Assertion (A) : The polynomial has two zeroes.
Reason (R) : A quadratic polynomial can have at most two zeroes.
(a) Both, Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(b) Both, Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true.
Assertion (A) : The polynomial has two zeroes.
Reason (R) : A quadratic polynomial can have at most two zeroes.
(a) Both, Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(b) Both, Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true.
Options
Official answer
(a) Both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A) — has exactly 2 zeroes , the maximum possible for a quadratic.
Marking scheme
- 1.1 mark: correctly selecting option (a); no partial credit in MCQ format.
Hint
Factorise to find the two zeroes, then recall that a degree-n polynomial has at most n zeroes.
Quick oral answer
factorises as , giving zeroes -1 and -3, so it has two zeroes — the Assertion is true. Since a quadratic can have at most two zeroes, the Reason is also true and correctly explains the Assertion, so the answer is option (a).
Common mistakes
- 1.Forgetting to actually factorise/solve and instead assuming the number of zeroes without verification.
- 2.Confusing 'at most two zeroes' (Reason, a general true rule) with 'exactly two zeroes' and doubting whether R explains A.
- 3.Choosing option (b) out of hesitation, not recognising that R is indeed the textbook justification for why a quadratic can have 2 zeroes.
Analysis and explanation
Tests the zero-count rule for quadratic polynomials.
Concept
- A degree-n polynomial has at most n real zeroes (Fundamental Theorem of Algebra, real case).
- factorises as .
Key points
- Zeroes of p(y): and — exactly two distinct real zeroes ⟹ A is TRUE.
- A quadratic (degree 2) can have at most 2 zeroes ⟹ R is TRUE, and it correctly explains why A holds (this quadratic attains the maximum of 2 zeroes).
- Hence option (a): both true, R is the correct explanation of A.
Common mistakes
- Wrongly assuming R must be unrelated just because it's a general statement — here it directly explains A.
- Confusing this with Q19's pattern, where R was true-sounding but factually wrong; here both statements are independently correct and logically linked.
Interesting facts
The rule 'a polynomial of degree n has at most n zeroes' is a direct consequence of the Fundamental Theorem of Algebra and is one of the most frequently tested one-line facts in CBSE Class 10 Algebra.
can also be solved instantly using the quadratic formula: , giving -1 and -3, matching the factorisation method.
Graphically, the two zeroes -1 and -3 are exactly the two points where the upward parabola crosses the x-axis.
Question details
Q20 of 56 questions
- Board
- CBSE
- Class
- Class 10
- Subject
- Mathematics (Standard)
- Year
- 2026
- Section
- Section A
- Marks
- 1
- Type
- MCQ
- Chapter
- Polynomials
- Topic
- Zeroes of a Quadratic Polynomial
- Bloom's level
- understand
- Syllabus topic
- Unit 2: Algebra — Polynomials: Zeroes of a quadratic polynomial and their number
- Author
- CBSE
- Reviewed by
- Padhantu Academic Team
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Frequently asked questions
How many zeroes can a polynomial of degree n have?
A polynomial of degree n can have at most n real zeroes; it may have fewer if some roots are repeated or complex/imaginary.
What are the zeroes of ?
Factorising gives , so the zeroes are and .