The two co-initial adjacent sides of a parallelogram are 2i-4j-5k and 2i+2j+3k. Find its diagonals and use them to find the area of the parallelogram.

Diagonals 4 i-2 j-2 k and -6 j-8 k; area =2√(101) sq. units

MathsCBSE Class 12 · Sample paper, session 2024-25 · 2 marks · Very short answer

CBSE Class 12 Mathematics sample paper 2024-25: question 25

From the official CBSE sample question paper for 2024-25, section B, question 25, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q25 · 2 marks · Very short answerVector Algebra

The two co-initial adjacent sides of a parallelogram are $2 \hat{i} - 4 \hat{j} - 5 \hat{k}$ and $2 \hat{i} + 2 \hat{j} + 3 \hat{k}$ . Find its diagonals and use them to find the area of the parallelogram.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

The diagonals of a parallelogram with sides $a, b$ are $a + b$ and $a - b$ .

Hint 2

Area in terms of diagonals is half the magnitude of their cross product.

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.

With $a = 2 \hat{i} - 4 \hat{j} - 5 \hat{k}$ , $b = 2 \hat{i} + 2 \hat{j} + 3 \hat{k}$ , the diagonals are $d_{1} = a + b = 4 \hat{i} - 2 \hat{j} - 2 \hat{k}$ and $d_{2} = a - b = - 6 \hat{j} - 8 \hat{k}$ .
½ mark
$d_{1} \times d_{2} = \hat{i} 40 \hat{j} - 2 - 6 \hat{k} - 2 - 8 = (16 - 12) \hat{i} - (- 32 - 0) \hat{j} + (- 24 - 0) \hat{k} = 4 \hat{i} + 32 \hat{j} - 24 \hat{k} = 4 (\hat{i} + 8 \hat{j} - 6 \hat{k})$ .
1 mark
Area $= \frac{1}{2} ∣ d_{1} \times d_{2} ∣ = \frac{1}{2} \cdot 4 1 + 64 + 36 = 2101$ sq. units.
½ mark

Answer: Diagonals $4 \hat{i} - 2 \hat{j} - 2 \hat{k}$ and $- 6 \hat{j} - 8 \hat{k}$ ; area $= 2101$ sq. units

Where the marks are

Finds both diagonals (½ mark)
Computes $d_{1} \times d_{2}$ correctly (1 mark)
Area $= \frac{1}{2} ∣ d_{1} \times d_{2} ∣ = 2101$ sq units (½ mark)

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

Practise this chapterBoard-style questions from Vector Algebra, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2024-25: every question, by section.

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CBSE Class 12 Mathematics sample paper 2024-25: question 25

Hints

Step-by-step solution

Revise the chapter

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