Maths Methods 3 2014 – Example Questions The MTM315114 course includes some material not previously not covered in the MTM315109 course. The following are some example questions designed to indicate the scope of these new materials. Note that this is not an exhaustive list. Marks listed are indicative only. Note: These questions are provided without working out space to save room. All exam questions will have sufficient room to provide an answer. C Standard Questions – typically Part 1 Critieria 4 – Demonstrate an understanding of polynomial, hyperbolic, exponential and logarithmic functions Question 1 (2 marks) Determine the equation of the relation show. (3 marks) Sketch the relationship showing all intercepts (x – 3)2 + (y + 7)2 = 25. Question 5 (3 marks) The curve C is defined by (x+1)2 + (y–3)2 = 36 (i) State the domain and range of C. (ii) Is C a function? Justify. Critieria 5 – Demonstrate an understanding of circular functions Question 6 (2 marks) π
Given that sinθ = 0.1 and 0 < θ < 2 then use an appropriate trigonometric identity to determine the exact value of cos2θ. Question 7 Question 2 Question 4 (2 marks) On the same graph draw the inverse to the relation shown. 3
(3 marks) π
Given that sinθ = 5 and 0<θ <2 then use an appropriate trigonometric identity to determine the exact value of sin2θ. Question 8 (2 marks) Use an appropriate trigonometric identity to determine π
π
π
π
the exact value of sin 4 cos12+ cos4 sin12. Question 9 (3 marks) π
(i) Express 12 in degrees. (ii) Use an appropriate trigonometric identity to determine the exact value of sin12. π
Question 10 11π
Question 3 (3 marks) Determine the centre and radius of the circle with formula x2 + y2 – 8x + 2y – 32 = 0. (3 marks) (i) Express 12 in degrees. (ii) Use an appropriate trigonometric identity to 11π
determine the exact value of tan 12 . A & B Standard Questions – typically Part 2 Critieria 4 – Demonstrate an understanding of polynomial, hyperbolic, exponential and logarithmic functions Question 11 (4 marks) (4 marks) (3 marks) (4 marks) (4 marks) Determine the centre and radius of the circle with formula 3x2 + 3y2 – 3x + 2y + 1 = 0. Question 12 Determine algebraically the points of intersection of the circle x2 + y2 = 4 and the line y = 2x + 2. Question 13 Write down the function for the lower semicircle with centre (3,-­‐2) and radius 4. Question 14 The relationship (x + 3)2 + (y – 4)2 = 4 shown here undergoes the following series of transformations: translation 4 units down followed by a dilation by a factor of 2 both vertically and horizontally followed by a reflection in the y–axis. Determine the equation of the resultant curve. Note: Intermediate equations need not be written down. Critieria 5 – Demonstrate an understanding of circular functions Question 15 3
5
Given than sinθ = 5 and cosφ = 13 use an appropriate trigonometric identity to determine cos(θ–φ). Question 16 (4 marks) (4 marks) 1
Use an appropriate trigonometric identity to solve for 0≤x≤2π: 2 – 2sinxcosx = 0. Question 17 Use the trigonometric identity tan2x = 2tanx
π
2 and the exact value tan = 1 to: 1 – tan x
4
π
π
(i) show algebraically that tan2 + 2tan –1 = 0 and, 8
8
π
(ii) hence algebraically determine the exact value of tan . 8