Q) An arch of a railway bridge, built on Chenab riverbed, is shown in the above diagram. It is a parabolic arch connecting two hills at P and Q. If the parabolic curve is represented by the polynomial p(x)= – 0.0025 x ² – 0.025 x + 136

Observe the diagram and based on above information, answer the following questions:
(i) Write the co-ordinates of point A.
(ii) Find the span of the arch.
(iii) Write the zeroes of the polynomial using diagram and verify the relationship between sum of zeroes and polynomials.
(iv) Find the values of p(x) at x = 100 and x = – 100. Are they same?

(Q 37 – 30/5/2 – CBSE 2026 Question Paper)

Ans:

(i) Co-ordinates of point A:

Point A is lies on y-axis

∴ its abscissa value, x = 0

The value of y is given by polynomial p(x)= – 0.0025 x ² – 0.025 x + 136

∴ at x = 0, value of y = p(0) = – 0.0025 (0) ² – 0.025 (0) + 136 = 136

Therefore, Co-ordinates of point A are (0, 136)

(ii) Span of the arch:

The span is calculated as the absolute difference between these x-coordinates:

∴ Span = |(-238.5) – (228.5)|

= |- 467| = 467

Therefore, Span of the arch is 467 m

(iii) Zeroes of the polynomial & relationship verification:

Zeroes of the polynomial:

The zeroes of a polynomial are the x-intercepts.

From diagram, zeroes are:

α = 228.5

and

β = – 238.5

Therefore, the zeroes of polynomial are 228.5 and – 238.5

Verification of the relationship:

Let’s compare polynomial – 0.0025 x ² – 0.025 x + 136 = 0

with standard quadratic equation a x ² + b x + c = 0, we get

a = -0.0025, b = – 0.025 and c = 136

Also, We calculated zeroes of this polynomial as: α = 228.5 and β = – 238.5

Let’s start verifying the relationships one by one:

First, we take relationship 1 for sum of zeroes: α + β =

In LHS: α + β

= (228.5) + (- 238.5) = -10

and RHS:

=  = = 10

Since LHS = RHS, the relation of sum of the zeros (α + β = ) is verified.

Therefore, 228.5 and – 238.5 are the zeroes of the polynomial. The relationship between the zeroes and coefficients is verified as α + β = ).

(iv) Values of p(x):

Given polynomial is p(x) = – 0.0025 x ² – 0.025 x + 136

for x = 100: p (100) = – 0.0025 (100) ² – 0.025 (100) + 136

= -25 – 2.5 + 136 = 108.5

and for x = – 100: p (- 100) = – 0.0025 (- 100) ² – 0.025 (- 100) + 136

= -25 + 2.5 + 136 = 113.5

Therefore, polynomial’s values for x = 100 and x = -100 are not the same (108.5 ≠ 113.5).
Do you know why it happens? This is because the polynomial contains a linear term (-0.025x). It makes the parabola slightly asymmetrical relative to the y-axis.

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