If one root of the quadratic equation 3x² - 8x - (2k + 1) = 0 is seven times the other, then find the value of k.

MathsCBSE Class 10 · Basic · Board exam 2023, set 430/6/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Basic 2023: question 23(b)

Set in the CBSE Class 10 board exam of 2023, paper 430/6/1, section B, question 23(b), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23(b) · 2 marks · Very short answerQuadratic Equations

If one root of the quadratic equation $3 x^{2} - 8 x - (2 k + 1) = 0$ is seven times the other, then find the value of k.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/6/1.

Hints

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

Hint 1

Call the smaller root α; what is the other root?

Hint 2

Use sum of roots = −b/a to find α, then product of roots = c/a to find k.

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 roots be $α$ and $7 α$ .
½ mark
Sum of roots: $α + 7 α = - \frac{- 8}{3} = \frac{8}{3} \Rightarrow 8 α = \frac{8}{3} \Rightarrow α = \frac{1}{3}$
½ mark
Product of roots: $α \cdot 7 α = \frac{- ( 2 k + 1 )}{3} \Rightarrow 7 α^{2} = - \frac{2 k + 1}{3}$
½ mark
$7 \cdot \frac{1}{9} = - \frac{2 k + 1}{3} \Rightarrow 2 k + 1 = - \frac{7}{3} \Rightarrow 2 k = - \frac{10}{3} \Rightarrow k = - \frac{5}{3}$
½ mark

Answer: $k = - \frac{5}{3}$

Where the marks are

Takes roots as α and 7α (½ mark)
Uses sum of roots to get α = 1/3 (½ mark)
Uses product of roots: $7 α^{2} = - (2 k + 1) /3$ (½ mark)
k = −5/3 (½ mark)

Revise the chapter

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

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 430/6/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2023: question 23(b)

Hints

Step-by-step solution

Revise the chapter

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