CBSE Class 10 Polynomials Sure Shot Questions

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PRACTICE QUESTIONS

CLASS X : CHAPTER - 2

POLYNOMIALS

1. If p(x) = 3x³ – 2x² + 6x – 5, find p(2).

2. Draw the graph of the polynomial f(x) = x² – 2x – 8.

3. Draw the graph of the polynomial f(x) = 3 – 2x – x² .

4. Draw the graph of the polynomial f(x) = –3x² + 2x – 1.

5. Draw the graph of the polynomial f(x) = x² – 6x + 9 .

6. Draw the graph of the polynomial f(x) = x³.

7. Draw the graph of the polynomial f(x) = x³ – 4x.

8. Draw the graph of the polynomial f(x) = x³ – 2x².

9. Draw the graph of the polynomial f(x) = –4x² + 4x – 1.

10. Draw the graph of the polynomial f(x) = 2x² – 4x + 5.

11. Find the quadratic polynomial whose zeroes are 2 +√3 and 2 –√3 .

12. Find the quadratic polynomial whose zeroes are 3-√3/5 and 3 +√3 /5

13. Find a quadratic polynomial whose sum and product of zeroes are √2 and 3 respectively.

14. Find the zeroes of the polynomial mx² + (m + n)x + n.
15. If m and n are zeroes of the polynomial 3x² + 11x – 4, find the value of m + n n m
16. If a and b are zeroes of the polynomial x² – x – 6, then find a quadratic polynomial whose zeroes are (3a + 2b) and (2a + 3b).
17. If p and q are zeroes of the polynomial t² – 4t + 3, show that 1 +1 - 2 pq + 14 = 0 . p q 3

18. If (x – 6) is a factor of x3 + ax² + bx – b = 0 and a – b = 7, find the values of a and b.

19. If 2 and – 3 are the zeroes of the polynomial x² + (a + 1)x + b, then find the value of a and b.

20. Obtain all zeroes of polynomial f(x) = 2x⁴ + x³ – 14x²– 19x – 6 if two of its zeroes are –2 and –1.

21. Find all the zeroes of the polynomial 2x³ -4x - x²+ 2 , if two of its zeroes are √2 and - √2 .

22. Find all the zeroes of the polynomial x⁴ -3x³ + 6 x - 4 , if two of its zeroes are √2 and -√ 2 .

23. Find all the zeroes of the polynomial 2x⁴ -9x³+ 5x2 + 3x -1 , if two of its zeroes are 2 + √3 and 2 - √3 .

24. Find all the zeroes of the polynomial 2x⁴ +7 x3 -19x² - 14x + 30 , if two of its zeroes are √2 and - √ .

25. Find all the zeroes of the polynomial x³ +3x² - 2x - 6 , if two of its zeroes are √2 and - √2 .

26. Find all the zeroes of the polynomial 2x³ -x² - 5x - 2 , if two of its zeroes are –1 and 2.

27. Find all the zeroes of the polynomial x³ +3x² - 5x -15 , if two of its zeroes are √5 and -√ 5 .

28. Find all the zeroes of the polynomial x³ -4x² - 3x + 12 , if two of its zeroes are √3 and - √3 .

29. Find all the zeroes of the polynomial 2x3 +x²- 6x - 3 , if two of its zeroes are √3 and - √3 .

30. Find all the zeroes of the polynomial x⁴ + x³ - 34x² - 4 x + 120 , if two of its zeroes are 2 and –2.

31. If the polynomial 6x⁴ + 8x³ + 17x² + 21x + 7 is divided by another polynomial 3x² + 4x + 1, the remainder comes out to be (ax + b), find a and b.

32. If the polynomial x⁴ + 2x³ + 8x² + 12x + 18 is divided by another polynomial x² + 5, the remainder comes out to be px + q, find the value of p and q.

33. Find the zeroes of a polynomial opposite in sign. x³ - 5x² -16x + 80 , if its two zeroes are equal in magnitude but

34. If two zeroes of the polynomial  x⁴ + 3x³ - 20x² - 6x + 36 are √2 and - √2 of the polynomial. find the other zeroes

35. On dividing x³ – 3x² + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4 respectively. Find g(x).

36. If the product of zeroes of the polynomial ax² – 6x – 6 is 4, find the value of ‘a’.

37. If one zero of the polynomial (a² + 9)x² + 13x + 6a is reciprocal of the other. Find the value of a.

38. Write a quadratic polynomial, sum of whose zeroes is 2√3 and their product is 2.

39. Find a polynomial whose zeroes are 2 and –3.

40. Find the zeroes of the quadratic polynomial the zeroes and the coefficients. x² + 5x + 6 and verify the relationship between

41. Find the sum and product of zeroes of p(x) = 2(x2 – 3) + x.

42. Find a quadratic polynomial, the sum of whose zeroes is 4 and one zero is 5.

43. Find the zeroes of the polynomial p(x) = √2x²  - 3x - 2√2 .

44. If a and b are the zeroes of 2x2 + 5(x – 2), then find the product of a and b .

45. Find a quadratic polynomial, the sum and product of whose zeroes are 5 and 3 respectively.

46. Find the zeroes of the quadratic polynomial f(x) = abx² + (b² – ac)x – bc and verify the relationship between the zeroes and its coefficients.

47. Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

      (i) 4x² – 3x – 1

      (ii) 3x² + 4x – 4 (iii) 5t² + 12t + 7 (iv) t³ – 2t² – 15t

      (v) 2x² + 7/2 x + 3/4

      (vi) 4x² + 5 √2 x – 3

      (vii) 2s2 – (1 + 2√2 )s +√2

     (viii) v² + 4√3 v – 15

     (ix) y2 + 3/2  √5 y – 5

     (x) 7y2 – 11/3y – 2/3

48. Find the zeroes of the quadratic polynomial 6x² - 7x - 3 and verify the relationship between the zeroes and the coefficients.

49. Find the zeroes of the polynomial x² + 1/6x – 2, and verify the relation between the coefficients and the zeroes of the polynomial.

51. Find the zeroes of the quadratic polynomial x² + 5 x + 6  and verify the relationship between the zeroes and the coefficients.
52.  
53. Find a quadratic polynomial, the sum and product of whose zeroes are √2 and - 3/2 ,respectively. Also find its zeroes.

52. If one zero of the quadratic polynomial x² + 3x + k is 2, then find the value of k

53. Given that two of the zeroes of the cubic polynomial ax³ + bx² + cx + d are 0, find the third zero.

54. Given that one of the zeroes of the cubic polynomial ax³ + bx² + cx + d is zero, then find the product of the other two zeroes.

55. If one of the zeroes of the cubic polynomial x³ + ax² + bx + c is –1, then the product of the other two zeroes

Answer the Questions from 28 to 32 and justify:

56. Can x²  – 1 be the quotient on division of x⁶ + 2x³ + x – 1 by a polynomial in x of degree 5?

57. What will the quotient and remainder be on division of ax²  + bx + c by px³ + qx² + rx + s, p ¹ 0?

58. If on division of a polynomial p (x) by a polynomial g (x), the degree of quotient is zero, what is the relation between the degrees of p (x) and g (x)?

59. If on division of a non-zero polynomial p (x) by a polynomial g (x), the remainder is zero, what is the relation between the degrees of p (x) and g (x)?

60. Can the quadratic polynomial x2 + kx + k have equal zeroes for some odd integer k > 1?

61. If one of the zeroes of the quadratic polynomial (k–1)x²  + k x + 1 is –3, then the value of k

62. If the zeroes of the quadratic polynomial x²  + (a + 1) x + b are 2 and –3, then find the value of a and b.
63. If a and b are zeroes of the quadratic polynomial x²  – (k + 6)x + 2(2k – 1). Find the value of k if α + β = 1/2 ab .

64. Obtain all the zeroes of 3x⁴ +6x³ - 2x² -10x + 5 , if two of its zeroes are √5/3 and - √5/3 .

65. Obtain all the zeroes of x4 -7 x3 + 17 x2 - 17x + 6 , if two of its zeroes are 3 and 1.

66. Obtain all the zeroes of x4 -7 x2 + 12 , if two of its zeroes are 3 and - 3 .

67. Two zeroes of the cubic polynomial ax3 + 3x² – bx – 6 are – 1 and – 2. Find the 3rd zero and value of a and b.

68. α , β and γ are the zeroes of cubic polynomial x³ + px² + qx + 2 such that a . b + 1 = 0. Find the value of 2p + q + 5.

69. Find the number of zeroes in each of the following:

70. If the remainder on division of x³ + 2x² + kx +3 by x – 3 is 21, find the quotient and the value of k. Hence, find the zeroes of the cubic polynomial x³ + 2x² + kx –18.

71. Find the zeroes of the polynomial f(x) = x³ – 5x² – 16x + 80, if its two zeroes are equal in magnitude but opposite in sign.

72. Find the zeroes of the polynomial f(x) = x³ – 5x² – 2x + 24, if it is given that the product of two zeroes is 12.

73. Find the zeroes of the polynomial f(x) = x³ – px² + qx – r, if it is given that the sum of two zeroes is zero.

74. If the zeroes of the polynomial x³ – 3x² + x + 1 are a – b, a, a + b, find a and b.

75. If the zeroes of the polynomial 2x³ – 15x² + 37x – 30 are a – b, a, a + b, find all the zeroes.

76. If the zeroes of the polynomial x³ – 12x2 + 39x – 28 are a – b, a, a + b, find all the zeroes.

77. If the polynomial x⁴ – 6x³ + 16x2 – 25x + 10 is divided by another polynomial x2 – 2x + k, the remainder comes out to be x + a, find k and a.

78. If the polynomial 6x4 + 8x³ – 5x2 + ax + b is exactly divisible by the polynomial 2x2 – 5, then find the values of a and b.

79. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively.

80. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 3, –1, –3 respectively.

81. Find a cubic polynomial whose zeroes are 3, 1/2 and –1.

82. Find a cubic polynomial whose zeroes are –2, –3 and –1.

83. Find a cubic polynomial whose zeroes are 3, 5 and –2.

84. Verify that 5, –2 and 1/3 are the zeroes of the cubic polynomial p(x) = 3x³ – 10x² – 27x + 10 and verify the relation between its zeroes and coefficients.

85. Verify that 3, –2 and 1 are the zeroes of the cubic polynomial p(x) = x³ – 2x² – 5x + 6 and verify the relation between its zeroes and coefficients.

86. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:

      (i) 2x³ + x² – 5x + 2;     1/2 , 1, – 2 (ii) x³ – 4x² + 5x – 2; 2, 1, 1

87. Find the quotient and remainder when 4x3 + 2x2 + 5x – 6 is divided by 2x2 + 3x + 1.

88. On dividing x⁴ – 5x + 6 by a polynomial g(x), the quotient and remainder were –x² – 2 and –5x + 10 respectively. Find g(x).

89. Given that √2 is a zero of the cubic polynomial 6x³ + √2 x² – 10x – 4√2 , find its other two zeroes.

90. Given that the zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.

91. For which values of a and b, are the zeroes of q(x) = x3 + 2x2 + a also the zeroes of the polynomial p(x) = x5 – x4 – 4x3 + 3x2 + 3x + b? Which zeroes of p(x) are not the zeroes of q(x)?

92. Find k so that x2 + 2x + k is a factor of 2x4 + x3 – 14 x2 + 5x + 6. Also find all the zeroes of the two polynomials.

93. Given that x –√5 is a factor of the cubic polynomial x3 – 3 5 x + 13x – 3 5 , find all the zeroes of the polynomial.

94. For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.

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