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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 zeroes.
Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for Assertion (A).
Assertion (A) is true but Reason (R) is false.
Assertion (A) is false but Reason (R) is true.
(4)
The polynomials of the form and has only equal roots and graphs of these polynomials cut x-axis at only one point. These polynomials are quadratic Thus, Assertion is false Reason is true.
Assertion (A): If the graph of a polynomial intersects the x-axis at exactly two points, then the number of zeroes of that polynomial is 2.
Reason (R): The number of zeroes of a polynomial is equal to the number of points where the graph of the polynomial intersects x-axis.
Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
Assertion (A) is true, but Reason (R) is false.
Assertion (A) is false, but Reason (R) is true.
(1) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
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.
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).
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.
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
Assertion (A): Degree of a zero polynomial is not defined.
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).
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.
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).
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.
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).
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.
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).
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