- 2 -
Useful Formulae
If u= (u1, u2, u3) and v= (v1, v2, v3) are two vectors then
u·v=|u||v|cos(θ) = ±|uk||v|=u1v1+u2v2+u3v3
|u×v|=|u||v||sin(θ)|=|u⊥||v| ; u×v= (u2v3−u3v2, u3v1−u1v3, u1v2 −u2v1) u =u⊥+uk ; uk =u·v
v·v
v ; |uk|= |u·v|
|v|
1. (a) The following augmented matrices in echelon form describe systems of equa- tions with variablesa, b,c and d. In each case interpret the matrix and if possible solve the system.
i.
1 2 1 1 4 0 1 1 9 2 0 0 0 1 0
ii.
1 2 1 4 1 0 1 1 2 9 0 0 0 0 1
iii.
1 2 1 4 1 0 0 1 5 9 0 0 0 0 0
(b) Solve the following system of equations using the Gauss-Jordan method
y −z = −2
3x +y −z = 1
x −y +z = 3
(c) Find the inverse of the matrix
2 1 3 2
, and show how it can be used to solve the system of equations
2x +y = −5
3x +2y = 7
2. (a) Find the point where the line (x, y, z) = (1,1,3) +λ(2,−1,1) intersects the plane 3x−y+z = 5.
(b) Write a vector parametric equation for the line with equations x−2
3 = y+ 1 1 = z
4
(c) Find the volume of the parallelepiped spanned by the vectors (1,0,3), (0,2,2) and (3,−1,0).
CONTINUED
- 3 -
3. Consider P = (1,2,−1), Q= (1,3,1) and origin O = (0,0,0).
(a) Find the vectors −→
OP, −→
P Q and −→
OQ.
(b) Find the equation of the line throughP and Q.
(c) Compute −→
OP ·−→
OQ and −→
OP ×−→
OQ.
(d) Find∠P OQ, the angle between−→
OP and −→
OQ.
(e) Compute the area of the triangle 4P OQ.
(f ) Find the Cartesian equation of the plane through O, P and Q.
4. (a) Find the equation of the line formed by the intersection of the two planes x+ 2y+ 3z= 7 and 2x+ 4y+z = 2.
(b) Find the angle at which the planes x+ 2y+ 3z = 7 and 2x+ 4y+z = 2 intersect.
(c) Findvk and v⊥, the components of v= (7,0,0) parallel and perpendicular to the vectoru= (1,2,3).
(d) Find the equation of the line through the origin which is normal to the planex+ 2y+ 3z = 7.
(e) Find the distance between the plane x+ 2y+ 3z = 6 and the origin.
5. (a) Compute
0 1 2 1 0 1
2 1 1 1 0 1
+
3 4 2 5
(b) Find the inverse of the matrixA=
0 1 1
2 0 −1
1 2 0
.
(c) Find the determinant of the following matrix.
M =
0 1 0 3 0 2 0 4 3 2 1 1 1 1 2 3
TURN OVER
- 4 -
6. (a) Write 3.2515 as a fraction, where the line over the last two digits indicates that these repeat.
(b) Use the Euclidean Algorithm to find gcd(1880,1516), the greatest common divisor of 1880 and 1516.
(c) Solve the linear Diophantine equation 1516x+ 1880y= 20 where x, y ∈Z. Give the general solution.
7. (a) What is the remainder when x4+ 3x3 −2x2−5 + 8 is divided by x−1?
(b) Divide x4 + 2x3 + 4x2 + 3x+ 2 by x2+x+ 1 and find the quotient and remainder.
(c) Simplify the complex numberz = (1−i)(2−4i) + 4
1 +i and write it in the formz =x+iy where x and y are real.
(d) Write the complex numberz = 4iin the formreiθ. Hence find all solutions to the equationz4−i= 0.
8. (a) If A and B are two sets, with |A| = 17, |B| = 23 and |A∪B| = 36, then how many elements are in the intersection ofA and B?
(b) Use the technique of mathematical induction to prove that 1
1×3+ 1
3×5 + 1
5×7 +· · ·+ 1
(2n−1)(2n+ 1) = n 2n+ 1
for all positive integersn. You will be marked on the quality of your proof.
