Q1-4: Essential Skills

Q4-8: Applied Skills

Q9-12: Pure Skills

Q13-15: Thinking Skills

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

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Mαths PS4

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(a) The driver of a car is travelling at 18𝑚/𝑠 along a straight road when she sees an obstruction

ahead. She applies the brakes and the brakes cause the car to slow down to rest with a constant

deceleration of 3𝑚/𝑠.

Find: i) the distance travelled as the car decelerates

ii) the time it takes for the car to decelerate from 18𝑚/𝑠 to rest

(b) Two particles, P and Q, of masses 5kg and 3kg respectively are connected by a light inextensible

string. Particle P is pulled by a horizontal force of magnitude 40N along a rough horizontal plane.

Particle P experiences a frictional force of 10N and particle Q experiences a frictional force of 6N.

Draw a force diagram to model this situation, labelling all of the forces involved.

i) How have you used the fact that the string is light and inextensible?

ii) How have you used the fact that the plane is rough?

Mαths PS4 Problem 2 A

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(a) Two fair spinners each have four sectors numbered 1 to 4.

The two spinners are spun together and the sum of the spinners indicated on each spinner is recorded.

Find the probability of the spinners indicating:

i) exactly 5

ii) more than 5

(b) A bag contains seven green beads and five blue beads.

A bead is taken from the bag at random and not replaced. A second bead is then taken from the bag.

Using a tree diagram, find the probability that:

i) both beads are green

ii) the beads are different colours

Mαths PS4 Problem 3 A

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The diagram shows the graph of the quadratic function 𝑦 = 𝑓(𝑥)

The graph meets the x-axis at (1, 0) and (3, 0) and the minimum point is (2, -1)

(a) find the equation of 𝑓(𝑥) in the form 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐

(b) Find the equations of these functions:

(i) 𝑔 𝑥 = 𝑓(𝑥 + 2)

(ii) ℎ 𝑥 = 𝑓(2𝑥)

(c) On separate axes, labeling all important points, sketch the graphs of

(i) 𝑦 = 𝑔(𝑥) (ii) 𝑦 = ℎ(𝑥)

Mαths PS4 Problem 4 A

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On the same axes, sketch the curves with equations 𝑦 =6

𝑥 and 𝑦 = 1 + 𝑥.

The graphs of 𝑦 =6

𝑥 and 𝑦 = 1 + 𝑥 intersect at the points A and B.

(a) Find the coordinates of A and B

The curve C with equation 𝑦 = 𝑥2 + 𝑝𝑥 + 𝑞, where 𝑝 and 𝑞 are integers, passes through 𝐴 and 𝐵.

(b) Find the values of 𝑝 and 𝑞

(c) Add 𝐶 to your sketch

Mαths PS4

The points A(-7, 7) B(1, 9) C(3, 1) and D(-7, 1) lie on a circle. (a)Find the equation of the perpendicular bisector of: i)AB ii) CD (b) Find the equation of the circle

Problem 5 A

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The points A(-7, 7) B(1, 9) C(3, 1) and D(-7, 1) lie on a circle.

(a) Find the equation of the perpendicular bisectors of:

(i) AB (ii) CD

(b) Find the equation of the circle

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Essential Skills A

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Convert these expressions to the 𝛼𝑥𝑛 form

(a) 𝑥

7 (b)

𝑥2+ 𝑥

𝑥 (c)

7−𝑥+2𝑥5

𝑥2

Solve these equations

(a) 𝑥− 1

2 = 6 (b) 𝑥− 2

3 = 9 (c) 𝑥3 =1

27

Find the equation of the line:

(a) through 2, 3 perpendicular to 𝑦 − 9𝑥 = 8

(b) through 50, 50 parallel to 100 = 𝑦 +1

5𝑥

(c) through −6, 2 parallel to 3𝑦 =1

3𝑥+2

(a) State the centre & radius of the circle 𝑥 + 3 2 + 𝑦 + 2 2 = 50

(b) State the centre & radius of the circle 𝑥 − 3 2 + 𝑦2 = 16

(c) State equation of the circle centre −7, 13 radius 2

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(d) State equation of the circle centre 3, 5 radius 5 7

1

3

4

2

Mαths PS4 Applied Skills A

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Draw a force diagram to model this situation:

A book of mass 500g being pushed at a constant speed from rest across a

horizontal table by a finger which is at an angle 30°

A particle moves in a straight line with constant velocity.

When t = 0 its velocity is 3 𝑚𝑠−1 and when t = 4 its velocity is 12 𝑚𝑠−1. Find its acceleration and total distance travelled using the following methods:

(a) By drawing st, vt and at graphs and using the vt graph

(b) Using suvat equations

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Mαths PS4 Applied Skills A

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This skill is REALLY important! Your task is to ‘resolve’ the vector

(which could represent a force) into its horizontal and vertical components.

(a) (b) (c)

Solve these simultaneous equations to find the values of T and R, which are

forces acting on a particle, giving your answers to 2sf and using the correct

units. g represents the acceleration due to gravity, 9.8 𝑚𝑠−2

𝑇 cos 30 − 0.2𝑅 = 15 and 𝑅 + 𝑇 sin 30 = 5𝑔

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Mαths PS4 Pure Skills A

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Sketch: (a) 𝑦 = 𝑥 − 2 2 + 2 (b) 𝑦 =5

𝑥+2 2 + 2

(c) 𝑦 = 𝑥 𝑥 + 1 3 (d) 𝑦 = 4𝑥 𝑥 + 2 1 − 𝑥

Solve by completing the square (a) 15 − 6𝑥 − 2𝑥2 = 0 (b) 16𝑥 + 4𝑥2 = 0

Solve by factorising (a) 4𝑥2 − 49 = 0 (b) 7𝑥2 − 19𝑥 − 6 = 0

Solve the simultaneous equations using substitution

(a) 2𝑥 + 18𝑦 − 21 = 0 (b) 𝑦 + 𝑥 = 3

−14𝑦 = 3𝑥 + 14 𝑥2 − 3𝑦 = 1

For each subset of the real line ℝ, sketch it then convert the interval

into set notation or vice versa. If possible, write it in two different ways

(a) (1, 6 (b) −∞, 3 ∩ −2,∞) (c) 𝑥: 𝑥 ≤ 7 ∪ 𝑥: 𝑥 ≥ 5

9

10

11

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Mαths PS4 Thinking Skills A

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Use the information given to find the missing value of k

(a) The line 5

4𝑥 + 2𝑦 +

5

𝑘= 0 crosses the x axis at −4, 0

(b) The line between 5, 𝑘 and 8, 3 has gradient −3

(c) The triangle 0,1 , −2, 𝑘 , 0, 𝑘 + 1 has area 6

(a) Prove that 𝑥

1+ 2≡ 𝑥 2 − 𝑥, ∀𝑥 ∈ ℝ (this means ‘for all real numbers)

(b) Disprove the statement that 𝑥2 + 𝑏 ≡ 𝑥 +𝑏

2

2−

𝑏2

4, ∀𝑥, 𝑏 ∈ ℝ by

finding a counter example

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Mαths PS4 Thinking Skills A

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Trig functions: sin 𝑥 =𝑂

𝐻 cos 𝑥 =

𝐴

𝐻 tan 𝑥 =

𝑂

𝐴

Reciprocal trig functions: cosec 𝑥 =𝐻

𝑂 sec 𝑥 =

𝐻

𝐴 cot 𝑥 =

𝐴

𝑂

cosec 𝑥 =1

sin 𝑥 sec 𝑥 =

1

cos 𝑥 cot 𝑥 =

1

tan 𝑥

Change the following equations from ‘reciprocal trig’ equations to ‘trig’

equations of the form sin 𝑥 = ⋯ or cos 𝑥 = ⋯ or tan 𝑥 = ⋯

(a) 2 sec 𝑥 = 3 (b) 5 + cosec 𝑥 = 10 (c) 5 cot 𝑥 = 1

15

Mαths PS4 Answers B

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(ai) 54 (aii) 6

(b) Teacher to check

Mαths PS4 Answers B

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(ai) 1

4 (aii)

3

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(bi) 7

22 (bii)

35

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Mαths PS4 Answers B

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(a) 𝑦 = 𝑥2 − 4𝑥 + 3

(bi) ( (bii)

Mαths PS4 Answers B

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m 4

(a) 𝐴 −3,−2 𝐵 2, 3

(b) 𝑦 = 𝑥2 + 2𝑥 − 5

Mαths PS4 Answers B

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(ai) 𝑦 = −4𝑥 − 4 (aii) 𝑥 = −2

(b) 𝑥 + 2 2 + 𝑦 − 4 2 = 34

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1 (a) 1

7𝑥

1

2 (b) 𝑥 + 𝑥− 1

2 (c) 7𝑥−2 − 𝑥−1 + 2𝑥3

2 (a) 𝑥 =1

36 (b) 𝑥 = ±

1

27 (c) 𝑥 =

1

3

3 (a) 𝑥 + 9𝑦 − 29 = 0 (b) 𝑥 + 5𝑦 − 300 = 0 (c) 𝑥 − 9𝑦 + 24 = 0

4 (a) Centre −3,−2 radius 5 2 (b) Centre 3, 0 radius 4

(c) 𝑥 + 7 2 + 𝑦 − 13 2 =20

9 (d) 𝑥 − 3 2 + 𝑦 − 5 2 = 175

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1

2 𝑎 =9

4 𝑚𝑠−2

𝑠 = 30 𝑚

3

4 T = 26 N (2 sf), R = 36 N (2 sf)

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10 (a) 𝑥 = −3

2±

1

239 (b) 𝑥 = −4, 0 (c) 𝑥 = ±

7

2 (d) 𝑥 = −

2

7, 3

11 (a) 𝑥 = −21, y =7

2 (b) −5, 8 , 2, 1

12 (a) 𝑥: 𝑥 > 1 ∩ 𝑥: 𝑥 ≤ 6 (b) 𝑥: 𝑥 ≥ −2 ∩ 𝑥: 𝑥 < 3 (c) (−∞,−7 ∪ 5,∞)

OR 𝑥: 1 < 𝑥 ≤ 6 OR 𝑥:−2 ≤ 𝑥 < 3

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13 (a) 𝑘 = 1 (b) 𝑘 = 12 (c) 𝑘 = 6

14 (a) The layout used should be LHS = …

= …

= RHS

Conclude your proof with a statement of what has been proved.

(b) State which values of x and b you are going to use: ‘let x = …, b = …’

Sub your values of x and b into the LHS only to get a value

Separately, sub them into the RHS to get another value

Conclude your proof with a statement of what has been proved.

15 (a) cos 𝑥 =2

3 (b) sin 𝑥 =

1

5 (c) tan 𝑥 = 5

Mαths PS4 A

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1 Convert these expression to the 𝛼𝑥𝑛 form

(a) 1

5 𝑥 (b)

𝑥2+2

2𝑥 𝑥 (c)

12−𝑥+𝑥3

3

2 Solve the equation

(a) 𝑥 2

3 = 16 (b) 𝑥− 1

2 = 5 (c) 𝑥− 2

3 =1

9

3 Find the equation of the line:

(a) through 0, 0 perpendicular to 𝑥 − 𝑦 = 13

(b) through −2, 2 parallel to 100 = 𝑦 +1

5𝑥

(c) through −6, 2 parallel to 3𝑦 =1

3𝑥+2

4 (a) State the centre & radius of the circle 𝑥 − 8 2 + 𝑦2 = 64

(b) State the centre & radius of the circle 𝑥 + 10 2 + 𝑦 − 7 2 = 121

(c) State equation of the circle centre 6, −8 radius 4

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(d) State equation of the circle centre −1, 0 radius 7 5

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Extra Applied Skills A

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5 Draw a force diagram to model this situation:

A heavy box, being dragged with acceleration of 0.2 𝑚𝑠−2 along a rough horizontal floor by a

rope which is at an angle of 45° to the horizontal

6 A car is moving on a straight road at 15 𝑚𝑠−1 approaching traffic lights.

The driver applies the brakes and comes uniformly to rest in 45 m

Find the deceleration of the car and the time taken to stop using the following methods:

(a) By drawing st, vt and at graphs and using the vt graph

(b) Using suvat equations

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Extra Applied Skills A

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7 This skill is REALLY important!

This skill is REALLY important! Your task is to ‘resolve’ the vector

(which could represent a force) into its horizontal and vertical components.

(a) (b) (c)

8 Solve these simultaneous equations to find the values of T and R, which are

forces acting on a particle, giving your answers to 2sf and using the correct

units. g represents the acceleration due to gravity, 9.8 𝑚𝑠−2

𝑇 cos 20 − 0.18𝑅 = 4.7 and 𝑅 + 𝑇 sin 20 = 0.3𝑔

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9 Sketch: (a) 𝑦 = 8 −

1

𝑥+1 (b) 𝑦 = − 𝑥 + 3 2

(c) 𝑦 = 𝑥 − 3 2𝑥 − 1 2𝑥 + 5 (d) 𝑦 = 2 𝑥 − 3 1 − 3𝑥 1 − 𝑥 2

10 Solve by completing the square (a) 8𝑥 − 2𝑥2 = 0 (b) 2𝑥2 − 8𝑥 + 7 = 0

Solve by factorising (a) 4 − 9𝑥2 = 0 (b) 5𝑥2 − 13𝑥 − 6 = 0

11 Solve the simultaneous equations using substitution

(a) 6𝑥 + 2𝑦 − 8 = 0 (b) 𝑦 = 6𝑥2 + 3𝑥 − 7

4𝑥 + 3 = −3𝑦 𝑦 = 2𝑥 + 8

12 For each subset of the real line ℝ, sketch it then convert the interval

into set notation or vice versa. If possible, write it in two different ways

(a) −3, 0) (b) 𝑥: 𝑥 ≤ 6 ∩ 𝑥: 𝑥 ≥ 2 (c) −∞,−7 ∪ −1,∞

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Mαths PS4 A

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Extra Thinking Skills T

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13 Use the information given to find the missing value of k

(a) The line 7k𝑥 − 2𝑘2𝑦 − 9 = 0 crosses the x axis at 3, 0

(b) The line between −1,2𝑘 and 1, 4 has gradient −1

4

(c) The triangle −2,4 , −2,−2 , 𝑘,−4 has area 12

14 (a) Prove that 𝑥 −2

𝑥

3≡ 𝑥3 − 6𝑥 +

12

𝑥−

8

𝑥3 , ∀𝑥 ∈ ℝ (this means ‘for all real numbers)

(b) Disprove the statement ‘the product of two odd numbers is even’ by finding a counter example

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Mαths PS4 A

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Extra Thinking Skills T

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Trig functions: sin 𝑥 =𝑂

𝐻 cos 𝑥 =

𝐴

𝐻 tan 𝑥 =

𝑂

𝐴

Reciprocal trig functions: cosec 𝑥 =𝐻

𝑂 sec 𝑥 =

𝐻

𝐴 cot 𝑥 =

𝐴

𝑂

cosec 𝑥 =1

sin 𝑥 sec 𝑥 =

1

cos 𝑥 cot 𝑥 =

1

tan 𝑥

15 Change the following equations from ‘reciprocal trig’ equations to ‘trig’ equations of the

form sin 𝑥 = ⋯ or cos 𝑥 = ⋯ or tan 𝑥 = ⋯

(a) 1 − 2 cosec 𝑥 = 3 (b) 4 sec 𝑥 − 3 = 5 (c) 9 cot 𝑥 = 1

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Mαths PS4 Answers E

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1 (a)

1

5𝑥−

1

2 (b) 1

2𝑥

1

2 + 𝑥− 3

2 (c) 4 −1

3𝑥 +

1

3𝑥3

2 (a) 𝑥 = 64 (b) 𝑥 =1

25 (c) 𝑥 = 27

3 (a) 𝑥 + 𝑦 = 0 (b) 3𝑥 − 2𝑦 + 10 = 0 (c) 5𝑥 + 𝑦 − 14 = 0

4 (a) Centre 8, 0 radius 8 (b) Centre −10, 7 radius 11

(c) 𝑥 − 8 2 + 𝑦 + 6 2 =48

25 (d) 𝑥 + 1 2 + 𝑦2 = 245

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Mαths PS4 A

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5 6 7 8 T = 5.2 N (2 sf), R = 1.2 N (2 sf)

𝑎 = −22.5 𝑚𝑠−2

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Mαths PS4 Answers A

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10 (a) 𝑥 = −4, 0 (b) 𝑥 = 2 ±1

22 (c) 𝑥 = ±

3

2 (d) 𝑥 = −

2

5, 3

11 (a) 𝑥 = 3, y = −5 (b) −5

2, 3 ,

3

2, 11

12 (a) 𝑥: 𝑥 ≥ −3 ∩ 𝑥: 𝑥 < 0 b) 2, 6 OR 𝑥:−3 ≤ 𝑥 < 0 OR (−∞, 6 ∩ 2,∞)

(c) 𝑥: 𝑥 < −7 ∪ 𝑥: 𝑥 > −1

Mαths PS4 A

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13 (a) 𝑘 =3

7 (b) 𝑘 =

9

4 (c) 𝑘 = 2

14 (a) The layout used should be LHS = …

= …

= RHS

Conclude your proof with a statement of what has been proved.

(b) State which values of x and y you are going to use: ‘let x = …, y = …’

Sub your values of x and y into the LHS only to get a value

Separately, sub them into the RHS to get another value

Conclude your proof with a statement of what has been proved.

15 (a) sin 𝑥 = 1 (b) cos 𝑥 =1

2 (c) tan 𝑥 = 9