Maths · Algebra · chapter 2 of 14
Class 10 Maths Chapter 2: Polynomials case study question with answers
One case-based question of the kind the board asks in Section E: a passage and a figure, then 3 parts worth 4 marks in all, with the answer and the marking scheme for each. Written from the NCERT chapter.
Case study · 4 marks
Students of a model-making club in Pune drew the outline of an arched gateway on graph paper, taking the ground line as the x-axis and 1 unit as 1 metre. The arch is part of the graph of y = p(x), where p(x) = −x2 + 4x + 5. They extended the curve a little below the ground line, as shown, to see where it crosses the x-axis. The highest point of the arch is at (2, 9).
Part 1 · 1 mark
The number of zeroes of p(x) is
Show answer
Answer: (a) 2
The curve meets the x-axis at two points, x = −1 and x = 5, so p(x) has 2 zeroes.
Part 2 · 1 mark
Write the zeroes of p(x).
Answer and marking scheme
The x-coordinates of the points where the curve meets the x-axis: −1 and 5.
- −1 and 5 (1)
Or
Find p(0) and say what it shows on the graph.
Answer and marking scheme
p(0) = −0 + 0 + 5 = 5: the curve meets the y-axis at (0, 5).
- p(0) = 5, the y-intercept (0, 5) (1)
Part 3 · 2 marks
Verify the relationship between the zeroes and the coefficients of p(x), and state why the arch opens downwards.
Answer and marking scheme
a = −1, b = 4, c = 5. Sum −1 + 5 = 4 = −4−1 = −ba. Product (−1)(5) = −5 = 5−1 = ca. The arch opens downwards because a = −1 < 0.
- sum 4 = −\frac{b}{a} and product −5 = \frac{c}{a} verified (1)
- a < 0, so it opens downwards (1)
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