## 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

= 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[ (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, 2 ]

(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.