Contents (Term II) Preface Latest Syllabus (v) (vii). Quadratic Equations... 5. Arithmetic Progressions... 1 7. Coordinate Geometry... 8 9. Some Applications of Trigonometry... 1 10. Circles... 53 11. Constructions... 7 1. Areas Related to Circles... 8 13. Surface Areas and Volumes... 10 15. Probability... 119 Value Based Questions... 133 Practice Papers (I III)... 11 xv
Each Chapter Contains: TIPS AND TRICKS FORMATIVE ASSESSMENT SUMMATIVE ASSESSMENT CBSE and Other Important Questions Objective Type Questions Higher Order Thinking Skills (HOTS) Questions NCERT Textbook Exercises
Quadratic Equations CHAPTER Tips and Tricks Standard form of a quadratic equation: The standard form of a quadratic equation in variable x is ax + bx + c, a 0 Roots of a quadratic equation: A real number α is said to be a root of the quadratic equation ax + bx + c = 0 if aα + bα + c = 0. Note: The roots of the quadratic equation ax + bx + c = 0 are the same as the zeroes of the quadratic polynomial ax + bx + c. Solving a quadratic equation (i) by factorisation: If we can factorise the quadratic polynomial ax + bx + c into two real linear factors, then the roots of the quadratic equation ax + bx + c = 0 can be found by equating each factor to zero. (ii) by completing the square: By adding and subtracting a suitable constant, as required, we club the x and x terms in the quadratic equation so that they become a complete square and then solve for x. (iii) by quadratic formula: The roots of the quadratic equation ax + bx + c = 0 are given by x = b± b ac, provided b ac 0 a Note: The expression b ac is called the discriminant of the quadratic equation. Nature of roots: Value of b ac Nature of roots (i) > 0 two distinct real roots (ii) = 0 two equal real roots (iii) < 0 no real roots Sol. 1. Solve for x: We have, ILLUSTRATIVE EXAMPLES CBSE Exam. 5 3 = x x + 3, x 0, 3 3 x = 5 x + 3 3 x = 5 x x + 3 ( 3x) (x + 3) = 5x 8x + 1 6x 9x = 5x 6x + 6x 1 = 0 x + x = 0 x + x x = 0 x(x + ) 1(x + ) = 0 (x + ) (x 1) = 0 x + = 0 or x 1 = 0 x = or x = 1 x =, 1.. Solve for x: x + 1 x x + = 3; x 1, 1 x + Sol. We have, x + 1 x x + 1 x + = 3 A-
QUADRATIC EQUATIONS 3 ( x + 1)( x + ) + ( x )( x 1) = 3 ( x 1)( x + ) x + x + x + + x x x + = 3 x + x x x + = 3 x + x x + = 3(x + x ) x + = 3x + 3x 6 x + 3x 10 = 0 x + 5x x 10 = 0 x(x + 5) (x + 5) = 0 (x + 5) (x ) = 0 x + 5 = 0 or x = 0 x = 5 or x = x = 5,. 3. Two water taps together can fill a tank in 9 3 hours. The 8 ( x 10) + x x( x 10) = 8 75 x 10 = 8 x 10x 75 ( x 5 ) x 10x = 8 75 x 5 = x 10x 75 (x 10x) = 75(x 5) x 0x = 75x 375 x 115x + 375 = 0 x 100x 15x + 375 = 0 x (x 5) 15(x 5) = 0 (x 5) (x 15) = 0 x 5 = 0 or x 15 = 0 x = 5 or x = 15 Sol. tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank. Let the tap of smaller diameter take x hours to fill the tank alone. Then, the tap of larger diameter will take (x 10) hours to fill the tank. Tank filled by the tap of smaller diameter in 1 hour = 1 x Tank filled by the tap of larger diameter in 1 hour 1 = x 10 Tank filled by both the tanks together in 1 hour = 1 + 1 x x 10 Tank filled by both the tanks together in 9 3 8 hours = 9 3 8 F HG F HG 1 1 + x x 10 I KJ I KJ = 75 1 1 + 8 x x 10 According to the question, 75 F 1 1 I + = 1 8 HG x x 10KJ 1 + 1 = 8 x x 10 75 x = 5, 15 x = 15 is inadmissible as then x 10 is negative which is not possible. [Time cannot be ve] x = 5 x 10 = 15 Hence, tap of smaller and larger diameter take respectively 5 hours and 15 hours to fill the tank alone.. Find value of p such that the quadratic equation (p 1)x (p 1)x + = 0 has equal roots. Sol. The given quadratic equation is (p 1)x (p 1)x + = 0...(1) Comparing it with Ax + Bx + C = 0, we get A = p 1 B = (p 1) C = For equal roots, Discriminant = 0 B AC = 0 B = AC { (p 1)} = (p 1) () (p 1) = (p 1) () (p 1) = (p 1)
CCE MATHEMATICS X Sol. (p 1) (p 1) = 0 (p 1) {(p 1) } = 0 (p 1) (p 1) = 0 p 1 = 0 or p 1 = 0 p = 1 or p = 1 p = 1, 1 p = 1 is inadmissible as then from the given equation = 0 which is impossible. p = 1 5. Solve: 1 1 x + x 7 The given equation is 1 1 x + x 7 ( x 7) ( x + ) ( x + )( x 7) 11 = ( x + )( x 7) = 11 30 ; x, 7 = 11 30 = 11 30 11 30 (x + ) (x 7) = 30 x + x 7x 8 = 30 x 3x + = 0 x x x + = 0 x (x 1) (x 1) = 0 (x 1) (x ) = 0 x 1 = 0 or x = 0 x = 1 or x = x = 1, 6. The length of a rectangular plot is greater than thrice its breadth by m. The area of the plot is 10 sq. m. Find the length and breadth of the plot. Sol. Let the breadth of the plot be x m. Then, length of the plot = (3x + ) m Area of the plot = length breadth = (3x + )x m According to the question, (3x + )x = 10 3x + x = 10 3x + x 10 = 0 3x + 0x 18x 10 = 0 x(3x + 0) 6(3x + 0) = 0 (3x + 0) (x 6) = 0 3x + 0 = 0 or x 6 = 0 3x = 0 or x = 6 x = 0 3 or x = 6 x = 0 3, 6 x = 0 is inadmissible as breadth cannot be ve. 3 x = 6 3x + = 3(6) + = 0 Hence, the length and breadth of the plot are 0 m and 6 m, respectively. Formative Assessment ORAL QUESTIONS (Conversation Type) 1. What is the standard form of a quadratic equation in variable x?. What is the expression for the discriminant of the quadratic equation ax + bx + c = 0? 3. What is the maximum number of roots of a quadratic equation?. In a quadratic equation ax + bx + c = 0, if a = 0, then how will you call this equation? 5. What is the condition that a root of the quadratic equation ax + bx + c = 0 is 1? TRUE OR FALSE 1. If the coefficient of x and the constant term have the same sign and if the coefficient of x term is zero, then the quadratic equation has no real roots.. If the coefficient of x and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots. 3. Every quadratic equation has at most two roots.. Every quadratic equation has at least two roots. 5. Every quadratic equation has exactly one root.
QUADRATIC EQUATIONS 5 Assignments Name:... Class:... Section:... Roll No.:... Grade:... Teacher s sign.:... CLASS ASSIGNMENT 1 1. Find the roots of the equation x 5 = 0.. Find the discriminant of the quadratic equation x + x 5 = 0. 3. Is 3 a root of the equation x 3 = 0?. What must be added and subtracted to solve the quadratic equation x 3x + = 0 by the method of completing the square? 5. If 1 is a root of the quadratic equation x + x + k = 0, then find the value of k. 6. The sum of a number and its reciprocal is 10 3. Form a quadratic equation for this situation. 7. Does the equation ( x 3) = 0 have two equal roots? 8. Is x 3 x = (x + 1) 3 a quadratic equation? 9. The sum of the squares of two consecutive natural numbers is 5. Form a quadratic equation for this situation. 10. What is the sum of the roots of the quadratic equation x 3 x + 1 = 0?
6 CCE MATHEMATICS X Name:... Class:... Section:... Roll No.:... Grade:... Teacher s sign.:... CLASS ASSIGNMENT (From CBSE Examination Paper) 1. If r = 3 is a root of the quadratic equation kr kr 3 = 0, find the value of k.. Find the nature of the roots of the equation 3x x + 3 =0. 3. If 1 is a root of the equation x + kx + 5 = 0, then find the value of k.. Prove that x = 3 is a root of the equation x 5x 3 = 0. 5. If one root of the equation x 10x + p = 0 is, then find the value of p. 6. Find the roots of the equation x + x p(p + 1) = 0, where p is a constant. 7. Find the roots of the equation x 3x m (m + 3) = 0, where m is a constant. 8. For what value of k, the quadratic equation 9x + 8kx + 16 = 0 has equal roots? 9. For the quadratic equation x x + 1 = 0, find the value of x + 1 x. 10. Find the nature of roots of the quadratic equation 3 1 x x + = 0.
QUADRATIC EQUATIONS 7 Name:... Class:... Section:... Roll No.:... Grade:... Teacher s sign.:... HOME ASSIGNMENT 1 1. The area of a triangle is 5 cm. The altitude to the base is cm less than its corresponding base. Form a quadratic equation for this situation. 6. Is x + x + 1 = ( x) + 1 a quadratic equation? 7. Find the discriminant of the quadratic equation 5x 3x + 1 = 0.. One diagonal of a rhombus is cm less than the other. The area of the rhombus is 36 cm. Form a quadratic equation for this situation. 8. What are the roots of the equation x x = 0? 3. Can you say that 0.1 is a root of the equation x 0.01 =0?. What must be added and subtracted to solve the qua- 3 dratic equation 9x + x = 0 by the method of completing the square? 5. If 1 is a root of the quadratic equation x + x + k = 0, then find the value of k. 9. What is the product of the roots of the quadratic equation x 5x + 1 = 0? 10. For the quadratic equation ax + bx + c = 0, what is the expression for the value of x using quadratic formula?
8 CCE MATHEMATICS X Name:... Class:... Section:... Roll No.:... Grade:... Teacher s sign.:... HOME ASSIGNMENT (From NCERT Exemplary Problems) 1. Is 0. a root of the equation x 0. = 0?. Does (x 1) + (x + 1) = 0 have a real root? 6. Find the roots of 6x x = 0 by the factorisation of the corresponding quadratic polynomial. 7. The square of a natural number diminished by 8 is equal to thrice of 8 more than the given number. Form a quadratic equation for this situation. 3. Which constant should be added and subtracted to solve the quadratic equation x 3 x 5 = 0 by the method of completing the square? 8. A natural number, when increased by 1, equals 160 times its reciprocal. Form a quadratic equation for this situation.. How many real roots does the equation (x + 1) x = 0 have? F H 9. Is x = (5 x) x 3 I K a quadratic equation? 5. Find the roots of the quadratic equation x 5 x = 0 by using the quadratic formula. 10. Find the value of k for which the quadratic equation x kx + k = 0 has equal roots.