UNIT 2 MATHEMATICAL METHODS 2013 MASTER CLASS PROGRAM WEEK 11 EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS

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1 UNIT MATHEMATICAL METHODS 01 MASTER CLASS PROGRAM WEEK 11 EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT SECTION 1 MULTIPLE CHOICE QUESTIONS QUESTION 1 QUESTION QUESTION Answer is B Answer is A Answer is C QUESTION 4 QUESTION 5 QUESTION Answer is E Answer is C (Remember to expand the brackets) Answer is C The School For Excellence 01 Unit Master Classes Maths Methods Exam Page 1

2 QUESTION 7 Answer is E cos cos sin 1 1 sin cos 4 5 As the angle lies in the nd Quadrant, cos 4 5 cos Is Negative. QUESTION 8 Answer is B x cos 0 a x cos a Basic Angle: 1 Cos x x n or n a a x 5 x 7 n or n a a 5a 7a x na or x na n 1: 5 a 7a x na or 7 a 5a x na n 1: 5 a 17a x na or 7 a 19a x na QUESTION 9 Answer is C Curves must be the mirror image of one another in the line y x. The School For Excellence 01 Unit Master Classes Maths Methods Exam Page

3 QUESTION 10 Answer is B f will have an inverse function if it is a 1-to-1 function. Use technology to draw a graph of f. The smallest value of x at which f has a turning point is x Therefore, f is a 1-to-1 function over the domain x QUESTION 11 Answer is D QUESTION 1 Answer is C Container is filled from the bottom up. Initial part of container height of water increases by the same amount per unit time. Therefore h vs t graph is an oblique line. Answer is B or C. Middle part of container is becoming wider. The height occupied by water will decrease per unit time. Gradient of curve is decreasing, therefore, shape of graph resembles: Top part of container is becoming narrower. The height occupied by water will increase per unit time. Gradient of curve is increasing, therefore, shape of graph resembles: QUESTION 1 Answer is B The School For Excellence 01 Unit Master Classes Maths Methods Exam Page

4 QUESTION 14 Answer is B dy dx ax b dy Substitute 4 dx Substitute (1, ) into when 1 y ax x : 4 a b. (1) bx : a b. () Solve equations (1) and 9) simultaneously: a and b 0. QUESTION 15 Answer is D QUESTION 1 Answer is D You cannot differentiate at the end points of a domain. QUESTION 1 QUESTION 17 Answer is A Answer is A The function g (x) is obtained from f (x) through the following three transformations: A dilation from the x axis (or parallel to the y axis) by a factor of 5 which results in the minimum value being at (, 5). A reflection in the y axis which now means that the minimum is at (, 5). Finally there is a translation of 1 unit to the right which results in the minimum now being at ( 1, 5) The School For Excellence 01 Unit Master Classes Maths Methods Exam Page 4

5 QUESTION 18 Answer is E If then dy dx =x 8x +7 y x x x At x, dy = dx Gradient of tangent is. Therefore, gradient of the normal is 1. As y 1 at x : Equation of the normal is: y 1= 1 (x ) y = x + y x 5 0 QUESTION 19 Answer is C QUESTION 0 Answer is E The School For Excellence 01 Unit Master Classes Maths Methods Exam Page 5

6 QUESTION 1 Answer is E QUESTION Answer is B The School For Excellence 01 Unit Master Classes Maths Methods Exam Page

7 SECTION EXTENDED ANSWER QUESTIONS QUESTION 1 a. (i) (ii) b. (i) 1 mark for letting the derivative equation = 0. No marks for derivative. 1 mark for the correct answer in exact form. The School For Excellence 01 Unit Master Classes Maths Methods Exam Page 7

8 (ii) c. d. This question can only be solved using technology. (i) Maximum value can occur at the stationary point or end point. When x 0, y. 44 When x 4, y. 45 Stationary point: x 1. 70, y 9. 8 Maximum value occurs at the end point and is equal to y. 45. (ii) Find f ( x) 0 : x. 94 (iii) f '() The School For Excellence 01 Unit Master Classes Maths Methods Exam Page 8

9 QUESTION a. (i) B represents the amplitude and reflection in the X axis (if it exists). Amplitude = half the distance between the maximum and minimum values 1.5 m No reflection exists, therefore, b (ii) Substitute the point ( 0, ) into the equation: a 1.5 cos a (iii) Period n hours b. Convert minutes to hours. Find h when t hrs: () h () cos h () cos 5.9 m 1.4 t c. (i) cos t 1.5cos t cos Equating with Acos( Bt) C results in A 1, B, C The School For Excellence 01 Unit Master Classes Maths Methods Exam Page 9

10 (ii) 1st Quadrant Angle: Cos 1 (0.5) Solutions are to lie in the quadrants where cos is negative i.e. Quadrants and. t, 1.4 t 4, 1.4 t 1.4, t 4.1, 8. t 4.1, 8.7 hours. (iii) The tide is.75 m high 4.1 hrs and 8.7 hrs after high tide. t 7 d. (i) h cos cos. 1m (ii) Let x y (, 5.9) 1, 1 Let x y (7,.1), y y m 0.4 m / hr x x 7 1 The School For Excellence 01 Unit Master Classes Maths Methods Exam Page 10

11 QUESTION a. x b. f ( x) 00e 500 c. (i) y x 00e 500 x 500 y e 00 x 500 y log e loge e 00 x 500 log e y f 1 ( x) log x e 00 Domain of f (x) ; x R Range of f (x) ; { y : y 500} Domain of f 1 ( x ) ; { x : x 500} Range of f 1 ( x 1 ) ; f ( x) R (ii) x' 0 x 0 y' 0 1 y 1 (iii) x ' y ' x ' y ' The School For Excellence 01 Unit Master Classes Maths Methods Exam Page 11

12 QUESTION 4 a. (i) (ii) b. Simplify the expression by substituting any known/given data: When x 0, the derivative 1 f : i.e. 0 1 x x x 0 b f 1 0 b 1 b x x Find an antiderivative: x c f x x x x c Solve for c by substituting the given values of x and y into the equation describing f ( x): 0 f i.e. When x 0, y c c The equation describing x f x is x x. The School For Excellence 01 Unit Master Classes Maths Methods Exam Page 1

13 c. (i) The School For Excellence 01 Unit Master Classes Maths Methods Exam Page 1

14 (ii) The School For Excellence 01 Unit Master Classes Maths Methods Exam Page 14

15 QUESTION 5 (a) (b) (c) (d) (e) (f) The School For Excellence 01 Unit Master Classes Maths Methods Exam Page 15

UNIT 2 MATHEMATICAL METHODS 2015 MASTER CLASS PROGRAM WEEK 11 EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS

UNIT 2 MATHEMATICAL METHODS 2015 MASTER CLASS PROGRAM WEEK 11 EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS UNIT MATHEMATICAL METHODS 015 MASTER CLASS PROGRAM WEEK 11 EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT WWW.TSFX.COM.AU/MC-UPDATES SECTION 1 MULTIPLE CHOICE QUESTIONS Where possible, students