Nova Scotia Eaminations Advanced Mathematics Web Sample Student Booklet
General Instructions - WEB SAMPLE* This eamination is composed of two sections with the following suggested time allotment: Selected-Response (Multiple-Choice) Questions Constructed-Response Questions Value 5 pts (appro. 40 min)* Value 75 pts (appro. 0 min) *note: there are 35 constructed response questions on the Math NSEs as of January 008 Total time: 3 hours (revision time included) Use these suggested times to guide you in the completion of the eamination; however, you might not find it necessary to spend the suggested time on each section. Plan your time to enable you to complete the eamination. You are not permitted to use your own graphing calculator unless your teacher has cleared the memory immediately prior to this eamination. The only graphing calculators permitted are TI-8, TI-83, TI-83 Plus, TI-84, or TI-84 Plus. If the question indicates that you are not to use a graphing calculator, you are still permitted to use a calculator to perform arithmetic operations. Calculators are not to be shared. Graph paper, scrap paper, and formula sheets are provided at the end of this booklet. These pages can be removed from the booklet for your use during the eamination. Note: Diagrams are not necessarily drawn to scale. Advanced Mathematics Page
Selected-Response Questions - WEB SAMPLE (Total Value: 5 points) In this part of the eamination, there are 5 selected-response questions*, each with a value of point. Read each question carefully, and decide which of the responses best answers the question being asked. You are provided a separate student answer sheet. In the selected-response section of the student answer sheet, fill in the bubble that corresponds to your choice as shown in the eample below. Use an HB pencil only. Eample. What are the roots of. A B D + 3 4= 0? A. 4 and B. 4 and 3 C. 4 and D. 4 and 3 (On student answer sheet) If you wish to change an answer, please ensure that you erase your first answer completely on the student answer sheet. Calculations or rough work on the selected-response pages of the eamination booklet will not be scored. *Note: As of January 008 there are 35 selected response questions on the NSE Math eams. Page Advanced Mathematics
. The function t = n + 3n is used to generate a quadratic sequence. What is the value of D? n A. B. C. 3 D. 4 4. The ais of symmetry for the parabola defined by y = 6+ is A. = 4 B. = 6 C. = 3 D. = 3. A cannon ball is shot into the air and its height, in metres, is represented by the function h=.5 + 3.t 4.9t where 't' is time in seconds. Which of the following best approimates how long it will take for the ball to hit the ground? A..34 seconds B..50 seconds C..78 seconds D. 0.5 seconds ± i 9 4. If the roots of a + b + c = 0 are = 0, then what is true of the graph of the function y = a + b+ c? A. has no -intercept B. C. is not parabolic D. has no y-intercept has distinct -intercepts 5. The value of c that will make the epression 9 + c a perfect square is A. 36 B. 4 C. 8 D. 6 6. Given 4 + 4+ k = 0, for what values of k will there be no real roots? A. k < B. k > C. k < 0 D. k > Advanced Mathematics Page 3
7. A basketball player is trying to increase her shot accuracy. She stays in the same position on the court and increases the arc of the flight path of the ball. y Path II Path I Her coach graphs the quadratic function a ( y k ) = ( h ) to model parabolic path I of the basketball. The coach then changes certain values in the given equation to graph path II. Which value(s) did the coach NOT change? A. a B. C. k D. h h and a 8. Which one of the following tables of values was generated by an eponential function? A. B. 3 5 3 5 y 4 y 3 9 7 C. D. 0 3 4 4 y 3 3 3 y 4 9 6 k k k 9. The epression 3 + 3 + 3 equals 3k 3 3 A. 3 k B. 9 k 3 k + 9 3k C. D. Page 4 Advanced Mathematics
0. Which graph represents y = a b, where a >0 and 0< b <? A. B. y y C. D. y y. After certain transformations, the image of y = 3 is y = (3). The negative sign in the eponent is a result of a A. reflection in the -ais B. C. horizontal translation to the left D. reflection in the y-ais vertical stretch downwards log 6. The epression is equal to log ( ) A. 4 B. 3 C. 5 D. 3 Advanced Mathematics Page 5
3. Which of the following graphs could represent the remaining amount, A, of a radioactive substance as it decays over time t in years? A. B. A A t t C. D. A A t t 4. Which of the following is FALSE? loga M loga M loga N log N = log a M c A. B. = loga M c a log N C. logm N = D. log N( MN) = log N M + log M 5 5. What is the value of in log8 =? 3 A. 3.48 B. C. 3 D. 6 40 3 Page 6 Advanced Mathematics
6. The segment below is to be cut into 4 equal pieces. A(-4, 5) B C D E(8, -3) What are the coordinates of B? A. (,) B. (,) C. (, ) D. (,3) 7. If two chords of a circle intersect, then their perpendicular bisectors A. don't intersect B. C. bisect each other D. are parallel intersect at the centre of the circle 8. Which is the general form of ( 3) + ( y+ ) =? A. + y 6+ y+ = 0 B. C. + y + 6 y+ 9= 0 D. y y + 6 + + 9= 0 + y + 9= 0 9. The equation of an ellipse is + 4y = 36. Its domain is A. 3 3 B. C. 6 6 D. 8 8 0. C is the centre of the following circle. The value of is A. 90 B. 45 C. 55 D. 35 Advanced Mathematics Page 7
. Given the following figure: y B (3, 5) A (-, ) C (7, 0) D (3, -3) Which of the following statements is false? AB = DC T A. B. The midpoint of AD is (, ). C. The diagonals AC and BD are of equal D. lengths. 500!. The value of is 499! A. B..00 C. 500 D. undefined 3. Consider the following Venn diagram. The diagonals AC and BD bisect each other. Event K Event L Which of the following is correct? A. P(K or L) = P(K) + P(L) B. P(K and L) = P(K) + P(L) C. P(K or L) = P(K) P(L) D. P(K and L) = P(K) P(L) Page 8 Advanced Mathematics
4. Your math teacher gives your class a list of eight questions to study. Five of the eight questions will be randomly selected for the net test. If you study only the first five questions from the list, the probability that all of those five questions will be on the test is A. B. 8C5 8P5 5 8 8 C. D. 5. There are 5 red marbles and 7 blue marbles in a bag. Two marbles are chosen randomly without replacement. The probability that a red and then a blue marble will be chosen is 5 7 A. B. 5 7 + C. D. C C 5 7 C Advanced Mathematics Page 9
Constructed-Response Questions (Total Value: 75 points) Read each question carefully, and be sure to write your response in the bo and space provided. If the answer bo indicates that you are to show your work, then points will be awarded for your correct work and your correct final answer. The method used to solve a problem must clearly be shown even when using a graphing calculator. If the answer bo requires that just a final answer be provided, then points will be awarded for the correct answer only. When working with decimal values, you may round off to the hundredths place in your final answer only. If any decimal values are rounded prior to the final step of the solution, at least 4 decimal places must be kept. With the eception of the probability unit, all answers must be given in simplified form. Page 0 Advanced Mathematics
6. The severity of an automobile crash increases significantly as the speed increases. The table shows the relationship between the speed and a crash severity inde. Speed (km/h) 0 0 30 40 50 Crash severity inde.0 4.40 9.60 6.80 6.00 (a) Jimmy claims that a quadratic function would best model this situation. Is Jimmy's claim correct? Eplain. ( points) (b) What speed would have a crash severity inde of 50.40? (3 points) Advanced Mathematics Page
7. Solve for in each of the following equations using a different algebraic method for each. Epress your answers in simplified form. (a) = + 6 (3 points) (b) + + 5= 0 (3 points) Page Advanced Mathematics
8. A football is kicked into the air. The equation h= 4.9t + 9.8t+ epresses the relationship between height, h, in metres and time, t, in seconds. (a) Determine the maimum height reached by the football. (3 points) (b) For how long was the ball at a height of at least 5 m above the ground? (3 points) Advanced Mathematics Page 3
9. A golf ball is hit from ground level in a flat field and reaches a maimum height of 5 m. The ball first hits the ground 00 m away while following a parabolic path. (a) Draw a diagram and include all important information needed to model this problem. Remember to label your aes. ( points) (b) How high is the golf ball above the ground when it is at a horizontal distance of 0 m from where it was hit? (4 points) Page 4 Advanced Mathematics
30. Prove that there are no real values of m for which the quadratic equation has equal roots. 5= m( ) (4 points) 3+ 5 3. Find the zero(s) of the function f( ) = 7. (3 points) Advanced Mathematics Page 5
3. Algebraically solve for. (a) ( ) = 6 3 ( points) (b) log 8 log + log 3 = log 5 5 ( points) (c) log ( + 3) log = 4 4 ( points) Page 6 Advanced Mathematics
( ) 3 33. Given 4 y =. (a) Sketch the graph of the given function. Clearly state the coordinates of the y-intercept and two other points on the graph. Sketch the asymptote and provide the equation of the asymptote. (3 points) (b) Indicate whether the function above represents a growth curve or a decay curve. Eplain how you know. ( point) (c) State the domain and range of the function. ( point) Range: Domain: Advanced Mathematics Page 7
34. The population of country A is 40 million on January, 990 and increases by 6% each year thereafter. The population of country B is 60 million on January, 990 and increases by 4% each year thereafter. When will the populations be the same? (5 points) Page 8 Advanced Mathematics
35. (a) Evaluate: ( point) (i) log 8 = (ii) - log 8 = (b) Evaluate: ( point) (i) log3 9 = (ii) - log 9 = 3 (c) Based on the answers obtained in parts (a) and (b), write an epression equivalent to log b N. ( point) 36. Describe a situation that could be modelled by the function P =000(.03) t. (3 points) Advanced Mathematics Page 9
37. Complete each of the following statements. ( points) (a) Chords in the same circle are of equal length iff (b) A line is a perpendicular bisector of a chord iff 38. Write the equation of the following graph in transformational form. (3 points) y 0 6 Page 0 Advanced Mathematics
39. The endpoints of a diameter of a circle are A(-, 3) and B(3, 5). Determine if the point C(, 9) is on the circumference of the circle. Clearly show your reasoning. (4 points) 40. Given: PO SO (5 points) OQ OR Prove: ΔPQR ΔSRQ P O S Q R Advanced Mathematics Page
4. There are 5 jellybeans randomly distributed in a jar; 5 are yellow and 0 are orange. You reach into the jar and, without looking, remove jellybeans. What is the probability that you will remove yellow jellybeans? (3 points) Page Advanced Mathematics
4. In a group of 5 people, 4 are left-handed and are right-handed. Seven people are selected at random from this group. (a) What is the probability that all 4 left-handed people will be selected? ( points) (b) If Sarah and Mike, two of the left handers, have already been chosen, what is the probability that all the other members selected will be right-handed? ( points) Advanced Mathematics Page 3
43. Create a real-life problem that demonstrates P(A or B) = P(A) + P(B) P(A and B) when the events A and B are NOT mutually eclusive. (You don't have to solve the problem.) ( points) You have reached the end of the ADVANCED WEB SAMPLE Eamination. Please check your work to ensure you have completed all questions. Page 4 Advanced Mathematics
Formula Sheet Advanced Mathematics Quadratic Unit General form: Standard form: y = + + a b c y= a( h) + k Transformational form: If a + b+ c= 0, then ( k) ( h) a y = ± = b b 4ac a Eponential Growth Unit y = ab A( y C) = b B( D) log ( y) = log + log y a a a log a( y) = loga loga y or log = log log y b log = b(log ) a a a a a y
Circle Geometry Unit d = ( ) + ( y y ) The coordinates of M are: +, y + y General form: A + Ay + D+ Ey+ F= 0 A + By + D+ Ey+ F= 0 Standard form: ( h) + ( y k) = r Transformational form: ( h) + ( y k) = r r ( h) + ( y k) = a b Δy m = Δ Probability Unit P(A and B) = P(A) P(B) P(A or B) = P(A) + P(B) P(A and B) P(A B) = P(A and B) P(B) n P r n! = ( n r)! n C r n! = r!( n r )!
Nova Scotia Eaminations Advanced Mathematics Web Sample Marking Guide Teacher Name:
Selected Response Answers. D. C 3. C 4. A 5. B 6. B 7. B 8. B 9. C 0. B. B. A 3. C 4. A 5. C 6. D 7. D 8. B 9. C 0. C. C. C 3. A 4. A 5. A
Question 6 (a) ( points) Yes. D is constant. (D =, see table below) OR Yes. R = using quadratic regression. Question 6 (b) (3 points) = + + y a b c Quad Reg on TI-83 00a = a = 0.0 c = 0 0.0 0.0 y = + Using Table ( 0,.0 ) ( ).0 = 0.0 0 + 0b + 0.0 = + 0b 0.0 = 0b 0.0 = b 69.99 50.39 70.00 50.4 70.0 50.4 = 70 0.0 + 0.0= 50.4 0.0 + 0.0 50.4 = 0 Solve for using any method to get = 70. OR 0 0 30 40 50 60 70. 4.4 9.6 6.8 6 37. 50.4 3. 5. 7. 9.. 3. = 70
Question 7 (a) (3 points) ( )( ) = + 6 6= 0 3 + = 0 3= 0 + = 0 = 3 = = + 6 6= 0 ( ) ( ) ( 4)( )( 6) ( )( ) ± = = ± 5 both answers must be correct + 5 = = 3 5 = = OR = + 6 = 6 + = 6 + 4 4 5 = 4 5 =± 4 5 = ± 5 = + = 3 5 = = 3
Question 7 (b) (3 points) + + 5= 0 ( )( )( ) ( )( ) ± 4 5 = ± 36 = 4 ± 6i = 4 ± 3i = OR 5 0 + + = 5 + + = 0 5 + = 5 + + = + 4 4 9 + = 4 9 + =± 4 3 ± 3i = ± i or 4
Question 8 (a) (3 points) h= t + t+ 4.9 9.8 h = t + t 4.9 9.8 ( ) ( ) h = 4.9 t t h 4.9= 4.9 t t+ ( h 5.9) = ( t ) 4.9 The maimum height is 5.9 m. OR b t = a 9.8 = 4.9 = ( ) ( ) ( ) ( ) h = 4.9 + 9.8 + = 5.9 The maimum height is 5.9 m. y = + + 4.9 9.8 y (, 5.9) pts The maimum height is 5.9 m. 5
Question 8 (b) (3 points) + + = 4.9t 9.8t 5 4.9t + 9.8t 4 = 0 y y t t = 4.9 + 9.8 4 ( ) ( )( )( ) ( )( 4.9) 9.8 ± 9.8 4 4.9 4 t = = 0.57 and.43.5 pts Time above 5 m =.43 0.57 = 0.86 (0.57, 0) (.43, 0) Time above 5 m =.43 0.57 = 0.86 OR y = + + y = 5 4.9 9.8 y (0.57, 5) (.43, 5).5 pts Time above 5 m =.43 0.57 = 0.86 6
Question 9 (a) ( points) h (50, 5) (0, 0) (00, 0) t For ais labels Question 9 (b) (4 points) Using regression on TI-83 Using regression on TI-83 y 0.0 = + y 0.0 = + ( ) y = 0.0 0 + 0 = 6 The golf ball is 6 m high. y 9 5.39 0 6 6.59 The golf ball is 6 m high. OR a 5 50 ( y ) = ( ) 5 50 a 5 a = 500 a = 0.0 ( 0, 0) = ( ) 0.0 = 0.0 = 00 5 = 900 ( y 5) ( 50) ( y 5) ( 0 50) ( y ) y 5 = 9 y = 6 The golf ball is 6 m high. 7
Question 30 (5 points) ( ) m 5= 0 m m ( ) 5= m + 4 5= 0 If equal roots ( m) ( )( m ) 4m 6m 0 0 4ac= 0 4 4 5 = 0 m b + = 4m+ 5= 0 ()( ) 4± 6 4 5 m = 4± 4 = 4± i = = ± i OR = b Discriminant 4 ac ( 4) ( 4)( )( 5) = = 6 0 = 4 The roots are not real. There are no real numbers for which b 4ac = 0; therefore there are no values of m such that the equation has equal roots. 8
Question 3 (3 points) 3+ 5 7= 0 3+ 5 = 7 3+ 5 log = log 7 ( ) 3 + 5 log = log 7 3 log + 5log = log 7 log 7 5log = 3log = 0.73 3+ 5 7= 0 3+ 5 = 7 3 + 5= log 7 log 7 5 = 3 = 0.73 OR y = 3+ 5 7 y y = y = 7 3+ 5 y pts ( 0.73, 0) ( 0.73, 7) = 0.73 = 0.73 9
Question 3 (a) ( points) ( ) 3 = 6 3 = 3 4 ( ) 4 8 = 3 = 4 8 3= 8 6 5 = 6 = 6 5 OR ( ) ( ) 3 3 ( ) = 6 + log = log6 ( ) + 3log = log6 3log log6 = log6 3log log6 = log6 log6 = 3log log6 = 3. Question 3 (b) ( points) log 8 log + log 3 = log 5 ( 8)( 3) log = 3 log 7 = 3 3 = 7 = 3 5 OR log log + log 3 = log 5 log8 log log 3 + = 3 log log log log8 log + log3 = 3log.433 = 3log.433 = log 3 0.477 0 = 3 5 = 0
Question 3 (c) ( points) ( ) log4 + 3 log4 = + 3 log4 = + 3 = 4 + 3= 6 3= 5 5 = ( ) ( ) log4 + 3 log4 = log4 + 3 log4 = log46 + 3 log4 = log46 + 3 = 6 + 3= 6 3= 5 5 = OR ( ) ( + ) log4 + 3 log4 = log 3 log = log 4 log 4 log ( + 3) log = log 4 + 3 log = log 4 + 3 = 0 log4 log4 + 3= 0 3= 5 5 =
Question 33 (a) (3 points) y y-intercept (0, 4.5) (, 5) (3, 6) y = 4 is asymptote Question 33 (b) ( point) The function is a growth curve because, for y = ab + c, a > 0 and b >. Question 33 (c) ( point) Domain: Range: ( ) or, { y y > 4, y } or ( 4, )
Question 34 (5 points) ( ) Country A : P = 40.06 ( ) Country B : P 60.04 A n B = n n n ( ) = ( ) n ( ) = ( ) 40.06 60.04 n log 40.06 log 60.04 log 40 + nlog.06 = log 60 + nlog.04 nlog.06 nlog.04 = log 60 log 40 n( log.06 log.04) = log 60 log 40 log 60 log 40 n = log.06 log.04 n.9 The populations will be the same.9 years later. ( ) ( ) y = 40.06 60.04 y OR n ( ) = ( ) 40.06 60.04 n.06 60 = n.04 40 n.06 =.5.04.06 n log = log.5.04 log.5 n =.06 log.04 n.9 n The populations will be the same.9 years later. y ( ) ( ) y = 40.06 y = 60.04 (.9, 38.7) (.9, 0) pts The populations will be the same.9 years later. The populations will be the same.9 years later. 3
Question 35 (a) ( point) (i) log 8 = 3 (ii) log 8 = 3 Question 35 (b) ( point) (i) log3 9 = (ii) log 9 = 3 Question 35 (c) ( point) log N = log b N b Question 36 (3 point) These are eamples of acceptable answers. Others are possible. The value, P, of an investment of $000 which earns 6% per year compounded semi-annually for t years. The number of bacteria, P, in a Petri dish increases by 3% every half day for t days. The initial number of bacteria is 000. 4
Question 37 (a) ( point) Eamples of acceptable answers (each worth ):...they are equidistant from the centre....they subtend arcs of equal length....they subtend equal angles. Question 37 (b) ( point) Eamples of acceptable answers (each worth ):...it passes through the centre....it bisects the chord at right angles. Question 38 (3 points) 5 y 4 + = 9 5 Question 39 (3 points) + 3 3+ 5 Midpoint of diameter :, ( 6, 4) ( ) ( ) D radius = 6+ + 4 3 = 50 Distance from centre to (, 9): D = ( 6 ) + ( 4 9) = 50 (, 9) is on the circle, because the circle is the set of all points from (6, 4) 50 5
Question 40 (5 points) Statement(s) Reason(s) PO SO POQ SOR OQ OR PQO PQ SR PR SQ QR QR PQR SRO SRQ Given Vertically opposite angles Given SAS CPCTC Segment addition Common side SSS OR Statement(s) Reason(s) PO SO OQ OR PR SQ OQR is an isosceles triangle OQR ORQ QR QR PQR SRQ Given Given Segment addition OQ OR Base angles of an isosceles triangle Common side SAS 6
Question 4 (3 points) C C = 5 5 0 or 05 OR 5 4 0 = or 5 4 0 Question 4 (a) ( points) C C 4 4 3 5 7 C = 65 or or 0.03 6435 39 Question 4 (b) ( points) C C = 5 3 5 46 4 or or 0.36 87 39 7
Question 43 ( points) Eamples follow, other responses may be accpetable: What is the probability of being 6 3 or left-handed? What is the probability of becoming a teacher or a mother? 8