9
Do not write
outside the
3k box
10 The radius of a sphere, in cm, is
2
The volume of the sphere, in cm3, is 972π
4 3
Volume of a sphere = πr where r is the radius
3
Work out the value of k.
[3 marks]
Answer
11 Expand and simplify fully (5x + 3y2)(4x – y2)
[3 marks]
Answer
10
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*09*
IB/M/Jun21/8365/2
10
Do not write
outside the
box
12 A and B are points on the line y = 3x + 2
B, C and D (5, 0) are points on the line L.
OA : AC = 1 : 4
Work out the x-coordinate of B.
[5 marks]
Answer
*10*
IB/M/Jun21/8365/2
11
Do not write
outside the
box
13 P is the point on the curve y = ax3 + 10x2 where x=2
1
The gradient of the normal to the curve at P is –
4
Work out the value of a.
[4 marks]
Answer
Turn over for the next question
Turn over ►
*11*
IB/M/Jun21/8365/2
12
Do not write
outside the
1 0 box
14 (a) A=
0 –1
Describe geometrically the single transformation represented by A.
[1 mark]
Answer
0 1
14 (b) B=
–1 0
Describe geometrically the single transformation represented by B2
[2 marks]
Answer
*12*
IB/M/Jun21/8365/2
13
Do not write
outside the
box
15 A, B and C are points on a circle, centre O.
ACD is a straight line.
Angle BCD = w
Prove that w = x + 90°
[5 marks]
Turn over ►
*13*
IB/M/Jun21/8365/2
14
Do not write
outside the
box
16 The coefficient of x4 in the expansion of (a + 2 x )6 is 1500
Work out the two possible values of a.
[3 marks]
Answer and
*14*
IB/M/Jun21/8365/2
15
Do not write
outside the
box
17 ABCDEFGH is a cube with side length 32 cm
M and N are points on DH and CG respectively.
Work out the size of the angle that the line BM makes with the plane ABCD.
[5 marks]
Answer degrees 8
Turn over ►
*15*
IB/M/Jun21/8365/2
16
Do not write
outside the
3 box
18 y = 12x +
x
Show that y has a minimum value when x = 0.5
[5 marks]
*16*
IB/M/Jun21/8365/2
17
Do not write
outside the
box
19 (a) f(x) = (x + 2)3
g is a function such that gf(x) = (x + 2)12
Work out an expression for g(x)
[1 mark]
Answer
19 (b) h(x) = x2 + 5
k is a function such that hk(x) = 4x2 + 5
Work out an expression for kh(x)
[2 marks]
Answer
Turn over for the next question
Turn over ►
*17*
IB/M/Jun21/8365/2
18
Do not write
outside the
2sin x + cos x 1 box
20 Show that – can be written in the form acos x + bsin x
tan x sin x
where a and b are integers.
[4 marks]
*18*
IB/M/Jun21/8365/2
19
Do not write
outside the
box
21 3x2 + 2bx + 8a can be written in the form 3(x + a)2 + b + 2
Work out the two possible pairs of values of a and b.
[6 marks]
a= b=
a= b=
END OF QUESTIONS
10
*19*
IB/M/Jun21/8365/2
2
Answer all questions in the spaces provided.
1 Factorise fully 6𝑎! 𝑏 − 15𝑎𝑏 "
[2 marks]
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Answer _____________________________________
2 Work out the values of 𝑝, 𝑞 and 𝑟 such that 2𝑥 # − 8𝑥 + 𝑝 ≡ 𝑞 𝑥 + 𝑟 #
+ 19
[3 marks]
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𝑝 = ____________ 𝑞 = ____________ 𝑟 = ____________
3
3 2 6 8 1
Work out
4 5 10 3
[2 marks]
Answer ________________________
4 How many numbers satisfy all of the following conditions?
• The number is a four-digit integer..
• The second digit is 8.
• The other digits are all odd numbers.
[3 marks]
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Answer ________________________
4
5 Simplify 8( 98 + 32 − 50 ) writing your answer as an integer.
[3 marks]
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Answer ________________________
6 Solve 𝑥 # − 𝑥 < 12
[2 marks]
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Answer __________________________________
5
7 Use matrix multiplication to show that, in the 𝑥−𝑦 plane,
• a reflection in the line 𝑦 = 𝑥, followed by
• a reflection in the line 𝑦 = −𝑥
is equivalent to a rotation of 180º about the origin.n in the x-axis.
[3 marks]
6
8 The curve shown has equation 𝑦 = 𝑎𝑏 $% where 𝑎 and 𝑏 are positive integers.
The point (2, 0.75) lies on the curve. Find the values of 𝑎 and 𝑏.
[2 marks]
𝑎 = ____________ 𝑏 = ____________
7
9 The curve with equation 𝑦 = 𝑥 & + 4𝑥 " − 6𝑥 # − 5𝑥 + 13 has a turning point at (1,7).
Determine whether this point is a maximum or a minimum.
[2 marks]
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Answer ________________________
10 f(𝑥) = 2𝑥 − 1
g 𝑥 = 𝑥# + 3
Work out fg(𝑥)
[2 marks]
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Answer ________________________
8
11 (a) On the axes, sketch 𝑦 = cos 𝑥 for − 360° ⩽ 𝑥 ⩽ 360°
[3 marks]
(b) You are given that 180° < 𝑢 < 360° and that tan 𝑢 = 1
Find the value of 𝑢.
[1 mark]
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Answer ________________________
9
12 The 𝑛th term of a sequence is 𝑈'
4𝑛 − 11
𝑈' =
5𝑛
(a) Work out the least value of 𝑛 for which 𝑈' ≥ 0.7
[3 marks]
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Answer ________________________
(b) Write down the limiting value of 𝑈' as 𝑛 → ∞
[1 mark]
Answer ________________________
10
13 (a) Show that (2sin 𝜃 + cos 𝜃)(cos 𝜃 − 2sin 𝜃) + 2 ≡ 5 cos# 𝜃 − 2
[3 marks]
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(b) Hence, or otherwise, find the value of (2sin 60° + cos 60°)(cos 60° − 2sin 60°)
[3 marks]
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Answer ________________________
11
14 The table lists the equations of three straight lines.
Equation of line Gradient 𝑦-intercept
𝑦 = 7 − 3𝑥
5𝑥 + 2𝑦 = 20
𝑦 − 21
=3
𝑥−5
Fill in the gradients and 𝑦-intercepts of each line.
[6 marks]
