Further Mathematics代考 IFYFM004代写

IFYFM004 Further Mathematics

Examination Exemplar

Further Mathematics代考 All working must be shown. Just giving the answer, even the correct one, will score no marks if this working is not seen.

INSTRUCTIONS TO STUDENTS

Answer ALL questions

The marks for each question are indicated in square brackets [ ].

A formula booklet and graph paper will be provided. Further Mathematics代考
An approved calculator may be used in the examination.
Show ALL workings in your answer booklet.
Examination materials must not be removed from the examination room.

DO NOT OPEN THIS QUESTION PAPER UNTIL INSTRUCTED BY THE INVIGILATOR

Question 1

The quadratic equation 3x²− 6x − 2 = 0 has roots α and β.

Without working out the values of and , find the equation with roots α³and β³.

Give your answer in the form ax²+ bx + c = 0 where a, b and c are integers.[ 4 ]

Question 2 Further Mathematics代考

Solve cosh 2x +sinh²x − 13 sinh= −3.

Give your answers in logarithmic form.[ 4 ]

Question 3

a)By differentiating a suitable number of times, obta in the Taylor expansion of

sin in ascending (increasing) powers of x up to the term in x².[ 3 ]

b)Use your result in part a) to find an approximate value of sin 36° giving your answer in terms of π.

Question 4 Further Mathematics代考

a)If (a + 3i)(b − i) = 31 + 17i, find the possible values of a and b.[ 5 ]

b)The complex number w is defined as w = 9 +40i.

Find the modulus of w and the argument of w*. Give the argument of w* to 2 decimal places.

In this question, 1 mark will be given for the correct use of decimal places.[ 4 ]

Question 5

Further Mathematics代考

In this question you may use the results and

a)Expand and simplify your answer.[ 2 ]

b)Hence express sin68 in the form a cos 6θ + b cos 4θ + c cos 2θ + dwhere

a, b, c and d are rational numbers. [ 3 ]

c)Hence evaluate

Give your answer in exact form.

All working must be shown. Just giving the answer, even the correct one, will score no marks if this working is not seen.[ 3 ]

Question 6 Further Mathematics代考

A curve has parametric equations x = √t + 1 and y = t² − 1 (t ≥ 0)

a)Write down a Cartesian equation of the curve.

Give your answer in the form y = ƒ(x). [ 2 ]

b)Write down the equation of the normal to the curve when .

Give your answer in the form ax + by + c = 0 where a, b and c are integers. [ 3 ]

c)Integral I is definedas

Write the integral in terms of t and hence evaluate I.

All working must be shown. Just giving the answer, even the correct one, will score no marks if this working is not seen.[ 5 ]

Question 7

An ellipse has parametric equations x = θ cos θ and y = 6 sin θ.

a)Derivethe equation of the tangent to the ellipse, and show that it can be written as 4y sin θ + 3x cos θ =24 [ 4 ]

b)The tangent crosses the x − axis at point P and the y − axis at point Q. Point M is the mid-point of PQ.

Write down the coordinates of point M in terms of θ. [ 2 ]

c)Find the Cartesian equation of the locus of point M as8 [ 2 ]

Question 8 Further Mathematics代考

In this question, all working must be shown.

a)Vectorsa and b are defined as a = 2i + 4j − 5k and b = −3i + j + =2k.

Find a × b. [ 2 ]

b)Find the coordinates of the point of intersection of the line withequation r = 3i − 4j + 2k + µ(i + 2j − 3k) and the plane which has equation (−2i + 3j − 4k) = 22.[ 3 ]

c)The position vector of a particle at time t seconds is given by

r = (−4t3 + 3t2 + 9t)i + (6t2 + 5)j + (2t + 6)k

where i, j and k are 3 mutually perpendicular vectors of length 1 metre.

Find the magnitude of the velocity of the particle when . [ 4 ]

Question 9 Further Mathematics代考

A second order differential equation is defined as

a)Find the complementary function and aparticular integral.[ 5 ]

b)Find the particular solution given y = 5 and ^dy= 2 when x =0. [ 4 ]

Question 10

Figure 1 shows two particles A and B which are connected by a light inextensible string over a smooth pulley.

Particle A has mass M kg and particle B has mass (M + 2) kg. The system is released from rest.

Find the acceleration of the particles and the tension in the string.

Give your answers in terms of M and g, and in their simplest form. [ 5 ]

Figure 2 shows a crate of mass 20 kg being pulled along a rough horizontal surface by a rope. The tension in the rope is 60 Newtons and it is inclined at θ° to the horizontal where tan θ = .

The crate starts from rest and accelerates at 0.8 ms^-2.

Find the coefficient of friction between the crate and the surface and how

long, in seconds, it takes the crate to travel 10 metres. [ 5 ]

Question 11

Figure 3

Figure 3 shows two spheres X and Y on a smooth horizontal surface. X has mass 4 kg and is travelling at 5 ms^-1. Y has mass 6 kg and is travelling at 4 ms^-1 in the same direction as X. Further Mathematics代考

The spheres collide. After the collision, both spheres continue to move in the same direction as before the collision with X travelling at u ms-1 and Y travelling at v ms^-1.

The coefficient of restitution between the spheres is .