Practise 7 selected Polynomials Assertion–Reason questions from CBSE Class 10 previous-year board papers (2023–2026). Check each statement, then decide whether the Reason explains the Assertion. Two true statements do not always form a correct explanation.
7 distinct questions · 21 verified board-paper appearances Solutions prepared by Team TutorVertex · Last checked: 25 October 2026
Selected board years: 2023, 2024 and 2026 · Basic and Standard · Degree, factorisation and graphical zeroes.
The related questions found in the screened 2025 papers focus on real roots of quadratic equations and are classified under Quadratic Equations rather than included here.
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
Choices shown as in 30/5/1, Q20. The A–D statements and their order are identical in the listed appearances; only layout, capitalisation and punctuation may differ.
Show Answer & Explanation
Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. The zero polynomial has no term with a non-zero coefficient, so there is no highest power to determine its degree.
Reason: True. A non-zero constant c can be written as cx⁰, so its degree is 0.
Link: R describes a non-zero constant polynomial. It does not explain why the zero polynomial has no defined degree. Answer: B.
Question 2 · Basic
Assertion (A): Polynomial x² + 4x has two real zeroes.
Reason (R): Zeroes of the polynomial x² + ax (a ≠ 0) are 0 and a.
Choices shown as in 430/2/1, Q19. The A–D statements and their order are identical in the listed appearances; only layout, capitalisation and punctuation may differ.
Show Answer & Explanation
Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. x² + 4x = x(x + 4). Hence its zeroes are 0 and −4, which are real and distinct.
Reason: False. x² + ax = x(x + a), so its zeroes are 0 and −a, not 0 and a.
Link: A is true and R is false. The sign comes from solving x + a = 0. Answer: C.
Graphs and zeroes
Key results
The zeroes are the x-coordinates where the graph y = p(x) meets the x-axis.
A quadratic polynomial can have zero, one or two distinct real zeroes. “At most two” does not mean “exactly two”.
A graph touching the x-axis at a single point can represent a quadratic polynomial.
Question 3 · Standard
Assertion (A): If the graph of a polynomial intersects the x-axis at exactly two points, then the number of zeroes of that polynomial is 2.
Reason (R): The number of zeroes of a polynomial is equal to the number of points where the graph of the polynomial intersects x-axis.
Choices shown as in 30/S/2, Q20. The A–D statements and their order are identical in the listed appearances; only layout, capitalisation and punctuation may differ.
Show Answer & Explanation
Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. The two distinct points of intersection have two distinct x-coordinates at which the polynomial is zero.
Reason: True. At each point on the x-axis, the y-coordinate is 0. For the graph y = p(x), this means p(x) = 0.
Link: R directly gives the number of zeroes from the two points of intersection, so it explains A. Answer: A.
Question 4 · Basic
Assertion (A): Zeroes of a polynomial p(x) = x² − 2x − 3 are −1 and 3.
Reason (R): The graph of polynomial p(x) = x² − 2x − 3 intersects x-axis at (−1, 0) and (3, 0).
Q4 retains CBSE’s printed wording: “explains Assertion (A) completely / does not explain Assertion (A)”. The A–D meanings and order match the decision table. In 430/1/3, the paper labels this Q20 while the marking scheme reproduces it under Q19 and gives A.
Show Answer & Explanation
Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. x² − 2x − 3 = (x + 1)(x − 3). Therefore the zeroes are −1 and 3.
Reason: True. p(−1) = 1 + 2 − 3 = 0 and p(3) = 9 − 6 − 3 = 0. Thus the graph meets the x-axis at (−1, 0) and (3, 0).
Link: Zeroes are the x-coordinates of the points where the graph meets the x-axis. Hence R explains A. Answer: A.
Question 5 · Standard
Assertion (A): The polynomial p(y) = y² + 4y + 3 has two zeroes.
Reason (R): A quadratic polynomial can have at most two zeroes.
Choices shown as in 30/1/1, Q20. The A–D statements and their order are identical in the listed appearances; only layout, capitalisation and punctuation may differ.
Show Answer & Explanation
Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. y² + 4y + 3 = (y + 1)(y + 3), so the zeroes are −1 and −3.
Reason: True. A polynomial of degree 2 can have at most two distinct real zeroes.
Link: R gives only the maximum possible number. It does not establish that this particular polynomial has two zeroes; the factorisation does. Answer: B.
Question 6 · Standard
Assertion (A): If the graph of a polynomial touches x-axis at only one point, then the polynomial cannot be a quadratic polynomial.
Reason (R): A polynomial of degree n (n > 1) can have at most n zeroes.
Choices shown as in 30/1/1, Q20. The A–D statements and their order are identical in the listed appearances; only layout, capitalisation and punctuation may differ.
Show Answer & Explanation
Official CBSE answer · Explanation by Team TutorVertex
Assertion: False. The quadratic polynomial p(x) = x² has only one distinct zero, 0. Its graph touches the x-axis only at (0, 0).
Reason: True. A non-zero polynomial of degree n has at most n distinct real zeroes.
Link: An upper limit of two permits a quadratic to have only one distinct zero. Thus A is false and R is true. Answer: D.
Question 7 · Standard
Assertion (A): The polynomial p(x) = x² + 3x + 3 has two real zeroes.
Reason (R): A quadratic polynomial can have at most two real zeroes.
Choices shown as in 30/2/1, Q20. The A–D statements and their order are identical in the listed appearances; only layout, capitalisation and punctuation may differ.
Show Answer & Explanation
Official CBSE answer · Explanation by Team TutorVertex
Assertion: False. x² + 3x + 3 = (x + )² + . A square is non-negative for real x, so p(x) > 0 and has no real zeroes.
Reason: True. A polynomial of degree 2 can have at most two real zeroes.
Link: “At most two” allows zero, one or two distinct real zeroes. This polynomial has none, so A is false and R is true. Answer: D.
Common questions
Can a quadratic polynomial have only one zero?
Yes. For example, p(x) = x² has only one distinct real zero, 0. Its graph touches the x-axis at (0, 0).
Does “at most two zeroes” mean exactly two?
No. A quadratic polynomial can have zero, one or two distinct real zeroes. Check the particular polynomial rather than assuming that its degree gives the exact count.
What is the difference between a zero and an x-intercept?
A zero is a number k satisfying p(k) = 0. The corresponding point on the graph is (k, 0). For example, −1 is a zero and (−1, 0) is an x-intercept.
How do I choose between A and B?
Both require the Assertion and Reason to be true. Choose A only when the Reason explains the Assertion. A general upper bound alone need not explain an exact count.
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, Q19 and Q20, in Section A, worth one mark each across the paper. This does not promise a question from every chapter.
What if both statements seem false?
None of the four standard choices used here represents both statements being false. Recheck the exact wording, conditions and calculations. If both remain false, flag a possible question or transcription error rather than forcing an answer.
Official sources and marking schemes
Source labels identify the original paper, set and question number. ZIP links open official archives containing the named paper.
This selection contains 7 distinct questions across 21 verified board-paper appearances: 2023, 2024 and 2026. Exact repeats are merged and their original set codes and question numbers are listed. The order and meaning of the four choices are checked for each listed appearance. This is a selection, not every previous-year question.
Question statements and mathematical symbols are checked against official papers; answers are compared with the corresponding marking schemes and verified mathematically. Explanations are written by Team TutorVertex in three labelled parts: Assertion, Reason and Link. No incorrect-choice explanation blocks are added.
NCERT Chapter 2 and the applicable syllabus guide the scope, terminology and working. Degree and factorisation recall earlier learning; the degree question is labelled as prerequisite practice. Distinct real zeroes are counted, following NCERT’s graphical interpretation. The selected questions do not require a supplied diagram or data table, so none is invented.
The available official board-paper corpus and Basic and Standard sample papers for sessions 2022–23 through 2026–27 were screened. Sample-paper questions needing a real-zero domain clarification are held outside scored practice. Questions centred on solving quadratic equations or the nature of their roots are reserved for the Quadratic Equations resource. Coverage does not claim every language edition or every repeated appearance.
Left out and why
These items are not counted in the 7 questions or 21 appearances above.
CBSE 2026 second board examination · Standard · accessible paper 30(B) · Q19
Left out because opposite real zeroes would have a non-positive product, but this polynomial’s product is 7; CBSE’s answer is D.
CBSE 2026–27 official sample paper · Standard · Q20
Left out because the calculated value k = −1 gives a polynomial with no real zeroes, so the meaning of “zeroes” needs clarification; CBSE’s answer is A.