THE NCUK INTERNATIONAL FOUNDATION YEAR

IFYFM002 Further Mathematics

End of Semester One Test

2020-21

Examination Session

Semester One

Time Allowed

3 hours 10 minutes


INSTRUCTIONS TO STUDENTS

SECTION A           Answer ALL questions. This section carries 40 marks.

SECTION B           Answer 4 questions ONLY. This section carries 60 marks.

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

•    You must handwrite your answers on paper. Once complete, your handwritten work must be clearly scanned or photographed and then inserted into a word-processed document for submission.

•    Work must be submitted in a single word-processed file.

•    You MUST show ALL of your working.

•    An approved calculator may be used in the assessment.

•    All work must be completed independently. The penalty for collusion is a mark of zero.

•    Due to the nature of the questions, there should be no need to use external sources of information to answer them. If you do use external sources of information you must ensure you reference these. Plagiarism is a form of academic misconduct and will be penalised.

•    Work must be submitted by the deadline provided. Your Study Centre can be contacted only for guidance on submission of work and cannot comment on the contents of the assessment.


Section A

Answer ALL questions. This section carries 40 marks.

Question A1

The complex number z is defined as z = 5 - 3j.

Find | 20 ÷ z* | where z* is the complex conjugate of z.

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

Give your answer to 4 significant figures.                      [ 4 ]

In this question, 1 mark will be given for the correct use of significant figures.


Question A2

Find the values of a if the determinant of the matrix below is 15.                     [ 4 ]



Question A3

(a) On the same axes sketch the graphs of (This must not be done on graph paper.)                 [ 2 ]

(b) Find the x - values where intersects with y = 2x + 1.                 [ 1 ]

(c) Use your sketch, or otherwise, to write down the range of values of x which satisfy the inequality                [ 2 ]

Question A4

Find the smallest value of n for the series

to exceed 2000.                                          [ 3 ]



Question A5

The quadratic equation has roots and .

Without working out the values of and , find the quadratic equation with roots

Give your answer in the form where a, b and c are integers.            [ 4 ]


Question A6

A car of mass 1200 kg is being driven up a smooth slope which is inclined at to the horizontal where The car’s acceleration is 2

(a) Find the driving force of the car’s engine.                        [ 2 ]

(b) When the car reaches a speed of 15 it passes a tree.

     It passes a second tree 3 seconds later.

     Find the distance between the two trees.                           [ 2 ]


Question A7

Solve the equation

Give your answer in exact logarithmic form.                   [ 4 ]


Question A8

A curve has parametric equations x = sinh t and y = cosh 2t.

Find a Cartesian equation of the curve, giving your answer in the form y = f(x).                   [ 3 ]


Question A9

Point P lies on the parabola with Cartesian equation .

The directrix of the parabola and the line y = 2a  intersect at point Q.

The tangent to the parabola at point P passes through point Q.

Find the possible values of p.                           [ 4 ]


Question A10

(a) Use the Taylor expansion to express ln in ascending powers of x up to the term in .           [ 4 ]

(b) Hence write down the first three terms of the expansion of ln .                 [ 1 ]


Section B

Answer 4 questions ONLY. This section carries 60

marks.

Question B1

(a) p and q are real numbers such that

     Find the value of p and the value of q.                              [ 4 ]

(b) i. Use the Factor Theorem to confirm that (x +2) is a factor of                      [ 1 ]

     ii. Divide                          [ 2 ]

     iii. Solve                                [ 2 ]

     iv. Sketch the roots found in part iii on an Argand diagram.                             [ 2 ]

(c)     The complex number w is defined as w = (a -2) + (a + 3)j where

     i.   Find the value of a, given that a > 0. [ 3 ]

     ii.  If the argument of w is , write down the value of tan .                       [ 1 ]


Question B2

(a)   

          Matrix A =

     i.   Find the eigenvalues of matrix A.

          All working must be shown. Just quoting the eigenvalues, even the correct ones, will score no marks if this working is not seen.           [ 3 ]

     ii.   For each eigenvalue found in part i, find a corresponding eigenvector.                     [ 4 ]

          Matrix B =

     iii.  Find and .

          What do you notice?                        [ 5 ]


(b)

     i.   Find the inverse of the matrix                       [ 2 ]


     ii.  Explain why the matrix has no inverse.                 [ 1 ]



Question B3
(a) A curve C has equation

     i. Write down the equations of the asymptotes to curve C.                   [ 2 ]

     ii. Write down the coordinates of the points where curve C crosses the x - axis.                  [ 2 ]
     iii. Show that curve C has no stationary values.                     [ 3 ]
     iv. Sketch curve C (this must not be done on graph paper). Show clearly the asymptotes and the coordinates where the curve crosses the x - axis.                  [ 4 ]

(b) You are given cosh x = 3.

     Find the exact values of

     i.  sinh x                                 [ 1 ]


     ii. sinh 2x                                [ 1 ]

     iii. tanh 2x                               [ 2 ]


Question B4

(a) An ellipse has parametric equations and

     i. Find the equation of the normal to the ellipse when                      [ 4 ]

     The rectangular hyperbola has a focus with positive coordinates at point K.

     The normal to the ellipse in part i passes through point K.

     ii. Find the value of c.                       [ 3 ]

(b) A curve has parametric equations

     i. Find the value of t when the tangent to the curve is parallel to the x - axis.                   [ 3 ]

     The integral I is defined as



     ii. Write I in terms of t.                     [ 2 ]

     iii. Hence evaluate I.                       [ 3 ]

          All working must be shown. An answer, even the correct one, will receive no marks if this working is not seen.



Question B5

(a) i. Show that


         Each stage of your working must be clearly shown.                                [ 4 ]

     ii.  Hence find the value of

          Give your answer in full with no rounding off.                    [ 3 ]


(b) i. By differentiating a suitable number of times, derive the Maclaurin expansion of up to the term in .

        All working must be shown.                [ 3 ]


     ii. Use your result to find an approximate value of giving your answer in the form where m and n are integers.              [ 4 ]

     iii. Explain why, when using the Maclaurin expansion on a better approximation is obtained than the one on                  [ 1 ]


Question B6

(a)

Figure 1 shows two blocks P and Q. P has mass 3 kg and rests on a rough horizontal surface where the coefficient of friction between P and the surface is Q has mass 2 kg and hangs freely. A light inextensible string connects P and Q over a smooth pulley.

The system is released from rest.

     i.  Copy the diagram and show all of the forces acting on P and all of the forces acting on Q.             [ 2 ]

     ii.  Find the acceleration of the blocks and the tension in the string.                  [ 5 ]

(b)

Figure 2 shows a particle of mass 5 kg which is initially at rest on a smooth slope inclined at to the horizontal. A horizontal force of Newtons is applied to the particle and it starts to move up the slope.

     i.  Find the acceleration of the particle.[ 3 ]

After 10 seconds the horizontal force of Newtons is removed.

     ii. How much further up the slope does the particle continue to move before coming to rest? [ 3 ]

     iii. The particle then starts to move back down the slope. Find how long it takes to travel 11.25 metres down the slope. [ 2 ]


- This is the end of the test. -