IFYFMOOl Further Maths o NCUK THE NORTHERN CONSORTIUM THE NCUK INTERNATIONAL FOUNDATION YEAR (IFY) Further Mathematics Examination Session Time Allowed Summer 2009 3 hours 10 minutes (Including 10 minutes reading time) INSTRUCTIONS TO STUDENTS SECTION A Answer ALL questions. This section carries 40 % of the exam marks. SECTION B Answer FOUR questions. This section carries 60 % of the exam marks. The marks for each part of the question are indicated in square brackets [ ]. • No answers must be written during the first 10 minutes. • Write your Candidate Number clearly on the Answer Book in the space provided. • Write your answers in the Answer Book provided. Additional sheets will be provided on request. • Clearly write the number and parts of questions attempted at the start of each answer. • No written material is allowed in the examination room. • No mobile phones are allowed in the examination room. • An approved calculator may be used in the examination. • State the units where necessary. • Where appropriate, working should be carried out to 4 significant figures and answers given to 3 significant figures. • Full marks will only be given for full and detailed answers. • Students will receive a formula book. Page 1 of 7 V30809
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IFYFMOOl Further Maths
o NCUK
THE NORTHERN CONSORTIUM
THE NCUK INTERNATIONAL FOUNDATION YEAR (IFY)
Further Mathematics
Examination Session Time Allowed Summer 2009 3 hours 10 minutes
(Including 10 minutes reading time)
INSTRUCTIONS TO STUDENTS
SECTION A Answer ALL questions. This section carries 40% of the exam marks.
SECTION B Answer FOUR questions. This section carries 60% of the exam marks.
The marks for each part of the question are indicated in square brackets [ ].
• No answers must be written during the first 10 minutes. • Write your Candidate Number clearly on the Answer Book in the space
provided. • Write your answers in the Answer Book provided. Additional sheets will
be provided on request. • Clearly write the number and parts of questions attempted at the start
of each answer. • No written material is allowed in the examination room. • No mobile phones are allowed in the examination room. • An approved calculator may be used in the examination. • State the units where necessary. • Where appropriate, working should be carried out to 4 significant
figures and answers given to 3 significant figures. • Full marks will only be given for full and detailed answers. • Students will receive a formula book.
Page 1 of 7 V30809
IFYFMOOl Further Maths
Section A Answer ALL questions.
This section carries 40 marks.
Question A1
Let z = 5 - 3i and w =-2 + 4i .
(i) Find the value of 3z - w. [ 1 ]
w (ii) Express in the form a + bi, where z* is the complex conjugate [4 ]
w+z* of z.
Question A2
-12 -10 -10J
Let the matrix B = 5 3 5 . (
10 10 8
(i) Find B 2 • [ 2 ]
(ii) Hence show that B 2 - B is a multiple of the identity matrix, I. [ 1 ]
(iii) Hence find B-1 • [ 2 ]
Question A3
A uniform rod, AB, of length 70 cm and mass 5 kg, is freely hinged at a point A on a vertical wall. The rod is held in equilibrium horizontally by a light inextensible string attached to the wall at C and to a point P on the rod 45 cm from A as shown in Figure 1. The string makes an angle of 30° with the horizontal.
C
PA ~=---'----------,-'--~"------ B
5g
Figure 1
Page 2 of 7 V30809
IFYFMOOl Further Maths
(i) Find the tension in the string. [ 2 ]
(i i) Show that the magnitude of the reaction force, F, at the hinge A is [ 3 ]
5Ji5l --g.
9
Question A4
Solve the differential equation
d 2 y dy--9--lOy =60 dt 2 dt
.completely, given that y =2 and
dy-d1
= 14 when 1 =O.
[ S ]
Question AS
The lines I[
II : 12 :
and 12 have equations
r =(5,-4,6)+ s(2,- 5,4) r =(4,10, - 4)+ 1(3,4, - 2)
(i)
(i i)
Show that II and 12 intersect and find the coordinates of the point of
intersection.
Find the Cartesian equation of the plane in which II and 12 lie.
[ 3 ]
[ 2 ]
Question A6
A particle of mass 1.6 kg is attached to the end B of a light inelastic string AB of length 1.5 m, the other end of which is fixed at A. The particle moves in a horizontal circle whose centre is vertically below A with a constant angular speed. The particle takes 1.2 s to complete one revolution.
Let the acceleration due to gravity, g, be 9.8 ms -2, and take 1i as 3.14.
(i)
(i i)
Calculate the tension in the string.
Find the radius of the circular path of B.
[ 2 ]
[ 3 ]
Page 3 of 7 V30809
IFYFMOOl Further Maths
Question A7
(i)
(i i)
2 A B Express in the form - +- .
(r+1Xr+3) r+1 r+3
13 2 Hence find the exact value of I ( X ).
r=3 r + 1 r + 3
[ 2 ]
[ 3 ]
Question AS
The curve, C, has equation y = 13 cosh x + 12 sinh x .
(i)
(ii)
( iii)
Find the exact value of the x -coordinate of the turning point on terms of a natural logarithm.
Hence find the exact value of y.
Determine the nature of the turning point.
C in [ 3 ]
[ 1 ]
[ 1 ]
Page 4 of 7 V30809
IFYFMOOl Further Maths
Section B Answer 4 questions. This section carries 60 marks.
Question 81
B
Figure 2
Figure 2 shows two particles A and B, of mass 5 kg and 6 kg respectively. They are connected by a light inextensible string which passes over a light smooth pulley P. Particle A rests on a smooth plane inclined at an angle of 30° to the horizontal. Particle B rests on a rough horizontal plane. The string is parallel to the line of greatest slope of the inclined plane. Let the acceleration due to gravity, g be 9.8 ms-2
•
(i) Draw a diagram to show the forces acting on the bodies A and B. [ 2 ]
(ii) If the coefficient of friction, j.i, is 0.15, calculate the acceleration of [ 5 ]
body B.
(iii) Find in Newtons, the tension in the string. [ 1 ]
(iv) After travelling from rest for 1s, the string breaks. Calculate the time [ 4 ] taken for B to come to rest given that it does not reach the pulley.
(v) What is the total distance that B travels? [ 3 ]
Question 82
(i) Use integration to find the centre of mass of a uniform semicircular [ 6 ] lamina of radius 4 cm. You may use the formula for the area of a circle.
lOcm BA r-o::::::::----------.....-_ I I I I
I 4cm: ,
I I : c
---01--------- ,------------------~--------- _.I I ,,I ,I I II , ,I
II
E D
Figure 3
Page 5 of 7 V30809
IFYFMOOI Further Maths
(ii)
(iii)
(iv)
A semicircular section AFE is removed from a uniform rectangular lamina ABDE and placed at BCD to form the uniform lamina ABCDEF as shown in Figure 3. If the semicircles have radii 4 cm find the position of the centre of mass of the lamina.
the lamina ABCDEF is freely suspended from A. Find the angle AB makes with the vertical.
A particle of mass pm, where m is the mass of the lamina, is added at
E. Find the value of J1 so that AB makes an angle of 45° with the
vertical.
[ 3 ]
[ 2 ]
[ 4 ]
Question 83
(i)
(ii)
(iii)
(iv)
(v)
x2 y2 Show that the point P = (4cost, 7sint) lies on the ellipse -+ = 1.
16 49
Find the equation of the tangent at P.
Find the equation of the normal at P.
The normal meets the axes at the points Q and R. Lines are drawn
parallel to the axes through the points Q and R;. these lines meet at
the point V . Find the coordinates of V .
x2 y2 Find the equation of the locus of V as t varies in the form -2 +-2 = 1.
a b
[ 1 ]
[ 4 ]
[ 3 ]
[ 2 ]
[ 3 ]
(vi) Find the eccentricity and foci of the locus of V . [ 2 ]
Question 84
(i)
(ii)
(a)
(b)
In an Argand diagram, the point P represents the complex number z, where z =x + iy. Given that z 2 =Ai(z 8), where
A is a real parameter, show that as A varies the locus of P is a circle, and find its centre and radius.
If in (i) above z=J1(4-3i), where J1 is real, prove that there is
only one possible position for the point P and find its coordinates.
Use De Moivre's theorem to solve (z - 2Y=-8 for z .
[ 7 ]
[ 4 ]
[ 4 ]
Page 6 of 7 V30809
IFYFMOOI Further Maths
Question 85
(a) The roots of the cubic x 3 + bx2 + ex + d are a, f3 and r.
( i) Show that a 3 + /33 +r3 =_b 3 + 3bc -3d. [ 5 ] 3(ii) Given the cubic x - 4x2 + X + 5 evaluate [ 4 ]
a f3 ~ /3 r a.
r a
2x(b) Differentiate j(x) =e cosx a suitable number of times and hence find [ 6 ] the first four terms of its Maclaurin series.
Question 86
(a) ( i) Express 4x2 -12x + 25 in the form (px + qY+ r . [ 2 ]
( ii) Hence find 3f.J
dx in terms of a natural logarithm. [ 8 ]
3/2 4x 2 -12x + 25
(b) Find the surface area of a parabolic mirror obtained by rotating the [ 5 ] (2 (
parabola X=-, y=- from (0,0) to (4,2) about the x-axis, giving4 2
your answer correct to one decimal place.
Page 7 of 7 V30809
IFYFM001 Further Maths
NCUK THE NORTHERN CONSORTIUM
THE NCUK INTERNATIONAL FOUNDATION YEAR (IFY)
Further Mathematics
Mark Scheme
V30809 Page 1 of 10
IFYFMOOl Further Maths
Level of accuracy: If a question specifies how many decimal places or significant figures are required, there is a mark for this. Otherwise accept any reasonable level of accuracy and alternative form.
Error carried forward: Where numerical errors have been made, students lose a mark/marks at that stage but may be awarded marks for using correct methods subsequently if the student demonstrates basic understanding.
Section A
A1(i) z = 5 - 3i, W = -2 + 4i, 3z - W = 17 -13i . I
A4 The auxiliary equation is n" - 9n -10 = 0 . This factorises as (n + 1Xn -10) = 0 with roots n = -1, 10.
+ Be lOIThe complementary function is Ae-I • I
For a particular integral try y = C . Then - 10C = 60 , so C = -6.
+ Be lOIThe general solution is y = Ae -I - 6. I
When 1 = 0, A + B - 6 = 2, A + B = 8 I
Differentiating: dy = -Ae-' + 10Be lol
dl Whenl=0,-A+IOB=14 I
Solving, A = 6, B = 2
So y = 6e-1 + 2e lOI - 6 I
A5(i) II: r=(5,-4,6)+s(2,-5,4)
12 : r=(4,10,-4)+1(3,4,-2) If the lines intersect, then at the point of intersection (5, - 4,6) + s(2, - 5,4) = (4,10, - 4) + 1(3,4, - 2) giving the equations:
5 + 2s = 4 + 31 2s - 31 = -I - 4 - 5s = 10+ 41 5s + 41 = -14 6 + 4s = -4 - 21 4s + 21 = -10 I Solving the first two equations, s = -2, I = -1, and these values satisfy the third equation also, so the lines intersect. I There r = (5,-4,6)-2(2,-5,4)= (1,6,-2) I
(ii) The vector equation of the plane is r = (1,6, - 2) + s(2, - 5,4) + 1(3, 4, - 2)
The normal is (2, - 5,4)x (3,4, - 2) = (- 6,16,23) I
so the equation of the plane is - 6x + 16y + 23z = 44 I
(iv) The velocity after l] s when the string breaks is
u2 = all] = 1.425 xl = 1.425 ms- 1 1
Now 6a2 = -f.16g, so a2 = -f.1g = -1.47ms-2 1
The time to rest is l2 = _!!.l- = 1.425 = 0.9697 :::::: 0.970 s 2 a2 1.47
(v) The distance travelled before the string breaks is Sf = ~a)lj2 = 0.7127m 1 2
The distance travelled after the string breaks is S2 = -~a2l/ = 0.6911 m 1
The total distance travelled is Sj + S2 = 1.4038 :::::: 1.40 m 1
V30809 Page 5 of 10
B2(i)
(ii)
(iii)
(iv)
IFYFM001 Further Maths
Place the semicircular lamina to the right of the y -axis, with its centre at the
origin. By symmetry the centre of gravity will lie on the x -axis. I Divide the lamina up with vertical strips of width &. The mass of the strip at
distance x from the origin is 2p .J16 - x 2 & where p is the density. I
The mass of the lamina is -.!.. Jl"4 2 P = 8np I 2
Taking moments about the origin we see that, letting the centre of gravity be (x, 0)
8np X = 4J2 p x.J16 - x 2 dx o
=Hp(16-X')'" I 128
=-p MIAI 3
So x = 128p = 16 I 3 x 8np 3n
Once again, by symmetry, the centre of gravity lies on the x -axis. Call this point (x, 0). (not the same x) The mass of the lamina is 8 xl 0P =80 p. I Taking moments about the origin,
80pi = 80p x 5 - 8np x ~ + 8np x (10 + ~J = (400 + 80n)p3n 3n
So x = 5 + n. MIAI If the side of the lamina hangs at angle {} to the vertical, then
4tan{) =-- ~ 0.4913 I
5+n So {} = 26.17° ~ 26.2° I
After the particle is added let the new centre of gravity be (X, Y) Taking moments:
- ( ) - 5 + n(m + j.1Jn )X = m 5 + n so X =-- I 1+ ,u
(m+ j.1Jn)Y = j.1Jn(-4) so Y = -4,u I 1+ ,u
If the system now hangs at 45° to the vertical then X = 4 - Y. SO 5 + n =4 _ - 4,u
1+,u 1+,u
5 + n = 4(1 + ,u)+ 4,u = 4 + 8,u 1+ n
,u =-- = 0.5177 ~ 0.518 MIAI 8
V30809 Page 6 of 10
B3(i)
(ii)
(iii)
(iv)
(v)
(vi)
IFYFMOOl Further Maths
2 216cos t 49sin t • 2 17 + =cos-t+sm t= 1
16 49 Differentiating implicitly
2x + 2y dy = 0 16 49 dx
So dy = _ 49x = _ 49x 4cost = _ 7cost 1 dx 16y 16x7sint 4sint
... h . f' h . 7 cos tTherelore t e equatIOn 0 t e tangent IS y = - -,- x + c 1 4smt
where, since the tangent must pass through P
. 7 cos t 4 7 sin 2 t + 7 cos 2t 7 c = 7 sm t +-- cos t = = 1
4 sin t sin t sin t .. 7cost 7
So t he equatIOn IS Y = --,-x + -. 4smt smt
or 4(sint)y+7(cost)x=28 1
f hlP' 4 sin tTh d· t at - e gra lent 0 e norma IS 1 7 cost
. f h 1 . 4 sin tSo t he equatIOn 0 t e norma IS Y = --x + c 7cost
. 4sint 4 49-16. 33.h -- sm t = - sm t 1were c = 7 sm t - cos t =
7 cost 7 7 ... h .. 4sint 33.
Therelore t e equatIOn IS Y =-- x + - sm t 7 cost 7
or 7(cost)Y - 4(sint)x = 33 sin t cost 1
33 At Q, Y = 0, so x = --cost 1
4
At R, x = 0, so y = 33 sin t 7
So the coordinates of V are [ - 3: cos t, 3; sin t) 1
At V cost = __x_ sint = -y- 2 , 33/4' 33/7
x2 y2 so the equation of the locus of V is + = 1 1
(33/4)" (33/7)222 g2For the ellipse ':;-+~=1, b 2 =a 2 (1-e 2 ),so e= 1--