MathsCBSE Class 10 · Standard · Board exam 2025, set 30/4/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2025: question 32(B)

Set in the CBSE Class 10 board exam of 2025, paper 30/4/1, section D, question 32(B), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(B) · 5 marks · Long answerQuadratic Equations

Express the equation $\frac{x - 2}{x - 3} + \frac{x - 4}{x - 5} = \frac{10}{3}$ ; $(x \neq = 3, 5)$ as a quadratic equation in standard form. Hence, find the roots of the equation so formed.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/4/1.

Hints

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

Hint 1

Take the LCM $(x - 3) (x - 5)$ on the left side.

Hint 2

Cross-multiply and bring all terms to one side; then split $- 19 x$ using factors of $2 \times 42 = 84$ .

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.

$\frac{( x - 2 ) ( x - 5 ) + ( x - 4 ) ( x - 3 )}{( x - 3 ) ( x - 5 )} = \frac{10}{3}$
$\Rightarrow \frac{( x ^{2} - 7 x + 10 ) + ( x ^{2} - 7 x + 12 )}{x ^{2} - 8 x + 15} = \frac{10}{3}$
1½ marks
$3 (2 x^{2} - 14 x + 22) = 10 (x^{2} - 8 x + 15)$
$6 x^{2} - 42 x + 66 = 10 x^{2} - 80 x + 150 \Rightarrow 4 x^{2} - 38 x + 84 = 0$
$\Rightarrow 2 x^{2} - 19 x + 42 = 0$ (standard form)
1½ marks
Splitting the middle term: $2 x^{2} - 12 x - 7 x + 42 = 0 \Rightarrow 2 x (x - 6) - 7 (x - 6) = 0 \Rightarrow (x - 6) (2 x - 7) = 0$
1 mark
$x = 6$ or $x = \frac{7}{2}$ (both allowed since $x \neq = 3, 5$ )
1 mark

Answer: $2 x^{2} - 19 x + 42 = 0$ ; roots $x = 6, \frac{7}{2}$

Where the marks are

Combines LHS into a single fraction (1½ marks)
Simplifies to $2 x^{2} - 19 x + 42 = 0$ (1½ marks)
Factorises as $(x - 6) (2 x - 7) = 0$ (1 mark)
Roots $6$ and $\frac{7}{2}$ (1 mark)

Revise the chapter

Chapter: Quadratic Equations (notes, formulas, where marks are lost and every past question)
Topic: Standard Form of a Quadratic Equation

Practise this chapterBoard-style questions from Quadratic Equations, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 30/4/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2025: question 32(B)

Hints

Step-by-step solution

Revise the chapter

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