Find the zeroes of the polynomial p(x) = 2x² - 7x - 15 and verify the relationship between its coefficients and zeroes.

Zeroes: -3/2 and 5; sum = 7/2, product = -15/2 (relationships verified).

MathsCBSE Class 10 · Basic · Board exam 2023, set 430/2/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2023: question 30

Set in the CBSE Class 10 board exam of 2023, paper 430/2/1, section C, question 30, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30 · 3 marks · Short answerPolynomials

Find the zeroes of the polynomial $p (x) = 2 x^{2} - 7 x - 15$ and verify the relationship between its coefficients and zeroes.

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

Split the middle term: find two numbers with product $2 \times (- 15) = - 30$ and sum $- 7$ .

Hint 2

After finding the zeroes, compare their sum with $- b / a$ and their product with $c / a$ .

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.

$2 x^{2} - 7 x - 15 = 2 x^{2} - 10 x + 3 x - 15 = 2 x (x - 5) + 3 (x - 5) = (2 x + 3) (x - 5)$
1 mark
Setting $p (x) = 0$ : zeroes are $α = - \frac{3}{2}$ and $β = 5$ .
1 mark
Sum: $α + β = - \frac{3}{2} + 5 = \frac{7}{2}$ and $- \frac{coefficient of x}{coefficient of x ^{2}} = - \frac{( - 7 )}{2} = \frac{7}{2}$ . Verified.
½ mark
Product: $α β = - \frac{3}{2} \times 5 = - \frac{15}{2}$ and $\frac{constant term}{coefficient of x ^{2}} = \frac{- 15}{2}$ . Verified.
½ mark

Answer: Zeroes: $- \frac{3}{2}$ and 5; sum $= \frac{7}{2}$ , product $= - \frac{15}{2}$ (relationships verified).

Where the marks are

Factorises as $(2 x + 3) (x - 5)$ (1 mark)
Zeroes $- \frac{3}{2}$ and 5 (1 mark)
Verifies sum of zeroes $= - b / a$ (1/2 mark)
Verifies product of zeroes $= c / a$ (1/2 mark)

Revise the chapter

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

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 30

Hints

Step-by-step solution

Revise the chapter

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