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“If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to
keep happy”
Alfr&eacute;d R&eacute;nyi
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AS Maths Assignment  (mu)
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Remember - you need to complete this assignment, so start early and use the help available!
DRILL
Section A Solve these inequalities for x:
2 x 2  x  15  0
(1)
(2)

x  x2  2  0
Section B Without using a calculator, work out the values of these expressions:
log 2
(1)
1
4
(2)
log 4 2
(3)
log 2 128
Section C Sketch the following curves of y = f (x), showing the coordinates of the turning
point and any points of intersection with the coordinate axes:
f ( x)  4 x  x2
(1)
(2)
f ( x)  2x2  6x  8
Expensive Error
There are expensive errors in this student’s work. Copy out the students work,
highlight the mistakes and explain what they did wrong. You do not need to do the
question yourself.
1
Q)
If
i)
A) i)
p  log2 5, express the following in terms of p
log2 140  log2 7
ii)
log2 125
log2 10
 140 
log2 140  log2 7  log2 

 7 
 log2 20
 log2 4  5
 log2 4  log2 5
 2p
 2p
ii)
log2 125
125 
 log2 

log2 10
 10 
25 
 log2 

 2 
 log2 25  log2 2
 log2 5 2  1
 2 log2 5  1
 2p  1
MECHANICS
2
Dynamics
Draw labelled diagrams and form equations for the following situations
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
(a)
A 30kg weight is sitting on a slanted rough roof. The angle between the roof and the
horizontal is 30𝑜 . Given that the weight is at the point of slipping down the roof, what
is the reaction force and the coefficient of friction?
(b)
A particle of weight 5 kg is resting on a rough slope inclined at 15𝑜 to the horizontal
with a coefficient of friction of 0.45. A horizontal force P is also applied to the particle.
The particle is at the point of moving up the slope.
(i) What is the magnitude of the force P?
(ii) The force P is removed. Does the particle move? Explain your answer? (Draw
another model, and make new equations)
(c)
Two particles A and B of mass 5 kg and m kg respectively (5 &lt; m), are connected by a
light inextensible string. A is held resting on a rough fixed plane inclined at 20&ordm; to the
horizontal. The string passes over a smooth pulley P fixed at the top of the plane. The
portion AP of the string lies along a line of greatest slope of the plane and B hangs
freely from the pulley. Given that the coefficient of friction is 0.2, and when the system
is released from rest, the acceleration is 5𝑚𝑠 −2. Find the Tension in the string , and m.
3
(a)
Kinematics
A particle is projected upwards with a speed of 14 m s–1. Find for how long it is above
2 m.
(b)
A stone is dropped from the top of a tower. One second later another stone is thrown
vertically downwards from the same point with a velocity of 14 m s–1. If they hit the
ground together find the height of the tower.
(c)
A particle moves along a straight line. It accelerates from rest to a speed of 20 m s –1 in
25 s. It then moves at a constant speed of 20 m s–1 for a period of time. Then the
particle decelerates uniformly to rest in a half of the time for which it was travelling at
a constant speed.
(i)
Sketch a speed-time graph to illustrate the motion of the particle.
Given that the total distance travelled by the particle is 500 m,
(ii)
find the total time for which the particle is moving,
(iii)
sketch an acceleration-time graph illustrating the motion of the particle.
PURE MATHS
3
4
Given that y  2 x 2  1
(b)
d2y
Find
( this means the second derivative, not the derivative squared!)
dx 2
2
d2y
2 d y
By substituting expressions for y and
,
into
the
4
x
 3 y show
dx 2
dx 2
d2y
 3 y  k where k is an integer to be found.
that 4 x 2
dx 2
(c)
Find
(a)
5
 
d 2
y
dx
The second and fifth terms of an arithmetic series are 26 and 41 respectively.
(a)
Show that the common difference of this series is 5.
(b)
Find the 12th term of this series
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Another series has first term –12 and common difference 7. Given that the sum of the
first n terms of these two series are equal
(c)
find the value of n.
log 2  x 2  4   2  log 2 x
3  log 2  2 x  5  log 2  5 x  7 
6
Solve: (a)
7
(a)
Show that (2x – 1) is a factor of 2x3 + x2 + x – 1.
(b)
When 2x3 + ax2 + x + 1 is divided by x + 2 the remainder is -29. Find a.
8
(b)
The points A and B have coordinates (4, 6) and (12, 2) respectively. The straight line
l1 passes through A and B.
(a)
Find an equation for l1 in the form ax + by = c, where a, b and c are integers.
The straight line l2 passes through the origin and has gradient –4.
9
(b)
Write down an equation for l2.
(c)
The lines l1 and l2 intersect at the point C. Find the exact coordinates of the
mid-point of AC.
The sequence u1 , u2 , u3 ,  is defined by un1  un 2  1, n  1
Given that u1  k , where k is a constant,
10
(a)
find expressions for u2 and u3 in terms of k.
(b)
Given also that u2  u3  11 , find the possible values of k (hint: you can
Solve these equations to find  on 0  θ  2π giving your answers in terms of π
(a)
11
3 tan θ  1  0
(b) 4cos2 θ  1
A curve has equation
y  x3  x 
2 sin   12 cos   1  0
(c)
x  4x
x2
2
(a) Find the equation of the tangent to the curve at the point where x = 1.
(b) This tangent cuts the x-axis at the point A and cuts the y-axis at the point B.
Find the area of the triangle OAB, where O is the origin.
12
Sketch the following graphs, stating any intercepts with the axes:
(a)
y  ( x  3) 2
(b)
(c)
y
4
x2
(d)
5
A: (1)   x  3
2
(2) x  1 or x  2
B: (1) -2
y
1
4
x
y  x 2  6 x  13
(2) &frac12;
(3) 7
 3 25 

2 2 
(2) (–1,0), (4,0), (0,–8) min  ,
C: (1) (0,0), (4,0) max (2,4)
(2)(a) 𝑅 = 254.6, 𝜇 = 0.577 (b) i) 40N ii) No as 5𝑔𝑠𝑖𝑛15 &lt; 𝐹𝑅 (c) 𝑚 = 10.62𝑘𝑔, 𝑇 = 50.97 𝑁
1
(3) (a) 2.56s (b) 23m (c) ii) 40s
3 2
(4a)
x
2
1
(4b) k = 3
(4c) 12 x 2  6 x 2 (5b) 76
(5c) 34 (6a) x = 2
(6b) x = 3
3
2
(7a) Hint: if (2x – 1) is a factor of 2x + x + x – 1 then when you divide by (2x – 1) the
remainder would be…… Check whether this is true!
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(7b)  3
(8a) x + 2y – 16 = 0 (8b) y = – 4x
(9a) u2  k 2  1 , u3  k 4  2k 2
(10b)
π 2π 4π 5π
π 5π 2π 4π
(9c) , , ,
,
,
,
3 3 3 3
6 6 3 3
 6 53 
, 
7 7 
(8c) 
(9b) k  2
(10a)
π 7π
,
6 6
(11a) x  2 y  15  0
(11b)
225
4
(12)
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