# 5HL Past common Test ```Past common Test
5AAHL Term 3
GDC required
1. [Maximum mark: 6]
Do not use calculator for this question
Find the exact value of
∫
3
2
1


2
+ sin ( 3πx )  dx
 (2 x − 3) +
x +1


2. [Maximum mark: 7]
(a) Show that
5𝑥𝑥−1
𝑥𝑥 2 −4𝑥𝑥−5
≡
𝐴𝐴
𝑥𝑥+1
+
𝐵𝐵
𝑥𝑥−5
,where 𝐴𝐴 and 𝐵𝐵 have to be found.
2
(b) Hence find the value of k such that ∫1
3. [Maximum mark: 7]
(a) Evaluate ∫ 𝑥𝑥tan2 𝑥𝑥 d𝑥𝑥.
5𝑥𝑥−1
𝑥𝑥 2 −4𝑥𝑥−5

d𝑥𝑥 = ln𝑘𝑘 .


m
(b) Determine the value of m if ∫0 𝑥𝑥tan2 𝑥𝑥 d𝑥𝑥 = 0.5, where m &gt; 0 .
4. [Maximum mark: 6]
Find ∫
ln(ln 𝑥𝑥)
𝑥𝑥

𝑑𝑑𝑑𝑑 , where 𝑥𝑥 &gt; 1.
5. [Maximum mark: 7]
By using the substitution 𝑥𝑥 3 = 4 cosec θ , show that ∫
6. [Maximum mark: 5]
( x)
Find from first principles the derivative of f =
7. [Maximum mark: 7]
d𝑥𝑥
𝑥𝑥√𝑥𝑥 6 −16
=−
1
12
4
arc sin � 3 � + 𝑐𝑐
𝑥𝑥
2x − 3
π
3
A curve has equation arcsin( x 2 ) + arcsin( y 2 ) =
.
dy
in terms of x and y .
dx
1
dy
(b) Find the value of
when x =
, y&lt;0 .
dx
2
(a) Find


8. [Maximum mark: 4]
x
.
2x +1
1
Using first principles, show that f ′( x) =
.
(2 x + 1) 2
The function f is defined by f ( x) =
Page 1 of 8
9. [Maximum mark: 7]
2
The curve C is defined by equation x ln y = 2 x − 1, y &gt; 0.
(a) Find
dy
in terms of x and y.
dx
(b) Find the value of

dy
at the point on C where y = e and x &gt; 0.
dx

10. [Maximum mark: 7]
Differentiate the following with respect to x
( )
(a) y = x 5 x
(b) y =
x2
arctan kx
11. [Maximum mark: 5]
By using differentiation from first principles, obtain the derivative of f ( x) =
x
.
2x −1
12. [Maximum mark: 8]
The curve below shows the graph of y = f ( x) .
(a) Find the value of the following:
(i)

f ′(−2)
(ii) f ′(1.5)
Page 2 of 8
13. [Maximum mark: 11]
x
.
 y
A curve C is defined implicitly by the equation xy
= y 2 + ln 
dy
.
dx
(a)
Find an expression for
(b)
Find the equation of the normal to the curve C at the point ( 2, 2 ) .


Set P contains the x-coordinates of all the intersection points between the graph of C
and y = x .
(c)
By showing clear reasoning, determine the elements of set P.

(d)
Find the asymptotes of the graph of C.

14. [Maximum mark: 6]
Find the coordinates of the points on the curve 3𝑥𝑥 2 + 𝑥𝑥𝑥𝑥 + 𝑦𝑦 2 = 33 at which the tangent
is parallel to the 𝑥𝑥-axis.
15. [Maximum mark: 9]
3
x dx
∫ tan x sec =
sec3 x
+c.
3
(a)
Show that
(b)
Using the substitution 𝑥𝑥 = 2 tan 𝑢𝑢 , find ∫ 𝑥𝑥√4 + 𝑥𝑥 2 𝑑𝑑𝑑𝑑 .
16. [Maximum mark: 7]


𝜋𝜋
Using integration by parts, find ∫02 e−2𝑥𝑥 sin 𝑥𝑥 d𝑥𝑥 in exact form.
17. [Maximum mark: 10]
Find an expression for
(a)
dy
for the following:
dx
y = x sin y

1
x

2
(b) arctan  =
 y + ln( xy )
18. [Maximum mark: 5]
Let y = cos x . Show that
(a) 2 y
dy
+ sin x =
0,
dx
(b) 2 y
d2 y
 dy 
+ 2   + y2 =
0.
2
dx
 dx 

2

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19. [Maximum mark: 7]
Diagram not to
scale
In the diagram, POQ is a rail where OQ is horizontal and ∠𝑃𝑃𝑃𝑃𝑃𝑃 =
2𝜋𝜋
3
.
AB is a straight rod of length 3 metres which is free to slide on the rail with one end A on OP
and the other end B on OQ, till end B is at point O.
End A is initially at point O and end B is pushed towards O at a constant speed of
√3
3
𝑚𝑚𝑠𝑠 −1 .
After 𝑡𝑡 seconds, B is 𝑥𝑥 metres away from point O and the rod makes angle 𝜃𝜃 radians with
the horizontal.
(a)
(b)
Express 𝑥𝑥 in terms of 𝜃𝜃 .

Let 𝑆𝑆 𝑚𝑚2 be the area of ∆𝐴𝐴𝐴𝐴𝐴𝐴 .
Show that
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝜋𝜋
= 3√3 sin � − 2𝜃𝜃� .
20. [Maximum mark: 4]
3

The equation of the gradient at any point of a curve is given by:
dy
= e2 x + x 2 .
dx
If the y-intercept of the curve is 6, find its equation.
21. [Maximum mark: 7]
A
B
3 − x2
= 2
+ 2
(a) Given that
, find the values of A and B
2
2
( x + 1)( x + 9 ) x + 1 x + 9
where 𝐴𝐴, 𝐵𝐵 ∈ ℚ .
3 − x2
(b) Hence obtain the indefinite integral of ∫ 2
dx
( x + 1)( x 2 + 9 )


Page 4 of 8
22. [Maximum mark: 11]
Consider the function f ( x) = xe − x for x ≥ 0.
(a) (i) Find an expression for f ′( x) .
(ii) Determine the coordinates of the point A where f ′( x) = 0 .

(b) (i) Find an expression for f ′′( x) .
(ii) Hence show that A is a maximum point.

(c) Find the coordinates of B, the point of inflexion on f ( x) .

23. [Maximum mark: 7]
Using integration by substitution with x =
sin u
, find
3
k
Hence or otherwise, find k such that
1
∫ 1− 9x
0
2
1
∫ 1− 9x
2
dx .
dx = 0.4 .
24. [Maximum mark: 7]
Using integration by parts, show that
π
∫x
2
( −1)
cos nx dx =
n
0
n
2π
2
for n ∈  + .
25. [Maximum mark: 6]
Find
dy
 y
if ln ( x 2 + y 2 ) =
arctan  
dx
x
26. [Maximum mark: 5] (Do not use a calculator for this question)
2
−x
The graph of the function defined by f ( x) = xe has a local minimum at the
point A and a local maximum at the point B.
Find the coordinates of the points A and B.
27. [Maximum mark: 6]
Find the equation of the normal to the curve whose equation is given by
x2 + 2 y − 2 =
e xy at the point where y = 0 and x &gt; 0.
28. [Maximum mark: 6]
Consider the curve y =
points of inflection.
d2 y
1+ x
.
Find
. Hence show that the curve has three
dx 2
1 + x2
Page 5 of 8
29. [Maximum mark: 8]
x cm
B
A
10 cm
C
In the diagram above, triangle ABC has AC = 10 cm, AB = x cm, and angle ACB = θ
π
radian. Line AC is horizontal and the point B is moving away
6
π
from point A at a constant rate of 2 cms − 1 , making an angle of
6
horizontal.
(a) Find the value of x when θ =
π
2
 5π

10sin θ .
(b) Show that x sin  − θ  =
 6

(c) Using parts (a) and (b), find the rate of change of θ when θ =


π
2

30. [Maximum mark: 11]
Find
(a)
∫
ln(e 2 x )
dx
x2

(b)
∫ cot α dα

(c)
∫3
dx

(d)
∫
ln x
dx
x2

2
2 −3 x
31. [Maximum mark: 5]
Find the remainder when x 2 + 6 is divided by x 2 + 2 x + 10 .
Hence, find
x2 + 6
∫ x 2 + 2 x + 10 dx
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32. [Maximum mark: 2]
Given that
∫
4
1
f ( x) dx = 6, find the value of
∫
4
1
f (5 − x) dx .
33. [Maximum mark: 6]
Using the substitution x = sin y , find the exact value of
1
2
0
∫
x
dx .
1− x
34. [Maximum mark: 6]
Find
∫e
sin x
sin 2 x dx .
35. [Maximum mark: 14]
The graph of the first derivative f ′ of a function f is shown below.
(a) Find all possible x -coordinate(s) of the maximum point(s). Justify your answer.

(b) State the range of values of x for which the function is decreasing.

(c) State the x -coordinates of the inflection points of f .

(d) State the range of values of x for which the function f is
(i) Concave upwards
(ii) Concave downwards
(e) Sketch the graph of f ′′( x) for 0 ≤ x ≤ 5 .


Page 7 of 8
36. [Maximum mark: 7]
A cone is formed by joining the two straight edges of a sector from a circle of radius r.
Let θ be the angle between the two straight edges.
r
θ
r
(a) Show that the radius of the cone is
rθ
.
2π
(b) Find an expression for the height of the cone in terms of r and θ .


(c) Given that θ varies,
find the value of θ which makes the volume of the cone a maximum.

37. [Maximum mark: 8]
(a) Find the root of the equation e 2− 2 x = 2e − x .
(b) The curve
=
y e 2− 2 x − 2e − x has a minimum point. Find the coordinates of
this minimum.
(c) The curve
=
y e 2− 2 x − 2e − x is shown below.
Write down the coordinates of the points A, B and C.
k
(d) Hence state the set of values of k for which the equation e 2− 2 x − 2e − x =
has two distinct positive roots.




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