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QUESTION 7 The diagram shows a slope field. | / / / \ \ \ \ The differential equation represented by the slope field is d 5 A Y2 dx % dy 5y° B) L _2V dx ¥ dy Sy C) === ©) B 2 dy 5)° (D) Y _2V dx x2
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 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 5 The gradient of the tangent at point A on the curve y2 =4x1s 1.36 The x-coordinate of A is A) 0.12 (B) 0.46 (C) 0.54 (D) 147
QUESTION 12 (9 marks) For a certain experiment the number of yeast cells VN after # hours in a test tube can be modelled by the differential equation d—N:LN(IOOO—N) fort >0 dr 1000 1000 1 1 a) Given ————=—+———— show that the general solution of the differential N(1000—N) N 1000—N equation can be expressed as N —— |=t+4c 1000N| ln| [2 marks] A scientist commenced this experiment at 9:00 am on a certain day and observed that 100 yeast cells were present at this time. b) Show that the solution of the differential equation can be expressed as ~ 1000 1+9¢” [3 marks] c) Determine the time of day when 900 yeast cells were present. [2 marks] The scientist predicted that the number of yeast cells would eventually exceed 1200. d) Evaluate the reasonableness of the scientist’s prediction. [2 marks]
QUESTION 17 (7 marks) An object is released from rest at a height of 100 m above the ground. The motion of the vertical descent of the object is modelled by v 9 810 004 (v>0) dx where v is the velocity (m s) and x is the displacement from the ground (m). Determine the velocity of the object when it strikes the ground.
QUESTION 9 — e— e— e— e— e— e— e— — — e— e— o— e— e— e— a—ll a—ll a—ll a—ll a—ll a—l e—l e— - /| /S e e = — = = = — — — ) ~N\\\ /) /s rrem——— = = = W ~ ~ ~ ~ N \ N a_ \_A ~ ~ \ _‘_p ~ N N\ 8 \ /| /|- — = = = — S o o o o | e o e e e e — o s e e e am— | — om— c— c— c— c— — c— The slope field for the differential equation — The solution curve to the differential equation that passes through the point (3 —5) also passes through point (A) P B) Q C) R (D) S
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 15 (8 marks) Water 1s poured into a cone-shaped cup at a rate of 2 em® s71. The cup has a height of 12 cm and a radius of 6 cm as shown. ] 12 cm h Not to scale J As the cup fills the ratio of the height of the water / to the surface radius of the water » remains constant. a) Given that 7 =2r determine a function for the volume of water in the cup ¥ in terms of 4. Express your answer in simplified form. [1 mark] b) Use the results from Question 15a) to show that the rate at which the height of water in the cup is increasing with respect to time is given by — [3 marks] h c) Determine the rate at which the height of water in the cup is increasing with respect to time when the volume of water in the cup reaches half of the total capacity of the cup. [4 marks]
QUESTION 17 (7 marks) An object with a mass of 2 kg is released from rest at the top of a 1 metre long frictionless plane inclined at 30° to the horizontal. A force of P newtons acting parallel to the plane opposes the motion of the object as it travels down the plane. When the object is x metres from the top of the plane its velocity is v m s 4 Va4 -—x? ’ Given ‘P| = determine x when v = 2.
QUESTION 14 (4 marks) —0.5(y—4 The slope field for the differential equation Y = w x#0using—6<x<6and-6<y<6 is shown. ax * a) Determine the value of the slope at point A. [2 marks] : dy —0.5(y-4) b) Use the slope field to sketch the solution curve for . given that X when x =-6 y=3.5 & [2 marks]
QUESTION 3 2 Determine the solution of the differential equation — & M : T given y =0 when x = —. dx cos(2x) 5 (A) y= —2ln‘cos(2x)|—2_35 B) y= —2ln‘cos(2x)|+ 2.35 C) y= —%1n‘cos(2x)|— 0.59 (D) y= —%ln‘cos(2x)|+ 0.59
QUESTION 8 Determine the gradient of the tangent to the curve y2 —3x =5 at the point (1 22 ) (A) 041 (B) 0.53 (C) 1.06 (D) 8.49
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 19 (7 marks) A research organisation plans to use a drone to drop a scientific instrument vertically from a stationary position above the ocean surface. The acceleration (m s_z) of the falling instrument can be modelled by 9.8—0.1v where v is its velocity (m s~ ) In order for the instrument sensors to activate its speed as it hits the ocean surface must reach at least 20 m s~ L. However if it hits with a speed above 50 m s7! the sensors will be damaged. Determine the range of the drone’s flying height above the ocean surface to ensure that the sensors are activated but not damaged.
QUESTION 6 NS — 1S 2y AA /1 1/ [ 1 dy _ x dx_ (B) A - o L -~ A A ~ - 7 P 7 - 7~ 7~ ~ = 7 NN IS RN NS s (A) The slope field that represents the differential equation 1 0 VNN NN D ) C (
QUESTION 17 (7 marks) The curve shown below represents the solution to the differential equation ay _ ¥ dx X +1 x>0 < A(e? 0) Use y = wx (where w is a function of x) to determine the equation of the curve in simplest form.
QUESTION 3 Use implicit differentiation to determine the value of the gradient of 2e* — 3y? + 372 = 2e at (1 7). (A) —24.17 (B) —13.41 (C) 0.29 (D) 2.85
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 7 The differential equation for which the solution is a logistic equation of the form y = and C are constants is —— where a b b+Ce dy A) ~—=0.25(1—0.01¢ (a) =0.25(1-0.011 (B) ® _.25(1-0.01y) dt (C) ® _0.251(1-0.011) ame (D) “ = 0.25y(1—0.01y)
QUESTION 16 (5 marks) A curve is defined by the parametric equations x = 2 tan(@) and y =3sin(26) where 0< 0 < > Given that dy can be expressed in the form acos* (0)+ bcos” (0) where a b € R determine the values he of a and b.
QUESTION 18 (6 marks) ; ; a: ee A particular solution to the differential equation dx (¥ n 1) tan( y) through the origin. Determine this solution in the form x = f(y). Leave your answer in simplified form. where x = 0 and = < y <0 passes
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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