IFYFM002 Further Mathematics 2019-20
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
IFYFM002 Further Mathematics
2019-20
Section A
Answer ALL questions. This section carries 40 marks.
Question A1
(a) A complex number, , has modulus 24 and argument .
Write in the form + where is an integer and is in surd form.
(b) Show that can be simplified to +
Question A2
3 21 |
2 |
The determinant of matrix M is 12 .
Find the values of .
Question A3
Solve the inequality
<
Question A4
Find the sum of 2 + 16 + 54 + 128 + … .. + 43904 .
Give your answer in full with no rounding off.
Question A5
The roots of the equation 22 − 5 − 1 = 0 are and .
Find the equation with roots (2 + ) and ( 2 + ).
Give your answer in the form 2 + + = 0 where , and are integers.
Question A6
A missile is fired vertically upwards at 500 ms-1 .
Find its height after 1 seconds. Give your answer to 2 significant figures.
In this question, 1 mark will be given for the correct use of significant figures.
Question A7
Without using exponentials, solve
2 sinh2 − 9 cosh + 6 = 0.
Give your answers in exact logarithmic form.
Question A8
A parabola has parametric equations = 82 and = 16 .
(a) Find the equation of the tangent to the parabola in terms of .
This tangent meets the directrix of the parabola on the − axis.
(b) Find the values of .
Question A9
A curve has equation
=
Show that the curve has no stationary values.
Question A10
Point A with position vector (3 − 2 + ) lies in a plane.
The vector (2 + + 3) is perpendicular to the plane.
Find an equation of the plane in the form . = .
Question A11
Solve the differential equation
− 3 = 5
(where is a constant) subject to = when = 0 . Give your answer in the form = () where () is in terms of and .
Section B
Answer THREE questions ONLY. This section carries 60 marks.
Question B1
(a)
P
Q
Figure 1
Figure 1 shows two particles P and Q which are connected by a light
inextensible string over a smooth pulley.
P has mass kg and Q has mass 1.8 kg.
The system is released from rest and Q moves downwards with acceleration 0.28 ms-2 .
Find the value of and the tension in the string.
20 N
30o
|
|
|
A B
Figure 2
Figure 2 shows a block of mass 16 kg being pulled along a smooth horizontal surface by a light inextensible rope which is inclined at 30o to the horizontal. The tension in the rope is 20 Newtons. The block starts from rest at point A, and it reaches point B after 8√3 seconds.
i. Find the speed of the block when it reaches point B.
The surface beyond point B is rough. The tension in the rope is kept at 20 Newtons but the block now travels at a constant speed.
ii. Find the coefficient of friction between the block and the surface beyond point B.
Parts (c) and (d) are on the next page.
(c)
Question B1 – (continued)
K
L
|
|
|
Figure 3
m
Figure 3 shows two particles K and L which are at the top of two slopes.
K has mass 7 kg and is metres vertically above the horizontal. L has mass 4 kg and is metres vertically above the horizontal. All surfaces
are smooth.
Both particles are released from rest.
i. Find the speeds of K and L when they reach the bottom of their slopes.
The particles then travel along a horizontal surface until they collide. After the collision, K is brought to rest.
iii. Find the coefficient of restitution between the particles.
(d) A car is travelling at a constant speed of 20 ms-1 up a smooth slope which is inclined at ° to the horizontal where sin = .
The power output of the engine is 13720 Watts.
Find the mass of the car.
1 |
0 |
1 |
0 |
1 |
− 1 |
1 |
− 1 |
|
i. Find, in terms of , the characteristic polynomial of Matrix A.
You do not need to simplify your answer.
ii. Given that one of the eigenvalues of A is 3, find the value of
iii. Find the other two eigenvalues of A.
.
ii. For each eigenvalue found in part iii find a corresponding eigenvector
(b) If = tanh−1 ( ), prove that (9 − 2 ) = 3 .
(c) Evaluate
ln 2
∫ sinh2 .
0
Give your answer in the form − ln√ where , and are integers.
Question B3
(a) The hyperbola with Cartesian equation − = 1 has parametric
equations = 4 sec and = 3 tan where is a parameter.
i. Find the eccentricity.
ii. Explain why your answer to part i is sensible.
iii. Find the equation of the tangent to the hyperbola when = .
Give your answer in the form = + .
The tangent found in part iii crosses the − axis at point Y, and the directrix of the hyperbola with positive − value at point X.
iv. Find the length of XY, giving your answer in the form where and are integers.
(b) A curve has parametric equations = + 1 and = √( + 1) where
is a parameter.
i. State the range of values of where the curve is defined.
ii. Write a Cartesian equation of the curve.
iii. The integral is defined as
5
= ∫ 2 .
3
Write in terms of .
iv. Hence evaluate giving your answer in exact form.
v. Find, for > 0, the equation of the normal to the curve when = 3.
Give your answer in the form + + = 0 where , and are integers.
Question B4
(a) i. Write down the quadratic equation with roots (4 + 2) and (4 − 2).
roots (4 + 2) and (−4 + 2).
(b) Solve the equation
642 − = 20.
(c) Find the values of and if ( + )(3 + 3) = −33 + 15 .
(d) The locus of a complex number, , is defined as
| + 6 − 4| = | − 2 + 4|
Find the Cartesian equation of the locus.
Give your answer in the form + + = 0 where , and are integers.
(e) i. Expand (cos + sin )3 .
ii. Express cos 3 in terms of cos 3 and cos .
iii. Hence find the exact value of
3
∫ cos 3 .
6
Question B5
(a) i. By differentiating a suitable number of times, obtain a Taylor expansion
of cos ( + ) up to the term in 2 .
ii. Hence find an approximate value of cos 125° in terms of .
(b) i. Show that
∑ (32 − + 1) = (2 + + 1)
=1
ii. Hence find
(3 × 100 − 9) + (3 × 121 − 10) + (3 × 144 − 11) + ⋯ + (3 × 900 − 29).
Give the answer in full with no rounding off.
(c) A second order differential equation is given by
− 6 + 9 = 182 + 3 + 9 − 14
where is a constant.
ii. Find a particular integral.
iii. Find the particular solution, given when = 0, = 2 + and = 10.
2022-05-26