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






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“Logic,
like whiskey, loses its beneficial
effects when 
taken in too
large quantities.”


   
    
 Lord
 
    
Dunsany
 
A2 Maths with Statistics Assignment  (rho)
15 questions including drill, exam style questions and challenge
due in w/b 2/2/15
Drill

Part A Integrate the following functions with respect to x:
cos2 2x
(a)
(b)
tan 2 3x
(c)
2
3x  1
Part B Solve the following equations giving x to 2dp:
2e x  3e x
(a)
(b)
log 2 x  4 log x 2
(c) log 2 1  3x   log 2 2 x  1  1
Part C Solve the following equations:
x2 4
(a)
(b)
2 3x  1  1  0
(c)
x  2  3 x 1
Part D Given that the following vectors are perpendicular, find the value of p:
(a)
a = pi + j + k,
b = 3i – 4j – 2k
(b)
a =3i + j + k,
b = 4i + pj – k
(c)
a = 3i + j – 2k,
b = 3i – 2j + pk
S2: Current work Continuous Random Variables
1.
Explain why the following cannot represent probability density functions:


2.

(a)
k 3 2x 1  x  2
f (x)  
otherwise
 0
(b)
x 2
f (x)  
0
(c)
x 12  x 1  x  2
f (x)  
otherwise
0

1  x  1
otherwise
Sketch the following probability density functions:
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

2 4  x 1  x  4
f (x)  9
otherwise

 0

1 x 3 0  x  2
f (x)  4
otherwise

 0

 1 9  x 2  3  x  3
f (x)  36
otherwise

0

(a)
(b)

(c)

3.

The continuous random variable X has the probability density function f (x) given by:
kx2
f (x)  
 0
0x3
otherwise
(a)
Find the value of k
(b)
Find P
 x  1
(c)
Find and specify the cumulative distribution function F (x)
C3 Consolidation
4.
Find the exact solutions to the equations
(a) ln x + ln 3 = ln 6,
(b) ex + 3e–x = 4
5.
f(x) =
2x  3
9  2x
–
,
2
x2
2 x  3x  2
x>
1
2
.
4x  6
.
2x  1
(b) Hence, or otherwise, find f (x) in its simplest form.
(a) Show that f (x) =
6.
A curve C has equation y = x2ex.
(a) Find
dy
, using the product rule for differentiation.
dx
(b) Hence find the coordinates of the turning points of C.
(c) Find
d2 y
.
dx 2
(d) Determine the nature of each turning point of the curve C.
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7.
f(x) = – x3 + 3x2 – 1.
(a) Show that the equation f(x) = 0 can be rewritten as
 1 

.
3 x
(b) Starting with x1 = 0.6, use the iteration
x=
xn + 1 =
 1

 3  xn



to calculate the values of x2, x3 and x4, giving all your answers to 4 decimal
places.
(c) Show that x = 0.653 is a root of f(x) = 0 correct to 3 decimal places.
8.
The functions f and g are defined by
f : x  ln (2x – 1),
g: x 
2
,
x3
x  ℝ, x >
1
2
,
x  ℝ, x  3.
(a) Find the exact value of fg(4).
(b) Find the inverse function f –1(x), stating its domain.
(c) Sketch the graph of y = |g(x)|. Indicate clearly the equation of the vertical
asymptote and the coordinates of the point at which the graph crosses the y-axis.
(d) Find the exact values of x for which
2
= 3.
x3
S2 consolidation
9. An engineering company manufactures an electronic component. At the end of the
manufacturing process, each component is checked to see if it is faulty. Faulty
components are detected at a rate of 1.5 per hour.
(a) Suggest a suitable model for the number of faulty components detected per
hour.
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(b) Describe, in the context of this question, two assumptions you have made in
part (a) for this model to be suitable.
(c) Find the probability of 2 faulty components being detected in a 1 hour period.
(d) Find the probability of at least one faulty component being detected in a 3
hour period.
10. (a) Write down the conditions under which the Poisson distribution may be used
as an approximation to the Binomial distribution.
A call centre routes incoming telephone calls to agents who have specialist
knowledge to deal with the call. The probability of the caller being connected to
the wrong agent is 0.01.
(b) Find the probability that 2 consecutive calls will be connected to the wrong
agent.
(c) Find the probability that more than 1 call in 5 consecutive calls are connected
to the wrong agent.
The call centre receives 1000 calls each day.
(d) Find the mean and variance of the number of wrongly connected calls.
(e) Use a Poisson approximation to find, to 3 decimal places, the probability that
more than 6 calls each day are connected to the wrong agent.
11. In
a large population of animals, 5% have a genetic disorder. By using a suitable
approximation in each case, find the probability that
(a) in a random sample of 50 animals, the number with the disorder is greater than 3
(b) in a random sample of 500 animals, the number with the disorder is greater than 30.
12.
For this question decide which of the responses given is (are) correct then
choose
A if 1, 2 and 3 are correct
B if only 1 and 2 are correct
C if only 2 and 3 are correct
D if only 1 is correct
E if only 3 is correct
OP  2i  3 j  k
OQ  3i  2 j  k
1. PQ  5i  5 j
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2. OP.OQ  11

3. POQ  arccos  

11 

14 
Challenge
The curvy shape ABD shown here is called
a Reuleaux triangle ( after French engineer
Franz Reuleaux (1829-1905)). Its perimeter
consists of three equal arcs AB, BC, CA;
each with the same radius and centered at
the opposite vertex. In the Reuleaux
triangle shows, each arc has a radius 3cm.
What is the area (in cm2) of the inscribed
circle?
Preparation: Solving differential equations and
Connected Rates of Change: new C4 pages 42-46 and 114 -118 old C4
pages 38 -42 and 108-112.
Drill Answers
Part A
1
1

(a)  x  sin 4 x   c
2
4

Part B
1 3
(a) ln
2 2
Part C
(a) x = 6 or –2
(b)
1
tan 3x  x  c
3
2
ln 3x 1  c
3
(c)
(b) 4 or 1/4
(c)
3
7
1
1
or
6
2
(c)
5
1
or
2
4
(c)
7
2
(b)
Part D
(b) – 11
(a) 2
Answers
(1),(2) discuss in class (consider the area under the curve)
(3a) k  19
(3b) 271
(3c) discuss in class
(4a) x = 2
(4b) x = ln 3, x = 0
(5b)
8
( 2 x  1) 2
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(6a)
dy
 x 2 e x  2 xe x
dx
(6b) x = 0, y = 0 and x = –2, y = 4e–2
d2 y
 x 2e x  2 xe x  2 xe x  2e x (6d) x = 0 is a minimum, x = –2 is a maximum
(6c)
2
dx
(7b) x2 = 0.6455, x3 = 0.6517, x4 = 0.6526
(8a) ln 3
(8b) f–1(x) =
1 x
(e + 1), Domain x  ℝ
2
(8d) x = 3
2
1
,x=2
3
3
(9a) P0 (1.5)
(9b)
faults occur independently or randomly
faults occur at constant rate
faults occur singly
(9c) 0.2510
(9d) 0.9889
(10a) n large, p small
(10d) X~B (1000, 0.01), 10, 9.9
(10b) 0.0001
(10c) 0.00098
(10e) 0.8699
(11a) 0.2424 (Poisson approx) (11b) 0.1292 (Normal approx)
(12) B
Challenge
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MATHEMATICS
DEPARTMENT
ASSIGNMENT rho COVER SHEET
Name
Current Maths Teacher
Please tick honestly:
Yes
No - explain why.
Have you ticked/crossed
your answers using the
answers given?
Have you corrected all the
questions which were
wrong?
How did you find this assignment?
Use this space to outline any problems you’ve had, how you got help as well as the
things which went well or which you enjoyed/learned from.
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