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EXCELLENCE IN EDUCATION, HOME ASSIGNMENT PRACTICE PAPER, QUADRATIC EQUATION, SUBMISSION DUE DATE- 15th MARCH 2022., , 1.Find the roots of the equation x2 – 3x – m (m + 3) = 0, where m is a constant., 2. If 1 is a root of the equations ay2 + ay + 3 = 0 and y2 + y + b = 0, then find the value of ab., 3. If x = – 1/2 , is a solution of the quadratic equation 3x2 + 2kx – 3 = 0, find the value of k, 4. Find the value of p so that the quadratic equation px(x – 3) + 9 = 0 has two equal roots., (2011D, 2014OD), 5. Find the value of p so that the quadratic equation px(x – 3) + 9 = 0 has two equal roots., (2011D, 2014OD), 6. Find the roots of 4x2 + 3x + 5 = 0 by the method of completing the squares. (2011D), , 7. 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) 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. What is the speed of the train?, 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., 8., , 9. The 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., 10. 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, 11. Is it possible to design a rectangular park of perimeter 80 and area 400, sq.m.? If so find its length and breadth.

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12. Find the discriminant of the equation 3x2– 2x +1/3= 0 and hence find the, nature of its roots. Find them, if they are real., Solution:, 13. In a flight of 600 km, an aircraft was slowed due to bad weather. Its, average speed for the trip was reduced by 200 km/hr and the time of flight, increased by 30 minutes. Find the original duration of the flight., 14. If x = 3 is one root of the quadratic equation x2 – 2kx – 6 = 0, then find, the value of k., 15. Find the value of p, for which one root of the quadratic equation px 2 – 14x, + 8 = 0 is 6 times the other., 16. Solve for x: [1/(x + 1)] + [3/(5x + 1)] = 5/(x + 4); x ≠ -1, -⅕, -4, 17. 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, find the value of k., 18. Write all the values of p for which the quadratic equation x2 + px + 16 = 0, has equal roots. Find the roots of the equation so obtained., 19.A plane left 30 minutes late than its scheduled time and in order to reach, the destination 1500 km away in time, it had to increase its speed by 100, km/hr from the usual speed. Find its usual speed., 20.If ad ≠ bc, then prove that the equation (a2 + b2)x2 + 2(ac + bd)x + (c2 + d2) =, 0 has no real roots., 21.A train travels at a certain average speed for a distance of 63 km and then, travels at a distance of 72 km at an average speed of 6 km/hr more than its, original speed. If it takes 3 hours to complete the total journey, what is the, original average speed?, 22.Solve for x: [1/(x – 1)(x – 2)] + [1/(x – 2)(x – 3)] = ⅔; x ≠ 1, 2, 3, 23.If the quadratic equation px2 – 2√5 px + 15 = 0 has two equal roots, then, find the value of p., 24.Solve the following quadratic equation for x:, 4x2 + 4bx – (a2 – b2) = 0

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25.Solve for x: √3 x2 – 2√2 x – 2√3 = 0, 26.Solve the quadratic equation 2x2 + ax – a2 = 0 for x., , CONCEPT, , Graphical Representation of a Quadratic Equation, The graph of a quadratic polynomial is a parabola. The roots of a quadratic, equation are the points where the parabola cuts the x-axis i.e. the points, where the value of the quadratic polynomial is zero., Now, the graph of x2+5x+6=0 is:, , In the above figure, -2 and -3 are the roots of the quadratic equation, x2+5x+6=0., For a quadratic polynomial ax2+bx+c,

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If a>0, the parabola opens upwards., If a<0, the parabola opens downwards., If a = 0, the polynomial will become a first-degree polynomial and its graph is, linear., The discriminant, D=b2−4ac, , Nature of graph for different values of D., , If D>0, the parabola cuts the x-axis at exactly two distinct points. The roots are, distinct. This case is shown in the above figure in a, where the quadratic, polynomial cuts the x-axis at two distinct points., If D=0, the parabola just touches the x-axis at one point and the rest of the, parabola lies above or below the x-axis. In this case, the roots are equal., This case is shown in the above figure in b, where the quadratic polynomial, touches the x-axis at only one point., If D<0, the parabola lies entirely above or below the x-axis and there is no, point of contact with the x-axis. In this case, there are no real roots., This case is shown in the above figure in c, where the quadratic, polynomial neither cuts nor touches the x-axis., , 27. Solve the following quadratic equation for x: 4x2– 4a2x + (a4 – b4) = 0. (2015D), 28. Solve the following quadratic equation for x: x2 – 2ax – (4b2 – a2) = 0) (2015OD), 29. solve for x : √(6x + 7) - (2x - 7) = 0.

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30. For what values of k, the roots of the quadratic equation (k + 4)x2 + (k + 1)x + 1 = 0 are, equal? (2013D), 31. For what value of k, are the roots of the quadratic equation: (k – 12)x2 + 2(k – 12)x + 2 = 0, equal? (2013OD), 32., 32. Find that non-zero value of k, for which the quadratic equation kx2 + 1 – 2(k – 1)x + x2 = 0, has equal roots. Hence find the roots of the equation. (2015D), 33. Find that value of p for which the quadratic equation (p + 1)x2 – 6(p + 1)x + 3 (p + 9) = 0, p ≠ 1 had equal roots. (2015D, 34. Solve for x: √3x2-2√2x-2√3=0, 35. If the roots of the quadratic equation (a – b)x2 + (b – c)x + (c – a) = 0 are equal, prove that, 2a = b + c. (2016OD), 36. Solve for x : (x + 1)/(x - 1) + (x - 2)(x + 2) = 4 - (2x + 3)/(x - 2); x ≠ 1, -2, 2, 37. Solve the following quadratic equation for x : x2 + [a/(a + b) + (a + b)/a]x + 1 = 0., 38. Three consecutive natural numbers are such that the square of the middle number, , exceeds the difference of the squares of the other two by 60. Find the numbers., (2016OD), 39. If the sum of two natural numbers is 8 and their product is 15, find the numbers. (2012OD), , 40. Solve the following for x:, 41. A shopkeeper buys some books for 80. If he had bought 4 more books for the same amount,, each book would have cost ₹1 less. Find the number of books he bought. (2012D), 42. The diagonal of a rectangular field is 16 metres more than the shorter side. If the longer side, is 14 metres more than the shorter side, then find the lengths of the sides of the field. (2015OD), 43. The perimeter of a right triangle is 60 cm. Its hypotenuse is 25 cm. Find the area of the, triangle. (2016D), 44. The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator,, the fraction is decreased by 1/15. Find the fraction. (20120D), 45. The difference of two natural numbers is 5 and the difference of their reciprocals is 1/10., Find the numbers. (2014D)

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46. A rectangular park is to be designed whose breadth is 3 m less than its length. Its, , area is to be 4 square metres 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 12 m. Find the length and breadth of the rectangular park. (2016OD), 47. A train travels 180 km at a uniform speed. If the speed had been 9 km/hour more, it would, have taken 1 hour less for the same journey. Find the speed of the train. (2011OD), 48. In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed is, reduced by 100 km/h and time increased by 30 minutes. Find the original duration of the flight., 49. While boarding an aeroplane, a passenger got hurt. The pilot, showing promptness and, concern, made arrangements to hospitalise the injured and so the plane started late by 30, minutes. To reach the destination, 1500 km away in time, the pilot increased the speed by 100, km/hour. Find the original speed/hour of the plane. (2013OD), 50. A truck covers a distance of 150 km at a certain average speed and then covers, , another 200 km at an average speed which is 20 km per hour more than the first speed., If the truck covers the total distance in 5 hours, find the first speed of the truck., 51. Two pipes running together can fill a tank in 11 1/9 minutes. If one pipe takes 5 minutes, more than the other to fill the tank separately, find the time in which each pipe would fill the tank, separately. (2016OD), 52. A motorboat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream, than to return downstream to the same spot., Find the speed of the stream. (2014OD), 53. To fill a swimming pool two pipes are to be used. If the pipe of larger diameter is used for 4, hours and the pipe of smaller diameter for 9 hours, only half the pool can be filled. Find, how, long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes, 10 hours more than the pipe of larger diameter to fill the pool. (2015D), 54. The time taken by a person to cover 150 km was 2 hours more than the time taken in the, return journey. If he returned at a speed of 10 km/hour more than the speed while going, find the, speed per hour in each direction. (2016D), ********************************** GOOD LUCK ********************************, PREPARED BY:, Er. AMRITESH KUMAR ,MIT(2K13), CONTACT – 7488313116, [email protected]