Year 12 Maths Ark Globe Academy Phase V Assessment Monday … 12 Maths... · 2020-07-08 · [Edexcel C4 June 2017 Q2a] 𝑓(𝑥)=(2+𝑘𝑥)−3, |𝑘𝑥|

Year 12 Maths Ark Globe Academy Phase V Assessment

Monday 13 July - Friday 17 July

Phase V Assessment

Question 1

It is given that

(1 + 𝑥)𝑛 = 1 + 𝑛𝑥 +𝑛(𝑛 − 1)

1 × 2𝑥2 + ⋯ +

𝑛(𝑛 − 1) … (𝑛 − 𝑟 + 1)

1 × 2 × … × 𝑟𝑥𝑟 + ⋯ (|𝑥| < 1, 𝑛 ∈ ℝ)

Find the binomial expansion in ascending powers of 𝑥, up to and including the term in 𝑥3, of

(1 + 2𝑥)−4

..........................

Question 2

[Edexcel C4 June 2017 Q2a]

𝑓(𝑥) = (2 + 𝑘𝑥)−3, |𝑘𝑥| < 2, where 𝑘is a positive constant

The binomial expansion of 𝑓(𝑥), in ascending powers of 𝑥, up to and including the term in 𝑥2 is

𝐴 + 𝐵𝑥 +243

16𝑥2

where 𝐴and 𝐵are constants.

Write down the value of 𝐴.

..........................

(1 mark)

Question 3

[Edexcel C4 June 2015 Q1a]

Find the binomial expansion of

(4 + 5𝑥)12

|𝑥| <

4

5, in ascending powers of 𝑥, up to and including the term in 𝑥2.

Give each coefficient in its simplest form.

..........................

(5 marks)

Question 4

[Edexcel C4 June 2012 Q3c Edited]

𝑓(𝑥) =6

√9−4𝑥, |𝑥| <

9

4

The binomial expansion of 𝑓(𝑥) in ascending powers of 𝑥, up to and including the term in 𝑥3 is:

𝑓(𝑥) = 2 +4

9𝑥 +

4

27𝑥2 +

40

729𝑥3

Use this answer to find the binomial expansion in ascending powers of 𝑥, up to and including the

term in 𝑥3, of

ℎ(𝑥) =6

√9−8𝑥 |𝑥| <

9

8

..........................

(2 marks)

Question 5

[OCR C4 June 2016 Q7]

Given that the binomial expansion of (1 + 𝑘𝑥)𝑛 is 1 − 6𝑥 + 30𝑥2 + ⋯ find the values of 𝑛 and 𝑘.

State the set of values of 𝑥 for which this expansion is valid.

𝑛 = ..........................

𝑘 = ..........................

|𝑥| < ..........................

(6 marks)

Question 6

[OCR C4 June 2012 Q3i]

Expand 1+𝑥2

√1+4𝑥 in ascending powers of 𝑥, up to and including the term in 𝑥3.

..........................

(6 marks)

Question 7

[OCR C4 June 2013 Q10iii Edited]

It can be shown that 𝑥

(1−𝑥)3 = −1

𝑥2 (1 −1

𝑥)

−3

Hence find the first three terms of the binomial expansion of 𝑥

(1−𝑥)3 in powers of 1

𝑥.

..........................

(4 marks)

Question 8

[Edexcel C4 Jan 2011 Q5c Edited]

The binomial expansion of(2 − 3𝑥)−2, up to and including the term in 𝑥3 is given as:

(2 − 3𝑥)−2 =1

4+

3

4𝑥 +

27

16𝑥2 +

27

8𝑥3, |𝑥| <

2

3,

𝑓(𝑥) =−1+3𝑥

(2−3𝑥)2, |𝑥| <2

3

In the binomial expansion of 𝑓(𝑥), in ascending powers of 𝑥, the coefficient of 𝑥is 0 and the

coefficient of 𝑥2 is 9

16.

Find the coefficient of 𝑥3, giving your answer as a simplified fraction.

..........................

(3 marks)

Question 9

Work out the area of the sector 𝐴𝑂𝐵.

Give your answer correct to 3 significant figures.

.......................... cm 2

Question 10

Work out the length of the arc 𝐴𝐵.

.......................... cm

Question 11

[OCR C2 June 2012 Q3i]

The diagram shows a sector AOB of a circle, centre O and radius 𝑟 cm. Angle AOB is 72°.

Express 72° exactly in radians, simplifying your answer.

.......................... radians

(1 mark)

Question 12

[OCR C2 June 2011 Q3i]

The diagram shows a sector AOB of a circle, centre O and radius 8 cm. The perimeter of the sector is

23.2 cm.

Find angle AOB in radians.

.......................... radians

(3 marks)

Question 13

[Edexcel C2 May 2015 Q4b Edited]

Figure 1 shows a sketch of a design for a scraper blade. The blade AOBCDA consists of an

isosceles triangle COD joined along its equal sides to sectors OBC and ODA of a circle with

centre O and radius 8 cm. Angles AOD and BOC are equal. AOB is a straight line and is

parallel to the line DC. DC has length 7 cm.

The angle COD is 0.906 radians, correct to 3 significant figures.

Find the perimeter of AOBCDA, giving your answer to 3 significant figures.

.......................... 𝑐𝑚

(3 marks)

Question 14

[Edexcel C2 May 2014 Q5a]

The shape ABCDEA, as shown in Figure 2, consists of a right-angled triangle EAB and a

triangle DBC joined to a sector BDE of a circle with radius 5 cm and centre B.

The points A, B and C lie on a straight line with BC = 7.5 cm.

Angle EAB = 𝜋

2radians, angle EBD = 1.4 radians and CD = 6.1 cm.

Find, in cm2, the area of the sector BDE.

.......................... 𝑐𝑚2

(2 marks)

Question 15

[Edexcel C2 Jan 2013 Q7d Edited]

The triangle 𝑋𝑌𝑍in Figure 1 has 𝑋𝑌 = 6 cm, 𝑌𝑍 = 9 cm, 𝑍𝑋 = 4 cm and angle 𝑍𝑋𝑌 = 𝛼.

The point 𝑊lies on the line 𝑋𝑌.

The circular arc 𝑍𝑊, in Figure 1 is a major arc of the circle with centre 𝑋and radius 4 cm.

𝛼 = 2.22, correct to 3 significant figures.

The region enclosed by the major arc 𝑍𝑊of the circle and the lines 𝑊𝑌and 𝑌𝑍is shown shaded

in Figure 1.

Calculate the perimeter 𝑍𝑊𝑌𝑍 of this shaded region.

Give your answer correct to 1 decimal place.

.......................... cm

(4 marks)

Question 16

[OCR C2 Jan 2013 Q7iii Edited]

The diagram shows two circles of radius 7 cm with centres A and B. The distance AB is 12 cm and the

point C lies on both circles. The region common to both circles is shaded.

It can be shown that angle CAB = 0.5411 radians, correct to 4 significant figures.

Find the area of the shaded region.

Input note: give your answer correct to 3 significant figures.

.......................... 𝑐𝑚2

(5 marks)