MATH 201

[12] 1. (a) Write an equation of the line passing through the point (2, −3) and
parallel to the line 3y + 6x − 1 = 0.
(b) Write an equation of the line passing through the point (−3, 0) and
orthogonal to the line that contains the points (−4, 1) and (2, −2).
(c) Find the coordinates of the center and the radius of the circle
x2 + y 2 − 6y = −6
√
(d) Find the domain and the range of f (x) = x2 − 16 − 5 .
[9]

2.

Consider the quadratic function f (x) = 3 + 2x − x2 .
(a) Express f (x) in standard form.
(b) Find the coordinates of the vertex and indicate whether it corresponds
to the maximum or the minimum of f .
(c) Find the x− and y − intercepts.
(d) Sketch the graph of f (x) using the information above.

[9]

3.

(a) Let f (x) = 53x−4 − 2. Find the inverse function f −1 (x).
(b) Let f (x) = ex+1 and g (x) = ln(x − 1). Find g ◦ f and determine its domain.

[12] 4. Find the solutions of the following equations:
(a) 22x − 2x+3 − 20 = 0
(b) log3 x + log3 (x + 2) − 2 = 0

MATH 201
[12] 5.

Final Examination

December 2012

Page 2 of 2

Find the solutions of the following equations:
(a) 2 · 2x − 8x = 0
(b) log5 (x + 1) − log5 (x − 1) = 2

[9] 6.

(a) Find the radius of the circle if its sector with a central angle
1
θ = radian has an area A = 9 m2 .
2
(b) A car’s wheels are 70 cm in diameter. What is the speed of the car,
in km/hour, if the wheels rotate at 180 revolutions per minute ?

[12] 7.

Solve the triangle ABC (i.e. find the missing sides and angles)
(a) ∠A = 30◦ , ∠B = 70◦ , b = 30 cm
(b) ∠A = 53◦ b = 15 cm, c = 20 cm

[9]

8.

1
(a) Find the amplitude, period, and phase shift of y = 3 sin[π (x − 3 )]

(b) A ladder leans against a vertical wall of a building so that the angle
between the ground and the ladder is 72◦ and its bottom on the ground
is at 3 m from the wall. How long is the ladder? How high does it reach?
[6]

9.

Verify the identities
sin x
1 − cos x
−
=0
sin x
1 + cos x
cot x
(b) csc x − sin x =
sec x

(a)

[10] 10.

Solve the following trigonometric equations in [0, 2π ]
(a)
(b)

[5] 11.

sin2 x + sin x = cos2 x
sin 2x cos x + cos 2x sin x = 1

Bonus Question
If a function f (x) is defined for all real x and has an inverse f −1 (x), does it
necessarily follow that also g (x) = [f (x)]2 has an inverse g −1 (x) ?
Explain why it does, or give an example when it does not.