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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 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 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 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 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 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 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 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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