Practise 4 selected Quadratic Equations Assertion–Reason questions from CBSE Class 10 previous-year board papers (2023 and 2025). Check each statement, then decide whether the Reason explains the Assertion. Two true statements do not always form a correct explanation.
4 distinct questions · 12 verified board-paper appearances Solutions prepared by Team TutorVertex · Last checked: 25 October 2026
Selected board years: 2023 and 2025 · Basic and Standard · Real roots, discriminants, reciprocal roots and conjugate surds.
Coverage review: 2022–2027. Questions currently represented: 2023 and 2025. See the year-by-year coverage notes, including the 2026–27 sample-paper session.
How to solve Assertion–Reason questions
In the official 2026–27 CBSE Class 10 Mathematics sample-paper format, Section A contains two Assertion–Reason questions worth one mark each across the whole paper. Basic format · Standard format.
Check whether the Assertion is true.
Check whether the Reason is true, separately.
If both are true, decide whether the Reason explains the Assertion.
Match your conclusion to the printed answer choices.
Decision table for the standard choices used below
Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. Reciprocal roots have product 1. Hence = 1, giving k − 4 = 4 and k = 8. At k = 8, 4x² − 10x + 4 = 2(2x − 1)(x − 2), so the real roots are and 2.
Reason: False. For x² − x + 1 = 0, b² − 4ac = (−1)² − 4(1)(1) = −3 < 0. Therefore its roots are not real.
Link: A correctly gives k = 8, whereas R makes a false claim about a different equation. A is true and R is false. Answer: C.
Question 4 · Standard
Assertion (A): If 5 + √7 is a root of a quadratic equation with rational co-efficients, then its other root is 5 − √7.
Reason (R): Surd roots of a quadratic equation with rational co-efficients occur in conjugate pairs.
Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. The root conjugate to 5 + √7 is 5 − √7.
Reason: True. With rational coefficients, an irrational square-root part in the quadratic formula occurs with opposite signs in the two roots, giving the conjugate pair.
Link: R applies directly to the given irrational root 5 + √7 and explains why the other root is 5 − √7. Answer: A.
Why the conjugate is also a root (optional proof)
Let f(x) = ax² + bx + c have rational coefficients, a ≠ 0. Substituting 5 + √7 gives (32a + 5b + c) + (10a + b)√7 = 0. Since √7 is irrational, b = −10a and c = 18a. Thus f(x) = a[(x − 5)² − 7], whose other root is 5 − √7.
Common questions
Does every quadratic equation have real roots?
No. For example, x² + 1 = 0 has discriminant −4, so it has no real roots.
What happens when b² − 4ac equals zero?
The roots are real and equal. Both values from the quadratic formula coincide.
What is the product of reciprocal roots?
It is 1. If α and β are reciprocal, αβ = 1; use αβ = c/a when the equation is quadratic.
Why must rational coefficients be stated for conjugate surd roots?
The condition matters: a quadratic equation with irrational coefficients need not have the same conjugate-root property. Preserve the rational-coefficient condition in the question.
How many Assertion–Reason questions come in the CBSE Class 10 Maths board paper?
The official 2026–27 Basic and Standard sample-paper format has two Assertion–Reason questions in Section A, worth one mark each across the whole paper. It does not guarantee a question from this chapter.
What if both statements seem false?
None of the four standard choices used here represents both statements being false. Recheck the wording, restrictions and calculations before flagging a possible source error.
Coverage: 2022–2027
The review range and the years represented by selected questions are different. A year is not added to the question count unless a relevant Assertion–Reason item has been verified.
Year-by-year status for this Quadratic Equations selection
Year
Coverage note
2022
The 2021–22 Term II Mathematics format used written questions rather than the current Assertion–Reason section. See the official Basic and Standard formats. No 2022 board appearance is counted here.
2023
Two distinct questions included: reciprocal roots and conjugate surd roots; six verified set appearances.
2024
No qualifying Quadratic Equations Assertion–Reason item found in the 40 available Basic and Standard main and supplementary papers rechecked. Polynomial graph and zeroes questions are covered in the Polynomials resource.
2025
Two distinct questions included: existence of real roots and the discriminant; six verified set appearances.
2026
No additional qualifying item identified in the screened board selection. Related polynomial-zeroes items are handled with Polynomials.
2027
The 2027 board examination has not taken place as of 25 October 2026. The released 2026–27 Basic and Standard sample papers were screened; neither contributes a question to this selection.
Basic and Standard official sample papers for sessions 2022–23, 2023–24, 2024–25, 2025–26 and 2026–27 were also checked; none contributes a scored question here. Their paper and marking-scheme links are listed below. These notes describe the screened files, without claiming every language edition or every repeated appearance.
Official sources and marking schemes
Source labels identify the original paper, set and question number. ZIP links open official archives containing the named paper.
Choices shown as in 430/3/1, Q20. The meanings and A–D order are the same in the listed appearances; layout, capitalisation and punctuation may differ.
Choices shown as in 430/4/1, Q20. The meanings and A–D order are the same in the listed appearances; layout, capitalisation and punctuation may differ.
This selection contains 4 distinct questions across 12 verified appearances from the 2023 and 2025 board papers. Exact repeats are merged. Full set codes, original question numbers and choice-wording notes are kept in the expandable source details.
Question text, mathematical symbols and the four choices are checked against the original papers. Marking-scheme answers are matched by question content as well as question number. Explanations are written by Team TutorVertex as Assertion, Reason and Link, using real-root methods, factorisation and the discriminant. The 2023 conjugate-surd item is an earlier board application involving rational coefficients.
The coverage review spans 2022–2027, with the year-specific outcomes shown above. The available board corpus and Basic and Standard sample papers for sessions 2022–23 through 2026–27 were checked for relevant items; the 2022 Term II format and the future 2027 board examination are distinguished from completed examinations using the current format. No sample-paper item is counted in this scored selection. Polynomial-zeroes and graph questions are covered separately in the Polynomials Assertion–Reason resource. This selection does not claim every language edition or every repeated appearance.
None of the four selected questions supplies a graph, diagram or data table. Figures belonging to nearby questions have not been included. The small selection uses one question section and one Key results box, without a topic filter.