Topic Question Set
Q 1
:
Assertion (A): The polynomial p(x) = x² – 2x + 1 has exactly one zero.
Reason (R): The discriminant of the polynomial is zero.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false.
A is false, but R is true.
(1)
The polynomial can be rewritten as (x – 1)², indicating a repeated zero at x = 1.
For p(x) = x² – 2x + 1, we have [a = 1, b = –2, c = 1].
Here, D = b² – 4ac = 4 – 4 = 0.
Q 2
:
Assertion (A): The polynomial s(x) = x³ – x has three real zeros.
Reason (R): The polynomial can be factored into x(x – 1)(x + 1), which means its 3 zeros are –1, 0, and 1.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A
A is true, but R is false.
A is false, but R is true.
(1)
The zeros of s(x) derived from its factored form x(x – 1)(x + 1) are x = 0, 1, and –1, which indeed are three real zeros.
Therefore, both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Q 3
:
Assertion (A): The graph of q(x) = x³ – 3x² + 3x – 1 does not intersect the x-axis.
Reason (R): The polynomial can be factored as (x – 1)³.
Both A and R are true, and R is the correct explanation of A
Both A and R are true, but R is not the correct explanation of A
A is true, but R is false
A is false, but R is true
(4)
Given: q(x) = x³ – 3x² + 3x – 1 = (x – 1)³.
We know that if a polynomial p(x) contains a factor of the form (x – a)?, then the graph of p(x) will touch the x-axis and turn at (a, 0) if k is even, and will cross the x-axis at (a, 0) if k is odd.
Here, k = 3, so the graph of q(x) will cross the x-axis at x = 1.
Q 4
:
Assertion (A): If the graph of a polynomial touches x-axis at only one point, then the polynomial cannot be a quadratic polynomial.
Reason (R): A polynomial of degree n (n > 1) can have at most n zeros.
Both A and R are true, and R is the correct explanation of A
Both A and R are true, but R is not the correct explanation of A
A is true, but R is false
A is false, but R is true
(4)
Assertion (A) is false as the graph of a polynomial that touches x-axis at one point can be a quadratic polynomial having real and equal zeros. Reason is always true
Q 5
:
Assertion (A): Degree of a zero polynomial is not defined.
Reason (R): Degree of a non-zero constant polynomial is 0
Reason (R): Degree of a non-zero constant polynomial is 0
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A
A is true, but R is false
A is false, but R is true.
(2)
Since, degree of a zero polynomial is not defined and degree of a non-zero constant polynomial is 0. Hence, Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Q 6
:
Assertion (A): If 2, 3 are the zeros of a quadratic polynomial, then the polynomial is x² - 5x + 6.
Reason (R): If α, β are the zeros of a quadratic polynomial, the polynomial is x² - (α + β)x + αβ.
Reason (R): If α, β are the zeros of a quadratic polynomial, the polynomial is x² - (α + β)x + αβ.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false.
A is false, but R is true.
(1)
The polynomial having 2, 3 as zeros is x² - (2 + 3)x + 2×3 = x² - 5x + 6.
Hence, Assertion (A) and Reason (R) both are true and Reason (R) is the correct explanation of Assertion (A).
Q 7
:
Assertion (A): If α, β are the zeros of the polynomial x² - ax + 1, then 1/α + 1/β = a.
Reason (R): If α, β are the zeros of the polynomial ax² + bx + c, then α + β = -b/a and αβ = c/a.
Reason (R): If α, β are the zeros of the polynomial ax² + bx + c, then α + β = -b/a and αβ = c/a.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false
A is false, but R is true
(1)
Given polynomial is x² - ax + 1.
Now, 1/α + 1/β = (β + α)/αβ = a/1 = a.
Also, for polynomial ax² + bx + c,
Sum of zeros = -b/a and Product of zeros = c/a.
∴ Assertion (A) and Reason (R) both are true and Reason (R) is the correct explanation of Assertion (A).
Q 8
:
Assertion (A): If the sum of the zeros of the quadratic polynomial x² - 2kx + 8 is 2 then value of k is 1.
Reason (R): Sum of zeros of a quadratic polynomial ax² + bx + c is -b/a.
Reason (R): Sum of zeros of a quadratic polynomial ax² + bx + c is -b/a.
Both A and R are true, and R is the correct explanation of A
Both A and R are true, but R is not the correct explanation of A
A is true, but R is false
A is false, but R is true.
(1)
Given polynomial is x² - 2kx + 8.
Sum of zeros = -(-2k)/1 = 2k = 2 ⇒ k = 1.
Hence, Assertion (A) and Reason (R) both are true and Reason (R) is the correct explanation of Assertion (A).
