MathsCBSE Class 10 · Basic · Board exam 2023, set 430/5/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2023: question 35(a)

Set in the CBSE Class 10 board exam of 2023, paper 430/5/1, section D, question 35(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35(a) · 5 marks · Long answerQuadratic Equations

The difference of two numbers is 5 and the difference of their reciprocals is $\frac{1}{10}$ . Find the numbers.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/5/1.

Hints

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

Hint 1

Call the numbers x and x + 5; which reciprocal is bigger?

Hint 2

Take the LCM of the two reciprocals to get a quadratic equation.

Hint 3

Solve the quadratic by splitting the middle term and find both pairs.

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 smaller number be $x$ ; then the larger is $x + 5$ .
½ mark
Since $\frac{1}{x + 5} < \frac{1}{x}$ (for positive numbers): $\frac{1}{x} - \frac{1}{x + 5} = \frac{1}{10}$
1 mark
$\frac{x + 5 - x}{x ( x + 5 )} = \frac{1}{10} \Rightarrow 50 = x^{2} + 5 x \Rightarrow x^{2} + 5 x - 50 = 0$
1 mark
Factorising: $x^{2} + 10 x - 5 x - 50 = 0 \Rightarrow (x + 10) (x - 5) = 0$
1 mark
$x = - 10$ or $x = 5$
½ mark
If $x = 5$ , the numbers are 5 and 10; if $x = - 10$ , the numbers are −10 and −5. (Check: $\frac{1}{5} - \frac{1}{10} = \frac{1}{10}$ ; $- \frac{1}{10} + \frac{1}{5} = \frac{1}{10}$ .)
1 mark

Answer: The numbers are 5 and 10, or −10 and −5

Where the marks are

Takes numbers as x and x + 5 (½ mark)
Forms the reciprocal equation $\frac{1}{x} - \frac{1}{x + 5} = \frac{1}{10}$ (1 mark)
Reduces to $x^{2} + 5 x - 50 = 0$ (1 mark)
Factorises as $(x + 10) (x - 5) = 0$ (1 mark)
$x = - 10, 5$ (½ mark)
States both pairs of numbers (1 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 paper2023 board paper, set 430/5/1 (Basic): 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 Basic 2023: question 35(a)

Hints

Step-by-step solution

Revise the chapter

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