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PLC Papers

Created For:

PiXL PLC 2017 Certification

Algebra and proof 2 Grade 8

Objective: Use algebra to construct proofs Question 1

a) If n is a positive integer explain why the expression 2n + 1 is always an odd number.

(1)

b) Use algebra to prove that the product of two odd numbers is also odd.

(4)

Question 2

a) If x > 3 and prove that F > 1

(4)

b) Explain what happens if x = 3

(1)

Total /10


Approximate solutions to equations using iteration 2 Grade 9 Objective: Find approximate solutions to equations using iteration.

Question 1.

Find the first four iterations of each iterative formulae. Start each one with �1 = 6.

a) ��+1 = 5�1 − 4

……………………………….…………………….……………….………………………

(1)

b) ��+1 =��2 + 5

……………………………….…………………….……………….………………………

(1)

c) ��+1 =14��+1

……………………………….…………………….……………….………………………

(1)

(Total 3 marks)

Question 2.

Starting with �1 = 5.3 verify that 5.37 is a solution, correct to 2 decimal places, of the quadratic equation �2 − 5� − 2 = 0 using iteration.

………………………

(Total 3 marks)


Question 3.

a) Show that � =4� − 9 can be rearranged into the equation �2 + 9� − 4 = 0.

(1)

b) Use the iterative formula ��+1 =4�� − 9 and a starting value of �1 = 0.5 to obtain the

solution to the equation correct to 2 decimal places.

………………………

(3)

(Total 4 marks)

TOTAL /10


Equation of a circle 2 Grade 9

Objective: Recognise and use �� + �� = ��.

Question 1

(a) Write down the equation of a circle with centre (0, 0) and radius 1.5.

……………………………

(3)

(b) Write down the centre and radius of the circle �2 + �2 = 81.

Centre =……………………………

Radius =……………………………

(2)

(Total 5 marks)

Question 2

On the grid, draw the graph of �2 + �2 = 72.

(3)

(Total 3 marks)


Question 3

A graph has been drawn for you on the grid below.

Write down the equation of this graph.

……………………………

(2)

(Total 2 marks)

TOTAL /10


Equation of a tangent to a circle 2 Grade 9

Objective: Find the equation of a tangent to a circle at a given point.

Question 1

The grid below shows a circle with equation �2 + �2 = 8.

There are two tangents to this circle with gradient 1.

(a) Draw these tangents on the graph above.

(2)

(b) Write down the equation of these tangents.

y =……………………………

y =……………………………

(2)

(Total 4 marks)


Question 2

Here is a circle, �2 + �2 = 13, and a tangent to the circle.

The tangent goes through the point B(2, -3) on the circle.

Find the equation of the tangent at point B.

……………………………

(4)

(Total 4 marks)


Question 3

The equation of a circle is �2 + �2 = �.

The line � = 7 is a tangent to the circle.

Work out the value of k.

……………………………

(2)

(Total 2 marks)

TOTAL /10


Gradients and area under a graph 2 Grade 8

Objective: Calculate or estimate the gradient of a graph and the area under a graph

Question 1

A straight line has been drawn on a grid.

Calculate the gradient of the line.

…………………………

(2)

(Total 2 marks)

Question 2

Work out the gradient of the line 5� − 3� = 20

……………………………

(2)

(Total 2 marks)


Question 3

The graph of � = �3 + 3�2 − 2� − 1 is drawn on the grid below.

Calculate an estimate to the gradient of the curve at the point Q(-1, 3).

……………………………

(3)

(Total 3 marks)


Question 4

The scatter graph shows the cost of cars in a used car showroom.

(a) Draw a line of best fit and calculate the gradient of this line.

……………………………

(2)

(b) Give an interpretation of this gradient.

(1)

(Total 3 marks)

TOTAL /10


Quadratic equations (completing the square) 2 Grade 8 Objective: Solve quadratic equations by completing the square.

Question 1.

Rewrite �2 + 6� + 7 in the form (� + �)2 − �

………………………

(Total 1 mark)

Question 2.

Solve �2 − 10� + 9 = 0 by completing the square.

………………………

(Total 2 marks)


Question 3.

Solve �2 − 8� − 12 = 0 by completing the square.

Leave your answers in surd form.

………………………

(Total 3 marks)

Question 4.

Solve 4�2 + 28� − 24 = 0 by completing the square.

Give your answers to 3 significant figures.

………………………

(Total 4 marks)

TOTAL /10


Trigonometric Graphs 2 Grade 8

Objective: Recognise, sketch, and interpret graphs of trigonometric functions

Question 1

Sketch the graph of y = tan x for 0 ≤ � ≤ 360°

(3)

(Total 3 marks)

Question 2

Here is the graph of y = cos x for 0 ≤ � ≤ 360°

On the axes above, sketch the graph � = cos(2�) − 2 for 0 ≤ � ≤ 360°

(3)

(Total 3 marks)


Question 3

The graph of y = sin x for 0 ≤ � ≤ 360° is shown below.

What are the coordinates of the 4 points labelled on the graph?

(………, ………)

(………, ………)

(………, ………)

(………, ………)

(4)

(Total 4 marks)

TOTAL /10

PLC Papers

Created For:


Algebra and proof 2 Grade 8 Solutions

Objective: Use algebra to construct proofs Question 1

a) If n is a positive integer explain why the expression 2n + 1 is always an odd number.

2n is a multiple of 2 so it must be even so 2n + 1 is the number after an even number so it must

be odd.

(1)

b) Use algebra to prove that the product of two odd numbers is also odd.

(2n + 1) (2m + 1)

= 4mn + 2n + 2m + 1

= 2 ( 2mn + n + m) + 1

2 ( 2mn + n + m) must be even so

2 ( 2mn + n + m) + 1 must be odd

(4)

Question 2

a) If x > 3 and prove that F > 1

x + 2 > x so numerator is bigger than denominator hence F > 1

(4)

b) Explain what happens if x = 3

If x = 3 then x – 3 = 0

If you divide by x – 3 you are dividing by 0 so F is undefined

(May write you can’t divide by 0)

(1)

Total /10

• Expand and simplify brackets • Factorise • Explain why factorised part is even • State result must be odd

• Factorise numerator • Factorise denominator • Simplify fraction • Explain why F > 1


Approximate solutions to equations using iteration 2 Grade 9 SOLUTIONS Objective: Find approximate solutions to equations using iteration.

Question 1.

Find the first four iterations of each iterative formulae. Start each one with �1 = 6.

a) ��+1 = 5�1 − 4 �2 = 26, �3 = 126, �4 = 626, �5 = 3126, (A1)

(1)

a) ��+1 =��2 + 5 �2 = 8, �3 = 9, �4 = 9.5, �5 = 9.75, (A1)

(1)

b) ��+1 =14��+1 �2 = 2, �3 =

143 , �4 =4217 , �5 =

23859 (A1)

(1)

(Total 3 marks)

Question 2.

Starting with �1 = 5.3 verify that 5.37 is a solution, correct to 2 decimal places, of the quadratic equation �2 − 5� − 2 = 0 using iteration. ��+1 = �5�� + 2 (M1) �2 = 5.33853 … (M1) �3 = 5.35655 … �4 = 5.36495 … �5 = 5.36887 … �6 = 5.37069 … �5 = �6 to 2dp (C1)

………………………

(Total 3 marks)


Question 3.

a) Show that � =4� − 9 can be rearranged into the equation �2 + 9� − 4 = 0.

� =4� − 9 �2 = 4 − 9� �2 + 9� − 4 = 0 (M1)

(1)

b) Use the iterative formula ��+1 =4�� − 9 and a starting value of �1 = 0.5 to obtain a

solution to the equation correct to 2 decimal places. �2 = −1 (M1) �3 = −13 �4 = −9.30769 … �5 = −9.42975 … �6 = −9.42418 … �7 = −9.42443 … �6 = �7 to 2dp (C1) � = −9.42 (A1)

………………………

(3)

(Total 4 marks)

TOTAL /10


Equation of a circle 2 Grade 9 Solutions

Objective: Recognise and use �� + �� = ��.

Question 1

(a) Write down the equation of a circle with centre (0, 0) and radius 1.5. �2 + �2 = � where c>0 (M1)

1.52 = 2.25 (M1) �2 + �2 = 2.25 (A1)

(3)

(b) Write down the centre and radius of the circle �2 + �2 = 81.

Centre = (0, 0) (A1)

Radius = 9 (A1)

(2)

(Total 5 marks)

Question 2

On the grid, draw the graph of �2 + �2 = 72.

(3)

(Total 3 marks)

Centre at (0, 0) and attempt of circle in

more than 3 quadrants (M1)

Radius approx. 6√2 ≈ 8.5 (M1)

Fully correct graph (G1)


Question 3

A graph has been drawn for you on the grid below.

Write down the equation of this graph.

Sight of 42 = 16 (M1) �2 + �2 = 16 (A1)

(2)

(Total 2 marks)

TOTAL /10


Equation of a tangent to a circle 2 Grade 9 Solutions

Objective: Find the equation of a tangent to a circle at a given point.

Question 1

The grid below shows a circle with equation �2 + �2 = 8.

There are two tangents to this circle with gradient 1.

(a) Draw these tangents on the graph above.

(2)

(b) Write down the equation of these tangents.

y = x – 4 (B1)

y = x + 4 (B1)

(2)

(Total 4 marks)

Any line with gradient 1 (B1)

y = x - 4 and y = x + 4 drawn (B1)


Question 2

Here is a circle, �2 + �2 = 13, and a tangent to the circle.

The tangent goes through the point B(2, -3) on the circle.

Find the equation of the tangent at point B.

Grad of OB = -3/2 (M1)

Grad of tangent = 2/3 (M1) � + 3 = 23 (� − 2) Use of this or y = mx + c (M1)

3� + 9 = 2� − 4

3� = 2� − 13 o.e. (A1)

� = 23 � − 133

(4)

(Total 4 marks)


Question 3

The equation of a circle is �2 + �2 = �.

The line � = 7 is a tangent to the circle.

Work out the value of k.

Radius of circle = 7 from diagram or explanation (M1)

�2 + �2 = 49 or k = 49 (A1)

(2)

(Total 2 marks)

TOTAL /10


Gradients and area under a graph 2 Grade 8 Solutions

Objective: Calculate or estimate the gradient of a graph and the area under a graph

Question 1

A straight line has been drawn on a grid.

Calculate the gradient of the line. �� = −42 (M1)

m = -2 (A1)

(2)

(Total 2 marks)

Question 2

Work out the gradient of the line 5� − 3� = 20

Correct attempt to make y the subject: � =35 � + 20 (M1) � =

35 (A1)

(2)

(Total 2 marks)


Question 3

The graph of � = �3 + 3�2 − 2� − 1 is drawn on the grid below.

Calculate an estimate to the gradient of the curve at the point Q(-1, 3).

Consider points just above and just below, i.e. x = -1.1 and x = -0.9 (M1)

(-1.1, 3.499) and (-0.9, 2.501) �� = 2.501−3.499−0.9+1.1 (M1)

= −4.99

m = -4.99 (or -5) (A1)

(3)

(Total 3 marks)


Question 4

The scatter graph shows the cost of cars in a used car showroom.

(a) Draw a line of best fit and calculate the gradient of this line.

Using their line, �� =

−80008 or use of any other points (M1)

m = -1000 (A1)

(2)

(b) Give an interpretation of this gradient.

The value of a car goes down by £1000 every year it gets older (or similar explanation) (C1)

(1)

(Total 3 marks)

TOTAL /10


Quadratic equations (completing the square) 2 Grade 8 SOLUTIONS Objective: Solve quadratic equations by completing the square.

Question 1.

Rewrite �2 + 6� + 7 in the form (� + �)2 − � �2 + 6� + 9 − 9 + 7

(� + 3)2 − 2 (A1)

………………………

(Total 1 mark)

Question 2.

Solve �2 − 10� + 9 = 0 by completing the square. �2 − 10� + 25 − 25 + 9 = 0

(� − 5)2 − 16 = 0 (M1)

(� − 5)2 = 16 � − 5 = ±4 � = 5 ± 4 � = 9 �� = 1 (A1)

………………………

(Total 2 marks)


Question 3.

Solve �2 − 8� − 12 = 0 by completing the square.

Leave your answers in surd form. �2 − 8� + 16− 16 − 12 = 0

(� − 4)2 − 28 = 0 (M1)

(� − 4)2 = 28 � − 4 = ±√28 � = 4 ± √28 (M1) � = 4 ± 2√7 (A1)

………………………

(Total 3 marks)

Question 4.

Solve 4�2 + 28� − 24 = 0 by completing the square.

Give your answers to 3 significant figures. �2 + 7� − 6 = 0 (M1) �2 + 7� + 12.25− 12.25− 6 = 0

(� + 3.5)2 − 18.25 = 0 (M1)

(� + 3.5)2 = 18.25 � + 3.5 = ±√18.25 � = −3.5 ± √18.25 (M1) � = 0.772 �� = −7.77 (A1)

………………………

(Total 4 marks)

TOTAL /10


Trigonometric Graphs 2 Grade 8 Solutions

Objective: Recognise, sketch, and interpret graphs of trigonometric functions

Question 1

Sketch the graph of y = tan x for 0 ≤ � ≤ 360°

(3)

(Total 3 marks)

Question 2

Here is the graph of y = cos x for 0 ≤ � ≤ 360°

On the axes above, sketch the graph � = cos(2�) − 2 for 0 ≤ � ≤ 360°

(3)

(Total 3 marks)

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7

y = cos x y = cos(2x) -2


Question 3

The graph of y = sin x for 0 ≤ � ≤ 360° is shown below.

What are the coordinates of the 4 points labelled on the graph?

(……0…, …0…)

( 90…, ……1…)

(……270…, …-1……)

(……360…, ……0…)

(4)

(Total 4 marks)

TOTAL /10

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