~UTS. University of Technology, Sydney TO BE RETURNED AT THE END OF THE EXAMINATION. THIS PAPER MUST NOT BE REMOVED FROM THE EXAM CENTRE.

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1 ~UTS Cover Type B University of Technology, Sydney TO BE RETURNED AT THE END OF THE EXAMINATION. THIS PAPER MUST NOT BE REMOVED FROM THE EXAM CENTRE. SURNAME: FIRST NAME: STUDENTNUMBER: COURSE: AUTUMN SEMESTER 2010 SUBJECT NAME FOUNDATION MATHEMATICS SUBJECT NO DAYIDATE Saturday 19 June 2010 TIME ALLOWED STARTIEND TIME: Three hours plus ten minutes reading time 9:30am / 12:40pm NOTESIINSTRUCTIONS TO CANDIDATES: Attempt all questions All questions are of equal value Calculators MAY be used Answer each question in a separate booklet CLEARLY indicate the question number on the front of the booklet Examiner: Dr B J Moore Assessor: Dr M Coupland

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3 Question 1 ( = 20 marks) Page One x x (a) Solve for x: = 1, 2 3 (b) The quantities A, B, and C are related by the formula =-, ABC Find the exact value of B if A = 3 and C = 5, (c) Find the equation ofthe tangent at the point (-1,5) on the curve y = 4x 2 - x 3, (d) Solve, by factorisation, the quadratic equation 5x x - 6 =0, (e) Differentiate f(x) =In(3x - 4); Differentiate f (x) = ~; x+1 (iii) Evaluate 07C16 sec 2 (2x), 1 (f) Solve, to one decimal place, the equation In(2x)= 5, Express in terms of lnx, lny and Inz : Over!...

4 Page Two Question 2 ( = 20 marks) Begin a new booklet and write Q2 on the cover (a) A garden bed is rectangular in shape, having width x + 1 metres, and length 2x + 3 metres. Find x (to 2 decimal places) ifthe garden bed has an area of 11 sq. metres. (b) The diagram shows a circle of radius r cm and a sector in which the angle is e radians. Find a formula in terms of rand e for the ratio A. Here A is the area I of the sector, and I is the length ofthe arc ofthe sector. Find A if I is 5 cm, and r is 15 cm. F d 1 x 2 -x-6 (c) In Im--- x--+3 X - 3 (d) Consider the geometric series 5 + 1Ox + 20x x For what values of x does this series have a limiting sum? Find the value of x ifthe limiting sum ofthis series is 100. (e) Find, without a calculator, the exact value of tan(150 ). Solve the equation 2 sin(e) - J3 = 0 for 0 0 < e < (f) Find the amplitude and period ofy = 3cos(2x), and sketch the graph for one period. Overl...

5 Page Three Question 3 ( = 20 marks) Begin a new booklet and write Q3 on the cover e- 3x (a) Show that y = satisfies -f-22-15y = 0 d 2 d F d dy 'f 1 (b) m -1 y=--. x+2 Find a point (one only required) on the curve y = _1_ where the x+2 tangent is parallel to the line x + 9y =6. (c) You are told that, for the curve below, dy = 0 at a stationary point, and that d 2 ;, = 0 at a point of inflection. Consider the curve where a and b are real numbers. y = x 3 + ax 2 + bx - 7, Find the values of a and b ifthe given curve has a stationary point when x =2 and a point of inflection when x = -3. (d) Below is a sketch ofthe curve y = x 3-4x 2-3x + 25 in which A is a maximum turning point and B is a minimum turning point. Find the coordinates of A and B; (iii) (iv) Find those values of x for which the function increases; Find the x value of the point of inflection; Find those values of x for which the curve is concave up. yaxis Not to scale A B x axis Over/...

6 ... Page Four Question 4 ( = 20 marks) Begin a new booklet and write Q4 on the cover (a) Differentiate each ofthe following. f(x) = (lnxy; f(x) =e 2x sin- 1 x ; (iii) f(x) = 5x-2. 4x-1 (b) Between midnight and 3.00 am, if t is the number minutes after midnight, the depth of water, h metres, in an estuary is given by the formula h = 9 + 6sin( ;0). Find the rate, dh, at which depth of water is changing at 1.10 am (give your dt answer to three decimal places). (c) The variables Qand t are related by the formula where t and Q are real numbers. For which values of t is Q a real number? Find dq when t = 2, expressing your answer as a fraction. dt (d) The diagram below shows a sketch of a cardboard box that is open at the top. The base is a square of side x cm. The height is y cm. The volume of the box is 32 sq. cm. Show thatx 2 y = 32. open Write down a formula for the area of cardboard used to make the box. (iii) Use calculus to find the values of x and y that minimise the area in. xcm ycm Over!...

7 Page Five Question 5 ( = 20 marks) Begin a new booklet and write Q5 on the cover (a) Find the following: f 1 3x+2 ' (iii) fcos x(sinx + 1)3 (Hint: let u = sinx +1). (b) (c) For a particular curve, dy =4e 2x and the curve passes through the point (0,5). Find the equation ofthe curve. You are given the expansion cos(a + fj) =cosa cos fj - sin a sin fj. Without a calculator, find the exact value ofcos(75 ). Simplifycos(2A)cos A - sin(2a)sin A. (d) The diagram below shows a sketch (not to scale) ofthe curve y = x and the line y = 4x + 1 on the same set of axes. The curve and the line intersect at the point A in the first quadrant. yaxis x axis Show algebraically that the coordinates of A are (2,9). Find the area in the first quadrant that is bounded by the curve and the line. (iii) Find the volume when the area between curve y = x 3 +1 and the x axis from x = to x = 2 is rotated through one revolution about the x axis. Over/...

8 \ Page Six TABLE OF INTEGRALS AND FORMULAE f n+1 Xn = _x_+c, n+1 n :;t:-1 J~ =lnlxl+c fcos x = sin x + C fsinx = -cosx + C fsec 2 x = tan x + C J ~ = sm - + a 2 _x2 a. -l(x) C -- =!tan-1 (x) + C J a 2 +X2 a a Arc length when sector angle is () radians is 1= r() Area of sector when sector angle is () radians is