A Level Mathematics Homework Assignment 12
Proof, Algebraic Division, Factor Theorem
In each of the following cases, choose one of the statements
1
P ⇒Q
P ⇐Q
P⇔Q
to describe the relationship between P and Q.
2
a)
P: 𝑥 is an integer that ends in 5
Q: 𝑥 is an integer that is divisible by 5
b)
P: 𝑛 is odd
Q: 𝑛2 is odd
c)
P: 𝑦 > 0
Q: 𝑦 2 > 0
Prove that if you add the squares of three consecutive numbers and then subtract two, you always
get a multiple of 3.
3
Find the quotient and remainder when 7𝑥 3 + 37𝑥 2 + 9𝑥 − 17 is divided by 𝑥 + 5
4
Find the quotient and remainder when 3𝑥 3 − 7𝑥 + 2 is divided by 𝑥 + 2
5
Find the quotient and remainder when 12𝑥 3 − 14𝑥 2 − 5𝑥 + 9 is divided by 3𝑥 − 2
𝑓(𝑥) = 𝑥 3 − 4𝑥 2 − 11𝑥 + 30
6
a) Calculate
i) 𝑓(0)
ii) 𝑓(1)
1
iii) 𝑓(2)
iv) 𝑓(−2)
v)𝑓(−3)
vi) 𝑓(5)
vii) 𝑓 (2)
iii) 𝑓(2)
iv) 𝑓(−2)
v)𝑓(−3)
vi) 𝑓(5)
vii) 𝑓 (2)
iv) 𝑓(−2)
v)𝑓(−3)
vi) 𝑓(5)
vii) 𝑓 (2)
b) Factorise 𝑓 completely
7
𝑓(𝑥) = 2𝑥 3 + 𝑥 2 − 5𝑥 + 2
a) Calculate
i) 𝑓(0)
ii) 𝑓(1)
1
b) Factorise 𝑓 completely
8
𝑓(𝑥) = 9𝑥 3 − 54𝑥 2 + 47𝑥 − 10
a) Calculate
i) 𝑓(0)
ii) 𝑓(1)
iii) 𝑓(2)
1
b) Factorise 𝑓 completely
continued…
Mixed Practice
9
Find the point of intersection of the lines
10
Solve
11
Make 𝑎 the subject of the formula 𝑥 =
2𝑥+3
𝑥
𝑥 + 4𝑦 – 7 = 0 and 2𝑥 – 5𝑦 + 8 = 0
= −2
3+2𝑎
𝑎−5
5
4
3
15𝑥 2
in the form 𝑘𝑥 𝑛 .
12
a) Simplify (3 + √5)(2 − √5)
13
State the transformation that takes the graph of 𝑦 = 𝑥 2 onto the graph of
a) 𝑦 = 𝑥 2 − 4
b) Evaluate 8
b) 𝑦 = (𝑥 − 4)2
c) Write
3√𝑥
c) 𝑦 = (𝑥 + 3)2 + 7
14
Find the centre and radius of the circle 𝑥 2 + 𝑦 2 + 4𝑥 − 8𝑦 = 5.
15
Solve 𝑥 2 + 8𝑥 + 10 = 0, giving your answers in simplified surd form.
16
Prove that the sum of squares of two consecutive odd numbers is never a multiple of 8
17
The numbers 𝑎, 𝑏, 𝑐 are such that
𝑎 < 0,
𝑏 > 1,
−1 < 𝑐 < 1
Decide whether he following statements are Always True, Sometimes True, or Never True.
a) 𝑎3 < 0
b) 𝑏 < 10𝑎2
c) 𝑎𝑏 > 0
d) 𝑏 − 𝑐 > 1
18*
(i)
(ii)
(iii)
(iv)
The fixed positive integers a,b,c,d are such that exactly two of the following four statements are valid:
𝑎≤𝑏<𝑐≤𝑑
𝑎+𝑏 =𝑐+𝑑
𝑎 = 𝑐 𝑎𝑛𝑑 𝑏 = 𝑑
𝑎𝑑 = 𝑏𝑐
Which of the following is a pair of valid statements?
(i) and (ii)
(i) and (iii)
(i) and (iv)
(iii) and (i)