Prelim revision
Gaussian Elimination
Make sure you are happy with
 a system of equations that give a unique solution ( x= , y= ,z= )
 a system that gives no solution (inconsistency)
 a system that gives an infinite family of solutions
1)
Use Gaussian elimination to solve the following system of equations
x
-z = 2
2y - 3z = 6
2x + y + z = 1
(5)
2)
A car manufacturer is planning future production patterns. Based on estimates of
time, cost and labour, he obtains a set of three equations for the numbers x, y, z of three new
types of car.
These equations are
x + 2y + z = 60
2x + 3y + z = 85
3x + y + (λ + 2) z = 105
Where the integer λ is a parameter such that 0 < λ < 10.
(a) Use Gaussian elimination to find an expression for z in terms of λ.
(5)
(b) Given that z must be a positive integer, what are the possible values for z?
(2)
(c) Find the corresponding values of a and y for each value of z.
(2)
3)
For what value of t does the system of equations
x + 2y – 3z = -7
4x – y + 2z = 9
3x – 2y + tz = 13
have no solution?
(5)
4) Use Gaussian Elimination to solve the system of linear equations
x+y+z=0
2x - y + z = -1.1
x + 3y + 2z = 0.9
(5)
5) Use Gaussian Elimination to solve the following system of equations
3x + 2y + 5z = 0
2x + y – 2z = 5
7x + 4y + z = 10
(5)
6) Use Gaussian Elimination to solve the following system of equations
2x – y + 3z = 6
x + y + 2z = 7
4x + y + 7z = 9
7) Use Gaussian Elimination to solve the equations
x+y=4
x + 2y – z = 5
3y -2z = 6
(5)
Prelim Revision
Functions
1)
2)
The function f is defined by f  x  
(a)
(b)
(c)
4x
where x  R , x  1 .
x 1
Find the equations of the asymptotes of the graph of
Prove that the graph of
Sketch the graph of
f x  .
f x  has no turning points or points of inflection.
(3)
(4)
f x  indicating all the important features.
(2)
3)
The function f is defined by f  x  
(a)
6
x  1x  3
where x  R , x  1, 3 .
Find the equations of the asymptotes of the graph of f x  .
(3)
(b) Prove that the graph of f x  has a turning point and determine its nature
and coordinates.
(5)
(c)
Sketch the graph of f x  indicating all the important features.
(2)
4) The function f is defined by
f(x) = 2x – x2 for 1≤ x ≤ 3.
(a) Sketch the graph of f and state the range of f.
(b) Sketch the graph of the inverse function f- -1 and state the domain of f -1.
(c) Obtain a formula for the inverse function f -1
(3)
(3)
(4)
5) Determine whether the function f(x) = x2cosx + x3 is odd, even or neither.
Justify your answer.
3
Prelim Revision
Sequences
1)
Let u1, u2, …,un…be an arithmetic sequence and v1, v2, …, vn…be a geometric
sequence. The first terms u1 and v1 are both equal to 45, and the third terms u3 and v3
are both equal to 5.
(a) Find u11
(3)
∞
(b) Given that v1,v2,… is a sequence of positive numbers calculate n=1
Σ vn
(3)
2)
(a)
The sum of the first n terms of a series is given by
Sn  n2  5 n
Find an expression for the nth term, un .
(3)
Prove that the series is arithmetic and write down the first four terms.
(3)
(b)
Find the value of  , 0   

, such that
2
1  cos 2   cos 4   cos6   . . . = 2
(5)
3)
4)
(a)
x  1, 3x  1, 6 x  2 are the first three terms of an arithmetic sequence.
For what value of n does S n , the sum of the first n terms, first exceed
100?
(4)
(b)
The sum of the first three terms of a positive geometric sequence is
315 and the sum of the 5th, 6th and 7th terms is 80640. Identify the
sequence.
(6)
The first two terms of a series are 1 +
(a)
2 and 1 +
1
2
If the series is arithmetic, show that the common difference is Show also that the sum of the first ten terms is
(b)
.


5
45 2 .
2
1
2
2.
4
If the series is geometric, show that the sum to infinity exists.
Show also that S  = 4 + 3 2 .
5
Prelim revision
Binomial expansion
5
1)
4

Obtain the binomial expansion of  3a 2   .
b

2)
The coefficients of x and x in the expansion of a  bx  are 218750 and
262500 respectively. Find the value of the constants a and b.
2
(3)
7
(5)
12
3)
 3 1
Find the term independent of x in the expansion of  x  
x

4)
Find the term in x7 in ( x + 2/x)9
(4)
4

2
5) Find the term independent of p in the expansion of  3 p 3   .
p

4
Proof
(refer also to exam questions given in the unit notes)
1) Consider the following two statements S and T.
S: If p and q are two odd prime numbers then p + q is not prime.
T: If p and q are two odd prime numbers then p – q is not prime.
For each of S and T, give a proof if it is true, or give a counter-example if it is false.
(3)
2) Use the method of proof by contradiction to show that √3 is irrational
(4)
3)
Prove that if n is odd then n – 1 is divisible by 8.
4
(3)
4)
5)
Prove by induction that n(n + 1)(n + 2) is divisible by 6 for all positive
integers n.
(6)
Prove by induction that 4n-1 is divisible by 3 for all positive integers n.
(5)
6) Prove by induction that for all natural numbers n, 23n  1 is divisible by 7.
(5)
Partial Fractions
Make sure you are happy with
 linear factors
 repeated linear factors
 irreducible quadratics
 improper fractions (which first need simplified through long
division before applying partial fractions)
Revisit the mixed exercise Ex 5 p21 and also p22 Ex 6 unit 1 text-book to ensure you
are confident with all types.
1)
2)
Express
1
x  2x  8
2
in partial fractions.
(2)
Hence evaluate

1
1
1
dx , giving your answer correct to
x  2x  8
2
3 decimal places.
3)
(a)
(4)
2 x 2  x  10
Express 3
in partial fractions.
x  2x 2  4x  8
(3)
(b)
Hence evaluate

2
2 x  x  10
x  2x 2  4x  8
2
3
0
dx
(5)
4)
Express
1
in partial fractions
( x 1)( x  2)
(2)
Hence find a simple expression for the sum of the series
1
1
1
+
+….+
(n 1)( n  2)
23
3 4
in terms of n.
Evaluate
(3)


k 1
1
(k 1)( k  2)
(1)
Prelim revision
Complex Numbers
1) Plot the complex number z = √3 + i on an Argand diagram and find the modulus
and argument of z
(2)
z z
Calculate
where z denotes the complex conjugate of z.
(2)
z z
Use de Moivres theorem to evaluate z6.
(2)
2) The point A represents -5 + 5i on an Argand diagram and ABCD is a square with
centre -2 + 2i. Find the complex numbers represented by the points B, C and
D, giving your answers in the form x + iy.
(4)
1  3i
3)
Express the complex number
in the form x + iy, where x and y are real
1  2i
numbers.
(3)
Determine the modulus and argument of this complex number
(3)
4)
4  2i a  ib   6  2i .
Express a + ib in the form r cos  i sin   and
hence, or otherwise, calculate a  ib  .
15
(8)
5)
Given that z  2i is a root of the equation 2 z 3  3 z 2  8 z  k  0 find the
value of k.
(2)
Hence find all the roots of this equation
6)
Given that 4 – 3i is a root of the equation z  12 z  62 z  140 z  125  0 ,
find all the solutions.
(3)
4
3
2
(6)
7)
(a) Verify that z = 2 is a solution of the equation z3 – 8z2 + 22z – 20 = 0. (1)
(b) Express z3 – 8z2 + 22z – 20 as a product of a linear factor and a quadratic
factor with real coefficients.
Hence find all the solutions of z3 – 8z2 + 22z – 20 = 0.
(4)
8)
Given that z = -2 + i is one root of z4 + z3 – z2 + 9z + 30 = 0 find all remaining
roots.
(6)
9)
Let z = cosθ + isinθ.
(a) Use the binomial theorem to show that the real part of z4 is
cos4θ - 6cos2θsin2θ + sin4θ
Obtain a similar expression for the imaginary part of z4 in terms of θ
(b) Use de Moivre’s theorem to write down an expression for z4 in terms of 4θ
(c) Use your answers to (a) and (b) to express cos4θ in terms of cosθ and sinθ.
(d) Hence show that cos4θ can be written in the form k(cosmθ – cosnθ) + p where
k,m,n,p are integers. State the values of k,m,n,p.
(5)
(1)
(1)
(4)
10) a) State the binomial expansion of (a + b)5
(1)
b) de Moivre’s theorem states that
cos nθ + isin nθ = (cos θ + isinθ)n for any integer n.
(i) By using de Moivre’s Theorem and by equating imaginary parts, show that sin5θ
can be expressed in the form
kcos4θsinθ + mcos2θsin3θ + nsin5θ for some real values of k, m, n.
(5)
(ii) Hence find an expression for sin5θ entirely in terms of sinθ
11) Let z =
(3)
1
.
cos   i sin 
(a)Use de Moivre’s theorem to express z5 in the form cos p  - isin p  , where p is a
natural number.
2
(b)
Use the binomial theorem to express sin5  in the form
qsin  + rsin3  + tsin5  ,
and state the values of q, r and t.
5
12) Use the binomial theorem to find constants A, Band C such that
5
1
1

 5 1
 3 1

 z  z   A  z  z5   B  z  z3   C  z  z 








(3)
(b) Let z = cosθ + isinθ. Use de Moivre’s theorem to show that
1
zk  k  2i sink
z
For all positive integers k.
(3)
(c) Use your answers to (a) and (b) to deduce that
16 sin5θ = sin 5θ – 5 sin 3θ + 10 sinθ
Hence evaluate


0
2
sin5 d
(5)
13) Identify the locus in the complex plane given by:
z  3i  z  2
(3)
14) Identify the locus in the complex plane given by /z + i/ = 2
(3)