Practice Exam
Chapter
Chapter 7
Section
Practice Exam
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Solutions 51 Videos

Evaluate \displaystyle (-\frac{3}{5})^4 . Express your answer in the simplest possible fraction form.

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0.20mins
Q1

Simplify \displaystyle (-7)^{43} \times (-7)^{52} \div (-7)^{30}

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Q2

Express \displaystyle (\frac{x^4}{y^6})^7 as a quotient of powers.

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Q3

Express \displaystyle 5^{-9} using a positive exponent.

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0.11mins
Q4

Express \displaystyle \sqrt[5]{48} as a product of power of prime base.

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Q5

Evaluate \displaystyle (\sqrt[5]{-32})^3 .

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Q6

What type of function is represented by y =8^x?

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0.12mins
Q7

State the y—intercept of the graph of \displaystyle y =(\frac{1}{5})^x .

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0.12mins
Q8

State the equation of the horizontal asymptote of the graph of y = 9^x.

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0.52mins
Q9

State the range of the graph of \displaystyle y = (\frac{3}{4})^x .

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Q10

State the equation that represents the graph of y = 2^x after it is translated 3 units to the right.

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Q11

State the equation that represents the graph of \displaystyle f(x) = 6^x after it is compressed horizontally by a factor of \displaystyle \frac{1}{2} .

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Q12

Is the graph of y = 5^x increasing or decreasing?

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Q13

Express \displaystyle y = (\frac{1}{8})^x in terms of an equivalent equation using base 2.

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0.20mins
Q14

State the equation of the horizontal asymptote for \displaystyle y = 3^{x +2} -1

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0.32mins
Q15

For each graph, write a corresponding equation in terms of sine.

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1.32mins
Q16

Simplify \displaystyle \frac{9x}{x^2 -5x + 6} - \frac{4x}{x^2 +x- 12} and state any restrictions on the variable.

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1.30mins
Q17

Determine whether \displaystyle f(x) = (2x + 3)^2 -(4x + 5)(x-7) is equivalent to g(x) \displaystyle = \frac{11x^2 -33x -44}{x + 1} . Justify your answer.

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1.20mins
Q18

Identify the parameters and use them to describe the transformations that must be applied to the graph of each base function to obtain the transformed function. Write the transformed equation in simplified form.

a) f(x) = \sqrt{x}, to obtain the transformed function y = -f[4(x+3)] - 1

b) f(x) = \frac{1}{x}, to obtain the transformed function y = \frac{1}{2}f[-(x + 5)] +2.

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Q19

Each graph represents a transformation of one of the base functions f(x) = x, f(x) = x^2, f(x) =\sqrt{x}, or f(x) =\frac{1}{x}. State the base function and the equation of the transformed function.

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0.42mins
Q20

Consider these functions:

  • i) f(x) = \frac{1}{2-x}
  • ii) \displaystyle f(x) = 3x^2+1

a) state the domain and range off f(x)

b) evaluate f(-5)

c) determine the equation of the inverse function.

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Q21

Determine the vertex of each quadratic function by completing the square. Verify your answer by using partial factoring. State if the vertex is a maximum or minimum.

a) \displaystyle f(x) = -3x^2 + 6x + 4

b) \displaystyle f(x) = 2x^2 - 8x + 7

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1.25mins
Q22

Find an equation in standard form for the quadratic function with zeros x = 3 \pm \sqrt{5} and containing the point (2, -8).

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1.19mins
Q23

State the value of the discriminant and determine the number of points of intersection of each. Justify your answers.

a) the line y = 2x + 1 and the quadratic function f(x) = x^2-3x+3

b) the line y = -3x+ 2 and the quadratic function f(x) =x^2-4x + 9

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1.13mins
Q24

Andrew and David are on a canoe trip between three campsites located at different points on a lake. Starting at campsite A, they canoe 11 km to campsite B. From that site, they canoe 20 km to campsite C. If the angle from campsite A to campsite C to campsite B is 30°. how far must Andrew and David canoe in order to return directly to campsite A?

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1.58mins
Q25

A squash player hits a ball 2.3 m to the side wall. The ball rebounds at an angle of 100° and travels 3.1 m to the front wall. How far is the ball from the player when it hits the front wall? Assume the player does not move after her shot.

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Q26

State the amplitude, the period, the phase shift. and the range of the graph of each sinusoidal function. Then, sketch a graph of the function.

a) \displaystyle y = 5\sin[4(x - 30^o)] -2

b) \displaystyle y = -2\sin[3(x + 45^o)] + 5

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Q27

Sarah is standing on a balcony that is 10 m above the ground. From her position, the angle of elevation to the top of a tower across the street is 45°. Liz is standing on a balcony that is 20 m higher than Sarah‘s. From Liz’s position. the angle of elevation to the top of the same tower is 30°. Determine the height of the tower. Express your answer using exact values.

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Q28

Prove each.

a) \displaystyle \frac{\csc \theta + \cot \theta}{\csc \theta - \cot \theta} = \frac{1 + 2\cos \theta + \cos^2 \theta}{\sin^2 \theta}

b) \displaystyle \frac{\cos \theta}{\sec \theta} - \frac{\sin \theta}{\cot \theta} = \frac{\cot \theta \cos \theta - \tan \theta}{\csc \theta}

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Q29

The equation T = 18 \sin[30(t -4)] + 11 models the average monthly temperature for a city in southwestern Ontario. In this equation, I denotes the number of months, with January represented by 1, and T is the temperature, in degrees Celsius.

a) Determine the city’s maximum average monthly temperature. When does this occur?

b) Determine the city’s minimum average monthly temperature. When does this occur?

c) What is the difference between the maximum and minimum average monthly temperatures?

d) What is the relationship between your answer in part c) and the coefficient of the sine term in the equation?

e) What is the sum of the maximum and minimum average monthly temperatures?

f) What is the relationship between your answer in part e) and the value of the constant term in the equation?

g) What is the average monthly temperature in October?

h) When is the average monthly temperature 20 °C?

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Q30

Write t_{18, 8}, as the difference of two terms, each in the form t_{n, r}.

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Q31

Use Pascal’s triangle to expand (4x - y)^2.

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Q32

Determine the number of terms in each arithmetic sequence.

16, 23, 30, ..., 583

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Q33a

Determine the number of terms in each arithmetic sequence.

x + 2, x - 1, x -4, ..., x -52

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Q33b

Determine a and d and then write the formula for the nth term of each arithmetic sequence with t_8=79 and t_{21} = 235.

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Q34

Determine the number of terms in the geometric sequence

\displaystyle -36, -18, -9, ..., -\frac{9}{128} .

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Q35

Determine an arithmetic series such that the sum of the first five terms of the series is 85 and the sum of the first six terms of the series is 123.

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Q36

A jogger runs 300 m in the first minute. The distance the jogger covers decreases by 20 m in each succeeding minute. What distance does the jogger cover in the 7th minute?

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Q37

A ball bounces to \frac{5}{6} of its height when dropped on a hard surface. Suppose the ball is dropped from a height of 40 m.

a) What height does the ball bounce back up to after the 7th bounce?

b) What is the total distance travelled by the ball after 10 bounces?

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Q38

There is a legend that the inventor of chess chose the following for his reward: 1 grain of wheat on the first square. 2 grains of wheat on the second square, 4 on the third, 8 on the fourth, and so on, for all 64 squares on the chessboard. Find an expression for the amount of wheat required to fulfill his request.

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Q39

Identify whether each series is arithmetic or geometric. Justify your answer. Then, determine the sum of each series.

100 + 90 + 80 + ... - 220

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Q40a

Identify whether each series is arithmetic or geometric. Justify your answer. Then, determine the sum of each series.

1 + 3+ 9 + ... + 2187

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Q40b

In a geometric series, t_2 = 10 and t_5 =1250. Determine t_4 and S_6.

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Q41

Matteo is very excited about his new sports car! Although he paid $24 800 for the car, its resale value will depreciate (decline) by 22% of its current value every year. The equation relating the car’s depreciated value, v, in dollars, to the time, t, in years, since its purchase is v(t) = 24 800(078^t).

a) Explain the significance of each part of this equation.

b) How much will Matteo's car be worth in

  • i. 2 years?
  • ii. 5 years?

c) Explain why this is an example of exponential decay.

d) How long will it take for the value of Matteo’s sports car to decline to 40% of its original cost?

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Q42

Timothy invests $2500 in an 18-month term deposit that pays simple interest at an annual rate of 3.5%. How much interest does Timothy earn?

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Q43

Amanda buys a $5000 guaranteed investment certificate (GIC) that will earn simple interest at the rate of 5.8% per year for 7 years.

a) Determine the amount of interest that Amanda will earn on the GIC.

b) What is the amount of the GIC at the end of 7 years?

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Q45

Victoria invests $5000 in a registered retirement savings plan (RRSP) that earns interest at the rate of 6.95% per annum, compounded monthly. What is the value of the RRSP at the end of 7 years?

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Q45

Kathy wants to have $16 000 in 5 years in order to buy a new motorcycle. If Kathy has $10 000 to invest today, what rate of interest, to the nearest hundredth of a percent, compounded quarterly, does he need to achieve his goal?

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Q46

On his 21st birthday, Carol receives $5000 from her grandparents, the accumulated amount of an investment they made for her when she was born. What was the original amount of the investment if it earned interest at the rate of 8.75% per year, compounded monthly?

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Q47

How long does it take $100 to grow to $500 if the amount is invested at a rate of 13.5% per annum, compounded semi-annually?

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Q48

John invests $25 000 in an annuity that earns 7.5% interest per year, compounded semi-annually. If the annuity is to pay John twice a year for 10 years, starting 6 months from now, what is the amount of each semi-annual payment?

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Q49