End Of Unit-Algebraic Expression
Total Marks: 30
Question 1
Express
3
2
−
2𝑥 + 1 𝑥 + 1
as a single algebraic fraction in its simplest form.
..........................
(2 marks)
Question 2
Solve the equation (2 + √5)𝑥 = 6 − √5, giving 𝑥 in the form 𝑎 + 𝑏√5 where 𝑎 and 𝑏 are
integers.
..........................
(4 marks)
Question 3
The function 𝑓 is defined by
2
6
2
60
𝑓(𝑥) = 2𝑥+5 + 2𝑥−5 + 4𝑥 2 −25
𝑥>4
𝐴
Show that 𝑓(𝑥) = 𝐵𝑥+𝐶 where 𝐴, 𝐵 and 𝐶 are constants to be found.
..........................
(4 marks)
Question 4
The function 𝑓 is defined by
𝑓(𝑥) =
6
2
60
+
+ 2
2𝑥+5
2𝑥−5
4𝑥 −25
𝑥 > 4It can be shown that 𝑓(𝑥) =
8
2𝑥−5
Find 𝑓 −1 (𝑥).
𝑓 −1 (𝑥) = ..........................
(3 marks)
Question 5
Given 𝑦 = 2𝑥 , express the following in terms of 𝑦.
3
1
42𝑥−3
Write your expression in its simplest form.
..........................
(2 marks)
Question 6
Solve the equation
22𝑥+5 − 7(2𝑥 ) = 0
giving your answer to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
..........................
(4 marks)
Question 7
Solve the equation
4
10 + 𝑥√8 =
6𝑥
√2
Give your answer in the form 𝑎√𝑏 where 𝑎 and 𝑏 are integers.
𝑥 = ..........................
(4 marks)
Question 8
Factorise fully
81 − 16𝑥 4
..........................
(3 marks)
5
Question 9
Figure 4 shows the plan view of the design for a swimming pool.
The shape of this pool 𝐴𝐵𝐶𝐷𝐸𝐴 consists of a rectangular section 𝐴𝐵𝐷𝐸 joined to a
semicircular section 𝐵𝐶𝐷 as shown in Figure 4.
Given that 𝐴𝐸 = 2𝑥 metres, 𝐸𝐷 = 𝑦 metres and the area of the pool is 250 m 2 , show that the
perimeter, 𝑃 metres, of the pool is given by
𝑃 = 2𝑥 +
𝑎 𝜋𝑥
+
𝑥
2
where 𝑎 is a constant to be found.
..........................
(4 marks)
6
Answers
Question 1
1−𝑥
(2𝑥+1)(𝑥+1)
Question 2
𝑎 = −17, 𝑏 = 8
Question 3
𝐴 = 8, 𝐵 = 2, 𝐶 = −5
Question 4
𝑓 −1 (𝑥) =
8+5𝑥
2𝑥
Question 5
64
𝑦4
7
Question 6
𝑥 = −2.19
Question 7
𝑥 = 5√2
Question 8
(9 + 4𝑥 2 )(3 − 2𝑥)(3 + 2𝑥)
Question 9
𝑎 = 250