Find the value of 'c' for which the quadratic equation (c + 1)x² – 6(c + 1)x + 3(c + 9) = 0; c ≠ –1 has real and equal roots.

MathsCBSE Class 10 · Standard · Board exam 2024, set 30/4/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2024: question 34(b)

Set in the CBSE Class 10 board exam of 2024, paper 30/4/1, section D, question 34(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34(b) · 5 marks · Long answerQuadratic Equations

Find the value of 'c' for which the quadratic equation $(c + 1) x^{2} -6 (c + 1) x + 3 (c + 9) = 0$ ; $c \neq = -1$ has real and equal roots.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/4/1.

Hints

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

Hint 1

Real and equal roots means the discriminant $b^{2} - 4 a c$ is zero.

Hint 2

Take $12 (c + 1)$ common instead of expanding everything.

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.

Here $a = c + 1$ , $b = - 6 (c + 1)$ , $c_{0} = 3 (c + 9)$ . For real and equal roots, $D = b^{2} - 4 a c_{0} = 0$ : ${- 6 (c + 1)}^{2} - 4 (c + 1) \times 3 (c + 9) = 0$
2 marks
$36 (c + 1)^{2} - 12 (c + 1) (c + 9) = 0 \Rightarrow 12 (c + 1) [3 (c + 1) - (c + 9)] = 0 \Rightarrow 12 (c + 1) (2 c - 6) = 0$
2 marks
Since $c \neq = - 1$ , $c + 1 \neq = 0$ , so $2 c - 6 = 0 \Rightarrow c = 3$ .
1 mark

Answer: c = 3

Where the marks are

Sets discriminant {−6(c+1)}² − 4(c+1)·3(c+9) = 0 (2 marks)
Simplifies to 12(c + 1)(2c − 6) = 0 (2 marks)
Uses c ≠ −1 to get c = 3 (1 mark)

Revise the chapter

Chapter: Quadratic Equations (notes, formulas, where marks are lost and every past question)
Topic: Nature of Roots and Discriminant

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 30/4/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2024: question 34(b)

Hints

Step-by-step solution

Revise the chapter

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