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Student Name:………………………………………

SPECIALIST MATHEMATICS

Calculus: CAS Active

Writing time: 75 minutes

Total Marks: 50

SECTION 1

Question 1

The antiderivative of is:

A

B

C

D

E

Question 2

By using the appropriate substitution, e2x dx is equal to:

A (u + u) du

B (2u + u) du

C (u + 2u + u) du

D (u + 2u) du

E (u + u) du

Question 3

The values of the constants, a and b, if y = a sin2x + b cos2x satisfies the differential equation

– 2 + 3y = 9 sin2x + 2 cos2x, are:

Aa = –3, b = 1

Ba = –1, b = 2

Ca = –2, b = –1

Da = 1, b = 2

Ea = 3, b = 2

Questions 4 and 5 refer to the following information.

A tank containing 500L of water has 40 kg of sugar dissolved in it. Fresh water is pumped in at a rate of 5L/min and the resulting solution is pumped out at the same rate. The mixture is kept uniform.

Question 4

If x g is the amount of sugar in the container at any time, t minutes, then the differential equation describing the rate of change of sugar in the container, in g/min, is:

A

B

C

D

E

Question 5

The concentration of sugar in the container after 10 minutes is nearest to:

A0.07 g/L

B36.20 g/L

C362 g/L

D30 g/L

E0.35 g/L

Question 6

is equal to:

A

B

C

D

E

where c is a constant.

Question 9

Using a suitable substitution, can be expressed as

A.

B.

C.

D.

E.

Question 10

Point is any point that lies on a particular curve.

The gradient of a line joining point and P is two more than the gradient of the tangent to the particular curve at P.

The coordinates of point P satisfy the differential equation given by

A.

B.

C.

D.

E.

Question 11

Using an appropriate substitution, can be expressed as

A.

B.

C.

D.

E.

Question 12

The graph of the function with the rule is shown above.

The antiderivative function has the rule . For the graph of , it is true to say that it

A.has no points of inflection

B.has no stationary points of inflection

C.has no stationary points

D.has two stationary points

E.has one local maximum and one local minimum

Question 13

The equation of the normal to the curve at the point where

is given by

A.

B.

C.

D.

E.

SECTION 2

Question 1

Let where f has a maximal domain.

  1. Sketch the graph of on the set of axes below. Indicate clearly on your graph
  • the location of any x-intercepts
  • the coordinates of any stationary points expressed as an exact value
  • the equation of any asymptotes

4 marks

  1. If there are two solutions to the equation where , find a. Express your answer as an exact value.

1 mark

That part of the graph of f for which , is rotated around the x-axis to form a volume of revolution.

  1. i. Write down a definite integral that could be evaluated to find this volume of revolution.

ii.Find the exact value of the definite integral in part i.

  1. Find the rule and the domain of the function

2 marks

  1. On the set of axes below, sketch the graph of .

Indicate clearly on your graph the

  • coordinates of any axes intercepts
  • equation of any asymptotes

2 marks

  1. Find and hence explain why the graph of does not have a point of inflection.

1 mark

Total 13 marks

Question 2:

A gelatine mixture of concentration 1.4 kg/L is added to a 120-litre tank of water at a constant rate of 3 L/min. At the same time, the solution is pumped out at 3 L/min, while the mixture is kept uniform by continuous stirring.

Let G = amount of gelatine (kg) at time t minutes

(i)Calculate the input rate of the gelatine in kg/min

______

______

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1 mark

(ii)Calculate the output rate of the gelatine in kg/min

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1 marks

(iii)Construct a differential equation to show the rate of change of concentration of gelatine

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1 mark

(iv)Assume that the water can set to jelly if the gelatine concentration reaches 1.2 kg/L.

When does the water set to jelly?

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3 marks

(v)Find an expression for the amount of gelatine, G (kg), in terms of the time (t) in minutes

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3 marks [3 marks]

(vi)Sketch a graph of the mass of gelatin G (kg) against time t (minutes).

3 marks

Total Marks: 12 marks

QUESTION 2

The graph of f: [0, 1] R, where: f(x) = is drawn below.

(i) Write down the range of f(x)

______

1 mark [1 mark]

(ii)Use calculus to show that is the point of greatest slope of f(x)

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3 marks [4 marks]

Joshua wants to build an ornamental pond for his garden. He uses as a model, the function:

f: [0, 1] R, where: f(x) =

and rotates it about the y-axis. All lengths are in metres.

(iii) Using calculus find the volume of the pond when filled to capacity. Give your answer in cubic metres (m3) correct to 3 decimal places.

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4 marks

[4 marks]

(iv)Once constructed, Joshua fills the pond with water at the rate of 0.012 m3/min.

Show that the rate at which the water level is rising at any time t is given by:

m/min

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[4 marks]

4 marks

Total Marks: 12 marks