Question

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) = –0.0025x2 – 0.025x + 136.

Observe the diagram and based on above information, answer the following questions:

(i) Write the co-ordinates of point A.   [1]

(ii) Find the span of the arch.   [1]

(iii) (a) Write the zeroes of the polynomial using diagram and verify the relationship between sum of zeroes and polynomials.   [2]

OR

(iii) (b) Find the values of p(x) at x = 100 and x = –100. Are they same?   [2]

Answer 1

(i) Point A lie on y-axis that means x-coordinate is 0.

p(x) = –0.0025x² – 0.025x + 136

Put x = 0

p(0) = 0 – 0 + 136 = 136

That means coordinates of A(0, 136).

(ii) P(228.5, 0) and Q(–238.5, 0)

Span = X_P – X_Q

= 228.5 – (–238.5)

= 228.5 + 238.5

= 467 units

(iii) (a) Zeroes of polynomial are x₁ = 228.5 and x₂ = –238.5

Sum of zeroes = 228.5 + (–238.5) = –10

Verification: Now given polynimial p(x) = –0.0025x² – 0.025x + 136

Sum of zeroes = `(-b)/a`

= `(-(-0.025))/(-0.0025)`

= `(0.025)/(-0.0025)`

= –10

Hence Verified.

OR

(iii) (b) p(x) = –0.0025x² – 0.025x + 136

Put x = 100

p(100) = –0.0025(100)² – 0.025(100) + 136

= `-25/10000 xx 10000 - 25/1000 xx 100 + 136`

= –25 – 2.5 + 136

= 108.5

Put x = –100

p(–100) = –0.0025(–100)² – 0.025(–100) + 136

= –0.0025(10000) + 2.5 + 136

= –25 + 2.5 + 136

= 113.5

No, p(100) and p(–100) are not same.