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x -axis. f withthe θ ,giventhat0 ‹ θ ‹ 2 π ,(b)thecoordinatesofthepointofintersectionofthegraphof x -axisexactlyonce.Findthe:(a)thepossiblevaluesof f ( x )= x −√ 8cos θx +3cos θ − 1intersectsthe 2 1.(6points)Giventhatthegraphofthefunction Nazaret1IBHLTest

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Nazaret 1IB HL Test 2 September 24, 2018

1. (6 points) Given that the graph of the function f (x) = x2 −√

8 cos θx + 3 cos θ − 1 intersects the x-axis exactly once. Find the:

(a) the possible values of θ, given that 0 ¬ θ ¬ 2π,

(b) the coordinates of the point of intersection of the graph of f with the x-axis.

(2)

Nazaret 1IB HL Test 2, page 2 of 4 September 24, 2018

2. (7 points) When a person is at rest, the blood pressure, P millimetres of mercury at any time t seconds can be approximately modelled by the equation

P (t) = −20 cos(5π

3 t) + 100, t ­ 0 (a) Determine the amplitude and period of P .

(b) What is the maximum blood pressure reading that can be recorded for this person?

(c) Sketch the graph of P (t), showing one full cycle.

(d) Find the first two times when the pressure reaches a reading of 110 mmHg.

(3)

Nazaret 1IB HL Test 2, page 3 of 4 September 24, 2018

3. (8 points) (a) Prove that:

sin 2x + sin x

cos 2x + cos x + 1 = tan x (b) Hence solve the equation:

sin 2x + sin x

cos 2x + cos x + 1 = −1 for 0 ¬ x ¬ π.

(c) The equation

sin 2x + sin x + cos 2x + cos x + 1 = 0 has two solutions in the interval 0 ¬ x ¬ π.

i. state one of the solutions, ii. find the other solution.

(4)

Nazaret 1IB HL Test 2, page 4 of 4 September 24, 2018

4. (7 points) If α is an obtuse angle and tan α = −12, find the exact value of:

(a) cos 2α, (b) sin 2α, (c) sin 3α.

5. (4 points) Solve the inequality |x + 2| > |3x − 1|.

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