Instructions
· Use black ink or ball-point pen.
· Fill in the boxes at the top of this page with your name,
· centre number and candidate number.
· Answer all questions.
· Answer the questions in the spaces provided
– there may be more space than you need.
· Calculators must not be used.
Information
· There are 9 questions on this paper; the total mark is 38
· The marks for each question are shown in brackets
– use this as a guide as to how much time to spend on each question.
· All questions are labelled with an asterisk (*) and are ones where the quality of your written communication will be assessed.
Advice
· Read each question carefully before you start to answer it.
· Keep an eye on the time.
· Try to answer every question.
· Check your answers if you have time at the end.
GCSE Mathematics (Linear) 1MA0
Formulae: Higher Tier
You must not write on this formulae page.
Anything you write on this formulae page will gain NO credit.
Volume of prism = area of cross section × length Area of trapezium = (a + b)h
Volume of sphere πr3 Volume of cone πr2h
Surface area of sphere = 4πr2 Curved surface area of cone = πrl
In any triangle ABC The Quadratic Equation
The solutions of ax2+ bx + c = 0
where a ≠ 0, are given by
x =
Sine Rule
Cosine Rule a2 = b2+ c2– 2bc cos A
Area of triangle = ab sin C
Answer ALL NINE questions.
Write your answers in the spaces provided.
You must write down all stages in your working.
You must NOT use a calculator.
1*. Here is part of Gary’s electricity bill.
Electricity billNew reading 7155 units
Old reading 7095 units
Price per unit 15p
Work out how much Gary has to pay for the units of electricity he used.
(Total for Question 1 is 4 marks)
______
2*. Bill uses his van to deliver parcels.
For each parcel Bill delivers there is a fixed charge plus £1.00 for each mile.
You can use the graph to find the total cost of having a parcel delivered by Bill.
(a) How much is the fixed charge?
£ ......
(1)
Ed uses a van to deliver parcels.
For each parcel Ed delivers it costs £1.50 for each mile.
There is no fixed charge.
(b) Compare the cost of having a parcel delivered by Bill with the cost of having a parcel delivered by Ed.
(3)
(Total for Question 2 is 4 marks)
______
3*. Railtickets and Cheaptrains are two websites selling train tickets.
Each of the websites adds a credit card charge and a booking fee to the ticket price.
RailticketsCredit card charge: 2.25% of ticket price
Booking fee: 80 pence / Cheaptrains
Credit card charge: 1.5% of ticket price
Booking fee: £1.90
Nadia wants to buy a train ticket.
The ticket price is £60 on each website.
Nadia will pay by credit card.
Will it be cheaper for Nadia to buy the train ticket from Railtickets or from Cheaptrains?
(Total for Question 3 is 4 marks)
______
4*.
CDEF is a straight line.
AB is parallel to CF.
DE = AE.
Work out the size of the angle marked x.
You must give reasons for your answer.
(Total for Question 4 is 4 marks)
______
5*. Talil is going to make some concrete mix.
He needs to mix cement, sand and gravel in the ratio 1 : 3 : 5 by weight.
Talil wants to make 180 kg of concrete mix.
Talil has
15 kg of cement
85 kg of sand
100 kg of gravel
Does Talil have enough cement, sand and gravel to make the concrete mix?
(Total for Question 5 is 4 marks)
______
6*.
B, C and D are points on the circumference of a circle, centre O.
AB and AD are tangents to the circle.
Angle DAB = 50°
Work out the size of angle BCD.
Give a reason for each stage in your working.
(Total for Question 6 is 4 marks)
______
7*. Prove algebraically that the difference between the squares of any two consecutive integers is equal to the sum of these two integers.
(Total for Question 7 is 4 marks)
______
8.
OAYB is a quadrilateral.
= 3a
= 6b
(a) Express in terms of a and b.
......
(1)
X is the point on AB such that AX : XB = 1 : 2
and = 5a – b
*(b) Prove that =
(4)
(Total for Question 8 is 5 marks)
______
9.
APB is a triangle.
N is a point on AP.
= a = 2b = b
(a) Find the vector, in terms of a and b.
......
(1)
B is the midpoint of AC.
M is the midpoint of PB.
*(b) Show that NMC is a straight line.
(4)
(Total for Question 9 is 5 marks)
TOTAL FOR PAPER IS 38 MARKS
BLANK PAGE
14
1MA0 1H – Practice Paper (Set G) QWC /Question / Working / Answer / Mark / Notes /
1* / 9 / 4 / M1 for 7155 – 7095 or 60 seen or 7155×15 (or .15) or 7095×15 (or .15) or 107325 or 106425 or 1073.25 or 1064.25
M1 for ‘60’ ×15 or 7155 ×15 – 7095 × 15 [or .15 instead of 15]
A1 for 9 or 9.00 or 900
C1 (ft ) for answer with correct units (money notation) identified as the answer.
2* / (a) / 10 / 1 / B1 cao
(b) /
Miles / 0 / 10 / 20 / 30 / 40 / 50
Ed / 0 / 15 / 30 / 45 / 60 / 75
Bill / 10 / 20 / 30 / 40 / 50 / 60
/ Ed is cheaper up to 20 miles, Bill is cheaper for more than 20 miles / 3 / M1 for correct line for Ed intersecting at (20,30) ±1 sq tolerance or
10 + x = 1.5x oe
C2 (dep on M1) for a correct full statement ft from graph
eg. Ed cheaper up to 20 miles and Bill cheaper for more than 20 miles
(C1 (dep on M1) for a correct conclusion ft from graph
eg. cheaper at 10 miles with Ed ; eg. cheaper at 50 miles with Bill
eg. same cost at 20 miles; eg for £5 go further with Bill OR
A general statement covering short and long distances eg. Ed is cheaper for shorter distances and Bill is cheaper for long distances)
OR
M1 for correct method to work out Ed's delivery cost for at least 2 values of n miles where 0 < n ≤ 50 OR
for correct method to work out Ed and Bill's delivery cost for n miles where 0 < n ≤ 50
C2 (dep on M1) for 20 miles linked with £30 for Ed and Bill with correct full statement
eg. Ed cheaper up to 20 miles and Bill cheaper for more than 20 miles
(C1 (dep on M1) for a correct conclusion
eg. cheaper at 10 miles with Ed; eg. cheaper at 50 miles with Bill
eg. same cost at 20 miles; eg for £5 go further with Bill OR
A general statement covering short and long distances eg. Ed is cheaper for shorter distances and Bill is cheaper for long distances)
SC : B1 for correct full statement seen with no working
eg. Ed cheaper up to 20 miles and Bill cheaper for more than 20 miles
QWC: Decision and justification should be clear with working clearly presented and attributable
3 / 2.25 × 60 ÷ 100 = 1.35
1.35 + 0.80 = 2.15
1.5 × 60 ÷ 100 = 0.90
0.90 + 1.90 = 2.80
OR
2.25 – 1.5 = 0.75
0.075 × 60 ÷ 100 = 0.45
0.80 + 0.45 = 1.25
1.25 < 1.90 / Railtickets with correct calculations / 4 / NB. All work may be done in pence throughout
M1 for correct method to find credit card charge for one company
eg. 0.0225 × 60(=1.35) oe or 0.015 × 60 (=0.9) oe
M1 (dep) for correct method to find total additional charge or total price for one company
eg. 0.0225×60 + 0.80 or 0.015×60 + 1.90 or
2.15 or 2.8(0) or 62.15 or 62.8(0)
A1 for 2.15 and 2.8(0) or 62.15 and 62.8(0)
C1 (dep on M1) for a statement deducing the cheapest company, but figures used for the comparison must also be stated somewhere, and a clear association with the name of each company
OR
M1 for correct method to find percentage of (60+booking fee)
eg. 0.0225 × 60.8(=1.368) oe or 0.015 × 61.9(=0.9285)
M1 (dep) for correct method to find total cost or total additional cost
eg. '1.368' + 60.8(=62.168) or '1.368' + 0.8 (=2.168) or
'0.9285' + 61.9 (=62.8285) or '0.9285' +1.9 (=2.8285)
A1 for 62.168 or 62.17 AND 62.8285 or 62.83 OR
2.168 or 2.17 AND 2.8285 or 2.83
C1 (dep on M1) for a statement deducing the cheapest company, but figures used for the comparison must also be stated somewhere, and a clear association with the name of each company
OR
M1 for correct method to find difference in cost of credit card charge
eg. (2.25 – 1.5) × 60 ÷ 100 oe or 0.45 seen
M1 (dep) for using difference with booking fee or finding difference between booking fees
eg. 0.80 + “0.45”(=1.25) or
1.90 – “0.45” (=1.45) or 1.90 – 0.8 (=1.1(0))
A1 1.25 and 1.9(0) or 0.45 and 1.1(0)
C1 (dep on M1) for a statement deducing the cheapest company, but figures used for the comparison must also be stated somewhere, and a clear association with the name of each company
QWC: Decision and justification should be clear with working clearly presented and attributable
4* / x / –2 / –1 / 0 / 1 / 2
y / (1) / 3 / (5) / 7 / 9
/ 3, 7, 9 / 2 / B2 for all three values correct in the table
(B1 for 2 values correct)
/ graph of
y = 2x + 5 / 2 / (From their table of values)
M1 ft for plotting at least 2 of their points (any points from their table must be correctly plotted)
A1 for correct line from x = –2 to x = +2
(Use of y = mx + c)
M1 for line drawn with gradient of 2 or line drawn with a y intercept of 5 and a positive gradient)
A1 for correct line from x = –2 to x = +2
5* / 180÷9×1:180÷9×3:180÷9×5
=20:60:100
Not enough cement
(but enough sand and enough gravel)
OR
1×15:3×15:5×15
=15:45:75
15+45+75=135 (<180)
Not enough cement (to make 180kg of concrete) / No + reason / 4 / M1 for 180 ÷ (1+3+5) ( = 20) or 3 multiples of 1: 3: 5
M1 for 1×”20” or 3×”20” or 5×”20” or 20 seen or 60 seen or 100 seen
A1 for (Cement =) 20, (Sand =) 60, (Gravel) = 100
C1 ft (provided both Ms awarded) for not enough cement oe
OR
M1 for (1×15 and) 3×15 and 5×15 or 9×15 or sight of the numbers 15, 45, 75 together.
M1 for ‘15’ + ‘45’ + ‘75’
A1 for 135 (<180)
C1 ft (provided both Ms awarded) for not enough cement oe
6* / ABO = ADO = 90°
(Angle between tangent and radius is 90°)
DOB = 360 – 90 – 90 – 50
(Angles in a quadrilateral add up to 360°)
BCD = 130 ÷ 2
(Angle at centre is twice angle at circumference)
OR
ABD = (180 – 50) ÷ 2
(Base angles of an isosceles triangle)
BCD = 65
(Alternate segment theorem) / 65o / 4 / B1 for ABO = 90 or ADO = 90 (may be on diagram)
B1 for BCD = 65 (may be on diagram)
C2 for BCD = 65o stated or DCB = 65o stated or angle C = 65o stated with all reasons:
angle between tangent and radius is 90o;
angles in a quadrilateral sum to 360o;
angle at centre is twice angle at circumference
(accept angle at circumference is half (or ) the angle at the centre)
(C1 for one correct and appropriate circle theorem reason)
QWC: Working clearly laid out and reasons given using correct language
OR
B1 for ABD = 65 or ADB = 65 (may be on diagram)
B1 for BCD = 65 (may be on diagram)
C2 for BCD = 65o stated or DCB = 65o stated or angle C = 65o stated with all reasons:
base angles of an isosceles triangle are equal;
angles in a triangle sum to 180o;
tangents from an external point are equal;
alternate segment theorem
(C1 for one correct and appropriate circle theorem reason)
QWC: Working clearly laid out and reasons given using correct language
7* / (n + 1)2 – n2
= n2 + 2n + 1 – n2 = 2n + 1
(n + 1) + n = 2n + 1
OR
(n + 1)2 – n2
= (n + 1 + n)(n + 1 – n)
= (2n + 1)(1) = 2n + 1
(n + 1) + n = 2n + 1
OR
n2 – (n + 1)2 = n2 – (n2 + 2n + 1) =
–2n – 1 = – (2n + 1)
Difference is 2n + 1
(n + 1) + n = 2n + 1 / proof / 4 / M1 for any two consecutive integers expressed algebraically
eg n and n +1
M1(dep on M1) for the difference between the squares of ‘two consecutive integers’ expressed algebraically eg (n + 1)2 – n2
A1 for correct expansion and simplification of difference of squares, eg 2n + 1
C1 (dep on M2A1) for showing statement is correct,
eg n + n + 1 = 2n + 1 and (n + 1)2 – n2 = 2n + 1 from correct supporting algebra
8* / 6b – 3a / 1 / B1 for 6b – 3a oe
4 / M1 for AX = AB or ’(6b – 3a)’ or ft to 2b – a
M1 for OY = OB + BY = 6b + 5a – b (= 5b + 5a ) oe
M1 for OX = 3a + ‘2b – a’ = 2a + 2b oe
Or
OX = 6b – ‘(6b – 3a)’ (= 2a + 2b) oe
C1 for OY =×5(a + b) = 2(a + b) = OX
9* / (a) / a – 3b / 1 / B1 for a – 3b oe
*(b) / 4 / M1 for (NC =) oe
M1 for (NM =)
A1 for oe and oe
C1 for NC is a multiple of NM (+ common point)
OR
M1 for (NC =) oe
M1 for (MC =)
A1 for oe and oe
C1 for NC is a multiple of MC (+ common point)
OR
M1 for (NM =)
M1 for (MC =)
A1 for oe and oe
C1 for NM is a multiple to MC (+ common point)
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