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QUESTION 1 The graphs of f(x) = ¢* and g(x) =x~— 1 are shown. The area of the shaded section bounded by these graphs between the linesx=0and x =1 is A) 1-e (B) e-2 5 ©) B 1 (D) e=3
QUESTION 12 (5 marks) An object is moving in a straight line from a fixed point. The object is at the origin initially. The acceleration a (in m s ™) of the object is given by a(f) =m cos(mt) t=>0 where ¢ is time in seconds. The velocity at =115 0.5 m s ! a) Determine the initial acceleration. [1 mark] b) Determine the initial velocity. [2 marks] c) Determine the displacement after one second. [2 marks]
QUESTION 6 Oil is leaking from a tanker at the rate of »(¢) = 9000¢ % litres per hour where ¢ is in hours. Determine how much oil leaks from the tanker (to the nearest litre) from time # = 0 to time ¢ = 10. (A) 38910 litres (B) 8756 litres (C) 7782 litres (D) 1556 litres
QUESTION 19 (7 marks) Consider the following information when completing this question. 2 The length of a curve y = f(x) over the interval [a b] = Ib 1+ (%j dx a \f 67 You are driving along a road with a vertical distance above sea level D (in metres) given by the function D) =300 + In(x> -3x + ¢) where x 1s the horizontal distance from an initial point of measurement (in kilometres) at sea level. Assume that if x 1s positive you are east of the initial point of measurement and if x is negative you are west of the 1nitial point of measurement. You start your drive along the road at a horizontal distance of 10 kilometres west of the initial point of measurement and drive until you are at a horizontal distance of 10 kilometres east of the initial point. Determine the time you spend driving downhill if you drive downhill at an average speed of 40 km/h.
QUESTION 13 (5 marks) Consider the functions f(x)=x”and g(x)= 4x. a) Determine the x-coordinates of the points of intersection of the graphs of the two functions. [2 marks] b) Use the results from Question 13a) to calculate the area enclosed by the graphs of f(‘c‘) and g(‘c) [3 marks]
QUESTION 16 (4 marks) A tangent is drawn at the point (1 e) on the graph of the function y = ¢` as shown. Not to scale Determine the area of the shaded triangle.
QUESTION 18 (4 marks) The graph of y = f(x) where f(x) is the quadratic function f(x)= ax” + bx +4 is shown. Two regions of the area between the graph of y = /(x) and the x-axis are shaded. Not to scale : .2 43 . 5 Region P has an area of %umts“ and Region Q has an area of ?) units”. Determine the values of a and b.
QUESTION 11 (5 marks) Consider the function f(x)=¢ sin(x) 0<x<27 a) State the exact values of the x-intercepts of the graph of f`(x). [2 marks] b) Write an expression for the area enclosed between the graph of f (r) and the x-axis. [2 marks] ¢) Determine the area enclosed between the graph of f(x) and the x-axis to the nearest square unit. [1 mark]
QUESTION 3 The area between the curve y =9 — x” and the x-axis is (A) 12 units® (B) 18 units® (C) 36 units? (D) 54 units?
QUESTION 17 (4 marks) A snail is travelling along a straight path from point 4. The snail`s velocity (cm min ) is modelled by v(t)= 1.4ln(1 e ) where ¢ is time (in minutes) for 0 <z <15. An ant passes point 4 12 minutes after the snail and follows the snail`s path. The ant moves with a constant acceleration of 2 cm min~2 and passes the snail at # =15 minutes. Determine the ant`s velocity at point A4.
QUESTION 4 The area of the region enclosed by the graphs of y = xvVx + 1 and y = 2x 1s (A) 1.267 units® (B) 0.467 units® (C) 0.200 units` (D) 0.029 units®
QUESTION 17 (3 marks) During one 30-day period the rate at which pollution passes into a nearby lake is measured every six days and results are given in the table below. Rate of pollution in units per day p(t) This information can be modelled as a quadratic function. Determine the total amount of pollution entering the lake during this 30-day period.
QUESTION 9 Determine fj zsin Exax (A) 3 (B) 6 (Cc) -3 (D) -6
QUESTION 14 (6 marks) The rate of change in the number of bacteria in a science experiment is represented by < =e 420 where ¢ represents the time (hours) since starting the experiment and P represents the number of bacteria present (thousands). Initially there are 60 000 bacteria present i.e. P(O) = 60. a) Determine the equation for P(t). [2 marks] b) Determine the change in the number of bacteria during the third hour. Express your answer in terms of e. [2 marks] c) Determine how long it will take for the number of bacteria present to double after starting the experiment. [2 marks]
QUESTION 12 (4 marks) The graph shows the water level under a bridge over a 12-hour period. a) b) Water level (metres) 12:00 am 4:00 am 8:00 am 12:00 pm Time Determine the equation of the cosine function that models the water level as a function of time after 12:00 am. [1 mark] How long in the 12-hour period shown is the rate of change of water level more than 0.55 metres per hour? [3 marks]
QUESTION 13 (4 marks) The curved lines represent graphs of the equations y = x7 —4x+8 and y= 10cos(x +10). y a) Determine the coordinates of the points of intersection A and B. [1 mark] b) State an integral expression representing the area enclosed by the two graphs. [2 marks] c) Determine the area enclosed by the two graphs. [1 mark]
QUESTION 16 (6 marks) A particle is moving in a straight line. The velocity (ms!) of the particle is given by 20sin (2t v(t) =e) 6 —5cos(2r) The initial position of the particle is +6.0 m from the origin. t > 0 where ¢ is time (s) after moving from its initial position. a) Use calculus methods to determine an equation for the position of the particle from the origin at any time ¢. b) Determine the position of the particle relative to the origin when it first reaches maximum velocity. [3 marks] [3 marks]
QUESTION 18 (5 marks) A company makes windows using glass that has a mass of 5.6 kg per square metre. A customer orders an unusual window in a partial parabolic shape as shown. Not to scale 8m Determine the mass of the window.
QUESTION 19 (6 marks) Over a suitable domain a hill has a cross-sectional area given by fr (x) dx = a +c where: ¢ a b and c are constants b+0 ¢ h(x) represents vertical distance (m) x represents horizontal distance (m). It is known that 4(0) = 1.22 and h(40) = 25. Where the gradient of the hill is 0.86 there is a tree stump. A second tree stump is located further up the hill. The difference in hill gradient between the two tree stumps is 0.44. A surveyor predicts that the vertical distance separating the two tree stumps is between 7.5 m and 8.5 m. Evaluate the reasonableness of this prediction.
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