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AQA A-LEVEL MATHEMATICS. PAPER 1
2024 |ACTUAL PAPER|100% VERIFIED
ANSWES|A+GRADE
Answer all questions in the spaces provided.
1 Given that a > 0 , determine which of these expressions is not equivalent to theothers.
Do not write outside the box
Circle your answer.
[1 mark]
!2 log
2 log ( a ) log ( a^2 )
2 Given y e
kx
, where k is a constant, find
d y
d x
Circle your answer.
[1 mark]
d y kx
d x
¼ e
d y
d x
¼ kx e
kx !1 d y
d x
e
kx
k
3 The diagram below shows a sector of a circle.
The radius of the circle is 4 cm and y ¼ 0:8 radians.Find
the area of the sector.
Circle your answer.
1.28 cm 2 3.2 cm (^2) 12.8 cm^2
[1 mark]
d x
¼ k e
d y kx
6.4 cm 2
! 4 log 10 (
p
a
ffiffiffi )
10 10 10
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4 The point A has coordinates (!1, a ) and the point B has coordinates (3, b )The
line AB has equation 5 x þ 4 y ¼ 17
Find the equation of the perpendicular bisector of the points A and B.
[
marks]
Do not write outside the box
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s
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5 (c) Sn is the sum of the first n terms of the sequence.
Explain why the value you found in part (b) is the maximum value of Sn
[
marks]
Do not write outside the box
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s
6 The function f is defined by
f ( x ) ¼
( x
2
þ 1), x ≥ 0
Do not write outside the box
6 (a) Find the range of f.
[1 mark]
6 (b) (i) Find f!^1 ( x )
[3 marks]
6 (b) (ii) State the range of f!^1 ( x )
[1 mark]
Do not write outside the
7 (a) By sketching the graphs of y ¼
x
and y ¼ sec 2 x on the axes below, show that the
equation
box
x
¼ sec 2 x
has exactly one solution for x > 0
[3 marks]
7 (b) By considering a suitable change of sign, show that the solution to the equation lies
between 0.4 and 0.
[2 marks]
7 (c) Show that the equation can be rearranged to give
x
cos
!
x 2
[2 marks]
y
O л 2 л x
Do not write outside the
box
7 (d) (i) Use the iterative formula
x ¼
cos !
x
n þ 2
n
with x 1 0:4 , to find x 2 , x 3 and x 4 , giving your answers to four decimal places.
[2 marks]
7 (d) (ii) On the graph below, draw a cobweb or staircase diagram to show how convergencetakes
place, indicating the positions of x 2 , x 3 and x 4.
[2 marks]
0.0 0.1^ 0.2^ 0.3^ 0.4^ 0.5^ 0.6^ 0.
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Do not write outside the 9 Prove that the sum of a rational number and an irrational number is always irrational.
[5 marks]
box
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10 The volume of a spherical bubble is increasing at a constant rate.
Show that the rate of increase of the radius, r , of the bubble is inversely proportionalto r
2
box
Volume of a sphere
p r^3
[4 marks]
Do not write outside the
2 h 2 h
11 Jodie is attempting to use differentiation from first principles to prove that the gradientof y
¼ sin x is zero when x ¼
p
box
Jodie’s teacher tells her that she has made mistakes starting in Step 4 of her working.Her
working is shown below.
y
sin
p
þ h
! sin
p
Step 1 Gradient of chord AB
h
sin
p
cos ( h ) þ cos
p
sin ( h )! sin
p
Step 2
h
Step 3 ¼ sin
p
cos ( h )! 1
þ cos
p
sin ( h )
Step 4 For gradient of curve at A ,
let h ¼ 0 then
cos ( h )! 1
¼ 0 and
sin ( h )
Step 5 Hence the gradient of the curve at A is given by
sin
p
× 0 þ cos
p
× 0 ¼ 0
A
B
л
л
+ h
x
h h
Do not write outside the Complete Steps 4 and 5 of Jodie’s working below, to correct her proof.
Step 4 For gradient of curve at A ,
[
marks]
box
Step 5 Hence the gradient of the curve at A is given by
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12 (b) Hence, given x is obtuse and
2 cot 2
x þ 2 cosec
2
x ¼ 1 þ 4 cosec x find
the exact value of tan x
Fully justify your answer. [5 marks]
box
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Do not write outside the 13 A curve, C , has equation
e
3 x!
y ¼
x
2
box
Show that C has exactly one stationary point.
Fully justify your answer.
[7 marks]
Do not write outside the
A
2 x^3
14 The graph of y ¼
x
2 þ 1
is shown for 0 ≤ x ≤ 4
y
box
0 2 4 x
Caroline is attempting to approximate the shaded area, A , under the curve using the
trapezium rule by splitting the area into n trapezia.
14 (a) When n ¼ 4
14 (a) (i) State the number of ordinates that Caroline uses.
[1 mark]
14 (a) (ii) Calculate the area that Caroline should obtain using this method.
Give your answer correct to two decimal places.
[3 marks]
Do not write outside the box
14 (b) Show that the exact area of A is
Fully justify your answer.
16! ln 17
[5 marks]
Question 14 continues on the next page
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