MathsCBSE Class 10 · Standard · Sample paper, session 2025-26 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard sample paper 2025-26: question 31(B)

From the official CBSE sample question paper for 2025-26, section C, question 31(B), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31(B) · 3 marks · Short answerPair of Linear Equations in Two Variables

Solve the following system of equations graphically:
$2 x + y = 6$ , $2 x - y - 2 = 0$ . Find the area of the triangle so formed by two lines and $x$ -axis.

For Visually Impaired candidates: Five years hence, fathers age will be three times the age of son. Five years ago, father was seven times as old as his son. Find their present ages.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

For each line, find at least two points, such as where it meets the axes.

Hint 2

The point where the lines cross is the solution.

Hint 3

The triangle's base lies on the x-axis between the two x-intercepts, and its height is the y-coordinate of the meeting point.

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.

Table for $2 x + y = 6$ (i.e. $y = 6 - 2 x$ ): points $(0, 6)$ , $(1, 4)$ , $(3, 0)$ .
Table for $2 x - y - 2 = 0$ (i.e. $y = 2 x - 2$ ): points $(0, - 2)$ , $(1, 0)$ , $(2, 2)$ .
1 mark
Plot these points and draw both lines on graph paper. The first line cuts the x-axis at C(3, 0), the second at B(1, 0), and the two lines meet at A(2, 2).
1 mark
From the graph, the lines intersect at $(2, 2)$ , so the solution is $x = 2$ , $y = 2$ .
½ mark
The triangle formed with the x-axis has vertices A(2, 2), B(1, 0), C(3, 0). Base $B C = 2$ units and height = 2 units (the y-coordinate of A).
Area $= \frac{1}{2} \times 2 \times 2 = 2$ sq. units.
½ mark
For Visually Impaired candidates (alternative): Let the present ages of the father and son be $x$ and $y$ years. $x + 5 = 3 (y + 5)$ gives $x - 3 y = 10$ . (1 mark in VI version)
(VI) $x - 5 = 7 (y - 5)$ gives $x - 7 y = - 30$ . (1 mark)
(VI) Subtracting: $4 y = 40$ , so $y = 10$ and $x = 40$ . The father is 40 years old and the son is 10 years old. (1 mark)

Answer: x = 2, y = 2; area = 2 sq. units (VI version: father 40 years, son 10 years)

Where the marks are

Draws both lines correctly from tables of values (2 marks)
Reads the solution x = 2, y = 2 (½ mark)
Finds the area of the triangle with the x-axis as 2 sq. units (½ mark)
VI: forms $x - 3 y = 10$ (1), forms $x - 7 y = - 30$ (1), solves to 40 and 10 years (1)

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 paperSample paper 2025-26 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard sample paper 2025-26: question 31(B)

Hints

Step-by-step solution

Revise the chapter

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