MCR 3U Exam Review Introduction: This review is composed of possible test questions. The BEST way to study for math is to do a wide
selection of questions. This review should take you a total of 6 hours of work, provided you can refer to your notes easily for questions you have difficulty with. Once you are done, you will have an inventory of all possible types of exam questions. f = {( 2,1) , ( 3,1) , ( 4,1) }
1. Cons Consid ider er the the fol follo lowi wing ng set sets: s: g = {(1,2) , (1,3) , (1,4)} h
=
{(1,5) , ( 2,6) , ( 3,7) , ( 4,2) }
(a) Explain why h is a 1:1 function. (b) Which Which of the sets sets are are functions functions?? (c) State the range of g . (d) State the value of h (2) (2) 1 g ? Explain (e) Is it true that f (f) h (x ) = 7. State the value of x (g) State the value of h − ( 2) (h) True or false: the domain of f −1 is the same as the range of f. (i) Creat Createe a mappi mapping ng diagr diagram am for for h. 2. State the the domain domain and range range for the the function function whose whose defining defining equati equation on is y = −2( x −1) 2 + 6 . Explain how you know. 3. Dete Determ rmin inee the the inve invers rsee of y = −2 x +4 . State the domain and range of both the original and its inverse. 4. A re relation f is shown on the graph below −
=
1
6
y
4
2
x -5
5
-2
-4
-6
(a) Is the relati relation on a function function?? Explain. Explain. (b) Evaluate f ( 0 ) (c) State the domain and range of f (d) Graph f −1 5. Sup Suppose ose that hat f ( x ) 2 x 1 and g ( x ) = x 2 + 3 (a) Determine g ( −3) (b) Determine g ( x − 3) , and fully simplify your answer. 6. Make a reasonably accurate graph of the following on the grid paper below =
(a) y =
−
1
(b)
x
y
=
x
7. Describe Describe in in words, words, using using vocabula vocabulary ry suitable suitable for for this course course,, how (a) the graph of y = 3 ( − ( x − 4 ) ) + 2 is transformed from the graph of
y
1 1 + 4 is transformed from the graph of y = x 3x
(b) The graph of y = −
8. Sketch Sketch the the follow following ing graphs graphs on the the grid grid paper paper..
=
x
(a) y = 2( x + 4 ) 2
−
1 + 3 x − 1
y = −
(b)
3
9. The graph of y sin( x ) is reflected in the y-axis, vertically stretched by a factor of 3 and translated 4 units up and 2 units right. What is the equation of the new graph? =
10.
Graph the following functions on graph paper and complete the chart that follows. y = sinθ and y = 3sin2(θ – 60˚) – 2. Function
Amplitude
Period
Max value for y
Min value for y
Phase shift
Vertical Displacement
y = sinθ y = 3sin2(θ – 60˚)– 2
11. The graph below depicts the depth of the water on a typical day at ocean port.
a) Determine the maximum and minimum depths of the water. b) State the amplitude, period, and vertical displacement for this relationship. Assuming the graph is a transformed cosine graph, state an equation for the graph. 12. Solve the following triangles. a) B 109° 6cm 8cm
b)
E f 25°
A
b
C
D
d 62° F
4.8cm 13. A surveyor is on one side of a river. On the other side is a cliff of unknown height. To determine its height, the surveyor lays out a baseline AB of length 150m. From point A, she selects point C at the base of the cliff and measures CAB to be 51˚. She selects point D on the top of the cliff directly above point C and measures an angle of elevation of 32˚. She moves to point B and measures CBA as 62˚. Find the height of the cliff. D
h C
32˚ 51˚ A
62˚ 150m
B
14. Solve {don’t forget to check for the AMBIGUOUS CASE}
a) ∆ DEF, where d=3.2cm, f=5.6cm and ∠ D=22°. b) ∆ ABC, where a = 3cm, c = 5cm and ∠ A = 30°.
15. Sketch each angle in standard position and state the coordinates of P(x, y). Determine two other angles that are coterminal with it. Write the formula to determine and coterminal angle. a) θ = 16° b) θ = -25° c) θ = 152° d) θ = 212° e) θ = 284°
17. 18.
16. For each angle below. i) Sketch each angle in standard positions. ii) State the coordinates of point P(x, y) on the terminal arm of the unit circle using exact values. iii) State its exact value of the trigonometric function for each of the 6 trig functions. Do not use a calculator. Use your special triangles. a) 150˚ b) 240° c) 315° Find the values to three decimal places of the 6 trig ratios for angle θ = 48°. Determine each value of θ for 0° ≤ θ ≤ 360° to the nearest degree. a) cscθ = 4.039 b) secθ = 1.745 c) cotθ = 0.299 d) sin θ = 0.227 19. Prove the following trigonometric identities. a) csc θ − 1 = csc θ cos θ b) tan θ + cot θ = sec θ csc θ 2
c)
2
1 1 + cos θ
=
2
2
csc θ −
cot θ
1 + csc θ
d)
sin θ
cot θ
20. Simplify as much as possible 2 2 2 2x − 5x − 3 2x − 11x − 6 12ab − 4b ÷ (a) (b) x − 3 − 6ab x 2 − 3x (d)
2
y + 1
−6
(e)
5 x
−3
− sec θ =
tan θ
2
(c)
+ 12a a 2 − 4a − 5 × a − 5 a + 6
2a
3x
− x
2
− 4x + 3
21. State all restrictions on the variable in #13(b) and #13(e) 22. Solve the following quadratic equations. f ( x ) = x 2 − 2 x − 7 (a) y = x 2 − 5 x + 6 (b) 23. Determine the defining equation of the quadratic function that has zeros 1 + 2 2 and 1 −2 2 , and passes through the point (1, 8). NOTE: express your final answer in standard form. 24. Determine the value for k, where y = -2x + k is tangent to the parabola y = − x 2 + 6 x − 7 . 25. Evaluate. Be sure to show intermediate steps to prove you know the meanings of each kind of exponent. 1
(a)
3
−3
(b)
−
9 (e) 4
2
27 3
2
26. Simplify to a single power (one base, one exponent) 1
(a)
a
1 3
a
3 2
1 6 3 x (b) 1 x − 2
(n
x +3 y
(c)
)( n
n3 x
2 x − 4 y
)
y
−4
27. The population of a city is increasing by 1.3 % each year. In the year 1980, its population was 750 000. (a) Write an equation that will give the city’s population for any year. (b) Use your equation to calculate the population today (2006).
a) b)
28. An oil painting is worth $400. Its value increases by 7% each year. What is it worth after 1, 3, and 6 years? Draw a graph to show how its value increases during the first 6 years. c) How would the graph be different if the initial value of the painting was more than $400? d) How would the graph change if the value increased each year by more than 7%? 29. The population of a colony of bacteria doubles every 4 hours. There are 200 bacteria at the beginning of the experiment. (a) Determine an equation to model the relationship where N is the population of bacteria, in hundreds of thousands, and t is the time from the beginning of the experiment, in hours. (b) What is the population 2 days after the start of the experiment? 30. Complete the following table Sequence
Next 3 terms
Type of Sequence (A, G, N)
1 4 9 16 , , , 5 8 11 14 2 2 , ,2,6 9 3 2 2 10 14 , , , 9 3 9 9
31. State the first 4 terms for each of the following sequences
General Term
t 1 = 5 f (n ) = 2n 3 − n (a) t 2 = 2 (b) t n = t n −1 ×t n −2 32. Determine the explicit term and the sum of the first 250 terms for the arithmetic sequence whose 10th term is 250 and whose 14 th term is 230. 33. Calculate the number of terms in the sequence 15, 22, 29, …, 771 34. By using a formula for the sum of a series, determine the sum of the first 8 terms of
+ 30 + 45 + 67.5 + 35. Determine the sum of 2 + 11 + 20 + 29 + + 722 20
36. Calculate the following (a) the amount of a $ 2500 investment after 7 years with 5.3 % interest p.a. compounded semiannually. (b) the annual interest rate required for an investment of $ 4000 to double in 8 years if it is compounded monthly. (to the nearest tenth of a percent) (c) the present value of an annuity of $ 200 per month for 15 years if interest is 6 % p.a. compounded monthly. (d) the monthly payment for a $12 000 loan over 5 years at a rate of 8.2 % p.a. compounded monthly. (e) to the nearest year, the number of years required for a $2500 investment to grow to $ 4000 if interest is 7.5 % p.a. compounded quarterly (f) the income you would get every 6 months for the next 15 years if you have $ 750 000 in an investment now, assuming 8 % p.a. interest rate compounded semiannually. 37. Sherri would like to have $12000 after 5 years. She will make a regular deposit every month into an investment account that pays 5.25% compounded monthly. Calculate the amount she will need to deposit every month. 38. Terry wants to make an investment so that he can withdraw $500 every month for 8 years. How much should he invest now at 4.25% compounded monthly?
MCR3UR
EXAM REVIEW ANSWER KEY
1)
a) h is 1:1 because each x value produces only one y value and each y value has only one x value. b) both f and h. c) R: {2, 3, 4} d) h(2) = 6 e) yes – (x, y) pairs in f are switched in g. f) x = 3 g) h-1(2) = 4 h) true i) x y 1 2 3 4
2 5 6 7
2) D: {x Є R}, R: {y ≤ 6, yЄR} 3) f-1(x) = x2 - 4 , f(x) – D: {x ≥ -4, x Є R}, R: {y ≤ 0, y Є R}, f -1(x) – D: {x ≥ 0, x Є R}, R: {y ≥ -4, y Є R} 4 4) a) no – does not pass the VLT b) f(0) = -2 c) D:{0 ≤ x ≤ 5, xЄR}, R:{-2 ≤ y ≤ 5, yЄR} d)
5) a) g(-3) = 12 6) a)
b) g(x – 3) = x2 – 6x + 12
c) 13 b)
7) a) 3 – vertical expansion factor 3, ‘ - ‘ – reflection over the y-axis, 4 – horizontal translation R4, 2 – vert.translation U2 b) ‘-‘ – reflection over the x-axis, 3 – horizontal compression factor 1/3, Vertical translation U4 8) a) b)
9) y = 3sin(-(x – 2)) + 4 10) Function
y = sinθ y=3sin(2θ-120˚)-2 11)
Amplitude
1
a) 28m and 9m b) a = 9.5
Period
Max value for y
360˚ 3
d = 18.5
1 180˚
period = 13.6
Min value for y
-1 1
Phase Shift
0˚ -5
Vertical Displacement
0 60˚
equation: y = 9.5 cos360˚(x – 6) + 18.5 13.6
-2
12) a) b ≈ 11.5cm, C ≈ 30˚, A ≈ 41˚ b) E ≈ 93˚, d ≈ 2.0cm, f ≈ 4.2cm 13) h = 89.9m 14) a) Acute - F ≈ 41˚, E ≈ 117˚, e ≈ 7.6cm, Obtuse - F≈ 139˚, E ≈ 19˚, e ≈ 2.8cm b) Acute – C ≈ 56˚, B≈ 94˚, b ≈ 6.0cm, Obtuse - C ≈ 124˚, B ≈ 26˚, b ≈ 2.6cm 15) a) P(0.961, 0.275), 376˚ , -344˚, …, 16˚+ 360˚n, b) P(0.906, -0.423), 335˚, -385˚, …., -25˚+360˚n, c) P(-0.883, 0.469), 512˚, -208˚,….., 152˚+ 360˚n, d) P(-0.848, -0.530), 572˚, -148˚, …, 212˚+360˚n e) P(0.242, -0.970), 644˚, -76˚, …., 284˚+360˚n 16) a) i) ii) P(- √3/2, ½) iii) sin 150° = ½ , cos 150°= -√3 /2 , tan 150° = -1/ √3 csc150°= 2 , sec 150° = -2/√3, cot 150° =-√3
ii) P(-1/2, - √3/2)
b) i)
c)
ii) P(1/ √2, -1/√2)
i)
iii) sin240° = -√3/2, cos240°= -1/2 , tan 240° = √3 csc240° = -2/√3, cos240° = -2, cot 240°= 1/√3
iii) sin315° = -1/√2, cos315°= 1/√2, tan315° = -1 csc315°= -√2, cos315° = √2, cot315° = -1
17) sin 48˚= 0.743, cos 48˚= 0.669, tan 48˚= 1.111, csc 48˚=1.346, sec 48˚= 1.494, cot 48˚= 0.900 18) a) θ ≈ 14˚, 166˚ b) θ ≈ 55˚, 305˚ c) θ ≈ 73˚, 253˚ d) θ ≈ 13°, 167° 19) Answers may vary, see teacher to check your proofs. 20) a) -2(3a –b) b) x – 3 c) 2a (a + 1) d) -2(3y + 2) e) 2x – 5 3 x–6 y+ 1 (x-3)(x-1) 21) 21b) x ≠ 3, -1/2, 6, 0 21e) x ≠ 3, 1 22) a )x = 2, 3 b) x = 1 + 2√2, 1 - 2 √2 23) y = -x2 +2x +7 24) b = 9, …eq’n of tangent is y = -2x + 9 25) a) 1/27 b) 9 e) 2/3 26) a) a11/6 b) x5/36 c) n3y 27) a) P = 750000(1.013)n where n is the number of years since 2006. 28) a) 1 year - $428, 3 years – $490.12, 6 years - $600.29 b)
c) graph would translate vertically up (greater y intercept 29) a) N = 200(2) t/4 b) 819200 30)
b) ≈ 1049317 people
d) graph would get steeper at a faster rate
Type of Sequence (A, G, N) Neither
Sequence
Next 3 terms
General Term
1 4 9 16 , , , 5 8 11 14
25 , 36, 49 17 20 23
2 2 , ,2,6 9 3 2 2 10 14 , , , 9 3 9 9
18, 54, 162
Geometric
tn = 2(3)n-1 9
2, 22, 26 9 9
Arithmetic
tn = 4n -2 9
tn =
_n2 3n + 2
31) a) 5, 2, 10, 20 b) 1, 14, 51, 124 32) tn = -5n + 30 S250 = -81875 33) 109 terms 34) 985.15625 or 31525/32 35) S81 = 29322 Financial Applications
36) a) $3605.50 37) $175.33 38) $40630.87
b) 8.7%
c) $23700.70
d) $244.47
e) 6.25 years
f) $43372.57
