Page No 43:
Question 1:
Find the zeros of the quadratic polynomial (x2 + 3x − 10) and verify the relation between its zeros and coefficients.
Answer:
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Question 2:
Find the zeros of the quadratic polynomial (6x2 − 7x − 3) and verify the relation between its zeros an coefficients.
Answer:
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Question 3:
Find the zeros of the quadratic polynomial 4x2 − 4x − 3 and verify the relation between the zeros and the coefficients.
Answer:
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Question 4:
Find the zeros of the quadratic polynomial 5x2 − 4 − 8x and verify the relationship between the zeros and the coefficients of the given polynomial.
Answer:
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Question 5:
Find the zeros of the quadratic polynomial 6x2 − 3 − 7x and verify the relationship between the zeros and the coefficients of the given polynomial.
Answer:
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Question 6:
Find the zeros of the quadratic polynomial 2x2 − 11x + 15 and verify the relation between the zeros and the coefficients.
Answer:
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Question 7:
Find the zeros of the quadratic polynomial (x2 − 5) and verify the relation between the zeros and the coefficients.
Answer:
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Question 8:
Find the zeros of the quadratic polynomial (8 x2 − 4) and verify the relation between the zeros and the coefficients.
Answer:
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Question 9:
Find the zeros of the quadratic polynomial (5u2 + 10u) and verify the relation between the zeros and the coefficients.
Answer:
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Question 10:
Find the quadratic polynomial whose zeros are 2 and −6. Verify the relation between the coefficients and the zeros of the polynomial.
Answer:
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Question 11:
Find the quadratic polynomial whose zeros are and . Verify the relation between the coefficients and the zeros of the polynomial.
Answer:
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Question 12:
Find the quadratic polynomial, sum of whose zeros is 8 and their product is 12. Hence, find the zeros of the polynomial.
Answer:
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Question 13:
Find the quadratic polynomial, the sum of whose zeros is −5 and their product is 6. Hence, find the zeros of the polynomial.
Answer:
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Question 14:
Find the quadratic polynomial, the sum of whose zeros is and their product is 1. Hence, find the zeros of the polynomial.
Answer:
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Question 15:
Find the quadratic polynomial, the sum of whose zeros is 0 and their product is −1. Hence, find the zeros of the polynomial.
Answer:
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Question 16:
Find the quadratic polynomial, the sum of whose zeros is and their product is −12. Hence, find the zeros of the polynomial.
Answer:
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Question 17:
If α, β are the zeros of a polynomial, such that α + β = 6 and αβ = 4, then write the polynomial.
Answer:
Page No 52:
Question 1:
Verity that 3, −2, 1 are the zeros of the cubic polynomial p(x) = x3 − 2x2 − 5x + 6 and verify the relation between its zeros and coefficients.
Answer:
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Question 2:
Verify that 5, −2 and are the zeros of the cubic polynomial p(x) = 3x3 − 10x2 − 27x + 10 and verify the relation between its zeros and coefficients.
Answer:
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Question 3:
Find a cubic polynomial whose zeros are −2, −3 and −1.
Answer:
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Question 4:
Find a cubic polynomial whose zeros are 3, and −1.
Answer:
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Question 5:
When f(x) = 4x3 − 8x2 + 8x + 1 is divided by a polynomial g(x), we get (2x − 1) as quotient and (x + 3) as remainder. Find g(x).
Answer:
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Question 6:
Divide (2x2 + x − 15) by (x + 3) and verify the division algorithm.
Answer:

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Question 7:
Divide (12 − 17x − 5x2) by (3 − 5x) and verify the division algorithm.
Answer:
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Question 8:
Divide (3x3 − 4x2 + 7x − 2) by (x2 − x + 2) and verify the division algorithm.
Answer:
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Question 9:
Divide (6 + 19x + x2 − 6x3) by (2+ 5x − 3x2) and verify the division algorithm.
Answer:
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Question 10:
It being given that 2 is one of the zeros of the polynomial x3 − 4x2 + x + 6. Find its other zeros.
Answer:
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Question 11:
It is given that −1 is one of the zeros of the polynomial x3 + 2x2 − 11x − 12. Find all the given zeros of the given polynomial.
Answer:
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Question 12:
If 1 and −2 are two zeros of the polynomial (x3 − 4x2 − 7x + 10), find its third zero.
Answer:
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Question 13:
If 3 and −3 are two zeros of the polynomial (x4 + x3 − 11x2 − 9x + 18), find all the zeros of the given polynomial.
Answer:
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Question 14:
If 2 and −2 are two zeros of the polynomial (x4 + x3 − 34x2 − 4x + 120), find all the zeros of given polynomial.
Answer:
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Question 15:
Find all the zeros of (x4 + x3 − 23x2 − 3x + 60), if it is given that two of its zeros are and .
Answer:
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Question 16:
Find all the zeros of (2x4 − 3x3 − 5x2 + 9x − 3), it being given that two of its zeros are and .
Answer:
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Question 17:
Find all the zeros of the polynomial (2x4 − 11x3 + 7x2 + 13x), it being given that two if its zeros are and .
Answer:
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Question 18:
Obtain all other zeros of (x4 + 4x3 − 2x2 − 20x − 15) if two of its zeros are and .
Answer:
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Page No 56:
Question 1:
Which of the following is a polynomial?
(a)
(b)
(c)
(d) None of these
Answer:
(d) none of these
A polynomial in x of degree n is an expression of the form p(x) =ao +a1x+a2x2 +...+an xn, where an 0.
Page No 57:
Question 2:
Which of the following is not a polynomial?
(a)
(b)
(c)
(d)
Answer:
It is because in the second term, the degree of x is −1 and an expression with a negative degree is not a polynomial.
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Question 3:
The zeros of the polynomial x2 − 2x − 3 are
(a) −3, 1
(b) −3, −1
(c) 3, −1
(d) 3, 1
Answer:
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Question 4:
The zeros of the polynomial are
(a)
(b)
(c)
(d)
Answer:
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Question 5:
The zeros of the polynomial are
(a)
(b)
(c)
(d) none of these
Answer:
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Question 6:
The zeros of the polynomial are
(a) −3, 4
(b)
(c)
(d) none of these
Answer:
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Question 7:
The zeros of the polynomial are
(a)
(b)
(c)
(d) none of these
Answer:
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Question 8:
A quadratic polynomial whose zeros are 5 and −3, is
(a) x2 + 2x − 15
(b) x2 − 2x + 15
(c) x2 − 2x − 15
(d) none of these
Answer:
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Question 9:
A quadratic polynomial whose zeros are and ,is
(a) 10x2 + x + 3
(b) 10x2 + x − 3
(c) 10x2 − x + 3
(d)
Answer:
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Question 10:
The sum and product of the zeros of a quadratic polynomial are 3 and −10 respectively. The quadratic polynomial is
(a) x2 − 3x + 10
(b) x2 + 3x −10
(c) x2 − 3x −10
(d) x2 + 3x + 10
Answer:
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Question 11:
How many polynomials are there having 4 and −2 as zeros?
(a) One
(b) Two
(c) Three
(d) More than three
Answer:
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Question 12:
The zeros of the quadratic polynomial x2 + 88x + 125 are
(a) both positive
(b) both negative
(c) one positive and one negative
(d) both equal
Answer:
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Question 13:
If α and β are the zero of x2 + 5x + 8, then the value of (α + β) is
(a) 5
(b) −5
(c) 8
(d) −8
Answer:
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Question 14:
If α and β are the zero of 2x2 + 5x − 8, then the value of (αβ) is
(a)
(b)
(c)
(d)
Answer:
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Question 15:
If one zero of the quadratic polynomial kx2 + 3x + k is 2, then the value of k is
(a)
(b)
(c)
(d)
Answer:
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Question 16:
If one zero of the quadratic polynomial (k − 1) x2 + kx + 1 is −4, then the value of k is
(a)
(b)
(c)
(d)
Answer:
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Question 17:
If −2 and 3 are the zeros of the quadratic polynomial x2 + (a + 1) x + b, then
(a) a = −2, b = 6
(b) a = 2, b = −6
(c) a = −2, b = −6
(d) a = 2, b = 6
Answer:
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Question 18:
If one of the zeroes of the quadratic polynomial x2 + bx + c is negative of the other, then
(a) b = 0 and c is positive
(b) b = 0 and c is negative
(c) b ≠ 0 and c is positive
(d) b ≠ 0 and c is negative
Answer:
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Question 19:
If the zeros of the quadratic polynomial ax2 + bx + c, where a ≠ 0 and c ≠ 0, are equal then
(a) c and a have the same sign
(b) c and a have opposite signs
(c) c and b have the same sign
(d) c and b have opposite sign
Answer:
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Question 20:
The zeros of the quadratic polynomial x2 + kx + k, where k > 0
(a) are both positive
(b) are both negative
(c) are always equal
(d) are always unequal
Answer:
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Question 21:
If one zero of 3x2 + 8x + k be the reciprocal of the other, then k = ?
(a) 3
(b) −3
(c)
(d)
Answer:
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Question 22:
If the sum of the zeros of the quadratic polynomial kx2 + 2x + 3k is equal to the product of its zeros, then k = ?
(a)
(b)
(c)
(d)
Answer:
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Question 23:
If α, β are the zeros of f (x) = 2x2 + 6x − 6, then
(a) α + β = αβ
(b) α + β > αβ
(c) α + β < αβ
(d) α + β + αβ = 0
Answer:
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Question 24:
If α, β are the zeros of the polynomial x2 − 5x + c and α − β = 1, then c = ?
(a) 0
(b) 1
(c) 4
(d) 6
Answer:
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Question 25:
If α, β are the zeros of the polynomial x2 + 6x + 2, then
(a) 3
(b) −3
(c) 12
(d) −12
Answer:
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Question 26:
If α, β, γ are the zeros of the polynomial x3 − 6x2 − x + 30, then (αβ + βγ + γα) = ?
(a) −1
(b) 1
(c) −5
(d) 30
Answer:
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Question 27:
If α, β, γ are the zeros of the polynomial 2x3 + x2 − 13x + 6, then αβγ = ?
(a) −3
(b) 3
(c)
(d)
Answer:
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Question 28:
If α, β, γ be the zeros of the polynomial p(x) such that (α + β + γ) = 3, (αβ + βγ + γα) = −
10 and αβγ = −24, then p(x) = ?
(a) x3 + 3x2 − 10x + 24
(b) x3 + 3x2 + 10x −24
(c) x3 − 3x2 −10x + 24
(d) None of these
Answer:
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Question 29:
If two of the zeros of the cubic polynomial ax3 + bx2 + cx + d is 0, then the third zeros is
(a)
(b)
(c)
(d)
Answer:
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Question 30:
If one of the zeros of the cubic polynomial ax3 + bx2 + cx + d is 0, then the product of the other two zeros is
(a)
(b)
(c) 0
(d)
Answer:
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Question 31:
If one of the zeros of the cubic polynomial x3 + ax2 + bx + c is −1, then the product of the other two zeros is
(a) a − b − 1
(b) b − a − 1
(c) 1 − a + b
(d) 1 + a − b
Answer:
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Question 32:
If the zeros of the polynomial x3 − 3x2 + x + 1 are a −d, a and a + d, then a + d is
(a) a natural number
(b) an integer
(c) a rational number
(d) an irrational number
Answer:
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Question 33:
If α, β be the zeros of the polynomial x2 − 8x + k such that α2 + β2 = 40, then k = ?
(a) 6
(b) 9
(c) 12
(d) −12
Answer:
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Question 34:
If α, β be the zero of the polynomial 2x2 + 5x + k such that α2 + β2 + αβ = , then k = ?
(a) 3
(b) −3
(c) −2
(d) 2
Answer:
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Question 35:
On dividing a polynomial p(x) by a non-zero polynomial q(x), let g(x) be the quotient and r(x) be the remainder, than p(x) = q(x)⋅g(x) + r(x), where
(a) r(x) = 0 always
(b) deg r (x) <deg g(x) always
(c) either r(x) = 0 or deg r(x) <deg g(x)
(d) r(x) = g(x)
Answer:
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Question 36:
Which of the following is a true statement?
(a) x2 + 5x − 3 is a linear polynomial.
(b) x2 + 4x − 1 is a binomial.
(c) x + 1 is a monomial.
(d) 5x3 is a monomial.
Answer:
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Question 37:
If α, β are the zeros of the polynomial ax2 + bx + c, then (α2 + β2) = ?
(a)
(b)
(c)
(d)
Answer:
Page No 60:
Answer:
(c)
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Question 39:
Read the statements given below:
I. If α, β are the zeros of the polynomial x2 − p(x + 1) −c, then (a + 1)(β + 1) = 1 − c.
II. If α, β are the zeros of the polynomial x2 + px + q, then the polynomial having as zeros is qx2 + px + 1.
III. When x3 + 3x2 − 5x + 4 is divided by (x + 1), then the remainder is 9.
Which of the above statements is false?
(a) I only
(b) II only
(c) III only
(d) I and III both
Answer:
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Question 40:
Read the statements given below:
I. If the polynomial p(x) = 2x3 − kx2 + 5x + 2 is exactly divisible by (x + 2), then k = −6.
II. If the polynomial q(x) = x3 − 7x + k when divided by (x − 1) leaves the remainder 2, then k = 6.
III. If two zeros of the polynomial f(x) = x3 − 5x2 − 16x + 80 are equal in magnitude and opposite in sigh, then the third zero is 5.
Which of the above statements is not true?
(a) I only
(b) II only
(c) III only
(d) I as well as II
Answer:
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Question 41:
Assertion (A)
If one zero of the polynomial p(x) = (k2 + 4) x2 + 9x + 4k is the reciprocal of the other zero, then k = 2.
Reason (R)
If (x − α) is a factor of the polynomial p(x), then α is a zero of p(x).
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.
Answer:
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Question 42:
Assertion (A)
The polynomial p(x) = x3 + x has one real zero.
Reason (R)
A polynomial of nth degree has at most n zeros.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.
Answer:
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Question 43:
Assertion (A)
If on dividing the polynomial p(x) = x2 − 3ax + 3a − 7 by (x + 1), we get 6 as remainder, then a= 3.
Reason (R)
When a polynomial p(x) is divided by (x − α), then the remainder is p(α).
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.
Answer:
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Question 44:
Assertion (A)
A monic quadratic polynomial having 4 and −2 as zeroes is x2 − 2x − 8.
Reason (R)
The monic quadratic polynomial having α and β as zeroes is given by p(x) = x2 − (α + β) x + αβ.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.
Answer:
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Question 45:
If the zeros of a quadratic polynomial ax2 + bx + c are both negative, then a, b, c will have the same sign.
(a) True
(b) False
Answer:
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Question 46:
Match the following columns:
Column I |
Column II |
(a) If α and β be the zeros of the
polynomial x2 − 5x + k such that
(α − β) = 1, then k = ......... . |
(p) 10 |
(b) If one zero of 4x2 + 17x + p is
the reciprocal of the other, then
p = ......... . |
(q) −3 |
(c) If the zeros of x3 − 6x2 + 3x + m
are (a − d), a and (a + d), then
m = ......... . |
(r) 4 |
(d) If the zeros of x3 + 9x2 + 23x + 15
are (a − d), a and (a + d), then
a = ......... . |
(s) 6 |
Answer:
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Question 47:
Match the following columns:
Column I |
Column II |
(a) The polynomial whose zeros
are 2 and −3 is ......... . |
(p) x2 − 4x + 1 |
(b) The polynomial whose zeros are is ......... . |
(q) |
(c) The polynomial whose zeros
are is ......... . |
(r) x2 + x − 6 |
(d) The polynomial whose zeros
are |
(s) 4x2 − 4x − 3 |
Answer:
Page No 69:
Question 1:
Zeros of p(x) = x2 − 2x − 3 are
(a) 1, −3
(b) 3, −1
(c) −3, −1
(d) 1, 3
Answer:
(b) 3,-1
Here,
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Question 2:
If α, β, γ are the zeros of the polynomial x3 − 6x2 − x + 30, then the value of (αβ + βγ + γα) is
(a) −1
(b) 1
(c) −5
(d) 30
Answer:
(a) −1
Here,
Comparing the given polynomial with , we get:
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Question 3:
If α, β are the zeroes of kx2 − 2x + 3k such that α + β = αβ, then k = ?
(a)
(b)
(c)
(d)
Answer:
(c)
Here,
Comparing the given polynomial with , we get:
It is given that are the roots of the polynomial.
Also, =
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Question 4:
It is given that the difference between the zeroes of 4x2 − 8kx + 9 is 4 and k > 0. Then, k = ?
(a)
(b)
(c)
(d)
Answer:
(c)
Let the zeroes of the polynomial be .
Here, p
Comparing the given polynomial with , we get:
a = 4, b = −8k and c = 9
Now, sum of the roots
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Question 5:
Find the zeros of the polynomial x2 + 2x − 195.
Answer:
Here, p
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Question 6:
If one zero of the polynomial (a2 + 9)x2 + 13x + 6a is the reciprocal of the other, find the value of a.
Answer:
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Question 7:
Find a quadratic polynomial whose zeros are 2 and −5.
Answer:
It is given that the two roots of the polynomial are 2 and −5.
Let
Now, sum of the zeroes, = 2 + (−5) = −3
Product of the zeroes, = 2−5 = −10
∴ Required polynomial =
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Question 8:
If the zeroes of the polynomial x3 − 3x2 + x + 1 are (a − b), a and (a + b), find the values of a and b.
Answer:
The given polynomial and its roots are .
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Question 9:
Verify that 2 is a zero of the polynomial x3 + 4x2 − 3x − 18.
Answer:
Let p
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Question 10:
Find the quadratic polynomial, the sum of whose zeroes is −5 and their product is 6.
Answer:
Given:
Sum of the zeroes = −5
Product of the zeroes = 6
∴ Required polynomial =
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Question 11:
Find a cubic polynomial whose zeros are 3, 5 and −2.
Answer:
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Question 12:
Using remainder theorem, find the remainder when p(x) = x3 + 3x2 − 5x + 4 is divided by (x − 2).
Answer:
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Question 13:
Show that (x + 2) is a factor of f(x) = x3 + 4x2 + x − 6.
Answer:
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Question 14:
If α, β, γ are the zeroes of the polynomial p(x) = 6x3 + 3x2 − 5x + 1, find the value of
Answer:
Comparing the polynomial with , we get:
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Question 15:
If α, β are the zeros of the polynomial f(x) = x2 − 5x + k such that α − β = 1, find the value of k.
Answer:
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Question 16:
Show that the polynomial f(x) = x4 + 4x2 + 6 has no zeroes.
Answer:
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Question 17:
If one zero of the polynomial p(x) = x3 − 6x2 + 11x − 6 is 3, find the other two zeroes.
Answer:
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Question 18:
If two zeroes of the polynomial p(x) = 2x4 − 3x3 − 3x2 + 6x − 2 are and , find its other two zeroes.
Answer:
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Question 19:
Find the quotient when p(x) = 3x4 + 5x3 − 7x2 + 2x + 2 is divided by (x2 + 3x + 1).
Answer:
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Question 20:
Use remainder theorem to find the value of k, it being given that when x3 + 2x2 + kx + 3 is divided by (x − 3), then the remainder is 21.
Answer:
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