CBSE2026Class Class 10 · Mathematics4 Marks · Long✅ Verified
CASE STUDY : During a theatre drama, a backdrop of building arches was used. The shape of the curve shown can be represented by the polynomial $p(x) = -x^2 + 2x + 8$, where $x$ is the length (in feet) on stage level. In the graph, the curve meets the $x$-axis at $B$ and $A$, and $C(1, y)$ is the highest point of the arch.
Based on the figure given above, answer the following questions :
(i) Determine the height of the arch. [1]
(ii) (a) Find zeroes of the polynomial $p(x)$. Which points on the graph represent the zeroes ? [2]
OR
(ii) (b) Find the span of the arch on the stage floor. [2]
(iii) Write the coordinates of the point of intersection of the above curve with the $y$-axis. [1]
✅ Answer & Solution
(i) Height of the arch
Step 1: The highest point is $C(1, y)$, so put $x = 1$ in $p(x)$.
$$p(1) = -(1)^2 + 2(1) + 8 = -1 + 2 + 8 = 9$$
Hence the height of the arch $= 9$ feet.
(ii) (a) Zeroes of $p(x)$
Step 2: Put $p(x) = 0$.
$$-x^2 + 2x + 8 = 0 \Rightarrow x^2 - 2x - 8 = 0$$
Step 3: Split the middle term ($-4 \times 2 = -8$, $-4 + 2 = -2$) :
$$x^2 - 4x + 2x - 8 = 0 \Rightarrow x(x - 4) + 2(x - 4) = 0$$
$$(x - 4)(x + 2) = 0$$
Step 4: $$x = 4 \quad \text{or} \quad x = -2$$
Step 5: The zeroes are represented by the points where the curve cuts the $x$-axis, i.e. the points $A(4, 0)$ and $B(-2, 0)$.
OR
(ii) (b) Span of the arch
Step 6: The span is the distance between the two points where the arch meets the stage floor (the $x$-axis), i.e. between $B(-2, 0)$ and $A(4, 0)$.
$$\text{Span} = 4 - (-2) = 6\ \text{feet}$$
(iii) Intersection with the $y$-axis
Step 7: On the $y$-axis, $x = 0$.
$$p(0) = -(0)^2 + 2(0) + 8 = 8$$
Hence the point of intersection is $(0, 8)$.
