Year 10
Maths
Booklet
1
Year 10 – Topics covered (continues inside back cover)
Topics
Content
A
Formulae (manipulating,
rearranging & combining)
&
Functions
Sub numbers into a formula then solve equation.
Dealing with roots and powers & where x is on both sides and in fractions {eg y = 3√(x2–9),
a(x+b)=cx+d, y = (2+x)/(3-x)}
More complex composite functions (than in Y9) and use of rearranging formulae in finding an
inverse function
Test 1 on Topic A in Autumn Term
B
Rules of indices (& with
algebra)
C
Probability
(Tree + Venn)
D
Factorisation
0
1
b
c
b+c
b
c
b-c
-c
c
Revision/development of a = 1, a = a, a × a = a , a ÷ a = a , a = 1/a ,
(ab)c = abc, ab/c = (c√a)b, √(a/b)= (√a)/(√b)
Multiplying and dividing algebraic expressions using the rules on indices (eg 3x2y × 5xy, 30x4y
÷ 5x2y3, √(16x8), etc)
Venn diagrams {use of the rules: P(A∪B)=P(A)+P(B)-P(A∩B)} & set notation ξ, ∩, ∪, ∈ and
A′. Understand what mutually exclusive means. Understand that if events are independent then
P(A∩B)= P(A)×P(B). {Ch 24 is revision of KS3 material}
Ch 29 is conditional probability. Make sure students can construct a tree diagram from scratch
on their own and deal with the “no replacement” problems.
Revision of factorising by common factors
Factorising by pairing/grouping
Factorising quadratics (x2 + bx + c, ax2 + bx + c, Diff 2 squares, ax3 + bx2 +cx)
Test 2 on Topics B & C in Autumn Term
E
Trigonometry
F
Graphs of functions, real-life
graphs and solving equations
graphically
Revision of SOHCAHTOA & Pythagoras, exact trig values (sin/cos/tan 0/30/45/60/90 degrees)
have to be learnt, bearings, sin/cos/tan graphs for any angle (not just acute angles) and solving
trig equations when angles above 90°
Drawing accurate graphs of straight lines, quadratics, cubic, reciprocals (knowledge of
asymptote) and exponentials (y = abx) by plotting points.
Graphs that describe real-life situations (including applications to distance time & speed time
graphs, time series graphs; understanding of difference between displacement & distance
travelled)
Solving f(x) = k by plotting a graph of y = f(x)
Pages in
booklet
P3-7
P8
P9
2
131, 154, 188
P10 – 16
127, 151, 175,
185, 204
P17 - 19
94, 157, 158,
192
P19 – 20
P21 - 22
P23
P24 - 25
Test 3 on Topics D, E & F in Spring Term
MathsWatch
Clip Numbers
136, 190, 214,
215
124, 150, 168,
173, 195
96, 98, 153,
161, 194
Substitution
1)a) 𝑣 = 𝑢 + 𝑎𝑡, calculate 𝑣 if 𝑢 = 10, 𝑎 = 4, 𝑡 = 7 7) The formula for a circular based pyramid is 𝑉 =
b) 𝑣 = 𝑢 + 𝑎𝑡, calculate 𝑣 if 𝑢 = −5, 𝑎 = 3, 𝑡 = 6 1 𝜋𝑟 2 ℎ. Given that the radius, r = 2.6cm and the
c) 𝑣 = 𝑢 + 𝑎𝑡, calculate 𝑣 if 𝑢 = 13, 𝑎 = −5, 𝑡 = 2 3height, h = 6.7cm, find the value of V to 3
d)𝑣 = 𝑢 + 𝑎𝑡, calculate 𝑣 if 𝑢 = 9, 𝑎 = −3, 𝑡 = −5 significant figures.
2) a) 𝑎2 + 𝑏 2 = 𝑐 2 , calculate 𝑐 if 𝑎 = 4, 𝑏 = 8.
𝑃𝑅𝑇
8) Given the formula 𝐼 = 100 , substitute the values
b) 𝑎2 + 𝑏 2 = 𝑐 2 , calculate 𝑐 if 𝑎 = −5, 𝑏 = 3.
P=850, R=5.75 T=4, find the value of I.
3) a) 𝑎2 = 𝑐 2 − 𝑏 2 , calculate 𝑎 if 𝑏 = 5, 𝑐 = 9.
9) The formula to calculate the displacement of a
b) 𝑎2 = 𝑐 2 − 𝑏 2 , calculate 𝑎 if 𝑏 = 2, 𝑐 = −7.
particle, s, given the initial speed, u, the
4) a) 𝑥 = 𝑎 + 2𝑏 2 , calculate 𝑥 if 𝑎 = 3, 𝑏 = 5.
acceleration, a, and the time of movement, t, is
b) 𝑥 = 𝑎 + 2𝑏 2 , calculate 𝑥 if 𝑎 = −5, 𝑏 = 3.
1
calculated by the formula 𝑠 = 𝑢𝑡 + 𝑎𝑡 2 . Calculate
2
2
c) 𝑥 = 𝑎 + 2𝑏 , calculate 𝑥 if 𝑎 = −9, 𝑏 = −2.
the displacement if 𝑢 = −3, 𝑎 = −4, and 𝑡 = 5.
1
5) a) 𝑠 = 𝑢𝑡 + 2 𝑎𝑡 2 , calculate 𝑠
10) Using the formula 𝑐 = √𝑎2 + 𝑏 2 , calculate the
if 𝑢 = 3, 𝑡 = 2, 𝑎 = 4
length of the hypotenuse of a right angled triangle
1
b) 𝑠 = 𝑢𝑡 + 2 𝑎𝑡 2 , calculate 𝑠
with lengths a=3.7 and b=8.3 on the shorter sides.
if 𝑢 = −5, 𝑡 = 3, 𝑎 = −3
𝐿
11) 𝑇 = 2𝜋√ . Find the value of T if L = 23.4 and
1
2
𝑔
c) 𝑠 = 𝑢𝑡 + 2 𝑎𝑡 , calculate 𝑠
g = 9.8.
if 𝑢 = 5, 𝑡 = −3, 𝑎 = 6
𝑣2
12) Using the formula 𝐷 = 𝑣 + 20, find the value of
6) Given that the formula for the graph of a straight
D when v = 31.6.
line is y= mx + c, find the coordinate of y given
that the gradient, m = 3, the y-intercept, c = 5, and
the x-coordinate is -5.
13) If M = 3.8, m = 1.7 and g = 9.8, find the value
2𝑀𝑚𝑔
of T given that 𝑇 = 𝑀+𝑚 .
Solving Equations following a Substitution
1
1) 𝑣 = 𝑢 + 𝑎𝑡, find the value of 𝑢
if 𝑣 = 16, 𝑎 = 3, 𝑡 = 2.
9) 𝑠 = 𝑢𝑡 + 2 𝑎𝑡 2 , find the value of 𝑎
if 𝑠 = 52 , 𝑢 = 3 , 𝑡 = 4
2) 𝑣 = 𝑢 + 𝑎𝑡, find the value of 𝑎
if 𝑣 = 23, 𝑢 = 8, 𝑡 = 3.
10) 𝐴 = 𝑥𝑦. Find the value of 𝑥
if 𝐴 = 32.94, and 𝑦 = 6.1.
3) 𝑣 = 𝑢 + 𝑎𝑡, find the value of 𝑡
if 𝑣 = 3, 𝑢 = −13, 𝑎 = −8.
𝑃𝑅𝑇
11) 𝐼 = 100 . Find the value of 𝑅
if 𝑃 = 850, 𝐼 = 5.75 and 𝑇 = 4
4) 𝑎2 + 𝑏 2 = 𝑐 2 , find the value of 𝑏 if 𝑎 = 3, 𝑐 = 8
1
5) 𝐷 = 𝑏 2 − 4𝑎𝑐, find the value of 𝑏
if 𝐷 = 9, 𝑎 = 3, 𝑐 = 2.
12) 𝑠 = 𝑢𝑡 + 2 𝑎𝑡 2 . Find the value of 𝑎
if 𝑢 = −3, 𝑠 = −65, and 𝑡 = 4.
6) 𝐷 = 𝑏 2 − 4𝑎𝑐, find the value of 𝑎
if 𝐷 = 16, 𝑏 = 8, 𝑐 = 2.
13) 𝑐 = √𝑎2 + 𝑏 2 . Find the value of a
if 𝑐 = 29.9 and 𝑏 = 11.5.
7) 𝐷 = 𝑏 2 − 4𝑎𝑐, find the value of 𝑎
if 𝐷 = 3, 𝑏 = −1, 𝑐 = −4.
1
14) 𝑉 = 3 𝜋𝑟 2 ℎ. Find the value of ℎ
if 𝑉 = 100 and 𝑟 = 2.5.
1
8) 𝑠 = 𝑢𝑡 + 2 𝑎𝑡 2 , find the value of 𝑢
if 𝑠 = 54 , 𝑡 = 3, 𝑎 = 4
20
15) 𝑦 = 𝑥+1. If 𝑦 = 8, find the value of 𝑥.
3
Rearranging Formulae
1) Inversing BODMAS - Rearrange the following formulae, making 𝑥 the subject:
a) 𝑦 = 6𝑥 + 9
b) 𝑦 = 2𝑥 − 13
c) 𝑦 = 8𝑥 + 3
e) 𝑦 = 9(𝑥 + 4)
f) 𝑦 = 2(𝑥 + 9)
𝑥
𝑥
g) 𝑦 = 5𝑎
i) 𝑦 = 5 + 9
j) 𝑦 = 𝑝 + 𝑞
5𝑥𝑧+9
9𝑥𝑧−𝑏
m) 𝑦 =
q) 𝑦 =
2𝑎
𝑎(𝑏𝑥−𝑐)
𝑑
n) 𝑦 =
+ 𝑒2
r) 𝑦 =
8𝑎
𝑎𝑥𝑧 2
𝑏
d) 𝑦 = 5(𝑥 − 4)
𝑥−9
h) 𝑦 =
𝑥
k) 𝑦 = 6 − 2
+5
o) 𝑦 =
−𝑐
s) 𝑦 =
𝑥−8
𝑐
l) 𝑦 = 5𝑥𝑧 + 9
𝑎2 (𝑏𝑥−7)
p) 𝑦 =
𝑐
𝑧(𝑎2 𝑥+𝑏)
t) 𝑦 =
𝑐2
𝑎𝑥+6
+ 𝑐2
𝑏𝑑
𝑎2 𝑥 𝑧 2 −9
5𝑏
2) Inversing BODMAS - Rearrange the following formulae, making 𝑥 the subject:
a) 𝑦 = √𝑥
b) 𝑦 = √𝑥 + 𝑎
c) 𝑦 = (𝑥 − 𝑎)2
e) 𝑦 = (𝑎𝑥 − 𝑏)2
f) 𝑦 = 𝑎√𝑏 + 𝑥
g) 𝑦 =
𝑎+𝑏𝑥 2
i) 𝑦 = (
m) 𝑦 =
𝑥𝑧
j) 𝑦 = √6𝑎 + 𝑏
)
𝑐
√𝑥 2 −7
n) 𝑦 =
𝑏2
q) 𝑦 = √
𝑎𝑥 2 −𝑏
o) 𝑦 =
2𝑐
2
𝑥𝑧 2 −𝑏
𝑎2 𝑥+𝑏
k) 𝑦 =
𝑥 2 −𝑎
𝑥𝑧+𝑎
h) 𝑦 = √ 3𝑏
𝑏
√𝑥−𝑎
𝑏
l) 𝑦 = 𝑎√𝑥 2 − 𝑏
√𝑥𝑧−𝑎
𝑏
𝑎𝑥𝑧 2
p) 𝑦 =
𝑎𝑥 2 +𝑏
7𝑐
t) 𝑦 = √𝑎𝑥𝑧 + 𝑏 − 𝑐
s) 𝑦 = ( 𝑏 ) − 𝑐
r) 𝑦 = ( 2𝑎 )
𝑐
d) 𝑦 = √𝑎 + 𝑥𝑧
3) Moving a negative subject to the other side - Rearrange making 𝑥 the subject:
𝑎−7𝑥𝑧
a) 𝑦 = 9 − 𝑥
b) 𝑦 = 15 − 4𝑥
c) 𝑦 = 𝑎 − 5𝑥 2
d) 𝑦 =
e) 𝑦 = √𝑏 − 𝑎𝑥
f) 𝑦 = √𝑎 − 𝑏𝑥 2
g) 𝑦 = 𝑎 − 𝑏√𝑐 − 𝑥
h) 𝑦 =
k) 𝑦 = 𝑏 − √𝑎2 − 𝑥
l) 𝑦 = 𝑏 − 6
o) 𝑦 = 𝑎𝑏 − √9 − 7𝑥𝑧 2
p) 𝑦 = 𝑎 − √𝑏 − 9𝑥 2 𝑧
i) 𝑦 =
𝑎−3𝑏𝑥 2
j) 𝑦 = 𝑎 − (𝑏 − 𝑥)
5𝑐
m) 𝑦 = 𝑏 − √𝑎2 − 7𝑥𝑧
q) 𝑦 = 𝑎2 + (5 − √𝑥)
2
2
5𝑥𝑧
n) 𝑦 = √𝑏 − 6
𝑥2
2𝑥
r) 𝑦 = 𝑎2 − √5 − 3
s) 𝑦 = 𝑎 − √7 − 3
𝑏
𝑎−𝑥 2
𝑏
𝑎𝑥𝑧
9√𝑥
2
t) 𝑦 = (5 − 2 )
4) Tricky fractions - Rearrange making 𝑥 the subject:
7
𝑎
b) 𝑦 = 3𝑥 − 7
5
c) 𝑦 = 10 − 7𝑥
9
d) 𝑦 = 3 − 4𝑥
e) 𝑦 = 7 𝑥 + 5
f) 𝑦 = 5 − 3 𝑥
8
g) 𝑦 = 9 − 4𝑥 2
7
h) 𝑦 = 𝑏 − 𝑑 𝑥 2
i) 𝑦 = 9 − 5√3𝑥
j) 𝑦 = 2(7𝑥)2 + 5
k) 𝑦 = 3√8 − 5𝑥
l) 𝑦 = 𝑎 − 2√4 − 7𝑥
m) 𝑦 = 5(2𝑥 + 3)2
n) 𝑦 = 𝑎(𝑏 − 3𝑥)2
o) 𝑦 = 𝑎 − 4(𝑏 − 𝑐𝑥)2
p) 𝑦 = 3 + 5√2𝑥 + 𝑎
a) 𝑦 = 6𝑥 + 9
1
3
𝑎
q) 𝑦 = 7 √3𝑥 + 𝑐
2
𝑎 2
𝑎
r) 𝑦 = 5 (4𝑥 − 𝑏)
s) 𝑦 = 7 (9 − 𝑏 𝑥)
𝑎
2
𝑐
2
𝑎
t) 𝑦 = 2𝑏 (3𝑥 − 9)2
5) Pythagoras style - Rearrange making 𝑥 the subject:
a) 𝑥 2 + 𝑦 2 = 𝑧 2
b) 𝑦 2 − 𝑥 2 = 𝑧 2
c) 3𝑥 2 + 𝑦 2 = 𝑧 2
d) 𝑥 3 + 𝑦 3 = 𝑧 3
e) 𝑎2 𝑥 2 + 𝑦 2 = 𝑧 2
f) 𝑦 = √𝑧 2 − 𝑥 2
g) 𝑦 = 7√𝑧 2 − 3𝑥 2
h) 𝑦 2 =
4
2𝑥 2 −𝑧 2
6
6) Subject in the denominator - Rearrange making 𝑥 the subject:
𝑎
𝑎
8
b) 𝑦 = −
c) 𝑦 =
a) 𝑦 =
𝑥
𝑥+𝑏
𝑥
𝑎
𝑎
𝑎
g)
𝑦
=
e) 𝑦 =
f) 𝑦 =
𝑏−𝑐𝑥 2
√𝑥−𝑏
√4−𝑥
𝑎
𝑎
7
k) 𝑦 =
j) 𝑦 = −
i) 𝑦 = −
(𝑏𝑥−𝑐)2
𝑏−𝑐𝑥𝑧
𝑎−𝑏𝑥𝑧
d) 𝑦 = −
h) 𝑦 = −
l) 𝑦 = −
𝑎
𝑥−𝑏
𝑎
𝑏−𝑐𝑥 2
𝑎
√𝑏−𝑐𝑥
7) Subject appearing multiple times - Rearrange making 𝑥 the subject:
a) 𝑎𝑥 + 3𝑥 = 4
b) 𝑓𝑥 − 5𝑥 = 8
c) ℎ = 𝑝𝑥 + 𝑞𝑥
d) 5𝑥 + 𝑏 = 𝑎𝑥
e) 𝑐𝑥 = 5 − 𝑎𝑥
f) 𝑎𝑥 + 𝑏 = 𝑐 − 4𝑥
g) 6𝑎 − 3𝑥 = 𝑎𝑥 + 4𝑐
h) 𝑔2 − 7𝑥 = 𝑎𝑥 − 6
i) 2𝑥 + 𝑏𝑥 = 6𝑎 − 3𝑥
j) 5𝑥 + 9𝑎 = 𝑐𝑥 − 8𝑥
k) 𝑥𝑦 + 𝑓 = 7 − 𝑥
l) 7𝑎 − 𝑥 = 𝑝𝑥 + 8𝑎
m) 𝑦 =
2𝑥−5
3−4𝑥
7−8𝑥
q) 𝑦 =
𝑎𝑥+6
𝑥+5
u) 𝑦 = 6 −
𝑎−𝑥
n) 𝑦 =
r) 𝑦 =
8−3𝑥
4𝑥+9
4𝑎−𝑥
s) 𝑦 =
𝑎𝑥+𝑧
𝑏𝑥−𝑐
v) 𝑦 = 𝑎 −
7−𝑥
o) 𝑦 =
𝑥
9𝑥+7𝑏
6−𝑥
–5
𝑏−2𝑥
w) 𝑦 = 𝑧 −
7−𝑥
4𝑥−5
p) 𝑦 =
8𝑥−𝑐
t) 𝑦 =
𝑎−𝑥
𝑎−𝑥
+𝑏
8−𝑥
x) 𝑦 = 𝑎 −
7−3𝑥
𝑎−2𝑥
8) Cross multiplying - Rearrange making 𝑥 the subject:
a)
e)
3
=
𝑥−2
𝑥−4
𝑎+𝑏
=
𝑎
b)
𝑥
3−2𝑥
𝑐
f)
5
=
𝑥−𝑎
𝑦+𝑧
2𝑥−9
𝑎−1
=
c)
𝑥
𝑎−𝑏
g)
3𝑥−7
3−𝑎
𝑥
=
𝑥 2 +𝑎
𝑏
𝑎−3
𝑥−𝑎
=
8−𝑥 2
𝑎−𝑐
d)
h)
2
=
𝑥−1
𝑎
𝑥+𝑎
9
𝑏+𝑐
=
𝑎2 +𝑥 2 𝑦 2 −𝑥 2
9) Tricky Fractions with multiple x’s - Rearrange making 𝑥 the subject:
1
a) 𝑎 = 𝑏 + 𝑥
1
1
e) 𝑦 = 𝑥 + 𝑎
3
5
b) 5 = 𝑎 − 𝑥
5
6
c) 𝑎 = 𝑏 − 𝑥+3
6
7
7
8
5
𝑎2
𝑎
𝑏
g) 𝑦 − 𝑥 = 𝑧
f) 𝑥 + 𝑦 = 𝑧
3
5
1
1
d) 𝑎 = 4 + 7𝑥
1
h) 𝑥 + 𝑦 = 𝑧
10) Expanding a squared or cubed bracket
𝑎
𝑎
𝑐
𝑏
b) 𝑦 = 𝑏 √𝑥
𝑏
f) 𝑦 = 2𝑐 √𝑎𝑥 2
a) 𝑦 = 𝑎√𝑥
e) 𝑦 = 5 √𝑥 2
𝑏
c) 𝑦 = 𝑏 √ 𝑥
𝑑
g) 𝑦 = 3 √𝑥 3
5
6
d) 𝑦 = 2𝑎 √𝑥
𝑎
3
𝑐
h) 𝑦 = 𝑏 (√𝑥 2 )
Additional
Worksheet
with video
solutions
5
Functions
3
1) Inputting - For the following functions 𝑓(𝑥) = 5𝑥 2 − 1 , 𝑔(𝑥) = 𝑥 , ℎ(𝑥) =
a) 𝑓(2)
h) 𝑓(−1)
b) 𝑔(2)
i) 𝑔(−4)
c) ℎ(4)
j) ℎ(−1)
o) 𝑓(0)
p) 𝑓(√2)
q) 𝑓 (2)
1
d) 𝑓(4)
k) ℎ(3)
e) 𝑔(5)
l) 𝑓(−3)
1
𝑥
, evaluate:
f) ℎ(1)
m) 𝑔(−3)
1
r) 𝑔 (3)
2𝑥−1
g) ℎ(2)
n) ℎ(−2)
1
s) ℎ (2)
t) ℎ (− 6)
u) 𝑔(6)
1
2) For the function 𝑓(𝑥) = 𝑥, what value of 𝑥 cannot be an input for this function?
3) Solving Equations - For the following functions 𝑓(𝑥) = 𝑥 − 4, 𝑔(𝑥) = 3𝑥 + 2, ℎ(𝑥) =
a) 𝑓(𝑥) = 9
b) 𝑔(𝑥) = 20
c) 𝑓(𝑥) = −2
d) 𝑔(𝑥) = 32
e) ℎ(𝑥) = 2
1
𝑥+2
, solve
3
f) ℎ(𝑥) = 7
g) 𝑔(𝑥) = 2
𝑥
4) For the following functions 𝑓(𝑥) = √𝑥 + 1, 𝑔(𝑥) = 𝑥 , ℎ(𝑥) = 𝑥+1, solve the following equations:
a) 𝑓(𝑥) = 3
1
b) 𝑔(𝑥) = 5
c) 𝑓(𝑥) = 7
d) ℎ(𝑥) = 3
1
1
f) 𝑓(𝑥) = 3
e) 𝑔(𝑥) = 2
g) ℎ(𝑥) = 4
Composite Questions
1) Composite Inputting - For the following functions 𝑓(𝑥) = 2𝑥 + 3, 𝑔(𝑥) = 20 − 𝑥, ℎ(𝑥) = 3(𝑥 − 1),
evaluate the following:
a) 𝑔𝑓(2)
b) 𝑓𝑔(5)
c) 𝑓ℎ(3)
d) ℎ𝑔(11)
e) 𝑓ℎ(2)
f) 𝑔ℎ(2)
g) ℎ𝑔𝑓(3)
1
2) For the following functions 𝑓(𝑥) = √𝑥, (𝑥) = 𝑥 2 , ℎ(𝑥) = 𝑥 2 − 2𝑥, evaluate the following:
a) 𝑔𝑓(9)
b) ℎ𝑓(4)
c) 𝑓𝑔(4)
d) 𝑔ℎ(3)
e) ℎ𝑓(6)
f) 2𝑓ℎ(−2)
g) 3𝑔ℎ(1)
3) Inputting Algebra – for the following functions 𝑓(𝑥) = 2𝑥 − 5 , 𝑔(𝑥) = 7 − 3𝑥 2 , ℎ(𝑥) =
2𝑥+5
a) 𝑓(𝑎)
h) 𝑓(3𝑎 − 1)
g) ℎ(𝑎 + 1)
n) ℎ(2𝑎 − 5)
b) 𝑔(𝑎)
i) 𝑔(𝑎 − 4)
c) 𝑓(2𝑎)
j) ℎ(𝑎 − 7)
d) ℎ(𝑎)
k) 𝑓(5 − 2𝑎)
e) 𝑓(𝑎 + 1)
l) 𝑔(𝑎2 )
f) 𝑔(𝑎 + 1)
m) ℎ(𝑎2 )
𝑥
, evaluate
4) For the following functions 𝑓(𝑥) = 3𝑥 + 2 , 𝑔(𝑥) = 5𝑥 2 + 2 , ℎ(𝑥) = 𝑥 2 + 3𝑥, evaluate and simplify:
g) 𝑓(2𝑥 + 1)
a) 𝑓(2𝑥)
b) 𝑓(3𝑥)
c) 𝑓(𝑥 + 1) d) 𝑔(2𝑥)
e) 𝑔(𝑥 + 4) f) ℎ(5𝑥)
l) ℎ(3𝑥 + 2) m) 𝑔(3𝑥 + 2) n) 𝑓(5𝑥 2 + 2)
h) ℎ(𝑥 + 2) i) 𝑓(𝑥 2 )
j) 𝑔(𝑥 2 )
k) 𝑓(3𝑥 2 )
1
5) Creating Composite Functions - For the following functions 𝑓(𝑥) = 𝑥 + 5, 𝑔(𝑥) = 𝑥 , ℎ(𝑥) = √𝑥, write
out the following composite functions, and simplify your answer where possible.
a) 𝑔𝑓(𝑥)
b) 𝑓𝑔(𝑥)
c) 𝑔ℎ(𝑥)
d) ℎ𝑔(𝑥)
e) 𝑓ℎ(𝑥)
f) ℎ𝑓(𝑥)
1
6) For the following functions 𝑓(𝑥) = 5𝑥 − 2, 𝑔(𝑥) = 7 − 𝑥, ℎ(𝑥) = 2𝑥, write out the following composite
functions, and simplify your answer where possible.
a) 𝑔𝑓(𝑥)
b) 𝑓𝑔(𝑥)
c) 𝑔ℎ(𝑥)
d) 𝑓𝑓(𝑥)
e) 𝑓ℎ(𝑥)
f) 𝑔𝑔(𝑥)
𝑥
7) For the following functions 𝑓(𝑥) = 3𝑥 + 7, 𝑔(𝑥) = √𝑥 + 2 , ℎ(𝑥) = 7 − 2, write out the following
composite functions, and simplify your answer where possible.
a) 𝑔𝑓(𝑥)
b) 𝑓𝑔(𝑥)
c) ℎ𝑓(𝑥)
d) 𝑓𝑓(𝑥)
6
e) ℎ𝑔(𝑥)
f) 𝑓ℎ(𝑥)
𝑥
8) Solving Composite Equations - For the following functions 𝑓(𝑥) = 𝑥 + 5, 𝑔(𝑥) = 2 , ℎ(𝑥) = 7 − 2𝑥, by
writing out the composite function, solve the equation:
a) 𝑓𝑔(𝑥) = 7
b) 𝑔𝑓(𝑥) = 7
c) 𝑓ℎ(𝑥) = 6
d) ℎ𝑓(𝑥) = 5
9) For the following function 𝑓(𝑥) = 𝑥 2 , 𝑔(𝑥) = 𝑥 + 4 , ℎ(𝑥) = √𝑥 − 3, by writing out the composite
function, solve the equation:
a) 𝑔𝑓(𝑥) = 29
b) 𝑔ℎ(𝑥) = 6
c) ℎ𝑔(𝑥) = 7
d) ℎ𝑓(𝑥) = 1
10) 𝑓(𝑥) = √𝑥 , 𝑔(𝑥) = 13 − 2𝑥, solve:
a) 𝑓𝑔(𝑥) = 3
b) 𝑔𝑓(𝑥) = 3
c) 𝑓𝑓(𝑥) = 3
d) 𝑔𝑔(𝑥) = 3
11) 𝑓(𝑥) = 4(𝑥 − 2), 𝑔(𝑥) = √𝑥 + 5, solve:
a) 𝑓𝑔(𝑥) = 8
b) 𝑔𝑓(𝑥) = 5
c) 𝑓𝑓(𝑥) = 56
d) 𝑓𝑔(𝑥) = 20
𝑥
12) For the following functions 𝑓(𝑥) = 2, (𝑥) = 4𝑥 − 1 , ℎ(𝑥) = 𝑥 − 3, solve the following equations
a) 𝑔𝑓(𝑥) = ℎ(𝑥)
b) 𝑓ℎ(𝑥) = 𝑔(𝑥)
c) 6 + 𝑓(𝑥) + ℎ(𝑥) = 𝑔(𝑥)
Inverse Functions
1) Find the inverse of the following functions:
a) 𝑓(𝑥) = 2𝑥 − 5
𝑥
b) 𝑔(𝑥) =
𝑥−5
c) ℎ(𝑥) = 2(𝑥 + 7)
3
d) 𝑓(𝑥) = 2 − 3
e) 𝑔(𝑥) = 7 − 2𝑥
f) ℎ(𝑥) = −8 − 3𝑥
g) 𝑓(𝑥) = √𝑥 + 3
j) 𝑓(𝑥) = (𝑥 + 5)2
h) 𝑔(𝑥) = 𝑥 2 − 5
k) 𝑔(𝑥) = √3𝑥 − 7
i) ℎ(𝑥) = √4 − 𝑥
l) ℎ(𝑥) = 9 − 𝑥 2
2) Find the inverse of the following functions:
5
7
a) 𝑓(𝑥) = 𝑥
b) 𝑔(𝑥) = 𝑥+2
3
d) 𝑓(𝑥) = (𝑥+1)2
4
g) 𝑓(𝑥) = 7−𝑥
e) 𝑔(𝑥) =
c) ℎ(𝑥) =
7
6
√𝑥
4
f) ℎ(𝑥) = 3𝑥 2
√𝑥+2
4
5
h) 𝑔(𝑥) = 7− 𝑥
i) ℎ(𝑥) = 7−√𝑥+1
√
3) Find the inverse of the following functions:
a) 𝑓(𝑥) =
𝑥+1
𝑥
4𝑥
d) 𝑓(𝑥) = 3𝑥−2
𝑥
1
b) 𝑔(𝑥) = 𝑥−2
1
c) ℎ(𝑥) = 𝑥 − 3
2𝑥−3
6−𝑥
e) 𝑔(𝑥) = 7−𝑥
f) ℎ(𝑥) = 5−𝑥
4) The function 𝑓(𝑥) is defined by 𝑓(𝑥) = 2𝑥 − 5.
a) Find 𝑓 −1 (𝑥)
b) Evaluate 𝑓 −1 (7)
5) The function 𝑔(𝑥) is defined by 𝑔(𝑥) = √𝑥 − 4
a) Find 𝑔−1 (𝑥)
b) Evaluate 𝑔−1 (3)
3
6) The function ℎ(𝑥) is defined by ℎ(𝑥) = 𝑥+2.
a) Find ℎ−1 (𝑥)
b) Evaluate ℎ−1 (1)
7
Additional
Worksheet
with video
solutions
Problem Solving to find 𝒌
1) The function 𝑓(𝑥) is defined by 𝑓(𝑥) = 2𝑥 + 𝑘. Given that 𝑓(3) = 13, find the value of 𝑘.
2) The function 𝑔(𝑥) is defined by 𝑔(𝑥) =
𝑥−𝑘
2
. Given that 𝑔(7) = 2, find the value of 𝑘.
3) The function ℎ(𝑥) is defined by ℎ(𝑥) = √3𝑥 − 𝑘. Given that ℎ(8) = 4, find the value of 𝑘.
4) The function 𝑓(𝑥) = 𝑥 + 𝑘 and 𝑔(𝑥) = 2𝑥, and 𝑔𝑓(2) = 14, find the value of 𝑘.
5) The function 𝑓(𝑥) = 3𝑥 and 𝑔(𝑥) = 𝑥 − 𝑘, and 𝑔𝑓(3) = 5, find the value of 𝑘.
6) The function 𝑔(𝑥) = √𝑥 + 𝑘 and ℎ(𝑥) = 2𝑥 − 1, and 𝑔ℎ(3) = 3, find the value of 𝑘.
7) The function 𝑓(𝑥) is defined by 𝑓(𝑥) = 3𝑥 + 𝑘. Given that 𝑓𝑓(5) = 57, find the value of 𝑘.
8) The function 𝑔(𝑥) is defined by 𝑔(𝑥) = 𝑘 − 2𝑥. Given that 𝑔𝑔(3) = 5, find the value of 𝑘.
9) 𝑓(𝑥) = 5𝑥 − 𝑘. Given that 𝑓 −1 (18) = 5, find the value of 𝑘.
𝑥
10) 𝑔(𝑥) = 𝑘 + . Given that 𝑔−1 (7) = 6, find the value of 𝑘.
2
Functions – Hard Problem Solving
1) If 𝑓(𝑥) = 𝑥 2 and 𝑔(𝑥) = 3𝑥 + 1, find an expression for
a) 𝑔𝑓(𝑥)
b) 𝑓𝑔(𝑥)
c) 𝑔𝑔(𝑥)
d) Solve the equation 𝑔𝑔(𝑥) = 49
2) If 𝑓(𝑥) = 𝑥 2 − 4𝑥 + 1 and 𝑔(𝑥) = 𝑘𝑥 + 5, and 𝑔𝑓(1) = 2, find the value of 𝑘.
1
1
3) If 𝑓(𝑥) = 𝑥 and 𝑔(𝑥) = 3𝑥−1, find an expression for 𝑓𝑔(𝑥)
6
4) If 𝑔(𝑥) = 2 − 𝑥 and ℎ(𝑥) = 𝑥 + 3,
2𝑥
a) Show that 𝑔ℎ(𝑥) = 𝑥+3
b) Solve 𝑔ℎ(𝑥) = 1
5) If 𝑓(𝑥) = 4(𝑥 − 1) and 𝑔(𝑥) = 4(𝑥 + 1), find:
a) 𝑓 −1 (𝑥)
b) 𝑔−1 (𝑥)
c) 𝑓 −1 (𝑥) + 𝑔−1 (𝑥)
d) Find the value of 𝑎 if 𝑓 −1 (𝑎) + 𝑔−1 (𝑎) = 1
6) If ℎ(𝑥) = 2 − 3𝑥
a) Find ℎ−1 (𝑥)
b) Solve ℎ(𝑥) = ℎ−1 (𝑥)
2𝑥+3
7) If 𝑓(𝑥) = 𝑥−2 , show that 𝑓(𝑥) = 𝑓 −1 (𝑥)
8) If 𝑓(𝑥) = (𝑥 + 1)2 and 𝑔(𝑥) = 2(𝑥 − 1), show that 𝑔𝑓(𝑥) = 2𝑥(𝑥 + 2)
9) If 𝑓(𝑥) = 5𝑥 + 3 and 𝑔(𝑥) = 𝑎𝑥 + 𝑏 and 𝑔(3) = 20 and 𝑓 −1 (33) = 𝑔(1), find the values of 𝑎 and 𝑏.
10) 𝑓(𝑥) = 3𝑥 − 1 and 𝑔(𝑥) = 𝑥 2 + 4. If 𝑓𝑔(𝑥) = 2𝑔𝑓(𝑥) then show that 15𝑥 2 − 12𝑥 − 1 = 0
8
Indices with Algebra
1) Simplifying Algebra
a) 3𝑥𝑦 2 × 4𝑥 2 𝑦 4
b) 6𝑥𝑦 3 × 2𝑥 2 𝑦 2
3 5 )3
g) 7𝑥(𝑥𝑦)3
f) 3(𝑥 𝑦
c) (3𝑛𝑚3 )2
h) (𝑎3 𝑏 4 )2 × 𝑎2 𝑏
2) Simplifying Fractions
48𝑥 6 𝑦 3
6𝑥 5
b)
a) 2
12𝑥 4 𝑦
3𝑥
4𝑥 2 (𝑥+1)
14𝑦 5 (𝑦+2)3
f)
g)
6𝑥 3
18𝑦 2 (𝑦+2)
c)
3) Simplifying powers of powers
a) (4𝑥)3
b) (7𝑥 4 )3
f) √49𝑥 6
g) √81𝑥 8 𝑦 2
25𝑎4 𝑏
d) 𝑎𝑏 × 𝑎𝑏 2 𝑐 3
i) 7(3𝑥 4 )3
18𝑐 4 𝑑7
15𝑎5 𝑏3
15𝑎4 (𝑏+1)2
h)
25𝑏2 (𝑎+1)3
d)
c) (2𝑥 4 )6
h) √9𝑥 8 𝑦 4
d) (3𝑥 3 𝑦)4
i)
e) 2𝑥 2 × 3𝑦 3
j) (3𝑥 5 )4
e)
24𝑐 7 𝑑2
60𝑥 2 𝑦 7 𝑧 4
j)
50𝑥 5 𝑦3 𝑧
i) (25𝑥 4 𝑦 7 )
23𝑥 2 𝑦5
17𝑥 5 𝑦2
18𝑥 5 𝑦 9 𝑧 5
27𝑥 7 𝑦 7 𝑧 4
e) (5𝑥 3 𝑦 5 )3
1
2
1
j) (36𝑥𝑦 4 𝑧 8 )2
Values of Indices
1) Basic Powers, including fractions
a) 53
b) 62
c) 82
3 2
1 4
2 3
h) (4)
d) (−3)4
3 2
j) (3)
i) (5)
e) 44
f) (−10)4
7 3
k) (5)
g) 60
2 3
l) (8)
7 0
m) (9)
n) (5)
2) Fractional Powers
1
3
a) 92
4
b) 42
1
c) 83
2
d) 252
3
e) 643
3
f) 1002
g) 814
1
3
1
2
1
3
5
1 3
4 2
25 2
27 3
81 2
m) ( )
100
4 2
n) ( )
f) 6−3
g) 5−1
h) ( )
i) ( )
8
9
h) ( )
5
k) ( )
c) (−3)−3
d) 2−6
e) 1−3
36
3) Negative Powers
a) 5−2
b) 7−3
3 −3
j) ( )
49 2
l) ( )
81
64
1 −4
2 −2
3 −5
j) ( )
i) ( )
3 −3
l) ( )
k) ( )
2
7
1 −1
1 5
m) ( )
4
4
9
n) ( )
5
5
4) Negative Fractional Powers
1
3
a) 4−2
1
9 −2
h) ( )
4
4
3
b) 9−2
3
25 −2
i) ( )
49
2
j) (
27 −3
125
3
d) 27−3
c) 25−2
)
2
k) (
243 −5
32
)
1
e) 81−4
3
f) 49−2
g) 100−2
3
4 −2
l) ( )
m) (
e) 50
3
2
9
8
125
1
3
−
)
3
16 −4
n) ( )
81
5) Mixed
a) 4
1
2
1 −1
h) ( )
2
2 −1
o) ( )
3
b) 125
1
1
3
1 4
d) 3−3
c) ( )
1
2
j) 4
i) ( )
2
5
−
k) 9
2
3
−
2
l) 27
f) 4
5
−
3
9
m) ( )
25
4
2
9 −3
p) ( )
16
2
8 −3
q) ( )
27
5
r) 64−6
9
g) 8
1
s) 12964
1
2
1
27 3
n) ( )
64
2
t) (
343 −3
512
)
2
3
3
u) 25−2
Additional
Worksheet
with video
solutions
General Probability Questions
1) 10 cards are numbered from 1 to 10 inclusive. Find the probability that a card picked at random:
a) is a 4
b) is even.
c) is prime. d) is not prime.
2) Year 8 and year 9 girls are playing a netball match with a penalty shoot-out, so the teams cannot draw.
The probability of year 9 winning is 0.75.
a) What is the probability that the year 8 team wins?
b) If the two teams play each other 20 times, how many times would you expect the year 9 team to
win?
4
3) Alex plays a game of patience on the computer. The computer is programmed to let the player have a 9
chance of winning. If Alex plays 45 games, how many games would he expect to lose?
4) At a junction in the road there are 3 choices which a motorist can take, route 1, 2 or 3. The probability a
5
1
motorist takes route 1 is 9, the probability a motorist takes route 2 is 4. What is the probability a motorist
takes route 3?
5) In a class of thirty year 10 pupils, 9 play hockey, 12 play football, 5 play rugby, and 4 go swimming
during a games session. If a pupil is selected at random, what is the probability that the pupil will:
a) play football
b) play hockey or swim
c) not play rugby
d) not play rugby or swim
6) A fair spinner has four numbers 2, 2, 3 and 5. It is spun twice. The sum of the scores is noted. Draw a
possibility table and find the following probabilities:
a) The sum is 4
b) the sum is 6 or more
c) the sum is 9
d) the sum is a square number.
7) A company is trying to prove that most cats prefer chicken flavoured cat food. Unfortunately, Alfred the
cat doesn’t know this and selects his food at random. The company puts out 10 saucers of rabbit, 6 saucers
of sardine and only 4 saucers of chicken flavoured food. Calculate
a) P(Alfred selects sardine flavoured food) b) P(Alfred selects sardines or rabbit food)
c) P(Alfred does not select chicken flavoured food)
Given that an experiment is now repeated 100 times. An experimenter records that Alfred picked the
chicken saucer 18 times.
d) Comment on whether this proves the companies claims.
8) The following 8 dominoes are put into a bag. Riaz takes at random a domino from the bag.
a) Find the probability that he takes a domino with a total of 8 spots or a domino with 9 spots.
Helima takes at random 2 dominoes from the bag of 8 dominoes without replacement.
b) Work out the probability that:
i) the total number of spots on the
two dominoes is 18.
ii) the total number of spots on
the two dominoes is 17.
10
Probability Tree Questions
1) A bag contains 10 green counters and 5 yellow counters. Jane picks one counter at random, lays it on the
table, and then a second counter is picked from the bag. By considering drawing a tree diagram, find the
probability that:
a) Jane picks a green counter and then a yellow counter.
b) The two counters Jane picks are different colours.
c) The two counters Jane picks are the same colours.
2) A bag contains 12 counters, 8 are blue and the rest are red. Toby picks 2 counters at random. Use a tree
diagram to calculate the probability that the counters that Toby picks are the same colour.
3) Fiachra wakes up in the morning and opens his sock drawer. There are 9 white socks and 6 black socks in
the drawer. With the light still off, Fiachra picks 2 socks at random. Find the probability that Fiachra picks a
matching pair of socks.
4) In a sweet tin there are 3 coconut crunch, 7 coffee cream and 5 chewy toffees left. Two sweets are picked
at random. By drawing a tree diagram, calculate the probability that:
a) The two sweets are the same flavour.
b) The two sweets are different flavours.
5) In a pencil case there are 5 blue pens, 6 black pens and 3 pencils. Two items are picked from the pencil
case at random. By drawing a tree diagram, calculate the probability that:
a) The two items are the same.
b) The two items are different.
3
6) Sam and Atul are going to take a driving test. The probability that Sam will pass the test is . The
4
5
probability that Atul will pass the test is 8. By drawing a tree diagram, calculate the probability that:
a) Both pass the test.
b) Only one of them passes the test.
7) A bag contains 6 blue counters and 7 red counters. Three counters are picked, one after the other. By
drawing a tree diagram, find the probability of picking exactly 2 red counters and 1 blue counters (you must
consider the different orders that these can be picked).
8) A bag of sweets contain 4 strawberry flavoured sweets and 3 apple flavoured sweets. Three sweets are
picked at random, one after the other. By drawing a tree diagram, calculate the probability that at least two
of the sweets are strawberry flavour.
9) A pencil case contains 4 green, 5 blue and 6 red pens. Three pens are taken at random. By considering
drawing a tree diagram, calculate the probability that one of each coloured pen is picked from the pencil
case.
10) A bag of sweets contains 6 strawberry sweets and 7 apple flavoured sweets. Three sweets are taken at
random. Find the probability that at least one sweet of each flavour is taken from the bag.
11) A vending machine contains Cola, Lemonade and fruit juice. Two buttons are pressed at random and
two drinks are dispensed from the machine. Find the probability that the two drinks are different flavours.
11
Set Notation and Venn Diagrams
1) ξ = {1,2,3,4,5,6,7,8,9,10}, 𝐴 = {prime numbers} , 𝐵 = {multiples of 3}
a) Write out the elements in the set 𝐴 (use the squiggly brackets)
b) Write out the elements in set 𝐵.
c) Write out the elements in the set 𝐴 ∩ 𝐵.
d) Write out the elements in the set 𝐴 ∪ 𝐵.
e) Draw a Venn diagram for the following sets.
2) ξ = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔, ℎ}, 𝐴 = {𝑎, 𝑏, 𝑐} , 𝐵 = {𝑐, 𝑑, 𝑒, 𝑓}. Write down the elements in the set:
a) 𝐴 ∩ 𝐵
b) 𝐴 ∪ 𝐵
c) 𝐴’
d) 𝐵’
e) 𝐴’ ∩ 𝐵
f) 𝐴 ∩ 𝐵’
g) 𝐴 ∪ 𝐵’
h) 𝐴’ ∪ 𝐵
i) 𝐴’ ∪ 𝐵’
j) 𝐵 ∪ 𝐵’
3) ξ = {Integers in between 30 and 50 inclusive}. 𝐴 = {prime numbers} , 𝐵 = {square numbers}
a) Draw a Venn diagram representing the two sets.
b) i) Write out the elements of set 𝐴 ∩ 𝐵
ii) What does this mean about the two sets 𝐴 and 𝐵.
4) ξ = {𝑥: 10 ≤ 𝑥 ≤ 20}. 𝐴 = {Factors of 120}, 𝐵 = {Multiples of 2}
a) draw a Venn diagram to represent the two sets.
b) Find the value of:
i) 𝑃(𝐴)
ii) 𝑃(𝐴’)
iii) 𝑃(𝐴 ∩ 𝐵)
iv) 𝑃 (𝐴 ∪ 𝐵)
v) 𝑃(𝐴′ ∩ 𝐵)
5) ξ = {𝑥 : 𝑥 is a positive integer, 𝑥 < 30}.
𝐴 = {prime numbers}, 𝐵 = {square numbers}, 𝐶 = {cube numbers}, 𝐷 = {multiples of 4}.
List the elements of the following sets:
a) 𝐴 ∩ 𝐵
b) 𝐶 ∪ 𝐷
c) (𝐴 ∪ 𝐶)′
6) ξ = {positive whole numbers less than 13}.
𝐴 = {even numbers}, 𝐵 = {multiples of 3}, 𝐶 = {prime numbers}
a) List the members of the set
i) 𝐴 ∩ 𝐵
ii) 𝐵 ∪ 𝐶
b) Is it true that 14 ∈ A? Explain your answer.
7) ξ = {𝑥: 100 ≤ 𝑥 ≤ 115}
𝐴 = {odd numbers}, 𝐵 = {prime numbers}
List the members of the set
a) 𝐴 ∩ 𝐵
b) 𝐴 ∩ 𝐵 ′
c) 𝐴′ ∩ 𝐵
d) 𝐴′ ∩ 𝐵 ′
8) ξ = {𝑥: 40 ≤ 𝑥 ≤ 50}
𝐴 = {even numbers}, 𝐵 = {multiples of 3}
List the members of the set
a) 𝐴 ∩ 𝐵
b) 𝐴 ∩ 𝐵 ′
c) 𝐴 ∪ 𝐵
d) (𝐴 ∩ 𝐵)′
9) ξ = {𝑥: 0 ≤ 𝑥 ≤ 100}
𝐴 = {square numbers}, 𝐵 = {even numbers}
List the members of the set
a) 𝐴 ∩ 𝐵
b) 𝐴 ∩ 𝐵′
12
10) ξ = {1, 2, 3, 5, 7, 8, 9, 10, 12}
𝑉 = prime numbers in ξ .
𝑉 ∩ 𝑊 = {3,5,7}
𝑉 ∪ 𝑊 = {1,2,3,5,7,10,12}
a) Display this information in a Venn diagram.
b) List the members of the set 𝑉 ′ ∩ 𝑊
c) List the members of the set 𝑉 ∩ 𝑊 ′
11) ξ = {𝑥: 2 ≤ 𝑥 ≤ 20}
𝐴 = {prime numbers}, 𝐵 = {factors of 24}, 𝐶 = {square numbers}
a) Put all of this information into a three circle Venn diagram.
b) List the members of the set 𝐴 ∩ 𝐵 ∩ 𝐶′
Questions that involve Mutually Exclusivity
12) A = {p,r,a,g,u,e} , B = {p,a,r,i,s}, C = {b,u,d,a,p,e,s,t}.
a) List the members of the sets
i) A ∩ B
ii) B ∪ C
b) D = {r,o,m,e}, E = {l,i,s,b,o,n}, F = {b,e,r,l,i,n} . Which of these three sets are mutually exclusive with
set A.
13) ξ = {Students in Year 12},
G = {Students who study German}, F = {Students who study French}, M = {Students who study Maths}.
a) If G ∩ M = ∅. Use this information to write a statement about the students who study German in Year 12.
b) Preety is a student in Year 12. Preety ∉ F. Use this information to write a statement about Preety.
14) ξ = {even numbers}, 𝐴 = {2,4,6,8,10}.
a) 𝐵 is a set such that 𝐴 ∩ 𝐵 = {4,8}. Set B has 3 members. List the members of one possible set 𝐵.
b) 𝐶 is a set such that 𝐴 ∩ 𝐶 = ∅. Set C has 3 members. List the members of one possible set 𝐶.
15) If 𝐴 = {2,4,6,8,10}, 𝐴 ∩ 𝐵 = {2,4}, 𝐴 ∪ 𝐵 = {1,2,3,4,6,8,10}. Write down the elements in set B.
16) If ξ = {𝑥: 10 ≤ 𝑥 ≤ 20}, 𝐴 = {11, 12, 14, 16, 17}. 𝐵 = {even numbers}
a) Find the elements of the set 𝐴 ∩ 𝐵′.
It is given that 𝐴 and 𝐶 are mutually exclusive.
𝐴
1
b) Find the possible members of the set 𝐵 ′ ∩ 𝐶.
2
17) The elements {𝑥: 1 ≤ 𝑥 ≤ 12} are in one or more of
6
three sets in the diagram to the right.
a) List the elements of the set 𝐴 ∩ 𝐵
b) List the elements of the set 𝐴′ ∩ 𝐶
c) Is is true that 𝐴 and 𝐶 are mutually exclusive?
𝐵
13
11
12
5
7
9
Extension
Note: ∃ means “there exists”, ∄ means “there does not
exist”, ℕ = {1,2,3,4,…}.
Write down the elements of the following sets:
a) A = {x ∈ ℕ : ∃ y ∈ ℕ : x = 3y}
b) B = {x ∈ ℕ : ∄ y ∈ ℕ : x = 2y}
c) C = {x ∈ ℕ : ∃ y ∈ ℕ : x = y2}
d) D = {x ∈ ℕ : ∄ a, b ∈ ℕ, a,b > 1 : x = ab}
4
3
8
10
𝐶
Probability using Venn Diagrams
1) The following diagram shows how many people like to drink tea, coffee and
wine from a survey of 61 people.
a) Find the probability that they:
i) like tea
ii) like wine iii) like coffee iv) like tea and coffee
v) like tea but do not like wine
vi) like tea and coffee, but not wine
b) Given that they like tea, what is the probability that they like coffee?
c) Given that they do not like wine, what is the probability that they like tea?
d) Given that they do not like coffee, what is the probability that they don’t like tea
or wine?
2) In a group of 20 boys, 6 have blue eyes, 13 have fair hair and 4 have both blue eyes and fair hair.
a) Draw a Venn diagram to represent this information
b) What is the probability that a boy has fair hair but not blue eyes?
c) What is the probability that a boy does not have fair hair or blue eyes?
d) Given that a boy has fair hair, what is the probability that he has blue eyes?
3) In a group of 30 girls, 24 like oranges, 7 like bananas and 4 like neither oranges nor bananas. By drawing
a Venn diagram, find the probability that
a) A randomly selected girl likes both oranges and bananas.
b) Given that a girl likes bananas, find the probability that a girl does not like oranges.
4) In a group of 40 people, 24 played cricket and 30 played tennis. Find the least and greatest number of
people who played both?
5) 50 people were randomly selected to pick a favourite colour from a list of 8 colours. 20 people preferred
blue and 12 people preferred red. Given that the two events are mutually exclusive:
a) Draw a Venn diagram to represent this information
b) Find the probability that a randomly selected person does not like red or blue.
6) 35 members of a horticultural society were asked if they grew Alpines, Begonias and Cacti. The results
showed that: 4 grew all three plants, 6 grew Alpines and Begonias, 7 grew Begonias and Cacti, 6 grew
Alpines and Cacti, 19 grew Cacti, 14 grew Begonias and 11 grew Alpines.
a) Draw a Venn diagram to represent this information.
b) Find the probability that:
i) Someone grew Cacti
ii) Someone grew Alpines and Begonias.
iii) Given that someone grew Alpines, what is the probability that someone grows Cacti?
iv) Someone grew none of the plants above.
7) A survey of some adults who enjoy at least one of the activities Art, Music and Drama gave the following
information. 15% enjoy all three activities, 27% enjoy Art and Drama, 22% enjoy Music and Drama, 21%
enjoy Art and Music, 53% enjoy Art, 10% enjoy Drama only. Find the percentage of people who:
a) enjoy only Music. b) enjoy exactly one activity, c) enjoy exactly 2 activities,
14
8) During a meal, a waiter in a restaurant observed that, in a group of 40 people, 16 ate soup, 18 ate steak, 22
ate ice cream, 5 ate soup, steak and ice cream, 2 ate steak and ice-cream, but not soup. All the people who
ate soup also ate ice cream. Draw a diagram to represent this information.
a) Find the probability that a person:
i) ate soup, but not steak.
ii) ate ice cream, but not steak nor soup.
iii) ate soup or steak
iv) did not eat soup, nor steak nor ice cream.
b) Given that a customer ordered ice-cream, what is the probability they ordered the steak?
9) Mary has 55 buttons. Some of the buttons are two-holed, the remainder are four holed. There are 8 blue
two-holed buttons. There are 20 two-holed buttons that are not blue. Of the four-holed buttons, there are
twice as many that are not blue as there are that are blue. Find the number of buttons that are:
a) four-holed
b) four-holed and blue c) not blue.
10) In a group of 80 young people, 49 played table-tennis (𝑇), 49 played darts (𝐷), 50 played snooker (𝑆). 27
played table tennis and darts, 29 played darts and snooker, 28 played table-tennis and snooker. 16 people
played all three activities. Find:
a) P(𝑆 ∩ 𝑇) b) P(𝑆 ∪ 𝐷) c) P(𝑆 ∩ 𝐷’)
d) Given that a randomly selected person liked playing darts, what is the probability this person also liked
snooker?
e) Given that a randomly selected person liked playing snooker, what is the probability this person also liked
playing table-tennis.
f) Given that a randomly selected person liked playing snooker, what is the probability this person did not
like playing table-tennis.
11) There are 31 students in a class. The only languages available for the class to study are French and
Spanish. 17 students study French. 15 students study Spanish. 6 students study neither French nor Spanish.
Using a Venn diagram or otherwise, work out how many students study only one language.
12) A garage tests cars for faults. There are three types of faults – braking, steering and lighting. A car fails
the test if it has one or more of these three faults. Last week, 11 cars had break faults, 9 cars had steering
faults, 7 cars had lighting faults, no car had both steering faults and lighting faults, 2 cars had both breaking
faults and steering faults, and 3 cars had both braking faults and lighting faults. By drawing a Venn diagram,
or otherwise, find the number of cars which failed the test last week.
Difficult – Problem Solving:
1) In a group of students, 33 played football and 19 played hockey. Three times as many played both as
played neither. The number of people who played neither is x.
a) Find expressions, in terms of x, for the number of students who played:
i) Both
ii) one or the other, but not both.
b) Given that there are 44 students in the group, find the probability that a randomly selected person played:
i) Both
ii) only football
2) In a group of 39 students there are 23 girls. All the students are asked if they did maths or physics. The
responses are 19 do maths, 16 do physics, 5 boys do maths and physics, 9 girls do maths only, 1 boy does
physics only, 3 girls do maths and physics, twice as many boys do not do maths or physics as girls.
a) How many girls do physics?
b) How many students do maths or physics only?
15
Venn Diagram Questions Involving Algebra
1) In a group of 50 students, 30 take French GCSE, and 35 take Geography GCSE. 8 students take neither.
Find the probability that a randomly selected student studies French and Geography.
2) In a group of 40 students, 13 like chess, and 14 like playing bridge. 21 students don’t like playing either
game. Find the probability a randomly selected students likes playing chess and bridge.
3) In a group of 65 shoppers, 30 had bread in their basket, 24 had milk, and 18 had neither. Find the
probability that a randomly selected shopper had both bread and milk in their basket.
4) A group of 60 students were survey whether they had a PlayStation or x box. In the survey, there was
double the amount of students who just had an x box than had both consoles. There were 21 students who
just had a PlayStation and 12 students who had neither. Find the number of students who had both consoles.
5) Every member of a group of 28 women speaks Spanish or French or both. 20 speak Spanish. Of those
who speak French, 6 more speak only French than speak French and Spanish. Find the probability that a
randomly selected student speaks:
a) both Spanish and French
b) only Spanish
c) Given that a student speaks Spanish, what is the probability they speak French?
6) When 42 people were asked about the type of money they were carrying, 28 said they had some coins and
5 said they had neither coins or notes. Twice as many said they had only coins as those who said they had
only notes. Find the number of people who had: a) only notes,
b) both coins and notes.
Venn Diagram Questions Involving Independence and Mutually Exclusive
Events
1) In a group of 40 students, 20 like playing rugby, 13 like playing football, and 7 don’t like playing either.
Show that the two events are mutually exclusive.
2) In a group of 40 students, 30 study French, 20 study German, and 15 study both. Decide if the two events
are independent.
3
3) The probability that Joe wears a coat to school is 4. The probability that Joe has to hand in maths
2
homework is 5. Given that the two events are independent, work out the probability that
a) Joe wears a coat and has to hand in maths homework.
b) Joe wears a coat and does not have to hand in maths homework.
c) Joe doesn’t wear a coat or hand in maths homework.
4) Granny looks in her button tin. There are some blue and red buttons in the tin.
a) Describe why these two events are mutually exclusive.
1
1
3
Half of the buttons are “big”. 10 of the buttons are big red ones, 5 of the buttons are big blue ones, 10 of the
2
buttons are blue, 5 of the buttons are red. By drawing a Venn diagram, find the probability that a button
randomly picked is: b) a small red button,
c) a small non-red-or-blue button.
16
Additional
Worksheet
with video
solutions
Factorising into One Bracket
1) Easy - Factorise:
a) 5𝑥 − 20
b) 8𝑥 + 24
c) 12𝑥 + 18
d) 20𝑥 − 25
e) 8𝑥 2 − 24
f) 9𝑥 2 + 36
g) 9𝑥 2 + 36
h) 10𝑥 2 − 15
i) 𝑥 2 − 6𝑥
j) 𝑥 2 − 𝑥
k) 𝑝𝑥 − 3𝑝
l) 𝑞𝑥 − 𝑞 2
m) 5𝑥 2 − 15𝑥
n) 6𝑥 2 + 9𝑥
o) 15𝑥 2 − 35𝑥
p) 8𝑥 − 12𝑥 2
q) 𝑎𝑥 2 − 𝑎𝑥
r) 4𝑎𝑥 − 6𝑎
s) 6𝑎𝑥 2 − 15𝑎𝑥
t) −4𝑎𝑥 − 6𝑥 2
2) Middle – Factorise:
a) 8𝑥 2 + 4𝑥
b) 6𝑝2 + 3𝑝
c) 6𝑥 2 − 3𝑥
d) 3𝑏 2 − 9𝑏
e) 12𝑎 + 3𝑎2
f) 15𝑐 − 10𝑐 2
g) 21𝑥 4 + 14𝑥 3
h) 16𝑦 3 − 12𝑦 2
i) 6𝑑 4 − 4𝑑 2
j) 18𝑎4 − 12𝑎5
k) 𝑎𝑥 2 + 𝑎𝑥
l) 𝑝𝑟 2 − 𝑝𝑟
m) 𝑎𝑏 2 − 𝑎𝑏
n) 𝑞𝑟 2 − 𝑞 2
o) 𝑎2 𝑥 − 𝑎𝑥 2
p) 𝑏 2 𝑦 − 𝑏𝑦 2
q) 6𝑎3 − 9𝑎2
r) 8𝑥 3 − 4𝑥 4
s) 18𝑥 3 + 12𝑥 5
t) 8𝑎2 𝑏 − 6𝑎𝑏 2
3) Hard – Factorise:
a) 12𝑎2 𝑏 + 18𝑎𝑏 2
b) 4𝑥 2 𝑦 − 2𝑥𝑦 2
c) 4𝑎2 𝑏 + 8𝑎𝑏 2 + 12𝑎𝑏
d) 4𝑥 2 𝑦 + 6𝑥𝑦 2 − 2𝑥𝑦
e) 12𝑎𝑥 2 + 6𝑎2 𝑥 − 3𝑎𝑥
f) 𝑎2 𝑏𝑐 + 𝑎𝑏 2 𝑐 + 𝑎𝑏𝑐 2
g) 60𝑥 3 𝑦 2 − 48𝑥 2 𝑦 7
h) 90𝑥 4 𝑦 6 − 81𝑥 2 𝑦 4
i) 34𝑥 4 𝑦 + 51𝑥 7 𝑦 3
Factorising into Two Brackets
1) Factorise
a) 𝑥 2 + 7𝑥 + 10
b) 𝑎2 + 8𝑎 + 7
c) 𝑦 2 + 6𝑦 + 5
d) 𝑝2 + 4𝑝 + 3
e) 𝑥 2 − 12𝑥 + 11
f) 𝑎2 − 4𝑎 + 3
g) 𝑥 2 − 7𝑥 + 10
h) 𝑝2 + 4𝑝 − 21
i) 𝑥 2 − 3𝑥 − 10
j) 𝑎2 + 2𝑎 − 15
k) 𝑥 2 + 7𝑥 + 12
l) 𝑥 2 − 8𝑥 + 12
m) 𝑥 2 − 𝑥 − 12
n) 𝑎2 + 𝑎 − 12
o) 𝑝2 + 9𝑝 − 10
p) 𝑝2 − 10𝑝 + 16
q) 𝑥 2 − 2𝑥 − 24
r) 𝑎2 + 11𝑎 + 24
s) 𝑎2 − 25𝑎 + 24
t) 𝑝2 + 5𝑝 − 24
2) Factorise
a) 𝑎2 + 10𝑎 + 9
b) 𝑥 2 − 𝑥 − 2
c) 𝑝2 − 12𝑝 + 11
d) 𝑥 2 + 𝑥 − 20
e) 𝑎2 − 6𝑎 + 8
f) 𝑦 2 + 2𝑦 − 15
g) 𝑝2 − 5𝑝 − 36
h) 𝑥 2 − 9𝑥 − 36
i) 𝑎2 − 7𝑎 + 12
j) 𝑦 2 + 5𝑦 − 6
k) 𝑎2 + 2𝑎 − 15
l) 𝑥 2 + 17𝑥 + 42
m) 𝑎2 − 14𝑎 + 48
n) 𝑥 2 + 11𝑥 − 60
o) 𝑝2 + 98𝑝 − 200
p)𝑛2 + 24𝑛 + 144
q) 𝑥 2 − 39𝑥 + 360 r) 𝑥 2 − 19𝑥 + 60
s) 𝑥 2 − 17𝑥 − 60
t) 𝑥 2 − 27𝑥 + 140
3) Difference of Two Squares - Factorise
a) 𝑥 2 − 9
b) 𝑎2 − 49
c) 𝑏 2 − 81
d) 𝑦 2 − 25
e) 4𝑥 2 − 25
f) 9𝑥 2 − 1
g) 2𝑥 2 − 50
h) 3𝑥 2 − 27
i) 36𝑥 2 − 49
j) 8𝑥 2 − 50
k) 4𝑥 2 − 𝑦 2
l) 4𝑥 2 − 9𝑦 2
m) 𝑥 4 − 16
n) 𝑥 4 − 𝑦 4
o) 𝑥 6 − 𝑦 6
17
Harder Factorising into Two Brackets
1) Factorise
a) 2𝑥 2 + 9𝑥 + 10
b) 2𝑥 2 − 𝑥 − 21
c) 3𝑥 2 − 5𝑥 − 2
d) 2𝑥 2 + 9𝑥 + 7
e) 3𝑥 2 + 2𝑥 − 16
f) 2𝑥 2 − 11𝑥 + 14
g) 3𝑥 2 + 25𝑥 + 28
h) 5𝑥 2 + 22𝑥 + 21
i) 2𝑥 2 + 9𝑥 + 9
j) 3𝑥 2 + 5𝑥 − 28
k) 5𝑥 2 + 31𝑥 + 6
l) 3𝑥 2 + 13𝑥 − 30
m) 2𝑥 2 + 15𝑥 + 27
n) 3𝑥 2 + 10𝑥 + 7
o) 7𝑥 2 + 23𝑥 + 6
p) 5𝑥 2 + 43𝑥 + 24
q) 2𝑥 2 − 19𝑥 + 45
r) 3𝑥 2 − 19𝑥 + 28
s) 7𝑥 2 − 19𝑥 − 6
t) 5𝑥 2 − 14𝑥 − 24
a) 6𝑥 2 + 19𝑥 + 10
b) 10𝑥 2 + 21𝑥 + 9
c) 4𝑥 2 + 16𝑥 + 15
d) 4𝑥 2 + 15𝑥 + 14
e) 6𝑥 2 + 23𝑥 + 15
f) 8𝑥 2 + 26𝑥 + 15
g) 8𝑥 2 + 23𝑥 + 21
h) 6𝑥 2 − 13𝑥 − 15
i) 6𝑥 2 + 5𝑥 − 21
j) 10𝑥 2 − 13𝑥 − 14
k) 10𝑥 2 + 39𝑥 − 27
l) 6𝑥 2 + 31𝑥 + 33
m) 6𝑥 2 − 𝑥 − 22
n) 10𝑥 2 − 29𝑥 + 21
o) 14𝑥 2 − 27𝑥 − 20
p) 12𝑥 2 − 29𝑥 − 21
q) 12𝑥 2 − 8𝑥 − 15
r) 12𝑥 2 + 19𝑥 − 18
s) 12𝑥 2 − 29𝑥 + 10
t) 12𝑥 2 − 44𝑥 + 35
2) Factorise
3) Two Step Factorising - Factorise
a) 2𝑥 2 + 10𝑥 + 12
b) 𝑎3 + 7𝑎2 + 12𝑎
c) 3𝑥 2 + 24𝑥 + 21
d) 2𝑥 2 + 16𝑥 + 24
e) 𝑥 3 + 5𝑥 2 + 4𝑥
f) 2𝑣 3 + 12𝑣 2 + 18𝑣
g) 3𝑥 3 + 12𝑥 2 + 9𝑥
h) 4𝑥 2 − 24𝑥 + 32
i) 2𝑥 3 − 14𝑥 2 + 20𝑥
j) 𝑥 2 − 5𝑥 + 4
k) 3𝑦 2 + 9𝑦 − 30
l) 𝑥 4 + 2𝑥 3 − 8𝑥 2
m) 𝑠 2 𝑡 + 3𝑠𝑡 − 18𝑡
n) 2𝑥 2 𝑦 − 4𝑥𝑦 − 30𝑦
o) 3𝑡 4 𝑠 2 − 12𝑡 3 𝑠 2 − 36𝑡 2 𝑠 2
p) 3𝑥 2 − 147
q) 𝑥 3 − 64𝑥
r) 𝑥 4 − 16
s) 𝑥 4 − 81
t) 𝑥 4 − 625
u) 16𝑥 4 − 81
v) 16𝑥 4 − 625
w) 81𝑥 4 − 16
Expanding Three Brackets
a) (𝑎 + 1)(𝑏 + 1)(𝑐 + 2)
b) (𝑑 − 2)(𝑒 + 3)(𝑓 + 2)
c) (4 + 𝑔)(ℎ + 2)(𝑖 − 3)
d) (𝑗 − 2)(𝑘 − 5)(𝑙 + 3)
e) (𝑚 − 5)(𝑛 − 1)(𝑝 − 3)
f) (2 − 𝑞)(𝑟 + 4)(𝑠 − 6)
g) (3𝑡 + 2)(1 + 𝑢)(4 + 𝑣)
h) (2𝑤 + 2)(𝑥 + 7)(𝑦 − 2)
i) (−3 + 𝑧)(5 − 2𝑎)(𝑏 + 3)
j) 2(𝑧 + 3)(𝑧 + 2)(𝑧 + 1)
k) 4(𝑤 − 4)(𝑤 − 5)(𝑤 − 2)
l) −(𝑥 − 7)(𝑥 + 5)(𝑥 + 1)
m) (𝑥 − 3)3
n) 3(𝑥 − 1)3
o) 4 (2𝑥 + 4)3
1
Factorising by Grouping Terms First
a) 𝑥𝑦 + 𝑧𝑦 + 𝑥𝑤 + 𝑧𝑤
b) 𝑐𝑦 + 𝑑𝑦 − 𝑐𝑧 − 𝑑𝑧
c) 𝑝𝑞 + 𝑟𝑞 − 𝑝𝑟 − 𝑟 2
d) 𝑝𝑞 − 𝑧𝑞 + 𝑝𝑧 − 𝑧 2
e) 𝑐𝑎 − 𝑏𝑑 − 𝑎𝑑 + 𝑏𝑐
f) 𝑥𝑧 + 𝑦𝑤 − 𝑦𝑧 − 𝑥𝑤
g) 𝑎𝑏 − 𝑥 2 − 𝑎𝑥 + 𝑏𝑥
h) 𝑎𝑥 − 3 + 𝑎 − 3𝑥
i) 𝑎𝑏 2 − 1 + 𝑎 − 𝑏 2
18
Mixed Factorising
a) 𝑥 2 − 3𝑥 − 10
b) 14𝑥𝑦 4 + 13𝑥 5 𝑦 3 − 5𝑥 6 𝑦 4
c) 6𝑧 2 − 27𝑧 + 12
d) 12𝑥 2 − 24𝑥 + 12
e) 5𝑡 3 − 20𝑡 2 − 60𝑡
f) 4𝑥 2 − 9
g) 𝑣 3 + 6𝑣 2 + 9𝑣
h) 13𝑥 2 𝑦 2 + 22𝑥 6 𝑦 3 + 20𝑥 5 𝑦 3
i) 7𝑥𝑦 2 + 4𝑦 6 + 17𝑥 5 𝑦 2
j) 𝑡 2 − 9𝑡 + 14
k) 15𝑥 2 − 𝑥 − 2
l) 2𝑥 2 − 13𝑥 + 6
m) 16𝑎5 𝑏 5 − 𝑎4 𝑏 5 − 𝑎𝑏 3
n) 81𝑡 3 − 121𝑡
o) 16𝑎6 𝑏 4 + 8𝑎4 𝑏 2
p) 5𝑥 3 − 17𝑥 2 + 6𝑥
q) 4𝑐 2 − 144
r) 21𝑝6 𝑞 2 − 14𝑝𝑞 2 + 7𝑝3 𝑞
s) 4𝑥 2 − 26𝑥 + 12
t) 16𝑐 6 𝑑 5 − 14𝑐 3 + 8𝑐 3 𝑑 6
u) 𝑧 3 − 15𝑧 2 + 56𝑧
Algebra and Factorising Problems (Extra)
1) a) Factorise 𝑥 2 – 9
b) Hence or otherwise, compute 997 × 1003
2) a) Write an expression that represents the following scenario using algebra:
“I add three consecutive numbers”
b) Use algebra to explain why adding three consecutive numbers will give you a multiple of 3.
3) a) Factorise 𝑛2 + 2𝑛 + 1
b) What type of number m is if 𝑚 = 𝑛2 + 2𝑛 + 1
4) a) Factorise 𝑛2 + 7𝑛 + 6
b) Given that 𝑚 = 𝑛2 + 7𝑛 + 6, Tim says “I can find a value for n that will make m a prime number”.
Is Tim correct?
5) a) Factorise 2𝑛2 + 7𝑛 + 3
b) Using the factorisation above, write 273 as the product of two 2 digit numbers.
c) Write out 273 as the product of prime factors.
6) a) Factorise 𝑛3 + 6𝑛2 + 5𝑛
b) Use this to calculate the product of 10 × 11 × 15
7) a) Factorise 𝑥 2 – 𝑦 2
b) Sam says that 221 can be expressed as the difference of two square numbers. Find these square
numbers, and what is the value of 𝑥 and 𝑦?
c) Sam also says that 221 can be expressed as the product of two prime numbers. By using parts (a) and
(b), show that Sam is correct find these prime numbers.
19
Year 10 Maths Test 2 – Revision Questions
Question 1
𝜉 = {positive whole numbers< 20}
Find the members of: (a) 𝑃 ∩ 𝑄
𝑃 = {factors of 18}
(b)
(𝑃 ∪ 𝑄)ꞌ
𝑄 ={Prime numbers}
(c) Does 15 ∈ 𝜉 ?
Question 2
Consider the Venn diagram on the right.
(a) Find P(Aꞌ)
(b) Find P(𝐴 ∩ 𝐵 ꞌ)
(c) Are A and B independent? Provide evidence to support your answer.
Question 3
The Venn diagram represents the number in a year group of 100 pupils that like
chess (event C) and/or darts (event D).
(a) Given that the events C and D are mutually exclusive, find x.
(b) Find the probability that a randomly selected pupil likes neither chess nor darts.
ξ
0.2
A
0.15
0.3
B
0.35
ξ
D
C
18
8-x
14 + x
Question 4
Parts made on a production line can have one of three defects: A, B or C.
1000 parts were inspected and the following results obtained:
• 31 had type A defect
• 37 had type B defect
• 42 had type C defect
• 11 had both type A and type B defects
• 13 had both type B and type C defects
• 10 had both type A and type C defects
• 920 had no defects
(a) What is the probability that a randomly chosen part has all three types of defect?
(b) Given that a randomly selected part has type A defect, what is the probability that it also has type B defect?
Question 5
A bag contains 10 red and 6 black counters. A counter is removed at random and not replaced. A second counter is
then removed at random and not replaced. Finally, a third counter is now randomly selected.
(a) What is the probability that the counters selected are all the same colour?
(b) What is the probability that the counters selected are not all the same colour?
(c) If the first two counters selected are black, what is the chance that the third counter is also black?
Question 6
Factorise fully:
(a) 20y3 + 10y
(b) 6m7np2 – 14m2np3
Question 7
Factorise fully:
(a) x2 + 14x – 15
(b) 4y2 + 4y – 15
Question 8
Factorise fully:
(a) 9x2 – 100
(b) cd – bc + d 2 – db
(c) 3y3 – 15y2 + 12y
Question 9
Write the mean average of these 4 expressions in the form (2x + p)(x + q) where p and q are whole numbers to be
found.
2(x + 5)2 – 61
4x2 + 13x + 21
6x2 + 10x + 23
9x - 4x2 + 27
20
Trigonometry Questions
1) Calculate the value of the side labelled 𝑥
a)
b)
10
𝑥
𝑥
d)
𝑥
8
10𝑜
20𝑜
e)
𝑥
9
50𝑜
60
7
70𝑜
f)
c)
g)
h)
4
𝑥
12
35𝑜
6
𝑜
i)
7
j)
15
40𝑜
𝑥
𝑥
12
10𝑜
𝑥
65𝑜
𝑥
𝑥
30𝑜
2) Calculate the value of the angle labelled 𝑥
a)
b)
14
c)
15
9
𝑥
10
12
e)
13
9
8
𝑥
𝑥
𝑥
f)
5
d)
𝑥
8
g)
7
h)
i)
j)
𝑥
21
12
10
4
8
13
12
𝑥
𝑥
𝑥
20
9
𝑥
3) Calculate the value of the side or angle labelled with an 𝑥
a)
𝑥
b)
70𝑜
c)
d)
𝑥
17
6
8
9
𝑥
g)
55
8
h)
i)
15
𝑥
12
𝑥
5
m)
n)
𝑥
9
13
10
7
𝑥
25𝑜
𝑥
7
l)
30𝑜
10
𝑥
j)
𝑥
k)
𝑥
𝑥
40𝑜
7
12
𝑜
8
f)
e)
8
60𝑜
o)
11
𝑥
10
9
𝑥
60𝑜
𝑥
9
21
7
Additional
Worksheet
with video
solutions
Bearings Questions
1) Andy is due west of Barney at a distance of
10) Amy goes travelling in a sailing boat. She first
13km. Camilla is 7km due south of Barney. What is travels 10km at a bearing of 130o to point B, and
the bearing of Camilla from Andy?
then from point B by a further 10km on a bearing of
o
2) Emily is 8km due east of Freddie. George is 5km 240 to a point C.
due north of Freddie. What is bearing the bearing of a) Draw a diagram of these events.
b) What type of triangle is ABC?
Emily from George?
c) What is the bearing of point C from point A?
3) Harry is 9km due north of Ian. James is 5km due
11) Belinda travels 6km on a bearing of 240o to
west of Ian. Find the bearing of James from Harry.
point K. Then Belinda travels 6km on a bearing of
4) Lisa is 10km due west of Kirstie. Miles is 3km
170o to point M.
due south of Lisa. Find the bearing of Miles from
a) Draw a diagram of these events.
Kirstie.
b) What type of triangle is BKM?
5) Oliver is 8km due east of Nia. Oliver is 4km due c) What is the value of the angle KBM?
d) What is the bearing that Belinda will have to
north of Pip. Find the bearing of Nia from Pip.
travel to return back to her starting point?
6) A pilot flies on a bearing of 82o. On what bearing
would she have to fly to return to her starting point? 12) Colette travels 7km on a bearing of 300o to
point X. Then a further 7km on a bearing of 80o to
7) A camel walks 25km on a bearing of 305o. On
what bearing would the camel walk to return to the point Y. Find the bearing that Colette has to travel
to return to her original position.
starting point?
8) A ship sails 70km on a bearing of 145o. On what
bearing would the ship sail to return to the starting
point?
9) Amy goes travelling in a sailing boat. She first
travels 10km at a bearing of 130o to point B, and
then from point B by a further 10km on a bearing of
240o to a point C.
a) Draw a diagram of these events.
b) What type of triangle is ABC?
c) What is the bearing of point C from point A?
13) Dan travels 11km on a bearing of 40o to W.
Then he travels a further 11km on a bearing of 150o
to point Z. Find the bearing that Dan has to travel to
return to his original position.
14) Mark starts at point R. Mark travels 12km on a
bearing of 290o to point P. Then Mark travels 16km
on a bearing of 200o to point Q. Find the straight
line distance from point R to point Q.
Trigonometric Equations
A) Solve the following equations
1) sin(𝑥) = 0.5
0 < 𝑥 < 360
2) sin(𝑥) = −0.3
0 < 𝑥 < 360
3) 3 cos(𝑥) = 1
0 < 𝑥 < 360
4) 5 cos(𝑥) = −1
0 < 𝑥 < 360
5) tan(𝑥) = 3
0 < 𝑥 < 360
B) Solve the following equations
1) sin(𝑥) + 0.5 = 0.1
0 < 𝑥 < 360
2) sin(𝑥) − 1 = −0.9
0 < 𝑥 < 360
3) cos(𝑥) + 0.2 = 0.1
0 < 𝑥 < 360
4) cos(𝑥) − 3 = −2.5
0 < 𝑥 < 360
5) 2 + tan(𝑥) = 0
0 < 𝑥 < 360
C) Solve the following equations
1) sin(𝑥) − 1 = −0.4
−180 < 𝑥 < 180
2) 4 sin(𝑥) = 0.2
0 < 𝑥 < 90
3) 1 + cos(𝑥) = 0.2
−360 < 𝑥 < 0
4) 0.5 cos(𝑥) = −0.12
−180 < 𝑥 < 180
5) tan(𝑥) + 2 = −0.23
90 < 𝑥 < 270
D) Solve the following equations
1) 2 sin(𝑥) − 1 = 0.5
−180 < 𝑥 < 180
2) 1 + 3 sin(𝑥) = −0.2
0 < 𝑥 < 360
1
3) 2 cos(𝑥) − 2 = −2.1
0 < 𝑥 < 360
4) 3 cos(𝑥) + 2 = 1
−180 < 𝑥 < 180
5) 5 + 2 tan(𝑥) = 4
−180 < 𝑥 < 180
22
Drawing Graphs and Solving 𝒇(𝒙) = 𝒌
1) a) Draw the graph of 𝑦 = 𝑥 2 – 5𝑥 + 6 for −4 ≤ 𝑥 ≤ 4.
b) Use your graph to solve the equation 𝑥 2 – 5𝑥 + 6 = 0.
c) Use your graph to solve the equation 𝑥 2 – 5𝑥 + 6 = 2
2) a) Draw the graph of 𝑦 = 𝑥 2 – 2𝑥 and the graph of 𝑦 = 3 for −4 ≤ 𝑥 ≤ 4
b) Use your graph to solve the equation 𝑥 2 – 2𝑥 = 3
c) Use your graph to solve the equation 𝑥 2 – 2𝑥 = 0
3) a) Draw the graph of 𝑦 = 𝑥 2 + 𝑥 and the graph of 𝑦 = 2𝑥 + 2 for −4 ≤ 𝑥 ≤ 4.
b) Use your graph to solve the equation 𝑥 2 + 𝑥 = 2𝑥 + 2
4) a) Draw the graph of 𝑦 = 𝑥 2 – 𝑥 – 1 and the graph of 𝑦 = 𝑥 + 2 for −4 ≤ 𝑥 ≤ 4.
b) Use your graph to solve the equation 𝑥 2 – 𝑥 – 1 = 𝑥 + 2.
5) a) Draw the graph of 𝑦 = 𝑥 2 + 3𝑥 + 1 for −4 ≤ 𝑥 ≤ 4.
b) Use your graph to solve the equation 𝑥 2 + 3𝑥 + 1 = 𝑥 + 4.
[Hint: you might want to draw another line on your graph]
6) a) Draw the graph of 𝑦 = 𝑥 2 + 𝑥 – 3 for −4 ≤ 𝑥 ≤ 4.
b) Use your graph to solve the equation 𝑥 2 + 𝑥 – 3 = 2𝑥 + 3
7) a) Draw the graph of 𝑦 = 𝑥 2 + 1 for −4 ≤ 𝑥 ≤ 4.
b) Use your graph to solve the equation 𝑥 2 + 1 = 2𝑥 + 5
8) a) Draw the graph of 𝑦 = 𝑥 2 + 4𝑥 – 2 for −7 ≤ 𝑥 ≤ 3
b) Use your graph to solve the equation 𝑥 2 + 4𝑥 – 5 = 0
9) a) Draw the graph of 𝑦 = 𝑥 2 – 2𝑥 – 2 for −4 ≤ 𝑥 ≤ 4.
b) Use your graph to solve the equation 𝑥 2 – 3𝑥 – 4 = 0
10) a) Draw the graph of 𝑦 = 𝑥 2 – 2𝑥 for −3 ≤ 𝑥 ≤ 5.
b) Use your graph to solve the equation 𝑥 2 – 4𝑥 + 3 = 0
11) a) Draw the graph of 𝑦 = 𝑥 2 + 3𝑥 – 7 for −4 ≤ 𝑥 ≤ 4.
b) Use your graph to solve the equation 𝑥 2 + 𝑥 – 2 = 0
12) a) Draw the graph of 𝑦 = 𝑥 2 + 𝑥 – 3 for −4 ≤ 𝑥 ≤ 4.
b) Use your graph to solve the equation 𝑥 2 – 2𝑥 – 8 = 0
13) a) Draw the graph of 𝑦 = 𝑥 2 + 4𝑥 – 9 for −7 ≤ 𝑥 ≤ 3.
b) Use your graph to find an estimate of the solution for the equation 𝑥 2 + 5𝑥 – 11 = 0.
14) a) Draw the graph of 𝑦 = 𝑥 2 – 𝑥 + 5 for −4 ≤ 𝑥 ≤ 4.
b) Use your graph to find an estimate of the solution for the equation 𝑥 2 – 3𝑥 + 1 = 0.
23
Revision Questions for Year 10 Test 3
Question 1
Please refer to the diagram on the right (which is
not accurately drawn).
𝑁
The bearing of B from A is 070°
Angle ABC is 50°
𝑁
AB = CB
𝐵
Work out the bearing
𝑜
50
of C from A.
𝐴
Question 4
Six sketch graphs, A to F, are shown above.
Which one of the six sketch graphs could represent
the curve with equation
2
(a) y = x3 (b) 𝑦 = 𝑥 (c) 𝑦 = 4𝑥 (d) y = -x2 + 4x +3
𝑁
𝐶
Question 2
Water is poured into each container at a constant
rate. Match each graph with the correct container.
Question 5
Find all the solutions to the following equations in
the ranges given to 1 d.p.
a) sin(𝑥) = 0.1
0𝑜 ≤ 𝑥 ≤ 360𝑜
b) 3 cos(𝑥) = 2
− 360𝑜 ≤ 𝑥 ≤ 360𝑜
Question 6
The graph below has equation 𝑦 = 𝑥 2 − 𝑥 − 2
Question 3
Calculate the size of angle y.
24cm
5cm
𝑦
10cm
39𝑜
a) Solve the equation 𝑥 2 − 𝑥 − 2 = 0
b) By drawing an appropriate line on the graph,
estimate to 1 decimal place the solutions of the
equation x2 – 3x – 3 = 0
c) Draw the graph 𝑦 = 𝑥 2 + 3
24
Direct Proportion Questions
1) 𝑌 is directly proportional to the square root of 𝑥.
When 𝑌 = 20, 𝑥 = 4.
a) Find a formula for 𝑌 in terms of 𝑥.
b) Calculate 𝑌 when 𝑥 = 36.
c) Calculate 𝑥 when 𝑌 = 100.
10) 𝐻 is directly proportional to the square root of
𝑔. When 𝑔 = 9, 𝐻 = 21.
Find the value of 𝐻 when 𝑔 = 16
2) 𝐷 is directly proportional to the square of 𝑡.
𝐷 = 80 when 𝑡 = 4.
a) Express 𝐷 in terms of 𝑡.
b) Work out the value of 𝐷 when 𝑡 = 7
c) Work out the positive value of 𝑡 when 𝐷 = 45.
11) 𝑇 is directly proportional to the cube of 𝑡. When
𝑇 = 136, 𝑡 = 2.
Find the value of 𝑇 when 𝑡 = 3
12) 𝑌 is directly proportional to the cube root of 𝑥.
When 𝑌 = 57, 𝑥 = 27.
Find the value of 𝑥 when 𝑌 = 95.
Harder Questions
10) 𝐹 is directly proportional to the square of m.
When 𝐹 = 441 , 𝑚 = 7.
a) Work out the value of 𝐹 when 𝑚 = 6.
b) Work out the percentage increase in 𝐹 if m
increases by 40%.
3) 𝑊 is directly proportional to the square root of 𝑢.
When 𝑊 = 18, 𝑢 = 36.
a) Express W in terms of 𝑢.
b) Work out the value of 𝑊 when 𝑢 = 49.
c) Work out the value of 𝑢 when 𝑊 = 12.
4) 𝑅 is directly proportional to the cube of 𝑠. When
𝑅 = 500, 𝑠 = 5.
a) Express 𝑅 in terms of 𝑠.
b) Work out the value of 𝑅 when 𝑠 = 3.
c) Work out the value of 𝑠 when 𝑅 = 1372.
5) 𝑦 is directly proportional to the cube of 𝑥. When
𝑥 = 2, 𝑦 = 64.
a) Find an expression for 𝑦 in terms of 𝑥.
1
b) i) Calculate the value of 𝑦 when 𝑥 = 2
ii) Calculate the value of 𝑥 when 𝑦 = 27.
6) 𝑇 is directly proportional to 𝑓. When 𝑇 = 65
𝑓 = 5.
a) Find an expression for 𝑇 in terms of 𝑓.
b) i) Calculate the value of 𝑇 when 𝑓 = 4
ii) Calculate the value of 𝑓 when 𝑇 = −39.
7) The distance, 𝐷 travelled by a particle is directly
proportional to the square of the time, 𝑡, taken.
When 𝑡 = 10, 𝐷 = 30.
a) Find a formula for 𝐷 in terms of 𝑡.
b) Calculate 𝐷 when 𝑡 = 64.
c) Calculate 𝑡 when 𝐷 = 12. Give your answer to
three significant figures.
8) 𝑇 is directly proportional to the cube root of 𝑏.
When 𝑇 = 77, 𝑏 = 343.
a) Express 𝑇 in terms of 𝑏.
b) Work out the value of 𝑇 when 𝑏 = 125.
c) Work out the value of 𝑏 when 𝑇 = 33.
9) 𝑦 is directly proportional to the cube of 𝑥. When
𝑥 = 2, 𝑦 = 50.
a) Calculate the value of 𝑦 when 𝑥 = 0.5
b) Calculate 𝑥 when 𝑦 = 27 (2 d.p.)
11) The kinetic energy, 𝐸 joules, of a moving
object is directly proportional to 𝑣 2 , where 𝑣 m/s is
the objects speed. When its speed is 5 m/s, the
kinetic energy is 10750 joules.
a) Find a formula for 𝐸 in terms of 𝑣.
b) Find the speed of the object when the kinetic
energy is 5267.5 joules.
c) Find the percentage increase in kinetic energy if
the speed increases by 50%.
12) Round a bend on a railway track the height
difference (h mm) between the outer and inner rails
must vary in direct proportion to the square of the
maximum permitted speed (S km/h).
a) When 𝑆 = 50, ℎ = 35. Calculate h when
𝑆 = 40.
b) The maximum speed on a bend is to be increased
by 30%. What is the percentage increase in the
height difference between the outer and inner rails?
13) 𝑌 is directly proportional to the square root of
𝑥. When 𝑌 = 20, 𝑥 = 16.
a) Work out the value of 𝑌 when 𝑥 = 9.
b) Work out the percentage change in 𝑌 if 𝑥
decreases by 20% (answer to 2 d.p.).
14) In a factory, chemical reactions are carried out
in spherical containers. The time, 𝑇 minutes, the
chemical reaction takes is directly proportional to
the square of the radius 𝑅 cm, of the spherical
container. When 𝑅 = 120, 𝑇 = 32.
a) Find the value of 𝑇 when 𝑅 = 150.
b) If the size of spherical container is reduced in
size by 50%, calculate the percentage change in the
time the reaction takes.
25
Inverse Proportion Questions
1) 𝐹 is inversely proportional to 𝑟. When 𝐹 = 20,
𝑟 = 5. Find:
a) A formula linking 𝐹 and 𝑟.
b) 𝐹 when 𝑟 = 4.
c) 𝑟 when 𝐹 = 50.
2) 𝐺 is inversely proportional to the square of ℎ.
When 𝐺 = 14, ℎ = 2. Find:
a) 𝐺 in terms of ℎ. b) 𝐺 when ℎ = 4.
c) ℎ when 𝐺 = 7 to 2 d.p.
3) 𝑌 is inversely proportional to the square root of
𝑥. When 𝑌 = 6, then 𝑥 = 4.
a) What is the value of 𝑌 when 𝑥 = 9?
b) What is the value of 𝑥 when 𝑌 = 10?
4) 𝑃 is inversely proportional to 𝑡. When 𝑃 = 12,
𝑡 = 5. Find:
a) 𝑃 in terms of 𝑡.
b) 𝑃 when 𝑡 = 3.
c) 𝑡 when 𝑃 = 15.
5) 𝐻 is inversely proportional to the square root of
𝑎. When 𝐻 = 50, 𝑎 = 9. Find:
a) An equation linking 𝐻 and 𝑎.
b) 𝐻 when 𝑎 = 25. c) 𝑎 when 𝐻 = 25.
6) 𝑅 is inversely proportional to the cube of 𝑢.
When 𝑅 = 60, 𝑢 = 1.5. Find:
a) A formula linking 𝑅 and 𝑢.
b) 𝑅 when 𝑢 = 1.25.
10) The time, 𝑇 seconds, that it takes a water heater
to boil some water is directly proportional to the
mass of water, 𝑚 kg, in the water heater. When
𝑚 = 250, 𝑇 = 600.
a) Find 𝑇 when 𝑚 = 400.
The time, T seconds, that it takes a water heater to
boil a constant mass of water is inversely
proportional to the power, P watts, of the water
heater. When 𝑃 = 400, 𝑇 = 360.
b) Find the value of 𝑇 when 𝑃 = 900.
11) In biology, the inverse square law states that the
light energy for photosynthesis (𝐿) is inversely
proportional to the square of the distance of the light
source from the plant (𝑑). Given that when the light
source is a distance 15cm away from the plant, the
rate of photosynthesis is 54 units.
a) Find a formula for 𝐿 in terms of 𝑑.
b) Find 𝐿 when 𝑑 = 25cm.
c) Find 𝑑 when 𝐿 = 120 units (3 s.f.)
12) The graphs of y against x represent four
different types of proportionality. Match the graphs
to the types of proportionality.
1 – y is directly proportional to x.
2 – y is inversely proportional to x.
3 – y is proportional to the square of x.
4 – y is inversely proportional to the square of x.
7) 𝐵 is inversely proportional to the square of 𝑐.
When 𝐵 = 75, 𝑐 = 3. Find:
a) A formula linking 𝐵 and 𝑐.
b) 𝐵 when 𝑐 = 6.
8) The gravitational force, 𝑓, between two objects
is inversely proportional to the square of the
distance, 𝑑, between them. When 𝑑 = 100,
𝑓 = 20.
a) Write down an equation connecting f and d.
b) Use this to calculate the gravitational force
when 𝑑 = 800.
9) The force, 𝐹, between two magnets is inversely
proportional to the square of the distance, 𝑥,
between them. When 𝑥 = 3, 𝐹 = 4.
a) Find an expression for 𝐹 in terms of 𝑥.
b) Calculate 𝐹 when 𝑥 = 2.
c) 𝑥 when 𝐹 = 64.
26
Mixed Proportion Questions
1) 𝑌 is inversely proportional to the square of 𝑡.
When 𝑌 = 7, 𝑡 = 4.
Find the value of 𝑌 when 𝑡 = 5.
2) 𝐻 is directly proportional to the square root of 𝑡.
When 𝑡 = 9, 𝐻 = 54.
Find the value of 𝑡 when 𝐻 = 162
3) 𝑀 is inversely proportional to the square root of
𝑡. When 𝑡 = 16, 𝑀 = 12.
Find the value of 𝑀 when 𝑡 = 64.
4) 𝑃 is inversely proportional to 𝑥.
When 𝑥 = 5, 𝑃 = 24.
Find the value of 𝑥 when 𝑃 = 8.
5) 𝐾 is directly proportional to the cube of 𝑡.
When 𝑡 = 5, 𝐾 = 12.5.
Find the value of 𝐾 when 𝑡 = 4.
6) 𝑌 is inversely proportional to the square root of
𝑥. When 𝑥 = 0.64, 𝑌 = 25.
5
Find the value of 𝑥 when 𝑌 = 3
7) 𝑃 is inversely proportional to 𝑡.
3
When 𝑡 = 4, 𝑃 = 24.
Find the value of 𝑡 when 𝑃 = 12
8) 𝑋 is inversely proportional to the square of
𝑡. When 𝑋 = 6.4, 𝑡 = 5.
Find the possible values of 𝑡 when 𝑋 = 2.5
9) 𝐹 is proportional to the square of 𝑎.
When 𝐹 = 12.8 when 𝑎 = 8.
Find the possible values of 𝑎 when 𝐹 = 28.8.
10) 𝑇 is inversely proportional to the square of 𝑡
When 𝑡 = 8, 𝑇 = 37.5.
Find the possible values of 𝑡 when 𝑇 = 9.375
11) 𝑌 is proportional to the square root of 𝑡.
When 𝑡 = 196, 𝑌 = 7.
Find the value of 𝑡 when 𝑌 = 6.
12) 𝑀 is inversely proportional to the cube of 𝑥.
When 𝑥 = 3, 𝑀 = 180.
Find the value of 𝑥 when 𝑀 = 38.88
Hard Composite Proportion Questions
1) ℎ is inversely proportional to 𝑝.
𝑝 is directly proportional to √𝑡.
Given that ℎ = 10 and 𝑡 = 144 when 𝑝 = 6, find a
formula for ℎ in terms of 𝑡.
5) 𝑌 is directly proportional to 𝑥.
𝑥 is inversely proportional to the square of 𝑧.
When 𝑧 = 5, 𝑥 = 8, and 𝑌 = 24.
Find a formula for 𝑌 in terms of 𝑧.
2) 𝑦 is inverserly proportional to 𝑑2 .
When 𝑑 = 10, 𝑦 = 4.
𝑑 is directly proportional to 𝑥 2 .
When 𝑥 = 2, 𝑑 = 24.
Find a formula for 𝑦 in terms of 𝑥. Give your
answer in its simplest form.
6) 𝑀 is directly proportional to √𝑛.
When 𝑛 = 64, 𝑀 = 152.
𝑛 is inversely proportional to the cube of 𝑥.
When 𝑥 = 5, 𝑛 = 20.
Find a formula for 𝑀 in terms of 𝑥. Give your
answer in the form 𝑀 = 𝑎𝑥 𝑏 , where 𝑎 and 𝑏 are
rational numbers to be found.
3) 𝐻 is inversely proportional to 𝑝2 .
𝑝 is proportional to the cube of 𝑥.
1
Given that 𝑥 = 3 and 𝐻 = 4 when 𝑝 = 54, find a
formula for 𝐻 in terms of 𝑥.
4) 𝑇 is inversely proportional to 𝑚.
When 𝑚 = 5, 𝑇 = 12.
𝑚 is directly proportional to √𝑛.
When 𝑛 = 16, 𝑚 = 56.
Find a formula for 𝑇 in terms of 𝑛. Give your
answer in its simplest form.
7) 𝑃 is inversely proportional to 𝑛2 .
𝑛 is proportional to 𝑡.
Given that when 𝑡 = 7, 𝑛 = 21 and 𝑃 = 3969
Find the value of 𝑃 when 𝑡 = 5 to 3 s.f.
Additional
8) 𝑋 is directly proportional to √𝑡.
Worksheet
When 𝑡 = 25, 𝑋 = 85.
with video
𝑡 is inversely proportional to 𝑝.
solutions
When 𝑝 = 18, 𝑡 = 50.
Find a formula for 𝑋 in terms of 𝑝. Give your
answer in its simplest form.
27
Solving Quadratics by Factorising
1) Factorise and Solve - Solve the following quadratic equations by factorising:
a) 𝑥 2 + 5𝑥 + 4 = 0
b) 𝑥 2 + 10𝑥 + 16 = 0
c) 𝑥 2 + 𝑥 − 6 = 0
d) 𝑥 2 − 𝑥 − 6 = 0
e) 𝑥 2 − 6𝑥 + 5 = 0
f) 𝑥 2 − 8𝑥 + 15 = 0
g) 𝑥 2 + 7𝑥 + 10 = 0
h) 𝑥 2 + 11𝑥 − 12 = 0
i) 𝑥 2 + 2𝑥 − 15 = 0
j) 𝑥 2 + 𝑥 − 12 = 0
k) 𝑥 2 − 5𝑥 − 6 = 0
l) 𝑥 2 − 7𝑥 + 12 = 0
m) 𝑥 2 + 11𝑥 + 24 = 0
n) 𝑥 2 − 11𝑥 + 24 = 0
o) 𝑥 2 + 5𝑥 − 24 = 0
p) 𝑥 2 + 12𝑥 + 35 = 0
q) 𝑥 2 − 7𝑥 + 10 = 0
r) 𝑥 2 − 9𝑥 + 20 = 0
s) 𝑥 2 − 12𝑥 + 11 = 0
t) 𝑥 2 − 3𝑥 − 40 = 0
u) 𝑥 2 + 4𝑥 − 96 = 0
v) 𝑥 2 − 13𝑥 + 42 = 0
w) 𝑥 2 + 9𝑥 − 36 = 0
x) 𝑥 2 − 5𝑥 − 84 = 0
2) Rearrange, Factorise and Solve – Solve the following quadratic equations:
a) 𝑥 2 − 2𝑥 = 8
b) 𝑥 2 + 𝑥 = 12
c) 𝑥 2 − 3𝑥 = 10
d) 𝑥 2 − 7𝑥 = 8
e) 𝑥 2 = 2𝑥 + 15
f) 𝑥 2 + 11𝑥 = −24
g) 𝑥 2 − 4𝑥 = 5
h) 𝑥 2 = 7𝑥 − 6
i) 4𝑥 + 21 = 𝑥 2
j) 12𝑥 = 𝑥 2 + 35
k) 7𝑥 = 𝑥 2 + 12
l) 5𝑥 + 14 = 𝑥 2
m) 𝑥 2 + 4 = 6𝑥 + 20
n) 2𝑥 − 𝑥 2 = 6𝑥 − 45
o) 3𝑥 − 30 = 40 − 𝑥 2
p) 3𝑥 − 𝑥 2 = 8𝑥 − 24
q) 5𝑥 − 77 = 9𝑥 − 𝑥 2
r) 15 − 𝑥 2 = 2(𝑥 − 10)
s) 6𝑥 − 12 = 15 − 𝑥 2
t) 100 − 𝑥 2 = 3𝑥 − 80
3) Harder Factorise and Solve – Solve the following quadratic equations by factorising:
a) 3𝑥 2 − 5𝑥 − 2 = 0
b) 2𝑥 2 − 11𝑥 + 14 = 0 c) 5𝑥 2 + 31𝑥 + 6 = 0
d) 2𝑥 2 + 9𝑥 + 7 = 0
e) 3𝑥 2 + 2𝑥 − 16 = 0
f) 2𝑥 2 − 𝑥 − 21 = 0
g) 7𝑥 2 + 23𝑥 + 6 = 0
h) 3𝑥 2 + 13𝑥 − 30 = 0
i) 2𝑥 2 + 15𝑥 + 27 = 0
j) 3𝑥 2 + 5𝑥 − 28 = 0
k) 2𝑥 2 + 9𝑥 + 10 = 0
l) 5𝑥 2 + 43𝑥 + 24 = 0
m) 5𝑥 2 + 22𝑥 + 21 = 0
n) 3𝑥 2 + 10𝑥 + 7 = 0
o) 7𝑥 2 − 19𝑥 − 6 = 0
p) 5𝑥 2 − 14𝑥 − 24 = 0
4) Very Hard Factorise and Solve – Solve the following quadratic equations by factorising:
c) 4𝑥 2 + 15𝑥 + 14 = 0
a) 6𝑥 2 + 19𝑥 + 10 = 0
b) 8𝑥 2 + 26𝑥 + 15 = 0
d) 10𝑥 2 + 39𝑥 − 27 = 0
e) 6𝑥 2 − 𝑥 − 22 = 0
f) 10𝑥 2 − 29𝑥 + 21 = 0
g) 4𝑥 2 + 16𝑥 + 15 = 0
h) 12𝑥 2 − 29𝑥 − 21 = 0
i) 8𝑥 2 + 26𝑥 + 21 = 0
j) 12𝑥 2 + 19𝑥 − 18 = 0
k) 14𝑥 2 − 27𝑥 − 20 = 0
l) 6𝑥 2 + 31𝑥 + 33 = 0
m) 12𝑥 2 − 8𝑥 − 15 = 0
n) 10𝑥 2 − 13𝑥 − 14 = 0
o) 12𝑥 2 − 29𝑥 + 10 = 0
p) 12𝑥 2 − 44𝑥 + 35 = 0
5) Rearrange, Factorise and Solve – Solve the following quadratic equations by factorising:
a) 25𝑥 + 35 = 7 − 3𝑥 2
b) 4 − 3𝑥 2 = 32 − 19𝑥
c) 45 − 7𝑥 = 12𝑥 − 2𝑥 2
d) 3𝑥 − 2𝑥 2 = 12𝑥 + 9
e) 13 − 6𝑥 2 = 5𝑥 − 8
f) 26𝑥 + 9 = 5𝑥 − 10𝑥 2
g) 27𝑥 − 6𝑥 2 = 50𝑥 + 15
h) 3 − 13𝑥 = 18 − 6𝑥 2
6) Mixed – Solve the following equations by factorising:
a) 2𝑘 2 − 13𝑘 + 11 = 0
b) 2𝑑 2 − 11𝑑 + 15 = 0
c) 3𝑣 2 + 14𝑣 − 5 = 0
d) 2𝑞 2 = 3 − 𝑞
e) 3𝑣 2 + 14𝑣 = 5
f) 5𝑘 2 = 9𝑘 + 2
g) 13𝑒 + 10 = 3𝑒 2
h) 20𝑦 − 4𝑦 2 = 25
i) 5𝑐 2 = 16𝑐 − 3
j) 7ℎ2 + 15 = 38ℎ
k) 3𝑓 2 = 𝑓 2 + 𝑓 + 28
l) 5𝑥 2 = 7 − 2𝑥
28
Solving Quadratics by Completing the Square
1) Easy - Complete the square of the following quadratics:
a) 𝑥 2 + 8𝑥 − 3
b) 𝑥 2 − 6𝑥 + 4
c) 𝑥 2 − 10𝑥 + 3
d) 𝑥 2 − 2𝑥 + 5
e) 𝑥 2 + 4𝑥 − 3
f) 𝑥 2 + 4𝑥 − 7
g) 𝑥 2 − 6𝑥 + 11
h) 𝑥 2 − 8𝑥 + 1
i) 𝑥 2 − 2𝑥
j) 𝑥 2 − 8𝑥 − 3
k) 𝑥 2 + 12𝑥 − 11
l) 𝑥 2 − 6𝑥 + 15
m) 𝑥 2 − 4𝑥 + 7
n) 𝑥 2 − 4𝑥 + 14
o) 𝑥 2 − 6𝑥 + 9
p) 𝑥 2 + 6𝑥 − 9
q) 𝑥 2 + 10𝑥 − 7
r) 𝑥 2 + 8𝑥 − 14
s) 𝑥 2 − 6𝑥 + 19
t) 𝑥 2 − 4𝑥 + 16
2) Medium – Complete the square of the following quadratics:
a) 𝑥 2 − 7𝑥 + 3
b) 𝑥 2 + 3𝑥 − 9
c) 𝑥 2 + 5𝑥 − 8
d) 𝑥 2 − 9𝑥 + 4
e) 𝑥 2 − 5𝑥 + 9
f) 𝑥 2 − 𝑥 + 11
g) 𝑥 2 − 3𝑥 + 7
h) 𝑥 2 − 7𝑥 + 13
i) 𝑥 2 − 9𝑥 + 11
j) 𝑥 2 − 5𝑥 + 13
k) 𝑥 2 + 11𝑥 + 14
l) 𝑥 2 − 5𝑥 + 12
m) 𝑥 2 + 3𝑥 − 6
n) 𝑥 2 + 5𝑥 − 7
o) 𝑥 2 + 7𝑥 − 1
p) 𝑥 2 − 11𝑥 + 18
q) 𝑥 2 − 𝑥 + 11
r) 𝑥 2 − 3𝑥 + 1
s) 𝑥 2 − 9𝑥
t) 𝑥 2 − 13𝑥 + 5
3) Hard – Complete the square of the following quadratics in the form 𝑎(𝑥 + 𝑏) + 𝑐
a) 2𝑥 2 − 12𝑥 + 7
b) 2𝑥 2 − 6𝑥 + 5
c) 3𝑥 2 + 12𝑥 − 5
d) 3𝑥 2 + 6𝑥 − 8
e) 5𝑥 2 − 15𝑥 + 7
f) 3𝑥 2 + 5𝑥 + 7
g) 4𝑥 2 − 7𝑥 + 5
h) 2𝑥 2 − 9𝑥 + 5
i)) 3𝑥 2 + 8𝑥 − 11
j) 2𝑥 2 − 11𝑥 − 8
k) 3𝑥 2 + 7𝑥 − 4
l) 5𝑥 2 − 7𝑥 + 2
m) 2𝑥 2 − 5𝑥 + 9
n) 3𝑥 2 − 7𝑥 + 9
o) 4𝑥 2 − 9𝑥 + 3
p) 2𝑥 2 − 6𝑥 + 8
4) Solving – Solve the following equations by completing the square
a) 𝑥 2 − 8𝑥 + 5 = 0
b) 𝑥 2 − 6𝑥 + 2 = 0
c) 𝑥 2 − 4𝑥 − 2 = 0
d) 𝑥 2 + 2𝑥 − 9 = 0
e) 𝑥 2 + 6𝑥 − 11 = 0
f) 𝑥 2 + 8𝑥 − 3 = 0
g) 𝑥 2 + 2𝑥 − 11 = 0
h) 𝑥 2 + 4𝑥 − 9 = 0
i) 𝑥 2 − 5𝑥 − 7 = 0
j) 𝑥 2 + 5𝑥 + 3 = 0
k) 𝑥 2 − 3𝑥 − 6 = 0
l) 𝑥 2 − 7𝑥 − 3 = 0
m) 𝑥 2 − 𝑥 − 8 = 0
n) 𝑥 2 − 5𝑥 − 8 = 0
o) 𝑥 2 + 3𝑥 − 11 = 0
p) 𝑥 2 + 5𝑥 − 19 = 0
q) 𝑥 2 − 3𝑥 − 13 = 0
r) 𝑥 2 + 7𝑥 − 5 = 0
s) 𝑥 2 + 3𝑥 − 10 = 0
t) 𝑥 2 − 4𝑥 − 12 = 0
5) Harder Solving – Solve the following equations by completing the square
a) 2𝑥 2 + 6𝑥 − 3 = 0
b) 3𝑥 2 − 7𝑥 − 4 = 0
c) 2𝑥 2 + 5𝑥 − 9 = 0
d) 3𝑥 2 − 8𝑥 − 5 = 0
e) 2𝑥 2 − 8𝑥 − 3 = 0
f) 3𝑥 2 + 8𝑥 − 2 = 0
g) 5𝑥 2 − 11𝑥 − 5 = 0
h) 2𝑥 2 + 6𝑥 − 2 = 0
i) 3𝑥 2 − 10𝑥 − 5 = 0
j) 4𝑥 2 + 11𝑥 − 3 = 0
k) 2𝑥 2 − 5𝑥 − 9 = 0
l) 3𝑥 2 + 7𝑥 − 11 = 0
m) 2𝑥 2 − 7𝑥 − 13 = 0
n) 5𝑥 2 − 11𝑥 + 2 = 0
o) 4𝑥 2 − 15𝑥 − 3 = 0
p) 2𝑥 2 − 17𝑥 − 21 = 0
6) Turning Points – Find the turning point for the following graphs, and sketch the graph.
a) 𝑦 = 𝑥 2 − 6𝑥 − 3
b) 𝑦 = 𝑥 2 + 4𝑥 − 9
c) 𝑦 = 𝑥 2 − 2𝑥 + 7
d) 𝑦 = 𝑥 2 + 6𝑥 − 9
e) 𝑦 = 𝑥 2 − 4𝑥 + 11
f) 𝑦 = 𝑥 2 − 6𝑥 + 1
g) 𝑦 = 𝑥 2 + 8𝑥 − 5
h) 𝑦 = 𝑥 2 − 12𝑥 + 17
i) 𝑦 = 𝑥 2 − 3𝑥 + 7
j) 𝑦 = 𝑥 2 + 5𝑥 − 8
k) 𝑦 = 𝑥 2 + 𝑥 − 9
l) 𝑦 = 𝑥 2 + 3𝑥 − 11
m) 𝑦 = 𝑥 2 − 7𝑥 − 2
n) 𝑦 = 𝑥 2 − 9𝑥 + 5
o) 𝑦 = 𝑥 2 + 5𝑥 − 17
p) 𝑦 = 𝑥 2 − 3𝑥 + 11
q) 𝑦 = 2𝑥 2 − 3𝑥 + 8
r) 𝑦 = 2𝑥 2 − 7𝑥 + 9
s) 𝑦 = 3𝑥 2 − 8𝑥 + 3
t) 𝑦 = 3𝑥 2 − 5𝑥 − 13
29
Solving Quadratics by using the Formula
1) Solve the following equations by using the quadratic formula
a) 𝑥 2 − 3𝑥 − 11 = 0
b) 𝑥 2 − 5𝑥 − 2 = 0
c) 𝑥 2 + 4𝑥 − 11 = 0
d) 𝑥 2 − 9𝑥 − 13 = 0
e) 𝑥 2 − 7𝑥 − 4 = 0
f) 𝑥 2 − 5𝑥 − 12 = 0
g) 𝑥 2 + 3𝑥 − 19 = 0
h) 𝑥 2 − 2𝑥 − 13 = 0
i) 𝑥 2 + 5𝑥 − 7 = 0
j) 𝑥 2 − 𝑥 − 1 = 0
k) 𝑥 2 − 8𝑥 + 2 = 0
l) 𝑥 2 − 7𝑥 + 12 = 0
m) 2𝑥 2 − 7𝑥 − 6 = 0
n) 5𝑥 2 − 8𝑥 + 1 = 0
o) 4𝑥 2 − 9𝑥 + 5 = 0
p) 3𝑥 2 + 4𝑥 − 6 = 7𝑥 + 4
q) 2𝑥 2 + 8𝑥 − 3 = 3𝑥 2 − 𝑥
r) 5𝑥 2 + 5𝑥 − 4 = 3𝑥 2 + 2
s) 2𝑥 2 + 3𝑥 − 3 = 3𝑥 + 3
t) 2𝑥 2 − 2𝑥 − 2 = 𝑥 2 + 9
u) 8𝑥 + 8 = −3𝑥 2
v) 2(𝑥 2 − 1) = 𝑥
w) 3𝑥 2 + 7 = 4(𝑥 + 1)
x) 𝑥 2 − 12𝑥 + 13 = −𝑥 2
Sequence Notation
1) Write down the first 5 terms of the following sequences:
a) 𝑎𝑛 = 𝑎𝑛−1 − 3 , 𝑎1 = 7
b) 𝑏𝑛 = 4𝑏𝑛−1 + 2 , 𝑏1 = 2
c) 𝑐𝑛 = 7 − 𝑐𝑛−1 , 𝑐1 = 4
d) 𝑎𝑛 = (𝑎𝑛−1 )2 , 𝑎1 = −2
e) 𝑏𝑛 = 4 − (𝑏𝑛−1 )2 , 𝑏1 = −2
f) 𝑐𝑛 = 𝑐
h) 𝑏𝑛 = 𝑏𝑛−1 + 𝑘 , 𝑏1 = 2
i) 𝑐𝑛 = √𝑐𝑛−1 , 𝑐1 = 2
g) 𝑎𝑛 = 𝑎
3
𝑛−1
+ 2 , 𝑎1 = 9
1
𝑛−1
, 𝑐1 = 6
2) Write out the first 5 terms in the following sequence:
𝑓𝑛 = 𝑓𝑛−1 + 𝑓𝑛−2 ,
𝑓1 = 0 ,
𝑓2 = 1
Do you recognise the sequence?
Solving Equations using an Iterative Formula
1) a) Rearrange 𝑥 2 − 𝑥 − 1 = 0 into 𝑥 = √𝑥 + 1
b) Using 𝑥𝑛+1 = √𝑥𝑛 + 1 , and using 𝑥0 = 1.5, find 𝑥1 , 𝑥2 , 𝑥3 and 𝑥4 to 4 s.f.
1
2) a) Rearrange 𝑥 2 − 8𝑥 − 1 = 0 into 𝑥 = 𝑥 + 8
1
b) Using 𝑥𝑛+1 = 𝑥 + 8 , and using 5 as the starting term, work out the next 4 approximations to 5 s.f.
𝑛
3) a) Rearrange 𝑥 2 − 7𝑥 + 3 = 0 into 𝑥 = √7𝑥 − 3
b) Using 𝑥𝑛+1 = √7𝑥𝑛 − 3, and using 𝑥0 = 1, find 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 and 𝑥5 to 4 s.f.
c) Apply the iteration formulae until your answer is stable to 3 d.p.. Write down this value.
d) Interpret the meaning of this value.
5
4) a) Rearrange 𝑥 3 − 2𝑥 2 − 5 = 0 into 𝑥 = 2 + 𝑥 2
5
b) Using 𝑥𝑛+1 = 2 + 𝑥 2 , and using 𝑥0 = 1, find 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 and 𝑥5 to 4 s.f.
𝑛
c) Apply the iteration formulae until your answer is stable to 3 d.p.. Write down this value.
d) Interpret the meaning of this value.
5) a) Rearrange 𝑥 3 − 5𝑥 2 + 7 = 0 into 𝑥 = √
𝑥 3 +7
5
30
b) Using 𝑥𝑛+1 = √
3 +7
𝑥𝑛
and using 𝑥0 = 0, find 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 and 𝑥5 to 4 s.f.
5
c) Apply the iteration formulae until your answer is stable to 3 d.p.. Write down this value.
2
6) a) Rearrange 𝑥 2 − 4𝑥 + 2 = 0 into 𝑥 = 4 − 𝑥
2
b) Using 𝑥𝑛 = 4 − 𝑥 , and using 𝑥0 = 1, find 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 and 𝑥5 to 4 s.f.
𝑛
c) Apply the iteration formulae until your answer is stable to 3 d.p.. Write down this value.
d) Interpret the meaning of this value.
3
7) a) Rearrange 𝑥 3 − 5𝑥 − 7 = 0 into 𝑥 = √7 + 5𝑥
3
b) Using 𝑥𝑛+1 = √7 + 5𝑥𝑛 , and using 𝑥0 = 1, find 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 and 𝑥5 to 4 s.f.
c) Apply the iteration formulae until your answer is stable to 3 d.p.. Write down this value.
d) Interpret the meaning of this value.
11
8) a) Rearrange 𝑥 3 − 5𝑥 2 − 11 = 0 into 𝑥 = 5 + 𝑥 2
11
b) Using 𝑥𝑛+1 = 5 + 𝑥 2 and using 𝑥0 = 1, find 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 and 𝑥5 to 4 s.f.
𝑛
c) Apply the iteration formulae until your answer is stable to 3 d.p.. Write down this value.
d) Interpret the meaning of this value.
5
9) a) Rearrange 𝑥 2 + 2𝑥 − 5 = 0 into 𝑥 = 𝑥+2
5
b) Using 𝑥𝑛+1 = 𝑥 +2 , and using 𝑥0 = 0, find 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 and 𝑥5 to 4 s.f.
𝑛
c) Apply the iteration formulae until your answer is stable to 3 d.p.. Write down this value.
d) Interpret the meaning of this value.
10) a) Let 𝑓(𝑥) = 𝑥 3 + 9𝑥 − 2. Show that when 𝑓(𝑥) = 0, we can derive the iterative formula
2−𝑥 3
𝑥𝑛+1 = 9 𝑛 .
b) We are told that there is a solution between 0 and 1. Suggest a suitable starting value for our iterative
process.
c) Using this starting value for 𝑥0 , calculate the first 3 estimations to a solution for 𝑓(𝑥) = 0 to 4 d.p.
3
d) 𝑓(𝑥) = 0 can also be rearranged to the iterative formula 𝑥𝑛+1 = √2 − 9𝑥 . Using 𝑥0 = 0, write out
3
the first 6 estimations for a solution to 𝑓(𝑥) = 0, and show that 𝑥𝑛+1 = √2 − 9𝑥 does not yield any
solutions.
11) a) 𝑓(𝑥) = 3𝑥 2 − 𝑥 , 𝑔(𝑥) = 𝑥 2 + 4𝑥 − 1. Write out 𝑓(𝑥) = 𝑔(𝑥) in terms of x.
b) Rearrange 𝑓(𝑥) = 𝑔(𝑥) into 2𝑥 2 − 5𝑥 + 1 = 0
Additional
Worksheet
with video
solutions
5𝑥𝑛 −1
c) Use this to show that one iterative formula is 𝑥𝑛+1 = √
d) Using 𝑥0 = 2, calculate 𝑥1 , 𝑥2 , 𝑥3 and 𝑥4 to 4 s.f.
e) Show that another iterative formula could be 𝑥𝑛+1 =
f) Using 𝑥0 = 0, calculate 𝑥1 , 𝑥2 , 𝑥3 and 𝑥4 to 4 s.f.
.
2
1+2𝑥𝑛 2
5
.
2
12) a) Show that 𝑥 3 + 8𝑥 − 2 = 0 can be rearranged into the iterative formula 𝑥𝑛+1 = 𝑥 2 +8
𝑛
b) Using the iterative formula and 𝑥0 = 1, calculate the first 8 estimation to the solution of the
equation 𝑥 3 + 8𝑥 − 2 = 0
7
13) a) Show that 𝑥 2 + 3𝑥 − 7 = 0 can be rearranged into the iterative formula 𝑥𝑛+1 = 𝑥 +3
𝑛
b) Using the iterative formula and 𝑥1 = 1, calculate the first 8 estimations to the solution of the
equation 𝑥 2 + 3𝑥 − 7 = 0
31
Problem Solving with Quadratics
𝑥
1) The following diagram shows 2 connected rectangles. All
measurements are in cm. The area of the shape is 95cm2.
a) Show that 2𝑥 2 + 6𝑥 − 95 = 0.
b) Solve the equation to 3 s.f.
2𝑥 − 7
3𝑥
2𝑥 + 1
5
𝑥
3𝑥 + 4
𝑥
2) The following diagram shows a 6 sided shape. All measurements
are given in cm. The area of the shape is 85cm2.
a) Show that 9𝑥 2 – 17𝑥 – 85 = 0.
b) Solve the equation to 3 s.f.
3) The diagram shows a trapezium ABCD with AD
parallel to BC. AB = 𝑥 cm, BC = (𝑥 + 5)cm, and
AD = (𝑥 + 8) cm. The area of the trapezium is 42cm2.
a) Show that 2𝑥 2 + 13𝑥 – 84 = 0.
b) Calculate the value of x and find the perimeter of the
trapezium.
𝐵
(𝑥 + 5) cm
𝐶
𝑥 cm
𝐴
(𝑥 + 8) cm
4) The diagram shows a rectangular playground of width
𝑥 metres and length 3𝑥 meters. The playground is extended, by adding 10 meters to its width and 20 meters
to its length, to form a larger rectangular playground. The
area of the larger rectangular playground is double the area
𝑥
of the original playground.
2
a) Show that 3𝑥 – 50𝑥 – 200 = 0
b) Find the original area of the playground.
3𝑥
5) Clare buys some shares for $50𝑥. Later, she sells the shares for $(600 + 5𝑥). She makes a profit of 𝑥%.
Show that 𝑥 2 + 90𝑥 – 1200 = 0. Solve this equation to 3 significant figures.
6) The diagram shows a rectangle. The length of the rectangle is 𝑥 cm.
The length of a diagonal of the rectangle is 8cm. The perimeter of the
rectangle is 20cm.
a) Show that 𝑥2 – 10𝑥 + 18 = 0.
b) Find the area of the rectangle.
8 cm
𝑥 cm
7) A rectangle lawn has a length of 3𝑥 metres and a width of
2𝑥 metres. The lawn has a path of width 1 metre on three of its
sides. The total area of the lawn and the path is 100m2.
a) Show that 6𝑥 2 + 7𝑥 – 98 = 0
b) Calculate the area of the lawn.
32
Additional
Worksheet
with video
solutions
𝐷
Year 10 Maths Test 4 – Revision Questions
Question 1
Sean drives from Manchester to Gretna Green.
He drives at an average speed of 50 mph for the
first 3 hours of his journey.
He then has 150 miles to drive to get to Gretna
Green.
Sean drives these 150 miles at an average speed of
30 mph.
Sean says,
“My average speed from Manchester to Gretna
Green was 40 mph.”
Is Sean right?
You must show how you get your answer.
Question 4
{This is a hard question}
A pendulum of length L cm has time period T
seconds.
T is directly proportional to the square root of L.
The length of the pendulum is increased by 40%.
Work out the percentage increase in the time
period.
Question 5
Solve these equations using factorisation. You must
show working.
(a)
x2 + 14x = 0
(b)
x2 – 25 = 0
(c)
x2 – 5x + 4 = 0
(d)
3x2 + 13x – 10 = 0
Question 2
Given that y is inversely proportional to the square
of x, and that y is 12 when x = 3,
(a)
Calculate the value of y when x = 6
(b)
Calculate the value of x when y = 27
Question 6
Use the quadratic formula to solve the following
equation. 3x2 – 5x – 4 = 0
Give your answers correct to 1 decimal place.
Question 3
In the following question, the units are centimetres.
The area of the rectangle is 2cm2 greater than the
triangle.
a) Show that 2𝑥 2 + 7𝑥 − 15 = 0
b) Hence, find the dimension of the following
shapes.
Question 7
Thelma spins a biased coin twice.
The probability that it will come down heads both
times is 0.09
Calculate the probability that it will come down
tails both times.
Question 8
Liz is asked to solve the equation
3x2 + 8 = 83
Here is her working.
3x2 + 8 = 83
7
3x2 = 75
(𝑥 + 5)
x2 = 25
x=5
Explain what is wrong with Liz’s answer.
Question 9
Given that:
2x − 1 : x − 4 = 16x + 1 : 2x − 1
find the possible values of x.
(𝑥 + 3)
(𝑥 + 4)
33
Simplifying Algebraic Fractions
1) Simplify the following fractions:
a)
e)
𝑥 2𝑦3
𝑥 5𝑦
8𝑥 3 𝑦 7 (𝑥+2)
24𝑦 2 (𝑥+2)(𝑥+1)
b)
f)
12𝑎𝑏7
5𝑥 2 𝑦 7
8𝑎4 𝑏3
9𝑥(𝑥+1)
c)
27(𝑥+1)(𝑥+2)
g)
𝑥 2 (𝑥+1)
18𝑎3 𝑏4 𝑐 7
d) 3
𝑥 (𝑥+1)
14(2𝑥+3)
h)
7(𝑥+3)
5𝑥 2 +25
𝑥 2 −25
15𝑥 2 𝑦 5
16𝑎7 𝑏4 𝑐 7
2) Factorise first, and then simplify:
a)
𝑥 2 +2𝑥𝑦
𝑥𝑦+2𝑦 2
𝑥+2
e) 2
𝑥 +5𝑥+6
i)
9𝑥 2 −3𝑥+12
6𝑥 2 −2𝑥+8
b)
3𝑦 2 +15𝑦
c)
10𝑦+2𝑦 2
4𝑥−4
f) 2
𝑥 +2𝑥−3
g)
𝑎2 −𝑎−6
d)
𝑥 2 +5
𝑥 2 −4𝑥+3
h)
2𝑥−6
2
𝑥 +𝑥𝑦
k) 2 2
𝑥 −𝑦
j) 2
𝑎 −8𝑎+15
l)
𝑥+5
9𝑥 2 −1
9𝑥+3
3𝑎2 +3𝑎𝑏
6𝑎𝑏+6𝑏2
Adding and Subtracting Algebraic Fractions
1) Express the following as a single fraction:
7
9
2
5
b)
+
a)
+
𝑥+2
𝑥+3
𝑥+3
𝑥+4
7
6
9
7
e)
+
f)
+
𝑥+3
𝑥−5
𝑥−5
𝑥+6
8
3
5
9
i)
−
j)
−
𝑥+5
2𝑥+7
𝑥+2
𝑥+5
9
𝑥
4
6
m)
−
n)
−
5𝑥−6
𝑥+1
𝑥−9
3𝑥−1
8
9
5
9
q)
+
q) −
2𝑥+3
𝑥
𝑥
𝑥−4
c)
g)
k)
o)
r)
5
𝑥+4
9
𝑥
+
3
𝑥+1
7
𝑥+2
h)
𝑥+3
−
5
−
+
l)
3𝑥+7
9
p)
𝑥−6
4
s)
𝑥−2
2) Common Factors - Express the following as a single fraction:
8
9
5
9
b)
−
a)
+
(𝑥−2)(𝑥+3)
(𝑥−2)(𝑥+5)
(𝑥+3)(𝑥+4)
(𝑥+3)(𝑥+5)
7
6
9
7
d)
−
e)
+
(𝑥+8)(2𝑥−5)
(2𝑥−5)(𝑥−3)
(𝑥+3)(2𝑥+5)
𝑥(𝑥+3)
7
2
𝑥
5
h)
+ (𝑥+3)(2𝑥+7)
g)
− (𝑥+3)(3𝑥+4)
(2𝑥+7)(𝑥−2)
𝑥+3
11
8
12
5
j)
−
k)
+
2
(𝑥+3)
(𝑥+3)(3𝑥+5)
2𝑥−9
𝑥(2𝑥−9)
2𝑥
3
7𝑥
2
n)
+
m)
−
(𝑥−8)(𝑥+2)
(2𝑥+3)(𝑥+2)
(𝑥−4)(2𝑥+5)
(2𝑥+5)2
34
d)
7
2𝑥−5
𝑥
𝑥−9
8
+
7
𝑥+9
5
𝑥−8
6
𝑥+4
𝑥
+
+
−
2𝑥−7
7
𝑥−11
4
𝑥+2
4
𝑥+2
7
𝑥−4
9
−
−
𝑥−6
8
3𝑥
13
7
+
(𝑥+3)(𝑥+5)
(𝑥+5)(𝑥+6)
5
7
f)
+
(𝑥−2)(2𝑥−3)
(𝑥−2)(𝑥+4)
9
7
i)
−
𝑥(𝑥+5)
𝑥+5
7
9
l)
−
(𝑥−6)(𝑥+2)
(𝑥−6)2
8
9
o)
+ 2
𝑥(𝑥−2)
𝑥
c)
3) Factorise to Spot Common Factors - Express the following as a single fraction:
6
7
11
3
5
6
b) 2
− 2
c) 2
+ 2
a) 2
+ 2
𝑥 +𝑥−12
𝑥 −𝑥−20
𝑥 −7𝑥−18
𝑥 −3𝑥−54
𝑥 +5𝑥+6
𝑥 +9𝑥+14
8
5
9
3
10
3
e) 2
+ (𝑥+3)(𝑥+7)
d) 2
− 2
f) 2
−
𝑥 +3𝑥
𝑥 −49
𝑥 +5𝑥−14
𝑥 −5𝑥+6
𝑥 2 −3𝑥
7
4
7
3
𝑥
3
g) 2
+ 2
h) 2
− 2
i) 2
+ 2
𝑥 −2𝑥−35
𝑥 −25
𝑥 −11𝑥+30
𝑥 −6𝑥
𝑥 −𝑥−12
𝑥 −11𝑥+28
9𝑥
3
13𝑥
2
8
7
j) 2
− 2
k) 2
+ 2
l) 2 + 2
𝑥 −3𝑥−10
𝑥 −4
𝑥 −36
𝑥 +8𝑥+12
𝑥
𝑥 +2𝑥
4) Harder - Express the following as a single fraction:
a)
d)
g)
5
7
+
2𝑥 2 +13𝑥+10
7
3𝑥 2 +19𝑥+20
6
5𝑥 2 +37𝑥+14
2𝑥 2 −5𝑥−12
+
−
4
3𝑥 2 +22𝑥+35
7
5𝑥 2 −13𝑥−6
b)
e)
h)
3
+
3𝑥 2 +10𝑥+8
9
−
2𝑥 2 −3𝑥−14
3𝑥
3𝑥 2 +23𝑥+40
+
2
3𝑥 2 −5𝑥−12
5
2𝑥 2 +7𝑥+6
5
3𝑥 2 +8𝑥−35
5
c)
2𝑥 2 +13𝑥+21
11
f)
i)
3𝑥 2 +7𝑥
−
2𝑥 2 +11𝑥+15
3
+
3𝑥 2 +26𝑥+35
7𝑥
3
−
3𝑥 2 +8𝑥+5
2
2𝑥 2 −9𝑥
Multiplying and Dividing Algebraic Fractions
1) Calculate the following:
𝑎
𝑥
𝑥
𝑢
a) ×
b) ÷
𝑏
𝑦
𝑦
𝑣
𝑥−2
𝑥+3
𝑥+9
𝑥+7
e)
×
f)
÷
𝑥+3
𝑥+5
𝑥+4
𝑥+4
𝑥+2
c)
g)
𝑥+3
𝑥−5
𝑥+2
×
×
𝑥+4
d)
𝑥+5
𝑥+4
h)
𝑥−5
2) Calculate the following, simplifying your answers:
4𝑎
𝑏
8𝑎2
15𝑎
10𝑥𝑦 3
21𝑎2
a)
×
b)
÷
c)
×
𝑏
6𝑐
9𝑏
10𝑏2
7𝑎
5𝑥 2
𝑥+4
𝑥−3
𝑥−3
𝑥−2
𝑥+3
𝑥+5
f)
÷
g)
×
e)
×
2𝑥+10
𝑥+5
2𝑥−4
5𝑥+1
𝑥+4
2𝑥+6
4𝑥+2
4𝑥+3
𝑥+9
2𝑥+18
5𝑥+15
2𝑥+12
i)
×
j)
÷
k)
×
𝑥−7
2𝑥+1
4𝑥+3
4𝑥−3
𝑥+6
𝑥+3
d)
h)
l)
𝑥+5
𝑥+2
𝑥+1
𝑥+4
𝑥+8
÷
𝑥+1
𝑥+1
÷
16𝑎𝑥 3
21𝑏
6𝑥−3
8𝑥
÷
÷
𝑥+9
6𝑥+16
2𝑥+4
𝑥−2
7𝑏2
2𝑥−1
÷
𝑥−5
3𝑥+8
𝑥+5
3) Factorising before Operating - Simplify the following fractions:
a)
d)
g)
j)
(𝑥+2)(𝑥+5)
(𝑥+3)(𝑥−8)
× (𝑥+2)(𝑥+1)
(𝑥+3)(𝑥+7)
𝑥 2 +8𝑥+15
𝑥 2 +5𝑥−14
𝑥 2 +12𝑥+35
𝑥 2 −16
𝑥 2 +2𝑥−24
𝑥 2 −8𝑥+15
m)
𝑥 2 −25
𝑥 2 −49
𝑥 2 −7𝑥
× 2
÷
×
𝑥 +7𝑥+12
𝑥 2 +7𝑥+12
𝑥 2 +3𝑥+2
𝑥 2 +14𝑥+45
𝑥 2 +7𝑥−18
𝑥 2 +7𝑥
÷ 2
𝑥 +5𝑥
b)
e)
h)
k)
n)
(𝑥+5)(𝑥−2)
𝑥(𝑥+4)
𝑥 2 +7𝑥−18
(𝑥+5)(𝑥−5)
𝑥(𝑥+3)
𝑥 2 +3𝑥−10
𝑥 2 +16𝑥+63
𝑥 2 +13𝑥+36
𝑥 2 +11𝑥+28
2𝑥 2 +13𝑥+15
𝑥 2 −5𝑥−14
𝑥 2 +11𝑥+24
𝑥 2 −64
÷
÷ 2
𝑥 +8𝑥+15
𝑥 2 +7𝑥
× 2
𝑥 +𝑥−6
÷
×
35
2𝑥 2 +17𝑥+21
𝑥 2 −49
2𝑥 2 +9𝑥+10
2𝑥 2 +7𝑥+6
c)
f)
i)
l)
o)
(𝑥+9)(𝑥−2)
𝑥(𝑥−6)
𝑥 2 +9𝑥
𝑥 2 +5𝑥+6
𝑥(𝑥+9)
× (𝑥−2)(𝑥+7)
𝑥 2 −5𝑥−14
× 2
𝑥 2 +13𝑥+36
𝑥 2 +7𝑥+12
𝑥 +2𝑥−63
𝑥 2 +9𝑥
÷ 2
3𝑥 2 +13𝑥+21
𝑥 2 −25
3𝑥 2 +14𝑥+16
2𝑥 2 −3𝑥−35
𝑥 +2𝑥
×
÷
𝑥 2 −5𝑥
3𝑥 2 −2𝑥−21
3𝑥 2 −𝑥−24
2𝑥 2 +𝑥−21
Mixed Fraction Simplification
Express the following in a simplified single fraction:
𝑥 2 +7𝑥+10
a) 2
𝑥 +2𝑥−15
𝑥 2 +4𝑥
d) 3
𝑥 +7𝑥 2 +12𝑥
𝑥 3 +4𝑥 2 +3𝑥
𝑥+4
g)
× 2
2
𝑥 +6𝑥+8
2𝑥 +6𝑥
3𝑥
𝑥
j) 2
+ 2
𝑥 −4𝑥+3
𝑥 −5𝑥+4
3𝑥−1
2𝑥+1
m)
+
4
6
𝑎+1
𝑎+1
p) 3
− 2
2
𝑎 −2𝑎 −3𝑎
𝑎 +4𝑎+3
𝑥 2 +2𝑥+1
𝑥 2 +4𝑥+3
s) 2
÷ 2
𝑥 −5𝑥+4
𝑥 −2𝑥−8
2𝑐+1
𝑐𝑑
v)
÷
2
3𝑑
18
b)
e)
𝑐+2
𝑐
+
12𝑡 5
6𝑡+3𝑡
𝑧
𝑥 3 −16𝑥
𝑥+3
c) 2
× 2
𝑥 +5𝑥+6
𝑥 +4𝑥
𝑐+1
2𝑐
3 ÷
18𝑡 2
9𝑡
10
h) 2
+
𝑧 +3𝑧+2
𝑧+1
2𝑥 2 −8𝑥
k) 3
𝑥 −5𝑥 2 +4𝑥
𝑥 2 −3𝑥−4
𝑥 2 +4𝑥+4
n) 2
×
𝑥 +5𝑥+6
𝑥−4
3
2
𝑥 −4𝑥 +4𝑥
2𝑥 2 −4𝑥
q)
÷
𝑥 2 +6𝑥+5
3𝑥+15
𝑥−3
3
t) 2
+
𝑥 +4𝑥
𝑥+4
𝑥
𝑥+2
w) 3
+ 2
𝑥 −9𝑥
𝑥 −5𝑥+6
f)
𝑥 3 −7𝑥 2 +12𝑥
𝑥 2 −2𝑥−8
𝑥 2 −4𝑥+3
𝑥 2 −𝑥−6
i) 2
÷ 2
𝑥 +9𝑥+20
𝑥 +7𝑥+12
1
3𝑧
l)
×
2𝑧−5
𝑧−1
o)
2𝑡 2 +𝑡−45
4𝑡 2 −81
2𝑦−1
𝑦−1
r) 3
−
𝑦 −3𝑦 2 +2𝑦
𝑦 2 −5𝑦+6
𝑥 2 −4𝑥+4
𝑥 3 +3𝑥 2 −10𝑥
u) 3
÷
𝑥 +7𝑥 2 +10𝑥
𝑥 2 −𝑥−6
𝑡 2 −9
𝑡 2 +6𝑡+9
x) 2
÷ 3 2
𝑡 +3𝑡+2
𝑡 +8𝑡 +7𝑡
Solving Algebraic Fraction Equations
1) Cross Multiply – Solve the following equations:
𝑥+3
3𝑥+4
2𝑥−3
2𝑥
a)
=
b)
=
𝑥+5
3𝑥+8
𝑥+5
𝑥+13
𝑥+9
𝑥+2
𝑥+2
𝑥−4
d)
=
e)
=
2𝑥−2
𝑥−1
2𝑥
𝑥−3
𝑥+4
2𝑥+2
𝑥+2
3𝑥+6
g)
=
h)
=
2𝑥−2
2𝑥+5
2𝑥−6
4𝑥
2) Simplify and Solve – Solve the following equations:
2
1
3
5
2
1
a)
+
=
b)
−
=
𝑥+2
𝑥−1
2
𝑥+3
𝑥+1
3
7
2
3
7
1
10
d)
−
=
e)
+
=
𝑥+3
𝑥−1
8
𝑥+3
𝑥−3
9
6
3
1
2
2
4
g)
−
=
h)
+
=
3𝑥−1
𝑥+3
4
𝑥+1
2𝑥−3
5
6
1
1
7
3
1
j)
+
=
k)
−
=
2𝑥+3
𝑥+4
2
3𝑥+2
𝑥+2
8
1
1
1
1
m)
+
=5
n)
−
=2
2𝑥−1
𝑥+3
𝑥+2
𝑥+5
3
2
2𝑥−3
1
1
p)
+
=
q)
=
4−𝑥
1−𝑥
4−5𝑥+𝑥 2
𝑥 2 +2𝑥−1
𝑥 2 −2𝑥+1
36
c)
f)
i)
c)
f)
i)
l)
o)
r)
3𝑥−7
𝑥+2
𝑥−3
𝑥
5
𝑥+6
4
𝑥+2
1
=
=
+
2𝑥+1
9
𝑥+3
3
𝑥+8
2𝑥−6
3𝑥−7
3𝑥+4
2𝑥 2
2
𝑥−1
2
+
−
3𝑥+1
1
2𝑥+2
=
𝑥+3
2
𝑥−1
3
+
𝑥
+
=
𝑥+2
𝑥 2 −3𝑥+5
1
1
𝑥+1
=
−
=
3
5
5
8
=
9
10
1
𝑥 2 +𝑥−3
2
𝑥+2
=0
Additional
Worksheet
with video
solutions
Probability Questions including Algebra
1) A bag contains 10 coloured counters. Two counters are picked at random from the bag. The probability of
2
them both being red is 15. Use algebra to work out how many of the counters in the bag are red.
2) A bag contains 12 coloured marbles. Two counters are picked at random from the bag. The probability of
5
them both being blue is 33. Work out how many of the counters in the bag are blue.
3) A coin purse contains 13 coins. Two coins are picked at random from the bag. The probability of them
5
both being £1 is 26. Use algebra to work out how many of the coins in the purse are not £1 coins.
4) A bag contains 𝑥 blue counters and 4 green counters. Two counters are picked at random from the bag.
1
The probability of them both being green is 11. Work out how many counters are blue.
5) A bag contains 𝑛 red marbles and 3 yellow marbles. Two counters are picked at random from the bag.
5
The probability of them both being red is 14.
a) Show that 3𝑛2 − 13𝑛 − 10 = 0
b) Find how many red marbles were in the bag.
6) A bag contains 𝑥 green counters and 3 blue. Two counters are picked at random from the bag. The
1
probability that you pull out one counter of each colour is 2. Work out the two possible values for the
amount of green counters in the bag originally.
7) There are 𝑛 counters in a bag, 4 of them are blue, and the rest are red. Two counters are picked at random
4
from the bag. The probability that you pull out two counters that are the same colour is 9. Work out the
value for the amount of counters in the bag originally.
8) There are 𝑛 marbles in a bag, 5 of them are red, and the rest are blue. Two marbles are picked at random
5
from the bag. The probability that at least one red marble was picked out of the bag is 6. Work out the value
for the amount of marbles in the bag originally.
9) A bowl contains 𝑛 pieces of fruit. Of these, 4 are oranges and the rest are apples. Two pieces of fruit are
going to be taken at random from the bowl. The probability that the bowl will then contain (𝑛 − 6) apples is
1
. Work out the value of 𝑛. Show your working clearly.
3
10) a) A bag contains (𝑛 + 7) tennis balls: n of the balls are yellow and the other seven are white. John will
take at random a ball from the bag. He will look at its colour and then put it back into the bag.
i) Write an expression in terms of n for the probability that John will take a white ball.
2
ii) Bill states that the probability that John will take a white ball in 5 ? Is Bill correct?
b) After John has put the ball back into the bag, Mary will then take at random a ball from the bag. She will
4
note its colour. Given that the probability that John and Mary will take different colours is 9, prove that
2𝑛2 – 35𝑛 + 98 = 0.
c) Using your answer to part a, calculate the probability that John and Mary will both take white balls.
37
Straight Line Graphs and Parallel Lines
1) For each of the following equations, write down the gradient and y-intercept
1
a) 𝑦 = 3𝑥 + 7
b) 𝑦 = −5𝑥 + 2
c) 𝑦 = 2 𝑥 + 8
d) 𝑦 = 𝑥 + 9
e) 2𝑥 + 𝑦 = 9
f) 3𝑥 − 2𝑦 = 11
g) 𝑥 + 2𝑦 = 12
h) 3𝑥 − 𝑦 = 7
i) 5𝑥 + 2𝑦 = 19
j) 4𝑥 − 5𝑦 = 11
k) −3𝑥 + 𝑦 = 12
l) −2𝑥 + 3𝑦 = 8
2) Intersecting axes - For each of the following equations, find the coordinates where the lines intersect the 𝑥
and 𝑦 axes.
a) 𝑦 = 2𝑥 − 9
b) 𝑦 = 3𝑥 + 5
c) 𝑦 = −2𝑥 + 11
d) 𝑦 = 5𝑥 − 9
e) 2𝑥 + 𝑦 = 7
f) 2𝑥 + 3𝑦 = 11
g) 3𝑥 − 5𝑦 = 14
h) 2𝑥 + 5𝑦 = 12
i) 3𝑥 + 2𝑦 = 14
j) 5𝑥 − 2𝑦 = −19
k) 𝑥 + 3𝑦 = −14
l) 4𝑥 − 2𝑦 = 14
3) Parallel Lines - For each of the following, find the equation of the line which is parallel to the given line
and passes through the given point. Give your answers in the form 𝑦 = 𝑚𝑥 + 𝑐
a) 𝑦 = 2𝑥 + 5
(5,7)
b) 𝑦 = 5𝑥 − 2
(6,3)
c) 𝑦 = −3𝑥 + 9
1
(2, −3)
d) 𝑦 = −4𝑥 + 9
(−3,8)
e) 𝑦 = 3𝑥 + 11
(1, −7)
f) 𝑦 = 2 𝑥 + 5
(5,3)
g) 𝑦 = −2𝑥 + 9
(−4,9)
h) 𝑦 = 5𝑥 − 8
(2, −5)
i) 𝑦 = 3𝑥 − 11
(−5, −5)
4) For each of the following, find the equation of the line which is parallel to the given line and passes
through the given point. Give your answers in the form 𝑦 = 𝑚𝑥 + 𝑐
a) −4𝑥 + 𝑦 = 11
(4, −2)
b) 3𝑥 + 𝑦 = 12
(9, −1)
c) 𝑥 + 2𝑦 = 15
(−3,5)
d) 2𝑥 + 4𝑦 = 13
(2, −9)
e) 3𝑥 − 𝑦 = 14
(0,7)
f) 2𝑥 + 4𝑦 = 13
(8,2)
g) 3𝑥 − 4𝑦 = −4
(6, −1)
h) 5𝑥 − 2𝑦 = −9
(3,8)
i) 3𝑥 + 5𝑦 = 11
(−3,7)
5) Calculating Gradients – Find the equation of the line intersecting the two coordinates:
a) (5,6) & (7, 10)
b) (3,5) & (5, 11)
c) (4,2) & (7, 14)
d) (7, 8) & (3, 0)
e) (9,2) & (5, −10)
f) (4,9) & (1, 3)
g) (−3, 5) & (1, 21) h) (5,2) & (2, 17)
i) (6,4) & (3, 19)
j) (10,4) & (7, 10)
k) (7,2) & (2, −8) l) (6,4) & (8, 10)
m) (−7, 3) & (−3, 15) n) (−1, 2) & (3, −10) o) (4,6) & (2, −2)
6) Calculating Midpoints – Find the midpoint between the two coordinates given:
a) (5,8) & (1, 14)
b) (3,7) & (1,19)
c) (6,1) & (−2, 9)
d) (−4, 5) & (10, −3) e) (−5, 2) & (9,1 )
f) (−4, 9) & (−1, 4) g) (−5,9) & (0,6)
h) (−7,2) & (−11, −3) i) (−9, 2) & (−2, −8) j) (−7, 2) & (10, 9)
7) Distance between two coordinates – Find the distance between the two coordinates given:
a) (7,3) & (9, −3)
b) (6,7) & (10,2)
c) (−3,5) & (12,3)
d) (9, −2) & (4,3)
e) (11,5) & (8,3)
f) (9,2) & (−4,2)
g) (3,9) & (−2,8)
h) (5,8) & (−3,2)
i) (−4,8) & (−2,4)
j) (2,4) & (−9, −5)
Problem Solving
8) 𝐴 is the coordinate (5,9). 𝐵 is the coordinate (𝑑, 15). The gradient of the line is 3. Work out 𝑑.
9) 𝐴 is the coordinate (3,8). 𝐵 is the coordinate (7, 𝑎) The line 𝐴𝐵 is a parallel to 𝑦 = 3𝑥 − 8. Work out the
value of 𝑎.
10) 3) Point C lies on the line segment 𝐴𝐵. Given that 𝐴 (−3,5) and 𝐵 (9,1), find the coordinate 𝐶 such that
𝐴𝐶: 𝐶𝐵 = 3: 1.
38
Perpendicular Lines
1) Find the equation of the line perpendicular to the given line below, intersecting the given point.
a) 𝑦 = 2𝑥 + 5
1
d) 𝑦 = 2 𝑥 − 3
1
g) 𝑦 = 3 𝑥 − 7
1
j) 𝑦 = 2 𝑥 + 3
1
(4,5)
b) 𝑦 = 3 𝑥 − 8
(2,8)
c) 𝑦 = −4𝑥 + 7
(1,5)
(6,2)
e) 𝑦 = 3𝑥 − 7
(−6,3)
f) 𝑦 = −2𝑥 + 11
(4,2)
(2, −9)
h) 𝑦 = 2𝑥 + 5
(8, −2)
i) 𝑦 = 𝑥 + 3
(−3,5)
(5, −2)
l) 𝑦 = 2𝑥 − 9
(−5,2)
(3, −7)
1
k) 𝑦 = 4 𝑥 + 2
2) Find the equation of the line perpendicular to the given line below, intersecting the given point.
a) 2𝑥 − 𝑦 = 8
(3,5)
b) 𝑥 + 3𝑦 = −8
(5, −3)
c) 2𝑥 − 𝑦 = 13
(6,3)
d) 3𝑥 + 2𝑦 = 15
(6,2)
e) 2𝑥 + 3𝑦 = 11
(−5,2)
f) 5𝑥 + 2𝑦 = 9
(5, −3)
g) 3𝑥 + 4𝑦 = −8 (−5,2)
h) −2𝑥 + 5𝑦 = 6
(−2,8)
i) 3𝑥 − 𝑦 = 17
(−7,3)
j) 2𝑥 + 8𝑦 = 13
k) 3𝑥 − 4𝑦 = 11
(−6,4)
l) 2𝑥 + 5𝑦 = 19
(6, −2)
(−7,3)
Problem Solving
4) a) Show that (4,5) intersects the line 𝑦 = 2𝑥 − 3.
b) Find the equation of the perpendicular line to 𝑦 = 2𝑥 − 3 that intersects (4,5).
c) The perpendicular line above intersects the 𝑥-axis at point 𝐴, and the 𝑦-axis at point 𝐵. Find the area of
the triangle 𝐴𝑂𝐵.
5) a) Show that (−3,2) intersects the line 𝑦 = 3𝑥 + 11
b) Find the equation of the perpendicular line to 𝑦 = 3𝑥 + 11 that intersects (−3,2)
c) The perpendicular line above intersects the 𝑥-axis at point 𝐴, and
the 𝑦-axis at point 𝐵. Find the area of the triangle 𝐴𝑂𝐵.
6) 𝐴𝐵𝐶𝐷 is a rectangle. 𝐴, 𝐸 and 𝐵 are points on the straight line 𝑳
with equation 𝑥 + 2𝑦 = 12. 𝐴 and 𝐷 are points on a straight line.
𝐴𝐸 = 𝐸𝐵.
Find an equation for 𝑴.
7) The point 𝑃 has coordinates (2,7).
The point 𝑄 has coordinates (5, 𝑎).
A line perpendicular to 𝑃𝑄 is given by the equation 3𝑥 + 2𝑦 = 7
Find the value of 𝑎.
8) The point 𝑆 has coordinates (5,2).
The point 𝑇 has coordinates (𝑎, 𝑏).
A line perpendicular to 𝑆𝑇 is given by the equation 𝑥 + 2𝑦 = −4
Find an expression for 𝑏 in terms of 𝑎.
9) The straight line 𝑳 has equation 3𝑥 + 2𝑦 = 17. The point 𝐴 has coordinates (0,2). The straight line 𝑴 is
perpendicular to 𝑳 and passes through 𝐴.
Line 𝑳 crosses the 𝑦-axis at the point 𝐵.
Lines 𝑳 and 𝑴 intersect at point 𝐶.
Work out the area of triangle 𝐴𝐵𝐶.
39
Linear Simultaneous Equations
Solve the following simultaneous equations:
a) 2𝑥 + 7𝑦 = 17
5𝑥 + 3𝑦 = −1
b) 7𝑥 + 2𝑦 = 5.5
3𝑥 − 5𝑦 = 17
c) 7𝑥 − 2𝑦 = 34
3𝑥 + 5𝑦 = −3
d) 4𝑥 + 2𝑦 = 9
𝑥 − 4𝑦 = 9
e) 𝑥 + 2𝑦 = −0.5
3𝑥 − 𝑦 = 16
f) 3𝑥 + 2𝑦 = 15
10𝑥 − 4𝑦 = 2
g) 𝑥 + 𝑦 = 15
7𝑥 − 5𝑦 = 3
h) 2𝑥 + 7𝑦 = 31
5𝑥 − 3𝑦 = 16
i) 3𝑥 − 4𝑦 = 8
5𝑥 − 2𝑦 = 11
j) 3𝑥 + 2𝑦 = 5.5
5𝑥 − 3𝑦 = −13
k) 7𝑥 − 2𝑦 = 41
4𝑥 + 3𝑦 = 11
l) 5𝑥 − 2𝑦 = 9.5
4𝑥 + 2𝑦 = 13
Graphical Simultaneous Equations
For the following questions, draw the graphs and find the intersection points of the graphs OR solve the
following simultaneous equations to find the intersection points of the graphs.
a) 𝑦 = 𝑥 2 – 4𝑥 + 8
𝑦 = 2𝑥
b) 𝑦 = 𝑥 2 – 𝑥 – 1
𝑦 = 2 – 3𝑥
c) 𝑦 = 𝑥 2 – 4𝑥 – 28
𝑦 = 3𝑥 + 2
d) 𝑦 = 𝑥 2 + 3𝑥 − 2
𝑦 = 3 − 𝑥
e) 𝑦 = 𝑥 2 – 4𝑥 + 2
𝑦 = 2𝑥 – 6
f) 𝑦 = 𝑥 2 – 2𝑥 – 3
𝑦 = 3𝑥 + 11
g) 𝑦 = 𝑥 2 – 𝑥 − 5
𝑦 = 2𝑥 + 5
h) 𝑦 = 𝑥 2 – 4𝑥 + 8
𝑦 = 𝑥 + 4
i) 𝑦 = 2𝑥 2 + 𝑥 – 2
𝑦 = 8𝑥 – 5
j) 𝑦 = 2𝑥 2 + 9𝑥 + 30
𝑦 = 9 – 8𝑥
k) 𝑦 = 4𝑥 2 – 5𝑥 + 2
𝑦 = 2𝑥 – 1
l) 𝑦 = 7𝑥 2 – 12𝑥 – 1
𝑦 = 8𝑥 + 2
Quadratic Simultaneous Equations
1) Solve the following simultaneous equations:
a) 𝑥 2 + 𝑦 2 = 34
𝑥 + 2𝑦 = 13
b) 𝑥 2 + 𝑦 2 = 53
2𝑥 + 𝑦 = 11
c) 𝑥 2 + 𝑦 2 = 17
3𝑥 + 𝑦 = 7
d) 𝑥 2 + 𝑦 2 = 25
𝑥 + 4𝑦 = 19
e) 𝑥 2 + 2𝑦 2 = 22
2𝑥 + 𝑦 = 7
f) 2𝑥 2 + 𝑦 2 = 59
𝑥 + 3𝑦 = 14
g) 𝑥 2 + 2𝑦 2 = 57
𝑥−𝑦 =5
h) 2𝑥 2 + 𝑦 2 = 57
2𝑥 − 𝑦 = 3
i) 2𝑥 2 − 𝑦 2 = 28
𝑥 + 2𝑦 = 28
j) 𝑥 2 − 2𝑦 2 = 23
𝑥 + 3𝑦 = 32
k) 2𝑥 2 − 𝑦 2 = 23
2𝑥 + 𝑦 = 19
l) 𝑥 2 − 3𝑦 2 = 6
2𝑥 + 𝑦 = 23
m) 𝑥 2 + 𝑥𝑦 = 63
𝑥 − 3𝑦 = 1
n) 𝑥 2 + 2𝑥𝑦 = 33
3𝑥 − 𝑦 = 5
o) 3𝑥𝑦 + 𝑦 2 = 16
𝑥 + 2𝑦 = 7
p) 2𝑥 2 + 𝑥𝑦 = 60
2𝑥 + 𝑦 = 12
q) 𝑥 2 + 3𝑥𝑦 − 𝑥 = 24
𝑥 − 2𝑦 = −1
r) 3𝑥 2 − 𝑥𝑦 = 28
3𝑥 + 𝑦 = 17
s) 2𝑥𝑦 − 𝑦 2 + 𝑥 = 8
𝑥 + 2𝑦 = 5
t) 2𝑥 2 + 3𝑥𝑦 = 36
3𝑥 − 𝑦 = 7
2) Extra Hard – Solve the following simultaneous equations:
a) 𝑥 2 + 𝑦 2 = 29
2𝑥 + 3𝑦 = 16
b) 2𝑥 2 + 𝑦 2 = 36
3𝑥 + 2𝑦 = 17
c) 𝑥 2 + 2𝑥𝑦 = 9
2𝑥 + 5𝑦 = 22
d) 𝑥 2 + 2𝑦 2 = 22
4𝑥 + 3𝑦 = 17
e) 𝑥 2 + 𝑦 2 = 73
3𝑥 − 2𝑦 = 18
f) 𝑥 2 − 𝑦 2 = 24
5𝑥 + 2𝑦 = 45
g) 𝑥 2 − 2𝑦 2 = 9
3𝑥 + 2𝑦 = 39
h) 𝑥 2 − 4𝑦 2 = 9
3𝑥 + 4𝑦 = 7
3) Eddie thinks of two numbers. The difference between the numbers is 4, and the sum of the squares of the
two numbers is 58. Find the two possible pairs of numbers that Eddie could have been thinking of. Check
your answers.
40
Additional
Worksheet
with video
solutions
Equation of a Circle
1) a) Draw the graph with equation 𝑥 2 + 𝑦 2 = 25
b) Show that the coordinate (3,4) is a coordinate on
the circle.
c) Show that the coordinate (−3,4) is also a
coordinate on the circle.
2) a) Draw the graph with equation
𝑥 2 + 𝑦 2 = 100.
b) Draw the graph with equation 𝑥 = 6.
c) Use your graph to show that the simultaneous
equations 𝑥 2 + 𝑦 2 = 100 and 𝑥 = 6 will have two
solutions, and find the coordinates of these
solutions.
2
2
3) Consider the graph of 𝑥 + 𝑦 = 45.
a) Show algebraically that the coordinates
(6,3), and (−3, −6) are coordinates on the circle.
b) Suggest 6 other coordinates that will intersect the
circle.
c) Use your calculator to work out the radius of the
circle to 2 d.p.
d) By plotting these 12 coordinates first, graph the
equation of the circle.
4) Use a similar technique as above to draw the
graph 𝑥 2 + 𝑦 2 = 53
5) Consider the graph of 𝑥 2 + 𝑦 2 = 25. The graph
is moved 4 units to the right. Draw the graph and
label all of the 𝑥 and 𝑦 axes intersections.
6) Consider the graph of 𝑥 2 + 𝑦 2 = 25. The graph
is moved 2 units down. Draw the graph and label
all of the 𝑥 and 𝑦 axes intersections to 2 d.p.
7) Consider the graph of 𝑥 2 + 𝑦 2 = 20. The graph
is moved 3 units to the left. Draw the graph and
label all of the 𝑥 and 𝑦 axes intersections. Write all
of your answers to 2 d.p..
10) a) Draw the graph of 𝑥 2 + 𝑦 2 = 40.
b) Show that the coordinate (2,6) is on the circle.
c) Draw a tangent to the circle at the point (2,6).
d) Calculate the equation of the tangent.
11) a) Draw the graph of 𝑥 2 + 𝑦 2 = 17.
b) Show that the coordinate (1,4) is on the circle.
c) Draw a tangent to the circle at the point (1,4)
d) Calculate the equation of the tangent.
12) Find the equation of the tangent to the circle
𝑥 2 + 𝑦 2 = 45 at the point (6,3).
13) Find the equation of the tangent to the circle
𝑥 2 + 𝑦 2 = 10 at the point (-1,3).
14) Find the equation of the tangent to the circle
𝑥 2 + 𝑦 2 = 68 at the point (6, -2).
15) a) Find the equation of the tangent to the circle
𝑥 2 + 𝑦 2 = 80 at the point (-8, -4).
b) Find the coordinates of intersection between the
tangent and the 𝑥 and 𝑦 axes.
16) a) Find the equation of the tangent to the circle
𝑥 2 + 𝑦 2 = 25 at the point (-3, -4).
b) Find the coordinates of intersection between the
tangent and the 𝑥 and 𝑦 axes.
17) a) Draw the curve with equation 𝑥 2 + 𝑦 2 = 40
that is translated right 3 units.
b) Find the equation of the tangent to the circle
drawn above at the point (1,6)
1
18) Prove algebraically that 𝑦 = 2 𝑥 − 5 is a
tangent to the circle 𝑥 2 + 𝑦 2 = 20.
19) Prove algebraically that 𝑦 = 3𝑥 − 20 is a
tangent to the circle 𝑥 2 + 𝑦 2 = 40.
8) Consider the graph of 𝑥 2 + 𝑦 2 = 40. The graph
is moved 5 units upwards. Draw the graph and label
all of the 𝑥 and 𝑦 axes intersections. Write all of
your answer to 2 d.p..
20) Prove algebraically that 2𝑥 + 𝑦 + 15 = 0 is a
tangent to the circle 𝑥 2 + 𝑦 2 = 45.
9) a) Draw the graph of 𝑥 2 + 𝑦 2 = 20
b) Show that the coordinate (2,4) is on the circle.
c) Draw a tangent to the circle at the point (2,4)
d) Calculate the equation of the tangent.
22) Prove algebraically that 4𝑥 + 𝑦 = 34 is a
tangent to the circle 𝑥 2 + 𝑦 2 = 68.
21) Prove algebraically that 2𝑥 − 3𝑦 = 26 is a
tangent to the circle 𝑥 2 + 𝑦 2 = 52.
41
Sine and Cosine Rule
1) Using the sine rule, calculate the missing sides marked with an 𝑥. Answers to 1 d.p.
a)
b)
c)
d)
e)
5
𝑥
11
𝑥
6
78𝑜
38𝑜
32𝑜
5
𝑥
𝑜
106
132𝑜
42𝑜
𝑥
73𝑜
𝑥
29𝑜
56
12
𝑜
16
𝑜
2) Using the sine rule, calculate the missing angles marked with an 𝑥. Answers to 1 d.p.
5
a)
b)
c)
d)
e)
11
9
6
𝑥𝑜
32𝑜
𝑥𝑜
5
8
106𝑜
132𝑜
73𝑜
9
𝑥𝑜
𝑥
16
42𝑜
16
𝑥𝑜
𝑜
3) Using the cosine rule, calculate the missing sides marked with an 𝑥. Answers to 1 d.p.
a)
b)
c)
d)
e)
5
𝑥
38
6
78𝑜
𝑥
5
11
𝑥
6
𝑜
132𝑜
73𝑜
𝑥
𝑥
16
9
56𝑜
8
8
4) Using the cosine rule, calculate the missing angles marked with an 𝑥. Answers to 1 d.p.
a)
b)
c)
d)
e)
4
12
9
5
6
10
𝑥
𝑥
8
68
8
𝑥
𝑜
38𝑜
8
5
4
e)
12
𝑥
13
17
10
8
𝑥
21
12
5) Find the missing side or angle using the sine or the cosine rule
a)
b)
c)
d)
56𝑜
𝑥
𝑥
38
19
73𝑜
𝑜
12
𝑥
35𝑜
8
7
f)
g)
h)
8
𝑜
26
18
8
𝑥
𝑥
l)
𝑥
68𝑜
𝑥
8
𝑥
16
m)
n)
9
26𝑜
23
5
32
o)
6
39𝑜
𝑥
8
12
q)
12
𝑥
17
17
𝑜 𝑥
112
7
29
r)
s)
𝑜
12 26
112𝑜
𝑥
35
42
17
25
𝑜
𝑥
𝑥
106𝑜
𝑥
13
5
p)
17
112𝑜 44𝑜
𝑥
38𝑜
17
𝑥
9
j)
7
38𝑜
14
k)
i)
11
7
12
𝑥
𝑥
7
7
12
t)
𝑥 6
39𝑜
5
Sine and Cosine Rule Mixture
Calculate the value of 𝑥 to 3 s.f.
x
x
x
x
x
x
x
x
x
x
x
x
Area of a Triangle
1) Work out the areas of the triangles.
a)
b)
c)
d)
e)
6
5
81𝑜
125𝑜
5
78𝑜
9
g)
16
11 22
8
11
f)
h)
10
51𝑜
𝑜
21
j)
8
73𝑜
18
7
19
4
13
i)
9
39 7
𝑜
11
38𝑜
26𝑜
13
73𝑜
2) Work out the areas of the triangles (you may need to use other triangle rules first).
a)
b)
c)
d)
e)
17
9
14
11
9
67𝑜
f)
36
8
g)
14
15
68𝑜
10
39𝑜
17
11
𝑜
h)
j)
8
41
5
72𝑜
67𝑜
i)
4
14 49𝑜
4
37𝑜
𝑜
67𝑜
12
21
7
15
k)
l)
m)
20
11
6
78𝑜
29
𝑜
n)
8
19
16
o)
127𝑜
9 25𝑜
18
43
15
18
12
Problem Solving with Advanced Trigonometry
1) Town B is 45km due north of town A, Town C is
38km from town A in a direction of 62o. Calculate
the distance of town B from town C.
6) A man prospecting for oil in the desert drives
46km on a bearing of 034o and then 38km on a
bearing of 164o.
a) How far from his base is he now?
b) On what bearing must he drive to return to his
base?
2) Two ships leave port at the same time. Ship A
sails on a bearing of 054o for 50km. Ship B sails on
a bearing of 110o for 75km. What is the distance
between ship A and ship B.
7) An explorer walks 28km on a bearing of 070o
and then she walks 32km on a bearing of 134o.
a) How far from the starting point is she?
b) On what bearing must she walks to return?
3) An adventurer travels from base camp on a
bearing of 50o a distance 1.7km. Another
adventurer travels from the same base camp on a
bearing of 290o a distance 4km. Find the distance
between the two adventurers.
8) A ship sails from a harbour A for 3 miles on a
bearing of 065o. It then sails on a bearing of 125o
until it reaches a landing stage at a place B, which
is due east of A.
a) Calculate the distance AB.
b) Find how far the ship sailed altogether.
4) Two aircraft leave the airport at the same time.
Aircraft A flies on a bearing of 12o for 𝑥 km.
Aircraft B flies on a bearing of 077o for 123km.
The distance between the two aircraft now is
160km. What is the distance 𝑥 that aircraft A has
flown?
9) A ship sails from a harbour X for 12 miles on a
bearing of 194o. It then sails on a bearing of 305o
until it reaches a landing stage at a place Y, which
is due west of X.
a) Calculate the distance XY.
b) Find how far the ship sailed altogether.
5) Two tanks start from base in a desert. Tank A
drives on a bearing of 230o for 12km and tank B
drives on a bearing of 268o for 𝑦 km. The tanks are
now 33km apart. Find the distance, 𝑦, that tank B
has driven.
10) P and Q are two points on a coast. P is due North of Q. A ship is
at the point S. PS = 2.9km. The bearing of the ship from P is 062o.
The bearing of the ship from Q is 036o. Calculate the distance QS to
3 significant figures.
11) The diagram shows a vertical flagpole AB. There
is a platform at the point D on the flagpole. B and C
are points on the horizontal ground. AD = 16.5m. The
angle of elevation of A from C is 69o, and the angle of
elevation of D from C is 59o.
Calculate the height, AB, of the flagpole. Give
your answer correct to 3 significant figures.
A
16.5m
D
10o
59o
B
44
C
12) In the following diagram, AC = 𝑥 cm, AB =
5cm, and BC = 2𝑥 cm. Angle CAB = 60𝑜 .
a) Show that 3𝑥 2 + 5𝑥 − 25 = 0
b) Solve the equation and find the value of x to 3
s.f.
c) D is a point on AC such that angle ADB = 104o.
Calculate the length of BD.
A
5cm
60o
B
16) For the triangle to the left with sides 𝑥, 𝑦 and 5,
the perimeter is 𝑘. Given that 𝑥 = 𝑦 − 1, find the
value of 𝑘.
5
60o
𝑥
𝑦
𝑥
17) The diagram to the right shows quadrilateral
ABCD. Find the area of the quadrilateral.
C
2𝑥
A
9.3cm
13) A surveyor wishes to measure the height of a
church. Measuring the angle of elevation, she finds
that the angle increases from 30o to 35o after
walking 20 metres towards the church. What is the
height of the church?
97o
D
58o
B
47o
11.2 cm
C
18) The diagram to the left show LMNP. Work out
the size of angle MLP to 1 d.p.
M
13.7cm
N
58o
4.3cm
72o
15.6cm
L
14) The roads between three villages A, B and C
are straight Roman road as shown in the diagram. B
is north-east of C. Find the bearing of A from C.
10.2km
P
19) This sketch represents two adjacent fields, A
and B. All measurements are in metres.
a) Find the area of field B.
b) Find the area of field A.
B
A
35
9.4km
15.6km
100
P
120o
8cm
85o
B
50
C
15) Below is the quadrilateral PQRS. Angle SRQ is
acute. Work out the size of and SQR, giving your
answer to 1 d.p.
12 cm
o
60
A
55
70m
Q
9cm
R
Additional
Worksheet
with video
solutions
27o
S
45
3D Trigonometry and Pythagoras
1) The diagram shows a cuboid ABCDEFGH.
AB = 5cm. BC = 7cm. AE = 3cm.
a) Calculate the length of AG. Give your answer to
3 s.f.
b) Calculate the size of the angle between AG and
the plane ABCD. Give your answer to 3 s.f.
4) A pyramid has a horizontal square base ABCD
with sides of length 230m. M is the midpoint of
AC. The vertex, T, is vertically above M. The slant
edges of the pyramid are of length 218m. Calculate
the height, MT, of the pyramid. Give your answer
correct to 3 significant figures.
𝑇
𝐻
𝐺
218m
𝐸
𝐹
𝐷
3cm
𝐷
𝐶
𝐶
7cm
𝐴
5cm
230m
𝑀
𝐵
2) The diagram shows a pyramid with a horizontal
rectangular base PQRS. PQ = 16cm, QR = 10cm,
M is the midpoint of the line PR, the vertex, T, is
vertically above M, MT = 15cm.
Calculate the size of the angle between TP and the
base PQRS to 1 decimal place.
𝐴
𝐵
230m
5) In the diagram to the left. AF = 10cm, AB =
24cm, and BC = 8cm. Calculate:
a) Angle FBA.
b) The angle BE makes with the base.
𝑇
𝐸
15cm
𝑅
𝑆
𝐶
10cm
𝑄
16cm
3) The diagram shows a cube ABCDEFGH. The
sides of a cube are of length 5cm. Calculate the size
of the angle between the diagonal AH and the base
EFGH to 1 decimal place.
𝐴
𝐵
𝐷
𝐷
10cm
𝑀
𝑃
𝐹
8cm
𝐴
𝐵
6) In the diagram to the right, M is the midpoint of
AD, and V is vertically above M. Calculate the
angle VC makes with the base ABCD. Give your
answer to 3 s.f.
𝐵
𝐶
𝑉
5cm
𝐹
7cm
𝐺
𝐶
𝐴
5cm
𝐸
10cm
5cm
18cm
𝐻
𝑀
𝐷
46
7) ABCDE is a square based pyramid. AB = 15cm
and DE = 12cm. Calculate the size of angle DEB
𝐸
to 3 s.f.
11) The diagram to the right shows a cube
ABCDEFGH. M is the midpoint of GH. Find the
size of the angle between the line MA and the plane
ABCD.
𝐸
12cm
𝐻
𝑀
𝐶
𝐹
𝐺
𝐷
𝐵
𝐷
15cm
𝐶
𝐴
8) In the shape to the left, PR = 20cm, MP = 12cm,
Angle PRQ = 30o. Calculate the angle that the line
MR makes with the plane RQLN.
𝑀
12cm
𝑃
𝐴
𝐵
12) The diagram to the right shows a cuboid
ABCDEDGH. M is the midpoint of EH. AB =
16cm, HG = 15cm. It is given that BM makes an
angle of 24o with the base EFGH. Calculate the
height of the cuboid.
20cm
𝐿
16cm
𝐴
𝐵
𝑄
30𝑜
𝑁
𝐷
𝐶
𝑅
9) The diagram to the right shows a square based
pyramid ABCDP. O is the centre of the square
ABCD, and P is vertically above O. In the diagram
PA = 11cm, and angle PBA = 72o. Work out the
height OP to 1 d.p.
𝑃
11cm
𝐷
𝐺
15cm
𝐸
𝐻
𝑀
13) A tripod with vertex T stands on level ground.
The three legs TA, TB and TC are each 60cm long.
The triangle ABC is equilateral with sides 50cm.
The point M is the midpoint of BC, and G lies on
MA such that MG:GA = 1 : 2. You are given that T
lies vertically above G.
Find the angle which the leg TA makes with the
ground
𝑇
𝐶
𝑂
72𝑜
𝐴
𝐹
𝐵
10) The diagram to the left shows a prism. E lies on
AD. Angle EBC = 60o, Angle ECB = 50o, and
ABCD is a rectangle. Work out the length AB.
𝐴
𝐷
𝐸
50𝑜
𝐴
60𝑜
𝐺
𝐵
𝐶
𝑀
12cm
𝐵
47
𝐶
Additional
Worksheet
with video
solutions
Surds
1) Simplify the following:
a) √5 × √2
b) √7 × √3
c)
g) 7√2 × 3√7
h) √6 × 3√6
i)
√15
√5
6√14
d)
j)
2 √7
√32
√8
9√15
e) (√5)
k)
3 √5
2
f) 4√3 × 3√2
12√18
l)
4 √6
3√5×4√8
6√10
2) Simplify the following
a) √12
b) √28
c) √18
d) √50
e) √48
f) √75
g) √20
h) √44
i) √108
j) √98
k) √80
l) √45
3) Simplify the following:
a) √8 + √2
b) √20 − √5
c) √3 + √12
d) √8 − √2
e) √27 + √12
f) √125 − √20
g) √48 + √75
h) √18 + √72
i) √75 − √27
j) √80 − √20
k) √108 − √27
l) √27 − √12
4) Simplify the following:
a) √147 − √108
b) √48 − √27
c) √98 + √8 + √2
d) √99 − √44 − √11
e) 3√2 − √18
f) √175 − 4√7
g) 3√8 + √50
h) 5√5 + √20
i) 2√45 + 3√20
j) 3√32 − 2√18
b) (√2 − 3)(1 + √2)
c) (7 + √3)(1 − √3)
d) √2(√3 − √2)
f) (√2 + 1)(√2 − 1)
g) (2 + √5)(2 − √5)
h) (3 − √3)(3 + √3)
5) Simplify the following:
a) (√3 + 1)(2 + √3)
e) (√3 + 2)
2
6) Rationalise the denominator for the following expressions:
a)
f)
k)
3
√5
3
1+√3
1−√2
√2−3
b)
g)
6
√2
5
2+√2
4−2√2
l)
2−√2
c)
h)
k)
8
2√2
1+√2
1+√3
1+√3
2√2−1
d)
i)
l)
15
3 √5
3−√5
1−√2
3−√5
2√10−1
e)
1+√2
√3
4−√3
j)
5−√3
4−3√2
m)
5√2−2
7) The following diagram (on the right) shows a patio area of 5√2 − √3m2. One of the dimensions of the
patio area is (3 + √2)m.. Calculate the length of the other dimension.
8) The diagram on the left shows an octagon, hexagon and equilateral triangle stacked together. Given that
all the sides of the shapes are 1cm, calculate the height of the combined shape, giving your answer in surds
in the simplest possible form.
48
Surds Problem Solving (Non-Calculator)
1) IGCSE – May 2014 – Paper 3H
A trapezium ABCD has an area of 5√6 cm2. AB = 4cm, BC = √3
cm, DC = kcm.
Calculate the value of k, giving your answer in the form 𝑎√𝑏 − 𝑐
where a, b and c are positive integers.
2) The following shape is a symmetric cross. Calculate the
area, leaving your answer in surd form.
2 + √3
1 + √3
3) The following diagram shows a right angled triangle.
Calculate the missing side, leaving your answer in exact
terms.
3 − √3
8 + √3
12 − 2√3
8 − 3√3
5 + 5√3
4) Dayaanan draws the outline of two houses on centimetre
square paper.
a) Calculate the perimeter of this shape.
b) Calculate the perimeter of a shape 3 houses wide.
c) Write down a rule for calculating the perimeter of a shape n
houses wide.
d) Use your answer to part c, or otherwise, calculate the
perimeter of a shape 120 houses wide.
3
5) The following tripods are complete by multiplying the two numbers in the circles to
make the number on the connecting lines, for example:
Find the missing values in the following tripods:
a)
b)
2+ 2
6) Find the sum of
9+5 3
3+ 2
1
√1+√2
?
27+11 7
2+2 3
1
?
1
+ 2+ 3 + 3+ 4 + ⋯ + 99+ 100
√
2
?
?
?
1
√
8
c)
?
?
6
4
2+ 3
1+ 2
?
12
√
√
√
49
√
-15+7 7
2
?
Additional
Worksheet
with video
solutions
Trigonometry with Standard Results (Non-Calculator)
Calculate the value for 𝑥 in each of these questions:
50
Topics
Content
G
Proportion
H
Solving quadratics
Direct & indirect proportion. Graphs relating to proportion.
Solve a quadratic equation by factorising
Completing the square and using this technique to solve a quadratic equation
Solving using the formula (which must be learnt)
Solving a quadratic using iteration
Test 4 on Topics G & H in Spring Term
I
CFGs and box plots revision + outliers – interpretation & comparisons very important.
Representing data
Histograms - constructing from frequency table & completing incomplete table/histogram
J
Simplifying via factorisation, operations with (+ / - / × / ÷)
Algebraic fractions
Equations with algebraic fractions
K
Revision of y = mx + c. Parallel & perpendicular lines. Midpoint of line segment.
y = mx + c
Distance between two points (2D coords only)
End of year exam based on whole year’s work up to and including Topic K (in Summer Term)
Use revision materials on O: drive!
M
One linear and one quadratic: graphical & algebraic method
Quadratic simultaneous
Equation of circle centre (0, 0); equation of tangent to circle centre (0, 0)
equations, equations of
Intersections of lines with circles (including tangent to a circle)
circles centre (0, 0), solving
Solving graphically x2 + y2 = r2 & y = mx + c or ax + by = c
equations graphically
More advanced solving (inc. non-simultaneous) equations graphically but main focus
appears to be on circles + straight line when solving graphically.
N
Area of triangle, sine rule (with ambiguous case) & cosine rule [Formulae must be learnt
Non-right-angled 2D trig, 3D by students]
Pythagoras’ and trig
Exact trig values (e.g. sin150° = sin30°= ½)
Trig & Pythag in 3D
O&P
Fraction to recurring or terminating decimal by long division
Converting fractions to/from Recurring decimal to fraction
decimals & surds
Use and knowledge of √(a×b)=√a × √b and √(a/b)= √a / √b
Rationalise a denominator
End of year 10!
51
Pages in
booklet
P26 - 27
MathsWatch
Clip Numbers
199
P28 – 31
157, 160, 180,
191, 209
P32 – 33
P34 – 37
P38 – 39
130, 186, 187,
205
71, 73, 74, 76,
210
96, 97, 133, 159,
208
P40 - 41
140, 162, 197,
211
P42 – 47
150, 201, 202,
203, 217, 218
P48 - 50
84, 177, 189,
207
52