Math Questions With Answers (2)

Math Questions With Answers (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:

Log 465 =

Solution

Use the change of base formula to write

Log 465 = 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 Log x9 = 2, then x =

Solution

Log x9 = 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 = -1280o is equal to

Solution

How many -360o are there in -1280o?

-1280/-360 = 3.5 (approximately)

Let us now find a coterminal angle to -1280o by adding 4*360o

-1280 + 4*360 = 160o

Angle of measure 160o has terminal side in quadrant II. Hence the reference angle is given by

180 - 160 = 20o



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 = sqrt((-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 = sqrt(3 + (-3)) = 3 sqrt(2)

sin(x) = -3 / 3 sqrt(2) = - 1 / sqrt(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.

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 Pi t) is equal to 2Pi / 3
D. the period of y = -5 tan (0.5 Pi t) is equal to 2


Solution

D.

If y = a tan(bx) then period = Pi/ |b| = Pi / (0.5 Pi) = 2



Questions 18:

The radian measure of an angle of 25o is equal to

Solution

25 * Pi / 180 = 5 Pi / 36



Questions 19:

The period of the function f(x) = sin(x Pi/6 + Pi/4) is equal to

Solution

2Pi / (Pi/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 Pi < x < 3 Pi/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 = sqrt((-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 Pi < x < 3 Pi/2, then sin(x) can be expressed in terms of tan(x) as follows

A. sin(x) = tan(x) / sqrt[ 1 + tan2(x) ]
B. sin(x) = - sqrt[ 1 + tan2(x) ] / tan(x)
C. sin(x) = - tan(x) / sqrt[ 1 + tan2(x) ]
D. sin(x) = sqrt[ 1 + tan2(x) ] / tan(x)


Solution

Start with the identity

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) = - sqrt[ tan2(x) / (1 + tan2(x))]

= - tan(x) / sqrt(1 + tan2(x))



Questions 27:

Angle x = 11 Pi / 3 is coterminal to angle y given by

A. y = Pi / 3
B. y = Pi / 6
C. y = 5 Pi / 3
D. y = 2 Pi / 3


Solution

A possible coterminal angle to x is

11 Pi / 3 - 2 Pi = 11 Pi / 3 - 6 Pi / 3 = 5 Pi / 3



Questions 28:

If x is such that 3 Pi/2 < x < 2 Pi and sin(x) = -1 / 2, then x =

Solution

We first solve sin(y) = 1 / 2

y = Pi / 6 , which can be used as a reference angle

The solution to sin(x) = -1 / 2 is equal to

x = 2 Pi - Pi/6 = 11 Pi/6

Questions 29:

The exact value of cos(127 Pi/3) is given by

Solution

Let us write 127 Pi/3 as follows

127 Pi/3 = 126 Pi / 3 + Pi / 3 = 42 Pi + Pi / 3

A coterminal angle to 127 Pi/3 is Pi/3. Hence

cos(127 Pi/3) = cos(Pi/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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