1.      i)  Find the following limits giving brief reasons for each.

a)   lim x2  − x − 6

x→3          x − 3

x(x2  − x − 6)

ii)  Find all real numbers a and b (if any) such that the function g : R → R

de  ned by

g(x) =  a(x)x(2)b(x) − 6

is dierentiable everywhere.

x ≥ 0

x < 0.

iii)  Does   (x2  − x − 6)sin(x)   attain a minimum for x ∈ [510, 521]?  Give

iv)  Find the critical points of

h(x) = |x2  − x − 6|

on the interval I = [0, 4]. Deduce the maximum and minimum values of the function h over I .

v)  Using the Maple session below or otherwise,   nd the area between the

curves y = (x2  − x − 6)ex  and y = 0 for x between 0 and 4.

f := x2  − x − 6

> int(f*exp(x), x=0..3);

3 − 3e3

> int(f*exp(x), x=3..4);

3e3 + e4

vi)  Let p : R → (0, ∞) and q : R → (0, ∞) be dierentiable and y =

vii)  Given that f  : R → R is dierentiable, f(1) = 5 and 0 1 f(x) dx = 2, nd 0 1 xf′ (x) dx.

2.      i)  Let

y = Zx2(1)  √ 1 + costdt.

Find  dy dx .

ii)   a)  State the Mean Value Theorem carefully, including all necessary hypotheses.

b)  By considering the function f(t) = tan−1 t  on an appropriate interval, use the Mean Value Theorem to show that

 < tan−1 x < x

for any positive number x > 0.

iii)  Determine which, if any, of the following improper integrals converge. Give reasons for your answers.

a)  Z1 ∞  dx,

b)  Z2 ∞ 1(lnx)3 dx.

[You may use the fact, without proof, that  (lnx)3  < x  for  x > 100.]

iv)   a)  State the de  nition of sinh x in terms of the exponential function.

b)  Use this to prove the identity

4 sinh3 x = sinh 3x − 3 sinh x.

v)  Consider the function f(x) = x2 + sin x de  ned on the interval (1, ∞).

a)  Show that f is an increasing function.

b)  What is the domain of g = f−1?

c)  By considering f(2),   nd g(42 ) and g′ (42 ).

3.      i)  Suppose that z = 1 + i and w = √3 + i.

a)  Find zw in Cartesian form.

b)  Show that Arg(zw) = 5

c)  Hence show that

cos  512 =  .

ii)  Consider the vectors

a =  , b =  4  and c = 1  .

a)  Calculate a × b.

b)  Hence or otherwise calculate (4b) × (2a).

c)  Show that c · (a × b) = 0.

d)  Is c is a linear combination of a and b? Give reasons for your answer.

iii)  A plane  passing through the point P(1, 2, 3) is perpendicular to the

line

x =   +  2 .

a)  Find a Cartesian equation of the plane  .

b)  Find a parametric vector form for the plane.

c)  Find the intersection of the line and the plane.

d)  Find the shortest distance from the point (1, 0, 1) to the plane.

iv)  Use the following Maple output to assist you in answering the questions below.

>     with(LinearAlgebra):

>     m:=<<1,-1,p>|<2,p,-4>|<p,-1,p>>;

   1

− 1

p

>      t:=<1,0,-1>;

1  

0  

>      <m|t>;

   1

p    − 1

 1

G  :=   0

2

p + 2

0

p

− 1 + p    3 p − p2 − 2

1    

1

Consider the following system of linear equations:

x   +   2y   +  pz   =     1

x(x)      4(p)y(y)     p z(z)      −1(0)

a)  How many solutions does this system have when p = 2? Give reasons.

b)  How many solutions does this system have when p = 1? Give reasons.

c)  Discuss the case where p = −2.

4.      i)  Let A and B be 2 × 2 matrices.

a)  Use a counterexample to show that det(A + B) does not equal det(A) + det(B) in general.

b)  Use the fact that det(AB) = det(A)det(B) to prove that if A is an invertible matrix then det(A−1) = det(A)−1 .

ii)  Consider the following maple output.

> p := z^5-7*z^4+21*z^3-33*z^2+28*z-10;

p(z) = z5  − 7z4 + 21z3  − 33z2 + 28z − 10

> subs(z=1+I, p);

0

> subs(z=2-I, p);

0

a)  Show that 1 is a root of the polynomial

p(z) = z5  − 7z4 + 21z3  − 33z2 + 28z − 10.

b)  Use the maple output to factorise p(z) into complex linear factors.

c)  Express p(z) as a product of real linear and quadratic factors.

iii)  Consider the matrices

C =  1(1)   0(1)   1  and  D =      .

Here C is a 2 × 3 matrix and D is a 3 × 2 matrix.

a)  Calculate CD .

b)  Hence   nd the 2 × 3 matrix Y which solves the matrix equation

DY =   0(3) −5

6

− 10

0(5)

−7 .

iv)  Consider the following system of equations:

2x   + 4y               =  4

x    − 4y    + 3z    =  11

a)  Write the system in augmented matrix form and reduce it to row echelon form.

b)  Solve the system, expressing your solution in vector form.

c)  If x,y and z solve the above system and are all greater than or equal to zero, what is the largest possible value of z?

v)  Consider the   xed points A(1, 0, 0) and B(− 1, 0, 0) and variable point X

centred at the origin if and only if the triangle AXB  is right angled

at X .