Worked Solutions

Pre Public Examination GCSE Mathematics (Edexcel style) November 2016 Higher Tier Paper 1H

Name ……………………………………………………………… Class ………………………………………………………………

TIME ALLOWED

1 hour 30 minutes

INSTRUCTIONS TO CANDIDATES

• Answer all the questions. • Read each question carefully. Make sure you know what you have to

do before starting your answer. • You are NOT permitted to use a calculator in this paper. • Do all rough work in this book.

INFORMATION FOR CANDIDATES

• The number of marks is given in brackets [ ] at the end of each question or part question on the Question Paper.

• You are reminded of the need for clear presentation in your answers.

• The total number of marks for this paper is 80. © The PiXL Club Limited 2016 This resource is strictly for the use of member schools for as long as they remain members of The PiXL Club. It may not be copied, sold nor transferred to a third party or used by the school after membership ceases. Until such time it may be freely used within the member school. All opinions and contributions are those of the authors. The contents of this resource are not connected with nor endorsed by any other company, organisation or institution.

Que

stio

n

Mar

k

out o

f

1 2

2 3

3 3

4 3

5 4

6 4

7 2

8 6

9 3

10 5

11 3

12 3

13 6

14 3

15 3

16 3

17 3

18 3

19 4

20 5

21 5

22 4

Total 80

Question 1.

Make t the subject of the formula w = 3t + 11

3t + 11 = w

è Work backwards

t = !!!!!

t = !!!!!

……………… (Total 2 marks)

Question 2.

Ashten chooses three different whole numbers between 1 and 50

The first number is a prime number. x The second number is 4 times the first number. 4x The third number is 6 less than the second number. 4x - 6 The sum of the three numbers is greater than 57

Find the three numbers. x + 4x + 4x – 6 > 57

9x – 6 > 57 x > 7

Next prime number above 7 is 11.

So 3 numbers are 11, 44 and 38

11, 44 and 38 ……………

(Total 3 marks)

Question 3.

Liam, Sarah and Emily shared some money in the ratio 2 : 3 : 7 Emily got £80 more than Liam.

How much money did Sarah get?

£80 “more” is 5 sections of the ratio.

So Sarah gets 3 x 16 = £48

£48 ……………

(Total 3 marks)

Question 4.

Work out an estimate for

𝟓+ 𝟐×𝟕

𝟏𝟗

So about 4.5

about 4.5

……………

(Total 3 marks)

1616161616

Question 5.

Denzil has a 4-sided spinner. The sides of the spinner are numbered 1, 2, 3 and 4 The spinner is biased. The table shows each of the probabilities that the spinner will land on 1, on 3 and on 4 The probability that the spinner will land on 3 is x.

Number 1 2 3 4

Probability 0.3 x 0.1

(a) Find an expression, in terms of x, for the probability that the spinner will land on 2. Give your answer in its simplest form.

P all of them adds to 1. So P(2) = 1 –0.3 - 0.1 - x) = 0.6 – x

.............................................................................................................................................. (2)

(b) Write down the probability that the spinner will land on either 1 or 4

0.1 + 0.3 = 0.4 ..............................................................................................................................................

(1) Denzil spins the spinner 300 times.

(c) Write down an expression, in terms of x, for the number of times the spinner is likely to land on 3

300 × x = 300x

.............................................................................................................................................. (1)

(Total 4 marks)

Question 6.

Gary drove from London to Sheffield. It took him 3 hours at an average speed of 80km/h.

Lyn drove from London to Sheffield. She took 5 hours.

Assuming that Lyn drove along the same roads as Gary and did not take a break, Distance is 240 km (a) Work out Lyn's average speed from London to Sheffield.

Lynn takes longer 48 km/h ........................................................... km/h

(3) (b) If Lyn did not drive along the same roads as Gary, explain how this could affect your answer to part (a). She might drive a different distance so her speed could be different................................................................

............................................................................................................................................. (1)

(Total 4 marks)

Question 7.

Mr Brown gives his class a test. The 10 girls in the class get a mean mark of 70% The 15 boys in the class get a mean mark of 80% Nick says that because the mean of 70 and 80 is 75 then the mean mark for the whole class in the test is 75% Nick is not correct. Is the correct mean mark less than or greater than 75%? You must justify your answer.

More than because

è There are more boys than girls è Real mean is !"×!"!!"×!"

!"= 76

(Total 2 marks)

𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒80×3

240𝑆𝑝𝑒𝑒𝑑×5

Question 8.

There are 80 students at a language school. All 80 students speak at least one language from French, German and Spanish.

9 of the students speak French, German and Spanish. 19 of the students speak French and German. 28 of the students speak French and Spanish. 17 of the students speak Spanish and German. 45 students speak French. 50 students speak Spanish.

(a) Draw a Venn diagram to show this information.

(3)

One of the 80 students is selected at random.

(b) Find the probability that this student speaks German but not Spanish. 2380

(1) Given that the student speaks German, (c) find the probability that this student also speaks French.

40 students speak German.

So answer is !"!"

(2)

1940

……………………

(Total 6 marks)

9

F G

S

10

19 8

7

14

13

Question 9.

Here is a cuboid.

All measurements are in centimetres. x is an integer. The total volume of the cuboid is less than 900 cm3 Show that x ≤ 5.

Volume is x × 2x × 3x = 6x3

So 6x3 ≤ 900 x3 ≤ 150 53 = 125

Using x = 5; 6x3 = 6×53=750 cm3 which is less than 900 cm3

Using x = 6; 6x3 = 6×63=1296 cm3 which more than 900 cm3

So x must be ≤ 5 because it is an integer

(Total 3 marks)

Question 10.

The diagram shows a square with perimeter 16 cm.

Work out the proportion of the area inside the square that is shaded.

Area square is 4 × 4 = 16 cm2

Area left triangle is !×!!

=2 cm2

Area bottom triangle is !×!!

=4 cm2

Proportion shaded is !"!"

of it or decimal/fraction equivalent

1016

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

(Total 5 marks)

2cm

1cm

Question 11.

(a) Write down the value of 60

1

............................1...............................

(1)

(b) Work out 64

64!! = 4

64!! = 16, so 64!

!!= !

!"

................................ !!"

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

(2) (Total 3 marks)

Question 12.

Given that

(2 × 10y) × (6 × 10y-1) = 1200000. Find the value of y.

12 × 102y-1= 12 × 105

So 2y – 1 = 5

y = 3

y = 3

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

(Total 3 marks)

Question 13.

(a) Solve !(!!!!)

!! = 10

32x – 8 = 30x 2x = 8

x = 4

x = 4 ...........................................................

(3)

(b) Write as a single fraction in its simplest form

2 𝑦 − 6 − 1(𝑦 + 3)(𝑦 + 3)(𝑦 − 6)

𝑦 − 15(𝑦 + 3)(𝑦 − 6)

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

(3) (Total 6 marks)

Question 14.

y is inversely proportional to the square of x. When x = 5, y = 15

Write a formula for y in terms of x.

𝒚 =𝒌𝒙𝟐

𝟏𝟓 =𝒌𝟓𝟐

k = 375

𝒚 =𝟑𝟕𝟓𝒙𝟐

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

(Total 3 marks)

Question 15.

Simplify completely

(2𝑥 + 1)(𝑥 − 5)2𝑥!(2𝑥 + 1)

(𝑥 − 5)2𝑥!

(Total 3 marks)

Question 16.

Here are the first 5 terms of a quadratic sequence.

1 3 7 13 21

Find an expression, in terms of n, for the nth term of this quadratic sequence. First difference is 2, 4, 6, 8 Second difference is 2.

So sequence is n2 n2 sequence is 1, 4, 9, 16

Subtract n2 sequence from original -> 0, -1, -2, -3 Nth term of this new sequence is 1 - n

So nth term of sequence is n2 - n + 1

n2 - n + 1

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

(Total 3 marks)

Question 17.

Solve x2 > 3x + 4 x2 – 3x – 4 > 0

(x – 4)(x + 1) > 0 x < -1 and x > 4

x < -1 and x > 4

........................................................... (Total 3 marks)

Question 18.

Simplify fully (√a + √4√b)(√a – 2√b)

a + 2 𝑎𝑏 - 2 𝑎𝑏 - 4𝑏

a – 4b

a – 4b

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

(Total 3 marks)

Question 19.

Work out the value of x in the triangle shown.

By Pythagoras, x2 + (x+1)2=(x+2)2

x2 + x2+2x + 1 = x2 + 4x + 4

x2 - 2x - 3 = 0

(x -3)(x + 1) = 0

So x = 3 (can’t have a negative length)

........................x = 3..................................

(Total 4 marks)

Question 20.

CAYB is a quadrilateral.

X is the point on AB such that AX : XB = 1 : 2

Prove that 𝐶𝑌 = 6b + 5a – b = 5a + 5b

𝐴𝐵 = -3a + 6b

𝐶𝑋 =3a + !!𝐴𝐵

So CX = 3a + !!(-3a + 6b) =3a - a + 2b

= 2a + 2b

= !!

𝐶𝑌

(Total 5 marks)

Question 21.

A(−2, 1), B(6, 5) and C(4, k) are the vertices of a right-angled triangle ABC. Angle ABC is the right angle.

Find an equation of the line that passes through A and C. Give your answer in the form ay + bx = c where a, b and c are integers.

Gradient of AB is 0.5. So gradient of BC is -2.

This means coordinate C must be (4,9).

Gradient of AC is 8/6 = 4/3, so equation of AC is y = 4/3x + c

AC is 3y = 4x + c

Sub in (4, 9): 27 = 16 + c, so C = 11

Final Equation is 3y – 4x = 11

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

(Total 5 marks)

A(-2,1)

B(6,5)

C(4,k)

Question 22.

Prove algebraically that the difference between the squares of any two consecutive integers is equal to the sum of these two integers.

Two consecutive integers are n and n + 1.

Sum of the two integers is 2n + 1

Difference between the squares is (n + 1)2 – n2

n2 + 2n + 1 – n2

Simplifies to 2n + 1 (Proved)

(Total 4 marks)

TOTAL FOR PAPER IS 80 MARKS