✦ NCF 2023 ALIGNED • CBSE CLASS X ✦
Chapter 4: Quadratic Equations
Assertion-Reason Question Bank
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Assertion (A)
The quadratic equation \(x^2 - 4x + 4 = 0\) has two equal real roots.
Reason (R)
The discriminant \(D = b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0\). When \(D = 0\), roots are real and equal.
Correct Option: (A)
\(D = (-4)^2 - 4(1)(4) = 16 - 16 = 0\). Since \(D = 0\), the roots are real and equal (\(x = 2, 2\)). Reason gives the exact reason.
\(D = (-4)^2 - 4(1)(4) = 16 - 16 = 0\). Since \(D = 0\), the roots are real and equal (\(x = 2, 2\)). Reason gives the exact reason.
Correct Answer: (A)
\(D = 0\) implies real and equal roots. Reason explains Assertion.
\(D = 0\) implies real and equal roots. Reason explains Assertion.
Assertion (A)
If \(D = 0\), the roots of a quadratic equation are real and distinct.
Reason (R)
When \(D = 0\), the roots are real and equal (\(x = -b / 2a\)).
Correct Option: (D)
Assertion (A) is false (roots are equal, not distinct). Reason (R) is true.
Assertion (A) is false (roots are equal, not distinct). Reason (R) is true.
Correct Answer: (D)
Assertion false (roots are equal); Reason true.
Assertion false (roots are equal); Reason true.
Assertion (A)
The equation \(x^2 + 3x + 2 = 0\) has real roots \(-1\) and \(-2\).
Reason (R)
Product of roots of \(x^2 + 3x + 2 = 0\) is \(-2\).
Correct Option: (C)
Assertion (A) is true (\((x+1)(x+2)=0\)). Reason (R) is false because product of roots is \(c/a = 2/1 = 2\), not \(-2\).
Assertion (A) is true (\((x+1)(x+2)=0\)). Reason (R) is false because product of roots is \(c/a = 2/1 = 2\), not \(-2\).
Correct Answer: (C)
Assertion true; Reason false (product of roots is 2).
Assertion true; Reason false (product of roots is 2).
Assertion (A)
The roots of \(x^2 - 16 = 0\) are \(4\) only.
Reason (R)
Solving \(x^2 = 16\) yields two roots \(x = +4\) and \(x = -4\).
Correct Option: (D)
Assertion (A) is false (it has two roots \(\pm 4\)). Reason (R) is true.
Assertion (A) is false (it has two roots \(\pm 4\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
A quadratic equation always has real roots.
Reason (R)
If discriminant \(D < 0\), the quadratic equation has imaginary (non-real) roots.
Correct Option: (D)
Assertion (A) is false. Reason (R) is true.
Assertion (A) is false. Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The equation \(x^2 + 4x + 5 = 0\) has real roots.
Reason (R)
Discriminant \(D = 16 - 20 = -4 < 0\), which means the roots are not real.
Correct Option: (D)
Assertion (A) is false (\(D = -4 < 0\) implies non-real roots). Reason (R) is true.
Assertion (A) is false (\(D = -4 < 0\) implies non-real roots). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The product of the roots of \(2x^2 - 8x + 6 = 0\) is \(3\).
Reason (R)
The sum of roots of \(2x^2 - 8x + 6 = 0\) is \(4\).
Correct Option: (B)
Both are true (\(c/a = 6/2 = 3\) and \(-b/a = 8/2 = 4\)), but sum of roots is not the explanation for product of roots.
Both are true (\(c/a = 6/2 = 3\) and \(-b/a = 8/2 = 4\)), but sum of roots is not the explanation for product of roots.
Correct Answer: (B)
Both true, but R does not explain A.
Both true, but R does not explain A.
Assertion (A)
If \(x = 3\) is a root of \(x^2 - 5x + k = 0\), then \(k = 6\).
Reason (R)
For \(x^2 - 5x + 6 = 0\), roots are \(x = 2\) and \(x = 3\).
Correct Option: (B)
Both A and R are true. \(3^2 - 5(3) + k = 0 \implies k = 6\). R gives the full solution.
Both A and R are true. \(3^2 - 5(3) + k = 0 \implies k = 6\). R gives the full solution.
Correct Answer: (B)
Both true, R gives full solution.
Both true, R gives full solution.
Assertion (A)
The roots of \(x^2 + 5x + 6 = 0\) are \(-2\) and \(-3\).
Reason (R)
The discriminant of \(x^2 + 5x + 6 = 0\) is \(D = -1\).
Correct Option: (C)
Assertion (A) is true (\((x+2)(x+3)=0\)). Reason (R) is false because \(D = 25 - 24 = 1\), not \(-1\).
Assertion (A) is true (\((x+2)(x+3)=0\)). Reason (R) is false because \(D = 25 - 24 = 1\), not \(-1\).
Correct Answer: (C)
Assertion true; Reason false (\(D=1\)).
Assertion true; Reason false (\(D=1\)).
Assertion (A)
If \(x = 1\) is a common root of \(ax^2 + ax + 3 = 0\) and \(x^2 + x + b = 0\), then \(ab = 3\).
Reason (R)
Root \(x = 1\) in \(ax^2 + ax + 3 = 0\) gives \(a = 3\).
Correct Option: (C)
Assertion (A) is true: \(a(1)^2 + a(1) + 3 = 0 \implies 2a = -3 \implies a = -3/2\). \(1^2 + 1 + b = 0 \implies b = -2\). \(ab = (-3/2)(-2) = 3\). Reason (R) is false because \(a = -3/2\), not \(3\).
Assertion (A) is true: \(a(1)^2 + a(1) + 3 = 0 \implies 2a = -3 \implies a = -3/2\). \(1^2 + 1 + b = 0 \implies b = -2\). \(ab = (-3/2)(-2) = 3\). Reason (R) is false because \(a = -3/2\), not \(3\).
Correct Answer: (C)
Assertion true; Reason false (\(a = -3/2\)).
Assertion true; Reason false (\(a = -3/2\)).
Assertion (A)
If the discriminant \(D > 0\) and a perfect square, the roots of \(ax^2 + bx + c = 0\) (with rational coefficients) are real, distinct, and rational.
Reason (R)
The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Correct Option: (B)
Both statements are true mathematical facts regarding roots.
Both statements are true mathematical facts regarding roots.
Correct Answer: (B)
Both true, but R is the general quadratic formula rather than the specific explanation for rationality.
Both true, but R is the general quadratic formula rather than the specific explanation for rationality.
Assertion (A)
If one root of \(x^2 - 2x + k = 0\) is \(3\), then \(k = -3\).
Reason (R)
The sum of roots of \(x^2 - 2x + k = 0\) is \(-2\).
Correct Option: (C)
Assertion (A) is true (\(3^2 - 2(3) + k = 0 \implies 9 - 6 + k = 0 \implies k = -3\)). Reason (R) is false because sum of roots is \(-(-2)/1 = +2\), not \(-2\).
Assertion (A) is true (\(3^2 - 2(3) + k = 0 \implies 9 - 6 + k = 0 \implies k = -3\)). Reason (R) is false because sum of roots is \(-(-2)/1 = +2\), not \(-2\).
Correct Answer: (C)
Assertion true; Reason false (sum of roots is 2).
Assertion true; Reason false (sum of roots is 2).
Assertion (A)
The equation \(x + \frac{1}{x} = 3\) (\(x \ne 0\)) can be converted into a quadratic equation \(x^2 - 3x + 1 = 0\).
Reason (R)
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a \ne 0\).
Correct Option: (B)
Multiplying by \(x\) gives \(x^2 + 1 = 3x \implies x^2 - 3x + 1 = 0\) (A is true). R is standard form definition (true), but R alone doesn't show the algebraic step of multiplying by \(x\).
Multiplying by \(x\) gives \(x^2 + 1 = 3x \implies x^2 - 3x + 1 = 0\) (A is true). R is standard form definition (true), but R alone doesn't show the algebraic step of multiplying by \(x\).
Correct Answer: (B)
Both true, but R defines standard form rather than explaining the algebraic conversion.
Both true, but R defines standard form rather than explaining the algebraic conversion.
Assertion (A)
The equation \(x^2 + x + 1 = 0\) has no real roots.
Reason (R)
Discriminant \(D = b^2 - 4ac = 1^2 - 4(1)(1) = -3 < 0\). When \(D < 0\), quadratic equation has no real roots.
Correct Option: (A)
\(D = 1 - 4 = -3 < 0\). Negative discriminant implies no real roots. Reason explains Assertion.
\(D = 1 - 4 = -3 < 0\). Negative discriminant implies no real roots. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The roots of \(x^2 - 9 = 0\) are \(+3\) and \(-3\).
Reason (R)
\(x^2 - 9 = (x - 3)(x + 3) = 0\).
Correct Option: (B)
Both A and R are true and R provides factoring explanation.
Both A and R are true and R provides factoring explanation.
Correct Answer: (B)
Both true, R provides factoring explanation.
Both true, R provides factoring explanation.
Assertion (A)
The quadratic equation \(2x^2 - 5x + 8 = 0\) has real roots.
Reason (R)
\(D = (-5)^2 - 4(2)(8) = 25 - 64 = -39 < 0\), so no real roots exist.
Correct Option: (D)
Assertion (A) is false. Reason (R) is true.
Assertion (A) is false. Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The sum of roots of \(x^2 - 5x + 6 = 0\) is \(-5\).
Reason (R)
For \(ax^2 + bx + c = 0\), sum of roots is \(-b/a = -(-5)/1 = 5\).
Correct Option: (D)
Assertion (A) is false (sum is \(+5\), not \(-5\)). Reason (R) is true.
Assertion (A) is false (sum is \(+5\), not \(-5\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If \(kx^2 - 4x + 4 = 0\) has equal roots, then \(k = 1\).
Reason (R)
Condition for equal roots is \(D = b^2 - 4ac > 0\).
Correct Option: (C)
Assertion (A) is true (\(16 - 16k = 0 \implies k = 1\)). Reason (R) is false because condition for equal roots is \(D = 0\), not \(D > 0\).
Assertion (A) is true (\(16 - 16k = 0 \implies k = 1\)). Reason (R) is false because condition for equal roots is \(D = 0\), not \(D > 0\).
Correct Answer: (C)
Assertion true; Reason false (\(D=0\)).
Assertion true; Reason false (\(D=0\)).
Assertion (A)
The sum of the roots of \(3x^2 - 9x + 5 = 0\) is \(3\).
Reason (R)
For \(ax^2 + bx + c = 0\), sum of roots is \(-b/a = -(-9)/3 = 3\).
Correct Option: (A)
Sum of roots \(= -b/a = 9/3 = 3\). Reason gives rule and explains Assertion.
Sum of roots \(= -b/a = 9/3 = 3\). Reason gives rule and explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If the product of roots of \(3x^2 - 6x + k = 0\) is \(2\), then \(k = 3\).
Reason (R)
Product of roots is \(c/a = k/3 = 2 \implies k = 6\).
Correct Option: (D)
Assertion (A) is false (\(k = 6\), not \(3\)). Reason (R) is true.
Assertion (A) is false (\(k = 6\), not \(3\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The roots of quadratic equation \(2x^2 - 7x + 3 = 0\) are \(3\) and \(1/2\).
Reason (R)
By quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{49 - 24}}{4} = \frac{7 \pm 5}{4}\), giving \(x = 3\) or \(1/2\).
Correct Option: (A)
\(D = 49 - 24 = 25\). \(x = (7 \pm 5)/4 \implies x = 3, 1/2\). Reason shows step-by-step formula and explains Assertion.
\(D = 49 - 24 = 25\). \(x = (7 \pm 5)/4 \implies x = 3, 1/2\). Reason shows step-by-step formula and explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
A quadratic equation can have at most two real roots.
Reason (R)
The degree of a quadratic equation is \(2\).
Correct Option: (B)
Both are true facts. By Fundamental Theorem of Algebra, degree 2 polynomial has at most 2 roots.
Both are true facts. By Fundamental Theorem of Algebra, degree 2 polynomial has at most 2 roots.
Correct Answer: (B)
Both true, R gives algebraic degree.
Both true, R gives algebraic degree.
Assertion (A)
The quadratic equation \(x^2 - 6x + 9 = 0\) has real and equal roots.
Reason (R)
Discriminant of \(x^2 - 6x + 9 = 0\) is \(D = 36\).
Correct Option: (C)
Assertion (A) is true (\((x-3)^2 = 0\)). Reason (R) is false because \(D = (-6)^2 - 4(1)(9) = 36 - 36 = 0\), not \(36\).
Assertion (A) is true (\((x-3)^2 = 0\)). Reason (R) is false because \(D = (-6)^2 - 4(1)(9) = 36 - 36 = 0\), not \(36\).
Correct Answer: (C)
Assertion true; Reason false (\(D=0\)).
Assertion true; Reason false (\(D=0\)).
Assertion (A)
If \(a\) and \(c\) have opposite signs, \(ax^2 + bx + c = 0\) always has real roots.
Reason (R)
Discriminant \(D = b^2 - 4ac\). If \(a\) and \(c\) have opposite signs, \(-4ac > 0\), so \(D = b^2 + \text{positive} > 0\).
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 algebraic proof.
Both true, R provides algebraic proof.
Assertion (A)
The quadratic equation \((x - 2)^2 + 1 = 2x - 3\) is a quadratic equation in \(x\).
Reason (R)
Expanding gives \(x^2 - 4x + 4 + 1 = 2x - 3 \implies x^2 - 6x + 8 = 0\), which is of the standard form \(ax^2 + bx + c = 0\) with \(a \ne 0\).
Correct Option: (A)
Simplifying yields degree 2 polynomial equation \(x^2 - 6x + 8 = 0\). Reason explains Assertion.
Simplifying yields degree 2 polynomial equation \(x^2 - 6x + 8 = 0\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The nature of roots of \(2x^2 - 4x + 3 = 0\) is non-real (imaginary).
Reason (R)
\(D = (-4)^2 - 4(2)(3) = 16 - 24 = -8 < 0\), indicating no real roots.
Correct Option: (A)
\(D = -8 < 0\) proves non-real roots. Reason explains Assertion.
\(D = -8 < 0\) proves non-real roots. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The roots of \(2x^2 - 2\sqrt{2}x + 1 = 0\) are equal.
Reason (R)
Discriminant \(D = (-2\sqrt{2})^2 - 4(2)(1) = 8 - 8 = 8\).
Correct Option: (C)
Assertion (A) is true (\((\sqrt{2}x - 1)^2 = 0\)). Reason (R) is false because \(D = 8 - 8 = 0\), not \(8\).
Assertion (A) is true (\((\sqrt{2}x - 1)^2 = 0\)). Reason (R) is false because \(D = 8 - 8 = 0\), not \(8\).
Correct Answer: (C)
Assertion true; Reason false (\(D=0\)).
Assertion true; Reason false (\(D=0\)).
Assertion (A)
If \(x = 2\) is a root of \(kx^2 + 2x - 3 = 0\), then \(k = -1/4\).
Reason (R)
Substituting \(x = 2\) gives \(k(2)^2 + 2(2) - 3 = 0 \implies 4k + 1 = 0 \implies k = -1/4\).
Correct Option: (A)
Root \(x=2\) satisfies \(4k + 4 - 3 = 0 \implies 4k + 1 = 0 \implies k = -1/4\). Reason explains Assertion.
Root \(x=2\) satisfies \(4k + 4 - 3 = 0 \implies 4k + 1 = 0 \implies k = -1/4\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The equation \((x+1)^2 - x^2 = 0\) has two real roots.
Reason (R)
Expanding gives \(2x + 1 = 0 \implies x = -1/2\), which is a linear equation with one root.
Correct Option: (D)
Assertion (A) is false (it simplifies to a linear equation with 1 root). Reason (R) is true.
Assertion (A) is false (it simplifies to a linear equation with 1 root). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The roots of \((x-1)(x+2) = 0\) are \(1\) and \(-2\).
Reason (R)
If product of two factors is zero, at least one factor must be zero.
Correct Option: (B)
Both A and R are true zero-product property facts.
Both A and R are true zero-product property facts.
Correct Answer: (B)
Both true, R gives zero-product property.
Both true, R gives zero-product property.