mathematics cm - crashmaths · 24 1 2 3 3 2 2 1 1 8 m 1 8 5 16 the mass a is held at rest on a...
TRANSCRIPT
MathematicsAS PAPER 1
March Mock Exam (AQA Version) Time allowed: 1 hour and 30 minutes
Instructions to candidates:
• In the boxes above, write your centre number, candidate number, your surname, other names
and signature.
• Answer ALL of the questions.
• You must write your answer for each question in the spaces provided.
• You may use a calculator.
Information to candidates:
• Full marks may only be obtained for answers to ALL of the questions.
• The marks for individual questions and parts of the questions are shown in square brackets.
• There are 16 questions in this question paper. The total mark for this paper is 75.
Advice to candidates:
• You should ensure your answers to parts of the question are clearly labelled.
• You should show sufficient working to make your workings clear to the Examiner.
• Answers without working may not gain full credit.
CM
AS/P1/M18© 2018 crashMATHS Ltd.
1 2 3 3 2 2 1 1 8 M 1 8 5
Surname
Other Names
Candidate Signature
Centre Number Candidate Number
Examiner Comments Total Marks
2
1
1 2 3 3 2 2 1 1 8 M 1 8 5
Find the gradient of a line perpendicular to 2x – 3y = 6.
Circle your answer.
[1 mark]
Answer all questions in the spaces provided.
2 Given that , find the value of p.
Circle your answer.
[1 mark]
3 p = 30.25 3
34
18
16
14
23
− 23
− 32
32
3 The curve C has the equation y = f(x).
The curve C has one turning point at (–2, 5).
Find the coordinates of the turning point on the curve with equation y = f(x – 4)?
Circle your answer.
[1 mark]
(2, 5) (–6, 5) (2, 9) (2, 1)
4 Find the coefficient of x4 in the expansion of (2x – 1)12.
Circle your answer.
[1 mark]
7920 –7920 495 –495
Section A
3
1 2 3 3 2 2 1 1 8 M 1 8 5
Turn over ►
5 The point A has position vector 3i – 4j and the point B has position vector ai + 7j, where a is a constant.
Given that , find the largest possible value of the constant a.
[4 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
AB! "!!
= 5 5
4
6
1 2 3 3 2 2 1 1 8 M 1 8 5
The curve C has the equation y = f(x), where f(x) = tan(x – 40o) for 0 ≤ x ≤ 360o.
Solve the equation f(x) = 0.
[2 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
6 (a)
6 (b) Find the coordinates where the curve C crosses the y-axis.
[1 mark]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
6 (c) Write down the equations of any asymptotes to the curve C.
[2 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
5
1 2 3 3 2 2 1 1 8 M 1 8 5
Turn over ►
6 (d) Sketch the curve C.
[2 marks]
6
7
1 2 3 3 2 2 1 1 8 M 1 8 5
7 (i)
The diagram above shows a sketch of the curve with equation y = f(x). The curve
crosses the y-axis at the point (0, 1).
Sketch the curve with equation y = 2f(x).
[2 marks]
x
y
O
y = f(x)
1
7
1 2 3 3 2 2 1 1 8 M 1 8 5
Turn over ►
7 (ii) Sketch the curve with equation .
[2 marks]y = f '(x)
Turn over for the next question
8
8
1 2 3 3 2 2 1 1 8 M 1 8 5
The triangle ABC is shown in the diagram above, where angle ACB = 60o,
AC = 12 cm, BC = 10 cm, AB = a cm and a is a constant.
Find the area of the triangle ABC.
[2 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
A
B
C
a cm
12 cm
10 cm
60°
8 (a)
8 (b) Calculate the value of a.
[2 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
9
1 2 3 3 2 2 1 1 8 M 1 8 5
Turn over ►
Given that the angle BAC = xo, show that .
[1 mark]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
8 (d) Edward says,
“There are two possible values of x. Either:
. .”
Edward’s teacher says he is wrong and only one of these values is correct in this case.
Identify the correct value of x.
[1 mark]
__________________________________________________________________
__________________________________________________________________
8 (c) sin x = 5 9262
x = sin−1 5 9362
⎛
⎝⎜
⎞
⎠⎟ or x = 180 − sin−1 5 93
62
⎛
⎝⎜
⎞
⎠⎟
8 (d) (i)
8 (d) (ii) Show that Edward’s incorrect angle does not work.
[1 mark]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
10
9
1 2 3 3 2 2 1 1 8 M 1 8 5
The curves C1 and C2 have the equations and y = 4k – 6x respectively, where k
is a constant.
Show that x coordinates of the points of intersection between C1 and C2 satisify
[4 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
y = 5x2
(log5)x2 + (6log4)x − k log4 = 0
9 (a)
11
1 2 3 3 2 2 1 1 8 M 1 8 5
Turn over ►
9 (b) Given that the curves C1 and C2 do not intersect, show further that
[3 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
k < 9log0.25log5
12
1 2 3 3 2 2 1 1 8 M 1 8 5
10 (i) Given that
and that when x = 1, y = 4, express y in terms of x.
[6 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
dydx
= x3 − 1x
, x > 0
13
1 2 3 3 2 2 1 1 8 M 1 8 5
Turn over ►
10 (ii) Jessie proposes that for any two functions f and g, .
By choosing suitable functions for f and g, show that Jessie’s claim is false.
[3 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
f(x)g(x)dx =0
1
∫ f(x)dx0
1
∫ g(x)dx0
1
∫
14
1 2 3 3 2 2 1 1 8 M 1 8 5
11 The curve C1 has the equation y = f(x), where .
The line l is a normal to the curve C1 when x = 4.
Find the equation of the line l. Give your answer in the form y = mx + c.
[5 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
f(x) = x2 − 3 x3 + 4, x > 0
11 (a)
15
1 2 3 3 2 2 1 1 8 M 1 8 5
Turn over ►
11 (b) The curve C2 has the equation y = g(x) where g(x) = 4x3 + qx2 – 2x + 10 and q is a constant.
Given that the line l is a tangent to C2 when x = 1, find the value of q.
[3 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
END OF SECTION ATURN OVER FOR SECTION B
16
1 2 3 3 2 2 1 1 8 M 1 8 5
12 A particle of mass 1 kg is moved along a rough horizontal surface due to a horizontal force of 10 N.
The acceleration of the particle is 6 m/s2.
Find the magnitude of the force due to friction acting on the particle.
Circle your answer.
[1 mark]
16 N 4 N 60 N 0.6 N
Answer all questions in the spaces provided.
Section B
17
1 2 3 3 2 2 1 1 8 M 1 8 5
Turn over ►
BLANK PAGE
18
1 2 3 3 2 2 1 1 8 M 1 8 5
13 A ball, with mass 0.5 kg, is thrown vertically upwards from a point P at 22 m s–1.
The point P is 10 m above a large water reservoir. The ball is modelled as a particle
that moves freely under the influence of gravity until it reaches the reservoir.
Find the maximum height reached by the ball above the reservoir.
[3 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
13 (b) Find the speed of the ball as it hits the reservoir.
[2 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
13 (a)
19
1 2 3 3 2 2 1 1 8 M 1 8 5
Turn over ►
13 (c) After the ball hits the reservoir, it decelerates uniformly and comes to rest in 3 s.
Calculate the deceleration of the ball in the reservoir.
[1 mark]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
13 (d) Hence, find the magnitude of the resistive forces acting on the ball in the reservoir.
[2 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
20
1 2 3 3 2 2 1 1 8 M 1 8 5
14 A particle P moves on the x-axis. At time t s, P is moving with a velocity v m s–1, where
and a and b are positive constants.
The magnitude of the acceleration of P at t = 2 is 4 m s–2.
Find the values of the constants a and b.
[3 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
14 (b) In the space below, sketch a velocity-time graph for the particle P.
[2 marks]
v = a − bt 2 0 ≤ t ≤ 50 otherwise⎧⎨⎩⎪
14 (a)
21
1 2 3 3 2 2 1 1 8 M 1 8 5
14 (c) Find the total distance travelled by the particle P.
[4 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
22
1 2 3 3 2 2 1 1 8 M 1 8 5
15 In this question, i and j are perpendicular unit vectors.
A particle P has a position vector (xi + yj) m relative to a fixed origin O. Two variable forces, F1 N and F2 N, act on the particle as it moves, where
F1 = (4ysin2x + x)i + eyj
F2 = (4ycos2x)i – 6j
The particle passes through the point Q, which has position vector (ai + bj) m relative to O.
When the particle passes through Q, it is moving at constant speed.
Find the exact value of a and b.
[4 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
23
1 2 3 3 2 2 1 1 8 M 1 8 5
BLANK PAGE
24
1 2 3 3 2 2 1 1 8 M 1 8 5
16
The mass A is held at rest on a rough horizontal table and is attached to one end of a string. The mass of A is 2 kg.
The string passes over a pulley P, which is fixed at the edge of the table. The other end of the string is attached to the mass B, which has mass 4.5 kg and hangs freely, vertically below P.
The magnitude of the frictional force between A and the table is modelled as having a constant value of 0.4R N, where R is the magnitude of the normal reaction force exerted by the table on A.
The string is released from rest, with the string taut, as shown in the diagram above.
The masses are modelled as particle, the string is modelled as light and inextensible, the pulley is modelled as small and the acceleration due to gravity, g, is modelled as being 9.8 m s–2. The pulley is not modelled as a smooth pulley and the difference in tension between the two sides, ∆T N, is modelled as
∆T = α + βa
where α = 3 N, β = 0.3 kg and a is the acceleration of the masses.
A P
B
25
1 2 3 3 2 2 1 1 8 M 1 8 5
16 (a) Given that the tension in the string at B is greater than the tension in the string at A, find the acceleration of the particles.
[4 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
26
1 2 3 3 2 2 1 1 8 M 1 8 5
16 (b) Find the magnitude of the resultant force acting on the pulley.
[3 marks]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
16 (c) Suggest one improvement that can be made to the model.
[1 mark]
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
END OF QUESTIONS
Copyright © 2018 crashMATHS Ltd.