Grade 11 Mathematics Page 1 of 6 Final Exam Review (updated 2013)

Transcription

1 Grade Mathematics Page of Final Eam Review (updated 0) REVIEW CHAPTER Algebraic Tools for Operating With Functions. Simplify ( 9 ) (7 ).. Epand and simplify. ( ) ( ) ( ) ( 0 )( ). Simplify each of the following. State any restrictions on the variables. ( )( ) d) c d 8 d c 9 e) 0. Simplify each of the following. State any restrictions on the variables. d) 8 s s s e) 7 w 9 w f) f) 8 v 0 7v 0 v 7 v. Lucy s hockey team is selling candy bars, at $ each, to raise money for a tournament. It costs the team 0 to purchase each bar, with a fied shipping cost of $. How many bars must be sold for the team to earn at least $00? REVIEW CHAPTER Quadratic Functions and Equations. Simplify Find the maimum or minimum value of the following functions by completing the square. d) 8 e) 0 y = 0 y = y = 7. Find the maimum product of two numbers whose sum is 7.. Solve by completing the square. Epress answers as eact roots and as approimate roots, to the nearest hundredth, if necessary. = 0 8 = 0 = 0. Solve 0 = by factoring. Check the solution.. Solve 7 9 = 0 using the quadratic formula. 7. Simplify Epand and simplify ( 0 ) 9. Simplify.. ( 7 )( 7 ) 0. Solve each system of equations. y = y = y = 0 d) y = ( ) y = y = y= 9y = 8

2 Grade Mathematics Page of Final Eam Review (updated 0) REVIEW CHAPTER Transformations of Functions. Determine if each relation is a function. y y State the domain and range of each relation. Determine if each relation is a function. {(7, ), (, ), (, ), (, ), (7, 8)} y = y =. If f() =, find f( ) f(0) f(0). How do the graphs of y = and y = compare with the graph of y =?. How do the graphs of y = and y = 8 compare with the graph of y =?. Sketch the graph of y =. 7. Given the graph of y = f(), as shown, graph y = f() and y = f( ) on the same aes. Describe how the graphs of y = f() and y = f( ) are related to the graph of y = f(). 8. If f() =, write an equation to represent each of the following functions, describe how the graph of each function is related to the graph of y = f(), sketch each graph, state the domain and range, and identify any invariant points. y = f() y = f( ) 9. Find the inverse f of the function f whose ordered pairs are {(, 0), (, 9), ( 8, )}. Graph both functions. 0. For f ( ) = and ( ) = g : find the inverse graph the function and its inverse function determine whether the inverse is a function d) determine the domain and the range of the function and its inverse function. Graph y =, y =, and y = on the same grid. Describe how the graphs of y = and y = are related to the graph of y =.. Given the graph of y = f(), sketch the graphs of y = f(), y = f(), y = f(), and y = f. Describe how the graph of each of the functions in part is related to the graph of y = f().

3 Grade Mathematics Page of Final Eam Review (updated 0). If f() =, sketch the graphs of y = f() and y = f( ).. The graph of y = is compressed vertically by a factor of, reflected in the -ais, and translated units to the right and units upward. Write the equation of the transformed function. REVIEW CHAPTER Trigonometry Don t do #. In XYZ, X = 90, Y =., and y =. cm. Solve the triangle by finding the unknown angle the unknown sides, to the nearest tenth of a centimetre. In PQR, Q = 90, p =.9 m, and r = 8. m. Solve the triangle by finding the unknown angles, to the nearest tenth of a degree unknown side, to the nearest tenth of a metre. Find the length of the parallel of latitude, to the nearest 0 km. Assume that the radius of the Earth is 80 km.. To calculate the height of a tree, Marie measures the angle of elevation from a point A to be. She then walks 0 m directly toward the tree, and finds the angle of elevation from the new point B to be. What is the height of the tree, to the nearest tenth of a metre?. The point (, 0) is on the terminal arm of an angle θ in standard position. Find sin θ and cos θ.. The point (, 8) is on the terminal arm of an angle θ in standard position. Find sin θ and cos θ. 7. Evaluate, to four decimal places. sin.7 cos 8. Find A, to the nearest tenth of a degree, if 0 A 80. cos A = 0. sin A = To measure the distance from a point A to an inaccessible point B, a surveyor picks out a point C and measures BAC to be 7. He moves to point C, a distance of m from point A, and measures BCA to be 9. How far is it from A to B, to the nearest metre? 0. In ABC, B =.9, a =. cm, and b =. cm. Solve the triangle, rounding the side length to the nearest tenth of a centimetre and the angles to the nearest tenth of a degree, if necessary.. In DEF, d =. m, e = 8. m, and f = 9. m. Solve the triangle. Round each angle measure to the nearest tenth of a degree.. In KLM, M =., k = 0.7 cm, and l =. cm. Solve the triangle, rounding the side length to the nearest tenth of a centimetre and the angles to the nearest tenth of a degree, if necessary.. Solve RST, if S =.0, s =. m, and t =.8 m. Round the side length to the nearest tenth of metre and the angles to the nearest tenth of a degree, if necessary.. Solve JKL, if J = 0., j =. cm, and k =.7 cm. Round the side length to the nearest tenth of a centimetre and the angles to the nearest tenth of a degree, if necessary. REVIEW CHAPTER Trigonometric Functions. Change each radian measure to degree measure. Round to the nearest tenth of a degree, if necessary. π.8. Find the eact radian measure, in terms of π, for each of the following The point P(, ) lies on the terminal arm of an angle θ in standard position. Determine the eact values of sin θ, cos θ, and tan θ.. Find the eact values of: sin 0 tan 0 sin 0 d) cos 0. Find the values of the sine, cosine, and tangent of an angle that measures π 9

4 Grade Mathematics Page of Final Eam Review (updated 0). Sketch one cycle of each function. y = sin y = cos ( ) y = 0.sin ( 90 ) 7. Prove the following identities. d) y = cos e) y = sin ( 80 ) 0 sin cos = sin = cos tan cos sin tan 8. Solve the following equations for 0 θ π. sin cos cos = cos sin cos θ = 0 7sin θ = sin θ 9. Solve the following equations on the interval 0 0. Round approimate solutions to the nearest tenth of a degree. sin sin = 0 sin 8sin 8 = 0 0. Solve cos = sin on the interval 0 π. REVIEW CHAPTER Sequences and Series. Given the formula for the nth term, write the first four terms of each sequence. t n = n f(n) = n t n = n d) f(n) = n e) t n = 7() n f) f(n) = ( ) n. Find the formula for the nth term that determines each sequence. 7,,, 8,, 7, 8, 9, 7, 9,,, d),, 8,, e), 0, 00, 00, f), 8,,. Find the indicated terms. t n =.n 8; t f(n) = 0.8n 0.; t 0 t n = ( ) n ; t 7 d) f(n) = 8(0.) n ; t. Find the number of terms in each of the following sequences. 8,,,, 0 9, 7,,, 9 9, 8,, 7,, d) 0.0, 0.,,,, 09. Find a and d, and write the formula for the nth term, t n, of arithmetic sequences with the following terms. t = and t 8 = 7 t = 9 and t 0 = 9. Find a, r, and t n for each geometric sequence. t = 7 and t = 9 t = 70 and t = 0 7. Determine whether each sequence is arithmetic, geometric, or neither. Then, find the net two terms.,, 7, 8, 9,, 7,,,,, 0., d) 0.8, 0., 0., 0., 8. Use the recursion formula to write the first five terms of each sequence. t = 8; t n = t n t = 9.; t n = t n. t = ; t n = t n d) t = 0; t n = 0.t n 9. Eplain why t n = t (n )k is an eplicit formula for the sequence with recursion formula t n = t n k and t = t. 0. Find the indicated sum for each arithmetic series. S 8 for 7 S for. Find the indicated sum for each geometric series. S for 7 9 S 7 for. A student paid $00 tuition for her first year attending a university. Tuition at the university is projected to increase $70 a year for the net four years. How much tuition should the student epect to pay for her fourth year at the university? How much should the student epect to pay in tuition for all four years?. In an arithmetic series t = 0 and t =. Find the sum of the first 0 terms.. Two friends started a telephone chain. Each person in the chain called three people. Thus, there were si telephone calls in the first round. How many telephone calls were made in the fifth round? After eight rounds, how many telephone calls had been made in all? REVIEW CHAPTER 7 Compound Interest and Annuities. To purchase a new computer, Prasanna borrows $000 at an interest rate of.% per annum, compounded annually. He has arranged to pay back the loan in years. How much will Prasanna owe after years? How much of this is interest?. Gwen wants to invest $ She must decide between a -year plan with an interest rate of 8% per annum, compounded quarterly, and a - year plan with an interest rate of 7.7% per annum, compounded monthly. Which plan earns Gwen more interest, and by how much?. What rate of interest, to the nearest hundredth of a percent, compounded semi-annually, would be required for an investment of $ to grow to $0 000 after 8 years?. Marc needs to save $0 000 for a home gym, which he would like to have in years. How much should he invest today at an interest rate of 7% per annum, compounded quarterly?. P. J. needs $ 000 in years to purchase a van. P. J. s bank has offered her two investment plans:.% per annum, compounded quarterly, or.% per annum, compounded monthly. Which plan requires a smaller investment, and by how much?

5 Grade Mathematics Page of Final Eam Review (updated 0). To save money for college, Ronna plans to deposit $0 into an account at the end of every three months for the net two years. She will begin making payments three months from now. If her account has an interest rate of % per annum, compounded quarterly, how much will Ronna have after she makes her last payment? 7. Benny has 0 months to pay off a loan of $87.. He plans to make a payment into an account at the end of every month. The interest rate is 7.% per annum, compounded monthly. How much will each of Benny s payments be? How much does the loan cost Benny? 8. Clarence has $0 000 in the bank. He wants to create an investment to pay his $0 monthly car insurance payments for four years, with the first payment due in one month. How much of his $0 000 should he invest now at 8.% per annum, compounded monthly? 9. Half of the $ raised in a charity raffle is invested in an account at 7.% per annum, compounded quarterly. The winner of the raffle is to receive payments from this account every three months for the net five years, beginning three months from now. How much are the payments? 0. Penny s parents have agreed to loan her $00 to pay her tuition. They are charging her an interest rate of % per annum, compounded monthly. Penny has arranged to pay them $0 per month to pay off the loan. how long it takes Penny to pay off the loan the amount of her final payment. Jack is purchasing a house that he plans to rent to students attending community college. The price of the house is $ 000. Jack makes a down payment of % of the price and agrees to a mortgage at.8%, amortized over years, for the balance of the price. How much is Jack s mortgage? How much are Jack s monthly payments?. Kun Wah has a $ mortgage, with an interest rate of 7.%, amortized over 0 years. He is making monthly payments. July also has a $ mortgage, with an interest rate of 7.%, but she is making biweekly payments. July s payments are half the amount of Kun Wah s monthly payments. Find the monthly payments for Kun Wah and the biweekly payments for July. How long does it take July to pay off her mortgage? REVIEW: Eponential Functions. Sketch the curve for each of the following, and state i) the equation of the horizontal asymptote, ii) whether the function is increasing or decreasing, iii) the y-intercept iv) domain and range y = y = y = d) y = e) y = ( ). Bacteria of a certain type are known to divide every hour, thus producing two bacteria for every previously eisting bacterium. Suppose that 00 of these bacteria are breathed into Paul s lung. How many bacteria will live in his lung after hours? How many after t hours?. The doubling period of a bacteria culture is min and it starts with 000 bacteria. How many bacteria will there be after min hour.. An antique vase was purchased in 000 for $8000. If the vase appreciates in value by % per year, what is its estimated value in the year 00, to the nearest thousand dollars?. A car depreciates by % per year. If you buy a car for $ 000, find the value of the car in three years.. The population of the world was billion in 999. This population is growing eponentially and doubles every years Estimate the world population in 00, to the nearest half billion. When will the population be billion? REVIEW : Pascal s Triangles & Binomial Theorem. Epand and simplify each of the following. ( a ( ) d) Eam Review Answers Chapter : Answers ) ( ), 0,, & d ( ) 9 d), c & d 0 e),,, f),,,,,,, c ( )( ) 0 0 s 7s d), s 0 s Chapter : Answers e), w ( w ) ( v 8) f), v,, ( v )( v )( v ) )

6 Grade Mathematics Page of Final Eam Review (updated 0) 7 d) 0 e) ) 09 ), ) no solutions 7 7 8) (-.,-.8) & (,7) (-,0) & (0,) (-,7) & (,) d) (,0) & (-9,) Chapter : Answers Yes No Yes D :{ 7, R} R :{ y 8 y, y R} D D :{ R} R :{ y y R}, - ± no solutions Yes, :{ R} R :{ y y, y R} Yes, ) up, down ) Right, Left 8 7 f(): reflect in -ais, f(-): reflect in y-ais 8 f ( ) = Inv. Pt: None f ( ) = Inv. Pt: None 9 (-0,),(-9,), (,-8) 0i) f ( ) = Yes d) f : R, y R f : R, y R ii) f ( ) = ± No d) f : R, y f :, y R v.s. by v.s. by v.s. by, v.s. by, h.s. by, h.s. by ) y = ( ) Chapter : Answers Z=.8, =., z=.9 P=9., R=0.8 q=. ) 9790 km ) 0. ) sin θ =,cosθ = ) sin θ =,cosθ = or 7 9).8 0) A=8..8, c=. ) D=.8, E=., F=.8 ) m=0., L=8., K=8. ) r=7., T=., R=. ) L =, L=., K=7. or L =7., L=.9, K=0.7 Chapter : Answers π π a ) 7 0 π ) sin θ =,cosθ =, tan θ = 9 9 d) Chapter : Answers,8,,8,0,, 7,,9, d)-,-,-9,- e)-7,-,-,-89 f),-8,,-8 t n =7n n t n =n t n =n d)t n =7-n e)t n =() n- f)t n = d) d)9 a=, d=9 a=-, d= a=-, r= or - a=0, r = or - 7 neither A.S. -, - G.S. 0., 0. d) A.S. 0, 0. 88,,,,0-9.,-8,-.,-,-. -,, -8,9,-78 d)0,,.,.8, ) ) sin 0 - cos - 0 tan 0 / Chapter 7: Answers.. )0.8 ).7% )88.7 )Plan by $.9 ) )97.0 9).8 0 yrs and months ) Kun Wah: 7. July:.7 Appro years & 0 months Eponential Functions Answers. Equation of Asymptote Function is y intercepts Domain Range a. y = - increasing - R y > - b. y = increasing R y > c. y = 0 decreasing R y < 0 d. y = decreasing R y > e. y = - increasing R y < - a. 00 b. N(t) = t 00 a b $ $900 a.. billion b. 09 Binomial Theorem Answers 8a a b ab b 0 0 d)