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“To be a scholar of mathematics you must be born with talent, insight, concentration, taste,
luck, drive and the ability to visualize and guess”
Halmos, Paul R.
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Y1 Double Maths Assignment  (zeta)
Half term review: 21 questions including an
exam style mechanics question
Due in w/b 4th November
 VJM section is now due in on
(initials are for shared group) please note
Mondays and TAG section on Tuesdays
You have now completed a quarter of year 1 of your Double Maths
course! Well done!
Your C1/C2 end of module mocks are in w/b 11th November in lesson time
Section A – Core 1 & Mechanics (VJM)
Mechanics (give your answers to an appropriate degree of accuracy)
Kinematics:
Write down Newton’s equations of motion from memory, state any assumptions
made. Model the following situations with a suvat table (you may need more than
one) and then solve the problems.
i.
(1a)
A particle is moving along a straight line. It passes point B, 3 seconds after passing
point A, and it passes point C, 5 seconds after passing point B. If AC is
80 m and the velocity of the particle at A is 5 m s–1 find the acceleration, assumed
constant of the particle and the distance AB.
ii.
A particle is moving along a straight line with constant acceleration. It passes through
points A, B and C. It takes 2 secs to travel from A to B, a distance of
14 m, and it takes 3 secs to travel from B to C, a distance of 36m. Find the
acceleration of the particle, and the speed as it passes through point A.
iii.
A man on top of a tower of height 40m holds his arm over the side and throws astone
of mass 200g vertically upwards with a speed of 15ms-1.
Find the time taken for the stone to reach the ground
Find the speed of the stone as it hits the ground
Statics
(1b) Draw a labelled diagram and form equations for each of the following
situations: then solve them.
A picture of mass 5 kg is suspended by two light inextensible, each inclined at 450 to
the horizontal as shown. By modelling the picture as a particle, find the tension in the
strings when the system is in equilibrium
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
M1 Exam practice
2. Andrew hits a tennis ball vertically upwards towards his sister Barbara who is leaning out
of a window 7.5 m above the ground to try to catch it. When the ball leaves Andrew’s racket,
it is 1.9 m above the ground and travelling at 21 m s-1 Barbara fails to catch the ball on its way
up but succeeds as the ball comes back down.
Modelling the ball as a particle and assuming that air resistance can be neglected,
(a) find the maximum height above the ground which the ball reaches. (4 marks)
(b) find how long Barbara has to wait from the moment that the ball first passes her until she
catches it.
(6 marks)
CORE 1
3.
(a)
By completing the square, find in terms of the constant k the roots of the
2
equation x  4kx  k  0
(b)
Hence find the set of values of k for which the equation has no real roots.
4.
A triangle has vertices P2, 3, Q 4, 9, R 5, 2
5.
(a)
Find the exact perimeter of the triangle
(b)
Show that the triangle is right angled
The points P and Q have coordinates 7, 4 and 9, 7 respectively. The straight
line m has gradient 8 and passes through the origin, O. The lines l and m intersect at
the point R
6.
(a)
Find an equation for the straight line l which passes through P and Q. give
your answer in the form ax  by  c  0 where a, b and c are integers.
(b)
Write down an equation for m.
(c)
Show that OP = OR (AB means the distance between the points A and B).
The graph below shows the curve with equation y  f (x) which crosses the x axis at
2
the origin and at the points A and B. Also, f ( x)  6  4 x  3x .
A
7.
B
(a)
find an expression for y, showing how you found the value of c
(b)
show that AB  k 7 , where k is an integer to be found (AB means the
distance between the points A and B).
A curve has the equation y  x 
3
,
x
x0.
The point P on the curve has x coordinate 1.
(a)
Show that the gradient of the curve at P is – 2.
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8.
(b)
Find an equation for the normal to the curve at P, giving your answer in the
form y = mx + c.
(c)
Find the coordinates of the point where the normal to the curve at P intersects
the curve again (DO NOT MISS OUT THIS PART OF THE QUESTION!!).
The line l1 has equation x  2 y  1  0 . The line l2 is perpendicular to l1 and passes
through the point A(1, 5).
(a)
Show that l1 and l2 cross at the point (–1, 1)
The points B(–3, 2) and C(3, –1) lie on l1.
(b)
9.
10.
How good is your GCSE algebra now?! Make x the subject of these formulae:
x2  y 2  r 2
(a)
ax  b  2 x
(d)
l
1 2
2 x
(e)
x 2  kx  0
(g)
x3  kx2  0
(h)
 x  a
(b)
2
(c)
ax  b
k
cx  d
(f)
x2  a2  0
b
The first term of an arithmetic series is -5 and the 9th term of the series is 1.
(a)
(b)
11.
Find the area of the triangle with vertices A, B, C.
Find the common difference
How many terms of the series are less than 50?
The sequence of terms u1 , u 2 , u 3 , is defined by u1  8, u n 1  au n  1 where a is
a negative constant. The third term of the sequence is 8. Find the value of a.
Section B – Core 2 (TAG)
12
Given that p = logq 16, express in terms of p,
(a)
13
14
logq 2,
(b)
logq (8q).
Given that log 3 x  k , find, in terms of k,

 x 12
log 3 
 3





(a)
log3  x
(d)
 x 12
Hence, or otherwise, solve log 3  x   log 3 
 3

(a)
Sketch, for 0  x  360, the graph of y = sin (x + 30).
(b)
Write down the coordinates of the points at which the graph meets the axes.
(c)
Solve, for 0  x < 360, the equation
2
(b)
log3  9x 
2
(c)

  log 3  9 x   0


sin (x + 30) =  12 .
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15
(a)
Find the value of x for which the curve y  800 x 
2
, x > 0, has a stationary
x
point.
(b)
16
Using the second derivative, determine whether this point is a maximum or
minimum point.
The diagram shows the cross-section of a ball, radius r cm floating in
water. The surface of the water touches the ball at A and B. AB
2π
subtends an angle of
at the centre of the ball.
3
(a)
Find the length of AB in terms of r.
(b)
Show that, in terms of r, the area of the cross section of the ball
which is above the surface of the water is
17
A


1 2
r 8  3 3 .
12
A flower bed is designed as follows: Triangle DEF is an equilateral triangle of side
8m. Arcs EF, FD and DE have radii 8m and centres D, E and F respectively. Find
the area of the flower bed correct to 1dp.
n.b. This diagram is NOT a circle with
an inscribed equilateral triangle!!
18
Solve the following equations for x:
 
32 x  5 3x  4  0
(a)
(b)
3
(c)
log x 2  log 2 x  2
19
Find the length of r and hence the total area of the shape,
2𝑥+1
+ 5 = 16(3𝑥 )
(d)
4cm
2 log 4 x  log 4  x  1 
1
2
95°
r
r
60°
40°
dx
 3t 2  2t  1 and that x = 2 when t = 1, find the value of x when t = 2
dt
20
Given that
21
The circle C has centre (5, 2) and passes through the point (7, 3).
(a)
(b)
(c)
Find the length of the diameter of C
Find an equation for C
Show that the line y  2 x  3 is tangent to C and find the coordinates of the
point of contact
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B
Mechanics answers
75
1. a)1) 𝑎 = 5, 2 = , ii) 𝑢 = 6.9, 𝑎 = 0.13, iii) 𝑡 = 5
2
1.b) 𝑇 = 35
2. (a) it reaches 24 m above ground level (b) Barbara waits for 3.7 seconds
Answers to Section A
1
4
(3a) x  2k 
(5a)
4k 2  k (3b)   k  0
3x  2 y  13  0
(5b)
y  8x
y  6x  2x  x
1
7
 13 
(7b) y  x 
(7c)  6,

2
2
 2
2
(6a)
(4a)
3
(5c)
3 10  5 2
OP  OR  65
(6b) k = 2
(8b) 15
b
a2
(9f) x  a
(10a) d = ¾
(9a) x 
(9b) x   r 2  y 2 (9c) x 
dk  b
a  ck
(9d) x 
1
2l 2
(9e) x  0, x  k
xa b
(9g) x  0, x  k
(9h)
(10b) 74
(11) a = – 1
Answers to Section B
(12a)
p
4
(12b)
(13c) 12 k  1
3
p+1
4
(13d) 9
(21a) 2 5
(21b)
(13b) k + 2
(14a) Sketch
(14b) (0, 0.5) (150, 0) (330, 0)
(16a) 3r cm
(17) 45.1m2
(18a) x  log 3 4 or x = 0
(19) 4.40cm, 18.9cm2
(20) x  t 3  t 2  t  1, x  7
(13a) 2k
(14c) x = 180 , 300
(18b) 𝑥 = −1, 𝑥 = 1.46
x  52   y  22  5
(15a) x = 1/20
(18c) x = 2 (18d) x  1  3
(21c) (3, 3), (3,1)
Updated 16/10/13
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M THS
ASSIGNMENT COVER SHEET zeta
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 overcame them as well
as the things which went well or which you enjoyed/learned from.
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