In the given figure, ABCD is a parallelogram. If AB = 2i - 4j + 5k and DB = 3i - 6j + 2k, then find AD and hence find the area of parallelogram ABCD.

AD=- i+2 j+3 k; Area =115 (i.e. √(605)) sq units

MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2024: question 25

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, 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

In the given figure, ABCD is a parallelogram. If $A B = 2 \hat{i} - 4 \hat{j} + 5 \hat{k}$ and $D B = 3 \hat{i} - 6 \hat{j} + 2 \hat{k}$ , then find $A D$ and hence find the area of parallelogram ABCD.

Figure in the paper: Parallelogram ABCD with A top-left, B top-right, C bottom-right, D bottom-left. Arrows mark vector AB along the top side (A to B), vector AD along the left side (D to A direction arrow), and diagonal DB from D to B. See it in the official paper.

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

Use the triangle law in triangle ABD to relate AD, DB and AB.

Hint 2

Area of a parallelogram is the magnitude of the cross product of two adjacent sides.

Hint 3

Adjacent sides at A are $A B$ and $A D$ .

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.

By triangle law, $A D + D B = A B \Rightarrow A D = A B - D B = (2 \hat{i} - 4 \hat{j} + 5 \hat{k}) - (3 \hat{i} - 6 \hat{j} + 2 \hat{k}) = - \hat{i} + 2 \hat{j} + 3 \hat{k}$
½ mark
$A D \times A B = \hat{i} - 1 2 \hat{j} 2 - 4 \hat{k} 35 = \hat{i} (10 + 12) - \hat{j} (- 5 - 6) + \hat{k} (4 - 4) = 22 \hat{i} + 11 \hat{j}$
1 mark
Area of parallelogram $= ∣ A D \times A B ∣ = 2 2^{2} + 1 1^{2} = 605 = 115$ sq units
½ mark

Answer: $A D = - \hat{i} + 2 \hat{j} + 3 \hat{k}$ ; Area $= 115$ (i.e. $605$ ) sq units

Where the marks are

Finds $A D = - \hat{i} + 2 \hat{j} + 3 \hat{k}$ (0.5 mark)
Computes cross product $22 \hat{i} + 11 \hat{j}$ (1 mark)
Area $= 115$ sq units (0.5 mark)

Revise the chapter

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 paper2024 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 25

Hints

Step-by-step solution

Revise the chapter

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