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

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Question 158

It is given that

f(θ)=5cos⁡θ+sin⁡θf(\theta) = 5\cos\theta + \sin\thetaf(θ)=5cosθ+sinθ

and that f(θ)=Rcos⁡(θ−α)f(\theta) = R\cos(\theta - \alpha)f(θ)=Rcos(θ−α), where R>0 R > 0\,R>0 and 0≤α≤π2\displaystyle 0 \leq \alpha \leq \frac{\pi}{2}0≤α≤2π​.

a.

Find the value of R R\,R and the value of α\alphaα, each to three decimal places.

[3]
b.

Hence solve, for 0≤θ≤2π0 \leq \theta \leq 2\pi0≤θ≤2π, the equation

5cos⁡θ+sin⁡θ=25\cos\theta + \sin\theta = 25cosθ+sinθ=2

Give your answers to three decimal places.

[3]
c.

Find the minimum value of

5cos⁡4x+sin⁡4x+155\cos 4x + \sin 4x + 155cos4x+sin4x+15

and the smallest positive value of x x\,x at which this minimum occurs. Give each answer to three decimal places.

[2]
Markscheme

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

  1. A Level
  2. /Maths
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