A-LevelMathematicsTrigonometry9709/12 · Oct/Nov 2020 · Q1111 Marks

A curve has equation y = 3 cos 2x + 2 for 0 ≤ x ≤ π.

(a)State the greatest and least values of y.[2]
(b)Sketch the graph of y = 3 cos 2x + 2 for 0 ≤ x ≤ π.[2]
(c)(i)By considering the straight line y = kx, where k is a constant, state the number of solutions of the equation 3 cos 2x + 2 = kx for 0 ≤ x ≤ πin each of the following cases. k = -3[1]
(c)(ii)k = 1[1]
(c)(iii)k = 3[1]
(d)Functions f, g and h are defined for x ∈ R by f(x) = 3 cos 2x + 2, g(x) = f(2x) + 4, h(x) = 2f(x + π/4). Describe fully a sequence of transformations that maps the graph of y = f(x) on to y = g(x).[2]
(e)Describe fully a sequence of transformations that maps the graph of y = f(x) on to y = h(x).[2]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

[B1B1] Sight of each value
[*B1] Needs to be a curve, not straight lines. One complete cycle starting and finishing at their largest value.
[DB1] One complete cycle starting and finishing at y = 5 and going down to y = -1 and starting to level off at least one end.
[B1] 0 solution
[B1] 2 solutions
[B1] 1 solution
[B1] Stretch by (scale factor) ½, parallel to x-axis or in x direction (or horizontally)
[B1] Translation of (0, 4) Accept translation/shift Accept translation 4 units in positive y-direction.
[B1] Translation of (-π/4, 0) Accept translation/shift Accept translation π/2 units in x-direction.
[B1] Stretch by (scale factor) 2 parallel to y-axis (or vertically).

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