Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : If α,β, γ are the zeroes of x^{3}  2x^{2 }+ qx  r and α + β = 0 , then 2q = r.
Reason : If α,β, γ are the zeroes of ax^{3} + bx^{2 }+ cx + d , then
Clearly, Reason is true. [Standard Result]
α + β + γ = ( 2) = 2
0 + γ = 2
γ = 2
αβγ = ( r) = r
αβ(2) = r
αβ = r/2
γ = 2q Assertion is true.
Since, Reason gives Assertion.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : P(x) = 4x^{3} – x^{2} + 5x^{4} + 3x – 2 is a polynomial of degree 3.
Reason : The highest power of x in the polynomial P(x) is the degree of the polynomial.
Therefore, the degree of the polynomial P(x) is 4.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion (A): If the zeroes of a quadratic polynomial ax2 + bx + c are both positive, then a, b and c all have the same sign.
Reason (R): If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
∴ Given statement is incorrect.
In case of reason:
Let β = 0, γ = 0
f(x) = (x – α) (x – β) (x – γ)
= (x – α) x · x
⇒ f(x) = x^{3} – αx^{2} which has no linear (coefficient of x) and constant terms.
∴ Given statement is correct.
Thus, assertion is incorrect but reason is correct.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : (2  √3) is one zero of the quadratic polynomial then other zero will be (2 + √3).
Reason : Irrational zeros (roots) always occurs in pairs.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : x^{3} + x has only one real zero.
Reason : A polynomial of nth degree must have n real zeroes.
[x^{2 }+ 1 ≠ 0 for all x ∈ R]
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion (A): The value of k for which the quadratic polynomial kx^{2} + x + k has equal zeroes are ± 1/2.
Reason (R): If all the three zeroes of a cubic polynomial x^{3} + ax^{2} – bx + c are positive, then at least one of a, b and c is nonnegative.
f(x) = kx^{2} + x + k (a = k, b = 1, c = k)
For equal roots b^{2} – 4ac = 0
⇒ (1)^{2} – 4(k) (k) = 0
⇒ 4k^{2} = 1
⇒ k^{2} = 1/4
⇒ k = ± 1/2
So, there are values of k so that
the given equation has equal roots.
∴ Given statement is correct.
In case of reason:
All the zeroes of cubic polynomial are positive only when all the constants a, b, and c are negative.
∴ Given statement is incorrect.
Thus, assertion is correct but reason is incorrect.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : Zeroes of f(x) = x^{2}  4x  5 are 5,  1
Reason : The polynomial whose zeroes are 2 + √3, 2  √3 is x^{2}  4x + 7.
Zeroes of f(x) = x^{2} − 4x − 5 are obtained by solving:
x^{2} − 4x − 5 = 0, which implies
x^{2} − 5x + x − 5 = 0 or, x(x − 5) + 1(x − 5) = 0
which means x = 5 or x = −1
Thus the assertion is correct.
However, the reason is incorrect.
The numbers given are not the zeroes of x^{2} − 4x + 7
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : If one zero of polynominal p(x) = (k^{2} + 4)x^{2} + 13x + 4k is reciprocal of other, then k = 2.
Reason : If (x – a) is a factor of p(x), then p(a) = 0 i.e. a is a zero of p(x).
Product of Zeroes
⇒ k^{2} – 4k + 4 = 0
⇒ (k – 2)^{2} = 0 ⇒ k = 2
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion (A): The graph of y = p(x), where p(x) is a polynomial in variable x, is as follows:
The number of zeroes of p(x) is 5.
Reason (R): If the graph of a polynomial intersects the xaxis at exactly two points, it need not be a quadratic polynomial.
∴ Given statement is correct.
In case of reason:
If a polynomial of degree more than two has two real zeroes and other zeroes are not real or are imaginary, and then graph of the polynomial will intersect at two points on xaxis.
∴ Given statement is correct:
Thus, both assertion and reason are correct but reason is not the correct explanation for assertion.
Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : x^{2} + 4x + 5 has two zeroes.
Reason : A quadratic polynomial can have at the most two zeroes.
Because the degree of the polynomial is 2. It is a quadratic polynomial. We know that the quadratic polynomial has at the most two zeroes.
∴ x^{2} + 4x + 5 has two zeroes.
∴ Assertion is true.
Reason: Clearly, Reason is true.
Since both the Assertion and Reason are true and Reason is a correct explanation of Assertion.
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