MATHS EDUCATION : WORKSHEET OF POLYNOMIAL CLASS-10

CLASS-10

100 IMPORTANT QUESTIONS OF POLYNOMIAL

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1. A quadratic polynomial, whose zeroes are –3 and 4, is :

(a) x² – x + 12

(b) x² + x + 12

(c) x²/2 - x/ 2 – 6

(d) 2x² + 2x – 24

2. The zeroes of the quadratic polynomial x² + 99x + 127 are :

(a) both positive

(b) both negative

(c) one positive and one negative

(d) both equal

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3. If the zeroes of the quadratic polynomial ax² + bx + c, c ≠ 0 are equal, then :

(a) c and a have opposite signs

(b) c and b have opposite signs

(c) c and a have the same sign

(d) c and b have the same sign

4. If the sum of the zeroes of the polynomial p(x) = 2x³ – 3kx² + 4x – 5 is 6, then the value of k is :

(a) 2

(b) 4

(c) –2

(d) –4

5. If the product of the zeroes of the polynomial p(x) = ax³ – 6x² + 11x – 6 is 4, then the value of a is :

(a) 1/2

(b) 5/3

(c) 3/2

(d) −1/5

6. If α and β are the zeroes of the polynomial p(x) = x² + px + q, then a polynomial having 1/α and 1/β as its zeroes is :

(a) x² + px/q + 1/q

(b) x² - px/q + 1/q

(c) x² - px/q - 1/q

(d) x² + px/q + 1/q

7. If α and β are the zeroes of p(x) = x² – 5x + b and α - β =1, then b is :

(a) 1

(b) 6

(c) 4

(d) 0

8. Given that one of the zeroes of a cubic polynomial 3x³ + 5x² – 9x + a is zero. The value of a is :

(a) −5/3

(b) –3

(c) 3

(d) 0

9. If a and b are zeroes of x² + 4x + 4 then

(a) a=b, a + b = –4

(b) a = –b, a + b = 4

(c) a = b, a + b = 4

(d) a = –b, a + b = –4

10. If the two zeroes of the cubic polynomial x³ + x² are zero each, then the third zero is :

(a) 0

(b) 1

(c) 2

(d) –1

11. If the graph of y = p(x), where p(x) is a quadratic polynomial cuts x-axis at different points, the p(x) has :

(a) no zero

(b) exactly one zero

(c) exactly two zeroes

(d) none of these

12. If α , β and γ are zeroes of a cubic polynomial ax³ + bx² + cx + d, then αβ +βγ + γα is :

(a) d/a

(b) −c/a

(c) c/a

(d) −a/b

13. If two zeroes of the polynomial x³ – 5x² – 16x + 80 are equal in magnitude but opposite in sign, then its zeroes are :

(a) 4, –4, 5

(b) 3, –3, 5

(c) 2, –2, 5

(d) 1, –1, 5

14. If α and β are the zeroes of the quadratic polynomial p(x) = x² – 5x + 4, then

1/α + 1/β - 2 α β equals:

(a) 27/4

(b) −27/4

(c) 4/27

(d) −4/27

15. If two zeroes of the polynomial x³ + x² – 9x – 9 are 3 and –3, then its third zero is :

(a) –1

(b) 2

(c) –2

(d) 3

16. If √5 and − √5 are two zeroes of the polynomial p(x) = x³ + 3x² – 5x – 15, then its other zero is :

(a) 2

(b) –3

(c) 4

(d) –4

17. If the polynomial p(x) = 3x⁴ + 5x³ – 7x² + 2x + 2 is divided by g(x) = x² + 3x + 1, then the value of quotient is :

(a) 2x² – 4x + 2

(b) 3x² – 4x + 2

(c) 3x² + 4x + 2

(d) 3x² – 4x – 2

18. On dividing 2x² + 3x – 2 by x + 2, the quotient is :

(a) 3x + 1

(b) 3x – 1

(c) 2x – 1

(d) 2x + 1

19. On dividing 3x³ + x² + 2x + 5 by 1 + 2x + x², the remainder is :

(a) 9x – 10

(b) 9x + 10

(c) 8x + 5

(d) 8x – 5

20. If zeroes of a polynomial p(x) are √2 and − √2 , then the polynomial which is a factor of p(x) is :

(a) x² + 2

(b) x² – 2

(c) x – 2

(d) x + 2

21. The graph of y = p(x), where p(x) is a polynomial in variable

x is as follows. The number of zeroes of p(x) is :

(a) 2

(b) 3.

(c) 5.

(d) 0.

22. The sum of the zeroes of the polynomial 2x² – 8x + 6 is :

(a) –3

(b) 3

(c) –4

(d) 4

23. If α and β are the zeroes of the polynomial px² – 2x + 3p and α + β = αβ, then p =

(a) 3/2

(b) 2/3

(c) 3

(d) 2

24. If α and β are the zeroes of ax² – bx + c = 0 (a ≠ 0), then α + β is :

(a) −b/a

(b) a/b

(c) b/a

(d) −c/a

25. If α and β are the zeroes of the polynomial x² + x + 1, then

1/α + 1/β is :

(a) -1

(b) 1

(c) 1/2

(d) -1/2

26. The graph of y = p(x), where p(x) is a polynomial, is shown.

The number of zeroes of the polynomial p(x) is :

(a) 3

(b) 2

(c) 1

(d) no zero

27. If α and β are the zeroes of the polynomial x² + 6x + 2,

then 1/α + 1/β is :

(a) 3

(b) –3

(c) 12

(d) –12

28. A quadratic polynomial whose zeroes are

3/ 5 and −1/2 is :

(a) 10x² + x + 3

(b) 10x² + x – 3

(c) 10x² – x + 3

(d) 10x² – x – 3

29. If 1 is a zero of the polynomial p(x) = ax² – 3(a – 1), then the value of a is :

(a) 3

(b) 2

(c) -2/3

(d) 3/2

30. If one zero of the quadratic polynomial 2x² – 3x + p is 3, then its other zero is :

(a) −3/2

(b) 3/2

(c) 1/2

(d) – 1/2

31. If α, β are the zeroes of a polynomial such that α + β = 6 and αβ = 4, then the polynomial is :

(a) x² – 6x + 4

(b) –x² + 6x – 4

(c) x² – 6x – 4

(d) none of these

32. If two zeroes of a quadratic polynomial are 5 – 3√2 and 5 + 3√2 , then the quadratic polynomial is :

(a) x² – 10x – 7

(b) x² – 10x + 6

(c) x² – 10x + 14

(d) x² – 10x + 7

33. If α and β are the zeroes of the quadratic polynomial p(x) = ax² + bx + c, then 1/α + 1/β is

(a) −b/a

(b) b)c

(c) −b/c

(d) −a/b

34. If α and β are the zeroes of the quadratic polynomial p(x) = ax² + bx + c, then 1/α + 1/β – 2αβ is :

(a) (−ab -2c²) /ac

(b) (ab+2c²) /ac

(c) (ab+b²) /2c

(d) (−c² +ab)/2ac

35. If α and β are the zeroes of the quadratic polynomial p(x) = ax² + bx + c, then α³β² + α²β³ =

(a) −bc/a³

(b) −bc² / a³

(c) −ac/b³

(d) −a²c /b³

36. If α and β are the zeroes of the quadratic polynomial x² – 5x + k such that α – β = 1, then k =

(a) 2

(b) 3

(c) 4

(d) 6

37. If α, β and γ are the zeroes of the polynomial px³ – 5x + 9 and α³ + β³ + γ³ = 27, then :

(a) p = 0

(b) p = 1

(c) p = 2

(d) p = –1

38. If α and β are the zeroes of the polynomial p(x) = 2x² + 5x + k satisfying the relation α² + β² + αβ = 21/4 , then the value of k is :

(a) 1

(b) 2

(c) 3

(d) 4

39. The value of k such that 3x² + 2kx – k – 5 has the sum of the zeores as half of their product is :

(a) 2/3

(b) 5/3

(c) 7/3

(d) 8/3

40. If the sum and the product of the zeroes of the polynomial p(x) = 4x² – 27x + 3k² are equal, then the value of k is :

(a) ±2

(b) ±3

(c) ±5

(d) ±1

41. If α, β are the zeroes of x² – 6x + k, then the value of k when 3α + 2β = 20 is :

(a) 13

(b) –14

(c). 15

(d) –16

42. If the degree of polynomial p(x) is n, then the maximum number of zeroes it can have

is :

(a) n

(b) n²

(c) n³

(d) none of these

43. The value of k, if –4 is a zero of polynomial x² – x – (2k + 2) is :

(a) 5

(b) 6

(c) 7

(d) 9

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44. Zeroes of the polynomial 4x² – 9 are :

(a) ±2 /3

(b) ± 3/2

(c) ±5/2

(d) none of these

45. Sum of the zeroes of the polynomial x³ – 4x is :

(a) 1/2

(b) – 1/2

(c) 0

(d) – 3/2

46. If p and q are the zeroes of ax² – bx + c, a ≠ 0, then the value of p + q is :

(a) b/a

(b) c/a

(c) d/a

(d) – b/a

47. If 2 and –3 are the zeroes of the quadratic polynomial x² + (a + 1)x + b, then the value of a + b is :

(a) –5

(b) 0

(c) 6

(d) –6

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

and –2x + 4 respectively, then g(x) is :

(a) x² + x – 1

(b) x² – x + 1

(c) x² + 2x – 2

(d) x² + 2x + 3

49. If p(x) = 3x⁴ + 5x³ – 7x² + 2x + 2 is divided by g(x) = x² + 3x + 1, then reminder is :

(a) 1

(b) 0

(c) –1

(d) 2

50. If 1 is a zero of the polynomial 7x – x3 – 6, then other zeroes are :

(a) 1 and 2

(b) 2 and –3

(c) –3 and 2

(d) –2 and –3

FILL IN THE BLANKS.

51. If one of the zeroes of polynomial x² + ax + 5 is –1, then find the value of a._____________.

52. If two zeroes of the polynomial are √3 and -√3, then find the polynomial _____________.

53. Find the sum of zeroes of the polynomial 2x² - x - 1.___________.

54. If x² +4x + 4 is diveded by x+2, find remiander ____________.

55. .If sum and product of zeroes of the polynomial are 5 and -5, then find the polynomial.

56. If a and b are two zeroes of a polynomial x² -x -6, then find the value of (a+b) / ab

57. If zeroes are 1 ,2 ,3 of the polynomial then find polynomial__________.

60. If the zeroes of the polynomial x² + px + q are double in value to the zeroes of 2x² – 5x – 3, then the value of p and q are respectively ___________ and ____________.

61. If α and β are the zeroes of the polynomial 2x² + 7x + 5, then find α + β + αβ _________.

62. Find the degree of 5x³ + 2 x³y⁴ + 3√2.

63. If α and β are the zeroes of the quadratic polynomial p(x) = kx² + 4x + 4 such that

α² + β² = 24, then the value of k = ____________.

64. If α and β are the zeroes of the quadratic polynomial p(x) = x² – 5x + k such that α – β = 1, then the value of k = ____________.

65. If α and β are the zeroes of the quadratic polynomial p(x) = x² – px + q, then

α² + β² = ____________.

III. VERY SHORT ANSWER QUESTIONS.

66.: Find the quadratic polynomial if its zeroes are 0, √5.

Ans .x2 – √5x

67. How many zeros does the polynomial (x – 3)2 – 4 have? Also, find its zeroes.

Solution: the roots are 1 and 5.

68. What is the zero of a polynomial?

69. What is the zero of the linear polynomial ax – b?

70. Graph of a quadratic polynomial may be a straight line. Is it true?

71. A cubic polynomial can have at most how many zeroes?

72. Find the sum of the zeroes of the polynomial 2x2 – x + 4.

73. What is the product of zeroes of the polynomial y² – 5y – 3?

74. Write the polynomial whose zeroes are – 1 and 2

75.the zeroes of the polynomial x³ – 3x² + x + 1 are a – b, a, a + b, then find the value of a and b.

Solution: a=1 ,b=√2

76. The zeroes of x2–2x –8 are:

(a) (2,-4)
(b) (4,-2)
(c) (-2,-2)
(d) (-4,-4)
Answer: (b) (4,-2)

77. What is the quadratic polynomial whose sum and the product of zeroes is √2, 1/3 respectively?

(a) 3x2-3√2x+1

(b) 3x2+3√2x+1

(c) 3x2+3√2x-1

(d) None of the above

Answer: (a) 3x2-3√2x+1•

78.

If the zeroes of the quadratic polynomial ax2+bx+c, c≠0 are equal, then

(a) c and b have opposite signs

(b) c and a have opposite signs

(c) c and b have same signs

(d) c and a have same signs

Answer: (d) c and a have same signs

79.

The degree of the polynomial, x4 – x2 +2 is
(a) 2
(b) 4
(c) 1
(d) 0
Answer: (b) 4

80.

If one of the zeroes of cubic polynomial is x3+ax2+bx+c is -1, then product of other two zeroes is:

(a) b-a-1

(b) b-a+1

(c) a-b+1

(d) a-b-1

Answer: (b) b-a+1

Explanation: Since one zero is -1, hence;

P(x) = x3+ax2+bx+c

P(-1) = (-1)3+a(-1)2+b(-1)+c

0 = -1+a-b+c

c=1-a+b

Product of zeroes, αβγ = -constant term/coefficient of x3

(-1)βγ = -c/1

c=βγ

βγ = b-a+1

81.

If p(x) is a polynomial of degree one and p(a) = 0, then a is said to be:

(a) Zero of p(x)

(b) Value of p(x)

(c) Constant of p(x)

(d) None of the above

Answer: (a) Zero of p(x)

82.

Zeroes of a polynomial can be expressed graphically. Number of zeroes of polynomial is equal to number of points where the graph of polynomial is:

(a) Intersects x-axis

(b) Intersects y-axis

(c) Intersects y-axis or x-axis

(d) None of the above

Answer: (a) Intersects x-axis

83•

A polynomial of degree n has:

(a) Only one zero

(b) At least n zeroes

(c) More than n zeroes

(d) At most n zeroes

Answer: (d) At most n zeroes

Explanation: Maximum number of zeroes of a polynomial = Degree of the polynomial

84.

The number of polynomials having zeroes as -2 and 5 is:

(a) 1

(b) 2

(c) 3

(d) More than 3

Answer: (d) More than 3

Explanation: The polynomials x2-3x-10, 2x2-6x-20, (1/2)x2-(3/2)x-5, 3x2-9x-30, have zeroes as -2 and 5.

.Zeroes of p(x) = x2-27 are:

(a) ±9√3

(b) ±3√3

(c) ±7√3

(d) None of the above

Answer: (b) ±3√3

86.

Given that two of the zeroes of the cubic polynomial ax3 + bx2 + cx + d are 0, the third zero is
(a) -b/a
(b) b/a
(c) c/a
(d) -d/a
Answer: (a) -b/a

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

(a) 10

(b) –10

(c) 5

(d) –5

Answer: (b) -10

.A quadratic polynomial, whose zeroes are –3 and 4, is

(a) x² – x + 12

(b) x² + x + 12

(c) (x²/2) – (x/2) – 6

(d) 2x² + 2x – 24

Answer: (c) (x²/2) – (x/2) – 6

89.

The zeroes of the quadratic polynomial x2 + 99x + 127 are

(a) both positive

(b) both negative

(c) one positive and one negative

(d) both equal

Answer: (b) both negative

90.

The zeroes of the quadratic polynomial x2 + 7x + 10 are

(a) -4, -3

(b) 2, 5

(c) -2, -5

(d) -2, 5

Answer: (c) -2, -5

91.

If the discriminant of a quadratic polynomial, D > 0, then the polynomial has

(a) two real and equal roots

(b) two real and unequal roots

(c) imaginary roots

(d) no roots

Answer: (b) two real and unequal roots

92.

The product of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is
(a) -b/a
(b) c/a
(c) -d/a
(d) -c/a
Answer: (c) -d/a

.If the graph of a polynomial intersects the x-axis at three points, then it contains ____ zeroes.

(a) Three

(b) Two

(c) Four

(d) More than three

Answer: (a) Three

94.. Which of the following is not the graph of quadratic polynomial?

Answer: d

95.If x3 + 1 is divided by x² + 5, then the possible degree of quotient is

(a) 0
(b) 1
(c) 2
(d) 3

Ans b

96. If x3 + 11 is divided by x² – 3, then the possible degree of remainder is

(a) 0
(b) 1
(c) 2
(d) less than 2

Ans d

97.. A quadratic polynomial whose one zero is 6 and sum of the zeroes is 0, is

(a) x2 – 6x + 2

(b) x2 – 36

(c) x2 – 6

(d) x2 – 3

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98. If x¹⁰¹ +1001 is divided by x + 1

Then reminder is:

(a) 0

(b) 1

(c) 1001

(d) 1000

99.

A quadratic polynomial whose one zero is 6 and sum of the zeroes is 0, is

(a) x2 – 6x + 2

(b) x2 – 36

(c) x2 – 6

(d) x2 – 3

100.

If p(x) = ax2 + bx + c, then is equal to

(a) 0

(b) 1

(c) sum of zeroes

(d) product of zeroes

101. Find the zeros of the polynomial x² + x - p(p-1)

102. Find zeros of the polynomial x² - 3x - m ( m + 3)

102. If 3 is a zero of the polynomial kx² + x+ k, then find the value of k.

®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️®️

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Thursday, 25 April 2024

WORKSHEET OF POLYNOMIAL CLASS-10

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