Instructions for Section 2

Transcription

1 200 MATHMETH(CAS) EXAM 2 0 SECTION 2 Instructions for Section 2 Answer all questions in the spaces provided. In all questions where a numerical answer is required an eact 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 a. Part of the graph of the function g: ( 4, ) R, g() = 2 log e ( + 4) + is shown on the aes below O i. Find the rule and domain of g, the inverse function of g. ii. On the set of aes above sketch the graph of g. Label the aes intercepts with their eact values. SECTION 2 Question continued

2 200 MATHMETH(CAS) EXAM 2 iii. Find the values of, correct to three decimal places, for which g () = g(). iv. Calculate the area enclosed b the graphs of g and g. Give our answer correct to two decimal places = 0 marks SECTION 2 Question continued TURN OVER

3 200 MATHMETH(CAS) EXAM 2 2 b. The diagram below shows part of the graph of the function with rule f () = k log e ( + a) + c, where k, a and c are real constants. The graph has a vertical asmptote with equation =. The graph has a -ais intercept at. The point P on the graph has coordinates (p, 0), where p is another real constant. 0 P(p, 0) O i. State the value of a. ii. Find the value of c. iii. 9 Show that k log ( p ). e SECTION 2 Question continued

4 3 200 MATHMETH(CAS) EXAM 2 iv. Show that the gradient of the tangent to the graph of f at the point P is 9 ( p )log ( p ). e v. If the point (, 0) lies on the tangent referred to in part b.iv., find the eact value of p = 7 marks Total 7 marks SECTION 2 continued TURN OVER

5 203 MATHMETH (CAS) EXAM 2 22 Question 4 (6 marks) Part of the graph of a function g: R R, g() = is shown below. C = g() A O B a. Points B and C are the positive -intercept and -intercept of the graph of g, respectivel, as shown in the diagram above. The tangent to the graph of g at the point A is parallel to the line segment BC. i. Find the equation of the tangent to the graph of g at the point A. 2 marks ii. The shaded region shown in the diagram above is bounded b the graph of g, the tangent at the point A, and the -ais and -ais. Evaluate the area of this shaded region. 3 marks SECTION 2 Question 4 continued

6 MATHMETH (CAS) EXAM 2 b. Let Q be a point on the graph of = g(). Find the positive value of the -coordinate of Q, for which the distance OQ is a minimum and find the minimum distance. 3 marks The tangent to the graph of g at a point P has a negative gradient and intersects the -ais at point D(0, k), where 5 k 8. C D(0, k) = g() P O B c. Find the gradient of the tangent in terms of k. 2 marks SECTION 2 Question 4 continued TURN OVER

7 203 MATHMETH (CAS) EXAM 2 24 d. i. Find the rule A(k) for the function of k that gives the area of the shaded region. 2 marks ii. Find the maimum area of the shaded region and the value of k for which this occurs. 2 marks iii. Find the minimum area of the shaded region and the value of k for which this occurs. 2 marks END OF QUESTION AND ANSWER BOOK

8 205 MATHMETH (CAS) EXAM 2 2 SECTION 2 Answer all questions in the spaces provided. Instructions for Section 2 In all questions where a numerical answer is required, an eact 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 (9 marks) 2 Let f : R R, f( ) = ( 2) ( 5 ). The point P 4, is on the graph of f, as shown below. 5 5 The tangent at P cuts the -ais at S and the -ais at Q. 4 S O P 4, 5 Q a. Write down the derivative f () of f (). mark SECTION 2 Question continued

9 3 205 MATHMETH (CAS) EXAM 2 b. i. Find the equation of the tangent to the graph of f at the point P, 4. 5 mark ii. Find the coordinates of points Q and S. 2 marks c. Find the distance PS and epress it in the form b c, where b and c are positive integers. 2 marks SECTION 2 Question continued TURN OVER

10 205 MATHMETH (CAS) EXAM S O P 4, 5 Q d. Find the area of the shaded region in the graph above. 3 marks SECTION 2 continued

11 2 205 MATHMETH (CAS) EXAM 2 Question 4 (9 marks) An electronics compan is designing a new logo, based initiall on the graphs of the functions f( ) = 2sin( ) and g( ) = sin( 2), for 0 2π. 2 These graphs are shown in the diagram below, in which the measurements in the and directions are in metres. 2 O The logo is to be painted onto a large sign, with the area enclosed b the graphs of the two functions (shaded in the diagram) to be painted red. a. The total area of the shaded regions, in square metres, can be calculated as a sin( ) d. What is the value of a? π 0 mark SECTION 2 Question 4 continued TURN OVER

12 205 MATHMETH (CAS) EXAM 2 22 The electronics compan considers changing the circular functions used in the design of the logo. Its net attempt uses the graphs of the functions f () = 2sin() and h ( ) = sin( 3 ), for 0 2π. 3 b. On the aes below, the graph of = f () has been drawn. On the same aes, draw the graph of = h(). 2 marks 2 3 O c. State a sequence of two transformations that maps the graph of = f () to the graph of = h(). 2 marks SECTION 2 Question 4 continued

13 MATHMETH (CAS) EXAM 2 The electronics compan now considers using the graphs of the functions k() = msin() and q ( ) = sin( n), where m and n are positive integers with m 2 and 0 2π. n d. i. Find the area enclosed b the graphs of = k() and = q() in terms of m and n if n is even. Give our answer in the form am + b n 2, where a and b are integers. 2 marks ii. Find the area enclosed b the graphs of = k() and = q() in terms of m and n if n is odd. Give our answer in the form am + b n 2, where a and b are integers. 2 marks SECTION 2 continued TURN OVER

14 205 MATHMETH (CAS) EXAM 2 24 Question 5 (5 marks) t 2t a. Let St () = 2e3 + 8e 3, where 0 t 5. i. Find S(0) and S(5). mark ii. The minimum value of S occurs when t = log e (c). State the value of c and the minimum value of S. 2 marks iii. On the aes below, sketch the graph of S against t for 0 t 5. Label the end points and the minimum point with their coordinates. 2 marks S O t SECTION 2 Question 5 continued

15 MATHMETH (CAS) EXAM 2 iv. Find the value of the average rate of change of the function S over the interval [0, log e (c)]. 2 marks t ( ) 2t Let V : [0, 5] R, V() t = de3 + 0 d e 3, where d is a real number and d ( 0, 0). b. If the minimum value of the function occurs when t = log e (9), find the value of d. 2 marks c. i. Find the set of possible values of d such that the minimum value of the function occurs when t = 0. 2 marks ii. Find the set of possible values of d such that the minimum value of the function occurs when t = 5. 2 marks SECTION 2 Question 5 continued TURN OVER

16 205 MATHMETH (CAS) EXAM 2 26 d. If the function V has a local minimum (a, m), where 0 a 5, it can be shown that 2 ( ) m = k d 3 0 d 2 3. Find the value of k. 2 marks END OF QUESTION AND ANSWER BOOK

17 206 MATHMETH EXAM 2 20 Question 4 (2 marks) a. Epress b in the form a +, where a and b are non-zero integers. 2 marks b. Let f : R\{ 2} R, f( )= + 2. i. Find the rule and domain of f, the inverse function of f. 2 marks ii. Part of the graphs of f and = are shown in the diagram below. 2 = = f () 2 O 2 2 Find the area of the shaded region. mark SECTION B Question 4 continued

18 2 206 MATHMETH EXAM 2 iii. Part of the graphs of f and f are shown in the diagram below. 2 = f () = f () 2 O 2 2 Find the area of the shaded region. mark SECTION B Question 4 continued TURN OVER

19 206 MATHMETH EXAM 2 22 c. Part of the graph of f is shown in the diagram below. 2 P(c, d) O = f () 2 2 The point P(c, d) is on the graph of f. Find the eact values of c and d such that the distance of this point to the origin is a minimum, and find this minimum distance. 3 marks SECTION B Question 4 continued

20 MATHMETH EXAM 2 k + Let g : ( k, ) R, g ( )=, where k >. + k d. Show that < 2 implies that g ( ) < g ( 2 ), where ( k, ) and 2 ( k, ). 2 marks SECTION B Question 4 continued TURN OVER

21 206 MATHMETH EXAM 2 24 e. i. Let X be the point of intersection of the graphs of = g () and =. Find the coordinates of X in terms of k. 2 marks ii. Find the value of k for which the coordinates of X are,. 2 marks 2 2 SECTION B Question 4 continued

22 MATHMETH EXAM 2 iii. Let Z(, ), Y(, ) and X be the vertices of the triangle XYZ. Let s(k) be the square of the area of triangle XYZ. X = g() = Y O Z Find the values of k such that s(k). 2 marks SECTION B Question 4 continued TURN OVER

23 206 MATHMETH EXAM 2 26 f. The graph of g and the line = enclose a region of the plane. The region is shown shaded in the diagram below. = = g() O Let A(k) be the rule of the function A that gives the area of this enclosed region. The domain of A is (, ). i. Give the rule for A(k). 2 marks ii. Show that 0 < A(k) < 2 for all k >. 2 marks END OF QUESTION AND ANSWER BOOK

24 2008 MATHMETH(CAS) EXAM 2 4 Question 2 The diagram below shows part of the graph of the function f : R + R, f () = 7. C A O b a The line segment CA is drawn from the point C(, f ()) to the point A(a, f (a)) where a >. a. i. Calculate the gradient of CA in terms of a. ii. At what value of between and a does the tangent to the graph of f have the same gradient as CA? + 2 = 3 marks SECTION 2 Question 2 continued

25 MATHMETH(CAS) EXAM 2 e b. i. Calculate f( ) d. ii. Let b be a positive real number less than one. Find the eact value of b such that f( ) d is equal to 7. b + 2 = 3 marks c. i. Epress the area of the region bounded b the line segment CA, the -ais, the line = and the line = a in terms of a. ii. For what eact value of a does this area equal 7? SECTION 2 Question 2 continued TURN OVER

26 2008 MATHMETH(CAS) EXAM 2 6 iii. Using the value for a determined in c.ii., eplain in words, without evaluating the integral, wh a f( ) d < 7. Use this result to eplain wh a < e. mn d. Find the eact values of m and n such that f( ) d = 3 and f( ) d = 2. m n = 5 marks 2 marks Total 3 marks SECTION 2 continued