Rainbow is an arch of colours that is visible in the sky after rain or when water droplets are present in the atmosphere. The colours of the rainbow are generally, red, orange, yellow, green, blue, indigo and violet. Each colour of the rainbow makes a parabola. We know that any quadratic polynomial p(x) = ax² + bx + c (a ≠ 0) represents a parabola on the graph paper. Based on the above, answer the following questions: (i) The graph of a rainbow y = f(x) is shown in the figure. Write the number of zeroes of the curve. (ii) If the graph of a rainbow does not intersect the x-axis but intersects y-axis at one point, then how many zeroes will it have? (iii)(a) If a rainbow is represented by the quadratic polynomial p(x) = x² + (a + 1)x + b, whose zeroes are 2 and -3, find the value of a and b. (iii)(b) The polynomial x² - 2x - (7p + 3) represents a rainbow. If -4 is a zero of it, find the value of p.

MathsCBSE Class 10 · Basic · Board exam 2023, set 430/5/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2023: question 36

Set in the CBSE Class 10 board exam of 2023, paper 430/5/1, section E, question 36, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36 · 4 marks · Case-basedCompetency-basedPolynomials

Rainbow is an arch of colours that is visible in the sky after rain or when water droplets are present in the atmosphere. The colours of the rainbow are generally, red, orange, yellow, green, blue, indigo and violet. Each colour of the rainbow makes a parabola. We know that any quadratic polynomial $p (x) = a x^{2} + b x + c$ ( $a \neq = 0$ ) represents a parabola on the graph paper.

Based on the above, answer the following questions:

(i)
The graph of a rainbow $y = f (x)$ is shown in the figure. Write the number of zeroes of the curve.
1 mark
(ii)
If the graph of a rainbow does not intersect the x-axis but intersects y-axis at one point, then how many zeroes will it have?
1 mark
(iii)(a)
If a rainbow is represented by the quadratic polynomial $p (x) = x^{2} + (a + 1) x + b$ , whose zeroes are 2 and -3, find the value of a and b.
2 marks
OR
(iii)(b)
The polynomial $x^{2} - 2 x - (7 p + 3)$ represents a rainbow. If -4 is a zero of it, find the value of p.
2 marks

Figure in the paper: Graph of a downward-opening parabola y = f(x) with vertex on the positive y-axis. The curve crosses the x-axis at two points, one on each side of the origin O, and the arms extend downward beyond the x-axis. The x-axis and y-axis are labelled. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/5/1.

Hints

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

Hint 1

The number of zeroes equals the number of points where the graph meets the x-axis.

Hint 2

For (iii), a zero makes the polynomial equal to 0 — substitute each given zero into p(x).

Hint 3

In (iii)(a) you get two linear equations in a and b; subtract them to eliminate b.

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.

(i) The parabola cuts the x-axis at two points (one on each side of O), so the curve has 2 zeroes.
1 mark
(ii) Zeroes are the x-coordinates where the graph meets the x-axis. If the graph does not meet the x-axis, it has no zero (0 zeroes).
1 mark
(iii)(a) Since 2 is a zero: $p (2) = 4 + 2 (a + 1) + b = 0 \Rightarrow 2 a + b = - 6$ …(1). Since $- 3$ is a zero: $p (- 3) = 9 - 3 (a + 1) + b = 0 \Rightarrow - 3 a + b = - 6$ …(2).
1 mark
(iii)(a) Subtracting (2) from (1): $5 a = 0 \Rightarrow a = 0$ . Then from (1), $b = - 6$ . (Check: sum of zeroes $= - (a + 1) = - 1 = 2 + (- 3)$ , product $= b = - 6 = 2 \times (- 3)$ .)
1 mark
(iii)(b) Since $- 4$ is a zero: $(- 4)^{2} - 2 (- 4) - (7 p + 3) = 0 \Rightarrow 16 + 8 - 7 p - 3 = 0 \Rightarrow 21 - 7 p = 0$ .
1 mark
(iii)(b) $7 p = 21 \Rightarrow p = 3$ .
1 mark

Answer: (i) 2 zeroes; (ii) 0 (no zero); (iii)(a) a = 0, b = −6; (iii)(b) p = 3

Where the marks are

(i) States 2 zeroes (1 mark)
(ii) States no zero (1 mark)
(iii)(a) Forms $2 a + b = - 6$ and $- 3 a + b = - 6$ (½ + ½ marks)
(iii)(a) Solves to get $a = 0$ and $b = - 6$ (½ + ½ marks)
(iii)(b) Substitutes $x = - 4$ to get $21 - 7 p = 0$ (1 mark)
(iii)(b) Obtains $p = 3$ (1 mark)

Revise the chapter

Chapter: Polynomials (notes, formulas, where marks are lost and every past question)
Topic: Forming a Quadratic Polynomial from Its Zeros

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 430/5/1 (Basic): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Basic 2023: question 36

Hints

Step-by-step solution

Revise the chapter

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