MathsCBSE Class 10 · Basic · Board exam 2025, set 430/3/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2025: question 28(b)

Set in the CBSE Class 10 board exam of 2025, paper 430/3/1, section C, question 28(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(b) · 3 marks · Short answerPair of Linear Equations in Two Variables

Find the value of k for which the following pair of linear equations will have infinitely many solutions:

$k x + 3 y - (k - 3) = 0$ and $12 x + k y - k = 0$

Hence, find any two solutions of the given pair of equations.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/3/1.

Hints

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

Hint 1

Infinitely many solutions means the two lines coincide. Write the condition on the ratios of coefficients.

Hint 2

Find k from each pair of ratios and keep only the value that satisfies both; then pick any x and find y.

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.

For infinitely many solutions, $\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}$ :
$\frac{k}{12} = \frac{3}{k} = \frac{- ( k - 3 )}{- k} = \frac{k - 3}{k}$
1 mark
From $\frac{k}{12} = \frac{3}{k}$ : $k^{2} = 36$ , so $k = \pm 6$ .
From $\frac{3}{k} = \frac{k - 3}{k}$ : $3 k = k^{2} - 3 k$ , i.e. $k^{2} - 6 k = 0$ , so $k = 0$ or $6$ .
The common value is $k = 6$ .
1 mark
For $k = 6$ the equations are $6 x + 3 y - 3 = 0$ and $12 x + 6 y - 6 = 0$ . Both reduce to $2 x + y = 1$ , i.e. $y = 1 - 2 x$ .
Solution 1: $x = 0 \Rightarrow y = 1$ , giving $(0, 1)$ .
½ mark
Solution 2: $x = 1 \Rightarrow y = - 1$ , giving $(1, - 1)$ .
½ mark

Answer: k = 6; two solutions: (0, 1) and (1, −1)

Where the marks are

Writes the condition a₁/a₂ = b₁/b₂ = c₁/c₂ correctly (1 mark)
Solves to get k = 6 (1 mark)
Gives any two correct solutions, e.g. (0, 1) and (1, −1) (½ + ½ marks)

Revise the chapter

Chapter: Pair of Linear Equations in Two Variables (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Pair of Linear Equations in Two Variables, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 430/3/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2025: question 28(b)

Hints

Step-by-step solution

Revise the chapter

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