Find the value of 'k' such that the polynomial p(x) = 3x² + 2kx + x - k - 5 has the sum of zeroes equal to half of their product.

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

CBSE Class 10 Mathematics Basic 2023: question 24

Set in the CBSE Class 10 board exam of 2023, paper 430/2/1, section B, question 24, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q24 · 2 marks · Very short answerPolynomials

Find the value of 'k' such that the polynomial $p (x) = 3 x^{2} + 2 k x + x - k - 5$ has the sum of zeroes equal to half of their product.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/2/1.

Hints

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

Hint 1

First collect the x terms: the coefficient of x is $2 k + 1$ and the constant term is $- (k + 5)$ .

Hint 2

Use sum $= - b / a$ and product $= c / a$ , then set sum equal to half the product.

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.

$p (x) = 3 x^{2} + (2 k + 1) x - (k + 5)$ , so $a = 3$ , $b = 2 k + 1$ , $c = - (k + 5)$ . Sum of zeroes $= - \frac{b}{a} = - \frac{2 k + 1}{3}$
½ mark
Product of zeroes $= \frac{c}{a} = - \frac{k + 5}{3}$
½ mark
Given sum $= \frac{1}{2} \times$ product: $- \frac{2 k + 1}{3} = - \frac{1}{2} \cdot \frac{k + 5}{3}$
½ mark
$2 (2 k + 1) = k + 5 \Rightarrow 4 k + 2 = k + 5 \Rightarrow 3 k = 3 \Rightarrow k = 1$
½ mark

Answer: k = 1

Where the marks are

Sum of zeroes $- \frac{2 k + 1}{3}$ (1/2 mark)
Product of zeroes $- \frac{k + 5}{3}$ (1/2 mark)
Forms the equation sum = half of product (1/2 mark)
Solves to k = 1 (1/2 mark)

Revise the chapter

Chapter: Polynomials (notes, formulas, where marks are lost and every past question)
Topic: Relationship Between Zeros and Coefficients

Practise this chapterBoard-style questions from Polynomials, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 430/2/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2023: question 24

Hints

Step-by-step solution

Revise the chapter

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