If vectors a=2i+2j+3k, b=-i+2j+k and c=3i+j are such that b+ is perpendicular to a, then find the value of λ.

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

CBSE Class 12 Mathematics sample paper 2024-25: question 24(a)

From the official CBSE sample question paper for 2024-25, section B, question 24(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q24(a) · 2 marks · Very short answerVector Algebra

If vectors $a = 2 \hat{i} + 2 \hat{j} + 3 \hat{k}$ , $b = - \hat{i} + 2 \hat{j} + \hat{k}$ and $c = 3 \hat{i} + \hat{j}$ are such that $b + λ c$ is perpendicular to $a$ , then find the value of $λ$ .

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

Two vectors are perpendicular when their dot product is zero.

Hint 2

First add $b$ and $λ c$ component-wise, then dot with $a$ .

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.

$b + λ c = (- 1 + 3 λ) \hat{i} + (2 + λ) \hat{j} + \hat{k}$ .
½ mark
Perpendicular to $a$ means $(b + λ c) \cdot a = 0$ : $2 (- 1 + 3 λ) + 2 (2 + λ) + 3 (1) = 0$ .
1 mark
$8 λ + 5 = 0 \Rightarrow λ = - \frac{5}{8}$ .
½ mark

Answer: $λ = - \frac{5}{8}$

Where the marks are

Writes $b + λ c$ in components (½ mark)
Sets dot product with $a$ equal to zero (1 mark)
Finds $λ = - \frac{5}{8}$ (½ 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 paperSample paper 2024-25: every question, by section.

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CBSE Class 12 Mathematics sample paper 2024-25: question 24(a)

Hints

Step-by-step solution

Revise the chapter

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