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Date May 2019 Marks available 6 Reference code 19M.1.AHL.TZ1.H_5
Level Additional Higher Level Paper Paper 1 (without calculator) Time zone Time zone 1
Command term Find Question number H_5 Adapted from N/A

Question

A camera at point C is 3 m from the edge of a straight section of road as shown in the following diagram. The camera detects a car travelling along the road at t = 0. It then rotates, always pointing at the car, until the car passes O, the point on the edge of the road closest to the camera.

A car travels along the road at a speed of 24 ms−1. Let the position of the car be X and let OĈX = θ.

Find dθdt, the rate of rotation of the camera, in radians per second, at the instant the car passes the point O .

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

let OX = x

METHOD 1

dxdt=24   (or −24)       (A1)

dθdt=dxdt×dθdx       (M1)

3tanθ=x       A1

EITHER

3sec2θ=dxdθ       A1

dθdt=243sec2θ

attempt to substitute for θ=0 into their differential equation       M1

OR

θ=arctan(x3)

dθdx=13×11+x29       A1

dθdt=24×13(1+x29)

attempt to substitute for x=0 into their differential equation       M1

THEN

dθdt=243=8  (rad s−1)       A1

Note: Accept −8 rad s−1.

 

METHOD 2

dxdt=24   (or −24)       (A1)

3tanθ=x       A1

attempt to differentiate implicitly with respect to t       M1

3sec2θ×dθdt=dxdt      A1

dθdt=243sec2θ

attempt to substitute for θ=0 into their differential equation       M1

dθdt=243=8 (rad s−1)       A1

Note: Accept −8 rad s−1.

Note: Can be done by consideration of CX, use of Pythagoras.

 

METHOD 3

let the position of the car be at time t be d24t from O       (A1)

tanθ=d24t3(=d38t)       M1

Note: For tanθ=24t3 award A0M1 and follow through.

EITHER

attempt to differentiate implicitly with respect to t       M1

sec2θdθdt=8       A1

attempt to substitute for θ=0 into their differential equation       M1

OR

θ=arctan(d38t)       M1

dθdt=81+(d38t)2       A1

at O, t=d24       A1

THEN

dθdt=8       A1

 

[6 marks]

Examiners report

[N/A]

Syllabus sections

Topic 5 —Calculus » AHL 5.14—Implicit functions, related rates, optimisation
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