_______________________
Name:
_
A: Proof
_______________________
Class:
_
_______________________
Date:
_
Time: 77 minutes
Marks: 65 marks
Comments:
Page 1 of 16
,Q1.
Prove that
n is a prime number greater than 5 ⇒ n4 has final digit 1
(Total 5 marks)
Q2.
p(x) = 30x3 − 7x2 − 7x + 2
(a) Prove that (2x + 1) is a factor of p(x)
(2)
(b) Factorise p(x) completely.
(3)
(c) Prove that there are no real solutions to the equation
(5)
(Total 10 marks)
Q3.
Which of these statements is correct?
Tick one box.
x = 2 ⇒ x2 = 4
x2 = 4 ⇒ x = 2
x2 = 4 ⇔ x = 2
x2 = 4 ⇒ x = −2
(Total 1 mark)
Q4.
Prove that 23 is a prime number.
(Total 2 marks)
Q5.
Prove by contradiction that is an irrational number.
Page 2 of 16
, (Total 7 marks)
Q6.
Jessica, a maths student, is asked by her teacher to solve the equation tan x = sin x ,
giving all solutions in the range 0° ≤ x ≤ 360°
The steps of Jessica’s working are shown below.
The teacher tells Jessica that she has not found all the solutions because of a mistake.
Explain why Jessica’s method is not correct.
(Total 2 marks)
Q7.
Prove that the function f(x) = x3 − 3x2 + 15x − 1 is an increasing function.
(Total 6 marks)
Q8.
(a) Given that n is an even number, prove that 9n2 + 6n has a factor of 12
(3)
(b) Determine if 9n2 + 6n has a factor of 12 for any integer n .
(1)
(Total 4 marks)
Q9.
p(x) = 2x3 + 7x2 + 2x − 3
(a) Use the factor theorem to prove that x + 3 is a factor of p(x)
(2)
(b) Simplify the expression
(4)
(Total 6 marks)
Page 3 of 16
Name:
_
A: Proof
_______________________
Class:
_
_______________________
Date:
_
Time: 77 minutes
Marks: 65 marks
Comments:
Page 1 of 16
,Q1.
Prove that
n is a prime number greater than 5 ⇒ n4 has final digit 1
(Total 5 marks)
Q2.
p(x) = 30x3 − 7x2 − 7x + 2
(a) Prove that (2x + 1) is a factor of p(x)
(2)
(b) Factorise p(x) completely.
(3)
(c) Prove that there are no real solutions to the equation
(5)
(Total 10 marks)
Q3.
Which of these statements is correct?
Tick one box.
x = 2 ⇒ x2 = 4
x2 = 4 ⇒ x = 2
x2 = 4 ⇔ x = 2
x2 = 4 ⇒ x = −2
(Total 1 mark)
Q4.
Prove that 23 is a prime number.
(Total 2 marks)
Q5.
Prove by contradiction that is an irrational number.
Page 2 of 16
, (Total 7 marks)
Q6.
Jessica, a maths student, is asked by her teacher to solve the equation tan x = sin x ,
giving all solutions in the range 0° ≤ x ≤ 360°
The steps of Jessica’s working are shown below.
The teacher tells Jessica that she has not found all the solutions because of a mistake.
Explain why Jessica’s method is not correct.
(Total 2 marks)
Q7.
Prove that the function f(x) = x3 − 3x2 + 15x − 1 is an increasing function.
(Total 6 marks)
Q8.
(a) Given that n is an even number, prove that 9n2 + 6n has a factor of 12
(3)
(b) Determine if 9n2 + 6n has a factor of 12 for any integer n .
(1)
(Total 4 marks)
Q9.
p(x) = 2x3 + 7x2 + 2x − 3
(a) Use the factor theorem to prove that x + 3 is a factor of p(x)
(2)
(b) Simplify the expression
(4)
(Total 6 marks)
Page 3 of 16
