MATHEMATICAL METHODS UNITS 1 & 2 TRIAL EXAMINATION 1

Transcription

1 THE HEFFERNAN GROUP P.O. Bo 1180 Surrey Hills North VIC 17 Phone Fa info@theheffernangroup.com.au MATHEMATICAL METHODS UNITS 1 & TRIAL EXAMINATION Reading Time: 15 minutes Writing time: 1hour Instructions to students This eam consists of 10 questions. All questions should be answered in the spaces provided. There is a total of 40 marks available. If a question requires a numerical answer then an eact value must be given unless a decimal approimation is specifically asked for. Where more than one mark is allocated to a question working must be shown. Students may not bring any notes or any calculators into this eam. Diagrams in this eam are not to scale ecept where otherwise stated. A formula sheet can be found on page 11 of this eam. This paper has been prepared independently of the Victorian Curriculum and Assessment Authority to provide additional eam preparation for students. The publication is in no way connected with or endorsed by the Victorian Curriculum and Assessment Authority. THE HEFFERNAN GROUP 017 These eam questions and solutions are licensed on a non transferable basis to the purchasing school. They may be copied by the school which has purchased them. This license does not permit distribution or copying of these eam questions and solutions by any other party.

2 Question 1 (5 marks) The graph of the function with rule f () = ( ) is shown below. y 4 y = f () O a. Write down i. the domain of f. 1 mark ii. the range of f. 1 mark b. On the set of aes above, sketch the graph of f 1, the inverse function of f. 1 mark c. Find the rule for f 1. marks THE HEFFERNAN GROUP 017 Maths Methods Units 1 & Trial Eam 1

3 Question (4 marks) Solve the following for. a = 0 marks b. log 10 () log 10 (5)+ log 10 () = 0 marks Question ( marks) Solve sin()+1= 0 for [0, π ]. THE HEFFERNAN GROUP 017 Maths Methods Units 1 & Trial Eam 1

4 4 Question 4 ( marks) a. Find the derivative of ( +1) with respect to. 1 mark 8 b. Let f ( ) = 4 +, 0. Evaluate f '(4 ). marks THE HEFFERNAN GROUP 017 Maths Methods Units 1 & Trial Eam 1

5 5 Question 5 (5 marks) a. Find the antiderivative of mark 1 b. Given that g '( ) = and g() = 5, find g( ). marks 1 c. Evaluate d. marks THE HEFFERNAN GROUP 017 Maths Methods Units 1 & Trial Eam 1

6 6 Question 6 (4 marks) The graph of y = a b +, where a and b are constants, passes through the point (1,) and has a stationary point located at the point where = 1. Find the values of a and b. THE HEFFERNAN GROUP 017 Maths Methods Units 1 & Trial Eam 1

7 THE HEFFERNAN GROUP 017 Maths Methods Units 1 & Trial Eam 1 7 Question 7 ( marks) The curve y = 1, > 0 undergoes a sequence of transformations defined by T such that = y y T R R T 0 0, :. a. Describe one of the transformations that the curve undergoes. 1 mark b. Find the equation of the image of the curve after it has undergone the sequence of transformations. marks

8 8 Question 8 (5 marks) The graph of y = + +8 is shown below. y y = O a. Find dy d. 1 mark b. Find the values of for which the gradient of the graph is positive. marks c. There are two points on the graph which have a gradient of 1. Find the -coordinates of these two points. marks THE HEFFERNAN GROUP 017 Maths Methods Units 1 & Trial Eam 1

9 9 Question 9 (4 marks) For events A and B, Pr(A) = 0.75, Pr(B') = 0.5 and Pr(A' B) = Find a. Pr(A B). marks b. Pr(A B). 1 mark c. Pr ( A B). 1 mark THE HEFFERNAN GROUP 017 Maths Methods Units 1 & Trial Eam 1

10 10 Question 10 (5 marks) A plant trellis in the shape of a cylinder consists of four circles of wire and five straight lengths of wire as shown below. h The radius of the cylinder is r metres and the height is h metres. The volume of the cylinder must equal 0.8 cubic metres. a. Find an epression for h in terms of r. 1 mark r b. The total length of wire used to make the trellis is L metres. Show that L = 8πr + 4 πr. marks c. The graph of the function in part b. has eactly one turning point and it is a minimum. Find the radius of the trellis if the total length of wire used to make it is a minimum. marks THE HEFFERNAN GROUP 017 Maths Methods Units 1 & Trial Eam 1

11 11 Mathematical Methods Units 1 & formulas Functions and graphs distance formula d = ( ) + ( y y 1 1) 1 + y1 + y midpoint formula midpoint =, Straight line graphs general equation y = m + c equation through point, ) y y = m ) ( 1 y 1 1 ( 1 gradient m = y y 1 1 Mensuration circumference of a circle π r area of a circle π r volume of a sphere 4 π r volume of a cylinder r π h Calculus d d n n 1 ( ) = n n 1 n+ 1 d = n c, n 1 Probability Pr( A) = 1 Pr( A') Pr( A B) = Pr( A) + Pr( B) Pr( A B) Pr( A B) Pr( A B) = Pr( B) THE HEFFERNAN GROUP 017 Maths Methods Units 1 & Trial Eam 1