The value of in the equation + = is: A) 5/3 B) 7/3 C) 9/3 D) 3 0. For what values of k, the equation + (k 4) + k = 0 has equal roots? A) 8, B) 6, 4 C), D) 0, 4 MULTIPLE ANSWER QUESTIONS. Which of the following is the graph of a cubic function: (A) (B) (C) (D) Just touching here. The value of k for which polynomial k + 4 has equal zeroes is (A) 4 (B) (C) 4 (D) 3. Sum of two zeroes of a polynomial c + d is, then value of third zeroes can not be (A) 8 (B) 4 (C) (D) 6. Quadratic Equation I 45

24 4. α and β are zeroes of polynomial 3 +, then product of zeroes of a polynomial having zeroes α and β is (A) αβ (B) αβ (C) 0 (D) 5. If αβ, and γ are the zeroes of a polynomial , such that αβγ = then difference of first two zeroes can be: (A) (B) (C) 0 (D) PARAGRAPH QUESTIONS P a ssa g e - I The graph of an equation is given below. (, 4) (, ) (, 4) (, ). What is the degree of the polynomial? (A) (B) (C) 3 (D) 4. Identify the polynomial satisfying the graph? (A) y = + (B) y = (C) y = (D)y = 3. Which of the following statements is true about the graph? (A) The graph in symmetric about ais (B) The graph in symmetric about y ais (C) Sum of -intercept and y-intercept is greater than zero (D) The polynomial has 3 terms 6. Quadratic Equation I 46

25 . 8. CHAPTER X = MATRICES BASIC PRACTICE 7 3 FURTHER PRACTICE 3. =, and y = 3 ± 3 5. a =, b = 4 6. k = =, 3. Given system has infinite number of solutions..given system of equations has no solution for (ii) a = (iii) a = λ= 3. (i) a or b 3 5. (AB) = = B A. C. C 3. D 4. D 5. A 6. A 7. D 8. B 9. A 0. B. A. C 3. D 4. A 5. D 6. C 7. A 8. A 9. C 0. B. B. D 3. B 4. D 5. A 6. C 7. D 8. B 9. A 30. B 3. C 3. D 33. D 34. C 35. A 36. C 37. A 38. A 39. B 40. A 4. C 4. B 43. B 44. D BRAIN WORKS.B. A 3. B 4. A 5. C 6. B 7. B 8. A 9. D 0. A. C. B 3. C 4. C MULTIPLE ANSWER QUESTIONS. A,C. B,C,D 3. C,D 4. A,C 5. C, D BRAIN WORKS k = 5. (i) 36 (ii) 4 6. = A n + 4n 0. n (n ) , 0, a ( a) ( + a) a PARAGRAPH QUESTIONS. A. A MULTIPLE ANSWER QUESTIONS. A,C,D. A,B 3. A,B,C PARAGRAPH QUESTIONS. A. A 3. A 4. B 5. B Key 35

26 Concept maps provided for every chapter l Set of objective and subjective questions at the end of each chapter l Previous contest questions at the end of each chapter w w. (F bm re a e ta Sa le n m t. pl co e) m Simple, clear and systematic presentation l l Designed to fulfill the preparation needs for international/national talent eams, olympiads and all competitive eams UNIQUE ATTRACTIONS CLASS X Cross word Puzzles Graded Eercise w Basic Practice Further Practice Brain Works Multiple Answer Questions Paragraph Questions Detailed solutions for all problems of IIT Foundation & Olympiad Eplorer are available in this book YOUR COACH India s FIRST scientifically designed portal for Olympiad preparation Olympiad & Talent Eams preparation packages Analysis Reports Previous question papers Free Demo Packages Free Android Mobile App Get 5% discount on all packages by using the discount coupon code: KR57N A unique opportunity to take about 50 tests per subject. IIT Foundation & Olympiad Eplorer - Mathematics Class - IX l Integrated Syllabus FOUNDATION & OLYMPIAD CLASS - IX