MATHEMATICAL METHODS

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1 Victorian Certificate of Education 018 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER MATHEMATICAL METHODS Written examination 1 Friday 1 June 018 Reading time:.00 pm to.15 pm (15 minutes) Writing time:.15 pm to 3.15 pm (1 hour) QUESTION AND ANSWER BOOK Number of questions Structure of book Number of questions to be answered Number of marks Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners and rulers. Students are NOT permitted to bring into the examination room: any technology (calculators or software), notes of any kind, blank sheets of paper and/or correction fluid/tape. Materials supplied Question and answer book of 13 pages Formula sheet Working space is provided throughout the book. Instructions Write your student number in the space provided above on this page. Unless otherwise indicated, the diagrams in this book are not drawn to scale. All written responses must be in English. At the end of the examination You may keep the formula sheet. Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 018

2 018 MATHMETH EXAM 1 (NHT) THIS PAGE IS BLANK

3 3 018 MATHMETH EXAM 1 (NHT) Answer all questions in the spaces provided. Instructions In all questions where a numerical answer is required, an exact value must be given, unless otherwise specified. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Question 1 (4 marks) a. Let f( x)= Find f (x). x e ( x 3 ). b. Let y = (x + 5) log e (x). Find dy dx when x = 5. marks marks TURN OVER

4 018 MATHMETH EXAM 1 (NHT) 4 Question (4 marks) Let f (x) = x + x + 4 and g(x) = x. a. Find g ( f (3)). marks b. Express the rule for f (g(x)) in the form ax 4 + bx + c, where a, b and c are non-zero integers. marks

5 5 018 MATHMETH EXAM 1 (NHT) Question 3 ( marks) 1 x x Evaluate e e dx. 0 Question 4 (3 marks) Solve log 3 (t) log 3 (t 4) = 1 for t. TURN OVER

6 018 MATHMETH EXAM 1 (NHT) 6 Question 5 (3 marks) Let h: R + 7 { 0} R, h( x) =. x + 3 a. State the range of h. 1 mark b. Find the rule for h 1. marks

7 7 018 MATHMETH EXAM 1 (NHT) Question 6 (4 marks) The discrete random variable X has the probability mass function kx Pr( X = x) = k 0 x {, 146, } x = 3 otherwise a. Show that k = 1 1. marks b. Find E(X ). 1 mark c. Evaluate Pr(X 3 X ). 1 mark TURN OVER

8 018 MATHMETH EXAM 1 (NHT) 8 Question 7 (9 marks) π Let f : 0, R, f (x) = 4cos(x) and g R : 0, π, g (x) = 3sin(x). a. Sketch the graph of f and the graph of g on the axes provided below. marks y O b. Let c be such that f (c) = g (c), where c 0, π. π 4 Find the value of sin(c) and the value of cos(c). π x 3 marks Question 7 continued

9 9 018 MATHMETH EXAM 1 (NHT) c. Let A be the region enclosed by the horizontal axis, the graph of f and the graph of g. i. Shade the region A on the axes provided in part a. and also label the position of c on the horizontal axis. 1 mark ii. Calculate the area of the region A. 3 marks TURN OVER

10 018 MATHMETH EXAM 1 (NHT) 10 Question 8 (3 marks) Let P be the random variable that represents the sample proportions of customers who bring their own shopping bags to a large shopping centre. From a sample consisting of all customers on a particular day, an approximate 95% confidence interval for the proportion p of customers who bring their own shopping bags to this large shopping centre was determined to be, a. Find the value of p that was used to obtain this approximate 95% confidence interval. 1 mark 49 b. Use the fact that = to find the size of the sample from which this approximate 95% 5 confidence interval was obtained. marks

11 MATHMETH EXAM 1 (NHT) Question 9 (8 marks) The diagram below shows a trapezium with vertices at (0, 0), (0, ), (3, ) and (b, 0), where b is a real number and 0 < b <. y O On the same axes as the trapezium, part of the graph of a cubic polynomial function is drawn. It has the rule y = ax(x b), where a is a non-zero real number and 0 x b. a. At the local maximum of the graph, y = b. Find a in terms of b. b 3 x 3 marks Question 9 continued TURN OVER

12 018 MATHMETH EXAM 1 (NHT) 1 The area between the graph of the function and the x-axis is removed from the trapezium, as shown in the diagram below. y O b 3 b b. Show that the expression for the area of the shaded region is b x square units. 3 marks Question 9 continued

13 MATHMETH EXAM 1 (NHT) c. Find the value of b for which the area of the shaded region is a maximum and find this maximum area. marks END OF QUESTION AND ANSWER BOOK

15 Victorian Certificate of Education 018 MATHEMATICAL METHODS Written examination 1 FORMULA SHEET Instructions This formula sheet is provided for your reference. A question and answer book is provided with this formula sheet. Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 018

16 MATHMETH EXAM Mathematical Methods formulas Mensuration area of a trapezium curved surface area of a cylinder 1 a+ b h ( ) volume of a pyramid 1 π rh volume of a sphere volume of a cylinder π r h area of a triangle 3 Ah 4 π r bc A sin ( ) volume of a cone 1 π r h 3 Calculus d dx x n ( )= nx n 1 d n (( ax + b) )= an ax+ b dx n ( ) 1 n ( ) n 1 n+ 1 xdx= x + c, n 1 n + 1 d dx e ax ae ax ax 1 ( )= e dx a e ax = + c ax b dx 1 ( ) ( ax b an ) n+ 1 + = + + c, n d ( log e() x )= dx x x dx = log e() x + c, x > d 1 ( sin( ax) )= a cos( ax) sin( ax) dx = cos( ax) + c dx a d dx 1 cos( ax) = a sin( ax) cos( ax) dx = sin ( ax) + c a ( ) d a ( tan( ax) ) = = a sec ( ax) dx cos ( ax) product rule d ( dx uv)= u dv v du dx + dx quotient rule v du u dv d u dx dx dx v = v chain rule dy dy du = dx du dx

17 3 MATHMETH EXAM Probability Pr(A) = 1 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 = E((X µ) ) = E(X ) µ Probability distribution Mean Variance discrete Pr(X = x) = p(x) µ = x p(x) σ = (x µ) p(x) continuous Pr( a< X < b) = f( xdx ) µ = a b xf ( xdx ) σ µ = ( x ) f( xdx ) Sample proportions P = X n mean E(P ) = p standard deviation (ˆ ) sd P = p( 1 p) n approximate confidence interval p z ˆ ˆ ( ˆ ) ˆ ( ˆ ) p 1 p p 1 p, ˆp+ z n n END OF FORMULA SHEET

MATHEMATICAL METHODS

Victorian Certificate of Education 2016 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER MATHEMATICAL METHODS Written examination 1 Wednesday 2 November 2016 Reading time: 9.00 am to 9.15