A-LevelMathematicsIntegration9709/22 · Feb/Mar 2021 · Q79 Marks
Express 5√3 cos x + 5 sin x in the form R cos(x – α), where R > 0 and 0 < α < ½π.
- (a)Express 5√3 cos x + 5 sin x in the form R cos(x – α), where R > 0 and 0 < α < ½π.[3]
- (b)As x varies, find the least possible value of 4 + 5√3 cos x + 5 sinx, and determine the corresponding value of x where –π < x < π.[3]
- (c)Find ∫(1/(5√3 cos 3θ + 5 sin 3θ)²) dθ.[3]
Mark scheme
Marking points from the official mark scheme (Feb/Mar 2021). The mark or mark type is shown in brackets.
- [B1] State R = 10
- [M1] Use appropriate trigonometry to find α
- [A1] Obtain α = 1/6 π
- [B1 FT] State -6
- [M1] Attempt to find x from their cos(x-α)=-1
- [A1] Obtain x - 1/6 π = -π and hence x = -5/6 π
- [M1*] State integrand of form k₁ sec² (3θ - 1/6 π)
- [DM1] Integrate to obtain form k₂ tan (3θ - 1/6 π)
- [A1] Obtain 1/300 tan (3θ - 1/6 π) + c
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