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What is the exact value of sin (120 degrees)?

Select one:

a. ½

b. .866

c. 3/2

d. -1/2

e. None of these

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If f(x) = 3x +2 and g(x) = x -4, (f/g)(4) is:

Select one:

a. Undefined

b. 0

c. 14

d. (3, 2, 4)

e. None of these

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What the equation of the line passing through (-1, 4) and parallel to y = 2x + 1?

Select one:

a. y = 2x + 6

b. y = 2x + 1

c. y = 2x

d. No such line exists

e. None of these

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Solve the system: x + y + z = 6, 2x – y + z = 3, x + y – z = 0

Select one:

a. (0, 0, 6)

b. (3, 2, 1)

c. 1

d. (1, 2, 3)

e. None of these

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What is the equation of a line through (-1, 4) and (-2, 7)?

Select one:

a. y = 3x + 1

b. y = -3x + 1

c. y = -3x + 7

d. y = 3x + 7

e. None of these

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If I invest $10,000 at 5% compounded monthly, how much will I have in 8 years? Choose the closest answer.

Select one:

a. $5,000

b. $10,000

c. $15,000

d. $20,000

e. $25, 000

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How much must be invested today at 8%, compounded continuously, to be worth $100,000 in 12 years? Pick the closest answer.

Select one:

a. $250,000

b. $100

c. $40,000

d. $4,000

e. No amount will be enough

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Solve for x: (1/2)^x = 16

Select one:

a. -4

b. -2

c. 2

d. 4

e. None of these

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What is the polar representation of z = 6i?

Select one:

a. 6cis i

b. 6 cis

c. cis 6

d. 6cis (/2)

e. 36

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The graph of y = 4sin (3x + 2) + 1 has amplitude of:

Select one:

a. 0

b. 1

c. 2

d. 3

e. 4

Solution

1 ) sin( 120⁰) = sin(90⁰ + 30⁰ )

= cos(30⁰) [ since, sin(90⁰ +   ) = cos ]

sin( 120⁰) = 3 / 2

3 ) Let ( x₁ , y₁ ) = ( -1 , 4 )

y = 2x + 1 [ since , y = mx + b ,where m= slope ]

slope m = 2

The equation of the line passing through (-1, 4) and parallel to y = 2x + 1 is ,

y - y₁ = m ( x - x₁ )

y - 4 = 2 ( x - (-1) )

y - 4 = 2x + 2

y = 2x + 2 + 4

y = 2x + 6

Therefore,

Equation of line is  y = 2x + 6

4 ) Given that

x + y + z = 6..............1

2x – y + z = 3..............2

  x + y – z = 0................3

Solve equations 1 and 3

  x + y + z = 6

i.e x + y = 6 - z...........4

x + y - z = 0

i.e x + y = z ....................5

Equating 4 and 5

6 - z = z

6 = 2z

z = 6 / 2

z = 3

Solving equations 2 and 3

  2x – y + z = 3

  x + y – z = 0

From 2 and 3

2x + x = 3

3x = 3

x = 3

Substitute x = 1 and z = 3 in equation 1

x + y + z = 6

1 + y + 3 = 6

y + 4 = 6

y = 6 - 4

y = 2

Therefore,

( x , y , z ) = ( 1 , 2 , 3 )

5 ) Given that

let ( x₁ ,y₁ ) =  (-1, 4)

(x₂ , y₂ ) = (-2, 7)

Slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )

m = ( 7 - 4 ) / ( -2 - (-1) )

= 3 / ( -2+1)

= 3 / -1

m = -3

The equation of line is ,

y - y₁ = m( x - x₁ )

y - 4 = -3( x - (-1) )

y -4 = -3( x + 1 )

y -4 = -3x - 3

y = -3x -3 + 4

y = -3x +1

Therefore,

The equation of line passing through the two points is ,    y = -3x +1

6 ) Given that

Principal amount p = $10,000

Interest rate r = 5% = 0 .05

Time t = 8 years

We know that

compound interest monthly is ,

A = p( 1 + r /n )^nt

Where,

A = Total amount

n = Number of compoundings per year

n = 12

Amount   A = p( 1 + r /n )^nt

= $10,000( 1 + 0.05/12 )^(12X8)

A = $15,000

7) Given that

Amount A = $100,000

Interest rate r = 8% = 0.08

Time t = 12

We know that

Compound interest continuously is ,

A = pe^rt

p = A / e^rt

p = $100,000 / e^{( 0.08x12)}

= $3828.92886

Therefore,

Investment amount = $ 3828.92886   

8 ) Given that

(1/2)x = 16

x = 16 . 2

x = 32

9 )   

Given that

z = 6i

We know that

The polar form of non-zero complex number z = a + bi is given by ,

z = r(cos + i sin )

Where,

a = rcos , b = rsin , r = (a² + b² ) ^1/2  

Given that

z = 6i

a = 0 , b = 6

r =   ( 0² + 6² )^1/2

= (6²)^1/2

r = 6

a = rcos

cos = a / r

cos = 0/6 = 0

= cos^-1(0)

= 90⁰

Hence,

Polar form of compex number is,  z = r(cos + i sin )

   z = 6 ( cos(90⁰) + i sin(90⁰) )

z =  6cis (/2)