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3 Unit A2 1: Pure Mathematics (A-level only)

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Question 611

(i) The cost function CCC, in thousands of pounds, for a specialized manufacturing process is modeled by

C(x)=(2x+5)23x−1,x≠13 C(x) = \frac{(2x + 5)^2}{3x - 1}, \quad x \neq \frac{1}{3} C(x)=3x−1(2x+5)2​,x=31​

where x x\,x represents the amount of catalyst used in kilograms.

a.

Find C′(x)C'(x)C′(x) in the form P(x)Q(x)\displaystyle \frac{P(x)}{Q(x)}Q(x)P(x)​ where P(x)P(x)P(x) and Q(x)Q(x)Q(x) are fully factorised quadratic expressions.

[5]
b.

Determine the range of values of x x\,x for which the cost function C(x)C(x)C(x) is increasing.

[2]
ii.

The vertical displacement h h\,h of a floating buoy, in metres, is modeled by the function

h(t)=tsin⁡6t,0≤t<π6 h(t) = t\sqrt{\sin 6t}, \quad 0 \le t < \frac{\pi}{6} h(t)=tsin6t​,0≤t<6π​

where t t\,t is the time in seconds after a wave passes. The buoy reaches its maximum height at a point MMM.

Show that the ttt-coordinate of M M\,M satisfies the equation tan⁡6t+kt=0\tan 6t + kt = 0tan6t+kt=0, where k k\,k is a constant to be found.

[5]
Markscheme

3 Unit A2 1: Pure Mathematics (A-level only) Questions

  1. A Level
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