# Solutions and explanations Maths Questions (2)

Solutions and explanations to the math questions (2) in this site are presented.

Questions 1:
If f(x) = Log(x), then f -1(x) =
Solution
y = Log(x) if and only if
x = 10y
Interchange x and y
y = 10x
Hence
f -1(x) = 10x

Questions 2:
Log4 65 =
Solution
Use the change of base formula to write
Log4 65 = ln(65) / ln(4)
Then use a calculator
= 3.01 (rounded to 2 decimal places)

Questions 3:
If 2 3x - 1 = 16, then x =
Solution
Rewrite equation using 16 = 24
2 3x - 1 = 24
The two bases of the exponential expressions in the above equation are equal to 2. Hence their exponents has also to be equal
3x - 1 = 4
Solve for x
x = 5/3

Questions 4:
If Logx 9 = 2, then x =
Solution
Logx 9 = 2 if and only if
x 2 = 9
x 2 = 32
For the above exponential expressions to be equal, their bases must be equal. Hence
x = 3

Questions 5:
If x 2 + kx - 6 = (x - 2)(x + 3), then k =
Solution
Expand the polynomial on the right side
x 2 + kx - 6 = x 2 + x - 6
For two polynomials to be equal, all their corresponding coefficients must be equal. Hence
k = 1

Questions 6:
The vertex of the graph of y = 2x 2 + 8x - 3 is given by
Solution
Write y in vertex form by completing the square
y = 2x 2 + 8x - 3 = 2(x 2 + 4x + 4 - 4) - 3
= 2(x 2 + 4x + 4) - 8 - 3
= 2(x + 2)2 - 11 which is of the form a(x - h)2 + k
vertex at (-2 , -11)

Questions 7:
The quadratic equation whose roots are at x = 3 and x = 5 is given by
A. (x - 3)(x - 5) = 1
B. (x + 3)(x + 5) - 9 = (x + 3)(x + 5) - 25
C. (x + 3)(x + 5) = 0
D. x 2 - 8x = -15
Solution
Checking the given roots by substituting in equations A., B., C. and D. shows that D. is the equation with roots 3 and 5.

Questions 8:
The roots of the equation x / (x + 2) + 3 / (x - 4) = (4x + 2) / (x 2 - 2x - 8) are
Solution
We first note that
(x + 2)(x - 4) = x 2 - 2x - 8
We now multiply all terms of the given equation by (x + 2)(x - 4) and simplify to get
x (x - 4) + 3 (x + 2)= (4x + 2)
Expand, group and solve
x 2 - x + 6 = 4x + 2
x 2 - 5x + 4 = 0
(x - 1)(x - 4) = 0
Roots: x = 1 and x = 4
Check answer: only x = 1 is a solution since the given equation is not defined at x = 4.

Questions 9:
If the domain of function f given by f(x) = -x 2 + 6x is given by the interval [0 , 6], then the range of f is given by the interval
Solution
We first note that the zeros of f are x = 0 and x = 6. The graph of f cuts the x axis at x = 0 and x = 6. Also since the leading coefficient is negative, the graph of f which is a parabola opens downward and the maximum value of f(x) is within the interval [0,6]. The maximum of f(x)is at
x = - b / (2a) = - 6 / -2 = 3
the maximum value of f(x) is equal to
f(3) = -(3) 2 + 6(3) = 9
The range of f is given by interval
[0 , 9]

Questions 10:
The x intercepts of the graph of y = -x 2 + 3x + 18 are given by
Solution
The x intercepts are found by solving the equation
- x 2 + 3x + 18 = 0
-(x 2 - 3x - 18) = 0
-(x - 6)(x + 3) = 0
solutions: x = 6 and x = -3
x intercepts: (6 , 0) and (-3 , 0)

Questions 11:
The domain of f(x) = 2 Log |x - 2| is given by the interval
Solution
The domain of f is the set of x values for which f is defined. f is defined for
|x - 2| > 0
|x - 2| is positive for all x except x = 2. Hence the domain is given by the interval
(-infinity , 2) U (2 , + infinity)

Questions 12:
If 1.56x = 2, then x =
Solution
Take the ln of both sides
ln(1.56x) = ln(2)
x ln(1.56) = ln(2)
Take the ln of both sides
x = ln(2) / ln(1.56)

Questions 13:
The reference angle to angle a = -1280° is equal to
Solution
How many -360° are there in -1280°?
-1280/-360 = 3.5 (approximately)
Let us now find a coterminal angle to -1280° by adding 4×360°
-1280 + 4×360 = 160°
Angle of measure 160° has terminal side in quadrant II. Hence the reference angle is given by
180 - 160 = 20°

Questions 14:
If x is an angle in standard position with point A(-3 , 4) on the terminal side, then sec x =
Solution
Let r be the distance from the (0,0) to (-3,4).
r = √((-3)2 + 42) = 5
sec x = 1/cos x = 1 / (-3/5) = -5/3

Questions 15:
If x is an angle in standard position and it terminal side is in quadrant IV and is given by y = - x, then sin x =
Solution
Select a point in quadrant IV and on the line y = - x.
(3,-3)
let r be the distance from (0,0) to point (3,-3).
r = √(32 + (-3)2) = 3 √(2)
sin x = -3 / 3 √(2) = - 1 / √(2)

Questions 16:
Which statement is NOT true?
A. cos(-x) = cos x
B. tan(-x) = tan x
C. the amplitude of y = -2 cos (t) is equal to 2
D. the range of y = -5 sin (t) is given by [-5 , 5]
Solution
B is not true.
tan (- x) = - tan x , odd function

Questions 17:
Which statement is true?
A. sec(-x) = - sec x
B. the range of y = tan x is given by (0 , + infinity)
C. the period of y = -2 cos (3 π t) is equal to 2π / 3
D. the period of y = -5 tan (0.5 π t) is equal to 2
Solution
D is true.
If y = a tan(bx) then period = π/ |b| = π / (0.5 π) = 2

Questions 18:
The radian measure of an angle of 25° is equal to
Solution
25 × π / 180 = 5 π / 36

Questions 19:
The period of the function f(x) = sin(x π/6 + π/4) is equal to
Solution
2π / (π/6) = 12

Questions 20:
Which of the following is an identity?
A. cos(2x) = 2 cos x
B. cos(x + y) = cos x + cos(y)
C. sin(x - y) = sin x - sin(y)
D. sin(2x) = 2 sin x cos x
Solution
D. For all x real, sin(2x) = 2 sin x cos x

Questions 21:
Which of the following is NOT an identity?
A. cot(x + y) = [ 1 + cot x cot(y) ] / [ cot x + cot(y) ]
B. tan(x + y) = [ tan x + tan(y) ] / [ 1 - tan x tan(y) ]
C. sin(x - y) = sin x cos(y) - cos x sin(y)
D. cos(2x) = 2 cos 2(x) - 1
Solution
A. is not an identity.

Questions 22:
If x is an angle such that tan x = 5/12 and π < x < 3 π/2, then sec x =
Solution
tan x = b/a, where a and b are the coordinates of a point on the terminal side. Since x is in quadrant III, a and b are negative. Hence
tan x = a / b = 5/12 = (-5) / (-12) , a = -12 and b = -5
If r is the distance from (0,0) to (-5,-12), then
r = √((-12)2 +(-5)2) = 13
sec x = 1 / cos x = 1 / (-12/13) = -13/12

Questions 23:
The real solutions to the equation cos 2(x) - 1.5 cos x = 1 are given by the solutions to the equation
A. cos x = 1
B. cos x = 2
C. cos x = 1/2
D. cos x = - 1/2
Solution
Let's first rewrite the given equation as follows
cos 2(x) - 1.5 cos x - 1 = 0
Factor the left side
(cos x - 0.5)(cos x - 2) = 0
cos x - 0.5 = 0 , cos x = 0.5
or cos x - 2 = 0 or cos x = 2 , no real x is solution to this equation.
The given equation has same solutions as the equation
cos x = 1/2

Questions 24:
Which point is on the graph of the inverse of the function f(x) = 10x + 2?
A. (100 , 0)
B. (0 , 100)
C. (10 , 0)
D. (0 , 10)
Solution
Note that f(0) = 100. Hence
f -1(100) = 0 , where f -1 is the inverse of f
and therefore the point (100,0) is the graph of the inverse of f

Questions 25:
If 10x / y = A / B, then
A. x / y = Log A / Log B
B. y = x / (Log A - Log B)
C. x = y / (Log A + Log B)
D. y = x Log (A / B)
Solution
Take the Log of both sides and simplify
Log(10x / y) = Log(A / B)
x / y = Log A - Log B
y / x = 1 / (Log A - Log B)
y = x / (Log A - Log B)

Questions 26:
If π < x < 3 π/2, then sin x can be expressed in terms of tan x as follows
A. sin x = tan x / √[ 1 + tan2(x) ]
B. sin x = - √[ 1 + tan2(x) ] / tan x
C. sin x = - tan x / √[ 1 + tan2(x) ]
D. sin x = √[ 1 + tan2(x) ] / tan x
Solution
tan x = sin x / cos x
Square both sides
tan2(x) = sin2(x) / cos2(x)
Use the identity cos2(x) = 1 - sin2(x)
tan2(x) = sin2(x) / (1 - sin2(x))
which gives sin2(x) = tan2(x) / (1 + tan2(x))
Since x is in quadrant III, sin x is negative and therefore
sin x = - √[ tan2(x) / (1 + tan2(x))]
= - tan x / √(1 + tan2(x))

Questions 27:
Angle x = 11 π / 3 is coterminal to angle y given by
A. y = π / 3
B. y = π / 6
C. y = 5 π / 3
D. y = 2 π / 3
Solution
A possible coterminal angle to x is
11 π / 3 - 2 π = 11 π / 3 - 6 π / 3 = 5 π / 3

Questions 28:
If x is such that 3 π/2 < x < 2 π and sin x = -1 / 2, then x =
Solution
We first solve sin(y) = 1 / 2
y = π / 6 , which can be used as a reference angle
The solution to sin x = -1 / 2 is equal to
x = 2 π - π/6 = 11 π/6

Questions 29:
The exact value of cos(127 π/3) is given by
Solution
Let us write 127 π/3 as follows
127 π/3 = 126 π / 3 + π / 3 = 42 π + π / 3
A coterminal angle to 127 π/3 is π/3. Hence
cos(127 π/3) = cos(π/3) = 1/2

Questions 30:
If Log(x - y) = 3 and Log(x + y) = 4, then x =
Solution
Log(x - y) = 3 is equivalent to
x - y = 103 = 1000
Log(x + y) = 4 is equivalent to
x + y = 104 = 10,000
We now solve the system
x - y = 1000 and x + y = 10,000
2x = 11,000
x = 5,500

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