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    • Class 10 Maths Chapter 2 - Polynomials

    Assertion and Reason Questions

    1. 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.
    1. Assertion (A): The polynomial f(x)=x4−2x2+1f(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)f(x) is of degree 4, so it is not quadratic.
    1. Assertion (A): The zeroes of the polynomial f(x)=x2−5x+6f(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)(x-2)(x-3), confirming the assertion.
    1. 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.
    1. 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.
    1. 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.
    1. Assertion (A): The polynomial f(x)=3x3+2x−4f(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 3x33x^3 confirms the degree is 3.
    1. 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.
    1. Assertion (A): The polynomial f(x)=x2+1f(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 x2+1x^2 + 1 has no real roots (the roots are imaginary), even though it is a degree 2 polynomial.
    1. 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.
    1. Assertion (A): The polynomial f(x)=4x2−12x+9f(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)f(x) can be expressed as (2x−3)2(2x - 3)^2 , confirming both statements.
    1. Assertion (A): The polynomial g(x)=0g(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)=0g(x) = 0 has no variable terms, confirming it is constant.
    1. Assertion (A): The polynomial f(x)=x3−6x2+11x−6f(x) = x^3 - 6x^2 + 11x - 6 has three real roots.
      Reason (R): The Fundamental Theorem of Algebra states that a polynomial of degree nn has nn 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.
    1. Assertion (A): The polynomial f(x)=2x2+3x+5f(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 b2−4acb^2 - 4ac is negative.
    1. Assertion (A): The sum of the coefficients of the polynomial f(x)=2x3+3x2−4x+1f(x) = 2x^3 + 3x^2 - 4x + 1 is zero.
      Reason (R): The sum of the coefficients can be found by evaluating f(1)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=22 + 3 - 4 + 1 = 2 , so the assertion is false.
    1. Assertion (A): The polynomial f(x)=5x4+0x3−3x2+7f(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 xx.
    • 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.
    1. Assertion (A): The polynomial f(x)=−3x2+4f(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.
    1. Assertion (A): A polynomial of degree nn can have at most nn 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 nn polynomial can have exactly nn roots.
    1. Assertion (A): The polynomial f(x)=x4+4f(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)f(x) has complex roots.
    1. Assertion (A): The polynomial f(x)=x5−1f(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.
    1. Assertion (A): The polynomial g(x)=6x2+11x+3g(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)g(x) is not a perfect square, confirming it cannot be factored over the integers.
    1. Assertion (A): The polynomial h(x)=x3−3x+2h(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.
    1. Assertion (A): The polynomial f(x)=2x3−8xf(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)f(x) can be factored as 2x(x2−4)2x(x^2 - 4) , confirming both statements.
    1. Assertion (A): The roots of the polynomial f(x)=x2+4f(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.
    1. Assertion (A): The polynomial f(x)=x4−5x2+4f(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 (x2−4)(x2−1)(x^2 - 4)(x^2 - 1) .

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    March 06, 2025

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