MathsCBSE Class 10 · Standard · Board exam 2024, set 30/3/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2024: question 32(b)

Set in the CBSE Class 10 board exam of 2024, paper 30/3/1, 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

The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is $2 \frac{16}{21}$ , find the fraction.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/3/1.

Hints

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

Hint 1

Let the numerator be x. Write the denominator in terms of x, and convert $2 \frac{16}{21}$ to an improper fraction.

Hint 2

Add the fraction and its reciprocal, cross-multiply to get a quadratic in x, and factorise it.

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 numerator be $x$ . Then the denominator is $2 x + 1$ .
½ mark
Fraction $= \frac{x}{2 x + 1}$ , and its reciprocal is $\frac{2 x + 1}{x}$ .
½ mark
Given: $\frac{x}{2 x + 1} + \frac{2 x + 1}{x} = 2 \frac{16}{21} = \frac{58}{21}$ .
1½ marks
$21 [x^{2} + (2 x + 1)^{2}] = 58 x (2 x + 1) \Rightarrow 105 x^{2} + 84 x + 21 = 116 x^{2} + 58 x \Rightarrow 11 x^{2} - 26 x - 21 = 0$ .
½ mark
$11 x^{2} - 33 x + 7 x - 21 = 0 \Rightarrow 11 x (x - 3) + 7 (x - 3) = 0 \Rightarrow (x - 3) (11 x + 7) = 0$ .
1 mark
$x = 3$ or $x = - \frac{7}{11}$ . The numerator must be a whole number here, so $x \neq = - \frac{7}{11}$ . Hence $x = 3$ .
½ mark
Fraction $= \frac{3}{2 ( 3 ) + 1} = \frac{3}{7}$ . Check: $\frac{3}{7} + \frac{7}{3} = \frac{9 + 49}{21} = \frac{58}{21}$ ✓
½ mark

Answer: The fraction is 3/7

Where the marks are

Takes the numerator as x and the denominator as 2x + 1 (½ mark)
Writes the fraction $\frac{x}{2 x + 1}$ (½ mark)
Forms the equation $\frac{x}{2 x + 1} + \frac{2 x + 1}{x} = \frac{58}{21}$ (1½ marks)
Simplifies to $11 x^{2} - 26 x - 21 = 0$ (½ mark)
Factorises as $(x - 3) (11 x + 7) = 0$ (1 mark)
Rejects $- \frac{7}{11}$ and takes $x = 3$ (½ mark)
States the fraction $\frac{3}{7}$ (½ mark)

Revise the chapter

Chapter: Quadratic Equations (notes, formulas, where marks are lost and every past question)
Topic: Word Problems on Quadratic Equations

Practise this chapterBoard-style questions from Quadratic Equations, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 30/3/1 (Standard): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Standard 2024: question 32(b)

Hints

Step-by-step solution

Revise the chapter

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