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MATHEMATICS 3C TEST Time Allowed 30 minutes
Total Marks 26
Calculator Free Section.
Equipment Allowed pencils, pens ruler
Question 1. [5 and 8 marks](a) Simplify each of the following.
(i)
€
x2 − 1x
÷ x2 + 2x + 1
3x2 − 6x[2]
(ii)
€
5
y2 − 9 +
6y − 3
− 7
y + 3 [3]
1
(b) Find
€
dydx
in each of the following.
(i) y = (x2 1)e1 3x (Do not simplify) [2]
(ii) y =
€
2x − x + 3π3 + 4
x2 (Leave with positive indices.) [2]
(iii) y =
€
2x3
5 − 3x4 ⎛ ⎝ ⎜
⎞ ⎠ ⎟2 (Do not simplify) [2]
(iv) y =
€
u2 − 3 using Leibniz notation
€
dydx
= dydu
× dudx
where u = 2e2x + 3 [2]
2
Question 2 [4 and 4 marks]
Let f(x) = 5 +
€
x and g(x) =
€
1x
(a) State the domain and range of g(f(x)). [4]
(b) State the domain and range of f(g(x)). [4]
3
Question 3. [1, 1, 2 and 1 marks]
The graph of
€
y = x
e x2 is drawn below over the domain 0 x 2.
(a) Approximate the values of x such that y is an increasing function. [1]
(b) Determine the approximate range for y. [1]
(c) When x = 2, is it possible to determine the tangent line? Explain your answer. [2]
(d) Explain, using first derivative principles, why a local maximum point occurs. [1]
4
MATHEMATICS 3C TEST Time Allowed 30 minutes
Total Marks 28
Calculator Section.
Equipment Allowed CAS Calculator, pencils, pens ruler
Question 1. [2 and 2 marks]
Consider the function f (x) = 2x3 3x2 23x + p where p is a constant.
(a) Determine where the local (relative) extrema points occur. [2]
(b) Find the value of p given that the three roots are 3, 0.5 and 4. [2]
5
Question 2. [1, 2, 1, 3, 2, 2 and 2 marks]
On your CAS calculator, sketch the curve f (x) = x4 – 2x3 over the domain –1 x 2.5.
Determine
(a) the roots of f (x). [1]
(b) the stationary points of f (x). [2]
(c) the points of inflection of f (x). [1]
Sketch f (x) on the axes below, highlighting each of the points found in (a), (b) and (c) on the previous page. [3]
6
Use this sketch, or otherwise, to determine the x value(s) where:
(d) f (x) > 0 [2]
(e) f (x) > 0 [2]
(f) f (x) > 0 [2]
7
Question 3. [6 marks]
A small box with a square base and open at the top is to occupy 40mm3 for a diamond display. Ignoring waste and material thickness, find the dimensions of the box if the least amount of material is required in its construction. [6]
Question 4. [3 and 2 marks]
An oil tanker has hit a reef during a cyclone. The impact tore a hole in the hull that resulted in
crude oil escaping into the ocean. The quantity of crude oil S(t), (in megalitres (ML)), remaining in
the ship at time t hours is approximated by the exponential relationship
S(t) = S0ekt
After 5 and 8 hours, 73.55ML and 40.37ML of crude oil respectively remained in the ship.
(a) Determine the values of S0 (nearest unit) and k (to 2 decimal places). [3]
(b) Determine the quantity of oil that has escaped from the tanker after 20 hours. [2]
8
Solutions
Insert School Logo
Insert School Name
MATHEMATICS 3C TEST Time Allowed 30 minutes
Total Marks 26
Calculator Free Section.
Equipment Allowed pencils, pens ruler
Question 1. [5 and 8 marks](a) Simplify each of the following.
(i)
€
x2 − 1x
÷ x2 + 2x + 1
3x2 − 6x[2]
€
(x − 1)(x + 1)x
× 3x(x − 2)(x + 1)(x − 1)
€
3(x − 2)(x − 1)(x + 1)
(ii)
€
5
y2 − 9 +
6y − 3
− 7
y + 3 [3]
€
5 + 6(y + 3) − 7(y − 3)
(y2 − 9)
€
44 − y
y2 − 9
9
(b) Find
€
dydx
in each of the following.
(i) y = (x2 1)e1 3x (Do not simplify) [2]
€
dydx
= 2x e1 3x + (x2 1). (3). e1 3x
(ii) y =
€
2x − x + 3π3 + 4
x2 (Leave with positive indices.) [2]
€
dydx
= 2 − 1
2 x − 8
x3
(iii) y =
€
2x3
5 − 3x4 ⎛ ⎝ ⎜
⎞ ⎠ ⎟2 (Do not simplify) [2]
€
dydx
= 6x. 5 − 3x4 ⎛
⎝ ⎜ ⎞ ⎠ ⎟2 − 2x3 .2.(5 − 3x4 ).(−12x 3 )
5 − 3x4 ⎛ ⎝ ⎜
⎞ ⎠ ⎟4
(iv) y =
€
u2 − 3 using Leibniz notation
€
dydx
= dydu
× dudx
where u = 2e2x + 3 [2]
€
dydx
= 2u
2 u2 − 3.4e 2x
€
dydx
= (2e 2x + 3)
2e 2x + 3 ⎛ ⎝ ⎜
⎞ ⎠ ⎟2 − 3
.4e 2x
10
Question 2 [4 and 4 marks]
Let f(x) = 5 +
€
x and g(x) =
€
1x
(a) State the domain and range of g(f(x)). [4]
€
g(5 + x ) = 1
5 + x Dx = {x | x ≥ 0) Ry {y | 0 < y < 0.2}
(b) State the domain and range of f(g(x)). [4]
€
f 1x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = 5 + 1
x Dx = {x | x > 0) Ry {y | y > 6}
11
Question 3. [1, 1, 2 and 1 marks]
The graph of
€
y = x
e x2 is drawn below over the domain 0 x 2.
(a) Approximate the values of x such that y is an increasing function. [1]
0 < x < 0.7
(b) Determine the approximate range for y. [1]
0 ≤ y ≤ 0.03
(c) When x = 2, is it possible to determine the tangent line? Explain your answer. [2]
No. Tangent is not possible at a closed point.
(d) Explain, using first derivative principles, why a local maximum point occurs. [1]
12
f’(x) > 0 to the left of x ≈ 0.7 and f’(x) < 0 to the right of x ≈ 0.7
max TP
MATHEMATICS 3C TEST Time Allowed 30 minutes
Total Marks 28
Calculator Section.
Equipment Allowed CAS Calculator, pencils, pens ruler
Question 1. [2 and 2 marks]
Consider the function f (x) = 2x3 3x2 23x + p where p is a constant.
(a) Determine where the local (relative) extrema points occur. [2]
x ≈ 1.5207 x ≈ 2.5207
(b) Find the value of p given that the three roots are 3, 0.5 and 4. [2]
f(x) = (x + 3)(2x 1)(x 4)
p = 12
13
Question 2. [1, 2, 1, 3, 2, 2 and 2 marks]
On your CAS calculator, sketch the curve f (x) = x4 – 2x3 over the domain –1 x 2.5.
Determine
(a) the roots of f (x). [1]
x = 0, x = 2
(b) the stationary points of f (x). [2]
x = 0, x = 1.5
(c) the points of inflection of f (x). [1]
x = 0
Sketch f (x) on the axes below, highlighting each of the points found in (a), (b) and (c) on the previous page. [3]
Bounds
Shape
Main points
14
Use this sketch, or otherwise, to determine the x value(s) where:
(d) f (x) > 0 [2]
1 < x < 0, 2 < x ≤ 2.5
(e) f (x) > 0 [2]
1.5 < x < 2.5
(f) f (x) > 0 [2]
0 < x < 2.5
15
Question 3. [6 marks]
A small box with a square base and open at the top is to occupy 40cm3 for a diamond display. Ignoring waste and material thickness, find the dimensions of the box if the least amount of material is required in its construction. [6]
V = x2h = 40 SA = x2 + 4xh
SA =
€
x2 + 4x.40
x2
Min at x = 4.3089
Verify minimum from first derivative or second derivative test or a sketch.
Dimensions are 4.3089cm by 4.3089cm by 2.1544cm
Question 4. [3 and 2 marks]
An oil tanker has hit a reef during a cyclone. The impact tore a hole in the hull that resulted in
crude oil escaping into the ocean. The quantity of crude oil (in megalitres (ML)), remaining in the
ship at time t hours is approximated by the exponential relationship
S(t) = S0ekt
After 5 and 8 hours, 73.55ML and 40.37ML of crude oil respectively remained in the ship.
(a) Determine the values of S0 (nearest unit) and k (to 4 decimal places). [3]
Use the simultaneous solver in calculator
€
73.55 = xe−5y
40.37 = xe−8y x, y
⎧ ⎨ ⎪ ⎩ ⎪
Hence S0 = 200 and k = 0.20
(b) Using the equation found in (a), determine the quantity of oil that has escaped from the
tanker after 20 hours. [2]
S(t) = 200 200e0.2020
= 196.3369 ML
16
17
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