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

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

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

Set in the CBSE Class 10 board exam of 2024, paper 30/5/1, section D, question 32(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(a) · 5 marks · Long answerQuadratic Equations

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

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/5/1.

Hints

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

Hint 1

Real and equal roots means the discriminant is zero.

Hint 2

Identify a, b, c carefully: $b = - 6 (k + 1)$ , so $b^{2} = 36 (k + 1)^{2}$ .

Hint 3

You will get a quadratic in k; remember the condition $k \neq = - 1$ .

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 $a = k + 1$ , $b = - 6 (k + 1)$ , $c = 3 (k + 9)$ . For real and equal roots, $D = b^{2} - 4 a c = 0$ :
$36 (k + 1)^{2} - 4 (k + 1) \cdot 3 (k + 9) = 0$
2 marks
$36 k^{2} + 72 k + 36 - 12 k^{2} - 120 k - 108 = 0 \Rightarrow 24 k^{2} - 48 k - 72 = 0 \Rightarrow k^{2} - 2 k - 3 = 0$
1½ marks
$(k - 3) (k + 1) = 0 \Rightarrow k = 3$ or $k = - 1$
1 mark
Since $k \neq = - 1$ , $k = 3$ .
½ mark

Answer: k = 3

Where the marks are

Uses D = 0 and substitutes correctly: $36 (k + 1)^{2} - 12 (k + 1) (k + 9) = 0$ (2 marks)
Simplifies to $k^{2} - 2 k - 3 = 0$ (1½ marks)
Factorises $(k - 3) (k + 1) = 0$ (1 mark)
Rejects $k = - 1$ and gives $k = 3$ (½ 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 30/5/1 (Standard): every question, by section.

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

Hints

Step-by-step solution

Revise the chapter

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