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

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

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

Q24(b) · 2 marks · Very short answerCompetency-basedThree Dimensional Geometry

A person standing at $O (0, 0, 0)$ is watching an aeroplane which is at the coordinate point $A (4, 0, 3)$ . At the same time he saw a bird at the coordinate point $B (0, 0, 1)$ . Find the angles which $B A$ makes with the x,y and z axes.

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

$B A$ = position vector of $A$ minus position vector of $B$ .

Hint 2

The components of the unit vector along $B A$ are the cosines of the required angles.

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.

$B A = O A - O B = (4 \hat{i} + 3 \hat{k}) - \hat{k} = 4 \hat{i} + 0 \hat{j} + 2 \hat{k}$ .
½ mark
$∣ B A ∣ = 16 + 4 = 25$ , so the unit vector is $\frac{2}{5} \hat{i} + 0 \hat{j} + \frac{1}{5} \hat{k}$ ; direction cosines are $\frac{2}{5}, 0, \frac{1}{5}$ .
½ mark
Angles with the $x$ , $y$ , $z$ axes: $cos^{- 1} \frac{2}{5}$ , $\frac{π}{2}$ , $cos^{- 1} \frac{1}{5}$ .
1 mark

Answer: $cos^{- 1} \frac{2}{5}, \frac{π}{2}, cos^{- 1} \frac{1}{5}$

Where the marks are

Finds $B A = 4 \hat{i} + 2 \hat{k}$ (½ mark)
Finds unit vector / direction cosines (½ mark)
States the three angles (1 mark)

Revise the chapter

Chapter: Three Dimensional Geometry (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Three Dimensional Geometry, 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(b)

Hints

Step-by-step solution

Revise the chapter

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