CBSE-Class-10-Maths-Assertion-and-Reason-Chapter-2-Polynomialss

Class 10^th Maths Chapter Assertion and Reason Questions

Assertion (A): A polynomial of degree 3 is called a cubic polynomial.
Reason (R): The degree of a polynomial is determined by the highest power of its variable.
- A) Both A and R are true, and R is the correct explanation of A.
- B) Both A and R are true, but R is not the correct explanation of A.
- C) A is true, but R is false.
- D) A is false, but R is true.
- Answer: A) Both A and R are true, and R is the correct explanation of A.
- Explanation: A cubic polynomial has a highest degree of 3, confirming the assertion.

Assertion (A): The polynomial $f(x) = x^4 - 2x^2 + 1$ is a quadratic polynomial.
Reason (R): A quadratic polynomial is defined as a polynomial of degree 2.
- A) Both A and R are true, and R is the correct explanation of A.
- B) Both A and R are true, but R is not the correct explanation of A.
- C) A is true, but R is false.
- D) A is false, but R is true.
- Answer: D) A is false, but R is true.
- Explanation: The polynomial $f(x)$ is of degree 4, so it is not quadratic.

Assertion (A): The zeroes of the polynomial $f(x) = x^2 - 5x + 6$ are 2 and 3.
Reason (R): The zeroes of a polynomial can be found using the quadratic formula.
- A) Both A and R are true, and R is the correct explanation of A.
- B) Both A and R are true, but R is not the correct explanation of A.
- C) A is true, but R is false.
- D) A is false, but R is true.
- Answer: A) Both A and R are true, and R is the correct explanation of A.
- Explanation: The polynomial factors to $(x-2)(x-3)$ , confirming the assertion.

Assertion (A): The degree of a polynomial is the same as the highest exponent of the variable.
Reason (R): A polynomial can have multiple variables.
- A) Both A and R are true, and R is the correct explanation of A.
- B) Both A and R are true, but R is not the correct explanation of A.
- C) A is true, but R is false.
- D) A is false, but R is true.
- Answer: B) Both A and R are true, but R is not the correct explanation of A.
- Explanation: While the degree is determined by the highest exponent, multiple variables don't affect this definition.

Assertion (A): A polynomial can have complex coefficients.
Reason (R): Coefficients in polynomials can only be real numbers.
- A) Both A and R are true, and R is the correct explanation of A.
- B) Both A and R are true, but R is not the correct explanation of A.
- C) A is true, but R is false.
- D) A is false, but R is true.
- Answer: C) A is true, but R is false.
- Explanation: Polynomials can indeed have complex coefficients, contrary to the reason.

Assertion (A): The sum of two polynomials is always a polynomial.
Reason (R): The sum of a polynomial and a constant is also a polynomial.
- A) Both A and R are true, and R is the correct explanation of A.
- B) Both A and R are true, but R is not the correct explanation of A.
- C) A is true, but R is false.
- D) A is false, but R is true.
- Answer: A) Both A and R are true, and R is the correct explanation of A.
- Explanation: Both statements are valid and the sum of polynomials maintains the polynomial nature.

Assertion (A): The polynomial $f(x) = 3x^3 + 2x - 4$ has a degree of 3.
Reason (R): The degree of a polynomial is determined by its leading term.
- A) Both A and R are true, and R is the correct explanation of A.
- B) Both A and R are true, but R is not the correct explanation of A.
- C) A is true, but R is false.
- D) A is false, but R is true.
- Answer: A) Both A and R are true, and R is the correct explanation of A.
- Explanation: The leading term $3x^3$ confirms the degree is 3.

Assertion (A): The product of two polynomials is a polynomial.
Reason (R): Polynomials are closed under addition, subtraction, and multiplication.
- A) Both A and R are true, and R is the correct explanation of A.
- B) Both A and R are true, but R is not the correct explanation of A.
- C) A is true, but R is false.
- D) A is false, but R is true.
- Answer: A) Both A and R are true, and R is the correct explanation of A.
- Explanation: The closure property of polynomials validates both statements.

Assertion (A): The polynomial $f(x) = x^2 + 1$ has real zeroes.
Reason (R): A polynomial of degree 2 must have real roots.
- A) Both A and R are true, and R is the correct explanation of A.
- B) Both A and R are true, but R is not the correct explanation of A.
- C) A is true, but R is false.
- D) A is false, but R is true.
- Answer: D) A is false, but R is true.
- Explanation: The polynomial $x^2 + 1$ has no real roots (the roots are imaginary), even though it is a degree 2 polynomial.

Assertion (A): A polynomial can have both rational and irrational roots.
Reason (R): The roots of a polynomial are determined by the coefficients.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Polynomials can indeed have both rational and irrational roots based on their coefficients.

Assertion (A): The polynomial $f(x) = 4x^2 - 12x + 9$ is a perfect square trinomial.
Reason (R): A perfect square trinomial can be expressed as the square of a binomial.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: $f(x)$ can be expressed as $(2x - 3)^2$ , confirming both statements.

Assertion (A): The polynomial $g(x) = 0$ is a constant polynomial.
Reason (R): A constant polynomial has no variable terms.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The polynomial $g(x) = 0$ has no variable terms, confirming it is constant.

Assertion (A): The polynomial $f(x) = x^3 - 6x^2 + 11x - 6$ has three real roots.
Reason (R): The Fundamental Theorem of Algebra states that a polynomial of degree $n$ has $n$ roots in the complex number system.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: B) Both A and R are true, but R is not the correct explanation of A.
Explanation: The polynomial has three real roots, but the reason is about the total number of roots, including complex.

Assertion (A): The polynomial $f(x) = 2x^2 + 3x + 5$ can be factored into linear terms.
Reason (R): A polynomial can be factored if its discriminant is non-negative.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: The polynomial cannot be factored into real linear terms since its discriminant $b^2 - 4ac$ is negative.

Assertion (A): The sum of the coefficients of the polynomial $f(x) = 2x^3 + 3x^2 - 4x + 1$ is zero.
Reason (R): The sum of the coefficients can be found by evaluating $f(1)$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: The sum of the coefficients $2 + 3 - 4 + 1 = 2$ , so the assertion is false.

Assertion (A): The polynomial $f(x) = 5x^4 + 0x^3 - 3x^2 + 7$ is a quartic polynomial.
Reason (R): A polynomial's degree is determined by the term with the highest power of $x$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The degree of the polynomial is 4, confirming it is quartic.

Assertion (A): The polynomial $f(x) = -3x^2 + 4$ opens downward.
Reason (R): The sign of the leading coefficient determines the direction of the opening.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The leading coefficient is negative, confirming the downward opening.

Assertion (A): A polynomial of degree $n$ can have at most $n$ distinct roots.
Reason (R): This is a result of the Fundamental Theorem of Algebra.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The theorem states that a degree $n$ polynomial can have exactly $n$ roots.

Assertion (A): The polynomial $f(x) = x^4 + 4$ has only complex roots.
Reason (R): The sum of the squares of two numbers is always positive.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: B) Both A and R are true, but R is not the correct explanation of A.
Explanation: While the assertion is true, the reason provided does not explain why $f(x)$ has complex roots.

Assertion (A): The polynomial $f(x) = x^5 - 1$ can be factored into linear factors.
Reason (R): A polynomial of odd degree must have at least one real root.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The polynomial can be factored into linear factors, and the reason supports this.

Assertion (A): The polynomial $g(x) = 6x^2 + 11x + 3$ is not factorable over the integers.
Reason (R): A polynomial is factorable over the integers if its discriminant is a perfect square.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The discriminant of $g(x)$ is not a perfect square, confirming it cannot be factored over the integers.

Assertion (A): The polynomial $h(x) = x^3 - 3x + 2$ has three real roots.
Reason (R): The polynomial can be graphed to determine the number of real roots.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: C) A is true, but R is false.
Explanation: The polynomial has two real roots, not three; while graphing helps identify roots, it does not validate the assertion.

Assertion (A): The polynomial $f(x) = 2x^3 - 8x$ has a common factor.
Reason (R): A polynomial can have a common factor if it can be expressed as a product of other polynomials.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: $f(x)$ can be factored as $2x(x^2 - 4)$ , confirming both statements.

Assertion (A): The roots of the polynomial $f(x) = x^2 + 4$ are both real.
Reason (R): The discriminant determines the nature of the roots.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: The discriminant is negative, indicating complex roots, making the assertion false.

Assertion (A): The polynomial $f(x) = x^4 - 5x^2 + 4$ can be factored.
Reason (R): A polynomial can be factored if it can be written as a product of polynomials of lower degree.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The polynomial can be factored into $(x^2 - 4)(x^2 - 1)$ .

Chapter 2 Polynomials

Class 10th Maths Chapter Assertion and Reason Questions

Class 10^th Maths Chapter Assertion and Reason Questions