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Chapter 1: Quadratic Equation Chapter-1: QUADRATIC EQUATION Level – I (QUESTIONS EXACTLY COPIED FROM NCERT) 1. Out of the following quadratic equations are: (I) ( x + 1) = 2 ( x – 3) (II) ( x – 2 )( x + 1) = ( x –1)( x + 3) 2 (III) x 2 + 3 x + 1 = ( x – 2 ) 2 3 ( (IV) ( x + 2 ) = 2 x x 2 –1 ) (V) x3 – 4 x 2 – x + 1 = ( x – 2 ) 3 (a) I, III, V (b) I, II, V (c) I, V (d) I, II, III 2. 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. Represent the following situations in the form of quadratic equations. If ’u is the speed of the train. (a) u 2 − 8u − 1280 = 0 (b) u 2 + 32u − 273 = 0 (c) u 2 + 8u − 1280 = 0 (d) u 2 − 8u + 1280 = 0 3. The two consecutive positive integers, sum of whose squares is 365 is (a) 14, 15 (b) 13, 14 (c) 16, 17 (d) 17, 18 4. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, then the other two sides, is (a) 5 cm & 7 cm (b) 5 cm &12 cm (c) 12 cm & 13 cm (d) 5 cm & 13 cm 5. The roots of the equation, 3x 2 − 2 6 x + 2 = 0 , solved by method of completing the square is 2 2 2 2 3 3 3 3 (a) , (b) − , (c) , (d) − , 3 3 3 3 2 2 2 2 1 6. The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is . Then 3 his present age, is (a) 5 years (b) 6 year (c) 7 years (d) 8 years 7. A pole has to be erected at a point on the boundary of a circular park of diameter 13 meters in such a way that the differences of its distances from two diametrically opposite fixed gates A and B on the boundary are 7 meters. It what distances the gate A from the two gates should the pole be erected? (a) 5 m (b) 6 m (c) 8 m (d) 12 m 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. Then the speed of the train, is (a) 10 km/hr (b) 20 km/hr (c) 30 km/hr (d) 40 km/hr 3 9. Two water taps together can fill a tank 9 in hours. The tap of larger diameter takes 10 hours less than 8 the smaller one to fill the tank separately. Then the time in which larger diameter tap can separately fill the tank, is (a) 25 hour (b) 15 hours (c) 10 hours (d) 20 hours Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Quadratic Equation 10. Sum of the areas of two squares is 468 m2. If the difference of their perimeters is 24 m, then the sides of the two squares, is (a) 12 m, 14 m (b) 14 m, 16 m (c) 16 m, 18 m (d) 18 m, 12 m 11. The values of k for which the quadratic equation kx ( x – 2 ) + 6 = 0, have two equal roots is (a) 0 (b) 6 (c) 0 & 6 (d) 4 1 12. The quadratic 3x 2 – 2 x + = 0 , the nature of the roots is 3 (a) unequal real roots (b) equal real roots (c) imaginary roots (d) irrational roots 13. A rectangular park is to be designed whose breadth is 3 m less than its length. Its area is to be 4 square meters more than the area of a park that has already been made in the shape of an isosceles triangle with its base as the breadth of the rectangular park and of altitude 12m. Then its length is (a) 4 m (b) 5 m (c) 6 m (d) 7 m 14. A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Then the speed of the stream is (a) –54 km/r (b) 6 km/h (c) –54 km/h & 6 km/h (d) 60 km/hr 1 15. The roots of x − = 3,x  0 is x −3  13 3  13 3  13 3  13 (a) (b) (c) (d) 3 6 2 3 1 1 11 16. The roots of − = ,x  −4,7 is x + 4 x − 7 30 (a) 2,1 (b) −1,2 (c) −2, −1 (d) 1, −2 17. The nature of the roots of the quadratic equations 3x 2 − 4 3 x + 4 = 0 (a) Irrational roots (b) Equal roots (c) Imaginary roots (d) Conjugate roots 18. The length of the rectangular mango grove whose length is twice its breadth and the area is 800 m2 ? (a) 20 m (b) 40 m (c) 60 m (d) 80 m 19. The breadth of a rectangular park of perimeter 80 m and area 400 m2, is (a) 10 m (b) 20 m (c) 30 m (d) 40 m 20. The roots of 5x 2 − 6 x − 2 = 0 solved by the method of completing the square. 3  38 −3  38 3  19 −3  19 (a) (b) (c) (d) 6 5 5 5 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Quadratic Equation LEVEL – II (QUESTIONS BASED ON NCERT) 1. Which of the following is a quadratic equation? (a) x + x −1 = 1 (b) x 4 + x 2 + 7 = 0 (c) x2 + x − 7 = 0 (d) 32 x − 10.3x + 9 = 0 2. If one root of 3x 2 + 11x + k = 0 be reciprocal of the other, then the value of k is. (a) 1 (b) 2 (c) 3 (d) 4 α 3. If α ,β are the roots of 2 x 2 + 3x − 1 = 0, then the equation whose roots are and β is. β α (a) x 2 − 3x + 7 = 0 (b) 2 x 2 + 13x + 2 = 0 (c) 2 x 2 − 13x + 2 = 0 (d) None of these 4. Two numbers whose sum is 10 and product is 20, then the quadratic equation whose roots are reciprocal of these numbers is. (a) x2 − 10 x + 2 = 0 (b) 10 x 2 − x + 5 = 0 (c) 20 x 2 − 10 x + 1 = 0 (d) None of these 5. The value of k, for which the equation 2 x − kx + x + 8 = 0 , will have real and equal roots. 2 (a) 9, − 7 (b) − 7 (c) 9, 7 (d) –9 and –7 6. The roots of x ( x 2 + 8 x + 16 ) ( 4 − x ) = 0 are. (a) 0 (b) 0, 4 (c) 0, 4, –4 (d) 0, 4, –4, –4 7. If the roots of ax + bx + c = 0 are α and β, then equation with roots α − 1 and β −1 is. 2 (a) ax 2 + ( b + 2a ) x + 2a + b + c = 0 (b) ax 2 + ( b − 2a ) x + a + b + c = 0 (c) ax 2 + ( b + 2a ) x + a − b + c = 0 (d) ax 2 + ( b − 2a ) x + a − b + c = 0 8. If 2 + 3 is a root of x + px + q = 0 , where p and q are rational numbers then. 2 (a) p = 2, q = 2 (b) p = 4, q = 1 (c) p = 1, q = −2 (d) None 9. If the difference of roots of x 2 − 3 px + 4 = 0 is 2 5 , then p is. (a) 4 (b) +3,−2 (c) 2 (d) None of these   10. If α , β are roots of the quadratic ax2 + bx + c = 0 . An equation whose roots are and is.   (a) cx 2 + bx + a = 0 (b) ax2 − bx + c = 0 2 2 ( ) (c) acx − b − 2ac x + ac = 0 (d) None of these 11. In the quadratic equation x ( x − 1) − ( m + 1) = x , the roots are equal, when. ( x − 1)( m − 1) m (a) m = 1/2 (b) m = –1/2 (c) m = 1 (d) m = –1 12. The value of ‘m’ for which 9 x + mx + 1 = 0 , has both roots equal. 2 (a) 3 or 5 (b) –3or 7 (c) +6 or –6 (d) 2 or 8 13. The sum of roots of quadratic equation is 6. Their product is − 16. The quadratic equation is. (a) x 2 + 6 x + 16 = 0 (b) x 2 − 16 + 6 = 0 (c) x 2 − 6 x − 16 = 0 (d) x 2 + 16 x − 6 = 0 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Quadratic Equation 14. The equation which has roots x = 2, 1/2 is. (a) 2 x 2 − 5x + 2 = 0 (b) 2 x 2 − 5x − 2 = 0 (c) 2 x 2 + 5x − 2 = 0 (d) None of these 15. If the equation x 2 + 2 ( K + 2 ) x + 9 K = 0 has equal roots the value of K are: (a) 1,4 (b) –1, 4 (c) 1, –4 (d) –1, 4 1 16. The roots of x + = 3, x  0 are: x 1 3 3 5 − 5 3+ 5 3− 5 (a) 3, (b) ,− (c) , (d) , 3 2 2 2 2 2 2 17. Sum of areas of two squares is 117 m2. If the difference of their perimeters is 12 m. What is the length of the smaller square? (a) 4 m (b) 5 m (c) 6 m (d) 7 m 18. For what value of p, the expression 2 x2 + 2 x + p be factorized into real linear factors: 1 1 1 1 (a) p  (b) p  (c) p  (d) p  2 2 2 2 19. If sum of roots of the equation kx2 + 2 x + 3k = 0 is equal to their product, then the value of k is. 1 −1 2 −2 (a) (b) (c) (d) 3 3 3 3 x + 4 x − 4 10 20. The roots of + = , are. x−4 x+4 3 (a) 4 (b) 6 (c) 8 (d) 2  3 21. If α , β the roots of the equation x − 8 x + p = 0 and α + β = 40 , then p is equal to. 2 2 2 (a) 8 (b) 10 (c) 12 (d) 14 22. If the roots of equation ax2 + bx + c = 0 are negative reciprocal of each other, the a + c equal to. b (a) b (b) –b (c) (d) 0 a+c 23. If the ratio of roots of 12 x 2 + mx + 5 = 0 is 3 : 2 then the positive value of m is. 5 5 12 (a) 5 10 (b) 10 (c) (d) 2 12 5 24. The roots of the equation 4 x − 3.2 x+2 + 32 = 0 would include. (a) 1, 2 and 3 (b) 1 and 2 (c) 1 and 3 (d) 2 and 3 ( ) ( ) x x 25. The solution of 2 +1 + 2 −1 − 6 = 0 . (a) x = 0 only (b) x = 1 only (c) x = 2 only (d) x = 2 and x = –2 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Quadratic Equation 26. The value of p, for which x 2 − px + 3 = 0 , will have one root common with x 2 − 3x + 2 = 0 (a) –4 (b) 4 (c) 3 (d) –3 x−3 2x − 1 2x + 5 27. The value of + − : 2 2 2 x − x−6 2x + 5x − 3 x + 5x + 6 (a) 0 (b) l (c) –l (d) 2 x 2 − 7 x + 12 x 2 − 13x + 42 x 2 + 7 x + 10 28.  2  2 =? x 2 − 2 x − 35 x − x−6 x − 10 x + 24 x −1 x−6 x−7 (a) (b) (c) (d) 1 x −3 x+7 x + 12 5 x − 11 A B 29. The function was obtained by adding the two fraction and the values of A 2x + x − 6 2 x+2 2x − 3 and B must be: (a) A = –1, B = 3 (b) A = –11, B = 3 (c) A = 5, B = –11 (d) A = 3, B = –1 30. If x 2 − 3x + 2 = 0 and 2 x 2 + px + q = 0 have both roots common, then. (a) p + q = 0 and pq = 1 (b) p = –6, p = 4 (c) p + q = 0 and pq = 1 (d) None of these 31. If α , β are roots of x 2 + x + 1 , then α + β = ? 4 4 (a) 0 (b) 1 (c) –1 (d) 2 32. The solution set of the equation 5x+1 + 52− x = 126 is. (a) {1, 2} (b) { − 1, 2} (c) {1, − 2} (d) { − 1, − 2} 1  33. The equality = is satisfied for: x −1 x − 2 (a) non real values of x (b) only x = 1 (c) only x = 2 (d) only x = 0 34. Sum of the areas of two squares is 468 m2. If the difference of their perimeters is 24 m, the sides of the two squares are: (a) 18 m, 12 m (b) 16 m, 12 m (c) 18 m, 16 m (d) 14 m, 10 m 35. In the equation 2 x 2 − hx + 2k = 0 , if the sum of roots is 4 and product of roots is ‘–3’ then h and k have the values respectively. (a) 8, –6 (b) 4, –3 (c) –3, 4 (d) 8, –3 36. If x − 1 − x + 1 + 1 = 0 , then 4x is equal to. 1 (a) 0 (b) 4 −1 (c) 1 (d) 5 4 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Quadratic Equation 15 2 37. The roots of − = 1 , are. x −4 x−2 2 (a) +2, –2 (b) –5, 3 (c) 2, 2 (d) –3, 5 38. 𝑥 2 − 2𝑥 − 1, factorizes as (a) ( x − 2 ) ( x + 1) (b) ( x + 2 )( x − 1) ( )( (c) x − 2 − 1 x + 2 − 1 ) (d) Cannot be factorized 39. The value of x, for which − x 2 + 3x − 2  0 is. (a) 1  x  2 (b) x  1or  2 (c) x  −1or x  2 (d) All values of x 1 1 1 1 40. The quadratic equation + = + has roots m and –m then: x x+b m m+b (a) b2 = m2 (b) b2 = 2m2 (c) 2b2 = m2 (d) 2b2 = m 41. Find the sum of real roots of the equation x2 + | x | –6 = 0 : (a) 0 (b) – l (c) – 4 (d) 1 42. Quadratic equation x2 + bx + c = 0 has a root 3 –2 3 , find the value of c: (a) 3 (b) 3 3 (c) –3 3 (d) –3 43. If the quadratic equation ax2 + bx + c = 0 , have roots equal but of opposite sign then. c c c c (a) b  0,  0 (b) b  0,  0 (c) b = 0,  0 (d) b = 0,  0 a a a a 44. If the roots of the quadratic equation ( a 2 – b 2 ) x 2 + 2b ( c – a ) x + c ( b 2 – c 2 ) = 0 are equal then: 2ac (a) 2b = a + c (b) b 2 = ac (c) b = (d) b = ac a+c 45. Consider graphs given below. (A) (B) (C) (D) Which of the above graph is correctly matched? (1) y = x (2) y = − x (3) x = y (4) x = − y 2 2 2 2 (a) 1 → A,2 → B,3 → C,4 → D (b) 1 → A,2 → C,3 → B,4 → D (c) 1 → A,2 → B,3 → D,4 → C (d) 1 → B,2 → D,3 → A,4 → B Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Quadratic Equation 46. The equation of the graph, given on the right side is: (a) y 2 = 4 x (b) 2 x = y 2 (c) y = − x (d) y = − x 2 2 2 47. The equation of graph given on the right is: (a) y = x (b) x = y 2 2 (c) y = − x 2 (d) y = x 2 − 4 b2 48. If  1 and a  0 , which of the following correct graph of y = ax 2 + bx + c. 4ac (a) (b) (c) (d) None of these b2 49. In the quadratic equation ax + bx + c = 0 if ab  0 , and 2  1 , then which of the following graph 4ac correctly represents it. (a) (b) (c) (d) 50. If is plotted as a function of x, the graph would look like. (a) (b) (c) (d) Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Quadratic Equation ANSWERS LEVEL – I 1. c 2. a 3. b 4. b 5. a 6.c 7. d 8.d 9.b 10. d 11.b 12.b 13. d 14.b 15.c 16.a 17.b 18.b 19.b 20. c LEVEL – II 1. c 2. c 3. b 4. c 5. a 6. d 7. a 8. b 9. c 10. c 11. b 12. c 13. c 14. a 15. a 16. d 17. c 18. c 19. d 20. c 21. c 22. d 23. a 24. d 25. d 26. b 27. b 28. d 29. d 30. b 31. c 32. b 33. d 34. a 35. d 36. d 37. b 38. c 39. a 40. b 41. a 42. d 43. d 44. d 45. b 46. b 47. d 48. b 49. d 50. c Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458