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DMO’L.St Thomas More
C2: Starters
Revise formulae and develop problem solving skills.
1 2 3 4 5 6 7 8 9
10
11
12
13
14
15
16
17
18
19
20 21
22 23 24 25 26
DMO’L.St Thomas More
Starter 1
A circle has diameter with end points at (-4,-1) and (0,1).
Find the equation of the circle.
DMO’L.St Thomas More
Starter 1
A circle has diameter with end points at (-4,-1) and (0,1).
Centre = (-4+0 , -1+1)2 2
= (-2 , 0)
DMO’L.St Thomas More
Starter 1
A circle has diameter with end points at (-4,-1) and (0,1).
Centre = (-4+0 , -1+1) 2 2= (-2 , 0)
Radius = =
22 )10()42(
DMO’L.St Thomas More
Starter 1
A circle has diameter with end points at (-4,-1) and (0,1).
Centre = (-4+0 , -1+1) 2 2= (-2 , 0)
Radius = =
22 )10()42(
5Equation of circle:
(x+2)2 + y2 = 5Back
DMO’L.St Thomas More
Starter 2
Find the mediator of these the points A(2,6) and B(8,0)
Midpoint of AB = (5,3)
DMO’L.St Thomas More
Starter 2
Find the mediator of these the points A(2,6) and B(8,0)
Midpoint of AB = (5,3)Gradient of AB = -6/6 = -1
Equation of Mediator is
y – 3 = 1(x – 5)
x – y -2 = 0
Back
DMO’L.St Thomas More
To divide 6x3+27x2+14x+8 by x+4
6x2 +3x
x+4 6x3 +27x2 +14x +8
6x3 +24x2
3x2 +14x
DMO’L.St Thomas More
To divide 6x3+27x2+14x+8 by x+4
6x2 +3x
x+4 6x3 +27x2 +14x +8
6x3 +24x2
3x2 +14x
3x2 +12x
DMO’L.St Thomas More
To divide 6x3+27x2+14x+8 by x+4
6x2 +3x
x+4 6x3 +27x2 +14x +8
6x3 +24x2
3x2 +14x
3x2 +12x
2x
DMO’L.St Thomas More
To divide 6x3+27x2+14x+8 by x+4
6x2 +3x
x+4 6x3 +27x2 +14x +8
6x3 +24x2
3x2 +14x
3x2 +12x
2x +8
DMO’L.St Thomas More
To divide 6x3+27x2+14x+8 by x+4
6x2 +3x +2
x+4 6x3 +27x2 +14x +8
6x3 +24x2
3x2 +14x
3x2 +12x
2x +8
DMO’L.St Thomas More
To divide 6x3+27x2+14x+8 by x+4
6x2 +3x +2
x+4 6x3 +27x2 +14x +8
6x3 +24x2
3x2 +14x
3x2 +12x
2x +8
2x +8
DMO’L.St Thomas More
To divide 6x3+27x2+14x+8 by x+4
6x2 +3x +2
x+4 6x3 +27x2 +14x +8
6x3 +24x2
3x2 +14x
3x2 +12x
2x +8
2x +8
0Back
DMO’L.St Thomas More
To divide 2x3+4x2-9x-9 by x+3
2x2 -2x
x+3 2x3 +4x2 -9x -9
2x3 +6x2
-2x2 -9x
-2x2 -6x
DMO’L.St Thomas More
To divide 2x3+4x2-9x-9 by x+3
2x2 -2x
x+3 2x3 +4x2 -9x -9
2x3 +6x2
-2x2 -9x
-2x2 -6x
-3x
DMO’L.St Thomas More
To divide 2x3+4x2-9x-9 by x+3
2x2 -2x
x+3 2x3 +4x2 -9x -9
2x3 +6x2
-2x2 -9x
-2x2 -6x
-3x -9
DMO’L.St Thomas More
To divide 2x3+4x2-9x-9 by x+3
2x2 -2x -3
x+3 2x3 +4x2 -9x -9
2x3 +6x2
-2x2 -9x
-2x2 -6x
-3x -9
DMO’L.St Thomas More
To divide 2x3+4x2-9x-9 by x+3
2x2 -2x -3
x+3 2x3 +4x2 -9x -9
2x3 +6x2
-2x2 -9x
-2x2 -6x
-3x -9
-3x -9
DMO’L.St Thomas More
To divide 2x3+4x2-9x-9 by x+3
2x2 -2x -3
x+3 2x3 +4x2 -9x -9
2x3 +6x2
-2x2 -9x
-2x2 -6x
-3x -9
-3x -9
0
Back
DMO’L.St Thomas More
Simplify each of the following:
4log8log2log3
2log6log5log
3log7log
31
222
55
aaa
4log8log2log3
2log6log5log
21log3log7log
31
222
555
aaa
4log8log2log3
15log2log6log5log
21log3log7log
31
2222
555
aaa 4log8log2log4log8log2log3
15log2log6log5log
21log3log7log
313
31
2222
555
aaaaaa
4log2log8log4log8log2log3
15log2log6log5log
21log3log7log
31
2222
555
aaaaaa
64log4log8log2log3
15log2log6log5log
21log3log7log
31
2222
555
aaaa
C2: Starter 5
Back
DMO’L.St Thomas More
Express each of the following in terms of p,q and/or r
zryqxp aaa log,log,log
zyx
xy
a
a
z
yxa
zxy
a
32
3
log
)log
)(log
)(log
3
2
zyx
xy
rqpzyx
a
a
z
yxa
aaazxy
a
32
3
log
)log
)(log
logloglog)(log
3
2
zyx
xy
rqpzyx
rqpzyx
a
a
aaaz
yxa
aaazxy
a
32
3
log
)log
32log3loglog2)(log
logloglog)(log
3
2
zyx
qpyxxy
rqpzyx
rqpzyx
a
aaa
aaaz
yxa
aaazxy
a
32
23
21
23
213
log
loglog)log
32log3loglog2)(log
logloglog)(log
3
2
rqpzyxzyx
qpyxxy
rqpzyx
rqpzyx
aaaa
aaa
aaaz
yxa
aaazxy
a
21
2132
23
21
23
213
32loglog3log2log
loglog)log
32log3loglog2)(log
logloglog)(log
3
2
C2: Starter 6
Back
DMO’L.St Thomas More
Solve the equations
C2: Starter 8
Back532
12
1
87
255
2564
xx
x
xx 3333.1x
2
1x
43.4x
DMO’L.St Thomas More
Find the equation of the circle passing through the points A(-1,1), B(-1,5), C(5,7)
C2: Starter 9
DMO’L.St Thomas More
Find the equation of the circle passing through the points A(-1,1), B(-1,5), C(5,7)
Draw diagram
And then find
the mediators
C2: Starter 9
DMO’L.St Thomas More
Find the equation of the circle passing through the points A(-1,1), B(-1,5), C(5,7)
Mediator of AB
C2: Starter 9
DMO’L.St Thomas More
Find the equation of the circle passing through the points A(-1,1), B(-1,5), C(5,7)
Mediator of AB
y = 3
C2: Starter 9
DMO’L.St Thomas More
Find the equation of the circle passing through the points A(-1,1), B(-1,5), C(5,7)
Mediator of BC
Mid pt of BC = (2,6)
Grad BC = 1/3
Grad = -3
C2: Starter 9
DMO’L.St Thomas More
Find the equation of the circle passing through the points A(-1,1), B(-1,5), C(5,7)
Mediator of BC
Mid pt of BC = (2,6)
Grad BC = 1/3
Grad = -3
y- 6 = -3(x – 2)
C2: Starter 9
DMO’L.St Thomas More
Find the equation of the circle passing through the points A(-1,1), B(-1,5), C(5,7)
Mediator of BC
Mid pt of BC = (2,6)
Grad BC = 1/3
Grad = -3
Y = -3x +12
C2: Starter 9
DMO’L.St Thomas More
Find the equation of the circle passing through the points A(-1,1), B(-1,5), C(5,7)
Find centre using mediators
y = 3
y = -3x + 12
C2: Starter 9
DMO’L.St Thomas More
Find the equation of the circle passing through the points A(-1,1), B(-1,5), C(5,7)
Find centre using mediators
y = 3
y = -3x + 12
Centre = (3,3)
C2: Starter 9
DMO’L.St Thomas More
Find the equation of the circle passing through the points A(-1,1), B(-1,5), C(5,7)
Find centre using mediators
y = 3
y = -3x + 12
Centre = (3,3)
R2 = 42 + 22 = 20
C2: Starter 9
DMO’L.St Thomas More
Find the equation of the circle passing through the points A(-1,1), B(-1,5), C(5,7)
Equation of circle is
(x-3)2+(y-3)2 = 20
C2: Starter 9
Back
DMO’L.St Thomas More
Find the sum to 10 terms of
a. 2 + 10 + 50 + 250 + …
b. 100 + 50 + 25 + 12.5 + …
Find the sum to infinity of
c. 4 + 2 + 1 + ½
C2: Starter 10
DMO’L.St Thomas More
Find the sum to 10 terms of
a. 2 + 10 + 50 + 250 + …
b. 100 + 50 + 25 + 12.5 + …
Find the sum to infinity of
c. 4 + 2 + 1 + ½
C2: Starter 10
Back
4 882 812
199.8047
8
DMO’L.St Thomas More
a. Expand (x + 2y)4
b. Find the term in x7 in the expansion of (3x - y)10
c. Find the term that is independent of x in the expansion of (x2 + 5/x)9
C2: Starter 11
x4 + 8x3y + 24x2y2 + 32xy3 + 16y4
-262 440x7y3
1 312 500
Back
DMO’L.St Thomas More
a. Expand (x - y)4
b. Find the term in x7 in the expansion of (x + 3y)9
c. Find the term that is independent of x in the expansion of (x2 + 5/x)12
C2: Starter 12
x4 - 4x3y + 6x2y2 - 4xy3 + y4
324x7y2
193 359 375
Back
DMO’L.St Thomas More
a. Expand (x - y)5
b. Find the term in x5 in the expansion of (x + 7y)15
c. Find the term that is independent of x in the expansion of (x3 + 4/x)8
C2: Starter 13
x5 - 5x4y + 10x3y2 - 10x2y3 + 5xy4 – y5
(8.48x1011)x5y10
114 688
Back
DMO’L.St Thomas More
If = /5 and the radius is
10 cm find, to 3 significant
figures, the area of the
shaded segment.
C2: Starter 14
Area of segment
Back
5
25)sin(2
21 r
)sin( 52
522
210
23.15 cm
DMO’L.St Thomas More
a. Use the remainder theorem to show that (x-1) is a factor of 4x3 – 3x2 – 1
b. If y = 2x3 – 3x2 find dy/dx
c. Write log37 -4log3(1/2) as a single logarithm
C2: Starter 15
Rem =f(1) = 4 x 13 – 3x12 – 1 = 4 – 3 – 1 = 0
(x-1) is factor.
dy/dx = 6x2 – 6x
log37 -log3(1/2)4 = log3(7 x 24)
= log3(448)
Back
log37 -log3(1/2)4 = log3(7 x 24)
= log3(448)
log37 -log3(1/2)4 = log3(7 x 24)
= log3(112)
DMO’L.St Thomas More
a. Find the remainder when 4x3 – 3x2 – 1 is divided by (x-3)
b. If y = 12x4 – 4x3 find dy/dx
c. Write 3log32 -4log3(2) as a single logarithm
C2: Starter 16Rem =f(3) = 4 x 33 – 3x32 – 1
= 108 – 27 – 1
= 80
dy/dx = 48x3 – 12x2
log37 -log3(1/2)4 = log3(7 x 24)
= log3(448)
Back
log37 -log3(1/2)4 = log3(7 x 24)
= log3(448)
log323 -log324 = log3(8 ÷ 16)
= log3(1/2)
DMO’L.St Thomas More
a. Find the remainder when 4x3 – 3x2 – 1 is divided by (x+1)
b. If y = 24x3 – 12x2 find dy/dx
c. Write in surd form sin 330o
C2: Starter 17
Rem =f(3) = 4 x (-1)3 – 3x(-1)2 – 1
= -4 – 3 – 1
= -8
dy/dx = 72x2 – 24xBack
sin 330o = -sin 30o = -1/2
DMO’L.St Thomas More
a. If y = 12x3 – 6x2 find dy/dx
b. Find the two turning points and use the second differential to determine which maximum and which is minimum.
C2: Starter 18
dy/dx = 36x2 – 12x
For turning points
36x2 – 12x = 0
12x(3x - 1) = 0
x = 0 or x = 1/3
For turning points
36x2 – 12x = 0
12x(3x - 1) = 0
x = 0 or x = 1/3
d2y/dx2 = 72x – 12
x = 0 d2y/dx2 < 0 Max at (0,0)
x = 1/3 d2y/dx2 > 0 Min at (1/3,-2/9)
d2y/dx2 = 72x – 12
x = 0 d2y/dx2 < 0 Max at (0,0)
x = 1/3 d2y/dx2 > 0 Min at (1/3,-2/9)
d2y/dx2 = 72x – 12
x = 0 d2y/dx2 < 0 Max at (0,0)
x = 1/3 d2y/dx2 > 0 Min at (1/3,-2/9)
Back
DMO’L.St Thomas More
Find the sum to 10 terms of
a. 2 + 10 + 18 + 26 + …
b. 200 + 100 + 50 + 25 + …
Without using a calculator find
a. cos 135o
b. sin 330o
c. tan 225o
C2: Starter 19
DMO’L.St Thomas More
Find the sum to 10 terms of
a. 2 + 10 + 18 + 26 + …380
b. 200 + 100 + 50 + 25 + …399.61
Without using a calculator find
a. cos 135o
b. sin 330o
c. tan 225o
C2: Starter 19
DMO’L.St Thomas More
Find the sum to 10 terms of
a. 2 + 10 + 18 + 26 + …380
b. 200 + 100 + 50 + 25 + …399.61
Without using a calculator find
a. cos 135o -1/2
b. sin 330o
c. tan 225o
C2: Starter 19
DMO’L.St Thomas More
Find the sum to 10 terms of
a. 2 + 10 + 18 + 26 + …380
b. 200 + 100 + 50 + 25 + …399.61
Without using a calculator find
a. cos 135o -1/2
b. sin 330o -1/2
c. tan 225o
C2: Starter 19
DMO’L.St Thomas More
Find the sum to 10 terms of
a. 2 + 10 + 18 + 26 + …380
b. 200 + 100 + 50 + 25 + …399.61
Without using a calculator find
a. cos 135o -1/2
b. sin 330o -1/2
c. tan 225o 1
C2: Starter 19
Back
DMO’L.St Thomas More
Find the sum to 10 terms of
a. 3 + 12 + 21 + 30 + …
b. 80 + 60 + 45 + 33.75…
Without using a calculator find
a. cos 300o
b. sin 150o
c. tan 120o
C2: Starter 20
DMO’L.St Thomas More
Find the sum to 10 terms of
a. 3 + 12 + 21 + 30 + … 435
b. 80 + 60 + 45 + 33.75… 301.98
Without using a calculator find
a. cos 300o
b. sin 150o
c. tan 120o
C2: Starter 20
DMO’L.St Thomas More
Find the sum to 10 terms of
a. 3 + 12 + 21 + 30 + … 435
b. 80 + 60 + 45 + 33.75… 301.98
Without using a calculator find
a. cos 300o 1/2
b. sin 150o
c. tan 120o
C2: Starter 20
DMO’L.St Thomas More
Find the sum to 10 terms of
a. 3 + 12 + 21 + 30 + … 435
b. 80 + 60 + 45 + 33.75… 301.98
Without using a calculator find
a. cos 300o 1/2
b. sin 150o 1/2
c. tan 120o
C2: Starter 20
DMO’L.St Thomas More
Find the sum to 10 terms of
a. 3 + 12 + 21 + 30 + … 435
b. 80 + 60 + 45 + 33.75… 301.98
Without using a calculator find
a. cos 300o 1/2
b. sin 150o 1/2
c. tan 120o 3
C2: Starter 20
Back
DMO’L.St Thomas More
C2: Starter 21
a. Find f’(x) given;
b. Find f(x) given
c. Solve sin(+60o) = 1/2
xxxxxf
524)('
xxxxf 223)(
DMO’L.St Thomas More
a.
b. Find f(x) given
c. Solve sin(+60o) = 1/2
C2: Starter 21
xxxxxf
524)('
222 26)('3)( xxxfxxf xx
DMO’L.St Thomas More
a.
b. Find f(x) given
c. Solve sin(+60o) = 1/2
C2: Starter 21
222 26)('3)( xxxfxxf xx
cxxxf
xxfx
xxx
52
2
5
5
2)(
4)('
DMO’L.St Thomas More
a.
b. Find f(x) given
c. Solve sin(+60o) = 1/2
= 45 - 60, 135 – 60, 405 – 60
= 75o, 345o
C2: Starter 21
222 26)('3)( xxxfxxf xx
cxxxf
xxfx
xxx
52
2
5
5
2)(
4)('
Back
DMO’L.St Thomas More
C2: Starter 22
a. Find f’(x) given;
b. Evaluate
c. Solve tan(+50o) = 1
dxxx )4(2
2
3
xxxxf
323)(
DMO’L.St Thomas More
a.
b. Find f(x) given
c. Solve tan(+50o) = 1
C2: Starter 22
xxx
xf
xxf xx
222
3)('
3)(
2
2 3
dxxx )4(2
2
3
DMO’L.St Thomas More
a. Find f’(x) given;
b. Find f(x) given
c. Solve tan(+50o) = 1
C2: Starter 22
xxx
xf
xxf xx
222
3)('
3)(
2
2 3
0)4(2
2
3 dxxx
DMO’L.St Thomas More
a.
b. Find f(x) given
Solve tan(+50o) = 1
= -5o, 175o, 355o
C2: Starter 22
xxx
xf
xxf xx
222
3)('
3)(
2
2 3
0)4(2
2
3 dxxx
Back
DMO’L.St Thomas More
C2: Starter 23
Find the shaded
area.
Area under the
line is a triangle
222
22cm
DMO’L.St Thomas More
C2: Starter 23
Find the shaded
area.
Area under the
curve
231
3
1
223
11)
2
1
2(
2
12cmdxx
xdx
xx
DMO’L.St Thomas More
C2: Starter 24
Find
a. the coordinates
of A, B and C
A = (1,0) B = (5,0)
C = (6,5)
b. the shaded area.
C
DMO’L.St Thomas More
C2: Starter 24
Find
a. the coordinates
of A, B and C
A = (1,0) B = (5,0)
C = (6,5)
b. the shaded area.
Triangle ACM
M
C
DMO’L.St Thomas More
C2: Starter 24
Find
a. the coordinates
of A, B and C
A = (1,0) B = (5,0)
C = (6,5)
b. the shaded area.
Triangle ACM
M
25.122
55cm
C
DMO’L.St Thomas More
C2: Starter 24
Find
a. the coordinates
of A, B and C
A = (1,0) B = (5,0)
C = (6,5)
b. the shaded area.
Under curve BC
M
6
5)5)(1( dxxx
C
DMO’L.St Thomas More
C2: Starter 24
Find
a. the coordinates
of A, B and C
A = (1,0) B = (5,0)
C = (6,5)
b. the shaded area.
Under curve BC
M
6
5
26
5)56()5)(1( dxxxdxxx
C
DMO’L.St Thomas More
C2: Starter 24
Find
a. the coordinates
of A, B and C
A = (1,0) B = (5,0)
C = (6,5)
b. the shaded area.
Under curve BC
M
26
5 3126
52)56()5)(1( cmdxxxdxxx
C
DMO’L.St Thomas More
C2: Starter 24
Find
a. the coordinates
of A, B and C
A = (1,0) B = (5,0)
C = (6,5)
b. the shaded area.
Shaded area
M
261
31
21 10212 cm
C
Back
DMO’L.St Thomas More
Evaluate:
10log
1000log
27
1log
log
1024log
125log
1000
10
3
2
2
5
aa
C2: Starter 25
Back10log
1000log
27
1log
log
1024log
125log
1000
10
3
2
2
5
aa
10log
1000log
27
1log
log
101024log
3125log
1000
10
3
2
2
5
aa
10log
1000log
27
1log
2log
101024log
3125log
1000
10
3
2
2
5
aa
10log
1000log
327
1log
2log
101024log
3125log
1000
10
3
2
2
5
aa
10log
31000log
327
1log
2log
101024log
3125log
1000
10
3
2
2
5
aa
31
1000
10
3
2
2
5
10log
31000log
327
1log
2log
101024log
3125log
aa