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QUESTION 1 3x— A4 — X 5 dx can be determined using the partial fractions 1 + 2 1—x I+x 1—x The indefinite integral f The value of 4 1s (A) -3 (B) -1 <€) 1 (D) 3
QUESTION 5§ Determine f 4x(3x2 +5)3 dx (A) %(3x2+5)4+c (B) %(3x2—|—5)4—|—c ©) 2(3x2—|—5)2—|—c (D) 72x> (3x2 + 5)2 te
QUESTION 13 (4 marks) The expected value of an exponential random variable X with parameter A > 0 can be determined using the rule E(X)= j;)oox)\e_)‘xdx Use integration by parts to determine £(X). Express your answer in simplest form.
QUESTION 14 (4 marks) The motion of an object that moves in a straight line is given by v(x) —cos ! (2x) where v 1s the velocity (m s™!) and x is the displacement (m) from the origin. a) Determine a(x) where a is the acceleration (m s2) of the object. [2 marks] b) Use the result from 14a) to determine a(0) given —2rn < a(0) < 0. Express your answer in simplest form. [2 marks]
QUESTION 17 (7 marks) Determine the smallest positive value of a given fa [sec(2x) + tan (2x) cosec(2x)+1 2 dx =1 —da
QUESTION 19 (7 marks) A circular-based bowl has been positioned symmetrically on a Cartesian plane as shown in the diagram. y 4 The bowl has a shape that can be generated by rotating the curve y = S x —l about the y-axis — X for 4 <x<7.6 cm. The bowl is being filled with a liquid at the rate of 77 cm’s L. Determine the rate at which the depth of liquid is increasing when the depth of liquid reaches one-third of the height of the bowl.
QUESTION 14 (5 marks) The time ¢ (months) that it takes before a phone owner cracks the screen on their phone can be modelled by an exponentially distributed random variable ) = 0.16e~ 1 t>0 0 otherwise a) Show that f(¢) is a probability density function. [1 mark] b) Determine the probability that a phone owner cracks the screen on their phone within 1 year. [2 marks] Three-quarters of phone owners take between 1 and m months before they crack the screen on their phone. c) Determine the value of m. [2 marks]
QUESTION 11 (5 marks) Let f(x)=tan (%j for suitable values of x where f(x)e [—% %j : a) Determine f(2). [1 mark] b) Determine f'(2). [2 marks] c) Use the results from Questions 11a) and 11b) to determine the equation of the tangent to the graph of y = f(x) at x = 2. [2 marks]
QUESTION 15 (4 marks) 4x—17 2 —x—6 Use partial fractions to determine J dx where xe R x # -2 x # 3. Express your answer in the form ln| f (x)‘ +c.
QUESTION 17 (7 marks) The area between the graphs of the functions y =4x and y = 2x? is rotated about the y-axis to form a solid of revolution with a volume of V units. Determine the exact value of V.
QUESTION 18 (6 marks) This differential equation can be used to determine the current / (amperes) at time ¢ (seconds) with voltage V (volts) in an electric circuit containing a resistance R (ohms): dl k—+RI=V dt where k& R and V are positive constants and ¢ > 0. Assuming that there is no current in the electric circuit initially show that the size of the current can never be greater than %
QUESTION 2 Determine the area of the shaded region between the graphs of the functions y = lsec[£] and y = ZCOS(EJ as shown. 3 3 2 -3 Not to scale (A) 5.29 units? (B) 5.5 units® (C) 5.65 units” (D) 5.71 units?
QUESTION 13 (6 marks) The area under the graph of the function f(x)= 0.2¢7%** for 1< x <9 is shaded. f(x) 1 9 & Not to scale a) Use Simpson’s rule with four intervals to determine an approximation for this area. [4 marks] b) Use a calculus approach to evaluate the reasonableness of your area approximation from Question 13a). [2 marks]
QUESTION 8 Use the substitution u = tan(x) to determine Itan(x)secz (x)dx. (A) %tan(x) +c (B) %tan2 (x)+c ©) tan(x) +c (D) tan’ (x)+c
QUESTION 13 (6 marks) : : : 22 a) Use partial fractions to determine j (2x-3)(x+4) ax 0 b) Use the result from Question 13a) to determine J. 22 Express your answer in simplest form. [4 marks] [2 marks]
QUESTION 17 (5 marks) The region between the x-axis and the curve of the function y =1+ sin(Zx) for 0 < x < Z is rotated about the x-axis to form a solid of revolution. 2 Determine the volume of this solid. Express your answer in simplest form.
QUESTION 19 (7 marks) The function f (x) passes through the origin. The gradient function of f (x) is defined as g (x) = sin_l(ex ). Determine f (x)
QUESTION 4 The time taken for students to answer questions in a class is assumed to be a random variable X with an exponential distribution that has the probability density function f(x):{/leix x>0 0 otherwise The mean of X 1s l The mean length of time taken for students to answer questions in this class is 15 seconds. The probability that the next question in this class is answered between 8 seconds and 17 seconds is (A) 0.05 (B) 0.12 (C) 0.22 (D) 0.26
QUESTION 11 (6 marks) An aerial view of the surface of a dam 6 km in length i1s symmetrically positioned on a Cartesian plane as shown. A dam wall 1s located along the y-axis. 2 2 -2 The surrounding edge of the dam can be modelled by the ellipse (x 7 ) + % =1 for0<x<6. y Dam wall X Not to scale a) Use Simpson’s rule with four strips to determine an approximate area of the surface of the dam. [4 marks] b) Evaluate the reasonableness of this approximation. [2 marks]
QUESTION 14 (5 marks) An object is moving in a straight line with an acceleration represented by the differential equation dv dt a) Determine the general solution of the differential equation. [3 marks] = —(4 +? ) where v is the object’s velocity (m 57! ) over time #(s) where ¢ > 0 until it comes to rest. The initial velocity of the objectis 1.5 m s b) Determine the time when the particle comes to rest. [2 marks]
QUESTION 1 Determine the value of y = arccos (sin (4?”)) in the domain y € [0 7r]. (A) (B) ©) NFENFRIERE (D)
QUESTION 9 1 An antiderivative of ———=———F73x 15 V4 — x? cos‘l(E) (A) —In |cos‘1 g) (B) In |cos‘1 (g) (C) —2In |cos‘ (D) —2In |sin‘1 (g)
QUESTION 10 2 4 The value of f 0 m dx is X (A) itan‘1 G) (B) tan! G) (C) 2tan! G) (D) tan 1(2)
QUESTION 14 (6 marks) a) Determine the values of A and B in the following equation. 4x-1 A B (2x—1)2 (2x-1) (2x—1)2 [3 marks] b) Use the results from 14a) to determine the value of the following definite integral. dx J-Z 4x—-1 1 (2x—1)2 [3 marks]
QUESTION 10 The velocity v (m s~ 1) of a particle at time ¢t (s) is given by v = sin 1(¢) (0 <t < 15). The acceleration of the particle when t = 0.2 s is (A) 0.20ms™2 (B) 1.02ms™? (C) 1.04ms™2 (D) 1.37ms™2
QUESTION 12 (6 marks) Consider the semicircle shown below. a) Determine the equation of the semicircle in the form y = f(x). b) Determine the volume formed by rotating the semicircle about the x-axis. c) Evaluate the reasonableness of your answer to 12b). [2 marks] [2 marks] [2 marks]
QUESTION 14 (5 marks) An exponential random variable X with parameter A > 0 has a probability density function given by _ [ re™™ x>0 1) {O otherwise 1 The mean of X 1s ;\ The length of time that car owners keep their first car is an exponentially distributed random variable with a mean of 4 years. a) Determine the probability that an individual keeps their first car for between 3 and 5 years. [3 marks] 50% of all individuals keep their first car for at most k years. b) Determine the value of k. [2 marks]
QUESTION 18 (7 marks) An object of mass 5 kg moves in a straight line while being acted on by a net force F (N) in terms of its position from a reference point x (5 m < x < 10 m) such that 60 + 5x T 12x— 7%+ 3 Given the object was at rest when x = 5 m determine the position of the object from the reference point when its speed is 0.5 m s™.
QUESTION 11 (5 marks) Determine the following definite integrals. a) b) ty 1+x 5 AX oer; 2sin? (x) dx oes [2 marks] [3 marks]
QUESTION 8 Given f(x) = tan! (2x) determine (3). (A) 0.05 (B) 0.15 (C) 2.17 (D) 3.10
QUESTION 9 The time in minutes between the arrival of customers at a certain shop is assumed to be a random variable X with an exponential distribution that has the probability density function 0.1207 0-12 x>0 0 otherwise f(x) = | A customer arrives at the shop. The probability that the next customer arrives within 30 to 60 seconds to the nearest percent is (A) 3% (B) 5% (C) 7% (D) 11%
QUESTION 11 (4 marks) The bounded region between the graphs of the functions y = —1+sec =| and y = 0. 1x” over a certain domain is shaded as shown. The two functions intersect at the origin and point A. 7 A 6 x a) Determine the coordinates of point A. [1 mark] b) Calculate the area of the shaded region. [1 mark] The shaded region is rotated about the x-axis to form a solid of revolution. c) Determine the volume of the solid formed. [2 marks]
QUESTION 14 (4 marks) At a certain location a biologist measures the width of a river to be 12 m. She also records the depth of the river at regular 2 m interval widths as shown. wien) | oo | 2 | 4 | 6 | s | wo | 2 The biologist estimates the cross-sectional area of the river at this location to be 15 m”. Use Simpson’s rule to evaluate the reasonableness of this estimation. Justify your area calculation and decision regarding reasonableness using mathematical reasoning.
QUESTION 16 (6 marks) A curve modelled by the relation xy —y+ cos /(2x) = 1 where —0.35<x<0.27 andO<y<l intersects the y-axis at point A. Determine the equation of the tangent to the curve at point A.
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