MathsCBSE Class 12 · Board exam 2023, set 65/5/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2023: question 25

Set in the CBSE Class 12 board exam of 2023, paper 65/5/1, 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 answerThree Dimensional Geometry

Find the vector and the cartesian equations of a line that passes through the point A(1, 2, −1) and parallel to the line $5 x - 25 = 14 - 7 y = 35 z$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/5/1.

Hints

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

Hint 1

Make the coefficients of $x$ , $y$ and $z$ equal to 1 in each part of the given line.

Hint 2

Divide all parts by 35 to read off direction ratios, then use the point 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.

Rewrite the given line: $5 (x - 5) = - 7 (y - 2) = 35 z$ .
Dividing by 35: $\frac{x - 5}{7} = \frac{y - 2}{- 5} = \frac{z}{1}$
So its direction ratios are $7, - 5, 1$ .
1 mark
The required line passes through $(1, 2, - 1)$ and is parallel to $7 \hat{i} - 5 \hat{j} + \hat{k}$ .
Vector equation: $r = (\hat{i} + 2 \hat{j} - \hat{k}) + λ (7 \hat{i} - 5 \hat{j} + \hat{k})$
½ mark
Cartesian equation: $\frac{x - 1}{7} = \frac{y - 2}{- 5} = \frac{z + 1}{1}$
½ mark

Answer: $r = (\hat{i} + 2 \hat{j} - \hat{k}) + λ (7 \hat{i} - 5 \hat{j} + \hat{k})$ ; $\frac{x - 1}{7} = \frac{y - 2}{- 5} = \frac{z + 1}{1}$

Where the marks are

Writes the given line in standard form with direction ratios 7, −5, 1 (1 mark)
Correct vector equation (½ mark)
Correct cartesian equation (½ mark)

Revise the chapter

Chapter: Three Dimensional Geometry (notes, formulas, where marks are lost and every past question)
Topic: Equation of a Line in Space

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 paper2023 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2023: question 25

Hints

Step-by-step solution

Revise the chapter

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