DIRECTIONS TO CANDIDATES
* Attempt ALL questions.
* ALL questions are of equal value.
* All necessary working should be shown in every question. Marks may be deducted for careless or badly arranged work.
* Standard integrals are printed at the end of the paper.
* Board approved calculators may be used.
* Answer each question on a separate page.
* You may ask for extra paper if you need it.
QUESTION 1. Use a separate page.
(a) The mean distance between the Sun and the Earth is 149 600 000 km. Express this distance in scientific notation correct to 3 significant figures. (1 mk)
(b) Factorise completely: 16a3 54b3 (1 mk)
(c) Simplify: |-4| - |6| (1 mk)
(d) If T = 2π √ L / g find T, correct to 1 decimal place when L = 1.5 and g = 9.8 (1 mk)
(e) Solve the equation: 4(p-2)2 = 4 (2 mks)
(f) Express 0.27 recurring i.e. 0.277777.... as a fraction in its simplest form (2 mks)
(g) Represent the inequation |x - 2| > 1 on the number line (2 mks)
(h) A bookseller buys books from a wholesaler and then retails them in his shop at a mark-up of 50% on the wholesale price. A teacher is allowed a discount of 10% on the retail price and pays $32-94 for a book. What was the wholesale price of the book? (2 mks)
QUESTION 2. Use a separate page.
ABCD is a parallelogram. Three of the vertices are A(0,0), B(4,2), C(6,4).
(a) Draw a number plane and mark the points A, B and C. (1 mk)
(b) Determine the co-ordinates of D, the fourth vertex of the parallelogram. (1 mk)
(c) Find the length AB. (2 mks)
(d) Determine the gradients of the sides AB and BC. (2 mks)
(e) Show that the mid-points of AC and BD are both (3,2). (1 mk)
(f) Find the equation of the line AB. (1 mk)
(g) Find the perpendicular distance of (3,2) to the line AB. (2 mks)
(h) Find the area of the parallelogram ABCD. (2 mks)
QUESTION 3. Use a separate page.
(a) Differentiate
(i) 3√ x2 (1 mk)
(ii) 2e3x (1 mk)
(iii) 2x logex (2 mks)
(b) Given that tan x = sin x / cos x , use the quotient rule to show d/dx tan x = sec2x (2 mks)
(c) Find
(i) 0∫π sin x dx (2 mks)
(ii) 0∫2 e2x dx (2 mks)
(iii) ∫ x/(x2 + 4) dx (2 mks)
QUESTION 4. Use a separate page.
(a) ABCD is a square. E, F, G, H are the mid-points of AB, BC, CD and DA respectively.
Show that EFGH is a square. (4 marks)
(b) ABC is a triangle and DE cuts the sides AB and AC in the ratios shown.
(i) Show that DE || BC. (1 mk)
(ii) Show ΔADE ||| ΔABC (1 mk)
(iii) Find the length of BC (1 mk)
(iv) Use the cosine rule to determine the angle DAE. (2 mks)
(c) The numbers 12, x, 6 are in arithmetic sequence.
(i) Determine the value of x. (1 mk)
(ii) Find the tenth term of the sequence. (1 mk)
(iii) Find the sum of the first 10 terms. (1 mk)
QUESTION 5. Use a separate page.
(a) Kulsoom and Shakeeba were playing a game of snakes and ladders. They each rolled the die to determine who would start first.
(i) What is the probability that they each roll the same number? (1 mk)
(ii) What is the probability that Shakeeba rolls a higher number than Kulsoom? (2 mks)
(iii) What is the probability that they each roll the same number twice in a row? (1 mk)
(b) The region of the graph y = x2 4 lying between x = -2 and x = 2 is rotated around the x-axis.
Find the volume of the solid so formed. (5 mks)
(c) Solve the equation 2 sin 2x = 1 for 0 ≤x ≤ 2π (3 mks)
QUESTION 6. Use a separate page.
(a) Sophie was standing on a cliff that was 60 metres high. Emma was in a boat in the water below the cliff and Sophie noted that the angle of depression of Emma from the top of the cliff was 200. Emma rowed the boat 200 metres towards the cliff and Sophie noted that the angle of depression was now 400.
(i) How high is the cliff? (3 mks)
(ii) How far was Emma originally from the base of the cliff? (1 mk)
(b) The diagram shows the graph of a piecemeal function of x.
Use the graph to answer the following questions.
(i) Evaluate 0∫10 f(x) dx. Justify your answer. (2 mks)
(ii) For what value of s, 0 ≤s ≤10, does 0∫s have the greatest value? (1 mk)
(iii) For what value of s, 0 ≤s ≤10, does 0∫s have the least value? (1 mk)
(c) An observer in a lighthouse L sees a ship S at a bearing of 0630T at a distance of 9 nautical miles and a yacht Y at a bearing of 3330T and a distance of 12 nautical miles.
(i) Draw a diagram showing the information supplied. (1 mk)
(ii) Find the distance between the ship S and the yacht Y. (1 mk)
(iii) Find the bearing of the ship S from the yacht Y to the nearest degree. (2 mks)
QUESTION 7. Use a separate page.
(a) The number of undecayed atoms of a radioactive isotope is given by N = N0e-kt where N = number of undecayed atoms remaining N0 = original number of undecayed atoms
t = time
k = decay constant. Each isotope has its own decay constant.
A sample of a certain isotope of Thorium contains 1010 atoms. After 10 hours it contains 7.573 x 109 atoms.
(i) Determine the decay constant for thorium. (2 mks)
(ii) Determine the time after which only half of the original thorium atoms remain. 2 mks)
(iii) Determine the time it would take for only one thorium atom to remain. (2 mks)
(b) If α and β are the roots of the equation 2x2 5x + 3 = 0, find the value of
(i) 1/α + 1/β (2 mks)
(ii) α2 + β2 (2 mks)
(iii) ( α - β)2 (2 mks)
QUESTION 8. Use a separate page.
(a) The rate of flow of water from a tap into a bucket is given by R = 3(t+10) where: R = rate of flow in millilitres per second.
t = time in seconds
(i) What is the rate of flow after 10 seconds? (1 mk)
(ii) Find a formula for the volume V of water in the bucket assuming it was empty when t = 0. (2 mks)
(iii) How long will it take to fill an 18 litre bucket? (2 mks)
(b) y = 2x + 4 is a tangent to the parabola y = ax2 + b, touching it at P(1, 6)
(i) What is the gradient of the parabola at the point p(1, 6)? (1 mk)
(ii) Find a and b for the parabola. (2 mks)
(iii) If the distance between the point P(1, 6) and the focus is d, what is the perpendicular distance, in terms of d, between the focus and the directrix? (1 mk)
(c) Jenalle withdrew balls at random from a bag containing 8 white balls and 4 black balls. After withdrawing each ball she would record its colour and replace it in the bag. She agreed to stop after she had withdrawn a black ball on two occasions.
(i) What is the probability that it takes her three draws to withdraw the two black balls? (2 mks)
(ii) What is the probability that she will not have withdrawn the second black ball after 5 draws? (1 mk)
QUESTION 9. Use a separate page.
(i) Find the horizontal asymptote for the curve
(1 mk)(ii) Explain why there is no vertical asymptote. (1 mk)
(iii) Find any turning points for the curve x/(x2 + 1) and determine their nature. (4 mks)
(iv) Find any points of inflexion for the curve x/(x2 + 1). (3 mks)
(v) Sketch the curve . (3 mks)
QUESTION 10. Use a separate page.
(a) (i) Sketch the function y = ln (2x + 1), showing its essential features. (3 mks)
(ii) Use Simpsons rule with 5 function values to find an approximation of (3 mks)
(c) Nathan borrowed $200 000 from the bank to buy an apartment. He agreed to pay back the loan over 20 years at an interest rate of 6% per annum, reducible monthly.
(i)Show that Nathans monthly repayments are $1433 (to nearest dollar) (2 mks)
(ii) How much of the principal has Nathan paid back after 10 years? (2 mks)
(iii) If Nathan increases his repayments by $50 per month, how long will it take him to pay off the loan? (2 mks)