Class 10 Maths MCQs Chapter 2 Polynomials
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91.. If one of the zeroes of the cubic polynomial x3 + ax² + bx + c is -1, then the product of the
other two zeroes is
(a) b – a + 1
(b) b – a – 1
(c) a – b + 1
(d) a – b – 1
92. 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
93. The zeroes of the quadratic polynomial x² + kx + k, k? 0,
(a) cannot both be positive
(b) cannot both be negative
(c) are always unequal
(d) are always equal
94. 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
95. If one of the zeroes of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it
(a) has no linear term and the constant term is negative.
(b) has no linear term and the constant term is positive.
(c) can have a linear term but the constant term is negative.
(d) can have a linear term but the constant term is positive.
96. Which of the following is not the graph of quadratic polynomial?
97. The number of polynomials having zeroes as 4 and 7 is
(a) 2
(b) 3
(c) 4
(d) more than 4
98 A quadratic polynomial, whose zeores are -4 and -5, is
(a) x²-9x + 20
(b) x² + 9x + 20
(c) x²-9x- 20
(d) x² + 9x- 20
99. The zeroes of the quadratic polynomial x² + 1750x + 175000 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
100. The zeroes of the quadratic polynomial
1. 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
2. Given that two of the zeroes of the cubic poly-nomial ax3 + bx² + cx + d are 0, the third zero is
A. -b/a. B. B/a. C. C/a. D. - d/a.
3. If one of the zeroes of the quadratic polynomial (k – 1) x² + kx + 1 is – 3, then the value of k is
A. 4/3. B. -4/3. C. 2/3. D. -2/3
4. A quadratic polynomial, whose zeroes are -3 and 4, is
(a) x²- x + 12
(b) x² + x + 12
(c) x22−x2−6
(d) 2x² + 2x – 24
5. If the zeroes of the quadratic polynomial x2 + (a + 1) x + b are 2 and -3, then
(a) a = -7, b = -1
(b) a = 5, b = -1
(c) a = 2, b = -6
(d) a – 0, b = -6
6. The number of polynomials having zeroes as -2 and 5 is
(a) 1
(b) 2
(c) 3
(d) more than 3
7. Given that one of the zeroes of the cubic polynomial ax3 + bx² + cx + d is zero, the product of the other two zeroes is
A. -c/a. B. C/a. C.0. D. -b/a
8. If one of the zeroes of the cubic polynomial x3 + ax² + bx + c is -1, then the product of the
other two zeroes is
(a) b – a + 1
(b) b – a – 1
(c) a – b + 1
(d) a – b – 1
9. 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
10. The zeroes of the quadratic polynomial x² + kx + k, k? 0,
(a) cannot both be positive
(b) cannot both be negative
(c) are always unequal
(d) are always equal
11. 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
12. If one of the zeroes of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it
(a) has no linear term and the constant term is negative.
(a) 10
(b) -10
(c) 5
(d) -5
2. Given that two of the zeroes of the cubic poly-nomial ax3 + bx² + cx + d are 0, the third zero is
A. -b/a. B. B/a. C. C/a. D. - d/a.
3. If one of the zeroes of the quadratic polynomial (k – 1) x² + kx + 1 is – 3, then the value of k is
A. 4/3. B. -4/3. C. 2/3. D. -2/3
4. A quadratic polynomial, whose zeroes are -3 and 4, is
(a) x²- x + 12
(b) x² + x + 12
(c) x22−x2−6
(d) 2x² + 2x – 24
5. If the zeroes of the quadratic polynomial x2 + (a + 1) x + b are 2 and -3, then
(a) a = -7, b = -1
(b) a = 5, b = -1
(c) a = 2, b = -6
(d) a – 0, b = -6
6. The number of polynomials having zeroes as -2 and 5 is
(a) 1
(b) 2
(c) 3
(d) more than 3
7. Given that one of the zeroes of the cubic polynomial ax3 + bx² + cx + d is zero, the product of the other two zeroes is
A. -c/a. B. C/a. C.0. D. -b/a
8. If one of the zeroes of the cubic polynomial x3 + ax² + bx + c is -1, then the product of the
other two zeroes is
(a) b – a + 1
(b) b – a – 1
(c) a – b + 1
(d) a – b – 1
9. 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
10. The zeroes of the quadratic polynomial x² + kx + k, k? 0,
(a) cannot both be positive
(b) cannot both be negative
(c) are always unequal
(d) are always equal
11. 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
12. If one of the zeroes of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it
(a) has no linear term and the constant term is negative.
b.has linear term and constant term positive
(c) can have a linear term but the constant term is negative.
(d) can have a linear term but the constant term is positive.
(c) can have a linear term but the constant term is negative.
(d) can have a linear term but the constant term is positive.
Ans a
13. Which of the following is not the graph of quadratic polynomial?
14. The number of polynomials having zeroes as 4 and 7 is
(a) 2
(b) 3
(c) 4
(d) more than 4
15. A quadratic polynomial, whose zeores are -4 and -5, is
(a) x²-9x + 20
(b) x² + 9x + 20
(c) x²-9x- 20
(d) x² + 9x- 20
16. The zeroes of the quadratic polynomial x² + 1750x + 175000 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
17. The zeroes of the quadratic polynomial x² – 15x + 50 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
18. The zeroes of the quadratic polynomial 3x² – 48 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
19. The zeroes of the quadratic polynomial x² – 18x + 81 are
(a) both negative
(b) one positive and one negative
(c) both positive and unequal
(d) both equal and positive
20. The zeroes of the quadratic polynomial x² + px + p, p ≠ 0 are
(a) both equal
(b) both cannot be positive
(c) both unequal
(d) both cannot be negative
21. If one of the zeroes of the quadratic polynomial (p – l)x² + px + 1 is -3, then the value of p is
A.3/4. B. 4/3. C. -3/4. D. -4/3
22. If the zeroes of the quadratic polynomial Ax² + Bx + C, C # 0 are equal, then
(a) A and B have the same sign
(b) A and C have the same sign
(c) B and C have the same sign
(d) A and C have opposite signs
23. If x3 + 1 is divided by x² + 5, then the possible degree of quotient is
(a) 0
(b) 1
(c) 2
(d) 3
24. 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
25. If x4 + 3x² + 7 is divided by 3x + 5, then the possible degrees of quotient and remainder are:
(a) 3, 0
(b) 4, 1
(c) 3, 1
(d) 4, 0
26. If x5 + 2x4 + x + 6 is divided by g(x), and quotient is x² + 5x + 7, then the possible degree of g(x) is:
(a) 4
(b) 2
(c) 3
(d) 5
27. If x5 + 2x4 + x + 6 is divided by g(x) and quo-tient is x² + 5x + 7, then the possible degree of remainder is:
(a) less than 1
(b) less than 2
(c) less than 3
(d) less than 4
28. What is the number of zeroes that a linear poly-nomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
29. What is the number(s) of zeroes that a quadratic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
30. What is the number(s) of zeores that a cubic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
31. If one of the zeroes of the cubic polynomial x3 + px² + qx + r is -1, then the product of the other two zeroes is
(a) p + q + 1
(b) p-q- 1
(c) q – p + 1
(d) q – p – 1
32. If one zero of the quadratic polynomial x² + 3x + b is 2, then the value of b is
(a) 10
(b) -8
(c) 9
(d) -10
33. If 1 is one of the zeroes of the polynomial x² + x + k, then the value of k is:
(a) 2
(b) -2
(c) 4
(d) -4
.34. The zeroes of the quadratic polynomial x2 + 99x + 127are
(A) both positive
(B) both negative
(C) one positive and one negative
(D) both equal
35.If the zeroes of the quadratic polynomial x2 + bx + c , c ≠ 0are 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
.36. The number of polynomials having zeroes as –2 and 5 is
(A) 1
(B) 2
(C) 3
(D) more than 3
37. The degree of the polynomial (x + 1)(x2 – x – x4 +1) is:
(A)2
(B) 3
(C) 4
D.5
.38. If the zeroes of the quadratic polynomial x2 + (a + 1)x + b are 2 and –3, then
(A) a = –7, b = –1
(B) a = 5, b = –1
(C) a = 2, b = – 6
(D) a = 0, b = – 6
.39. Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is
(A)'c/a
(B) c/a
(C) 0 (D) 3
40.. If one of the zeroes of the cubic polynomial x3 + ax2 + bx + c is –1, then the
Product of the other two zeroes is
(A) b – a + 1
(B) b – a – 1
(C) a – b + 1
(D) a – b –1
41.. If one of the zeroes of the quadratic polynomial (k – 1)x2 + kx + 1 is –3, then the value of k is
(A) 4/3
(B) – 4/3
(C) 2/3
(D) – 2/3
42.The value of p for which the polynomial x3 + 4x2 –px + 8 is exactly divisible by (x – 2) is:
(A) 0
(B) 3
(C) 5 (D) 16
43. If sum of the squares of zeroes of the quadratic polynomial 6x2 + x + k is 25/36, the value of k is:
(A) 4
(B) – 4
(C) 2
(D) – 2
44. If α and β are zeroes of x2 – 4x + 1, then 1/α + 1/β – αβ is
(A) 3
(B) 5
(C) –5
(D) –3
45.. If (x + 1) is a factor of x2− 3ax +3a − 7, then the value of a is:
(A) 1
(B) –1
(C) 0
(D) –2
46.If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it
(A) has no linear term and the constant term is negative.
(B) has no linear term and the constant term is positive.
(C) can have a linear term but the constant term is negative.
(D) can have a linear term but the constant term is positive
47. If α, β are zeroes of x2 –6x + k, what is the value of k if 3α + 2β = 20?
(A)–16
(B) 8
(C) 2
(D) –8
48.The zeroes of x2–2x –8 are:
(a) (2,-4)
(b) (4,-2)
(c) (-2,-2)
(d) (-4,-4)
49.. What is the quadratic polynomial whose sum and the product of zeroes is √2, ⅓ respectively?
(a) 3x2-3√2x+1
(b) 3x2+3√2x+1
(c) 3x2+3√2x-1
(d) None of the above
50. 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
51.. The degree of the polynomial, x4 – x2 +2 is
(a) 2
(b) 4
(c) 1
(d) 0
52. 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
53. 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)
Explanation: Let p(x) = mx+n
Put x = a
p(a)=ma+n=0
So, a is zero of p(x).
54.. 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
55. 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
56. The number of polynomials having zeroes as -2 and 5 is:
(a) 1
(b) 2
(c) 3
(d) More than 3
57.. Zeroes of p(x) = x2-27 are:
(a) ±9√3
(b) ±3√3
(c) ±7√3
(d) None of the above
58. 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
59.. 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
60. A quadratic polynomial, whose zeroes are
13. Which of the following is not the graph of quadratic polynomial?
14. The number of polynomials having zeroes as 4 and 7 is
(a) 2
(b) 3
(c) 4
(d) more than 4
15. A quadratic polynomial, whose zeores are -4 and -5, is
(a) x²-9x + 20
(b) x² + 9x + 20
(c) x²-9x- 20
(d) x² + 9x- 20
16. The zeroes of the quadratic polynomial x² + 1750x + 175000 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
17. The zeroes of the quadratic polynomial x² – 15x + 50 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
18. The zeroes of the quadratic polynomial 3x² – 48 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
19. The zeroes of the quadratic polynomial x² – 18x + 81 are
(a) both negative
(b) one positive and one negative
(c) both positive and unequal
(d) both equal and positive
20. The zeroes of the quadratic polynomial x² + px + p, p ≠ 0 are
(a) both equal
(b) both cannot be positive
(c) both unequal
(d) both cannot be negative
21. If one of the zeroes of the quadratic polynomial (p – l)x² + px + 1 is -3, then the value of p is
A.3/4. B. 4/3. C. -3/4. D. -4/3
22. If the zeroes of the quadratic polynomial Ax² + Bx + C, C # 0 are equal, then
(a) A and B have the same sign
(b) A and C have the same sign
(c) B and C have the same sign
(d) A and C have opposite signs
23. If x3 + 1 is divided by x² + 5, then the possible degree of quotient is
(a) 0
(b) 1
(c) 2
(d) 3
24. 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
25. If x4 + 3x² + 7 is divided by 3x + 5, then the possible degrees of quotient and remainder are:
(a) 3, 0
(b) 4, 1
(c) 3, 1
(d) 4, 0
26. If x5 + 2x4 + x + 6 is divided by g(x), and quotient is x² + 5x + 7, then the possible degree of g(x) is:
(a) 4
(b) 2
(c) 3
(d) 5
27. If x5 + 2x4 + x + 6 is divided by g(x) and quo-tient is x² + 5x + 7, then the possible degree of remainder is:
(a) less than 1
(b) less than 2
(c) less than 3
(d) less than 4
28. What is the number of zeroes that a linear poly-nomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
29. What is the number(s) of zeroes that a quadratic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
30. What is the number(s) of zeores that a cubic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
31. If one of the zeroes of the cubic polynomial x3 + px² + qx + r is -1, then the product of the other two zeroes is
(a) p + q + 1
(b) p-q- 1
(c) q – p + 1
(d) q – p – 1
32. If one zero of the quadratic polynomial x² + 3x + b is 2, then the value of b is
(a) 10
(b) -8
(c) 9
(d) -10
33. If 1 is one of the zeroes of the polynomial x² + x + k, then the value of k is:
(a) 2
(b) -2
(c) 4
(d) -4
.34. The zeroes of the quadratic polynomial x2 + 99x + 127are
(A) both positive
(B) both negative
(C) one positive and one negative
(D) both equal
35.If the zeroes of the quadratic polynomial x2 + bx + c , c ≠ 0are 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
.36. The number of polynomials having zeroes as –2 and 5 is
(A) 1
(B) 2
(C) 3
(D) more than 3
37. The degree of the polynomial (x + 1)(x2 – x – x4 +1) is:
(A)2
(B) 3
(C) 4
D.5
.38. If the zeroes of the quadratic polynomial x2 + (a + 1)x + b are 2 and –3, then
(A) a = –7, b = –1
(B) a = 5, b = –1
(C) a = 2, b = – 6
(D) a = 0, b = – 6
.39. Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is
(A)'c/a
(B) c/a
(C) 0 (D) 3
40.. If one of the zeroes of the cubic polynomial x3 + ax2 + bx + c is –1, then the
Product of the other two zeroes is
(A) b – a + 1
(B) b – a – 1
(C) a – b + 1
(D) a – b –1
41.. If one of the zeroes of the quadratic polynomial (k – 1)x2 + kx + 1 is –3, then the value of k is
(A) 4/3
(B) – 4/3
(C) 2/3
(D) – 2/3
42.The value of p for which the polynomial x3 + 4x2 –px + 8 is exactly divisible by (x – 2) is:
(A) 0
(B) 3
(C) 5 (D) 16
43. If sum of the squares of zeroes of the quadratic polynomial 6x2 + x + k is 25/36, the value of k is:
(A) 4
(B) – 4
(C) 2
(D) – 2
44. If α and β are zeroes of x2 – 4x + 1, then 1/α + 1/β – αβ is
(A) 3
(B) 5
(C) –5
(D) –3
45.. If (x + 1) is a factor of x2− 3ax +3a − 7, then the value of a is:
(A) 1
(B) –1
(C) 0
(D) –2
46.If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it
(A) has no linear term and the constant term is negative.
(B) has no linear term and the constant term is positive.
(C) can have a linear term but the constant term is negative.
(D) can have a linear term but the constant term is positive
47. If α, β are zeroes of x2 –6x + k, what is the value of k if 3α + 2β = 20?
(A)–16
(B) 8
(C) 2
(D) –8
48.The zeroes of x2–2x –8 are:
(a) (2,-4)
(b) (4,-2)
(c) (-2,-2)
(d) (-4,-4)
49.. What is the quadratic polynomial whose sum and the product of zeroes is √2, ⅓ respectively?
(a) 3x2-3√2x+1
(b) 3x2+3√2x+1
(c) 3x2+3√2x-1
(d) None of the above
50. 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
51.. The degree of the polynomial, x4 – x2 +2 is
(a) 2
(b) 4
(c) 1
(d) 0
52. 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
53. 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)
Explanation: Let p(x) = mx+n
Put x = a
p(a)=ma+n=0
So, a is zero of p(x).
54.. 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
55. 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
56. The number of polynomials having zeroes as -2 and 5 is:
(a) 1
(b) 2
(c) 3
(d) More than 3
57.. Zeroes of p(x) = x2-27 are:
(a) ±9√3
(b) ±3√3
(c) ±7√3
(d) None of the above
58. 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
59.. 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
60. 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
61. 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
62. The zeroes of the quadratic polynomial x2 + 7x + 10 are
(a) -4, -3
(b) 2, 5
(c) -2, -5
(d) -2, 5
63. 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 root
64 .If the discriminant of a quadratic polynomial, D > 0, then the polynomial has two real and unequal roots.
65. If on division of a polynomial p(x) by a polynomial g(x), the quotient is zero, then the relation between the degrees of p(x) and g(x) is
(a) degree of p(x) < degree of g(x)
(b) degree of p(x) = degree of g(x)
(c) degree of p(x) > degree of g(x)
(d) nothing can be said about degrees of p(x) and g(x)
Answer: (a) degree of p(x) < degree of g(x)
Explanation:
We know that, p(x)= g(x) × q(x) + r(x)
Given that, q(x) = 0
When q(x) = 0, then r(x) = 0
So, now when we divide p(x) by g(x),
Then p(x) should be equal to zero.
If r(x) = 0, then the degree of p(x) < degree of g(x).
66. By division algorithm of polynomials, p(x) =
(a) g(x) × q(x) + r(x)
(b) g(x) × q(x) – r(x)
(c) g(x) × q(x) × r(x)
(d) g(x) + q(x) + r(x)
Answer: (a) g(x) × q(x) + r(x)
By division algorithm of polynomials, p(x) = g(x) × q(x) + r(x).
67. 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
68. 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
69. The quadratic polynomial whose sum of zeroes is 3 and product of zeroes is –2 is :
(a) x2 + 3x – 2
(b) x2 – 2x + 3
(c) x2 – 3x + 2
(d) x2 – 3x – 2
70.If (x + 1) is a factor of 2x3 + ax2 + 2bx + 1, then find the values of a and b given that 2a – 3b = 4
(a) a = –1, b = –2
(b) a = 2, b = 5
(c) a = 5, b = 2
(d) a = 2, b = 0
71.The number of zeroes that polynomial f(x) = (x – 2)^2 + 4 can have is:
(a) 1
(b) 2
(c) 0
(d) 3
72. The zeroes of the polynomial f(x) = 4x2 – 12x + 9 are:
(a) 3/2. , 3/2
(b) -3/2. And 3/2
(c) 3, 4
(d) –3, –4
73. If p(x) is a polynomial of at least degree one and p(k) = 0, then k is known as
(a) value of p(x)
(b) zero of p(x)
(c) constant term of p(x)
(c) none of these
74. If p(x) = ax + b, then zero of p(x)
(a) a
(b) b
(c) - a/b
(d) - b/a
75. Graph of a quadratic polynomial is a
(a) straight line
(b) circle
(c) parabola
(d) ellipse
76. Zeroes of a polynomial can be determined graphically. No. of zeroes of a polynomial is equal to no. of points where the graph of polynomial
(a) intersects y-axis
(b) intersects x-axis
(c) intersects y-axis or intersects x-axis
(d) none of these
77. If graph of a polynomial does not intersects the x-axis but intersects y-axis in one point, then no, of zeroes of the polynomial is equal to
(a) 0
(b) 1
(c) 0 or 1
(d) none of these
78. A polynomial of degree n has
(a) only 1 zero
(b) exactly n zeroes
(c) atmost n zeroes
(d) more than n zeroes
79. If p(x) = ax2 + bx + c, then
is equal to c/a
(a) 0
(b) 1
(c) sum of zeroes
(d) product of zeroes
80. If p(x) = ax2 + bx + c, then -b/a
is equal to
(a) 0
(b) 1
(c) product of zeroes
(d) sum of zeroes
81. If p(x) = ax^2 + bx + c, and a + b + c = 0, then
one zero
(a) -b/a
(b) c/a
(c) b/c
(d) none of these
82. If p(x) = ax2 + bx + c and a + c = b, then one of the zeroes is
(a) -b/a
(b) - c/a
(a) x² – x + 12
(b) x² + x + 12
(c) (x²/2) – (x/2) – 6
(d) 2x² + 2x – 24
61. 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
62. The zeroes of the quadratic polynomial x2 + 7x + 10 are
(a) -4, -3
(b) 2, 5
(c) -2, -5
(d) -2, 5
63. 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 root
64 .If the discriminant of a quadratic polynomial, D > 0, then the polynomial has two real and unequal roots.
65. If on division of a polynomial p(x) by a polynomial g(x), the quotient is zero, then the relation between the degrees of p(x) and g(x) is
(a) degree of p(x) < degree of g(x)
(b) degree of p(x) = degree of g(x)
(c) degree of p(x) > degree of g(x)
(d) nothing can be said about degrees of p(x) and g(x)
Answer: (a) degree of p(x) < degree of g(x)
Explanation:
We know that, p(x)= g(x) × q(x) + r(x)
Given that, q(x) = 0
When q(x) = 0, then r(x) = 0
So, now when we divide p(x) by g(x),
Then p(x) should be equal to zero.
If r(x) = 0, then the degree of p(x) < degree of g(x).
66. By division algorithm of polynomials, p(x) =
(a) g(x) × q(x) + r(x)
(b) g(x) × q(x) – r(x)
(c) g(x) × q(x) × r(x)
(d) g(x) + q(x) + r(x)
Answer: (a) g(x) × q(x) + r(x)
By division algorithm of polynomials, p(x) = g(x) × q(x) + r(x).
67. 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
68. 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
69. The quadratic polynomial whose sum of zeroes is 3 and product of zeroes is –2 is :
(a) x2 + 3x – 2
(b) x2 – 2x + 3
(c) x2 – 3x + 2
(d) x2 – 3x – 2
70.If (x + 1) is a factor of 2x3 + ax2 + 2bx + 1, then find the values of a and b given that 2a – 3b = 4
(a) a = –1, b = –2
(b) a = 2, b = 5
(c) a = 5, b = 2
(d) a = 2, b = 0
71.The number of zeroes that polynomial f(x) = (x – 2)^2 + 4 can have is:
(a) 1
(b) 2
(c) 0
(d) 3
72. The zeroes of the polynomial f(x) = 4x2 – 12x + 9 are:
(a) 3/2. , 3/2
(b) -3/2. And 3/2
(c) 3, 4
(d) –3, –4
73. If p(x) is a polynomial of at least degree one and p(k) = 0, then k is known as
(a) value of p(x)
(b) zero of p(x)
(c) constant term of p(x)
(c) none of these
74. If p(x) = ax + b, then zero of p(x)
(a) a
(b) b
(c) - a/b
(d) - b/a
75. Graph of a quadratic polynomial is a
(a) straight line
(b) circle
(c) parabola
(d) ellipse
76. Zeroes of a polynomial can be determined graphically. No. of zeroes of a polynomial is equal to no. of points where the graph of polynomial
(a) intersects y-axis
(b) intersects x-axis
(c) intersects y-axis or intersects x-axis
(d) none of these
77. If graph of a polynomial does not intersects the x-axis but intersects y-axis in one point, then no, of zeroes of the polynomial is equal to
(a) 0
(b) 1
(c) 0 or 1
(d) none of these
78. A polynomial of degree n has
(a) only 1 zero
(b) exactly n zeroes
(c) atmost n zeroes
(d) more than n zeroes
79. If p(x) = ax2 + bx + c, then
is equal to c/a
(a) 0
(b) 1
(c) sum of zeroes
(d) product of zeroes
80. If p(x) = ax2 + bx + c, then -b/a
is equal to
(a) 0
(b) 1
(c) product of zeroes
(d) sum of zeroes
81. If p(x) = ax^2 + bx + c, and a + b + c = 0, then
one zero
(a) -b/a
(b) c/a
(c) b/c
(d) none of these
82. If p(x) = ax2 + bx + c and a + c = b, then one of the zeroes is
(a) -b/a
(b) - c/a
(C) c/a
(D) b/c
83 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
.84. 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
.85.Given that two of the zeroes of the cubic poly-nomial ax3 + bx² + cx + d are 0, the third zero is
86. If one of the zeroes of the quadratic polynomial (k – 1) x² + kx + 1 is – 3, then the value of k is
87. A quadratic polynomial, whose zeroes are -3 and 4, is
(a) x²- x + 12
(b) x² + x + 12
(c) x^2/2 - x/2. - 6
(d) 2x² + 2x – 24
88. If the zeroes of the quadratic polynomial x2 + (a + 1) x + b are 2 and -3, then
(a) a = -7, b = -1
(b) a = 5, b = -1
(c) a = 2, b = -6
(d) a – 0, b = -6
.89.The number of polynomials having zeroes as -2 and 5 is
(a) 1
(b) 2
(c) 3
(d) more than 3
90 Given that one of the zeroes of the cubic polynomial ax3 + bx² + cx + d is zero, the product of the other two zeroes is
(D) b/c
83 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
.84. 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
.85.Given that two of the zeroes of the cubic poly-nomial ax3 + bx² + cx + d are 0, the third zero is
86. If one of the zeroes of the quadratic polynomial (k – 1) x² + kx + 1 is – 3, then the value of k is
87. A quadratic polynomial, whose zeroes are -3 and 4, is
(a) x²- x + 12
(b) x² + x + 12
(c) x^2/2 - x/2. - 6
(d) 2x² + 2x – 24
88. If the zeroes of the quadratic polynomial x2 + (a + 1) x + b are 2 and -3, then
(a) a = -7, b = -1
(b) a = 5, b = -1
(c) a = 2, b = -6
(d) a – 0, b = -6
.89.The number of polynomials having zeroes as -2 and 5 is
(a) 1
(b) 2
(c) 3
(d) more than 3
90 Given that one of the zeroes of the cubic polynomial ax3 + bx² + cx + d is zero, the product of the other two zeroes is
(a) -b/a
(b) - c/a
(b) - c/a
(C) c/a
(D) b/c
(D) b/c
91.. If one of the zeroes of the cubic polynomial x3 + ax² + bx + c is -1, then the product of the
other two zeroes is
(a) b – a + 1
(b) b – a – 1
(c) a – b + 1
(d) a – b – 1
92. 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
93. The zeroes of the quadratic polynomial x² + kx + k, k? 0,
(a) cannot both be positive
(b) cannot both be negative
(c) are always unequal
(d) are always equal
94. 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
95. If one of the zeroes of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it
(a) has no linear term and the constant term is negative.
(b) has no linear term and the constant term is positive.
(c) can have a linear term but the constant term is negative.
(d) can have a linear term but the constant term is positive.
96. Which of the following is not the graph of quadratic polynomial?
97. The number of polynomials having zeroes as 4 and 7 is
(a) 2
(b) 3
(c) 4
(d) more than 4
98 A quadratic polynomial, whose zeores are -4 and -5, is
(a) x²-9x + 20
(b) x² + 9x + 20
(c) x²-9x- 20
(d) x² + 9x- 20
99. The zeroes of the quadratic polynomial x² + 1750x + 175000 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
100. The zeroes of the quadratic polynomial
x² – 15x + 50 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
101. The zeroes of the quadratic polynomial 3x² – 48 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
102. The zeroes of the quadratic polynomial x² – 18x + 81 are
(a) both negative
(b) one positive and one negative
(c) both positive and unequal
(d) both equal and positive
103.. The zeroes of the quadratic polynomial x² + px + p, p ≠ 0 are
(a) both equal
(b) both cannot be positive
(c) both unequal
(d) both cannot be negative
104.. If one of the zeroes of the quadratic polynomial (p – l)x² + px + 1 is -3, then the value of p is
105. If the zeroes of the quadratic polynomial Ax² + Bx + C, C # 0 are equal, then
(a) A and B have the same sign
(b) A and C have the same sign
(c) B and C have the same sign
(d) A and C have opposite signs
106. If x3 + 1 is divided by x² + 5, then the possible degree of quotient is
(a) 0
(b) 1
(c) 2
(d) 3
107. 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
108. If x4 + 3x² + 7 is divided by 3x + 5, then the possible degrees of quotient and remainder are:
(a) 3, 0
(b) 4, 1
(c) 3, 1
(d) 4, 0
109. If x5 + 2x4 + x + 6 is divided by g(x), and quotient is x² + 5x + 7, then the possible degree of g(x) is:
(a) 4
(b) 2
(c) 3
(d) 5
110. If x5 + 2x4 + x + 6 is divided by g(x) and quo-tient is x² + 5x + 7, then the possible degree of remainder is:
(a) less than 1
(b) less than 2
(c) less than 3
(d) less than 4
111. What is the number of zeroes that a linear poly-nomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
112. What is the number(s) of zeroes that a quadratic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
113. What is the number(s) of zeores that a cubic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
114. If one of the zeroes of the cubic polynomial x3 + px² + qx + r is -1, then the product of the other two zeroes is
(a) p + q + 1
(b) p-q- 1
(c) q – p + 1
(d) q – p – 1
115. If one zero of the quadratic polynomial x² + 3x + b is 2, then the value of b is
(a) 10
(b) -8
(c) 9
(d) -10
117.If 1 is one of the zeroes of the polynomial x² + x + k, then the value of k is:
(a) 2
(b) -2
(c) 4
(d) -4
118. If p(x) is a polynomial of at least degree one and p(k) = 0, then k is known as
(a) value of p(x)
(b) zero of p(x)
(c) constant term of p{x)
(d) none of these
119. If one of the zeroes of the quadratic polynomial (k -1)x² + kx + 1 the value of k is [NCERT Exemplar Problems]
120. If the zeroes of the quadratic polynomial x² + (a + 1) x + b are 2 and -3, then [NCERT Exemplar Problem, CBSE 2011]
(a) a = -7, b = -1
(b) a = 5, b = -1
(c) a = 2, b = -6
(d) a = 0, b = -6
121. Which of the following is not the graph of a quadratic polynomial?[NCERT Exemplar Problems]
122. Zeroes of a polynomial can be determined graphically. No. of zeroes of a polynomial is equal to no. of points where the graph of polynomial
(a) intersects y-axis
(b) intersects x-axis
(c) intersects y-axis or intersects x-axis
(d) none of these
123.. If graph of a polynomial does not intersects the x-axis but intersects y-axis in one point, then no. of zeroes of the polynomial is equal to
(a) 0
(b) 1
(c) 0 or 1
(d) none of these
124. A polynomial of degree n has
(a) only 1 zero
(b) at least n zeroes
(c) atmost n zeroes
(d) more than n zeroes
125.. If p(x) = axr + bx + c, then –\(\frac{b}{a}\) is equal to
(a) 0
(b) 1
(c) product of zeroes
(d) sum of zeroes
126.. If p(x) = ax² + bx + c one zero is and a + b + c = 0, then one zero is
(a) \(\frac{-b}{a}\)
(b) \(\frac{c}{a}\)
(c) \(\frac{b}{c}\)
(d) none of these
127. If p{x) = ax2 + bx + one of the zeroes is c and a + c = b, then
128.. The number of polynomials having zeroes as -2 and 5 is [NCERT Exemplar Problems]
(a) 1
(b) 2
(c) 3
(d) more than 3
129.. Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is [NCERT Exemplar Problems]
120.. If one of the zeroes of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it
(a) has no linear term and the constant term is negative.
(b) has no linear term and the constant term is positive.
(c) can have a linear term but the constant term is negative.
(d) can have a linear term but the constant term is positive
121. If 4x² – 6x – m is divisible by x – 3, the value of m is exact divisor of
(a) 9
(b) 45
(c) 20
(d) 36
122.. Which one of the following statements is correct(a) if x6 + 1 is divided by x + 1, then the remainder is -2.
(b) if x6 + 1 is divided by x – 1, then the remainder is 2.
(c) if x6 + 1 is divided by x + 1, then the remainder is 1.
(d) if x6 + 1 is divided by x – 1, then the remainder is -1.
123. Consider the following statements
(i) x – 2 is a factor of x3 – 3x² + 4x – 4.
(ii) x + 1 is a factor of 2x3 + 4x + 6.
(iii) x – 1 is a factor of x5 + x4 – x3 + x² -x + 1.
In these statements
(a) 1 and 2 are correct
(b) 1, 2 and 3 are correct
(c) 2 and 3 are correct
(d) 1 and 3 are correct
124.. If f(x) = 5x – 10 is divided by x – √2, then the remainder will be
(a) non zero rational number
(b) an irrational number
(c) 0
(d) \(f\left(\frac{1}{\sqrt{2}}\right)\)
125. Zeroes of p(z) = z² – 27 are ______ and ______ .
126. Verify that x = 3 is a zero of the polynomial. p(x) = 2x3 – 5x² – 4x + 3
127. The graph of y =f(x) is given below. How many zeroes are there of f(x)?
128.. The graph of y = f(x) is given, how many zeroes are there of f(x)?
129.. The graph of y = f(x) is given below, for some polynomial f(x). Find the number of zeroes of f(x).
A. -c/a. B. C/a. C. 0. D. -b/a
130. The graph of x = p(y) is given below, for some polynomial p(y). Find the number of zeroes of p(y).
131.
Graph of the polynomial p(x) =px² + 4x – 4 is given as above. Find the value of p.
132.. If the product of the zeroes of x2 – 3kx + 2k1 – 1 is 7, then values of k are _____ and _____ .
133 . If zeroes ofp(x) = 2x² -Ix + k are reciprocal of each other, then value of k is _____ .
134.Find the product of the zeroes of – 2x² + kx + 6.
135.. Find the sum of the zeroes of the given quadratic polynomial -3x² + k.
136. If one zero of the polynomial x² -4x+ 1 is 2 + √3, write the other zero.
137. Write the polynomial, the product and sum of whose zeroes are –\(\frac{9}{2}\) and –\(\frac{3}{2}\) respectively.
138. The value of m, in order that x² – mx – 2 is the quotient where x3 + 3x² – 4 is divided by x + 2 is ____ .
139.. If one factor of x3 + 7kx² – 4kx + 12 is (x + 3), then the value of k is ______ .
140.. A polynomial of degree five is divided by a quadratic polynomial. If it leaves a remainder, then find the degree of remainder.
141. Check whether 3x – 7 is a factor of polynomial 6x3 + x² – 26x – 25?
142.. If x3 + x² – ax + b is divisible by x² – x, write the values of a and b.
Question 143
The maximum number of zeroes that a polynomial of degree 4 can have is
(a) One
(b) Two
(c) Three
(d) Four
Question 144
The graph of the polynomial p(x) = 3x – 2 is a straight line which intersects the x-axis at exactly one point namely
(a) (−23, 0)
(b) (0, −23)
(c) (23, 0)
(d) 23, −23
Question 145
In fig. given below, the number of zeroes of the polynomial f(x) is
Graph meet x axis at 3 points
(a) 1
(b) 2
(c) 3
(d) None
Question 146
The graph of the polynomial ax² + bx + c is an upward parabola if
(a) a > 0
(b) a < 0
(b) a = 0
(d) None
Question 147
The graph of the polynomial ax² + bx + c is a downward parabola if
(a) a > 0
(b) a < 0
(c) a = 0
(d) a = 1
Question 148
A polynomial of degree 3 is called
(a) a linear polynomial
(b) a quadratic polynomial
(c) a cubic polynomial
(d) a biquadratic polynomial
Question 149
If α, β are the zeroes of the polynomial x² – 16, then αβ(α + β) is
(a) 0
(b) 4
(c) -4
(d) 16
Question 150
If α and 1α are the zeroes of the polynomial ax² + bx + c, then value of c is
(a) 0
(b) a
(c) -a
(d) 1
Question 151
Zeroes of the polynomial x² – 11 are
(a) ±17−−√
(b) ±3–√
(c) 0
(d) None
Question 152
If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d then α + β + γ is equal
(a) −ba
(b) ba
(c) ca
(d) da
A. 3/4. B. 4/3. C. -3/4. D. -4/3
153If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d then αβ + βγ + αγ is equal to
(a) −ba
(b) ba
(c) ca
(d) da
Question 154
If the zeroes of the polynomial x³ – 3x² + x – 1 are stthe zeroes of the polynomial x³ – 3x² + x – 1 are st, s and st then value of s is
(a) 1
(b) -1
(c) 2
(d) -3
Question 155
If the sum of the zeroes of the polynomial f(x) = 2x³ – 3kx² + 4x – 5 is 6, then the value of k is
(a) 2
(b) 4
(c) -2
(d) -4
Question 156
If a polynomial of degree 4 is divided by quadratic polynomial, the degree of the remainder is
(a) ≤ 1
(b) ≥ 1
(c) 2
(d) 4
Question 157
If a – b, a and a + b are zeroes of the polynomial fix) = 2x³ – 6x² + 5x – 7, then value of a is
(a) 1
(b) 2
(c) -5
(d) 7
Question 158
Dividend is equal to
(a) divisor × quotient + remainder
(b) divisior × quotient
(c) divisior × quotient – remainder
(d) divisor × quotient × remainder
Question 159
A quadratic polynomial whose sum of the zeroes is 2 and product is 1 is given by
(a) x² – 2x + 1
(b) x² + 2x + 1
(c) x² + 2x – 1
(d) x² – 2x – 1
Question 160
If one of the zeroes of a quadratic polynomial ax² + bx + c is 0, then the other zero is
(a) −ba
(b) 0
(c) ba
(d) −ca
Question 161
The sum and the product of the zeroes of polynomial 6x² – 5 respectively are
(a) 0, −65
(b) 0, 65
(c) 0, 56
(d) 0, −56
Question 162
What should be subtracted from x³ – 2x² + 4x + 1 to get 1?
(a) x³ – 2x² + 4x
(b) x³ – 2x² + 4 + 1
(c) -1
(d) 1
(b) one positive and one negative
(c) both positive
(d) both equal
101. The zeroes of the quadratic polynomial 3x² – 48 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
102. The zeroes of the quadratic polynomial x² – 18x + 81 are
(a) both negative
(b) one positive and one negative
(c) both positive and unequal
(d) both equal and positive
103.. The zeroes of the quadratic polynomial x² + px + p, p ≠ 0 are
(a) both equal
(b) both cannot be positive
(c) both unequal
(d) both cannot be negative
104.. If one of the zeroes of the quadratic polynomial (p – l)x² + px + 1 is -3, then the value of p is
105. If the zeroes of the quadratic polynomial Ax² + Bx + C, C # 0 are equal, then
(a) A and B have the same sign
(b) A and C have the same sign
(c) B and C have the same sign
(d) A and C have opposite signs
106. If x3 + 1 is divided by x² + 5, then the possible degree of quotient is
(a) 0
(b) 1
(c) 2
(d) 3
107. 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
108. If x4 + 3x² + 7 is divided by 3x + 5, then the possible degrees of quotient and remainder are:
(a) 3, 0
(b) 4, 1
(c) 3, 1
(d) 4, 0
109. If x5 + 2x4 + x + 6 is divided by g(x), and quotient is x² + 5x + 7, then the possible degree of g(x) is:
(a) 4
(b) 2
(c) 3
(d) 5
110. If x5 + 2x4 + x + 6 is divided by g(x) and quo-tient is x² + 5x + 7, then the possible degree of remainder is:
(a) less than 1
(b) less than 2
(c) less than 3
(d) less than 4
111. What is the number of zeroes that a linear poly-nomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
112. What is the number(s) of zeroes that a quadratic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
113. What is the number(s) of zeores that a cubic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
114. If one of the zeroes of the cubic polynomial x3 + px² + qx + r is -1, then the product of the other two zeroes is
(a) p + q + 1
(b) p-q- 1
(c) q – p + 1
(d) q – p – 1
115. If one zero of the quadratic polynomial x² + 3x + b is 2, then the value of b is
(a) 10
(b) -8
(c) 9
(d) -10
117.If 1 is one of the zeroes of the polynomial x² + x + k, then the value of k is:
(a) 2
(b) -2
(c) 4
(d) -4
118. If p(x) is a polynomial of at least degree one and p(k) = 0, then k is known as
(a) value of p(x)
(b) zero of p(x)
(c) constant term of p{x)
(d) none of these
119. If one of the zeroes of the quadratic polynomial (k -1)x² + kx + 1 the value of k is [NCERT Exemplar Problems]
120. If the zeroes of the quadratic polynomial x² + (a + 1) x + b are 2 and -3, then [NCERT Exemplar Problem, CBSE 2011]
(a) a = -7, b = -1
(b) a = 5, b = -1
(c) a = 2, b = -6
(d) a = 0, b = -6
121. Which of the following is not the graph of a quadratic polynomial?[NCERT Exemplar Problems]
122. Zeroes of a polynomial can be determined graphically. No. of zeroes of a polynomial is equal to no. of points where the graph of polynomial
(a) intersects y-axis
(b) intersects x-axis
(c) intersects y-axis or intersects x-axis
(d) none of these
123.. If graph of a polynomial does not intersects the x-axis but intersects y-axis in one point, then no. of zeroes of the polynomial is equal to
(a) 0
(b) 1
(c) 0 or 1
(d) none of these
124. A polynomial of degree n has
(a) only 1 zero
(b) at least n zeroes
(c) atmost n zeroes
(d) more than n zeroes
125.. If p(x) = axr + bx + c, then –\(\frac{b}{a}\) is equal to
(a) 0
(b) 1
(c) product of zeroes
(d) sum of zeroes
126.. If p(x) = ax² + bx + c one zero is and a + b + c = 0, then one zero is
(a) \(\frac{-b}{a}\)
(b) \(\frac{c}{a}\)
(c) \(\frac{b}{c}\)
(d) none of these
127. If p{x) = ax2 + bx + one of the zeroes is c and a + c = b, then
128.. The number of polynomials having zeroes as -2 and 5 is [NCERT Exemplar Problems]
(a) 1
(b) 2
(c) 3
(d) more than 3
129.. Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is [NCERT Exemplar Problems]
120.. If one of the zeroes of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it
(a) has no linear term and the constant term is negative.
(b) has no linear term and the constant term is positive.
(c) can have a linear term but the constant term is negative.
(d) can have a linear term but the constant term is positive
121. If 4x² – 6x – m is divisible by x – 3, the value of m is exact divisor of
(a) 9
(b) 45
(c) 20
(d) 36
122.. Which one of the following statements is correct(a) if x6 + 1 is divided by x + 1, then the remainder is -2.
(b) if x6 + 1 is divided by x – 1, then the remainder is 2.
(c) if x6 + 1 is divided by x + 1, then the remainder is 1.
(d) if x6 + 1 is divided by x – 1, then the remainder is -1.
123. Consider the following statements
(i) x – 2 is a factor of x3 – 3x² + 4x – 4.
(ii) x + 1 is a factor of 2x3 + 4x + 6.
(iii) x – 1 is a factor of x5 + x4 – x3 + x² -x + 1.
In these statements
(a) 1 and 2 are correct
(b) 1, 2 and 3 are correct
(c) 2 and 3 are correct
(d) 1 and 3 are correct
124.. If f(x) = 5x – 10 is divided by x – √2, then the remainder will be
(a) non zero rational number
(b) an irrational number
(c) 0
(d) \(f\left(\frac{1}{\sqrt{2}}\right)\)
125. Zeroes of p(z) = z² – 27 are ______ and ______ .
126. Verify that x = 3 is a zero of the polynomial. p(x) = 2x3 – 5x² – 4x + 3
127. The graph of y =f(x) is given below. How many zeroes are there of f(x)?
128.. The graph of y = f(x) is given, how many zeroes are there of f(x)?
129.. The graph of y = f(x) is given below, for some polynomial f(x). Find the number of zeroes of f(x).
A. -c/a. B. C/a. C. 0. D. -b/a
130. The graph of x = p(y) is given below, for some polynomial p(y). Find the number of zeroes of p(y).
131.
Graph of the polynomial p(x) =px² + 4x – 4 is given as above. Find the value of p.
132.. If the product of the zeroes of x2 – 3kx + 2k1 – 1 is 7, then values of k are _____ and _____ .
133 . If zeroes ofp(x) = 2x² -Ix + k are reciprocal of each other, then value of k is _____ .
134.Find the product of the zeroes of – 2x² + kx + 6.
135.. Find the sum of the zeroes of the given quadratic polynomial -3x² + k.
136. If one zero of the polynomial x² -4x+ 1 is 2 + √3, write the other zero.
137. Write the polynomial, the product and sum of whose zeroes are –\(\frac{9}{2}\) and –\(\frac{3}{2}\) respectively.
138. The value of m, in order that x² – mx – 2 is the quotient where x3 + 3x² – 4 is divided by x + 2 is ____ .
139.. If one factor of x3 + 7kx² – 4kx + 12 is (x + 3), then the value of k is ______ .
140.. A polynomial of degree five is divided by a quadratic polynomial. If it leaves a remainder, then find the degree of remainder.
141. Check whether 3x – 7 is a factor of polynomial 6x3 + x² – 26x – 25?
142.. If x3 + x² – ax + b is divisible by x² – x, write the values of a and b.
Question 143
The maximum number of zeroes that a polynomial of degree 4 can have is
(a) One
(b) Two
(c) Three
(d) Four
Question 144
The graph of the polynomial p(x) = 3x – 2 is a straight line which intersects the x-axis at exactly one point namely
(a) (−23, 0)
(b) (0, −23)
(c) (23, 0)
(d) 23, −23
Question 145
In fig. given below, the number of zeroes of the polynomial f(x) is
Graph meet x axis at 3 points
(a) 1
(b) 2
(c) 3
(d) None
Question 146
The graph of the polynomial ax² + bx + c is an upward parabola if
(a) a > 0
(b) a < 0
(b) a = 0
(d) None
Question 147
The graph of the polynomial ax² + bx + c is a downward parabola if
(a) a > 0
(b) a < 0
(c) a = 0
(d) a = 1
Question 148
A polynomial of degree 3 is called
(a) a linear polynomial
(b) a quadratic polynomial
(c) a cubic polynomial
(d) a biquadratic polynomial
Question 149
If α, β are the zeroes of the polynomial x² – 16, then αβ(α + β) is
(a) 0
(b) 4
(c) -4
(d) 16
Question 150
If α and 1α are the zeroes of the polynomial ax² + bx + c, then value of c is
(a) 0
(b) a
(c) -a
(d) 1
Question 151
Zeroes of the polynomial x² – 11 are
(a) ±17−−√
(b) ±3–√
(c) 0
(d) None
Question 152
If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d then α + β + γ is equal
(a) −ba
(b) ba
(c) ca
(d) da
A. 3/4. B. 4/3. C. -3/4. D. -4/3
153If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d then αβ + βγ + αγ is equal to
(a) −ba
(b) ba
(c) ca
(d) da
Question 154
If the zeroes of the polynomial x³ – 3x² + x – 1 are stthe zeroes of the polynomial x³ – 3x² + x – 1 are st, s and st then value of s is
(a) 1
(b) -1
(c) 2
(d) -3
Question 155
If the sum of the zeroes of the polynomial f(x) = 2x³ – 3kx² + 4x – 5 is 6, then the value of k is
(a) 2
(b) 4
(c) -2
(d) -4
Question 156
If a polynomial of degree 4 is divided by quadratic polynomial, the degree of the remainder is
(a) ≤ 1
(b) ≥ 1
(c) 2
(d) 4
Question 157
If a – b, a and a + b are zeroes of the polynomial fix) = 2x³ – 6x² + 5x – 7, then value of a is
(a) 1
(b) 2
(c) -5
(d) 7
Question 158
Dividend is equal to
(a) divisor × quotient + remainder
(b) divisior × quotient
(c) divisior × quotient – remainder
(d) divisor × quotient × remainder
Question 159
A quadratic polynomial whose sum of the zeroes is 2 and product is 1 is given by
(a) x² – 2x + 1
(b) x² + 2x + 1
(c) x² + 2x – 1
(d) x² – 2x – 1
Question 160
If one of the zeroes of a quadratic polynomial ax² + bx + c is 0, then the other zero is
(a) −ba
(b) 0
(c) ba
(d) −ca
Question 161
The sum and the product of the zeroes of polynomial 6x² – 5 respectively are
(a) 0, −65
(b) 0, 65
(c) 0, 56
(d) 0, −56
Question 162
What should be subtracted from x³ – 2x² + 4x + 1 to get 1?
(a) x³ – 2x² + 4x
(b) x³ – 2x² + 4 + 1
(c) -1
(d) 1
Polynomials mcqs
Class 10 Maths MCQs Chapter 2 Polynomials
1. 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
2. Given that two of the zeroes of the cubic poly-nomial ax3 + bx² + cx + d are 0, the third zero is
A. -b/a. B. B/a. C. C/a. D. - d/a.
3. If one of the zeroes of the quadratic polynomial (k – 1) x² + kx + 1 is – 3, then the value of k is
A. 4/3. B. -4/3. C. 2/3. D. -2/3
4. A quadratic polynomial, whose zeroes are -3 and 4, is
(a) x²- x + 12
(b) x² + x + 12
(c) x22−x2−6
(d) 2x² + 2x – 24
5. If the zeroes of the quadratic polynomial x2 + (a + 1) x + b are 2 and -3, then
(a) a = -7, b = -1
(b) a = 5, b = -1
(c) a = 2, b = -6
(d) a – 0, b = -6
6. The number of polynomials having zeroes as -2 and 5 is
(a) 1
(b) 2
(c) 3
(d) more than 3
7. Given that one of the zeroes of the cubic polynomial ax3 + bx² + cx + d is zero, the product of the other two zeroes is
A. -c/a. B. C/a. C.0. D. -b/a
8. If one of the zeroes of the cubic polynomial x3 + ax² + bx + c is -1, then the product of the
other two zeroes is
(a) b – a + 1
(b) b – a – 1
(c) a – b + 1
(d) a – b – 1
9. 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
10. The zeroes of the quadratic polynomial x² + kx + k, k? 0,
(a) cannot both be positive
(b) cannot both be negative
(c) are always unequal
(d) are always equal
11. 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
12. If one of the zeroes of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it
(a) has no linear term and the constant term is negative.
(c) can have a linear term but the constant term is negative.
(d) can have a linear term but the constant term is positive.
13. Which of the following is not the graph of quadratic polynomial?
14. The number of polynomials having zeroes as 4 and 7 is
(a) 2
(b) 3
(c) 4
(d) more than 4
15. A quadratic polynomial, whose zeores are -4 and -5, is
(a) x²-9x + 20
(b) x² + 9x + 20
(c) x²-9x- 20
(d) x² + 9x- 20
16. The zeroes of the quadratic polynomial x² + 1750x + 175000 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
17. The zeroes of the quadratic polynomial x² – 15x + 50 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
18. The zeroes of the quadratic polynomial 3x² – 48 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
19. The zeroes of the quadratic polynomial x² – 18x + 81 are
(a) both negative
(b) one positive and one negative
(c) both positive and unequal
(d) both equal and positive
20. The zeroes of the quadratic polynomial x² + px + p, p ≠ 0 are
(a) both equal
(b) both cannot be positive
(c) both unequal
(d) both cannot be negative
21. If one of the zeroes of the quadratic polynomial (p – l)x² + px + 1 is -3, then the value of p is
A.3/4. B. 4/3. C. -3/4. D. -4/3
22. If the zeroes of the quadratic polynomial Ax² + Bx + C, C # 0 are equal, then
(a) A and B have the same sign
(b) A and C have the same sign
(c) B and C have the same sign
(d) A and C have opposite signs
23. If x3 + 1 is divided by x² + 5, then the possible degree of quotient is
(a) 0
(b) 1
(c) 2
(d) 3
24. 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
25. If x4 + 3x² + 7 is divided by 3x + 5, then the possible degrees of quotient and remainder are:
(a) 3, 0
(b) 4, 1
(c) 3, 1
(d) 4, 0
26. If x5 + 2x4 + x + 6 is divided by g(x), and quotient is x² + 5x + 7, then the possible degree of g(x) is:
(a) 4
(b) 2
(c) 3
(d) 5
27. If x5 + 2x4 + x + 6 is divided by g(x) and quo-tient is x² + 5x + 7, then the possible degree of remainder is:
(a) less than 1
(b) less than 2
(c) less than 3
(d) less than 4
28. What is the number of zeroes that a linear poly-nomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
29. What is the number(s) of zeroes that a quadratic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
30. What is the number(s) of zeores that a cubic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
31. If one of the zeroes of the cubic polynomial x3 + px² + qx + r is -1, then the product of the other two zeroes is
(a) p + q + 1
(b) p-q- 1
(c) q – p + 1
(d) q – p – 1
32. If one zero of the quadratic polynomial x² + 3x + b is 2, then the value of b is
(a) 10
(b) -8
(c) 9
(d) -10
33. If 1 is one of the zeroes of the polynomial x² + x + k, then the value of k is:
(a) 2
(b) -2
(c) 4
(d) -4
.34. The zeroes of the quadratic polynomial x2 + 99x + 127are
(A) both positive
(B) both negative
(C) one positive and one negative
(D) both equal
35.If the zeroes of the quadratic polynomial x2 + bx + c , c ≠ 0are 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
.36. The number of polynomials having zeroes as –2 and 5 is
(A) 1
(B) 2
(C) 3
(D) more than 3
37. The degree of the polynomial (x + 1)(x2 – x – x4 +1) is:
(A)2
(B) 3
(C) 4
D.5
.38. If the zeroes of the quadratic polynomial x2 + (a + 1)x + b are 2 and –3, then
(A) a = –7, b = –1
(B) a = 5, b = –1
(C) a = 2, b = – 6
(D) a = 0, b = – 6
.39. Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is
(A) –c/a
(B) c/a
(C) 0
(D) 3
40.. If one of the zeroes of the cubic polynomial x3 + ax2 + bx + c is –1, then the
Product of the other two zeroes is
(A) b – a + 1
(B) b – a – 1
(C) a – b + 1
(D) a – b –1
41.. If one of the zeroes of the quadratic polynomial (k – 1)x2 + kx + 1 is –3, then the value of k is
(A) 4/3
(B) – 4/3
(C) 2/3
(D) – 2/3
42.The value of p for which the polynomial x3 + 4x2 –px + 8 is exactly divisible by (x – 2) is:
(A) 0
(B) 3
(C) 5
(D) 16
43. If sum of the squares of zeroes of the quadratic polynomial 6x2 + x + k is 25/36, the value of k is:
(A) 4
(B) – 4
(C) 2
(D) – 2
44. If α and β are zeroes of x2 – 4x + 1, then 1/α + 1/β – αβ is
(A) 3
(B) 5
(C) –5
(D) –3
45.. If (x + 1) is a factor of x2− 3ax +3a − 7, then the value of a is:
(A) 1
(B) –1
(C) 0
(D) –2
46.If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it
(A) has no linear term and the constant term is negative.
(B) has no linear term and the constant term is positive.
(C) can have a linear term but the constant term is negative.
(D) can have a linear term but the constant term is positive
47. If α, β are zeroes of x2 –6x + k, what is the value of k if 3α + 2β = 20?
(A)–16
(B) 8
(C) 2
(D) –8
48.The zeroes of x2–2x –8 are:
(a) (2,-4)
(b) (4,-2)
(c) (-2,-2)
(d) (-4,-4)
49.. What is the quadratic polynomial whose sum and the product of zeroes is √2, ⅓ respectively?
(a) 3x2-3√2x+1
(b) 3x2+3√2x+1
(c) 3x2+3√2x-1
(d) None of the above
50. 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
51.. The degree of the polynomial, x4 – x2 +2 is
(a) 2
(b) 4
(c) 1
(d) 0
52. 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
53. 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)
Explanation: Let p(x) = mx+n
Put x = a
p(a)=ma+n=0
So, a is zero of p(x).
54.. 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
55. 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
56. The number of polynomials having zeroes as -2 and 5 is:
(a) 1
(b) 2
(c) 3
(d) More than 3
57.. Zeroes of p(x) = x2-27 are:
(a) ±9√3
(b) ±3√3
(c) ±7√3
(d) None of the above
58. 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
59.. 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
60. 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
61. 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
62. The zeroes of the quadratic polynomial x2 + 7x + 10 are
(a) -4, -3
(b) 2, 5
(c) -2, -5
(d) -2, 5
63. 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 root
64If the discriminant of a quadratic polynomial, D > 0, then the polynomial has two real and unequal roots.
65. If on division of a polynomial p(x) by a polynomial g(x), the quotient is zero, then the relation between the degrees of p(x) and g(x) is
(a) degree of p(x) < degree of g(x)
(b) degree of p(x) = degree of g(x)
(c) degree of p(x) > degree of g(x)
(d) nothing can be said about degrees of p(x) and g(x)
Answer: (a) degree of p(x) < degree of g(x)
Explanation:
We know that, p(x)= g(x) × q(x) + r(x)
Given that, q(x) = 0
When q(x) = 0, then r(x) = 0
So, now when we divide p(x) by g(x),
Then p(x) should be equal to zero.
If r(x) = 0, then the degree of p(x) < degree of g(x).
66. By division algorithm of polynomials, p(x) =
(a) g(x) × q(x) + r(x)
(b) g(x) × q(x) – r(x)
(c) g(x) × q(x) × r(x)
(d) g(x) + q(x) + r(x)
Answer: (a) g(x) × q(x) + r(x)
By division algorithm of polynomials, p(x) = g(x) × q(x) + r(x).
67. 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
68. 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
69. The quadratic polynomial whose sum of zeroes is 3 and product of zeroes is –2 is :
(a) x2 + 3x – 2
(b) x2 – 2x + 3
(c) x2 – 3x + 2
(d) x2 – 3x – 2
70.If (x + 1) is a factor of 2x3 + ax2 + 2bx + 1, then find the values of a and b given that 2a – 3b = 4
(a) a = –1, b = –2
(b) a = 2, b = 5
(c) a = 5, b = 2
(d) a = 2, b = 0
71.The number of zeroes that polynomial f(x) = (x – 2)2 + 4 can have is:
(a) 1
(b) 2
(c) 0
(d) 3
72. The zeroes of the polynomial f(x) = 4x2 – 12x + 9 are:
(a) 3/2. , 3/2
(b) -3/2. And 3/2
(c) 3, 4
(d) –3, –4
73. If p(x) is a polynomial of at least degree one and p(k) = 0, then k is known as
(a) value of p(x)
(b) zero of p(x)
(c) constant term of p(x)
(c) none of these
64. If p(x) = ax + b, then zero of p(x)
(a) a
(b) b
(c) - a/b
(d) - b/a
75. Graph of a quadratic polynomial is a
(a) straight line
(b) circle
(c) parabola
(d) ellipse
76. Zeroes of a polynomial can be determined graphically. No. of zeroes of a polynomial is equal to no. of points where the graph of polynomial
(a) intersects y-axis
(b) intersects x-axis
(c) intersects y-axis or intersects x-axis
(d) none of these
77. If graph of a polynomial does not intersects the x-axis but intersects y-axis in one point, then no, of zeroes of the polynomial is equal to
(a) 0
(b) 1
(c) 0 or 1
(d) none of these
78. A polynomial of degree n has
(a) only 1 zero
(b) exactly n zeroes
(c) atmost n zeroes
(d) more than n zeroes
79. If p(x) = ax2 + bx + c, then
is equal to c/a
(a) 0
(b) 1
(c) sum of zeroes
(d) product of zeroes
80. If p(x) = ax2 + bx + c, then
is equal to
(a) 0
(b) 1
(c) product of zeroes
(d) sum of zeroes
81. If p(x) = ax2 + bx + c, and a + b + c = 0, then
- b/a one zero
(a)
(b)
(c)
(d) none of these
82. If p(x) = ax2 + bx + c and a + c = b, then one of the zeroes is
(a) -b/a
(b) c/a
83 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
.84. 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
.85.Given that two of the zeroes of the cubic poly-nomial ax3 + bx² + cx + d are 0, the third zero is
86. If one of the zeroes of the quadratic polynomial (k – 1) x² + kx + 1 is – 3, then the value of k is
87. A quadratic polynomial, whose zeroes are -3 and 4, is
(a) x²- x + 12
(b) x² + x + 12
(c) \(\frac{x^{2}}{2}-\frac{x}{2}-6\)
(d) 2x² + 2x – 24
88. If the zeroes of the quadratic polynomial x2 + (a + 1) x + b are 2 and -3, then
(a) a = -7, b = -1
(b) a = 5, b = -1
(c) a = 2, b = -6
(d) a – 0, b = -6
.89.The number of polynomials having zeroes as -2 and 5 is
(a) 1
(b) 2
(c) 3
(d) more than 3
90 Given that one of the zeroes of the cubic polynomial ax3 + bx² + cx + d is zero, the product of the other two zeroes is
91.. If one of the zeroes of the cubic polynomial x3 + ax² + bx + c is -1, then the product of the
other two zeroes is
(a) b – a + 1
(b) b – a – 1
(c) a – b + 1
(d) a – b – 1
92. 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
93. The zeroes of the quadratic polynomial x² + kx + k, k? 0,
(a) cannot both be positive
(b) cannot both be negative
(c) are always unequal
(d) are always equal
94. 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
95. If one of the zeroes of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it
(a) has no linear term and the constant term is negative.
(b) has no linear term and the constant term is positive.
(c) can have a linear term but the constant term is negative.
(d) can have a linear term but the constant term is positive.
96. Which of the following is not the graph of quadratic polynomial?
97. The number of polynomials having zeroes as 4 and 7 is
(a) 2
(b) 3
(c) 4
(d) more than 4
98 A quadratic polynomial, whose zeores are -4 and -5, is
(a) x²-9x + 20
(b) x² + 9x + 20
(c) x²-9x- 20
(d) x² + 9x- 20
99. The zeroes of the quadratic polynomial x² + 1750x + 175000 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
100. The zeroes of the quadratic polynomial x² – 15x + 50 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
101. The zeroes of the quadratic polynomial 3x² – 48 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
102. The zeroes of the quadratic polynomial x² – 18x + 81 are
(a) both negative
(b) one positive and one negative
(c) both positive and unequal
(d) both equal and positive
103.. The zeroes of the quadratic polynomial x² + px + p, p ≠ 0 are
(a) both equal
(b) both cannot be positive
(c) both unequal
(d) both cannot be negative
104.. If one of the zeroes of the quadratic polynomial (p – l)x² + px + 1 is -3, then the value of p is
105. If the zeroes of the quadratic polynomial Ax² + Bx + C, C # 0 are equal, then
(a) A and B have the same sign
(b) A and C have the same sign
(c) B and C have the same sign
(d) A and C have opposite signs
106. If x3 + 1 is divided by x² + 5, then the possible degree of quotient is
(a) 0
(b) 1
(c) 2
(d) 3
107. 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
108. If x4 + 3x² + 7 is divided by 3x + 5, then the possible degrees of quotient and remainder are:
(a) 3, 0
(b) 4, 1
(c) 3, 1
(d) 4, 0
109. If x5 + 2x4 + x + 6 is divided by g(x), and quotient is x² + 5x + 7, then the possible degree of g(x) is:
(a) 4
(b) 2
(c) 3
(d) 5
110. If x5 + 2x4 + x + 6 is divided by g(x) and quo-tient is x² + 5x + 7, then the possible degree of remainder is:
(a) less than 1
(b) less than 2
(c) less than 3
(d) less than 4
111. What is the number of zeroes that a linear poly-nomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
112. What is the number(s) of zeroes that a quadratic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
113. What is the number(s) of zeores that a cubic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
114. If one of the zeroes of the cubic polynomial x3 + px² + qx + r is -1, then the product of the other two zeroes is
(a) p + q + 1
(b) p-q- 1
(c) q – p + 1
(d) q – p – 1
115. If one zero of the quadratic polynomial x² + 3x + b is 2, then the value of b is
(a) 10
(b) -8
(c) 9
(d) -10
117.If 1 is one of the zeroes of the polynomial x² + x + k, then the value of k is:
(a) 2
(b) -2
(c) 4
(d) -4
118. If p(x) is a polynomial of at least degree one and p(k) = 0, then k is known as
(a) value of p(x)
(b) zero of p(x)
(c) constant term of p{x)
(d) none of these
119. If one of the zeroes of the quadratic polynomial (k -1)x² + kx + 1 the value of k is [NCERT Exemplar Problems]
120. If the zeroes of the quadratic polynomial x² + (a + 1) x + b are 2 and -3, then [NCERT Exemplar Problem, CBSE 2011]
(a) a = -7, b = -1
(b) a = 5, b = -1
(c) a = 2, b = -6
(d) a = 0, b = -6
121. Which of the following is not the graph of a quadratic polynomial?[NCERT Exemplar Problems]
122. Zeroes of a polynomial can be determined graphically. No. of zeroes of a polynomial is equal to no. of points where the graph of polynomial
(a) intersects y-axis
(b) intersects x-axis
(c) intersects y-axis or intersects x-axis
(d) none of these
123.. If graph of a polynomial does not intersects the x-axis but intersects y-axis in one point, then no. of zeroes of the polynomial is equal to
(a) 0
(b) 1
(c) 0 or 1
(d) none of these
124. A polynomial of degree n has
(a) only 1 zero
(b) at least n zeroes
(c) atmost n zeroes
(d) more than n zeroes
125.. If p(x) = axr + bx + c, then –\(\frac{b}{a}\) is equal to
(a) 0
(b) 1
(c) product of zeroes
(d) sum of zeroes
126.. If p(x) = ax² + bx + c one zero is and a + b + c = 0, then one zero is
(a) \(\frac{-b}{a}\)
(b) \(\frac{c}{a}\)
(c) \(\frac{b}{c}\)
(d) none of these
127. If p{x) = ax2 + bx + one of the zeroes is c and a + c = b, then
128.. The number of polynomials having zeroes as -2 and 5 is [NCERT Exemplar Problems]
(a) 1
(b) 2
(c) 3
(d) more than 3
129.. Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is [NCERT Exemplar Problems]
120.. If one of the zeroes of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it
(a) has no linear term and the constant term is negative.
(b) has no linear term and the constant term is positive.
(c) can have a linear term but the constant term is negative.
(d) can have a linear term but the constant term is positive
121. If 4x² – 6x – m is divisible by x – 3, the value of m is exact divisor of
(a) 9
(b) 45
(c) 20
(d) 36
122.. Which one of the following statements is correct(a) if x6 + 1 is divided by x + 1, then the remainder is -2.
(b) if x6 + 1 is divided by x – 1, then the remainder is 2.
(c) if x6 + 1 is divided by x + 1, then the remainder is 1.
(d) if x6 + 1 is divided by x – 1, then the remainder is -1.
123. Consider the following statements
(i) x – 2 is a factor of x3 – 3x² + 4x – 4.
(ii) x + 1 is a factor of 2x3 + 4x + 6.
(iii) x – 1 is a factor of x5 + x4 – x3 + x² -x + 1.
In these statements
(a) 1 and 2 are correct
(b) 1, 2 and 3 are correct
(c) 2 and 3 are correct
(d) 1 and 3 are correct
124.. If f(x) = 5x – 10 is divided by x – √2, then the remainder will be
(a) non zero rational number
(b) an irrational number
(c) 0
(d) \(f\left(\frac{1}{\sqrt{2}}\right)\)
125. Zeroes of p(z) = z² – 27 are ______ and ______ .
126. Verify that x = 3 is a zero of the polynomial. p(x) = 2x3 – 5x² – 4x + 3
127. The graph of y =f(x) is given below. How many zeroes are there of f(x)?
128.. The graph of y = f(x) is given, how many zeroes are there of f(x)?
129.. The graph of y = f(x) is given below, for some polynomial f(x). Find the number of zeroes of f(x).
A. -c/a. B. C/a. C. 0. D. -b/a
130. The graph of x = p(y) is given below, for some polynomial p(y). Find the number of zeroes of p(y).
131.
Graph of the polynomial p(x) =px² + 4x – 4 is given as above. Find the value of p.
132.. If the product of the zeroes of x2 – 3kx + 2k1 – 1 is 7, then values of k are _____ and _____ .
133 . If zeroes ofp(x) = 2x² -Ix + k are reciprocal of each other, then value of k is _____ .
134.Find the product of the zeroes of – 2x² + kx + 6.
135.. Find the sum of the zeroes of the given quadratic polynomial -3x² + k.
136. If one zero of the polynomial x² -4x+ 1 is 2 + √3, write the other zero.
137. Write the polynomial, the product and sum of whose zeroes are –\(\frac{9}{2}\) and –\(\frac{3}{2}\) respectively.
138. The value of m, in order that x² – mx – 2 is the quotient where x3 + 3x² – 4 is divided by x + 2 is ____ .
139.. If one factor of x3 + 7kx² – 4kx + 12 is (x + 3), then the value of k is ______ .
140.. A polynomial of degree five is divided by a quadratic polynomial. If it leaves a remainder, then find the degree of remainder.
141. Check whether 3x – 7 is a factor of polynomial 6x3 + x² – 26x – 25?
142.. If x3 + x² – ax + b is divisible by x² – x, write the values of a and b.
Question 143
The maximum number of zeroes that a polynomial of degree 4 can have is
(a) One
(b) Two
(c) Three
(d) Four
Question 144
The graph of the polynomial p(x) = 3x – 2 is a straight line which intersects the x-axis at exactly one point namely
(a) (−23, 0)
(b) (0, −23)
(c) (23, 0)
(d) 23, −23
Question 145
In fig. given below, the number of zeroes of the polynomial f(x) is
Graph meet x axis at 3 points
(a) 1
(b) 2
(c) 3
(d) None
Question 146
The graph of the polynomial ax² + bx + c is an upward parabola if
(a) a > 0
(b) a < 0
(b) a = 0
(d) None
Question 147
The graph of the polynomial ax² + bx + c is a downward parabola if
(a) a > 0
(b) a < 0
(c) a = 0
(d) a = 1
Question 148
A polynomial of degree 3 is called
(a) a linear polynomial
(b) a quadratic polynomial
(c) a cubic polynomial
(d) a biquadratic polynomial
Question 149
If α, β are the zeroes of the polynomial x² – 16, then αβ(α + β) is
(a) 0
(b) 4
(c) -4
(d) 16
Question 150
If α and 1α are the zeroes of the polynomial ax² + bx + c, then value of c is
(a) 0
(b) a
(c) -a
(d) 1
Question 151
Zeroes of the polynomial x² – 11 are
(a) ±17−−√
(b) ±3–√
(c) 0
(d) None
Question 152
If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d then α + β + γ is equal
(a) −ba
(b) ba
(c) ca
(d) da
A. 3/4. B. 4/3. C. -3/4. D. -4/3
153If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d then αβ + βγ + αγ is equal to
(a) −ba
(b) ba
(c) ca
(d) da
Question 154
If the zeroes of the polynomial x³ – 3x² + x – 1 are stthe zeroes of the polynomial x³ – 3x² + x – 1 are st, s and st then value of s is
(a) 1
(b) -1
(c) 2
(d) -3
Question 155
If the sum of the zeroes of the polynomial f(x) = 2x³ – 3kx² + 4x – 5 is 6, then the value of k is
(a) 2
(b) 4
(c) -2
(d) -4
Question 156
If a polynomial of degree 4 is divided by quadratic polynomial, the degree of the remainder is
(a) ≤ 1
(b) ≥ 1
(c) 2
(d) 4
Question 157
If a – b, a and a + b are zeroes of the polynomial fix) = 2x³ – 6x² + 5x – 7, then value of a is
(a) 1
(b) 2
(c) -5
(d) 7
Question 158
Dividend is equal to
(a) divisor × quotient + remainder
(b) divisior × quotient
(c) divisior × quotient – remainder
(d) divisor × quotient × remainder
Question 159
A quadratic polynomial whose sum of the zeroes is 2 and product is 1 is given by
(a) x² – 2x + 1
(b) x² + 2x + 1
(c) x² + 2x – 1
(d) x² – 2x – 1
Question 160
If one of the zeroes of a quadratic polynomial ax² + bx + c is 0, then the other zero is
(a) −ba
(b) 0
(c) ba
(d) −ca
Question 161
The sum and the product of the zeroes of polynomial 6x² – 5 respectively are
(a) 0, −65
(b) 0, 65
(c) 0, 56
(d) 0, −56
Question 162
What should be subtracted from x³ – 2x² + 4x + 1 to get 1?
(a) x³ – 2x² + 4x
(b) x³ – 2x² + 4 + 1
(c) -1
(d) 1
Class 10 Maths MCQs Chapter 2 Polynomials
1. 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
2. Given that two of the zeroes of the cubic poly-nomial ax3 + bx² + cx + d are 0, the third zero is
A. -b/a. B. B/a. C. C/a. D. - d/a.
3. If one of the zeroes of the quadratic polynomial (k – 1) x² + kx + 1 is – 3, then the value of k is
A. 4/3. B. -4/3. C. 2/3. D. -2/3
4. A quadratic polynomial, whose zeroes are -3 and 4, is
(a) x²- x + 12
(b) x² + x + 12
(c) x22−x2−6
(d) 2x² + 2x – 24
5. If the zeroes of the quadratic polynomial x2 + (a + 1) x + b are 2 and -3, then
(a) a = -7, b = -1
(b) a = 5, b = -1
(c) a = 2, b = -6
(d) a – 0, b = -6
6. The number of polynomials having zeroes as -2 and 5 is
(a) 1
(b) 2
(c) 3
(d) more than 3
7. Given that one of the zeroes of the cubic polynomial ax3 + bx² + cx + d is zero, the product of the other two zeroes is
A. -c/a. B. C/a. C.0. D. -b/a
8. If one of the zeroes of the cubic polynomial x3 + ax² + bx + c is -1, then the product of the
other two zeroes is
(a) b – a + 1
(b) b – a – 1
(c) a – b + 1
(d) a – b – 1
9. 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
10. The zeroes of the quadratic polynomial x² + kx + k, k? 0,
(a) cannot both be positive
(b) cannot both be negative
(c) are always unequal
(d) are always equal
11. 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
12. If one of the zeroes of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it
(a) has no linear term and the constant term is negative.
(c) can have a linear term but the constant term is negative.
(d) can have a linear term but the constant term is positive.
13. Which of the following is not the graph of quadratic polynomial?
14. The number of polynomials having zeroes as 4 and 7 is
(a) 2
(b) 3
(c) 4
(d) more than 4
15. A quadratic polynomial, whose zeores are -4 and -5, is
(a) x²-9x + 20
(b) x² + 9x + 20
(c) x²-9x- 20
(d) x² + 9x- 20
16. The zeroes of the quadratic polynomial x² + 1750x + 175000 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
17. The zeroes of the quadratic polynomial x² – 15x + 50 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
18. The zeroes of the quadratic polynomial 3x² – 48 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
19. The zeroes of the quadratic polynomial x² – 18x + 81 are
(a) both negative
(b) one positive and one negative
(c) both positive and unequal
(d) both equal and positive
20. The zeroes of the quadratic polynomial x² + px + p, p ≠ 0 are
(a) both equal
(b) both cannot be positive
(c) both unequal
(d) both cannot be negative
21. If one of the zeroes of the quadratic polynomial (p – l)x² + px + 1 is -3, then the value of p is
A.3/4. B. 4/3. C. -3/4. D. -4/3
22. If the zeroes of the quadratic polynomial Ax² + Bx + C, C # 0 are equal, then
(a) A and B have the same sign
(b) A and C have the same sign
(c) B and C have the same sign
(d) A and C have opposite signs
23. If x3 + 1 is divided by x² + 5, then the possible degree of quotient is
(a) 0
(b) 1
(c) 2
(d) 3
24. 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
25. If x4 + 3x² + 7 is divided by 3x + 5, then the possible degrees of quotient and remainder are:
(a) 3, 0
(b) 4, 1
(c) 3, 1
(d) 4, 0
26. If x5 + 2x4 + x + 6 is divided by g(x), and quotient is x² + 5x + 7, then the possible degree of g(x) is:
(a) 4
(b) 2
(c) 3
(d) 5
27. If x5 + 2x4 + x + 6 is divided by g(x) and quo-tient is x² + 5x + 7, then the possible degree of remainder is:
(a) less than 1
(b) less than 2
(c) less than 3
(d) less than 4
28. What is the number of zeroes that a linear poly-nomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
29. What is the number(s) of zeroes that a quadratic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
30. What is the number(s) of zeores that a cubic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
31. If one of the zeroes of the cubic polynomial x3 + px² + qx + r is -1, then the product of the other two zeroes is
(a) p + q + 1
(b) p-q- 1
(c) q – p + 1
(d) q – p – 1
32. If one zero of the quadratic polynomial x² + 3x + b is 2, then the value of b is
(a) 10
(b) -8
(c) 9
(d) -10
33. If 1 is one of the zeroes of the polynomial x² + x + k, then the value of k is:
(a) 2
(b) -2
(c) 4
(d) -4
.34. The zeroes of the quadratic polynomial x2 + 99x + 127are
(A) both positive
(B) both negative
(C) one positive and one negative
(D) both equal
35.If the zeroes of the quadratic polynomial x2 + bx + c , c ≠ 0are 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
.36. The number of polynomials having zeroes as –2 and 5 is
(A) 1
(B) 2
(C) 3
(D) more than 3
37. The degree of the polynomial (x + 1)(x2 – x – x4 +1) is:
(A)2
(B) 3
(C) 4
D.5
.38. If the zeroes of the quadratic polynomial x2 + (a + 1)x + b are 2 and –3, then
(A) a = –7, b = –1
(B) a = 5, b = –1
(C) a = 2, b = – 6
(D) a = 0, b = – 6
.39. Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is
(A) –c/a
(B) c/a
(C) 0
(D) 3
40.. If one of the zeroes of the cubic polynomial x3 + ax2 + bx + c is –1, then the
Product of the other two zeroes is
(A) b – a + 1
(B) b – a – 1
(C) a – b + 1
(D) a – b –1
41.. If one of the zeroes of the quadratic polynomial (k – 1)x2 + kx + 1 is –3, then the value of k is
(A) 4/3
(B) – 4/3
(C) 2/3
(D) – 2/3
42.The value of p for which the polynomial x3 + 4x2 –px + 8 is exactly divisible by (x – 2) is:
(A) 0
(B) 3
(C) 5
(D) 16
43. If sum of the squares of zeroes of the quadratic polynomial 6x2 + x + k is 25/36, the value of k is:
(A) 4
(B) – 4
(C) 2
(D) – 2
44. If α and β are zeroes of x2 – 4x + 1, then 1/α + 1/β – αβ is
(A) 3
(B) 5
(C) –5
(D) –3
45.. If (x + 1) is a factor of x2− 3ax +3a − 7, then the value of a is:
(A) 1
(B) –1
(C) 0
(D) –2
46.If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it
(A) has no linear term and the constant term is negative.
(B) has no linear term and the constant term is positive.
(C) can have a linear term but the constant term is negative.
(D) can have a linear term but the constant term is positive
47. If α, β are zeroes of x2 –6x + k, what is the value of k if 3α + 2β = 20?
(A)–16
(B) 8
(C) 2
(D) –8
48.The zeroes of x2–2x –8 are:
(a) (2,-4)
(b) (4,-2)
(c) (-2,-2)
(d) (-4,-4)
49.. What is the quadratic polynomial whose sum and the product of zeroes is √2, ⅓ respectively?
(a) 3x2-3√2x+1
(b) 3x2+3√2x+1
(c) 3x2+3√2x-1
(d) None of the above
50. 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
51.. The degree of the polynomial, x4 – x2 +2 is
(a) 2
(b) 4
(c) 1
(d) 0
52. 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
53. 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)
Explanation: Let p(x) = mx+n
Put x = a
p(a)=ma+n=0
So, a is zero of p(x).
54.. 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
55. 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
56. The number of polynomials having zeroes as -2 and 5 is:
(a) 1
(b) 2
(c) 3
(d) More than 3
57.. Zeroes of p(x) = x2-27 are:
(a) ±9√3
(b) ±3√3
(c) ±7√3
(d) None of the above
58. 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
59.. 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
60. 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
61. 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
62. The zeroes of the quadratic polynomial x2 + 7x + 10 are
(a) -4, -3
(b) 2, 5
(c) -2, -5
(d) -2, 5
63. 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 root
64If the discriminant of a quadratic polynomial, D > 0, then the polynomial has two real and unequal roots.
65. If on division of a polynomial p(x) by a polynomial g(x), the quotient is zero, then the relation between the degrees of p(x) and g(x) is
(a) degree of p(x) < degree of g(x)
(b) degree of p(x) = degree of g(x)
(c) degree of p(x) > degree of g(x)
(d) nothing can be said about degrees of p(x) and g(x)
Answer: (a) degree of p(x) < degree of g(x)
Explanation:
We know that, p(x)= g(x) × q(x) + r(x)
Given that, q(x) = 0
When q(x) = 0, then r(x) = 0
So, now when we divide p(x) by g(x),
Then p(x) should be equal to zero.
If r(x) = 0, then the degree of p(x) < degree of g(x).
66. By division algorithm of polynomials, p(x) =
(a) g(x) × q(x) + r(x)
(b) g(x) × q(x) – r(x)
(c) g(x) × q(x) × r(x)
(d) g(x) + q(x) + r(x)
Answer: (a) g(x) × q(x) + r(x)
By division algorithm of polynomials, p(x) = g(x) × q(x) + r(x).
67. 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
68. 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
69. The quadratic polynomial whose sum of zeroes is 3 and product of zeroes is –2 is :
(a) x2 + 3x – 2
(b) x2 – 2x + 3
(c) x2 – 3x + 2
(d) x2 – 3x – 2
70.If (x + 1) is a factor of 2x3 + ax2 + 2bx + 1, then find the values of a and b given that 2a – 3b = 4
(a) a = –1, b = –2
(b) a = 2, b = 5
(c) a = 5, b = 2
(d) a = 2, b = 0
71.The number of zeroes that polynomial f(x) = (x – 2)2 + 4 can have is:
(a) 1
(b) 2
(c) 0
(d) 3
72. The zeroes of the polynomial f(x) = 4x2 – 12x + 9 are:
(a) 3/2. , 3/2
(b) -3/2. And 3/2
(c) 3, 4
(d) –3, –4
73. If p(x) is a polynomial of at least degree one and p(k) = 0, then k is known as
(a) value of p(x)
(b) zero of p(x)
(c) constant term of p(x)
(c) none of these
64. If p(x) = ax + b, then zero of p(x)
(a) a
(b) b
(c) - a/b
(d) - b/a
75. Graph of a quadratic polynomial is a
(a) straight line
(b) circle
(c) parabola
(d) ellipse
76. Zeroes of a polynomial can be determined graphically. No. of zeroes of a polynomial is equal to no. of points where the graph of polynomial
(a) intersects y-axis
(b) intersects x-axis
(c) intersects y-axis or intersects x-axis
(d) none of these
77. If graph of a polynomial does not intersects the x-axis but intersects y-axis in one point, then no, of zeroes of the polynomial is equal to
(a) 0
(b) 1
(c) 0 or 1
(d) none of these
78. A polynomial of degree n has
(a) only 1 zero
(b) exactly n zeroes
(c) atmost n zeroes
(d) more than n zeroes
79. If p(x) = ax2 + bx + c, then
is equal to c/a
(a) 0
(b) 1
(c) sum of zeroes
(d) product of zeroes
80. If p(x) = ax2 + bx + c, then
is equal to
(a) 0
(b) 1
(c) product of zeroes
(d) sum of zeroes
81. If p(x) = ax2 + bx + c, and a + b + c = 0, then
- b/a one zero
(a)
(b)
(c)
(d) none of these
82. If p(x) = ax2 + bx + c and a + c = b, then one of the zeroes is
(a) -b/a
(b) c/a
83 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
.84. 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
.85.Given that two of the zeroes of the cubic poly-nomial ax3 + bx² + cx + d are 0, the third zero is
86. If one of the zeroes of the quadratic polynomial (k – 1) x² + kx + 1 is – 3, then the value of k is
87. A quadratic polynomial, whose zeroes are -3 and 4, is
(a) x²- x + 12
(b) x² + x + 12
(c) \(\frac{x^{2}}{2}-\frac{x}{2}-6\)
(d) 2x² + 2x – 24
88. If the zeroes of the quadratic polynomial x2 + (a + 1) x + b are 2 and -3, then
(a) a = -7, b = -1
(b) a = 5, b = -1
(c) a = 2, b = -6
(d) a – 0, b = -6
.89.The number of polynomials having zeroes as -2 and 5 is
(a) 1
(b) 2
(c) 3
(d) more than 3
90 Given that one of the zeroes of the cubic polynomial ax3 + bx² + cx + d is zero, the product of the other two zeroes is
91.. If one of the zeroes of the cubic polynomial x3 + ax² + bx + c is -1, then the product of the
other two zeroes is
(a) b – a + 1
(b) b – a – 1
(c) a – b + 1
(d) a – b – 1
92. 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
93. The zeroes of the quadratic polynomial x² + kx + k, k? 0,
(a) cannot both be positive
(b) cannot both be negative
(c) are always unequal
(d) are always equal
94. 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
95. If one of the zeroes of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it
(a) has no linear term and the constant term is negative.
(b) has no linear term and the constant term is positive.
(c) can have a linear term but the constant term is negative.
(d) can have a linear term but the constant term is positive.
96. Which of the following is not the graph of quadratic polynomial?
97. The number of polynomials having zeroes as 4 and 7 is
(a) 2
(b) 3
(c) 4
(d) more than 4
98 A quadratic polynomial, whose zeores are -4 and -5, is
(a) x²-9x + 20
(b) x² + 9x + 20
(c) x²-9x- 20
(d) x² + 9x- 20
99. The zeroes of the quadratic polynomial x² + 1750x + 175000 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
100. The zeroes of the quadratic polynomial x² – 15x + 50 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
101. The zeroes of the quadratic polynomial 3x² – 48 are
(a) both negative
(b) one positive and one negative
(c) both positive
(d) both equal
102. The zeroes of the quadratic polynomial x² – 18x + 81 are
(a) both negative
(b) one positive and one negative
(c) both positive and unequal
(d) both equal and positive
103.. The zeroes of the quadratic polynomial x² + px + p, p ≠ 0 are
(a) both equal
(b) both cannot be positive
(c) both unequal
(d) both cannot be negative
104.. If one of the zeroes of the quadratic polynomial (p – l)x² + px + 1 is -3, then the value of p is
105. If the zeroes of the quadratic polynomial Ax² + Bx + C, C # 0 are equal, then
(a) A and B have the same sign
(b) A and C have the same sign
(c) B and C have the same sign
(d) A and C have opposite signs
106. If x3 + 1 is divided by x² + 5, then the possible degree of quotient is
(a) 0
(b) 1
(c) 2
(d) 3
107. 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
108. If x4 + 3x² + 7 is divided by 3x + 5, then the possible degrees of quotient and remainder are:
(a) 3, 0
(b) 4, 1
(c) 3, 1
(d) 4, 0
109. If x5 + 2x4 + x + 6 is divided by g(x), and quotient is x² + 5x + 7, then the possible degree of g(x) is:
(a) 4
(b) 2
(c) 3
(d) 5
110. If x5 + 2x4 + x + 6 is divided by g(x) and quo-tient is x² + 5x + 7, then the possible degree of remainder is:
(a) less than 1
(b) less than 2
(c) less than 3
(d) less than 4
111. What is the number of zeroes that a linear poly-nomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
112. What is the number(s) of zeroes that a quadratic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
113. What is the number(s) of zeores that a cubic polynomial has/have:
(a) 0
(b) 1
(c) 2
(d) 3
114. If one of the zeroes of the cubic polynomial x3 + px² + qx + r is -1, then the product of the other two zeroes is
(a) p + q + 1
(b) p-q- 1
(c) q – p + 1
(d) q – p – 1
115. If one zero of the quadratic polynomial x² + 3x + b is 2, then the value of b is
(a) 10
(b) -8
(c) 9
(d) -10
117.If 1 is one of the zeroes of the polynomial x² + x + k, then the value of k is:
(a) 2
(b) -2
(c) 4
(d) -4
118. If p(x) is a polynomial of at least degree one and p(k) = 0, then k is known as
(a) value of p(x)
(b) zero of p(x)
(c) constant term of p{x)
(d) none of these
119. If one of the zeroes of the quadratic polynomial (k -1)x² + kx + 1 the value of k is [NCERT Exemplar Problems]
120. If the zeroes of the quadratic polynomial x² + (a + 1) x + b are 2 and -3, then [NCERT Exemplar Problem, CBSE 2011]
(a) a = -7, b = -1
(b) a = 5, b = -1
(c) a = 2, b = -6
(d) a = 0, b = -6
121. Which of the following is not the graph of a quadratic polynomial?[NCERT Exemplar Problems]
122. Zeroes of a polynomial can be determined graphically. No. of zeroes of a polynomial is equal to no. of points where the graph of polynomial
(a) intersects y-axis
(b) intersects x-axis
(c) intersects y-axis or intersects x-axis
(d) none of these
123.. If graph of a polynomial does not intersects the x-axis but intersects y-axis in one point, then no. of zeroes of the polynomial is equal to
(a) 0
(b) 1
(c) 0 or 1
(d) none of these
124. A polynomial of degree n has
(a) only 1 zero
(b) at least n zeroes
(c) atmost n zeroes
(d) more than n zeroes
125.. If p(x) = axr + bx + c, then –\(\frac{b}{a}\) is equal to
(a) 0
(b) 1
(c) product of zeroes
(d) sum of zeroes
126.. If p(x) = ax² + bx + c one zero is and a + b + c = 0, then one zero is
(a) \(\frac{-b}{a}\)
(b) \(\frac{c}{a}\)
(c) \(\frac{b}{c}\)
(d) none of these
127. If p{x) = ax2 + bx + one of the zeroes is c and a + c = b, then
128.. The number of polynomials having zeroes as -2 and 5 is [NCERT Exemplar Problems]
(a) 1
(b) 2
(c) 3
(d) more than 3
129.. Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is [NCERT Exemplar Problems]
120.. If one of the zeroes of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it
(a) has no linear term and the constant term is negative.
(b) has no linear term and the constant term is positive.
(c) can have a linear term but the constant term is negative.
(d) can have a linear term but the constant term is positive
121. If 4x² – 6x – m is divisible by x – 3, the value of m is exact divisor of
(a) 9
(b) 45
(c) 20
(d) 36
122.. Which one of the following statements is correct(a) if x6 + 1 is divided by x + 1, then the remainder is -2.
(b) if x6 + 1 is divided by x – 1, then the remainder is 2.
(c) if x6 + 1 is divided by x + 1, then the remainder is 1.
(d) if x6 + 1 is divided by x – 1, then the remainder is -1.
123. Consider the following statements
(i) x – 2 is a factor of x3 – 3x² + 4x – 4.
(ii) x + 1 is a factor of 2x3 + 4x + 6.
(iii) x – 1 is a factor of x5 + x4 – x3 + x² -x + 1.
In these statements
(a) 1 and 2 are correct
(b) 1, 2 and 3 are correct
(c) 2 and 3 are correct
(d) 1 and 3 are correct
124.. If f(x) = 5x – 10 is divided by x – √2, then the remainder will be
(a) non zero rational number
(b) an irrational number
(c) 0
(d) \(f\left(\frac{1}{\sqrt{2}}\right)\)
125. Zeroes of p(z) = z² – 27 are ______ and ______ .
126. Verify that x = 3 is a zero of the polynomial. p(x) = 2x3 – 5x² – 4x + 3
127. The graph of y =f(x) is given below. How many zeroes are there of f(x)?
128.. The graph of y = f(x) is given, how many zeroes are there of f(x)?
129.. The graph of y = f(x) is given below, for some polynomial f(x). Find the number of zeroes of f(x).
A. -c/a. B. C/a. C. 0. D. -b/a
130. The graph of x = p(y) is given below, for some polynomial p(y). Find the number of zeroes of p(y).
131.
Graph of the polynomial p(x) =px² + 4x – 4 is given as above. Find the value of p.
132.. If the product of the zeroes of x2 – 3kx + 2k1 – 1 is 7, then values of k are _____ and _____ .
133 . If zeroes ofp(x) = 2x² -Ix + k are reciprocal of each other, then value of k is _____ .
134.Find the product of the zeroes of – 2x² + kx + 6.
135.. Find the sum of the zeroes of the given quadratic polynomial -3x² + k.
136. If one zero of the polynomial x² -4x+ 1 is 2 + √3, write the other zero.
137. Write the polynomial, the product and sum of whose zeroes are –\(\frac{9}{2}\) and –\(\frac{3}{2}\) respectively.
138. The value of m, in order that x² – mx – 2 is the quotient where x3 + 3x² – 4 is divided by x + 2 is ____ .
139.. If one factor of x3 + 7kx² – 4kx + 12 is (x + 3), then the value of k is ______ .
140.. A polynomial of degree five is divided by a quadratic polynomial. If it leaves a remainder, then find the degree of remainder.
141. Check whether 3x – 7 is a factor of polynomial 6x3 + x² – 26x – 25?
142.. If x3 + x² – ax + b is divisible by x² – x, write the values of a and b.
Question 143
The maximum number of zeroes that a polynomial of degree 4 can have is
(a) One
(b) Two
(c) Three
(d) Four
Question 144
The graph of the polynomial p(x) = 3x – 2 is a straight line which intersects the x-axis at exactly one point namely
(a) (−23, 0)
(b) (0, −23)
(c) (23, 0)
(d) 23, −23
Question 145
In fig. given below, the number of zeroes of the polynomial f(x) is
Graph meet x axis at 3 points
(a) 1
(b) 2
(c) 3
(d) None
Question 146
The graph of the polynomial ax² + bx + c is an upward parabola if
(a) a > 0
(b) a < 0
(b) a = 0
(d) None
Question 147
The graph of the polynomial ax² + bx + c is a downward parabola if
(a) a > 0
(b) a < 0
(c) a = 0
(d) a = 1
Question 148
A polynomial of degree 3 is called
(a) a linear polynomial
(b) a quadratic polynomial
(c) a cubic polynomial
(d) a biquadratic polynomial
Question 149
If α, β are the zeroes of the polynomial x² – 16, then αβ(α + β) is
(a) 0
(b) 4
(c) -4
(d) 16
Question 150
If α and 1α are the zeroes of the polynomial ax² + bx + c, then value of c is
(a) 0
(b) a
(c) -a
(d) 1
Question 151
Zeroes of the polynomial x² – 11 are
(a) ±17−−√
(b) ±3–√
(c) 0
(d) None
Question 152
If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d then α + β + γ is equal
(a) −ba
(b) ba
(c) ca
(d) da
A. 3/4. B. 4/3. C. -3/4. D. -4/3
153If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d then αβ + βγ + αγ is equal to
(a) −ba
(b) ba
(c) ca
(d) da
Question 154
If the zeroes of the polynomial x³ – 3x² + x – 1 are stthe zeroes of the polynomial x³ – 3x² + x – 1 are st, s and st then value of s is
(a) 1
(b) -1
(c) 2
(d) -3
Question 155
If the sum of the zeroes of the polynomial f(x) = 2x³ – 3kx² + 4x – 5 is 6, then the value of k is
(a) 2
(b) 4
(c) -2
(d) -4
Question 156
If a polynomial of degree 4 is divided by quadratic polynomial, the degree of the remainder is
(a) ≤ 1
(b) ≥ 1
(c) 2
(d) 4
Question 157
If a – b, a and a + b are zeroes of the polynomial fix) = 2x³ – 6x² + 5x – 7, then value of a is
(a) 1
(b) 2
(c) -5
(d) 7
Question 158
Dividend is equal to
(a) divisor × quotient + remainder
(b) divisior × quotient
(c) divisior × quotient – remainder
(d) divisor × quotient × remainder
Question 159
A quadratic polynomial whose sum of the zeroes is 2 and product is 1 is given by
(a) x² – 2x + 1
(b) x² + 2x + 1
(c) x² + 2x – 1
(d) x² – 2x – 1
Question 160
If one of the zeroes of a quadratic polynomial ax² + bx + c is 0, then the other zero is
(a) −ba
(b) 0
(c) ba
(d) −ca
Question 161
The sum and the product of the zeroes of polynomial 6x² – 5 respectively are
(a) 0, −65
(b) 0, 65
(c) 0, 56
(d) 0, −56
Question 162
What should be subtracted from x³ – 2x² + 4x + 1 to get 1?
(a) x³ – 2x² + 4x
(b) x³ – 2x² + 4 + 1
(c) -1
(d) 1
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