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CBSE Class 10  >  Class 10 Notes  >  Assertion & Reason Type Questions: Polynomials

Assertion & Reason Type Questions: Polynomials

Q1: 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.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: d
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.

Q2: 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 
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: a
Assertion (A): If the sum of the zeroes of the quadratic polynomial x2 - 2kx + 8 is 2, then the value of
 k is 1.
Reason (R): The sum of the zeroes of a quadratic polynomial ax2 + bx + c is - (b/a).
 Let's evaluate these statements:
 For a quadratic equation ax2 + bx + c, the sum of the zeroes is given by - (b/a).
Given the quadratic polynomial x2 - 2kx + 8, if the sum of the zeroes is 2, then we have:
 - (- 2k / 1) = 2
 2k = 2
 k = 1
 So, both assertion (A) and reason (R) are true, and reason (R) correctly explains assertion (A).
 Therefore, the correct choice is:
 1. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Q3:  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.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: b
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.

Q4:  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).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: b
Reason is true.Let α, 1/α be the zeroes of p(x), then

Assertion & Reason Type Questions: Polynomials

k2 - 4k + 4 = 0
(k - 2)2 = 0
k = 2 Assertion is true Since, Reason is not correct for Assertion.

Q5:  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.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: a
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.

Q6: 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.

Assertion & Reason Type Questions: Polynomials

Reason : The number of zero of the polynomial f(x) is the number of point of which f(x) cuts or touches the axes.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: c
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.

Q7:  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.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: d
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.

Q8:  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.

Assertion & Reason Type Questions: Polynomials

(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: c
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.

Q9: 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. 
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: c
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

Q10:  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.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: c
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.

Q11: 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). 
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: b
Let α, 1/α be the zeroes of p(x) then we have
 Product of Zeroes
= a x (1/a) = 4k / (k2 + 4) = 1
⇒ k2 - 4k + 4 = 0
⇒ (k - 2)2 = 0 ⇒ k = 2

Q12:  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
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: b
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.

Q13: 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
​​(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: b
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.

Q14:  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 -1/4 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.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: a
Sum of zeros = -1/4 and product of zeros = 1/4

Assertion & Reason Type Questions: Polynomials

Quadratic polynomial be 4x2 + x + 1. So, both A and R are correct and R explains A.

Q15:  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.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Ans: d
x2 + 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.

The document Assertion & Reason Type Questions: Polynomials is a part of Class 10 category.
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FAQs on Assertion & Reason Type Questions: Polynomials

1. What are polynomials and how are they classified?
Ans. Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. They are classified based on their degree: - A polynomial of degree 0 is called a constant (e.g., 5). - A polynomial of degree 1 is a linear polynomial (e.g., 2x + 3). - A polynomial of degree 2 is a quadratic polynomial (e.g., x² - 4x + 4). - A polynomial of degree 3 is a cubic polynomial (e.g., x³ + 3x² - x + 5). - Higher degrees are referred to similarly (quartic for degree 4, quintic for degree 5, etc.).
2. What is the standard form of a polynomial?
Ans. The standard form of a polynomial is the expression written in descending order of the degree of its terms. For example, a polynomial like 3x² + 5x - 2 is in standard form because the terms are arranged from the highest degree (x²) to the lowest (constant term). This helps in identifying the leading term and degree easily.
3. How can we perform operations such as addition and subtraction on polynomials?
Ans. To perform addition and subtraction on polynomials, combine like terms. Like terms are those containing the same variable raised to the same power. For example, to add (3x² + 2x + 1) and (4x² + 3), you combine the like terms: (3x² + 4x²) + 2x + (1 + 3) = 7x² + 2x + 4. For subtraction, subtract the coefficients of like terms. For example, (5x² + 3x) - (2x² + x) results in (5x² - 2x²) + (3x - x) = 3x² + 2x.
4. What is the factorization of polynomials and why is it important?
Ans. Factorization of polynomials involves expressing a polynomial as a product of its factors. This is important because it simplifies polynomial equations, making them easier to solve. For example, the quadratic polynomial x² - 5x + 6 can be factored as (x - 2)(x - 3). Understanding factorization helps in identifying the roots of the polynomial and is crucial for solving equations in algebra.
5. What is the Remainder Theorem and how is it applied?
Ans. The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - a), the remainder of this division is f(a). This theorem is useful for quickly evaluating polynomials at specific points and for determining factors. For example, if you have f(x) = x³ - 4x + 1 and you want to find the remainder when divided by (x - 2), you simply calculate f(2) = 2³ - 4(2) + 1 = 8 - 8 + 1 = 1. Thus, the remainder is 1.
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Apr 19, 2026 Last updated
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