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A1 Maths HW Assignment 12

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A Level Mathematics Homework Assignment 12
Proof, Algebraic Division, Factor Theorem
In each of the following cases, choose one of the statements
1
P β‡’Q
P ⇐Q
P⇔Q
to describe the relationship between P and Q.
2
a)
P: π‘₯ is an integer that ends in 5
Q: π‘₯ 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 7π‘₯ 3 + 37π‘₯ 2 + 9π‘₯ βˆ’ 17 is divided by π‘₯ + 5
4
Find the quotient and remainder when 3π‘₯ 3 βˆ’ 7π‘₯ + 2 is divided by π‘₯ + 2
5
Find the quotient and remainder when 12π‘₯ 3 βˆ’ 14π‘₯ 2 βˆ’ 5π‘₯ + 9 is divided by 3π‘₯ βˆ’ 2
𝑓(π‘₯) = π‘₯ 3 βˆ’ 4π‘₯ 2 βˆ’ 11π‘₯ + 30
6
a) Calculate
i) 𝑓(0)
ii) 𝑓(1)
1
iii) 𝑓(2)
iv) 𝑓(βˆ’2)
v)𝑓(βˆ’3)
vi) 𝑓(5)
vii) 𝑓 (2)
iii) 𝑓(2)
iv) 𝑓(βˆ’2)
v)𝑓(βˆ’3)
vi) 𝑓(5)
vii) 𝑓 (2)
iv) 𝑓(βˆ’2)
v)𝑓(βˆ’3)
vi) 𝑓(5)
vii) 𝑓 (2)
b) Factorise 𝑓 completely
7
𝑓(π‘₯) = 2π‘₯ 3 + π‘₯ 2 βˆ’ 5π‘₯ + 2
a) Calculate
i) 𝑓(0)
ii) 𝑓(1)
1
b) Factorise 𝑓 completely
8
𝑓(π‘₯) = 9π‘₯ 3 βˆ’ 54π‘₯ 2 + 47π‘₯ βˆ’ 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 π‘Ž the subject of the formula π‘₯ =
2π‘₯+3
π‘₯
π‘₯ + 4𝑦 – 7 = 0 and 2π‘₯ – 5𝑦 + 8 = 0
= βˆ’2
3+2π‘Ž
π‘Žβˆ’5
5
4
3
15π‘₯ 2
in the form π‘˜π‘₯ 𝑛 .
12
a) Simplify (3 + √5)(2 βˆ’ √5)
13
State the transformation that takes the graph of 𝑦 = π‘₯ 2 onto the graph of
a) 𝑦 = π‘₯ 2 βˆ’ 4
b) Evaluate 8
b) 𝑦 = (π‘₯ βˆ’ 4)2
c) Write
3√π‘₯
c) 𝑦 = (π‘₯ + 3)2 + 7
14
Find the centre and radius of the circle π‘₯ 2 + 𝑦 2 + 4π‘₯ βˆ’ 8𝑦 = 5.
15
Solve π‘₯ 2 + 8π‘₯ + 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 π‘Ž, 𝑏, 𝑐 are such that
π‘Ž < 0,
𝑏 > 1,
βˆ’1 < 𝑐 < 1
Decide whether he following statements are Always True, Sometimes True, or Never True.
a) π‘Ž3 < 0
b) 𝑏 < 10π‘Ž2
c) π‘Žπ‘ > 0
d) 𝑏 βˆ’ 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:
π‘Žβ‰€π‘<𝑐≀𝑑
π‘Ž+𝑏 =𝑐+𝑑
π‘Ž = 𝑐 π‘Žπ‘›π‘‘ 𝑏 = 𝑑
π‘Žπ‘‘ = 𝑏𝑐
Which of the following is a pair of valid statements?
(i) and (ii)
(i) and (iii)
(i) and (iv)
(iii) and (i)
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