Williamwood High School
MATHEMATICS
ADVANCED HIGHER
GEOMETRY AND PROOF
HOMEWORK & CONSOLIDATION BOOKLET
S6 Daily Homework Geo & Proof Week 1
S6 Daily Homework Geo & Proof Week 2
S6 Daily Homework Geo & Proof Week 3
S6 Daily Homework Geo & Proof Week 4
S6 Daily Homework Geo & Proof Week 5
S6 Daily Homework Geo & Proof Week 6
S6 Daily Homework Geo & Proof Week 7
ADVANCED HIGHER GEOMETRY AND PROOF HOMEWORK 1
1.
2.
3.
ADVANCED HIGHER GEOMETRY AND PROOF HOMEWORK 2
1. Solve z2 + 2z + 5 = 0 and represent the solutions on an Argand diagram.
2. Verify that z = 1 + i is a root of the equation z4 + 3z2 – 6z + 10, and find the other roots.
3. Interpret geometrically in the complex plane the equation =
4. Expand using the binomial theorem and De Moivre’s theorem.
Use your expansion to express as a polynomial in .
5.
6.
7.
Geometry, Proof and Systems of Equations Assessment Standard 1.1
1. Use Gaussian elimination to obtain the solution of the system of equations
x + y + z = 3
2y - z = 4
x + 5y + 2z = 2. (5)
2. Consider the matrices
and
where is a constant.
Find the matrices
(a) (2)
(b) (3)
3. Two matrices are given by
and where is a constant.
(a) Find the matrix (2)
(b) H is a singular matrix. Find the value(s) of t. (3)
Geometry, Proof and Systems of Equations Assessment Standard 1.2
4. Calculate c d, where c and d are the vectors given by c = 2j – k and d = i – j + k. (3)
5. Obtain, in (a) Symmetric form
OR (b) Vector form
OR (c) Parametric form
an equation of the line which passes through the points P(1, -1, 4) and Q(3, 1, 7). (2)
6. The plane has normal vector n = 2i – j + k and passes through the point R(2, 3, 7).
(a) Find, in Cartesian form, the equation of the plane .
OR (b) Find, in vector form, the equation of the plane .
OR (c) Find, in parametric form, the equation of the plane . (3)
Geometry, Proof and Systems of Equations Assessment Standard 1.3
7. Find the modulus and argument of
Hence express in polar form. (3)
8. A complex number w has modulus 2 and principal argument
(a) Plot w on an Argand diagram. (1)
(b) Using exact values, express w in Cartesian form. (2)
Geometry, Proof and Systems of Equations Assessment Standard 1.4
9. Use the Euclidean algorithm to determine the value of d, the greatest common divisor of
851 and 1147. (3)
Geometry, Proof and Systems of Equations Assessment Standard 1.5
10. It is conjectured that for any real numbers a, b, c and d
a < b and c < d ac < bd.
By providing a counter-example, disprove this conjecture. (3)
11. Let n be a natural number.
Prove, by contradiction, that if 3n is even that n is even. (4)
12. Prove directly that if 16 is added to the cube of any even number then the answer is
divisible by 8. (3)