Assertion (A) : For matrix A = bmatrix 1 & θ & 1 \\ -θ & 1 & θ \\ -1 & -θ & 1 bmatrix, where θ [0, 2π], A [2, 4]. Reason (R) : θ [-1, 1], θ [0, 2π]. (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.

MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 1 mark · Assertion-reason

CBSE Class 12 Mathematics 2024: question 19

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section A, question 19, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q19 · 1 mark · Assertion-reasonCompetency-basedDeterminants

Assertion (A) : For matrix $A = 1 - cos θ - 1 cos θ 1 - cos θ 1 cos θ 1$ , where $θ \in [0, 2 π]$ , $∣ A ∣ \in [2, 4]$ .
Reason (R) : $cos θ \in [- 1, 1], \forall θ \in [0, 2 π]$ .

(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.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Expand the determinant along the first row.

Hint 2

Use the range of $cos^{2} θ$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Expanding along $R_{1}$ : $∣ A ∣ = 1 (1 + cos^{2} θ) - cos θ (- cos θ + cos θ) + 1 (cos^{2} θ + 1) = 2 + 2 cos^{2} θ$ .
½ mark
Since $cos θ \in [- 1, 1]$ , $cos^{2} θ \in [0, 1]$ , so $∣ A ∣ \in [2, 4]$ . Both statements are true, and R is what gives A.
½ mark

Answer: (A)

Where the marks are

Finds $∣ A ∣ = 2 + 2 cos^{2} θ$ and uses the range of $cos θ$ (1 mark)

Revise the chapter

Chapter: Determinants (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Determinants, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 19

Hints

Step-by-step solution

Revise the chapter

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