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# Further Mathematics代考 IFYFM004代写

2022-06-08 11:16 星期三 所属： 数学代写 浏览：98 ## 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

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

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

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 ]

b)

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 .

a)Find the value of u  and the valueof v.[ 5 ]

b)Find the total kinetic energy lost in the collision.[ 2 ]

### Question 12

a)Prove by induction that

[ 5 ]

Express Y in terms of n in its simplest form, and hence find the value of n if Y = 0.008.[ 4 ]

### Question 13   Further Mathematics代考

Matrix A is defined as a)Find the eigenvalues ofmatrix A. [ 5 ]

b)For each eigenvalue found in part a), find acorresponding eigenvector.[ 6 ]

### Question 14

a)Solve theinequality

b)You are given the curve has no stationary values.

Sketch the curve . This must not be done on graph paper.

On your sketch, show clearly any asymptotes and the coordinates where the

curve crosses the y − axis. [ 3 ]

c)Draw and label the line y = x + 3 on your sketch.

Show clearly the coordinates where the line y = x + 3 and the

curve intersect. [ 2 ]

### Question 15

Give your answer as a single logarithm and in exact form. [ 4 ]

c)Point P lies at on the curve Find the length of the arc of the curve between the origin and point P.

Give your answer in the form where p and q are integers. [ 5 ] 