MathsCBSE Class 10 · Basic · Sample paper, session 2025-26 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic sample paper 2025-26: question 33(A)

From the official CBSE sample question paper for 2025-26, section D, question 33(A), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33(A) · 5 marks · Long answerQuadratic Equations

The numerator of a fraction is 3 less than its denominator. If 2 is added to both of its numerator and denominator then the sum of the new fraction and original fraction is $\frac{29}{20}$ . Find the original fraction.

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

Let the denominator be $x$ ; write the numerator in terms of $x$ .

Hint 2

Add 2 to both parts to get the new fraction, add the two fractions and equate to $\frac{29}{20}$ .

Hint 3

You will get a quadratic; factorise by splitting $- 98 x$ as $- 110 x + 12 x$ .

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.

Let the denominator be $x$ . Then numerator $= x - 3$ and the fraction is $\frac{x - 3}{x}$ .
1 mark
New fraction $= \frac{x - 1}{x + 2}$ . Given: $\frac{x - 1}{x + 2} + \frac{x - 3}{x} = \frac{29}{20}$
1 mark
$\frac{x ( x - 1 ) + ( x - 3 ) ( x + 2 )}{x ( x + 2 )} = \frac{29}{20} \Rightarrow 20 (2 x^{2} - 2 x - 6) = 29 (x^{2} + 2 x) \Rightarrow 11 x^{2} - 98 x - 120 = 0$
1½ marks
$11 x^{2} - 110 x + 12 x - 120 = 0 \Rightarrow (x - 10) (11 x + 12) = 0 \Rightarrow x = 10$ or $x = - \frac{12}{11}$
1 mark
$x = - \frac{12}{11}$ is rejected (denominator must be a natural number), so $x = 10$ and the fraction is $\frac{7}{10}$ .
½ mark

Answer: $\frac{7}{10}$

Where the marks are

Assumes fraction as $\frac{x - 3}{x}$ (1 mark)
Forms the equation $\frac{x - 1}{x + 2} + \frac{x - 3}{x} = \frac{29}{20}$ (1 mark)
Simplifies to $11 x^{2} - 98 x - 120 = 0$ (1½ marks)
Solves to get $x = 10$ or $- \frac{12}{11}$ (1 mark)
Rejects negative value and states fraction $\frac{7}{10}$ (½ 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 paperSample paper 2025-26 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic sample paper 2025-26: question 33(A)

Hints

Step-by-step solution

Revise the chapter

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