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Victorian Certificate of Education2021SUPERVISOR TO ATTACH PROCESSING LABEL HERELetterSTUDENT NUMBERMATHEMATICAL METHODSWritten examination 1Wednesday 3 November 2021Reading time: 9.00 am to 9.15 am (15 minutes)Writing time: 9.15 am to 10.15 am (1 hour)QUESTION AND ANSWER BOOKStructure of bookNumber ofquestionsNumber of questionsto be answeredNumber ofmarks9940 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 orsoftware), notes of any kind, blank sheets of paper and/or correction fluid/tape.Materials supplied Question and answer book of 11 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 electronicdevices into the examination room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2021
2021 MATHMETH EXAM 12InstructionsAnswer all questions in the spaces provided.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.b.Evaluate f ′(4), where f ( x) x 2 x 1.Question 2 (2 marks)Let f ′(x) x3 x.Find f (x) given that f (1) 2.1 mark2 marksdo not write in this areaQuestion 1 (3 marks)a. Differentiate y 2e–3x with respect to x.
32021 MATHMETH EXAM 1do not write in this areaQuestion 3 (5 marks)Consider the function g : R R, g(x) 2 sin(2x).a.State the range of g.1 markb.State the period of g.1 markc.Solve 2 sin(2x) 3 for x R.3 marksTURN OVER
2021 MATHMETH EXAM 14Question 4 (4 marks)a.2on the axes below. Label asymptotes with their equations and axisx 2intercepts with their coordinates.Sketch the graph of y 13 marksy654210–6 –5 –4 –3 –2 –1–1123456x–2–3–4–5–6b.Find the values of x for which 123.x 21 markdo not write in this area3
52021 MATHMETH EXAM 1Question 5 (4 marks)Let f : R R, f (x) x2 – 4 and g : R R, g(x) 4(x – 1)2 – 4.a.The graphs of f and g have a common horizontal axis intercept at (2, 0).do not write in this areaFind the coordinates of the other horizontal axis intercept of the graph of g.b.2 marksLet the graph of h be a transformation of the graph of f where the transformations have been applied inthe following order:1 dilation by a factor of from the vertical axis (parallel to the horizontal axis)2 translation by two units to the right (in the direction of the positive horizontal axis)State the rule of h and the coordinates of the horizontal axis intercepts of the graph of h.2 marksTURN OVER
2021 MATHMETH EXAM 16It is known that, in the box:1 of the doughnuts are with custard27 of the doughnuts are not glazed101 of the doughnuts are glazed, with custard.10a.A doughnut is chosen at random from the box.Find the probability that it is not glazed, with custard.b.1 markThe 20 doughnuts in the box are randomly allocated to two new boxes, Box A and Box B.Each new box contains 10 doughnuts.One of the two new boxes is chosen at random and then a doughnut from that box is chosen at random.Let g be the number of glazed doughnuts in Box A.Find the probability, in terms of g, that the doughnut comes from Box B given that it is glazed.2 marksQuestion 6 – continueddo not write in this areaQuestion 6 (6 marks)An online shopping site sells boxes of doughnuts.A box contains 20 doughnuts. There are only four types of doughnuts in the box. They are: glazed, with custard glazed, with no custard not glazed, with custard not glazed, with no custard.
7c.2021 MATHMETH EXAM 1The online shopping site has over one million visitors per day.It is known that half of these visitors are less than 25 years old.Let P̂ be the random variable representing the proportion of visitors who are less than 25 years old ina random sample of five visitors.3 marksdo not write in this areaFind Pr Pˆ 0.8 . Do not use a normal approximation.TURN OVER
2021 MATHMETH EXAM 18Question 7 (3 marks)A random variable X has the probability density function f given byk f ( x) x 2 01 x 2elsewherewhere k is a positive real number.Show that k 2.b.Find E(X).1 mark2 marksdo not write in this areaa.
92021 MATHMETH EXAM 1Question 8 (5 marks)do not write in this areaThe gradient of a function is given bydyx 3x 6 .2 2dx29The graph of the function has a single stationary point at 3, .4 a. Find the rule of the function.3 marksb.2 marksDetermine the nature of the stationary point.TURN OVER
2021 MATHMETH EXAM 110Question 9 (8 marks)Consider the unit circle x2 y2 1 and the tangent to the circle at the point P, shown in the diagram below.yPOa.(1, 0)xShow that the equation of the line that passes through the points A and P is given by y2x.332 marks x 1 0 x Let T : R 2 R 2 , T , where q R\{0}, and let the graph of the function h be the y 0 q y transformation of the line that passes through the points A and P under T.b.i.Find the values of q for which the graph of h intersects with the unit circle at least once.ii.Let the graph of h intersect the unit circle twice.Find the values of q for which the coordinates of the points of intersection have only positivevalues.1 mark1 markQuestion 9 – continueddo not write in this areaA(2, 0)
11c.2021 MATHMETH EXAM 1For 0 q 1, let P′ be the point of intersection of the graph of h with the unit circle, where P′ isalways the point of intersection that is closest to A, as shown in the diagram below.yP'OθA(2, 0)(1, 0)xLet g be the function that gives the area of triangle OAP′ in terms of θ.i.Define the function g.2 marksii.Determine the maximum possible area of the triangle OAP′.2 marksEND OF QUESTION AND ANSWER BOOK
Victorian Certificate of Education2021MATHEMATICAL METHODSWritten examination 1FORMULA SHEETInstructionsThis 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 electronicdevices into the examination room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2021
MATHMETH EXAM2Mathematical Methods formulasMensurationarea of a trapezium1 a b h2volume of a pyramid1Ah3curved surface areaof a cylinder2π rhvolume of a sphere4 3πr3volume of a cylinderπ r 2harea of a triangle1bc sin A2volume of a cone1 2πr h3Calculusd nx nx n 1dx x dx n 1 xn 1dn 1(ax b) n an ax bdx 1(ax b) n 1 c, n 1a (n 1)d axe ae axdxe1n(ax b) n dxaxdxc, n 11 axe ca1dx log e ( x) c, x 0xdsin (ax) a cos (ax)dx1sin (ax)dx cos (ax) cadcos (ax) a sin (ax)dxcos (ax)dx a sin (ax) c1dlog e ( x) xdx1daa sec 2 (ax)tan (ax) 2dxcos (ax)product ruleddvduuv u vdxdxdxchain ruledy dy du dx du dxquotient ruledudv uvd udxdx dx v v2
3MATHMETH EXAMProbabilityPr(A) 1 – Pr(A′)Pr(A B) Pr(A B) Pr(A) Pr(B) – Pr(A B)Pr A BmeanPr Bvar(X) σ 2 E((X – µ)2) E(X 2) – µ2varianceµ E(X)Probability distributiondiscretePr(X x) p(x)continuousPr(a X b)MeanVarianceσ 2 (x – µ)2 p(x)µ x p(x)baf ( x)dx x f ( x)dx2 ( x ) 2 f ( x)dx Sample proportionsP̂ Xnstandarddeviationmeanˆ) sd ( Pp (1 p )napproximateconfidenceintervalE(P̂ ) p ˆp (1 ˆp ) , p z ˆp (1 ˆp ) ˆp zˆ nn END OF FORMULA SHEET
5 2021 MATHMETH EXAM 1 TURN OER D O N O T W R I T E I N T H I S A R E A D O N O T W R I T E I N T H I S A R E A Question 5 (4 marks) Let f: R R, f (x) x2 – 4 and g: R R, g(x) 4(x – 1)2 – 4. a. The graphs of f and g have a common horizontal axis intercept at (2, 0). Find the coordinates of the othe
