MATH 20

 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. nd 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 dened 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.