Q1:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself:Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: x2 +7x+12 has no real zeroes.
Reason: A quadratic polynomial can have at the most two zeroes.
Explanation
The assertion is false because the quadratic equation x2 + 7x + 12 can be factored as (x + 3)(x + 4), which means it has two real zeros (-3 and -4). The reason is true because a quadratic polynomial indeed can have at most two zeros, but it doesn't directly relate to the assertion being true or false.
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Q2:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself:Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: If the sum of the zeroes of the quadratic polynomial x2-2kx+8 is 2 then value of k is 1.
Reason: Sum of zeroes of a quadratic polynomial ax2+bx+c is -b/a
Explanation
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Q3:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself:Directions: 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) = 4x3-x2+5x4+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.
Explanation
The assertion is false because the degree of the polynomial is determined by the highest power of x, which is 4 in this case, making it a polynomial of degree 4. The reason is true because the highest power of x indeed determines the degree of a polynomial, but it incorrectly describes the polynomial in the assertion.
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Q4:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself: 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.
Explanation
As irrational roots/zeros always occurs in pairs therefore, when one zero is (2 - √3) then other will be (2 + √3). So, both A and R are correct and R explains A.
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Q5:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself:Directions: 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 polynomial p(x) = (k2+4)x2+13x+4k is reciprocal of the 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).
Explanation
The assertion and reason are both true, but the reason does not directly explain k=2 when one zero is the reciprocal of the other. The calculation to find k would involve using the relationship between the product of the roots and the constants in the polynomial, which is not mentioned in the reason.
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Q6:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself:Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: x2+4x+5 has two real zeroes.
Reason: A quadratic polynomial can have at the most two zeroes.
Explanation
The assertion is false because x2 +4x+5 has no real zeroes (its discriminant, b 2 −4ac, is negative). However, it has two complex zeroes. The reason is true, as a quadratic polynomial can have at most two zeroes (real or complex).
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Q7:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself: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) = x2 - 4x - 5 are 5, - 1
Reason : The polynomial whose zeroes are 2 + √3, 2 - √3 is x2 - 4x + 7.
Explanation
Zeroes of f(x) = x2 − 4x − 5 are obtained by solving:
x2 − 4x − 5 = 0, which implies
x2 − 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 x2 − 4x + 7
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Q8:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself: 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 poly-nominal p(x) = (k2 + 4)x2 + 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).
Explanation
Let α, 1/α be the zeroes of p(x) then we have
Product of Zeroes
⇒ k2 – 4k + 4 = 0
⇒ (k – 2)2 = 0 ⇒ k = 2
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Q9:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself:Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: If the product of the zeroes of the quadratic polynomial x2+3x+5k is -10 then value of k is -2.
Reason: Sum of zeroes of a quadratic polynomial ax2+bx+c is -b/a
Explanation
Both the assertion and the reason are true, but the reason provided is related to the sum of the zeroes, not their product. The correct explanation involves the formula for the product of the zeroes of a quadratic polynomial, c/a, which would lead to 5k/−10, hence determining the value of k.
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Q10:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself:Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : x2 + 4x + 5 has two real zeroes.
Reason : A quadratic polynomial can have at the most two zeroes.
Explanation
Assertion: Consider the given polynomial x2 + 4x + 5
as its discriminant is negative b2 -4ac = 16 - 20 = -4 so it dont have real roots ,
∴ x2 + 4x + 5 has no zeroes.
∴ Assertion is false.
Reason: Clearly, Reason is true.
Since Assertion (A) is false but reason (R) is true.
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Q11:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself:Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : x2 + 4x + 5 has two real zeroes.
Reason : A quadratic polynomial can have at the most two zeroes.
Explanation
p(x) = 0 ⇒ x2 + 4x + 5 = 0
Discriminant, D = b2 – 4ac
= 42 – 4 x 1 x 5
= 16 – 20 = – 4 < 0
Therefore, no real zeroes are there.
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Q12:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself: 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 the quadratic polynomial (k – 1) x2 + kx + 1 is – 3, then the value of k is 4/3.
Reason (R): If – 1 is a zero of the polynomial p(x) = kx2 – 4x + k, then the value of k is –2.
Explanation
In case of assertion:
Let p(x) = (k – 1)x2 + kx + 1
As –3 is a zero of p(x), then p (−3) = 0
⇒ (k – 1) (−3)2 + k (−3) + 1 = 0
⇒ 9k – 9 – 3k + 1 = 0
⇒ 9k – 3k = +9 – 1
⇒ 6k = 8
⇒ k = 4/3
∴ Given statement is correct.
In case of reason:
Since, – 1 is a zero of the polynomial
and p(x) = kx2 – 4x + k, then p (–1) = 0
∴ k (–1)2 – 4 (–1) + k = 0
⇒ k + 4 + k = 0
⇒ 2k + 4 = 0
⇒ 2k = – 4
Hence, k = – 2
∴ Given statement is correct.
Thus, both assertion and reason are correct but reason is not the correct explanation for assertion.
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Q13:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself: 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 poly-nominal p(x) = (k2 + 4)x2 + 13x + 4k is reciprocal of other, then k = 2.
Reason : If (x - α) is a factor of p(x), then p(α) = 0
i.e. α is a zero of p(x).
Explanation
Reason is true.
Let α, 1/α be the zeroes of p(x), then
k2 - 4k + 4 = 0
(k - 2)2 = 0
k = 2 Assertion is true Since, Reason is not correct for Assertion.
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Q14:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself:Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : The graph y = f(x) is shown in figure, for the polynomial f(x). The number of zeros of f(x) is 3.
Reason : The number of zero of the polynomial f(x) is the number of point of which f(x) cuts or touches the axes.
Explanation
As the number of zeroes of polynomial f(x) is the number of points at which f(x) cuts (intersects) the x –axis and number of zero in the given figure is 3.
So A is correct but R is not correct.
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Q15:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself: 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) = 14x3 - 2x2 + 8x4 + 7x - 8 is a polynomial of degree 3.
Reason : The highest power of x in the polynomial p(x) is the degree of the polynomial.
Explanation
The highest power of x in the polynomial p(x)
= 14x3 - 2x2 + 8x4 + 7x - 8x is 4.
Degree of p(x) is 4. So, A is incorrect but R is correct.
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Q16:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself: Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion : The graph y = f(x) is shown in figure, for the polynomial f (x). The number of zeros of f(x) is 3.
Reason : The number of zero of the polynomial f(x) is the number of point of which f(x) cuts or touches the axes.
Explanation
As the number of zeroes of polynomial f(x) is the number of points at which f(x) cuts (intersects) the x –axis and number of zero in the given figure is 3.
So A is correct but R is not correct.
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Q17:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself: Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion : x3 + x has only one real zero.
Reason : A polynomial of nth degree must have n real zeroes.
Explanation
Reason is false [a polynomial of nth degree has at most x zeroes.]
Again, x3 + x = x(x2 + 1)
which has only one real zero (x = 0)
[x2 + 1 ≠ 0 for all x ε R]
Assertion is true.
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Q18:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself: Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion: Degree of a zero polynomial is not defined.
Reason: Degree of a non-zero constant polynomial is 0
Explanation
We know that, the constant polynomial 0 is called a zero polynomial. The degree of a zero polynomial is not defined.
So, Assertion is true. Now, the degree of a non-zero constant polynomial is zero. So, Reason is true.
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Q19:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself: Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion: The sum and product of the zeros of a quadratic polynomial are and 1/4 respectively.
Then the quadratic polynomial is 4x2 + x + 1.
Reason : The quadratic polynomial whose sum and product of zeros are given is x2-(sum of zeros). x + product of zeros.
Explanation
product of zeros = 1/4
Quadratic polynomial be 4x2 + x + 1. So, both A and R are correct and R explains A.
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Q20:
Question for Assertion & Reason Type Questions: Polynomials
Try yourself: Direction: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion : x2 + 7x + 12 has no real zeroes.
Reason : A quadratic polynomial can have at the most two zeroes.
Explanation
x
2 + 7x + 12 = 0
⇒ x2 + 4x + 3x + 12 = 0
⇒ x(x + 4) + 3(x + 4) = 0
⇒ (x + 4) (x + 3) = 0
⇒ (x + 4) = 0 or (x + 3) = 0
⇒ x = −4 or x = −3
Therefore, x2 + 7x + 12 has two real zeroes.
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and 1/4 respectively.
