aim for a* in gcse maths - part 1
TRANSCRIPT
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GCSE Higher Tier Practice Questions 1
GCSE Higher Tier Practice Questions
Contents:
Topic Page
1 Solving Linear, Quadratic and Simultaneous Equations 2
2 Algebra: Factorising Practice and Fractions 3
3 Changing Subjects of Formulae & Solving Inequalities 4
4 Quadratic Functions: Completing the Square and Sketching 5
5 Forming and Solving Quadratic Equations 5
6 Proportionality orVariation 6
7 Number: Estimating and Factorising 6
8 Constructions, Loci and Bearings 7
9 Vectors 8
10 Probability 9
11 Calculations in Right-Angled Triangles 10
12 Calculations Needing Sine and Cosine Rules
10A
13 Upper and Lower Bounds and Standard Form 11
Answers 12
(C) E J Trickey and M E B George 2004
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GCSE Higher Tier Practice Questions 2
1 Solving Linear, Quadratic and Simultaneous Equations*Always try to CHECK your answers*
Exercise 1.1 Solve:
1. 2954 =+ x 2. 129 = x
3. 44
7 = x 4. 134315 = xx
5. ( ) 1432312 += xx 6. 612
54=
+x
7. 448
5
4
3=+
xx8. 3
10
7
5
4=
xx
Exercise 1.2 Solve the following sets ofsimultaneous equations:
1.
==+
1425
1823
ba
ba2.
==
823
1627
yx
yx
3.
==+
1523
2054
qp
qp4.
=+=+
9737
696
yx
yx
5.
==
2425
1852
ba
ba6.
=+=1536
5211
yx
yx
7.
=+
=
342
13
22
1
yx
yx
Exercise 1.3 Solve:
1. 0652 =+ xx 2. 0652 = xx
3. 02452 =+ xx 4. 024102 =+ xx
5. 024232 = xx 6. 024102 =+ xx
7. 036122 =+ xx 8. 03652 =+ xx
Exercise 1.4 Solve, to 3 sig fig, using the
formulaa
acbbx
2
42 =
1. 01192 =+ xx 2. 0753 2 =+ xx
3. 05172 2 =+ xx 4. 0935 2 =+ xx
5. 11152 2 += xx 6. 7542 = xx
7. 8213 2 = xx
Exercise 1.5 Solve by factorising or by usingthe formula
1. 865 2 =+ xx 2. 6112 2 =+ xx
3. xx 13152 2 =+ 4. yy 952 2 =
5. 2522 =+ yy
Exercise 1.6 Simultaneous equations - one linear and one quadratic:
*Expect to reach a quadratic equation in one letter which will factorise (but you may use the
quadratic formula instead of factorising if you prefer it)
*Checkyour answers in the original equations
1.
=+
=+
26
4
22 yx
yx2.
=+
=
3
113
2xxy
xy3.
=
=+
182
12
2 xyx
yx
4.
=
=+
154
132
22 yx
yx5.
=+
=
8
34
2xxy
xy6.
=
=
4494
232
22 yx
yx
7.
=+
=
182
32
22 yx
yx8.
=
=+
93
24
2 xyx
yx9.
=
=+
124
22
22 yx
yx
(C) E J Trickey and M E B George 2004
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GCSE Higher Tier Practice Questions 3
2 Algebra: Factorising Practice and Fractions
Exercise 2.1 Factorise fully:
1. xx 52 + 2. yy 32 2 +
3. 2915 xx 4. 22 xyyx +
5. 223 52 baba + 6. cbabc 239
7. 2452 + xx 8. 24142 + nn
9. 2422 aa 10. 48282 2 + nn
11. 1582 ++ xx 12. 45243 2 ++ xx
13. 42 x 14. 10025 2 x
15. 2142
bb 16. 1052052
bb
Exercise 2.2 Factorise fully:
1. ycybxcxb +++
2. bybxhyhx +
3. bnanbmam + 236
4. byaybxax +++ 326
5. qspsqrpr + 428
6. 352 2 ++ xx 7. 372 2 ++ xx
8. 253 2 xx 9. 152 2 xx
10. 28173 2 xx 11. 5412 2 + xx
12. 31916 2 ++ xx 13. 376 2 + yy
14. 614122 + yy 15. 6113 2 + xx
16. 494 2 m 17. 236 p
Without a calculator, evaluate:
18. 22 15.085.9
19. 22 2.78.12
20. 22 2.338.66
21. 22 5.65.65.325.3 ++
22. 22 2.02.08.928.9 ++
Exercise 2.3 Simplify:
1.3
22
15
12
mn
nm2.
2
4
3
x
xy
3.
x
yxxy 2+
4.x
x
6
189 +5.
x
xx
4
884 2++
6.x
xyx
2
28 +7.
x
xxy
9
36 +
8.xx
xx
3
22
2
+
9.32
32
2
xx
xx
10. xx
xx
102
42
2
+
11. 2
562
2
++
xx
xx
12.145
2142
2
xx
xx13.
4
1072
2
++
x
xx
14.14
22
2
+
x
xx15.
xx
xx
10
90192
2
+
Exercise 2.4 Express as a single fraction,fully simplified:
1. 22 3
2
4
3
pqqp 2.
22 5
2
3
5
ba+
3.32 8
5
4
32
aaa+ 4.
b
a
a
b
4
3
2
5
5.3
3
2
5
++
xx6.
2
3
1
7
+
+ xx
7.1
5
3
2
+ xx8.
1
4
2
3
+
xx
9.12
5
4
62
xx
x
x10.
9
4
3
22
++ x
x
x
Exercise 2.5 Solve andcheckyour solutions(all quadratics reached correctly do factorise)
1. 44
3
1
2=
xx2.
2
1
52
32
=++
m
m
m
m
3. 23
= aa
4.2
1
3
12=
+
aa
(C) E J Trickey and M E B George 2004
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GCSE Higher Tier Practice Questions 4
5.13
2
7
4
=
a
a
(C) E J Trickey and M E B George 2004
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GCSE Higher Tier Practice Questions 5
3 Changing Subjects of Formulae & Solving Inequalities
Exercise 3.1 Solve:1. In each case, makexthe subject of the
formula:
(a) cbax =+ (b) rq
p
x=
(c) mnxy = (d) tm
xl =
(e) )()( dxcbxa +=+ (f) sn
ax=
(g) cxnbax = (h) 0=++ cbyax
2. In each case, makexthe subject of the
formula:
(a) 3=+
+dcx
bax
(b) 2=
bx
axn
(c) 222 bxa =+ (d) nmax =2
(e) qp
x4
2
=
(f) ( ) 222 nlax +=
(g)g
xT 2= (h) cbax =+
(i)zyx
111+= (j) m
y
x=
4
3. rhrA 222 += Findh, ifA = 704
and r= 7 (give your answer to 2 sig. fig.)
4.3
3
4rV = Findr, ifV= 17.2
(give your answer to 2 sig. fig.)
5. 222 cbaA ++= Finda, ifA = 19.10,
b = 11 and c = 12 (give answer to 2 sig. fig.)
6.g
lT 2=
T= 6.95 and l= 12. Findg to 2 sig. fig.
7.a
c
dx
bx=
++
Findx, ifa = 7, b = 3, c = 6
and d= 5
8. In each case, make
Ror
rthe subject of
the formula:
(a) hrV2
3
1= (b) trs
121
+=
(c)3
3
4rV = (d) 24 rA =
(e) 222 42 bar += (f)100
NRAR +=
Exercise 3.2 Solve the following and showyour answer clearly on a number line:
1. 132
10 x
5. x42216 < 6. 310
11 x
7. 17234
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GCSE Higher Tier Practice Questions 6
4 Quadratic Functions: Completing
the Square and Sketching
1. 136)( 2 + xxxf
(a) Re-writef(x) in the form ( ) baxxf + 2)(
(b) Hence state the minimum value off(x) and thevalue ofx for which this minimum occurs.
(c) Sketch the curvey=f(x)
2. 74)( 2 + xxxf
(a) Re-writef(x) in the form ( ) dcxxf + 2)((b) Hence state the minimum value off(x) and the
value ofx for which this minimum occurs.
(c) Sketch the curvey=f(x)
3. 32)( 2 ++ xxxf
(a) Re-writef(x) in the form ( ) fexxf ++ 2)((b) Hence state the minimum value off(x) and the
value ofx for which this minimum occurs.
(c) Sketch the curvey=f(x)
4.2811)( xxxf
(a) Re-writef(x) in the form ( ) 2)( BxAxf +(b) Hence state the maximum value off(x) and the
value ofx for which this maximum occurs.
(c) Sketch the graph ofy=f(x)
5. 210)( xxxf (a)Re-writef(x) in the form ( ) 2)( xDCxf (b) Hence state the maximum value off(x) and the
value ofx for which this maximum occurs.
(c) Sketch the graph ofy=f(x)
6.248)( xxxf
(a) Re-writef(x) in the form ( ) 2)( bxaxf +(b) Hence state the maximum value off(x) and the
value ofx for which this maximum occurs.
(c) Sketch the graph ofy=f(x)
7. xxxf 6)( 2 (a) Sketchy=f(x)
(b) By consideringf(x) in the form
( ) baxxf 2)( , or otherwise, state thehorizontal and vertical transformations that would
map xxy 62 = onto the graph of 2xy =
5 Forming and Solving Quadratic
Equations
1. Show that the
information given
about the lengths in
this right-angled triangle satisfies the equation:0342 =+ xx . Hence findx.(Two solutions)
2.All measurements are in centimetres
Both of these
quadrilaterals
are rectangles
and the inner
one is cut out
of the outer
one. The REMAINING white area is 162cm2.
Show that this information can give the equation
165183 2 =+ xx and solve forx.
3. All measurements in cm
The shaded
right-angled
triangle is
removed from
the rectangle.
The remaining
area is 90 cm2. Show clearly that this gives the
equation 9022 2 =+xx and solve it.
4. Not drawn to scale.
The curve is 822 = xxy(a) State the coordinates of D.
(b) By completing the square
in the form baxy = 2)( orotherwise, give the coordinates
of M, the minimum point of the graph.
(c) By factorising 822 = xxy , find thecoordinates of A and B.
(d) Find the equation of the line CB and thecoordinates of C.
5. 16)( 2 + xxxf(a)Rearrangef(x) in the form dcxxf 2)()((b)Hence solve, in surd form, 0162 =+ xx(c)Now solve 0162 =+ xx using the formula
a
acbbx
2
42 = leaving your answers in
fully simplified surd form. Show that these two
answers are identical to your answers from (b).
(C) E J Trickey and M E B George 2004
x
y
B
C (0,4)
D M
A
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6 Proportionality orVariation
Exercise 6.11.p varies directly as q.
Fill in the following table:
p 6 8 14 20
q 0.25 25 35 40 200
2. The area of a circle is proportional to the
square of the radius.
What happens if:
(a) the radius is doubled?
(b) the radius is trebled?
(c) the radius is made 10 times as big?
3. m varies as the square root ofa. When a = 4,
m = 13.6
(a) Find the equation linking m and a.
(b) Find: m when a = 16a when m = 34
m when a = 9
4. No calculators in this question!
Pis propotional to h2. When h = 15,P= 45
(a) Find Pwhen h = 20
and when h = 100
(b) Find h whenP= 180
and whenP= 0.2
5. It is known thaty is proportional tox.
Fill in the following table:
x 10 20
y 16 54 128
also find: (a) y whenx = 50
(b) x wheny = 6.75
6. If a stone is dropped from the top of a building,
the time tit takes to reach the ground is
proportional to the square root of the height h of
the building.
A stone was timed and took 4.5 seconds to drop
100 feet.
If the stone only dropped 50 feet, how long would
it take to reach the ground?
Exercise 6.21.tis inversely proportional to v.
Complete the following table:
v 10 20 50 100
t 5 4 2.5
2. It is known thaty is inversely proportional to
x2. Fill in the following table:
x 0.1 0.2
y 50 12.5 8 2
3. No calculators in this question!
His inversely proportional to the square root ofT.
When T= 36,H= 2.5
(a) Find: Hwhen T= 100
and when T= 0.01
(b) Find: TwhenH= 3
and whenH= 0.3
4. The resistanceR in a fixed length of wire
varies inversely as the square of the diameterdof
the wire. If the diameter is 5mm, the resistance is
0.04 ohms. Find the resistance if the diameter is
4mm.
5. (Give your answers to 3 sig. fig.)
The air pressure available from a bicycle pump is
inversely proportional to the square of the
diameter of the pump. If a pressure of 16 units is
available from a diameter of 25mm,
(a) Find the pressure available from a pump with
a diameter of 12mm.
(b) Find the diameter of a pump whose pressure
is 20 units.
7 Number: Estimating and Factorising(NO calculators to be used!)
1. Given that 28000 x 0.0032 = 89.6
Find: (a) 28 x 320 (b)00032.0
896.0
(c)000280
8960
2. hrV2
3
1=
EstimateVwhen r= 19.8 and h = 10.1(give your answer to 1 sig. fig.)
3.g
lT 2= Estimate Twheng= 9.8
and l= 155. (try 2sf for this estimate)
4. Express the following numbers as products of
their prime factors: {e.g. 12 = 22 x 3}
(a) 192 (b) 264 (c) 168
Hence find their highest common factor and their
lowest common multiple infactor form.
5. Find an estimate to 1 sig. fig. for:
08.249.17
8.2698.33.189
++
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8 Constructions, Loci and BearingsIn questions 1, 2 and 6 use only a pencil, a ruler and a pair of compasses on PLAIN paper. Do NOT erase
any construction lines or arcs.
1. Construct the sketched triangle accurately:
AB = 8cm
angle B = 90
angle A = 30
Measure AC to the nearest mm.
2. (a) Construct
rectangle ABCD.
(b) Construct the
locus of all points
equidistant fromAD and DC.
(c) Construct the locus of all points 6cm from A,
inside ABCD.
(d) Shade the region inside ABCD where the
points are nearer DC than AD but less than 6cm
from A.
(e) Calculate the area of this shaded region, to 3
sig. fig.
3. {not drawn to scale CALCULATE your answers!}
A ship sails
from port P on a
bearing of 062 to Q.
The distance PQ is 11km.
At Q the ship changes direction
and sails to Ron a bearing of 175.
QR= 19km.
Find the direct distance PRand the bearing ofR
from P, giving your answers to 3 sig. fig.
4. not to scale
Bs bearing from A is 076.
Bs bearing from C is 340.
AB = BC.
Find the bearing ofA fromC.
5.
The bearing ofA from B is 122.
The bearing ofC from B is 231.
BA = 8cm CA = 14cm.
Find the bearing ofC from A and the distance BC
(each to 3 sig. fig.)
6.
A B
Draw a line AB 8cm long and construct
accurately the locus of all points 4cm from any
point on AB.
Find the area within this locus to the nearest cm2.
7.
(a) Evaluate the area of the quadrilateral ABCD.
(b) Find the lengths ofAB, BC, CD and AD to 3
sig. fig.
(c) Calculate the angle A to 3 sig. fig.
2
4
6
8
2 4 6 8 10 120 x
y
A
B
C
D
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9 Vectors
Exercise 9.11.
(a) Write a, b, c, p, q, r, s, and t each as column
vectors.(b) Give, in simplified surd form,
|a| |b| |c| |q| |t|
(c) As you can see, p = 2a
Write q, r, s, and t each as a multiple
ofa orb orc.
2. {Tip: When you have a vector question in an
exam or test, NEVER attempt to solve it without a
DIAGRAM even if there is no grid drawn for
you!}
O is the origin and Sis the point (0,3).
=
2
6SP . Tis the midpoint ofSP.
(a) Find, as a column vector, TO
(b) Find OT
(c)R is on the liney = 0. If OR = OT , find
the possible coordinates ofR.
3.In each set of vectors below, find the one
vector that is notparallel to all the othervectors in its row:
(a)
5
2,
10
4,
25
4,
2
12
1
,
2
15
1
(b)
3
2,
6
4,
30
20,
9
4,
12
8
4. Given that in each case p and q AREparallel, find k:
(a) p = 6a + 18b q = ka + 6b
(b) p = 12a + kb q = 9a 21b
Exercise 9.21. O is the centre of a regular hexagon,
ABCDEF.
x=ABn=BC
Find, in terms ofxor
n or both, the following vectors:
AC , AD , CD , FA , FB , AE
2. This diagram shows a rough sketch of 2
quadrilaterals,ABEFandBCDE.
(a)Ifa = 2b, what can you conclude aboutA,B
and C?
(b)Ifa = b = e = d, what type of figure is
ABCDEF?
(c)Ifg= 2c, what type of figure isEBCD?
(d) Ifd+ c = e +g, name the four points that
are vertices of a parallelogram.
3.ABCD is a parallelogram.
s3=ABt3=AD
(a) Find, in terms ofs ortor both:
(i)BD (ii) BP (iii) BQ (iv) AQ (v)QC
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(b) Show clearly, giving reasons, thatAPCQ is
a parallelogram.
10 ProbabilityIf you are asked to find any probability, remember that the answer:
1Pr0..
obabilityeiNEGATIVEBECANNOT
ONEEXCEEDCANNOT
1. Three coins are tossed at the same time.
List all the possible outcomes. Find the
probability of obtaining:
(a) 3 Heads (b) 2 Heads and one Tail
(c) no Heads (d) at least one Head.
2. Cards with the numbers 2 to 101 are placed in
a hat. Find the probability of selecting:
(a) an even number
(b) a number less than 14
(c) a square number(d) a prime number less than 40
(e) a prime number greater than 90.
3. Two dice, one red and one blue, are thrown
simultaneously. Show all the possible outcomes
on a sample space (probability space). Find the
probability of obtaining:
(a) a total of 10
(b) a total of 12
(c) a total of less than 6
(d) the same number on both dice
(e) a total more than 9.(f) Which is the most likely total?
4.In each of the following, state whether the
events A and B are mutually exclusive:
(a) Two coins are tossed
(i)
TailHeadBEvent
HeadsAEvent
1,1:
2:
(ii)
HeadleastatBEvent
HeadsAEvent
1:
2:
(b) A card is drawn from a pack
aceanBEvent
spadeaAEvent
:
:
(c) A counter is drawn from a box of red, blue
and white counters
counterblueanotBEvent
counterredaAEvent
:
:
5. In a European car factory, 85% of the cars
manufactured are left-hand drive, the rest
(obviously) are right-hand drive.
The probability that a car needs its steering
adjusted before leaving the factory is 0.2. The
percentage of cars that are left-hand drive and do
NOT need adjustment to their steering is 75.
Show that steering adjustment and right / left
hand drive are NOT independent.
6. What is the probability of selecting a King
from a full pack of cards?What is the probability of selecting a red card
from a full pack?
Show that the event selecting a King and the
event selecting a red card are independent.
7. A bag contains 6 red marbles and 4 blue
marbles. A marble is drawn at random and not
replaced. Two further draws are made, again
without replacement.
Find the probability of drawing:
(a) 3 red marbles
(b) 3 blue marbles(c) no red marbles
(d) at least one red marble.
8. Sally goes to school in the mornings. The
probability that her alarm clock works is 0.9. If
her alarm clock has worked the probability that
she is on time for registration is 0.85. If her alarm
clock has failed the probability that she is late for
registration is 0.95.
Draw a tree diagram to help you work out the
probability that:
(a) Her alarm clock worked and she is on time
(b) She is on time, regardless of her clock.
9. A box containsxmilk chocolates andy plain
chocolates. Two are selected one after the other,
WITHOUT replacement.
Find, in terms ofxandy, the probability of
choosing:
(a) a milk chocolate on the first choice
(b) milk chocolates on both the first and second
choices
(c) one of each sort of chocolate(d) two plain chocolates.
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11 Calculations in Right-Angled Triangles
Exercise 11.1 Find each angle x to 3 sig.fig.1. 2.
3. 4.
5. 6.
7.
Exercise 11.2 Find each sidexto 3 sig.fig.1. 2.
3. 4.
5. 6.
7.
12 Calculations Needing Sine and Cosine Rules
Remember:SinC
c
SinB
b
SinA
a== and
bc
acbCosACosAbccba
22
222222 +=+= or
In each case, find the side lettered x, or the angle markedx, to 3 sig. fig. (Triangles NOT drawn to scale)
1. 2. 3.
4. {two possible answers!} 5. 6.
7. 8. 9.
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ANSWERS
1 Solving EquationsExercise 1.11 5=x 2 4=x3 12=x 4 4=x
5 2=x 6 4=x7 32=x 8 30=x
Exercise 1.2
1
==
3
4
b
a2
==
1
2
y
x
3
==
0
5
q
p4
==
9
10
y
x
5
==
2
4
b
a6
==
3
1
y
x
7=
=14
9
y
x
Exercise 1.3
1 32 orx =2 16 = orx3 83 = orx4 64 orx =5 241 orx =6 122 = orx
7 6=x (repeated)8 94 = orx
Exercise 1.4
1 46.154.7 orx =2 57.2907.0 = orx3 305.019.8 orx =4 67.107.1 = orx5 673.017.8 = orx6 89.69.10 = orx
7 566.007.7 = orxExercise 1.5
1 8.02 orx =
22
16 orx =
32
115 orx =
4 52
1ory =
5 221 = ory
Exercise 1.6
1
==
==
5
1
1
5
y
xor
y
x
2
==
==
5
1
1
5
y
xor
y
x
3=
==
=5
1
1
5
y
xor
y
x
4
==
==
5
1
1
5
y
xor
y
x
5
==
==
5
1
1
5
y
xor
y
x
6
==
==
5
1
1
5
y
xor
y
x
7=
==
=5
11
5yxor
yx
8
==
==
5
1
1
5
y
xor
y
x
9
==
==
5
1
1
5
y
xor
y
x
2 Algebra Practice
Exercise 2.11 )5( +xx 2 )32( +yy3 )35(3 xx 4 )( yxxy +5 )52(2 baba +6 )3(3 babc 7 )3)(8( + xx8 )2)(12( nn9 )6)(4( + aa10 )2)(12(2 nn11 )5)(3( ++ xx12 )5)(3(3 ++ xx13 )2)(2( + xx14 )2)(2(25 + xx15 )7)(3( + bb16 )7)(3(5 + bb
Exercise 2.2
1 ))(( cbyx ++2 ))(( bhyx
3 )2)(3( banm 4 )2)(3( yxba ++5 )4)(2( qpsr
6 )1)(32( ++ xx7 )3)(12( ++ xx8 )2)(13( + xx9 )3)(52( + xx10 )7)(43( + xx
11 )12)(56( + xx12 )1)(316( ++ xx13 )13)(32( + yy14 )13)(32(2 + yy15 )3)(23( xx16 )72)(72( + mm17 )6)(6( pp +18 97 19 112
20 3360 21 100
22 100
Exercise 2.3
1n
m
5
42
x
y
4
3
3 xyy + 4x
x
2
63+
5x
xx 2221 ++6 y+4
73
12 +y8
3
2
+
x
x
91+x
x10
)5(24+
xx
112
5
+
x
x12
2
3
++
x
x
132
5
+
x
x14
12 xx
15x
x 9
Exercise 2.4
1 2212
89
qp
pq
222
22
15
625
ba
ab +
33
2
8
5616
a
aa +
4ab
ab
4
310 22
5 )3)(2(
98
++xx
x
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7/31/2019 Aim for A* in GCSE Maths - Part 1
15/17
GCSE Higher Tier Practice Questions 15
6)2)(1(
114
+++xx
x
7)1)(3(
173
+xx
x
8
)1)(2(
11
+
+
xx
x
912
182
+xx
x
10)3)(3(
)1(6
+xx
x
Exercise 2.5
14
313 orx =
2 4
1
=m3 13 = ora4 34 += ora
53
25 = ora
3 Changing Subjects &
InequalitiesExercise 3.11
(a)a
bcx = (b) )( qrpx +=
(c)n
myx
= (d) )( tlmx =
(e)ca
abcdx
= (f) ansx +=
(g)ca
bnx
++
= (h)a
bycx
=
Remember:
ac
cdab
ca
abcd
2
(a)ca
bdx
3
3
= (b)ab
nx
+=
2
(c) 22 abx =
(d)a
nmx
=
(e) qpx 2=
(f)a
nlx
22 +=
(g) xgT
=
2
2
(h)a
bcx
=
2
(i) yz
yzx
+=
(j)16
2ymx =
3 )2(0.9 sfh =4 )2(6.1 sfr=5 )2(10 sfa =6 )2(8.9 sfg= 7 9=x
8
(a)h
Vr
3= (b)
st
str
=
2
(c) 34
3
Vr= (d)
4
Ar =
(e)2
4 22 bar
+=
(f)N
AR
=
100
100
Exercise 3.2
1 6
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GCSE Higher Tier Practice Questions 16
5 (a) ( ) 2525)( xxf (b) max is 25 whenx = 5
(c)
6 (a) ( ) 224)( + xxf(b) max is -4 whenx = -2
(c)
7
( ) 936 22 xxx , so for
this curve to map onto2xy = ,
it needs to move
9
3or 3 to
the left and 9 up.
5 Quadratic Equations1By Pythagoras:
( ) 222 )14()2)4( +=++ xxx18164416 222 ++=+++ xxxxx
0342 =+ xx 0)3)(1( = xx
==
3
1
xor
x
2
162)2)(3()3)(14( =++ xxxx16)65()3134( 22 =+++ xxxx
1623183 2 =+ xx 0165183 2 =+ xx
0)556(3
2
=+ xx 0)5)(11(3 =+ xx511 = xx
3
90)12)(2(2
1)4)(6( += xxxx
90224 22 += xxx 9022 2 =+xx
09022 2 =+ xx0)2)(4522( =+ xx
2, =xsonegativebecannotx4 (a) D is at (0,-8), (b)
9)1( 2 = xyso M is at (1,-9)
(c)A is at(-2,0) and B is at (4,0)
(d) CB has equation
4+= xy and C is at (-3,7)5 (a)
8)3(16 22 + xxx(b) 223=x(c)
223
2
246
2
326
2
2366
=
+=
+=
+=
x
x
x
x
6 Proportionality or
VariationExercise 6.11
Missingp values: 0.1,10,16,80
Missing q vlaues: 15,20,50
2 (a) becomes 4 x as big
(b) becomes 9 x as big
(c) becomes 100 x as big
3 (a) am 8.6=
(b)m = 27.2, a = 25, m = 20.4
4 (a)P= 80, P= 2000
(b) 1,30 == hh5
Missingxvalues: 30,40
Missingy value: 2
(a)y = 250
(b)x = 15
6 time = 3.18 seconds (3sf)
Exercise 6.21
Missing vvalues: 40, 80
Missing tvalues: 20, 10, 2
2
Missingxvalues: 0.4, 0.5,
Missingy value: 200
3 (a)H= 1.5, H= 150
(b) T= 25, T= 2500
4 Resistance = ohms161 or
0.0625 ohms
5(a) pressure = 69.4 units (3sf)
(b) diameter = 22.4 mm (3sf)
7 Number: Estimating
and Factorising1 (a) 8960 (b) 2800 (c) 0.032
2 4000V 3 24T4(a) 326
(b) 11323
(c)
7323 HCF = 24323 =LCM = 117326 {=14784 is unnecessary}
5 approx. 20
8 Constructions, Loci
and Bearings1 AC = 9.2 cm
2
Area = 10.3cm2
3PR = 17.9km, bearing = 140
4 bearing = 298
5 bearing = 264, BC = 9.18cm
x
y
0
(5,25)
x
y
(0,8)
(2,4)
x
y
0
(3,9)
(6,0)
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GCSE Higher Tier Practice Questions 17
6
Area = 114 cm2
7 (a) Area = 332
1square units
(b)AB = 5.83 units, BC = 5
CD = 7.21 AD = 5.39
(c) 99.2
9 VectorsExercise 9.1
1 (a) a =
1
2b =
2
2
c =
3
2p =
2
4q =
1
1
r =
3
2s =
3
6t =
44
(b) |a| = 5 |b| = 22
|c| = 13 |q| = 2 |t| = 24
(c) q =2
1 b r = - c
s = -3a t = 2b
2 (a)
=4
3TO
(b) 5=OT(c) R is either (-5,0) or (+5,0)
3 (a)
25
4(b)
9
4
4 (a) k= 2 (b) k= -28
Exercise 9.2
1 nx+=AC n2=AD
x-n=CD nx=FA
nx= 2FB x-n2=AE
2 (a)A,B and Cwould be
collinear.
(b) It would be all one
parallelogram.
(c) It would be a trapezium.(d)Vertices would beE, C, B, F
3 (a) (i) st 33 =BD(ii) st=BP(iii) st 22 =BQ
(iv) st+= 2AQ
(v) ts += 2QC
(b) ts += 2AP
QCAP =
QCtoparallelisAPand
ramlogparalleaisAPCQ10 Probability
1 (a)8
1 (b)
8
3
(c)8
1 (d)
8
7
2 (a)2
1 (b)
25
3
(c)100
9 (d)
25
3
(e)50
1
3 (a)12
1 (b)
36
1
(c)18
5 (d)
6
1 (e)
6
1
(f) 7 is the most likely total
4 (a) (i) mutually exclusive
(ii) not mutually exclusive
(b) not mutually exclusive
(c) not mutually exclusive
5 P(LHD) x P(no adjustment)
75.0
68.08.085.0
==
6 P(King) =13
1
52
4=
P(Red) =2
1
P(Red) x P(King) =26
1
P(Red King) =261
522 =
7 (a)6
1 (b)
30
1
(c)30
1 (d)
30
29
8
(a)P(alarm ok & on time) = 0.765
(b) P(on time) = 0.77
9 (a)
P(Milk 1st) =yx
x
+(b)
P(M,M)=)1)((
)1(
++
yxyx
xx
(c)
P(one of each)= )1)((
2
++ yxyx
xy
(d) P(Plain,Plain)=
)1)((
)1(
++
yxyx
yy
11 Calculations in
Right-Angled TrianglesExercise 11.11 x = 47.7 2 x = 67.4
3 x = 55.8 4 x = 26.5
5 x = 71.6 6 x = 65.97 x = 120 {all to 3 s.f.}
Exercise 11.21 x= 5.30 cm 2 x= 7.68 cm
3 x= 6.43 cm 4 x= 10.4 cm
5 x= 34.9 cm 6 x= 30.8 cm
7 x= 16.9 cm
12 Sine and Cosine
Rules
1 x= 7.45 cm 2x= 14.2 cm3 x = 60.9
4 x = 54.4 or126
5 x= 8.13 cm 6 x= 5.06 cm
7 x = 32.5 8 x = 47.9
9 x= 6.68 cm