Page 1 :
Assignments in Mathematics Class X (Term II), 4. QUADRATIC EQUATIONS, Important Terms, Definitions and Results, , •, , An equation of the form ax2 + bx + c = 0, where a,, b, c are real numbers and a ≠ 0, is called a quadratic, equation in x., , A real number a is called a root of the quardratic, equation ax2 + bx + c = 0, a ≠ 0, if aa2 + ba + c = 0., Any quardratic equation can have at most two roots., Note : If a is a root of ax2 + bx + c = 0, then we say, that, (i) x = a satisfies the equation ax2 + bx + c =0, or (ii) x = a is a solution of the equation, ax2 + bx + c = 0, , •, , N, , A, , ER, , TH, , O, , R, , B, , , , , , L, , , , To solve a quadratic equation by factorisation :, (a) Clear fractions and brackets, if necessary., (b) Transfer all the terms to L.H.S. and combine, like terms., (c) Write the equation in the standard form, i.e.,, ax2 + bx + c = 0., (d) Factorise the L.H.S., (e) Put each factor equal to zero and solve., (f) Check each value by substituting it in the given, equation., , YA, , , , , (i) If – p  ≥ 5, then p ≤ – 5, (ii) If – p  ≥ –5, then p ≤ – 5, (iii) If p2  ≥ 4, then either p ≤ –2 or p ≥ – 5, , (Important), (iv) If p2 ≤ 4, then p lies between –2 and 2, i.e.,, –2 ≤ p ≤ 2, (Important), , •, , Quadratic equations can be applied to solve word, problems involving various situations., To solve problems leading to quadratic equations,, following steps may be used :, 1. Represent the unknown quantity in the problem, by a variable (letter)., 2. Translate the problem into an equation, involving this variable., 3. Solve the equation for the variable., 4. Check the result by satisfying the conditions, of the original problem., 5. A root of the quadratic equation, which does, not satisfy the conditions of the problem, must, be rejected., , O, , •, , •, , The roots of a quadratic equation can also be found, by using the method of completing the square., , G, , , , , , , SH, , If ax2 + bx + c can be factorised as (x – a) (x – b),, then ax2 + bx + c = 0 is equivalent to, (x – a) (x – b) = 0, Thus, (x – a) (x – b) = 0 x–a = 0 or x – b = 0, i.e., x = a or x = b., Here a and b are called the roots of the equation, ax2 + bx + c = 0., , A, , •, , The discriminant, usually denoted by D, decides, the nature of roots of a quadratic equation., (i) If D > 0, the equation has real roots and roots, are unequal, i.e., unequal-real roots., If D is a perfect square, the equation has, unequal-rational roots., (ii) If D = 0, the equation has real and equal roots, and each root is −b, 2a, (iii) If D < 0, the equation has no real roots., , K, , Solving a quadratic equation means finding its, roots., , •, , PR, , •, , 2, , b − 4 ac, , S, , The roots of a quadratic equation ax2 + bx + c = 0, are called the zeros of the polynomial ax2 + bx + c., , −b ±, , (Shridharacharya’s formula), 2a, The expression b2 – 4ac is called the discriminant, of the quadratic equation ax2 + bx + c = 0., x=, , •, , •, , The roots of the quadratic equation ax2 + bx, + c = 0, a, b, c ∈ R and a ≠ 0 are given by, , A, , •, , Summative Assessment, Multiple choice Questions, , [1 Mark], , A. Important Questions, 1. If p(x) = 0 is a quadratic equation, then p(x) is a, polynomial of degree :, (a) one, , (b) two, , (c) three, , (d) four, , 2. Which of the following is a quadratic equation, in x?, (a) x + 12 = 0, (b) 7x = 2x2, 1, 2, (c) x + 2 = 2, (d) 6 – x(x2 + 2) = 0, x, 1

Page 2 :
(a) 2(x – 1)2 = 4x2 – 2x + 1, , 3. Which of the following is a root of the equation, 2x2 – 5x – 3 = 0?, , (b) 2x – x2 = x2 + 5, , (b) x = 4, (d) x = –3, , (c), , (a) x2 – 4x + 5 = 0, , 5, 1, is a root of the equation x 2 + kx − = 0,, 4, 2, then the value of k is :, (a) 2 , (b) – 2, 1, 1, (c), , (d), 4, 2, Which of the following equations has the sum, of its roots as 3?, , 15., , , A, , 2x2 + 7 x + 5 2 = 0, , N, , , , A, , , , , 6. Which of the following is a quadratic equation in x?, 2, 1, 2, (a) x + = x, (b) x 2 − 2 = 5, x, x, 2, (c) x − 3 x − x + 4 = 0, , (b) – x2 + 3x – 3 = 0, (a) 2x2 – 3x + 6 = 0, 3, (c), 2x2 −, x + 1 = 0 (d) 3x2 – 3x + 3 = 0, 2, 16. The roots of 4 x 2 + 4 3 x + 3 = 0 are :, , A, , K, , 7. For what value of k, x = 2 is a solution of, kx2 + 2x – 3 = 0?, 1, 1, (b) k =, 2, 2, 1, 1, (c) k = –, (d) k =, 4, 4, 8. Which of the following is a root of the equation, , PR, , (a) k = –, , (b) real and unequal, ,    (c) not real , , (d) none of these, , S, , (a) real and equal , 17. Discriminant of, , ER, , –x+, , TH, , ?, , 1, 1, (b) 1, (c), (d) 4, 4, 2, 3, 9. If x = 2 is a root of the equation, 3x2 – 2kx + 5 = 0, then k is equal to :, , + px + 2q = 0 is :, , (a) p – 8q , , (b) p2 + 8q, , (c) p2 – 8q , , (d) q2 – 8p, , R, , (a) k < 4 , (c) k > 4 , , B, , 4, 1, 1, 17, (b), (c), (d), 17, 4, 17, 4, 10. Which of the following is a quadratic equation in, x?, 1, 2, 2, x+, = 4 , (a), (b) x + 2 = 3   , x, x, 4, 1, 2, (d) x + x = 0, (c) x + 2 = 3    , 3, x, 11. Which of the following is a quadratic equation?, , (b) k > 4, (d) k < 4, , 19. The quadratic equation ax2 + bx + c = 0, a ≠ 0, has two distinct real roots, if D is equal to :, , L, , (a), , x2, , 18. If the equation x2 + 4x + k = 0 has real and, distinct roots, then :, , O, , (a), , (d) 3x2 – 6x – 2 = 0, , 14. If, , x2 − 2x − 4 = 0, , (c) 3x2 + 5x + 2 = 0 (d) (a) and (b) both, , 2x2, , = 3x 2 − 5 x, , (b) x2 + 3x – 12 = 0, , (c) 2x2 – 7x + 6 = 0, , 5. x = 2 is a solution of the equation :, , (d), , 2, , 13. Which of the following equations has 2 as a, root?, , (b) x = –2, (d) x = –1, , (a) x 2 + 2 x − 4 = 0 (b), , 2, , (d) (x2 + 2x)2 = x4 + 3 + 4x3, , 4. Which of the following is a root of the equation, 3x2 – 2x – 1 = 0?, (a) x = –1, (c) x = 1, , ( 2x + 3) + x, , SH, , (a) x = 3, (c) x = 1, , O, , YA, , (a) b2 – 4ac > 0   (b) b2 – 4ac = 0  , (c) b2 – 4ac < 0   (d) none of these, , G, , 20. If ax2 + bx + c = 0 has equal roots, then c is, equal to :, (a) –, , (a) x2 + 2x + 1 = (4 – x)2 + 3, , 21., , , , 12. Which of the following is not a quadratic, equation?, , , , 2, , (b), , −b 2, , (d), 4a, If the equation 9x2 + 6kx + 4, then the roots are :, 2, (b), (a) ± , 3, (c) 0 , (d), , (c), , 2, , (b) −2 x 2 = (5 − x)  2 x − , , 5, 3, (c) (k + 1) x 2 + x = 7, where k = −1, 2, (d) x3 – x2 = (x – 1)3, , b, , 2a, , b, 2a, b2, 4a, = 0 has equal roots,, 3, 2, ±3, ±

Page 3 :
22. The roots of the equation 2x2 + 5x + 5 = 0 are:, (b) not real, (d) real and unequal, , (a 2, , (c) x 2 − 4 x − 3 2 = 0 (d) 3 x 2 + 4 3 x + 4 = 0, 31. (x2 + 1)2 – x2 = 0 has :, , b 2 )x 2, , 23. If the equation, +, – 2(ac + bd)x, 2, 2, + c + d = 0 has equal roots, then :, (a) ad = cd , , (a) four real roots, (b) two real roots  , (c) no real roots   (d) one real root, , (b) ad = bc, , (c) ad = bc , , 32. If kx2 – 5x + 3 = 0 and 2x2 – kx + 1 = 0 have, equal discriminants, then the value of k is :, , (d) ab = cd `, , (a) 1, , 24. If the roots of the equation (a2 + b2)x2 – 2b, (a + c) x + (b2 + c2) = 0 are equal, then :, , (b) 5 : 4, , (c) 10 : 4 , , (d) none of these, , SH, , A, , 34. The quadratic equation 2(p + q)2x2 + 2(p + q), x + 1 = 0 has :, , A, , (a) equal roots, (b) no real roots, (c) real but not equal roots  , (d) none of these, , (d) – 3, , K, , (c) 6, , (d) 3, , (a) 5 : 2 , , Values of k for which the quadratic equation, 2x2 – kx + k = 0 has equal roots is :, (a) 0 only , (b) 4, (c) 8 only , (d) 0, 8, , A, , 35. If p2x2 + (p2 + q2)x + q2 = 0 has equal roots, then, p2 – q2 is equal to :, , PR, , 26., , , , , (b) 3.5, , (c) – 2, , 33. The sum and product of the roots of 2x2 – 5x + 4, are in the ratio :, , (a) 2b = a + c , (b) b2 = ac, 2ac, (c) b =, , (d) b = ac, a+c, 25. If x = 1 is a common root of the equations, ax2 + ax + 3 = 0 and x2 + x + b = 0, then ab is, equal to :, (a) 3, , (b) 2, , N, , (a) real and distinct, (c) real and equal, , 2, (a) x + 4 x + 3 2 = 0 (b) x 2 + 4 x − 3 2 = 0, , (a) – 2q2, , 27. Which constant must be added and subtracted to, 3, 2, solve the quadratic equation 9 x + 4 x − 2 = 0, by the method of completing the square?, 1, 1, 9, 1, (a), (b), (c), (d), 4, 64, 64, 8, , (b) 2q2, , (c) 0, , (d) – 1, , ER, , S, , 36. I f x = – 7 and x = – 5 are roots of, x2 + mx + n = 0, then the values of m and n are :, , TH, , (a) 12 and 36 , (c) – 12 and – 35 , , (b) 12 and 35, (d) none of these, , O, , 37. A boy said, “the product of my age 5 years before, and after 5 years is 75,” then the present age of, the boy is :, , 28. The quadratic equation 4 x 2 + 4 3 x + 3 = 0 has :, (a) two distinct real roots, (b) two equal real roots, (c) no real roots , (d) more than 2 real roots, , B, , R, , (a) 8 years , (c) 12 years , , (b) 10 years, (d) 15 years, , YA, , L, , 38. If the price of a pen is increased by Rs 2, a person, can buy 1 pen less for Rs 40, then the original, price of one pen is :, , 29. Which of the following equations has two distinct, real roots?, 9, 2, (a) 2 x − 3 2 x + = 0 (b) x2 + x – 5 = 0, 4, (c) x 2 + 3 x + 2 2 = 0 (d) 5x2 – 3x + 1 = 0, , G, , O, , (a) Rs 10 , (c) Rs 6 , , (b) Rs 8, (d) Rs 4, , 39. If the sum of the squares of three consecutive, integers is 29, then one of the integer is :, , 30. Which of the following equations has no real, roots?, , (a) 1, , (b) 2, , (c) 5, , (d) 6, , B. Questions From CBSE Examination Papers, 1. The roots of the equation x2 –, are : , (a), , 3 , 1 , , (c) – 3 , –1 , , (b) –, (d), , 2. The roots of the quadratic equation, , 3x − x + 3 = 0, [2011 (T-II)], , [2011 (T-II)], , 3 x 2 − 2 x − 3 = 0 are :, , 3,1, , (a) − 3,, , 3 , –1, 3, , 1, , 3, , (b) 2, 3

Page 4 :
(d), , 3, , 13. The value of k for which the equation, 2x2 – (k – 1) x + 8 = 0 will have real and equal, roots are :, [2011 (T-II)], (a) 9 and –7 , (b) only 9, (c) only –7 , (d) –9 and –7, , , , 3. Which of the following is not a quadratic equation :, [2011 (T-II)], , , (a) (x – 2)2 + 1= 2x – 3 , , 14. Which of the following is a solution of the quadratic, equation x2 – b2 = a (2x – a)?, [2011 (T-II)], a, (a) a + b, (b) 2b – a, (c) ab, (d), b, 15. If one root of the equation 2x2 – 10x + p = 0 is 2,, then the value of p is :, [2011 (T-II)], , (b) x(x + 1) + 8= (x + 2) (x – 2), (c) x(2x + 3) = x2 + 1 , (d) (x + 2)3 = x3 – 4, The roots of the quadratic equation, x2 + 7x + 12 = 0 are :, [2011 (T-II)], (a) – 4, – 3 , (b) 4, – 3, (c) 4, 3 , (d) – 4, 3, , (a) –3, , (c) 3x2 – 5x + 2 = 0, , (d) x2 – 2 2 x – 6 = 0, , N, , 1, 5, is the root of the equation x2 + kx –, = 0,, 2, 4, then the value of k is :, [2011 (T-II)], , 17. If, , A, , 6. Which of the following equations has two distinct, real roots?, [2011 (T-II)], , PR, , (a) 2x2 – 3 2 x + 9 /4 = 0, , (a) 2, , (b) x2 + x – 5 = 0, , ER, , (a) b2 = 4a , , 10x + k = 0, then, [2011 (T-II)], , (c) b2 > 4a , , TH, , x2 –, , 7. If 8 is a root of the equation, the value of k is :, , (d) 16, , R, , B, , (c) 1, , (d) –1, , O, , YA, , 9. The positive root of 3 x 2 + 6 = 9 is :, , [2011 (T-II)], (a) 3, (b) 4, (c) 5, (d) 7, , G, 1, 2, , 1, 2, , 2, (d) –1, 3, 7, 11. For what value of k will be a root of, 3, 3x2 – 13x – k = 0? , [2011 (T-II)], (b) –, , (c) –, , (b), , (a) –1, , (b) –2, , (c) 2, , (b) b2 < 4a, 1, (d) ab =, 4, , (b) ±, , (a) k < 4 , , (b) k > 4, , (c) k, , (d) k, , 4 , , 4, , 22. Value of k for which quadratic equation 2x2 – kx +, k = 0 has equal roots is :, [2011 (T-II)], , 3, −7, (c), (d) –14, 7, 2, 12. If x2 + 2 kx + 4 = 0 has a root x = 2, then the value, of k is?, [2011 (T-II)], , (a) 14, , 1, 2, , 21. If the equation kx2 + 4x + 1 = 0 has real and, distinct roots, then :, [2011 (T-II)], , 10. One of the roots of the quadratic equation, [2011 (T-II)], 6 x 2 − x − 2 = 0 is :, (a), , (d), , 3, (c) 0, (d) ± 3, 2, 20. The roots of the equation 3x2 – 4x + 3 = 0 are :, , [2011 (T-II)], (a) real and unequal (b) real and equal, (c) imaginary, (d) none of these, (a) ± 2, , L, , (b) –2, , 1, 4, , 19. If the equation 9x2 + 6kx + 4 = 0 has equal roots,, then the value of k is :, [2011 (T-II)], , O, , (c) –8, , 8. If the discriminant of 3x2 + 2x + a = 0, is double the, discriminant of x2 – 4x + 2 = 0, then the value of, a is:, [2011 (T-II)], (a) 2, , (c), , S, , 2 = 0 , 2, (d) 5x – 3x + 1= 0 , , (b) 8, , (b) –2, , 18. The roots of the equation ax2 + x + b = 0 are equal, if :, [2011 (T-II)], , (c) x2 + 3x + 2, , (a) 2, , (d) 12, , SH, , (b) x2 – 4x + 4 = 0, , (c) 9, , 16. Which of the following is a root of the equation, 2x2 – 5x – 3 = 0 ?, [2011 (T-II)], (a) x = 3 , (b) x = 4, (c) x = 1 , (d) x = –3, , 5. The quadratic equation whose roots are real and, equal is :, [2011 (T-II)], (a) 2x2 – 4x + 3 = 0, , (b) –6, , A, , 4., , , , , A, , (c), , −1, , 3,, , K, , 3, 2, , −, , 2, 3, , (a) –4, , (b) 4, , (c) 8, , (d) –8, , 23. If r = 3 is a root of quadratic equation kr2 – kr – 3 = 0,, value of k is :, [2011 (T-II)], 1, 1, (a), (b) 2, (c) –2, (d) –, 2, 2, , (d) –4, 4

Page 5 :
24. The value of k, for which the quadratic equation, 4x2 + 4 3 x + k = 0 has equal roots is :, (a) k = 2 , , (c) p = 4, , 27. The value of k for which 3x2 + 2x + k = 0 has real, roots is :, [2011 (T-II)], 1, 1, (a) k > , (b) k, 3, 3, , (b) k = –2, , (c) k = –3 , (d) k = 3, 25. For what value of k the equation kx2 – 6x – 2 = 0, has equal roots ?, [2011 (T-II)], −9, (b), 2, , (a) 7, 2, , (c) –3, , (d), , (c) k, , −7, 2, , (d) k <, , 1, 3, , 4, , A, , N, , (b) p, , 1, , 3, , 28. For the quadratic equation x2 – 2x + 1 = 0 the, 1, value of x +, is :, [2011 (T-II)], x, (a) –1, (b) 1, (c) 2, (d) –2, , 26. The value of p for which the quadratic equation, x(x – 4) + p = 0 has real roots, is : [2011 (T-II)], (a) p < 4, , (d) none of these, , 1. Find the roots of the quadratic equation, x2 – 3x = 0., , [2 Marks], , A, , A. Important Questions, , SH, , SHORT ANSWER TYPE QUESTIONS, , K, , quadratic equation has real roots., , A, , 13. If b = 0, c < 0, is it true that the roots of the, quadratic equation x2 + bx + c = 0 are numerically, equal and opposite in sign? Justify., [HOTS], , PR, , 2. Are x = 0, x = 1 the solution of the equation, x2 + x + 1 = 0?, , 14. Does a quadratic equation with integral coefficient, has always integral roots. Justify your answer. , [HOTS], , S, , 3. Find the discriminant of the quadratic equation, 2x2 – 4x + 3 = 0., , ER, , 4. If 4a2x2 – 4abx + k = 0 has equal roots, then find, the value of k., , 6. If kx 2 − 2 5 x + 4 = 0 has real and equal roots,, then find the value of k., , 16. Does (x – 1)2 + 2(x + 1) = 0 have a real root? Justify, your answer., [HOTS], , 7. If 1 is a root of x 2 − ( 3 + 1) x + k = 0 and, 4x2 + 4kx + a = 0 has equal roots, then find the, value of a., ,    Solve the following quadratic equations by factorisation, (17-26):, , L, , B, , R, , O, , TH, , 5. Find the discriminant of the quadratic equation, bx 2 − 2 acx + b = 0, , 15. Is the following statement ‘true’ or ‘false’?, Justify your answer. If in a quadratic equation the, coefficient of x is zero, then the quadratic equation, has no real roots., [HOTS], , O, , YA, , 8. If 2 is a root of the equation x2 + bx + 12 = 0 and, the equation has equal roots, find the value of b., 3a2x2, , + 8abx +, 9. If one of the roots of the equation, −2b, 4b2 = 0 is, , then find the other root., a, c, 10. If x = is a root of abx2 + (b2 – ac)x – k = 0, then, b, find the value of k., , ( 2 + 1) x +, − ( 3 + 1) x +, , 17. x 2 −, , 2=0, , 2, 18. x, , 3=0, , G, , 19. 4x2 + 4bx – (a2 – b2) = 0, 20. ax2 + (4a2 – 3b)x – 12ab = 0, 21. a2x2 + (a2 + b2)x + b2 = 0, a ≠ 0, m 2 n, x + = 1 – 2x, n, m, 2, 23. (a + b) x2 – 4abx – (a – b)2 = 0, , 22., , 11. State, whether, the, quadratic, equation, ( x − 2) 2 − 2( x + 1) = 0 has two distinct real roots., Justify your answer., , 24. a(x2 + 1) – x(a2 + 1) = 0, , 12. Write whether the following statement is true or, false. Justify your answer., , 25. x2 – x – a(a + 1) = 0, 34, 26. ( x − 3)( x − 4) = (33) 2, , If the coefficient of x2 and the constant term of a, quadratic equation have opposite signs, then the, 5

Page 6 :
36. a2 – b2 = 4ax – 4x2, , Solve the following quadratic equations by completing, the squares (27-32):, , 37. (a – b)x2 + (b – c)x + (c – a) = 0, 38. x2 – 2(m + n)x + (m + n)2 = 1, , 27. 4 x 2 + 4 3 x + 3 = 0, , Find the values of k for which the roots are real and, equal in each of the following equations (39-42):, , 28., , 2 x 2 − 3x − 2 2 = 0, , 29., , 3 x 2 + 10 x + 7 3 = 0, , 30. x, , 2, , (, , 39. (k + 1)x2 + 2(k + 3)x + (k + 8) = 0, , ), , 40. x2 – 2kx + 7k – 12 = 0, , - 2 +1 x + 2 = 0, , 41. 5x2 – 4x + 2 + k(4x2 – 2x – 1) = 0, , 31. a2x2 – 3abx +2b2 = 0, 32. y 2 +, , 42. (4 – k)x2 + (2k + 4)x + (8k + 1) = 0, , 1, y −1 = 0, 2, , 43. Show that the equation x2 + ax – 4 = 0 has real and, distinct roots for all real values of a., 44. For what value of k, (4 – k)x2 + (2k + 4), x + (8k + 1) = 0, is a perfect square., , A, , N, , Solve the following equations by using the formula, (33-38):, 33. x2 – 2ax + 3x – 6a = 0, , SH, , 45. If the roots of the equations ax2 + 2bx + c = 0, and bx 2 − 2 acx + b = 0 are simultaneously real,, then prove that b2 = ac., , 34. (2p + q)x = x2 + 2pq, , K, , A, , 35. a2b2x2 + b2x – a2x – 1 = 0, , PR, , A, , B. Questions From CBSE Examination Papers, 1. Find the values of k for which roots of the equation, x2 – 8kx + 2k = 0 are equal., [2011 (T-II)], , 12. For what value of k the equation 4x2 – 2 (k + 1) x, + (k + 1) = 0 has real and equal roots?, , [2011 (T-II)], , S, , 2. Find the values of p such that the quadratic equation, (p – 12)x2 – 2 (p – 12)x + 2 = 0 has equal roots., , [2011 (T-II)], , ER, , 13. Using quadratic formula, determine the roots of, following equation :, [2011 (T-II)], , TH, , 3. Find the roots of the quadratic equation, 3x2 – 14x + 8 = 0., [2011 (T-II)], , x−, , , 14. Find the values of k for which the following, quadratic equation has two equal roots., 2x2 + kx + 3 = 0 , [2011 (T-II)], , R, , O, , 4. Find the value(s) of k for which the equation, x2 – 2x + k = 0 has equal roots., [2011 (T-II)], , B, , 5. For what value(s) of k, the equation, x2 – 2kx – k = 0 will have equal roots?, , [2011 (T-II)], , L, , 15. For what value of k, the quadratic equation, 9x2 + 8kx + 16 = 0 has equal roots?, , [2011 (T-II)], , YA, , 6. Solve the equation : 10ax2 + 15ax – 6x – 9 = 0,, a ≠ 0., [2011 (T-II)], , O, , 16. Find the roots of the quadratic equation, [2011 (T-II)], 3x2 – 2 6 x + 2 = 0, , 7. Write all the values of k for which the quadratic, equation 2x2 + 2kx + 8 = 0, has equal roots. Also,, find the roots., [2011 (T-II)], , G, , 17. Find the roots of the following quadratic, equation:, [2011 (T-II)], 1, , ( x + 3)( x − 1) = 3  x – , , 3, , 18. Find the value of k such that the quadratic equation, x(x – 2 k) + 6 = 0 has real and equal roots., , [2011 (T-II)], , 8. Find the value of k for which the equation :, kx (x – 2) + 6 = 0 has equal roots. [2011 (T-II)], 9. Solve : 2 x − 3 = 1, [2011 (T-II)], x, 10. Find the roots of the following quadratic equation, by factorisation method., [2011 (T-II)], , 2, 2x + 7x + 5 2 = 0, 5, 11. One root of the equation 2x2 – 8x – m = 0 is ., 2, Find the other root and the value of m., , , 1`, =3, x, , 19. Find the value of k such that the quadratic equation, x2 – 2 kx + (7k –12) = 0 has real and equal roots., , [2011 (T-II)], , [2011 (T-II)], 6

Page 7 :
20. Find the nature of roots of the quadratic equation, 3, 1, 2x2 −, x+, =0, [2011 (T-II)], 2, 2, 21. Find the roots of 6 x 2 − 2 x − 2 = 0 [2011 (T-II)], , [2004], , 28. Solve for x : 4x2 – 4a2x + (a4 – b4) = 0., , [2004], , 29. Using quadratic formula, solve the following, quadratic equation for x :, , 22. Find the roots of the quadratic equation, [2011 (T-II)], 2 x2 − 5x + 3 = 0, , p2x2 + (p2 – q2)x – q2 = 0., , [2004], , 30. Using quadratic formula, solve the following, quadratic equation for x :, , 23. Find the roots of the quadratic equation, [2011 (T-II)], 6 x2 + 5x − 6 = 0, , x2 – 2ax + (a2 – b2) = 0., , 24. If –4 is a root of the quadratic equation, x 2 + px − 4 = 0 and the equation 2 x 2 + px + k = 0, has equal roots, find the value of k. [2011 (T-II)], , [2004], , x2 – 4ax + 4a2 – b2 = 0, –, , 8a2x, , +, , (a4, , –, , A, , 32. Solve for x :, , [2011 (T-II)], , 16x2, , N, , 31. Using quadratic formula, solve the following, quadratic equation for x :, , 25. If 4a 2 x 2 − 4abx + k = 0 has equal roots of x, then, find the value of k., , 27. Solve for x : 4x2 – 2(a2 + b2)x + a2b2 = 0., , b4), , [2004], =0, , [2004], , 33. Solve for x : 36x2 – 12ax + (a2 – b2) = 0. [2004C], , A, , SH, , 26. For what value of k does (k – 12) x2 + 2 (k – 12), x + 2 = 0 have equal roots?, [2008C], , [3 Marks], , A, , K, , SHORT ANSWER TYPE QUESTIONS, , PR, , A. Important Questions, , 10. Prove that the equation x2(a2 + b2) + 2x(ac + bd), + (c2 + d2) = 0 has no real root, if ad ≠ bc, , 1, 2, 6, +, = , ( x ≠ 0), x − 2 x −1 x, , 11. If the roots of the equation x2 + 2cx + ab = 0, are real and unequal, prove that the equation, x2 – 2(a + b)x + a2 + b2 + 2c2 = 0 has no real, roots., , ER, , 1., , S, , Solve the following quadratic equations by factorisation, (1–6) :, , [HOTS], , TH, , 2., , O, R, , 13. The product of two successive integral multiples, of 5 is 300. Determine the multiples., , B, , a, b, +, = a+b, ax − 1 bx − 1, x−a x−b a b, 4., +, = +, x−b x−a b a, a, b, 2c, +, =, 5., x−a x−b x−c, , 14. The sum of the squares of two numbers is 233 and, one of the numbers is 3 less than twice the other, number. Find the numbers., , YA, , L, , 3., , 12. The sum of two numbers is 8 and 15 times the sum, of their reciprocals is also 8. Find the numbers., , O, , 15. Find two consecutive positive integers, the sum of, whose squares is 365., , 2, , G, ,  x ,  x , 6. ,  − 5 ,  + 6 = 0, ( x ≠ 0), x +1, x + 1, 7. If a, b, c, are real numbers such that ac ≠ 0, then show that at least one of the equations, ax2 + bx + c = 0 and – ax2 + bx + c = 0 has a real, root., , 16. The difference of two numbers is 8 and the sum of, their squares is 274. Find the numbers., 17. Find three consecutive positive numbers such, that the square of the middle number exceeds the, difference of the squares of the other two by 60., , 8. If the equation (1 + m2)x2 + 2mcx + (c2 – a2) = 0, has equal roots, prove that c2 = a2(1 + m2)., , 18. If a number is added to three times its reciprocal,, 3, the result is 5 . Find the number., 5, 19. The length of a rectangle is 3 cm more than its, width and its area is 40 cm2. Find the dimensions, of the rectangle., , 9. If p, q, r and s are real numbers such that, pr = 2(q + s), then show that at least one of the, equations x2 + px + q = 0 and x2 + rx + s = 0 has, real roots., 7

Page 8 :
20. The denominator of a fraction is 3 more than, its numerator. The sum of the fraction and its, 9, Find the fraction., reciprocal is 2, 10, , 21. A two-digit number is 4 times the sum of its digits, and also equal to twice the product of its digits., Find the number., , B. Questions From CBSE Examination Papers, 1. The sum of the squares of two consecutive natural, numbers is 421. Find the numbers. [2011 (T-II)], 2. Solve for x by using, 36x2 – 12ax + (a2 – b2) = 0, , stop at intermediate stations). If the average speed, of the express train is 11 km/hour more than that, of the passenger train, find the average speed of the, two trains., [2011 (T-II)], , quadratic formula, [2011 (T-II)], , 15. If two pipes function simultaneously, a reservoir, will be filled in twelve hours. First pipe fills the, reservoir 10 hours faster than the second pipe., How many hours will the second pipe take to fill, the reservoir?, [2011 (T-II)], , A, , N, , 3. The sum of a number and its reciprocal is 10 . Find, 3, the number., [2011 (T-II)], , K, , A, , [2011 (T-II)], , A, , [2011 (T-II], , 2, 2, 2, 2, a (a + b ) x + b x − a = 0, 1, 1, 1, 18. Solve for x :, −, =, x−3 x+5 6, , PR, , one, , 16. Solve the equation for x :, x + 3 1 − x 17, −, =, x−2, x, 4, 17. Solve for x :, , 1, 1, 11, −, =, x ≠ –4, 7, x, +, 4, x, −, 7, 30, , 6. If, , SH, , 4. Solve the following equation :, 2x, 1, 3x + 9, 3, +, +, = 0; x ≠ 3, –, 2 , x − 3 2 x + 3 ( x − 3)(2 x + 3), , [2011 (T-II], 5. Solve the equation :, , root, , of the quadratic equation, x − 5 x + 6k = 0 is reciprocal of other, find the, value of k. Also find the roots., [2011 (T-II)], , [2011 (T-II)], , S, , 2, , 1, 1, −, =3, x x−2, , ER, , TH, , 20. If the equations 5 x 2 + (9 + 4 p ) x + 2 p 2 = 0 and, 5x + 9 = 0 are satisfied by the same value of x, find, the value of p., [2011 (T-II)], , [2011 (T-II)], , R, , 8. Solve :, , [2011 (T-II)], , O, , 7. Solve the given equation for x :, 1 1 1, 1, + + =, a, b, x, a, +, b+ x, , , x +1 x − 2, +, = 3; x ≠ 1, −2, x −1 x + 2, , [2011 (T-II)], 19. Solve for x :, , 21. Solve the following quadratic equation :, x2 – 3x – 10 = 0 , [2011 (T-II)], , L, , B, , 9. Find the roots of the following quadratic equation, by the factorisation method., [2011 (T-II)], 4 3 x 2 + 5 x − 2 3 = 0, , YA, , 22. Solve for x :, , 10. Find the roots of the following quadratic equation., − x 2 + 7 x − 10 = 0, [2011 (T-II)], , x, x + 1 34, +, =, x +1, x, 15, , [2011 (T-II)], , O, , 23. Find the value of k for which the quadratic equation, (k + 4) x2 + (k + 1) x + 1 = 0 has equal roots. , [2011 (T-II)], , G, , 11. Find two natural numbers, which differ by 3 and, whose squares have the sum 149., [2011 (T-II)], , 24. The sum of the reciprocals of Rehman’s ages (in, 1, years) 3 years ago and 5 years from now is ., 3, Find his present age., [2011 (T-II)], , 4, 5, −3, , x ≠ 0,, 12. Solve for x : − 3 =, x, 2x + 3, 2, , [2011 (T-II)], 13. The sum of two natural numbers is 8. Determine, the numbers, if the sum of their reciprocals is, 8/15., [2011 (T-II)], , 25. Solve for x :, , , 14. An express train takes 1 hour less than a passenger, train to travel 132 km between stations A and B, (without taking into consideration the time they, , 1, 2, 4, +, =, ; x ≠ –1, −2, −4, x +1 x + 2 x + 4, [2011 (T-II)], , 26. Solve for x : 9 x 2 − 3(a + b) x + ab = 0, , 8, , [2011 (T-II)]

Page 9 :
27. Find two positive numbers whose squares have the, difference 48 and the sum of the numbers is 12. , [2011 (T-II)], , thrice its breadth by 2 m. The area of the plot is, 120 sq. m. Find the length and breadth of the plot. , [2011 (T-II)], , 28. Solve for x :, [2011 (T-II)], 1 3,  4x − 3 ,  2x + 1 , ,  − 10 ,  = 3, x ≠ − ,, 4x − 3 , 2 4, 2 x + 1, , 31. Solve for x :, , [2011 (T-II)], , 32. Solve the following quadratic equation for x :, [2011 (T-II)], p2 x + ( p2 − q2 ) x − q2 = 0, , 29. Find the roots of the quadratic equation :, 2 2 2, 2, 2, a b x + b x − a x − 1 = 0, , 7 x 2 − x − 13 7 = 0, , 33. Solve for x :, , [2011 (T-II)], , [2004], , 1,  2 x − 1,  x+3 , 2, − 3, = 5; given that x ≠ −3, x ≠ ., , ,  x+3 ,  2 x − 1, 2, , 30. The length of a rectangular plot is greater than, , [4 Marks], , N, , LONG ANSWER TYPE QUESTIONS , 1. The difference of the squares of two numbers is, 45. The square of the smaller number is 4 times the, larger number. Find the numbers., , A, , A. Important Questions, , SH, , m × 40 m, a rectangular pond has to be constructed, so that the area of the grass surrounding the pond, would be 1184 m2. Find the length and breadth of, the pond., , A, , 2. Out of a number of Saras birds, one fourth the, 1, number are moving about in lotus plants, th, 9, 1, coupled (along) with, as well as 7 times the, 4, square root of the number move on a hill, 56 birds, remain in Vakula trees. What is the total number of, birds?, (Mahavira around 850 A.D.), , PR, , A, , K, , 7. A sailor can row a boat 8 km downstream, and return back to the starting point in 1 hour, 40 minutes. If the speed of the stream is 2 km/hr,, find the speed of the boat in still water., , S, , 8. The length of a rectangle is thrice as long as the, side of a square. The side of the square is 4 cm, more than the width of the rectangle. If their areas, being equal, find their dimensions., , TH, , ER, , 3. A train, travelling at a uniform speed for 360 km,, would have taken 48 minutes less to travel the, same distance if its speed were 5 km/h more. Find, the original speed of the train., , O, , 4. If Zeba were younger by 5 years than what she, really is, then the square of her age (in years) would, have been 11 more than five times her actual age., What is her age now?, , 9. Out of a group of swans, 7/2 times the square root, of the total number are playing on the shore of a, pond. The two remaining ones are swinging in, water. Find the total number of swans., , B, , R, , 10. Is it possible to design a rectangular mango grove, whose length is twice its breadth and the area is 800, m2? If so, find its length and breadth., , 5. At present Asha’s age (in years) is 2 more than the, square of her daughter Nisha’s age. When Nisha, grows to her mother’s present age, Asha’s age, would be one year less than 10 times the present, age of Nisha. Find the present ages of both Asha, and Nisha., , YA, , L, , 11. Is the following situation possible? If so, determine, their present ages. The sum of the ages of two, friends is 20 years. Four years ago, the product of, their ages in years was 48., , G, , O, , 12. Is it possible to design a rectangular park of, perimeter 80 m and area 400 m2? If so, find its, length and breadth., , 6. In the centre of a rectangular lawn of dimensions 50, , B. Questions From CBSE Examination Papers, 1. Three consecutive positive integers are taken, such that the sum of the square of the first and the, product of the other two is 154. Find the integers., , [2011 (T-II)], , 3. The product of the digits of a two digit positive, number is 24. If 18 is added to the number, then, the digits of the number are interchanged. Find the, number., [2011 (T-II)], 3, 4. Two taps together can fill a tank in 9 hours. The, 8, 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. , [2011 (T-II)], , 2. The speed of a boat in still water is 11 km/hr. It can, go 12 km upstream and return downstream to the, original point in 2 hrs 45 min. Find the speed of the, stream., [2011 (T-II)], 9

Page 10 :
15. By increasing the speed of a bus by 10 km/hr, it, takes one and half hours less to cover a journey of, 450 km. Find the original speed of the bus., , 5. A person on tour has Rs 360 for his daily expenses., If he extends his tour for four days, he has to cut, down his daily expenses by Rs 3. Find the original, duration of the tour., [2011 (T-II)], , [2011 (T-II)], , , , 6. A train travels at a uniform speed for a distance, of 63 km and then travels a distance of 72 km at, an aveage speed of 6 km/h more than its original, speed. If it takes 3 hours to complete the total, journey, what is the original speed of the train? , [2011 (T-II)], , 16. A factory produces certain pieces of pottery in a, day. It was observed on a particular day that the, cost of production of each piece (in rupees) was 3, more than twice the number of articles produced in, the day. If the total cost of production on that day, was Rs 90, find the number of pieces produced and, cost of each piece., [2011 (T-II)], , 7. Sum of the areas of two squares is 468 m2. If the, difference of their perimeters is 24 m, find the, sides of the two squares., [2011 (T-II)], , A, , N, , 17. The hypotenuse of a right triangle is 3 5 cm., If the smaller side is tripled and the larger side is, doubled, the new hypotenuse will be 15 cm. Find, the length of each side., [2011 (T-II], , SH, , 8. The difference of squares of two numbers is 180., The square of the smaller number is 8 times the, larger number. Find the two numbers., , [2011 (T-II)], , K, , A, , 18. A man bought a certain number of toys for, Rs 180. He kept one for his own use and sold, the rest for one rupee each more than he gave for, them. Besides getting his own toy for nothing, he, made a profit of Rs 10. Find the number of toys, he, initially bought., [2011 (T-II)], , PR, , A, , 9. Two pipes running together can fill a tank in 6, minutes. If one pipe takes 5 minutes more than the, other to fill the tank, find the time in which each, pipe would fill the tank separately. [2011 (T-II)], , 19. The product of Tanay’s age (in years) five years, ago and his age ten years later is 16. Determine, Tanay’s present age., [2011 (T-II)], , ER, , S, , 10. A motorboat whose speed is 18 km/hr in still, water takes 1 hr more to go 24 km upstream than, to return downstream to the same spot. Find the, speed of stream., [2011 (T-II)], , TH, , 20. A plane left 30 minutes later than the schedule time, and in order to reach its destination 1500 km away, in time, it has to increase its speed by 250 km/hr, from its usual speed. Find its usual speed., , R, , O, , 11. Some students planned a picnic.The budget for, food was Rs 480. But 8 of them failed to go, the, cost of food for each member increased by Rs 10., How many students attended the picnic?, , [2011 (T-II)], , B, , , , [2011 (T-II)], , L, , 21. A train travels 300 km at a uniform speed. If the, speed of the train had been 5 km/hour more, it, would have taken 2 hours less for the same journey., Find the usual speed of the train., [2011 (T-II)], , O, , YA, , 12. A fast train takes 3 hours less than a slow train for, a journey of 600 km. If the speed of the slow train, was 10 km/hr less than that of the fast train, find, the speeds of the trains., [2011 (T-II)], , G, , 22. The sum of two natural numbers is 8. Determine, the numbers, if the sum of their reciprocals is, 8 ., [2011 (T-II)], 15, , 13. A person has a rectangular garden whose area is, 100 sq m. He fences three sides of the garden with, 30 m barbed wire. On the fourth side, the wall of, his house is constructed; find the dimensions of the, garden., [2011 (T-II)], , 23. A two digit number is four times the sum of its, digits. It is also equal to three times the product of, its digits. Find the number., [2011 (T-II)], , 1, minutes., 13, If one pipe takes 3 minutes more than the other to, fill it, find the time in which each pipe can fill the, tank., [2011 (T-II)], , 14. Two pipes can together fill a tank in 3, , 24. The speed of a boat in still water is 15 km/hr. It, can go 30 km upstream and return downstream to, the original point in 4 hours 30 minutes. Find the, speed of the stream., [2011 (T-II)], 10

Page 11 :
25. The sum of ages of father and his son is 45 years., 5 years ago, the product of their ages was 124., Determine their present ages., [2011 ( T-II)], , 33. In a class test, the sum of Kamal’s marks in Maths, and English is 40. Had he got 3 marks more in, Maths and 4 marks less in English, the product of, their marks would have been 360. Find his marks, in two subjects., [2008C], , 26. The numerator of a fraction is 3 less than its, denominator. If 2 is addd to both numerator as, , 34. A person on tour has Rs 4200 for his expenses. If, he extends his tour for 3 days, he has to cut down, his daily expenses by Rs 70. Find the duration of, the tour., [2008C], , well as denominator, the sum of new and original, fraction is, , [2011 (T-II)], , , find the frction., , 27. The hypotenuse of a right angled triangle is 6 cm, more than twice its shortest side. If third side is 2, cm less than the hypotenuse, find the sides of this, triangle., [2011 (T-II)], , A, , N, , 35. In a class test the sum of Gagan’s, marks in, Mathematics and English is 45. If he had 1 more, mark in Maths and 1 less in English, the product, of marks would have been 500. Find the original, marks obtained by Gagan in Maths and English, separately., [2008C], , A, , SH, , 28. A takes 6 days less tahn the time taken by B to, finish a piece of work. If both A and B together can, finish it in 4 days, find the time taken by B to finish, the work., [2011 (T-II)], , A, , K, , 36. Places A and B are 100 km apart on a highway., One car starts from A and another from B at the, same time. If the cars travel in the same direction, at different speeds, they meet in 5 hours. If they, travel towards each other, they meet in 1 hour., What are the speeds of the two cars?, [2009], , S, , PR, , 29. The differene of squares of two natural numbers is, 45. The square of the smaller number is four times, the larger number. Find the numbers., , ER, , 30. The denominator of a fraction is one more than, twice the numerator. If the sum of the fraction and, 16, its reciprocal is 2 , find the fraction., 21, , [2011 (T-II)], , O, , TH, , 37. If –5 is a root of the quadratic equation, 2x2 + px – 15 = 0 and the quadratic equation, p(x2 + x) + k = 0 has equal roots, then find the, values of p and k., [2009], , R, , 31. In a class test, the sum of the marks obtained by, P in Mathematics and Science is 28. Had he got 3, more marks in Maths and 4 marks less in Science,, the product of marks obtained in the two subjects, would have been 180. Find the marks obtained in, the two subjects separately., [2008], , O, , YA, , L, , B, , 38. A trader bought a number of articles for Rs 900,, five articles were found damaged. He sold each, of the remaining articles at Rs 2 more than what, he paid for it. He got a profit of Rs 80 on the, whole transaction. Find the number of articles he, bought., [2009], , G, , 32. A peacock is sitting on the top of a pillar which is 9, m high. From a point 27 m away from the bottom, of the pillar, a snake is coming to its hole at the, base of the pillar. Seeing the snake the peacock, pounces on it. If their speeds are equal, at what, distane from the hole is the snake caught? [2008], , 39. Two years ago a man’s age was three times the, square of his son’s age. Three years hence his age, will be four times his son’s age. Find their present, ages., [2009], , 11

Page 12 :
Formative Assessment, 1. Make a square of dimension x × x (Here, x = 5 cm), as shown in figure 1. Here, area of square is equal, to 1st term of polynomial x2 + bx + c, i.e., x2., , Activity, Objective :, , To find solution of quadratic equation, x2 + bx + c = 0 by completing the square., , 2. Add two strips of dimensions 3 × x to figure 1 (2nd term, of polynomial is +ve) to obtain figure 2. Here, area, , Materials Required : Coloured paper, a pair of scissors,, geometry box and fevistick., , of each strip is equal to, , Procedure :, , x2 + bx + c, i.e.,, , of 2nd term of polynomial, , × bx., , PR, , A, , K, , A, , SH, , A, , N, , Case I : Let us find the solution of x2 + 6x + 8 = 0., , 4., , , , , , , G, , O, , YA, , L, , B, , R, , O, , TH, , ER, , S, , 3. Add and subtract a square of dimension 3 × 3 to figure 2. We will get an arrangement as shown in figure 3., , From above three figures, we observe that, x2 + 6x = x2 + 3x + 3x + 9 – 9 = (x + 3)2 – 9, ⇒ x2 + 6x + 8 = (x + 3)2 – 9 + 8 = (x + 3)2 – 1, ∴ x2 + 6x + 8 = 0 ⇒ (x + 3)2 – 1 = 0, ⇒ (x + 3)2 = 1 ⇒ x + 3 = + 1, ⇒ x = – 4, –2., , Case II : Let us find the solution of x2 – 6x + 8 = 0., 1. Make a grid of dimension x × x [x = 10] as shown in, figure 4., 12

Page 13 :
⇒ x2 – 6x + 8 = 0 ⇒ (x – 3)2 – 1 = 0, ⇒ (x – 3)2 = 1 ⇒ x – 3 = + 1, ⇒ x = 4, 2., , 2. Cut out 60 strips (exactly shown in figure 5) from, the grid., [2nd term of x 2 − 6 x + 8 is –ve], Area of 60 strips = Area of two rectangles of dimensions 3 × x., [ Here, 3x =, , 1, of 6x, i.e., 2nd term of x 2 − 6 x + 8 ], 2, , G, , O, , YA, , L, , B, , R, , O, , TH, , ER, , S, , PR, , A, , K, , A, , SH, , A, , N, , 3., From figure 5, we observe that, , Area of unshaded portion, , = Area of square ABCD – Area of square EFGD, = (x – 3)2 – 32, ⇒ x2 – 6x = (x – 3)2 – 9, ⇒ x2 – 6x + 8 = (x – 3)2 – 9 + 8 = (x – 3)2 – 1, , 13

Page 14 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Exercise 4.1, Question 1:, Check whether the following are quadratic equations:, , Answer:, , It is of the form, , ., , Hence, the given equation is a quadratic equation., , It is of the form, , ., , Hence, the given equation is a quadratic equation., , It is not of the form, , ., , Hence, the given equation is not a quadratic equation., , It is of the form, , ., , Hence, the given equation is a quadratic equation., , Page 1 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 15 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , It is, of the form, , ., , Hence, the given equation is a quadratic equation., , It is not of the form, , ., , Hence, the given equation is not a quadratic equation., It is, not of the form, , ., , Hence, the given equation is not a quadratic equation., , It is of the form, , ., , Hence, the given equation is a quadratic equation., , Question 2:, Represent the following situations in the form of quadratic equations., (i) The area of a rectangular plot is 528 m2. The length of the plot (in, metres) is one more than twice its breadth. We need to find the length, and breadth of the plot., (ii) The product of two consecutive positive integers is 306. We need, to find the integers., (iii) Rohan’s mother is 26 years older than him. The product of their, ages (in years) 3 years from now will be 360. We would like to find, Rohan’s present age., , Page 2 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 16 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , (iv) A train travels a distance of 480 km at a uniform speed. If the, speed had been 8 km/h less, then it would have taken 3 hours more to, cover the same distance. We need to find the speed of the train., Answer:, (i) Let the breadth of the plot be x m., Hence, the length of the plot is (2x + 1) m., Area of a rectangle = Length × Breadth, ∴ 528 = x (2x + 1), , (ii) Let the consecutive integers be x and x + 1., It is given that their product is 306., ∴, (iii) Let Rohan’s age be x., Hence, his mother’s age = x + 26, 3 years hence,, Rohan’s age = x + 3, Mother’s age = x + 26 + 3 = x + 29, It is given that the product of their ages after 3 years is 360., , (iv) Let the speed of train be x km/h., Time taken to travel 480 km =, In second condition, let the speed of train =, , km/h, , Page 3 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 17 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , It is also given that the train will take 3 hours to cover the same, distance., , Therefore, time taken to travel 480 km =, , hrs, , Speed × Time = Distance, , Page 4 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 18 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Exercise 4.2, Question 1:, Find the roots of the following quadratic equations by factorisation:, , Answer:, , Roots of this equation are the values for which, ∴, , = 0 or, , =0, , =0, , i.e., x = 5 or x = −2, , Roots of this equation are the values for which, ∴, , = 0 or, , =0, , =0, , i.e., x = −2 or x =, Page 5 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 19 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Roots of this equation are the values for which, ∴, , = 0 or, , i.e., x =, , =0, , =0, , or x =, , Roots of this equation are the values for which, , =0, , Therefore,, i.e.,, , Page 6 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 20 :
Class X, , Chapter 4 – Quadratic Equations, , Roots of this equation are the values for which, , Maths, , =0, , Therefore,, i.e.,, , Question 2:, (i) John and Jivanti together have 45 marbles. Both of them lost 5, marbles each, and the product of the number of marbles they now, have is 124. Find out how many marbles they had to start with., (ii) A cottage industry produces a certain number of toys in a day. The, cost of production of each toy (in rupees) was found to be 55 minus, the number of toys produced in a day. On a particular day, the total, cost of production was Rs 750. Find out the number of toys produced, on that day., Answer:, (i) Let the number of John’s marbles be x., Therefore, number of Jivanti’s marble = 45 − x, After losing 5 marbles,, Number of John’s marbles = x − 5, Number of Jivanti’s marbles = 45 − x − 5 = 40 − x, It is given that the product of their marbles is 124., Page 7 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 21 :
Class X, , Either, , Chapter 4 – Quadratic Equations, , Maths, , = 0 or x − 9 = 0, , i.e., x = 36 or x = 9, If the number of John’s marbles = 36,, Then, number of Jivanti’s marbles = 45 − 36 = 9, If number of John’s marbles = 9,, Then, number of Jivanti’s marbles = 45 − 9 = 36, (ii) Let the number of toys produced be x., ∴ Cost of production of each toy = Rs (55 − x), It is given that, total production of the toys = Rs 750, , Either, , = 0 or x − 30 = 0, , i.e., x = 25 or x = 30, Hence, the number of toys will be either 25 or 30., , Page 8 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 22 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Question 4:, Which of the following are APs? If they form an A.P. find the common, difference d and write three more terms., (i) 2, 4, 8, 16 …, (ii), (iii) − 1.2, − 3.2, − 5.2, − 7.2 …, (iv) − 10, − 6, − 2, 2 …, (v), (vi) 0.2, 0.22, 0.222, 0.2222 …., (vii) 0, − 4, − 8, − 12 …, (viii), (ix) 1, 3, 9, 27 …, (x) a, 2a, 3a, 4a …, (xi) a, a2, a3, a4 …, (xii), (xiii), (xiv) 12, 32, 52, 72 …, (xv) 12, 52, 72, 73 …, Answer:, (i) 2, 4, 8, 16 …, It can be observed that, a2 − a1 = 4 − 2 = 2, Page 9 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 23 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , a3 − a2 = 8 − 4 = 4, a4 − a3 = 16 − 8 = 8, i.e., ak+1− ak is not the same every time. Therefore, the given, numbers are not forming an A.P., (ii), It can be observed that, , i.e., ak+1− ak is same every time., Therefore,, , and the given numbers are in A.P., , Three more terms are, , (iii) −1.2, −3.2, −5.2, −7.2 …, It can be observed that, a2 − a1 = (−3.2) − (−1.2) = −2, a3 − a2 = (−5.2) − (−3.2) = −2, a4 − a3 = (−7.2) − (−5.2) = −2, Page 10 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 24 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , i.e., ak+1− ak is same every time. Therefore, d = −2, The given numbers are in A.P., Three more terms are, a5 = − 7.2 − 2 = −9.2, a6 = − 9.2 − 2 = −11.2, a7 = − 11.2 − 2 = −13.2, (iv) −10, −6, −2, 2 …, It can be observed that, a2 − a1 = (−6) − (−10) = 4, a3 − a2 = (−2) − (−6) = 4, a4 − a3 = (2) − (−2) = 4, i.e., ak+1 − ak is same every time. Therefore, d = 4, The given numbers are in A.P., Three more terms are, a5 = 2 + 4 = 6, a6 = 6 + 4 = 10, a7 = 10 + 4 = 14, (v), It can be observed that, , i.e., ak+1 − ak is same every time. Therefore,, The given numbers are in A.P., Page 11 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 25 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Three more terms are, , (vi) 0.2, 0.22, 0.222, 0.2222 …., It can be observed that, a2 − a1 = 0.22 − 0.2 = 0.02, a3 − a2 = 0.222 − 0.22 = 0.002, a4 − a3 = 0.2222 − 0.222 = 0.0002, i.e., ak+1 − ak is not the same every time., Therefore, the given numbers are not in A.P., (vii) 0, −4, −8, −12 …, It can be observed that, a2 − a1 = (−4) − 0 = −4, a3 − a2 = (−8) − (−4) = −4, a4 − a3 = (−12) − (−8) = −4, i.e., ak+1 − ak is same every time. Therefore, d = −4, The given numbers are in A.P., Three more terms are, a5 = − 12 − 4 = −16, a6 = − 16 − 4 = −20, a7 = − 20 − 4 = −24, (viii), It can be observed that, Page 12 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 28 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , (xiii), It can be observed that, , i.e., ak+1 − ak is not the same every time., Therefore, the given numbers are not in A.P., (xiv) 12, 32, 52, 72 …, Or, 1, 9, 25, 49 ….., It can be observed that, a2 − a1 = 9 − 1 = 8, a3 − a2 = 25 − 9 = 16, a4 − a3 = 49 − 25 = 24, i.e., ak+1 − ak is not the same every time., Therefore, the given numbers are not in A.P., (xv) 12, 52, 72, 73 …, Or 1, 25, 49, 73 …, It can be observed that, a2 − a1 = 25 − 1 = 24, a3 − a2 = 49 − 25 = 24, , Page 15 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 29 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , a4 − a3 = 73 − 49 = 24, i.e., ak+1 − ak is same every time., Therefore, the given numbers are in A.P., And, d = 24, Three more terms are, a5 = 73+ 24 = 97, a6 = 97 + 24 = 121, a7 = 121 + 24 = 145, , Question 3:, Find two numbers whose sum is 27 and product is 182., Answer:, Let the first number be x and the second number is 27 − x., Therefore, their product = x (27 − x), It is given that the product of these numbers is 182., , Either, , = 0 or x − 14 = 0, , i.e., x = 13 or x = 14, If first number = 13, then, Other number = 27 − 13 = 14, If first number = 14, then, Page 16 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 30 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Other number = 27 − 14 = 13, Therefore, the numbers are 13 and 14., , Question 4:, Find two consecutive positive integers, sum of whose squares is 365., Answer:, Let the consecutive positive integers be x and x + 1., , Either x + 14 = 0 or x − 13 = 0, i.e., x = −14 or x = 13, Since the integers are positive, x can only be 13., ∴ x + 1 = 13 + 1 = 14, Therefore, two consecutive positive integers will be 13 and 14., , Question 5:, The altitude of a right triangle is 7 cm less than its base. If the, hypotenuse is 13 cm, find the other two sides., Answer:, Let the base of the right triangle be x cm., Its altitude = (x − 7) cm, , Page 17 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 31 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Either x − 12 = 0 or x + 5 = 0, i.e., x = 12 or x = −5, Since sides are positive, x can only be 12., Therefore, the base of the given triangle is 12 cm and the altitude of, this triangle will be (12 − 7) cm = 5 cm., , Question 6:, A cottage industry produces a certain number of pottery articles in a, day. It was observed on a particular day that the cost of production of, each article (in rupees) was 3 more than twice the number of articles, produced on that day. If the total cost of production on that day was, Rs 90, find the number of articles produced and the cost of each, article., Answer:, Let the number of articles produced be x., Therefore, cost of production of each article = Rs (2x + 3), It is given that the total production is Rs 90., , Page 18 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 32 :
Class X, , Chapter 4 – Quadratic Equations, , Either 2x + 15 = 0 or x − 6 = 0, i.e., x =, , Maths, , or x = 6, , As the number of articles produced can only be a positive integer,, therefore, x can only be 6., Hence, number of articles produced = 6, Cost of each article = 2 × 6 + 3 = Rs 15, , Page 19 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 33 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Exercise 4.3, Question 1:, Find the roots of the following quadratic equations, if they exist, by the, method of completing the square:, , Answer:, , Page 20 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 34 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Page 21 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 35 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Page 22 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 36 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Question 2:, Find the roots of the quadratic equations given in Q.1 above by, applying the quadratic formula., Answer:, , Page 23 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 37 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Page 24 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 38 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Question 3:, Find the roots of the following equations:, , Page 25 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 39 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Answer:, , Page 26 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 40 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Question 4:, The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago, and 5 years from now is . Find his present age., Answer:, Let the present age of Rehman be x years., Three years ago, his age was (x − 3) years., Five years hence, his age will be (x + 5) years., It is given that the sum of the reciprocals of Rehman’s ages 3 years, ago and 5 years from now is ., , However, age cannot be negative., Therefore, Rehman’s present age is 7 years., , Page 27 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 41 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Question 5:, In a class test, the sum of Shefali’s marks in Mathematics and English, is 30. Had she got 2 marks more in Mathematics and 3 marks less in, English, the product of their marks would have been 210. Find her, marks in the two subjects., Answer:, Let the marks in Maths be x., Then, the marks in English will be 30 − x., According to the given question,, , If the marks in Maths are 12, then marks in English will be 30 − 12 =, 18, If the marks in Maths are 13, then marks in English will be 30 − 13 =, 17, , Question 6:, The diagonal of a rectangular field is 60 metres more than the shorter, side. If the longer side is 30 metres more than the shorter side, find, the sides of the field., Page 28 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 42 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Answer:, Let the shorter side of the rectangle be x m., Then, larger side of the rectangle = (x + 30) m, , However, side cannot be negative. Therefore, the length of the shorter, side will be, 90 m., Hence, length of the larger side will be (90 + 30) m = 120 m, , Question 7:, The difference of squares of two numbers is 180. The square of the, smaller number is 8 times the larger number. Find the two numbers., Answer:, Let the larger and smaller number be x and y respectively., According to the given question,, , Page 29 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 43 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , However, the larger number cannot be negative as 8 times of the, larger number will be negative and hence, the square of the smaller, number will be negative which is not possible., Therefore, the larger number will be 18 only., , Therefore, the numbers are 18 and 12 or 18 and −12., , Question 8:, A train travels 360 km at a uniform speed. If the speed had been 5, km/h more, it would have taken 1 hour less for the same journey. Find, the speed of the train., Answer:, Let the speed of the train be x km/hr., Time taken to cover 360 km, , hr, , According to the given question,, , Page 30 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 44 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , However, speed cannot be negative., Therefore, the speed of train is 40 km/h, , Question 9:, Two water taps together can fill a tank in, , hours. The 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., Answer:, Let the time taken by the smaller pipe to fill the tank be x hr., Time taken by the larger pipe = (x − 10) hr, Part of tank filled by smaller pipe in 1 hour =, Part of tank filled by larger pipe in 1 hour =, , Page 31 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 45 :
Class X, , Chapter 4 – Quadratic Equations, , It is given that the tank can be filled in, , Maths, , hours by both the pipes, , together. Therefore,, , Time taken by the smaller pipe cannot be, , = 3.75 hours. As in this, , case, the time taken by the larger pipe will be negative, which is, logically not possible., Therefore, time taken individually by the smaller pipe and the larger, pipe will be 25 and 25 − 10 =15 hours respectively., , Question 10:, An express train takes 1 hour less than a passenger train to travel 132, km between Mysore and Bangalore (without taking into consideration, the time they stop at intermediate stations). If the average speeds of, Page 32 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 46 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , the express train is 11 km/h more than that of the passenger train,, find the average speed of the two trains., Answer:, Let the average speed of passenger train be x km/h., Average speed of express train = (x + 11) km/h, It is given that the time taken by the express train to cover 132 km is, 1 hour less than the passenger train to cover the same distance., , Speed cannot be negative., Therefore, the speed of the passenger train will be 33 km/h and thus,, the speed of the express train will be 33 + 11 = 44 km/h., , Question 11:, Sum of the areas of two squares is 468 m2. If the difference of their, perimeters is 24 m, find the sides of the two squares., , Page 33 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 47 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Answer:, Let the sides of the two squares be x m and y m. Therefore, their, perimeter will be 4x and 4y respectively and their areas will be x2 and, y2 respectively., It is given that, 4x − 4y = 24, x−y=6, x=y+6, , However, side of a square cannot be negative., Hence, the sides of the squares are 12 m and (12 + 6) m = 18 m, , Page 34 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 48 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Exercise 4.4, Question 1:, Find the nature of the roots of the following quadratic equations., If the real roots exist, find them;, (I) 2x2 −3x + 5 = 0, (II), (III) 2x2 − 6x + 3 = 0, Answer:, We know that for a quadratic equation ax2 + bx + c = 0, discriminant, is b2 − 4ac., (A) If b2 − 4ac > 0 → two distinct real roots, (B) If b2 − 4ac = 0 → two equal real roots, (C) If b2 − 4ac < 0 → no real roots, (I) 2x2 −3x + 5 = 0, Comparing this equation with ax2 + bx + c = 0, we obtain, a = 2, b = −3, c = 5, Discriminant = b2 − 4ac = (− 3)2 − 4 (2) (5) = 9 − 40, = −31, As b2 − 4ac < 0,, Therefore, no real root is possible for the given equation., (II), Comparing this equation with ax2 + bx + c = 0, we obtain, , Page 35 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 49 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Discriminant, = 48 − 48 = 0, As b2 − 4ac = 0,, Therefore, real roots exist for the given equation and they are equal to, each other., And the roots will be, , Therefore, the roots are, , and, , ., , and, , ., , (III) 2x2 − 6x + 3 = 0, Comparing this equation with ax2 + bx + c = 0, we obtain, a = 2, b = −6, c = 3, Discriminant = b2 − 4ac = (− 6)2 − 4 (2) (3), = 36 − 24 = 12, As b2 − 4ac > 0,, Therefore, distinct real roots exist for this equation as follows., , Page 36 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 50 :
Class X, , Chapter 4 – Quadratic Equations, , Therefore, the roots are, , or, , Maths, , ., , Question 2:, Find the values of k for each of the following quadratic equations, so, that they have two equal roots., (I) 2x2 + kx + 3 = 0, (II) kx (x − 2) + 6 = 0, Answer:, We know that if an equation ax2 + bx + c = 0 has two equal roots, its, discriminant, (b2 − 4ac) will be 0., (I) 2x2 + kx + 3 = 0, Comparing equation with ax2 + bx + c = 0, we obtain, a = 2, b = k, c = 3, Discriminant = b2 − 4ac = (k)2− 4(2) (3), = k2 − 24, For equal roots,, , Page 37 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 51 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Discriminant = 0, k2 − 24 = 0, k2 = 24, , (II) kx (x − 2) + 6 = 0, or kx2 − 2kx + 6 = 0, Comparing this equation with ax2 + bx + c = 0, we obtain, a = k, b = −2k, c = 6, Discriminant = b2 − 4ac = (− 2k)2 − 4 (k) (6), = 4k2 − 24k, For equal roots,, b2 − 4ac = 0, 4k2 − 24k = 0, 4k (k − 6) = 0, Either 4k = 0 or k = 6 = 0, k = 0 or k = 6, However, if k = 0, then the equation will not have the terms ‘x2’ and, ‘x’., Therefore, if this equation has two equal roots, k should be 6 only., , Question 3:, Is it possible to design a rectangular mango grove whose length is, twice its breadth, and the area is 800 m2?, If so, find its length and breadth., Answer:, Let the breadth of mango grove be l., Page 38 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 52 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Length of mango grove will be 2l., Area of mango grove = (2l) (l), = 2l2, , Comparing this equation with al2 + bl + c = 0, we obtain, a = 1 b = 0, c = 400, Discriminant = b2 − 4ac = (0)2 − 4 × (1) × (− 400) = 1600, Here, b2 − 4ac > 0, Therefore, the equation will have real roots. And hence, the desired, rectangular mango grove can be designed., , However, length cannot be negative., Therefore, breadth of mango grove = 20 m, Length of mango grove = 2 × 20 = 40 m, , Question 4:, Is the following situation possible? If so, determine their present ages., The sum of the ages of two friends is 20 years. Four years ago, the, product of their ages in years was 48., Answer:, Let the age of one friend be x years., Age of the other friend will be (20 − x) years., 4 years ago, age of 1st friend = (x − 4) years, Page 39 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 53 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , And, age of 2nd friend = (20 − x − 4), = (16 − x) years, Given that,, (x − 4) (16 − x) = 48, 16x − 64 − x2 + 4x = 48, − x2 + 20x − 112 = 0, x2 − 20x + 112 = 0, Comparing this equation with ax2 + bx + c = 0, we obtain, a = 1, b = −20, c = 112, Discriminant = b2 − 4ac = (− 20)2 − 4 (1) (112), = 400 − 448 = −48, As b2 − 4ac < 0,, Therefore, no real root is possible for this equation and hence, this, situation is not possible., , Question 5:, Is it possible to design a rectangular park of perimeter 80 and area, 400 m2? If so find its length and breadth., Answer:, Let the length and breadth of the park be l and b., Perimeter = 2 (l + b) = 80, l + b = 40, Or, b = 40 − l, Area = l × b = l (40 − l) = 40l − l2, 40l − l2 = 400, l2 − 40l + 400 = 0, Page 40 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)

Page 54 :
Class X, , Chapter 4 – Quadratic Equations, , Maths, , Comparing this equation with, al2 + bl + c = 0, we obtain, a = 1, b = −40, c = 400, Discriminate = b2 − 4ac = (− 40)2 −4 (1) (400), = 1600 − 1600 = 0, As b2 − 4ac = 0,, Therefore, this equation has equal real roots. And hence, this situation, is possible., Root of this equation,, , Therefore, length of park, l = 20 m, And breadth of park, b = 40 − l = 40 − 20 = 20 m, , Page 41 of 41, Website: www.vidhyarjan.com, , Email: [email protected], , Mobile: 9999 249717, , Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051, (One Km from ‘Welcome Metro Station)