Maths 4u 1974 HSC
Maths 4u 1974 HSC
Maths 4u 1974 HSC
x0
sin x
x
=1
evaluate
2
;
(a) lim sinx(3x)
2
x0
(b) lim
x0
1cos(4x)
.
x2
Question 2.
State the Fundamental Theorem of Arithmetic (concerning the factorisation of integers into primes) and give a proof of this theorem by rst showing that, if a prime p
divides the product ab of two integers and does not divide a, then it must divide b.
Question 3.
(i) If z is the complex number 1 + 2i indicate on the Argand diagram the points
1
z, z, z 2 , .
z
(ii) On the Argand diagram P represents the complex number z and Q the complex
number z1 . If P lies on the straight line x = 1 prove that Q will lie on a certain circle
and nd its centre and radius.
Question 4.
In the Cartesian plane indicate (by shading) the region R consisting of those points
whose coordinates (x, y) simultaneously satisfy the ve relations
0 x /2, y 0, y sin x, y cos x, y tan x.
Also prove that the area of R is
1
2
Question 5.
51
.
2 2 1 5 loge
2
26
x4 + 3
6x
2x
1 + t4 dt
f (x) =
0
x3
x5
x2n1
+
+ (1)n1
+
3
5
2n 1
cos
A=
sin
0
sin
, B=
cos
1
1
0
, C=
0
2
2
,
0
27
1
I=
0
0
3
x
x
, d=
, r=
, r =
y
1
0
y
describe in geometrical language (using terms such as reection, rotation, etc.) the
transformation r r in the Cartesian plane in the following cases:
(i) r = Ar;
(ii) r = Br;
(iii) r = Br + d;
(iv) r = Cr;
(v) r = (I + C)2 r.
Question 9.
The matrix A is given by A =
1
2
5
3
3
.
5
(i) Show that the eigenvalues of A are 4,1 and nd the corresponding eigenvectors.
(ii) Find a matrix P such that P1 AP is a diagonal matrix.
(iii) Solve, for X, the matrix equation X2 = A.
Question 10.
A large vertical wall stand on horizontal ground. The nozzle of a water hose is
positioned at a point C on the ground at a distance c from the wall and the water
jet can be pointed in any direction from C. Also the water issues from the nozzle
with speed V . (Air resistance may be neglected and the constant g denotes the
acceleration due to gravity.)
(i) Prove that the jet can reach the wall above ground level if and only if V > gc.
(ii) If V = 2 gc prove that the portion of the wall that can be reached by the jet is