✦ NCF 2023 ALIGNED • CBSE CLASS X ✦
Chapter 2: Polynomials
Assertion-Reason Question Bank
✨ Explore more at: www.akashmaths.online
Assertion (A)
The polynomial \(p(x) = x^2 + 4x + 4\) has two distinct real zeroes.
Reason (R)
Discriminant \(D = b^2 - 4ac = 4^2 - 4(1)(4) = 0\), so it has two equal real zeroes.
Correct Option: (D)
Assertion (A) is false because \(D = 0\) means equal zeroes (\(x = -2, -2\)), not distinct. Reason (R) is true.
Assertion (A) is false because \(D = 0\) means equal zeroes (\(x = -2, -2\)), not distinct. Reason (R) is true.
Correct Answer: (D)
Assertion false (\(D=0\) gives equal zeroes); Reason true.
Assertion false (\(D=0\) gives equal zeroes); Reason true.
Assertion (A)
A quadratic polynomial with zeroes \(2\) and \(-3\) is \(x^2 + x - 6\).
Reason (R)
A quadratic polynomial with zeroes \(\alpha\) and \(\beta\) is given by \(x^2 - (\alpha + \beta)x + \alpha\beta\).
Correct Option: (A)
Sum of zeroes \(\alpha + \beta = 2 + (-3) = -1\). Product \(\alpha\beta = 2 \times (-3) = -6\). Polynomial \(x^2 - (-1)x + (-6) = x^2 + x - 6\). Reason (R) is the exact formula.
Sum of zeroes \(\alpha + \beta = 2 + (-3) = -1\). Product \(\alpha\beta = 2 \times (-3) = -6\). Polynomial \(x^2 - (-1)x + (-6) = x^2 + x - 6\). Reason (R) is the exact formula.
Correct Answer: (A)
Sum \(= -1\), Product \(= -6\). Reason gives the exact polynomial formula.
Sum \(= -1\), Product \(= -6\). Reason gives the exact polynomial formula.
Assertion (A)
If the sum of zeroes of quadratic polynomial \(3x^2 - kx + 6\) is \(3\), then \(k = 9\).
Reason (R)
For \(ax^2 + bx + c\), the sum of zeroes is \(-b/a\), so \(-(-k)/3 = 3 \implies k/3 = 3 \implies k = 9\).
Correct Option: (A)
Sum of zeroes \(= -b/a = k/3 = 3 \implies k = 9\). Reason explains Assertion.
Sum of zeroes \(= -b/a = k/3 = 3 \implies k = 9\). Reason explains Assertion.
Correct Answer: (A)
Sum of zeroes \(= k/3 = 3 \implies k = 9\). Reason explains Assertion.
Sum of zeroes \(= k/3 = 3 \implies k = 9\). Reason explains Assertion.
Assertion (A)
The polynomial \(x^2 + 4x + 5\) has no real zeroes.
Reason (R)
Discriminant \(D = b^2 - 4ac = 4^2 - 4(1)(5) = 16 - 20 = -4 < 0\), so there are no real roots/zeroes.
Correct Option: (A)
For \(x^2 + 4x + 5\), \(D = 16 - 20 = -4 < 0\). Since \(D < 0\), there are no real zeroes. Reason explains Assertion.
For \(x^2 + 4x + 5\), \(D = 16 - 20 = -4 < 0\). Since \(D < 0\), there are no real zeroes. Reason explains Assertion.
Correct Answer: (A)
\(D < 0\) implies no real zeroes. Reason explains Assertion.
\(D < 0\) implies no real zeroes. Reason explains Assertion.
Assertion (A)
If \(1\) is a zero of \(p(x) = a^2x^2 - 3ax + 2\), then \(a = 1\) or \(a = 2\).
Reason (R)
The sum of zeroes of \(a^2x^2 - 3ax + 2\) is \(-3/a\).
Correct Option: (C)
A is true: \(p(1) = a^2 - 3a + 2 = 0 \implies (a-1)(a-2)=0 \implies a=1, 2\). R is false because sum of zeroes is \(-b/a = -(-3a)/a^2 = 3/a\), not \(-3/a\).
A is true: \(p(1) = a^2 - 3a + 2 = 0 \implies (a-1)(a-2)=0 \implies a=1, 2\). R is false because sum of zeroes is \(-b/a = -(-3a)/a^2 = 3/a\), not \(-3/a\).
Correct Answer: (C)
Assertion true; Reason false (sum of zeroes is \(3/a\)).
Assertion true; Reason false (sum of zeroes is \(3/a\)).
Assertion (A)
If one zero of \(x^2 - 4x + 1\) is \(2 + \sqrt{3}\), the other zero is \(2 - \sqrt{3}\).
Reason (R)
Irrational roots of a polynomial with rational coefficients occur in conjugate pairs.
Correct Option: (B)
Both A and R are true mathematical facts, but R is a general theorem statement.
Both A and R are true mathematical facts, but R is a general theorem statement.
Correct Answer: (B)
Both true, but R is a general theorem.
Both true, but R is a general theorem.
Assertion (A)
The graph of \(y = p(x)\) intersects the x-axis at \(3\) distinct points, so \(p(x)\) has \(3\) real zeroes.
Reason (R)
The number of real zeroes of a polynomial \(p(x)\) is equal to the number of points where its graph intersects the x-axis.
Correct Option: (A)
The number of x-intercepts of the graph \(y=p(x)\) corresponds directly to the number of real zeroes of \(p(x)\). Reason explains Assertion.
The number of x-intercepts of the graph \(y=p(x)\) corresponds directly to the number of real zeroes of \(p(x)\). Reason explains Assertion.
Correct Answer: (A)
Number of x-intercepts equals real zeroes. Reason explains Assertion.
Number of x-intercepts equals real zeroes. Reason explains Assertion.
Assertion (A)
The degree of the zero polynomial is \(0\).
Reason (R)
The degree of the zero polynomial is undefined because \(0 \cdot x^n = 0\) for any power \(n\).
Correct Option: (D)
Assertion (A) is false (degree of zero polynomial is undefined). Reason (R) is true.
Assertion (A) is false (degree of zero polynomial is undefined). Reason (R) is true.
Correct Answer: (D)
Assertion false (degree is undefined); Reason true.
Assertion false (degree is undefined); Reason true.
Assertion (A)
If \(\alpha, \beta\) are zeroes of \(x^2 - 6x + k\) and \(3\alpha + 2\beta = 20\), then \(k = -16\).
Reason (R)
\(\alpha + \beta = -6\) for polynomial \(x^2 - 6x + k\).
Correct Option: (C)
Assertion (A) is true: \(\alpha+\beta = 6\). \(2\alpha+2\beta = 12\). Subtracting gives \(\alpha = 20 - 12 = 8\). Then \(\beta = -2\). \(k = \alpha\beta = -16\). Reason (R) is false because \(\alpha+\beta = -(-6)/1 = +6\), not \(-6\).
Assertion (A) is true: \(\alpha+\beta = 6\). \(2\alpha+2\beta = 12\). Subtracting gives \(\alpha = 20 - 12 = 8\). Then \(\beta = -2\). \(k = \alpha\beta = -16\). Reason (R) is false because \(\alpha+\beta = -(-6)/1 = +6\), not \(-6\).
Correct Answer: (C)
Assertion true; Reason false (\(\alpha+\beta = 6\)).
Assertion true; Reason false (\(\alpha+\beta = 6\)).
Assertion (A)
If \(\alpha, \beta\) are zeroes of \(2x^2 + 5x + 1\), then \(\alpha + \beta + \alpha\beta = -2\).
Reason (R)
The product of zeroes of \(ax^2 + bx + c\) is \(c/a\).
Correct Option: (B)
\(\alpha+\beta = -5/2\), \(\alpha\beta = 1/2\). \(\alpha+\beta+\alpha\beta = -5/2 + 1/2 = -4/2 = -2\) (A is true). Reason R is true (\(c/a\)), but it is only part of the calculation.
\(\alpha+\beta = -5/2\), \(\alpha\beta = 1/2\). \(\alpha+\beta+\alpha\beta = -5/2 + 1/2 = -4/2 = -2\) (A is true). Reason R is true (\(c/a\)), but it is only part of the calculation.
Correct Answer: (B)
Both true, but R is only part of the formula used.
Both true, but R is only part of the formula used.
Assertion (A)
The zeroes of polynomial \(x^2 - 3\) are \(\sqrt{3}\) and \(-\sqrt{3}\).
Reason (R)
\(x^2 - 3 = (x - \sqrt{3})(x + \sqrt{3})\).
Correct Option: (B)
Both statements are true. Factoring gives \((x - \sqrt{3})(x + \sqrt{3}) = 0 \implies x = \pm \sqrt{3}\).
Both statements are true. Factoring gives \((x - \sqrt{3})(x + \sqrt{3}) = 0 \implies x = \pm \sqrt{3}\).
Correct Answer: (B)
Both true, R gives the factorisation.
Both true, R gives the factorisation.
Assertion (A)
The zeroes of \(p(x) = x^2 - 5x + 6\) are \(-2\) and \(-3\).
Reason (R)
\(x^2 - 5x + 6 = (x-2)(x-3) = 0\), so zeroes are \(2\) and \(3\).
Correct Option: (D)
Assertion (A) is false (zeroes are \(2\) and \(3\), not \(-2\) and \(-3\)). Reason (R) is true.
Assertion (A) is false (zeroes are \(2\) and \(3\), not \(-2\) and \(-3\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
A linear polynomial \(ax + b\) (\(a \ne 0\)) has exactly one zero.
Reason (R)
The graph of \(y = ax + b\) is a straight line.
Correct Option: (B)
Both statements are true. A linear polynomial has \(1\) zero at \(x = -b/a\). The graph of a linear polynomial is a straight line.
Both statements are true. A linear polynomial has \(1\) zero at \(x = -b/a\). The graph of a linear polynomial is a straight line.
Correct Answer: (B)
Both true, but straight line graph property is visual rather than algebraic explanation.
Both true, but straight line graph property is visual rather than algebraic explanation.
Assertion (A)
The quadratic polynomial \(x^2 - 2x - 8\) has zeroes \(4\) and \(-2\).
Reason (R)
The product of zeroes of \(x^2 - 2x - 8\) is \(8\).
Correct Option: (C)
Assertion (A) is true (\((x-4)(x+2)=0\)). Reason (R) is false because product of zeroes is \(c/a = -8/1 = -8\), not \(8\).
Assertion (A) is true (\((x-4)(x+2)=0\)). Reason (R) is false because product of zeroes is \(c/a = -8/1 = -8\), not \(8\).
Correct Answer: (C)
Assertion true; Reason false (product is -8).
Assertion true; Reason false (product is -8).
Assertion (A)
The degree of a non-zero constant polynomial is \(0\).
Reason (R)
For any non-zero real number \(c\), \(c = c \cdot x^0\), where the highest exponent of variable \(x\) is \(0\).
Correct Option: (A)
Any constant \(c \ne 0\) can be written as \(cx^0\), so its degree is \(0\). Reason explains Assertion.
Any constant \(c \ne 0\) can be written as \(cx^0\), so its degree is \(0\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The degree of a cubic polynomial is \(3\).
Reason (R)
A cubic polynomial can have at most \(3\) real zeroes.
Correct Option: (B)
Both are true statements, but maximum zeroes is a property resulting from degree, not the definition of degree itself.
Both are true statements, but maximum zeroes is a property resulting from degree, not the definition of degree itself.
Correct Answer: (B)
Both true, but R is a property rather than a definition.
Both true, but R is a property rather than a definition.
Assertion (A)
If product of zeroes of \(ax^2 - 6x - 6\) is \(4\), then \(a = 3/2\).
Reason (R)
Product of zeroes is \(c/a = -6/a = 4 \implies a = -6/4 = -3/2\).
Correct Option: (D)
Assertion (A) is false (\(a = -3/2\), not \(3/2\)). Reason (R) gives the true calculation.
Assertion (A) is false (\(a = -3/2\), not \(3/2\)). Reason (R) gives the true calculation.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
A polynomial cannot have more zeroes than its degree.
Reason (R)
A polynomial of degree \(3\) can have \(4\) real zeroes.
Correct Option: (C)
Assertion (A) is true. Reason (R) is false because a degree 3 polynomial can have at most 3 real zeroes.
Assertion (A) is true. Reason (R) is false because a degree 3 polynomial can have at most 3 real zeroes.
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The maximum number of zeroes a polynomial of degree \(n\) can have is \(n\).
Reason (R)
A quadratic polynomial has degree \(2\).
Correct Option: (B)
Both are true facts, but R is just an example of degree 2, not a proof for general \(n\).
Both are true facts, but R is just an example of degree 2, not a proof for general \(n\).
Correct Answer: (B)
Both true, but R is an example.
Both true, but R is an example.
Assertion (A)
A quadratic polynomial always has two distinct real zeroes.
Reason (R)
A quadratic polynomial can have two distinct real zeroes, two equal real zeroes, or no real zeroes depending on \(D\).
Correct Option: (D)
Assertion (A) is false. Reason (R) correctly outlines the three cases of discriminant \(D\).
Assertion (A) is false. Reason (R) correctly outlines the three cases of discriminant \(D\).
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If \(x = 2\) is a zero of polynomial \(p(x) = kx^2 + 3x + k\), then \(k = -6/5\).
Reason (R)
If \(\alpha\) is a zero of polynomial \(p(x)\), then \(p(\alpha) = 0\).
Correct Option: (A)
\(p(2) = k(2)^2 + 3(2) + k = 0 \implies 4k + 6 + k = 0 \implies 5k = -6 \implies k = -6/5\). Reason gives the fundamental condition for a zero and explains Assertion.
\(p(2) = k(2)^2 + 3(2) + k = 0 \implies 4k + 6 + k = 0 \implies 5k = -6 \implies k = -6/5\). Reason gives the fundamental condition for a zero and explains Assertion.
Correct Answer: (A)
\(p(2) = 0 \implies k = -6/5\). Reason explains Assertion.
\(p(2) = 0 \implies k = -6/5\). Reason explains Assertion.
Assertion (A)
The expression \(x + \frac{1}{x}\) is a polynomial of degree \(1\).
Reason (R)
An algebraic expression is a polynomial only if all powers of the variable are non-negative integers.
Correct Option: (D)
Assertion (A) is false (\(x + x^{-1}\) has negative exponent \(-1\), so it is not a polynomial). Reason (R) is the true definition of a polynomial.
Assertion (A) is false (\(x + x^{-1}\) has negative exponent \(-1\), so it is not a polynomial). Reason (R) is the true definition of a polynomial.
Correct Answer: (D)
Assertion false (not a polynomial); Reason true.
Assertion false (not a polynomial); Reason true.
Assertion (A)
If zeroes of \(x^2 - px + q\) are two consecutive integers, then \(p^2 - 4q = 1\).
Reason (R)
The difference of two consecutive integers is \(2\).
Correct Option: (C)
Assertion (A) is true: Let zeroes be \(\alpha, \alpha+1\). \((\alpha+1 - \alpha)^2 = 1 \implies (\alpha-\beta)^2 = (\alpha+\beta)^2 - 4\alpha\beta = p^2 - 4q = 1\). Reason (R) is false because difference of consecutive integers is \(1\), not \(2\).
Assertion (A) is true: Let zeroes be \(\alpha, \alpha+1\). \((\alpha+1 - \alpha)^2 = 1 \implies (\alpha-\beta)^2 = (\alpha+\beta)^2 - 4\alpha\beta = p^2 - 4q = 1\). Reason (R) is false because difference of consecutive integers is \(1\), not \(2\).
Correct Answer: (C)
Assertion true; Reason false (difference is 1).
Assertion true; Reason false (difference is 1).
Assertion (A)
If \(\alpha, \beta, \gamma\) are zeroes of \(x^3 - 3x^2 + x + 1\), then \(\alpha\beta\gamma = 1\).
Reason (R)
For cubic polynomial \(ax^3 + bx^2 + cx + d\), product of zeroes \(\alpha\beta\gamma = -d/a = -1/1 = -1\).
Correct Option: (D)
Assertion (A) is false (product is \(-1\), not \(1\)). Reason (R) is true.
Assertion (A) is false (product is \(-1\), not \(1\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If \(\alpha, \beta\) are zeroes of \(x^2 - 5x + 6\), then \(\frac{1}{\alpha} + \frac{1}{\beta} = \frac{5}{6}\).
Reason (R)
\(\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = \frac{-(-5)/1}{6/1} = \frac{5}{6}\).
Correct Option: (A)
Sum \(= 5\), product \(= 6\). \(\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = 5/6\). Reason explains Assertion.
Sum \(= 5\), product \(= 6\). \(\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = 5/6\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The polynomial \(p(x) = x^3 - x\) has \(3\) real zeroes.
Reason (R)
\(p(x) = x(x^2 - 1) = x(x-1)(x+1)\), giving zeroes \(0, 1, -1\).
Correct Option: (B)
Both A and R are true and R correctly solves \(p(x)=0\).
Both A and R are true and R correctly solves \(p(x)=0\).
Correct Answer: (B)
Both true, R gives factorisation.
Both true, R gives factorisation.
Assertion (A)
The zeroes of \(x^2 + 7x + 10\) are \(-2\) and \(-5\).
Reason (R)
The sum of zeroes of \(x^2 + 7x + 10\) is \(7\).
Correct Option: (C)
Assertion (A) is true (\(x^2+7x+10 = (x+2)(x+5)=0 \implies x=-2, -5\)). Reason (R) is false because sum of zeroes is \(-b/a = -7\), not \(7\).
Assertion (A) is true (\(x^2+7x+10 = (x+2)(x+5)=0 \implies x=-2, -5\)). Reason (R) is false because sum of zeroes is \(-b/a = -7\), not \(7\).
Correct Answer: (C)
Assertion true; Reason false (sum of zeroes is -7).
Assertion true; Reason false (sum of zeroes is -7).
Assertion (A)
The graph of \(y = x^2 + 1\) does not cross the x-axis.
Reason (R)
The zeroes of \(x^2 + 1\) are \(1\) and \(-1\).
Correct Option: (C)
Assertion (A) is true (\(x^2+1 > 0\) for all real \(x\)). Reason (R) is false because \(x^2+1=0 \implies x^2=-1\), which has no real zeroes.
Assertion (A) is true (\(x^2+1 > 0\) for all real \(x\)). Reason (R) is false because \(x^2+1=0 \implies x^2=-1\), which has no real zeroes.
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
A quadratic polynomial whose zeroes are reciprocals of each other has \(a = c\).
Reason (R)
If zeroes are \(\alpha\) and \(1/\alpha\), their product is \(\alpha \times (1/\alpha) = 1 = c/a \implies c = a\).
Correct Option: (B)
Both A and R are true and R provides the algebraic proof.
Both A and R are true and R provides the algebraic proof.
Correct Answer: (B)
Both true, R provides the proof.
Both true, R provides the proof.
Assertion (A)
If \(\alpha, \beta\) are zeroes of \(x^2 - 4x + 3\), then \(\alpha^2 + \beta^2 = 16\).
Reason (R)
\(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = 4^2 - 2(3) = 16 - 6 = 10\).
Correct Option: (D)
Assertion (A) is false (\(\alpha^2+\beta^2 = 10\), not \(16\)). Reason (R) is true.
Assertion (A) is false (\(\alpha^2+\beta^2 = 10\), not \(16\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.