MATHEMATICAL METHODS

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1 Victorian Certificate of Eucation 207 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER MATHEMATICAL METHODS Written examination Wenesay 8 November 207 Reaing time: 9.00 am to 9.5 am (5 minutes) Writing time: 9.5 am to 0.5 am ( hour) QUESTION AND ANSWER BOOK Number of questions Structure of book Number of questions to be answere Number of marks Stuents are permitte to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners an rulers. Stuents are NOT permitte to bring into the examination room: any technology (calculators or software), notes of any kin, blank sheets of paper an/or correction flui/tape. Materials supplie Question an answer book of 2 pages Formula sheet Working space is provie throughout the book. Instructions Write your stuent number in the space provie above on this page. Unless otherwise inicate, the iagrams in this book are not rawn to scale. All written responses must be in English. At the en of the examination You may keep the formula sheet. Stuents are NOT permitte to bring mobile phones an/or any other unauthorise electronic evices into the examination room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 207

2 207 MATHMETH EXAM 2 THIS PAGE IS BLANK

3 3 207 MATHMETH EXAM Answer all questions in the spaces provie. Instructions In all questions where a numerical answer is require, an exact value must be given, unless otherwise specifie. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise inicate, the iagrams in this book are not rawn to scale. Question (4 marks) a. Let f : ( 2, ) R, f (x) = x x + 2. Differentiate f with respect to x. 2 marks b. Let g(x) = (2 x 3 ) 3. Evaluate g (). 2 marks TURN OVER

4 207 MATHMETH EXAM 4 Question 2 (4 marks) Let y = x log e (3x). a. Fin y x. 2 marks 2 b. Hence, calculate (log e( 3x) + ) x. Express your answer in the form log e (a), where a is a positive integer. 2 marks

5 5 207 MATHMETH EXAM Question 3 (4 marks) Let f : [ 3, 0] R, f (x) = (x + 2) 2 (x ). a. Show that (x + 2) 2 (x ) = x 3 + 3x 2 4. mark b. Sketch the graph of f on the axes below. Label the axis intercepts an any stationary points with their coorinates. 3 marks y O 2 x TURN OVER

6 207 MATHMETH EXAM 6 Question 4 (2 marks) In a large population of fish, the proportion of angel fish is 4. Let ˆP be the ranom variable that represents the sample proportion of angel fish for samples of size n rawn from the population. Fin the smallest integer value of n such that the stanar eviation of ˆP is less than or equal to 00.

7 7 207 MATHMETH EXAM Question 5 (4 marks) For Jac to log on to a computer successfully, Jac must type the correct passwor. Unfortunately, Jac has forgotten the passwor. If Jac types the wrong passwor, Jac can make another attempt. The probability of success on any attempt is 2. Assume that the result of each attempt is inepenent 5 of the result of any other attempt. A maximum of three attempts can be mae. a. What is the probability that Jac oes not log on to the computer successfully? mark b. Calculate the probability that Jac logs on to the computer successfully. Express your answer in the form a, where a an b are positive integers. b mark c. Calculate the probability that Jac logs on to the computer successfully on the secon or on the thir attempt. Express your answer in the form c, where c an are positive integers. 2 marks TURN OVER

8 207 MATHMETH EXAM 8 Question 6 (3 marks) ( )( )( + ) = Let tan( θ) sin( θ) 3 cos( θ) sin( θ) 3 cos( θ ) 0. a. State all possible values of tan(θ ). mark 2 2 ( )( ) = where 0 θ π. 2 marks b. Hence, fin all possible solutions for tan( θ) sin ( θ) 3cos ( θ ) 0,

9 9 207 MATHMETH EXAM Question 7 (5 marks) Let f :[ 0, ) R, f( x) = x +. a. State the range of f. mark b. Let g : (, c] R, g(x) = x 2 + 4x + 3, where c < 0. i. Fin the largest possible value of c such that the range of g is a subset of the omain of f. 2 marks ii. For the value of c foun in part b.i., state the range of f (g(x)). mark c. Let h : R R, h(x) = x State the range of f (h(x)). mark TURN OVER

10 207 MATHMETH EXAM 0 Question 8 (5 marks) For events A an B from a sample space, Pr( A B)= anpr( B A) =. Let Pr ( ). 5 4 A B = p a. Fin Pr(A) in terms of p. mark b. Fin Pr ( A B ) in terms of p. 2 marks c. Given that Pr( A B), state the largest possible interval for p. 2 marks 5

11 207 MATHMETH EXAM Question 9 (9 marks) The graph of f : [ 0, ] R, f( x = x( x) is shown below. y y = x x) 0 x a. Calculate the area between the graph of f an the x-axis. 2 marks b. For x in the interval (0, ), show that the graient of the tangent to the graph of f is 3 x 2 x. mark Question 9 continue TURN OVER

12 207 MATHMETH EXAM 2 The eges of the right-angle triangle ABC are the line segments AC an BC, which are tangent to the graph of f, an the line segment AB, which is part of the horizontal axis, as shown below. Let θ be the angle that AC makes with the positive irection of the horizontal axis, where 45 θ < 90. y C y = x x) A θ 0 B x c. Fin the equation of the line through B an C in the form y = mx + c, for θ = marks. Fin the coorinates of C when θ = marks END OF QUESTION AND ANSWER BOOK

13 Victorian Certificate of Eucation 207 MATHEMATICAL METHODS Written examination FORMULA SHEET Instructions This formula sheet is provie for your reference. A question an answer book is provie with this formula sheet. Stuents are NOT permitte to bring mobile phones an/or any other unauthorise electronic evices into the examination room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 207

14 MATHMETH EXAM 2 Mathematical Methos formulas Mensuration area of a trapezium curve surface area of a cyliner 2 a+ b h ( ) volume of a pyrami 2π rh volume of a sphere volume of a cyliner π r 2 h area of a triangle 3 Ah 4 π r bc A sin ( ) volume of a cone 2 π r h 3 Calculus x x n ( )= nx n n (( ax + b) )= an ax+ b x n ( ) n ( ) n n+ xx= x + c, n n + x e ax ae ax ax ( )= e x a e ax = + c ax b x ( ) ( ax b an ) n+ + = + + c, n + ( log e() x )= 0 x x x x = log e() x + c, x > ( sin( ax) )= a cos( ax) sin( ax) x = cos( ax) + c x a x cos( ax) = a sin( ax) cos( ax) x = sin ( ax) + c a ( ) a 2 ( tan( ax) ) = = a sec ( ax) x cos 2 ( ax) prouct rule ( x uv)= u v v u x + x quotient rule v u u v u x x x v = 2 v chain rule y y u = x u x

15 3 MATHMETH EXAM Probability Pr(A) = Pr(A ) Pr(A B) = Pr(A) + Pr(B) Pr(A B) ( ) ( ) Pr(A B) = Pr A B Pr B mean µ = E(X) variance var(x) = σ 2 = E((X µ) 2 ) = E(X 2 ) µ 2 Probability istribution Mean Variance iscrete Pr(X = x) = p(x) µ = x p(x) σ 2 = (x µ) 2 p(x) continuous Pr( a< X < b) = f( xx ) µ = a b xf ( xx ) σ 2 µ 2 = ( x ) f( xx ) Sample proportions P = X n mean E(P ) = p stanar eviation (ˆ ) s P = p( p) n approximate confience interval p z ˆ ˆ ( ˆ ) ˆ ( ˆ ) p p p p, ˆp+ z n n END OF FORMULA SHEET

MATHEMATICAL METHODS (CAS) Written examination 1

Victorian Certificate of Eucation 2006 SUPERVISOR TO ATTACH PROCESSING LABEL HERE STUDENT NUMBER Letter Figures Wors MATHEMATICAL METHODS (CAS) Written examination 1 Friay 3 November 2006 Reaing time: