Q20
1 markMCQSection A

  1. Assertion (A) : The polynomial p(y)=y2+4y+3p(y) = y^2 + 4y + 3 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.

Polynomials
Assertion-Reason: zeroes of a quadratic polynomial

Options

(A)Both A and R are true and R is the correct explanation of A
(B)Both A and R are true, but R is not the correct explanation of A
(C)A is true, but R is false
(D)A is false, but R is true
Official Answer

(b) Both A and R are true, but R is not the correct explanation of A — (y+1)(y+3)(y+1)(y+3) gives zeroes −1-1 and −3-3 (A true), while 'at most two zeroes' is a true general bound (R true) that does not explain why this particular polynomial has exactly two.

y² + 4y + 3 factorise(y+1)(y+3)zeroes −1 and −3at most two zeroesdiscriminant positiveboth true R not explanationoption B

Marking Scheme

  • 11 mark: Correct option (b) selected.
  • 2No partial marks in MCQ.
  • 3Justification expected (if asked): (y+1)(y+3) gives zeroes −1, −3 (A true); 'at most two' is true (R true) but is a bound, not the explanation → not correct explanation.

Hint

Both statements are true, so the question is only whether 'at most two zeroes' is the reason this polynomial has exactly two. A maximum bound is not an explanation of the exact count.

Quick Oral Answer

Factorising y2+4y+3y^2 + 4y + 3 gives (y+1)(y+3)(y+1)(y+3), so it has two zeroes −1-1 and −3-3, making the Assertion true; the Reason that a quadratic has at most two zeroes is also true but it is only an upper limit, not the reason this one has exactly two — so the correct answer is (b).

Analysis & Explanation

This item tests both factorising a quadratic and distinguishing a bound from an explanation.


Concept

  • Factorise p(y)=y2+4y+3=(y+1)(y+3)p(y) = y^2 + 4y + 3 = (y + 1)(y + 3), giving zeroes y=−1,−3y = -1, -3 — two distinct real zeroes, so Assertion A is true.
  • Reason R states a quadratic can have at most two zeroes — a true general theorem, so R is also true.

Key points

  • R only sets an upper limit (0, 1, or 2 zeroes possible); it does not explain why THIS polynomial has exactly two — that is due to its positive discriminant (D=16−12=4>0D = 16 - 12 = 4 > 0).
  • Since both statements are true but R does not causally explain A, the correct option is (b).

Common mistakes

  • Picking (a) just because both statements 'sound related' without checking whether R is the actual cause of A.
  • Wrongly assuming 'at most two' automatically explains 'exactly two'.

Common Mistakes

  1. 1Choosing (a) because both statements are true and 'seem related' — failing to test whether R is the actual explanation of A.
  2. 2Marking R false by confusing 'at most two zeroes' with 'exactly two zeroes' — the theorem states a maximum, which is correct.
  3. 3Sign errors while factorising, e.g. writing zeroes as +1+1 and +3+3 instead of −1-1 and −3-3.

Interesting Facts

The 'at most n zeroes' rule is a special case of the Fundamental Theorem of Algebra, proved by Carl Friedrich Gauss in his 1799 doctoral thesis.

The number of REAL zeroes of a quadratic is decided entirely by the sign of the discriminant b2−4acb^2 - 4ac: positive gives 2, zero gives 1 (repeated), negative gives 0 real zeroes.

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Frequently Asked Questions

What are the zeroes of the polynomial y2+4y+3y^2 + 4y + 3?

Factorise as (y+1)(y+3)(y + 1)(y + 3). Setting each factor to zero gives y=−1y = -1 and y=−3y = -3, so the polynomial has two distinct real zeroes.

Why is the Reason true but not the correct explanation here?

The Reason states a quadratic has 'at most two zeroes', which is a correct upper bound. But this bound does not explain why this polynomial has EXACTLY two zeroes — that is because its discriminant (16−12=416 - 12 = 4) is positive. So R is a true fact that fails to explain A.

How is this different from an Assertion-Reason where the answer is (a)?

The answer is (a) only when R directly causes or justifies A. Here R merely sets a maximum; a correct explaining reason would have been 'because its discriminant is positive, the quadratic has two distinct real zeroes.'