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.
MATHS TRIAL 2
Question 1. Use a separate page
(a) Find the value of 3.62/√(5.17 - 4.28) . Give your answer correct to 2 decimal places. (2 mks)
(b) Simplify: 5x 3x(1 2x) (2 mks)
(c) Differentiate 1/5x2 (2 mks)
(d) Solve | x+1| ≤ 5 (2 mks)
(e) Factorise: 3x2 x 2 (2 mks)
(f) Erin bought a DVD for $31.79 which included 10% GST on the original price. What was the original price? (2 mks)
Question 2. Use a separate page
The points A(1,1), B(6, 3) and C(5, 0) lie on the number plane.
(a) Draw a number plane with the points A, B and C clearly labeled. (1 mk)
(b) ABCD is a parallelogram. Write down the coordinates of D. (1 mk)
(c) Find the length of the line AB. (2 mks)
(d) Find the gradient of AB. (2 mks)
(e) Find the general form of the equation of AB. (2 mks)
(f) Find the perpendicular distance of the point C from the line AB. (2 mks)
(g) Find the area of parallelogram ABCD. (2 mks)
Question 3. Use a separate page
(a.) Differentiate the following:
(i) 1/√x (2 mks)
(ii) xe3x (2 mks)
(iii) 5 tan 2x (2 mks)
(b) With each application, an all purpose cleanser removes 65% of the grease and leaves the other 35%.
(i) Show that after 2 applications of the cleanser, 12¼ % of the grease remains. (1 mk)
(ii) How many applications of the cleanser are necessary to reduce the grease level to below 1% of its original value? (1 mks)
(c) Differentiate xex (1 mk)
Hence find the exact value of 0∫1ex(x + 1) dx (1 mk)
(d) Evaluate 0∫π/6 2 sec2dx (2 mks)
Question 4. Use a separate page
This question refers to the following diagram.
ABCD is a parallelogram with E the a point on the side DC. F is a point such that EF || BC and FC || EB. ∠ADE = 60o and ∠BED = 135o
i) Show that ΔEBC = ΔECF (2 mks)
ii) Show that ∠ECF = ∠ABE (2 mks)
iii) Calculate the value of ∠FEC (2 mks)
(b)
The population P, of a country town is declining at a rate that is proportional to its current population. The population at time t years is given by P = Poe-kt where Po is the original population and k is a constant. The population of the town was 60 000 on 1st Jan. 2000 and was 45 000 on 1st Jan. 2005.
(i) What is the value of k? (2 mks)
(ii) What will the population of the town be on 1st Jan. 2020? (2 mks)
(iii) Will the population approach a limiting sum as a G.P. or will it keep declining to less than one person? Justify your answer. (2 mks)
Question 5. Use a separate page
(a)
(i) Sketch the function y = 2 √(x2 - 16) (2 mks)
(ii) State its domain and range. (2 mks)
(b)
(i) Find log3 2 to 3 decimal places. (2 mks)
(ii) Hence solve 3x = 210 (1 mk)
(c) AOB is a sector of a circle with ∠AOB =30o.
Calculate the shaded area i.e. the area enclosed by the arc AB and the straight line AB. (2 mks)
(d) The diagram shows the cross section of a river with the depths shown in metres. The width of the river is 12 metres.
(i) Use Simpsons rule to find an approximate value for the area of cross section. (2 mks)
(ii) The river is flowing at 0.25 m/s. Calculate the volume of water that would flow past this section of the river in a day. Give your answer in scientific notation. (1 mk)
Question 6. Use a separate page
(a) A bee is flying west and a graph of its displacement (x) against time (t) is shown below.
(i) At what time does the bee first change direction? (1 mk)
(ii) At approximately what time is the acceleration zero? Give a reason for your answer. (2 mks)
(iii) At what time is the distance from the starting point greatest? (1 mk)
(b) The equation 5x2 (2k + 1)x + k =0 has real roots and its discriminant is 49. Find the value of k and find all possible roots of the equation. (4 mks)
(c) A circle with a circumference of 500m has its centre at O. AOB is a sector of the circle that makes an angle θ at O. The chord AB is 100m long.
(i) Find the radius of the circle. (1 mk)(ii) Calculate the angle &theta in radians to 2 decimal places. (1 mk)
(iii) Find the perimeter of the sector AOB. (2 mks)
Question 7. Use a separate page
(a)
The diagram below shows the parabola y = x2 + c
A tangent to the parabola cuts the x-axis when x = -2 and touches the parabola when x = +2.
(i) What is the gradient of the tangent? (1 mk)
(ii) What is the equation of the tangent? (1 mk)
(iii) What are the coordinates of the point where the tangent touches the parabola? (1 mk)
(iv) What is the equation of the parabola? (1 mk)
(v) What are the coordinates of the vertex of the parabola? (1 mk)
(vi) What is the area between the parabola and the x-axis between the points where x = 0 and x = 2? (1 mk)
(vii) The parabola is now rotated about the y-axis between the vertex and the line y = 8. What is the volume of the solid of revolution so formed? (2 mks)
(b)
The locus of a point moves such that its distance from the point (2, 1) is twice the distance from the point (5, 4)
(i) Show that the locus of the point traces out a circle? (2 mks)
(ii) What is the radius of the circle? (1 mk)
(iii) What are the coordinates of the centre of the circle? (1 mk)
Question 8. Use a separate page
(a)
(i) On the same axes, draw the graphs of y = 3cos x and y = 2 (-π≤ x ≤ π) (2 mks)
(ii) From your graphs or otherwise, find solutions to the equation 3cos x = 2. (-π≤ x ≤ π) (2 mks)
(iii) Find the area enclosed by the graphs y = 2 and y = 3cos x (-π≤ x ≤ π) (2 mks)
(b).
A cylindrical water tank, open at the top, is to be constructed out of sheet metal and is to have a capacity of 4.0 m3.
(i) If the radius of the tank is x m and the area of sheet metal is A m2
Show that A = πr2 + 8/r (1 mk)
(ii) Find the least value of A. (2 mks)
(c). The sum of the first 7 terms of an arithmetic series is 94 and the sum of the first 10 terms is 185.
(i) Find the first term and common difference for the series. (2 mks)
(ii) Is 382 a member of the series? Justify your answer. (1 mk)
Question 9. Use a separate page
(a).
The numbers 1, 2, 3, 4 & 5 are written, one to a card, on 5 separate cards.
Two cards are selected at random. The first card selected forms the first digit of a two digit number, and the second card selected forms the second digit.
What is the probability that the number so formed is
(i) Odd (1 mk)
(ii) greater than 32 (1 mk)
(iii) exactly 41 (1 mk)
(iv) less than 11 (1 mk)
(v) a perfect square (1 mk)
(vi) one where the sum of the two digits is 6? (1mk)
(b).
Solve the equations:
(i) 4x + 2x 2 = 0 (3 mks)
(ii) 2 sin2x = √2 (-2π ≤ x ≤ 2 π) (3 mks)
Question 10. Use a separate page
(a).
Shakeeba retired with a superannuation of $500 000. She calculated that she was likely to live another 30 years. She invested her $500 000 in an account that returns 6.0% p.a. compounded monthly.
Shakeeba withdraws $M from the account each month, immediately after the interest has been paid. She plans to do this for the next 30 years so that at the end of 30 years the account will be empty.
(i) How much money is in the account after the first withdrawal? (2 mks)
(ii) At the end of 30 years the amount remaining in the account will be zero. Write an equation in terms of M for the amount of money in the account immediately after the last withdrawal. (2 mks)
(iii) Calculate the value of M that leaves the account empty at the end of 30 years. (3 mks)
(b).
(i) Find the gradient of the tangent to the curve y = 2 loge (x2 + 4) at the point where x = 2 (2 mks)
(ii) Find the equation of the normal to the curve y = 2 loge (x2 + 4) at the point where x = 2. (3 mks)