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C3: StartersRevise 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
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Starter 1Solve the equation
for 01sin2cos3 20
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Starter 1Solve the equation
for 01sin2cos3 20
01sin)sin21(3 2
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Starter 1Solve the equation
for 01sin2cos3 20 01sin)sin21(3 2 02sinsin6 2
0)1sin2)(2sin3(
21sin
32sin or
65
6 ,,55.5,87.3 Back
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Starter 2Prove the identity
2244 tansectansec
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Starter 2Prove the identity
44 tansec LHS
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Starter 2Prove the identity
44 tansec LHS)tan)(sectan(sec 2222
)tantan1)(tan(sec 2222 )tan(sec 22
Back
RHS
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Starter 3Prove the identity
tan
cos3cossin3sin
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Starter 3
cos3cossin3sin
LHS
cos3cos)sin()cos(2 2
32
3
)cos()cos(2)sin()cos(2
23
23
23
23
)cos()2cos(2)sin()2cos(2
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Starter 3
cos3cossin3sin
LHS
cos3cos)sin()cos(2 2
32
3
)cos()cos(2)sin()cos(2
23
23
23
23
RHS tan
)cos()2cos(2)sin()2cos(2
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Starter 4Given that andwhere A is acute and B is obtuse, find
43tan A 13
5sin B
)(cos BAec
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Starter 4
By Pythagoras
43tan A 13
5sin B
54
53
cossin
AA
1312
125
costan
BB
1312
125
costan
BB
A is acute
B is obtuse
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Starter 4
)sin(1)(cosBA
BAec
BABA sincoscossin1
135
54
1312
53
1
6520
6536
1
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Starter 4
)sin(1)(cosBA
BAec
BABA sincoscossin1
135
54
1312
53
1
16651
6520
6536
Back
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Starter 5Differentiate
xey x sin2
xey 2sin
xxxy
lncossin
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Starter 5Differentiate
xey x sin2 xeu 2 xv sinx
dxdu e22 xdx
dv cos
xexedxdy xx cossin2 22
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Starter 5Differentiate
xey 2sin xu 2sin uey
xdxdu 2cos2 u
dudy e
xexedxdy xu 2cos22cos2. 2sin
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Starter 5Differentiate
xxxy
lncossin
xxu cossin xv ln
xxdxdu 22 sincos xdx
dv 1
2
1
)(lncossin2cosln
xxxxx
dxdy x
xdxdu 2cos
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Starter 5Differentiate
xxxy
lncossin
xxu cossin xv ln
xxdxdu 22 sincos xdx
dv 1
221
)(ln2sin2cosln
xxxx
dxdy x
xdxdu 2cos
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Starter 5Differentiate
xxxy
lncossin
xxu cossin xv ln
xxdxdu 22 sincos xdx
dv 1
2)(ln22sin2cosln2
xxxxxx
dxdy
xdxdu 2cos
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Starter 6Differentiate
)ln(sec xy
7)3sin1( xy
xexy sin
2cos
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Starter 6Differentiate
1)(cossec xxu uy ln
xxdxdu sin)(cos 2 udu
dy 1
xx
udxdy
2cossin.1
xy ln(sec)
xx
dxdu
2cossin
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Starter 6Differentiate
1)(cossec xxu uy ln
xxdxdu sin)(cos 2 udu
dy 1
xxxx
xx
udxdy tan
coscossin
cossin.1
22
)ln(sec xy
xx
dxdu
2cossin
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Starter 6Differentiate
xu 3sin1 7uy
xdxdu 3cos3 67udu
dy
xxxudxdy 3cos)3sin1(213cos3.7 66
7)3sin1( xy
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Starter 6Differentiate
xu 2cos xev sinxdx
du 2sin2 xdxdv ex sin.cos
x
xx
eexxxe
dxdy
sin2
sinsin ).(cos2cos)2sin2(
xexy sin
2cos
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Starter 6Differentiate
xu 2cos xev sinxdx
du 2sin2 xdxdv ex sin.cos
)cos2cos2sin2(sin xxxedxdy x
xexy sin
2cos
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Starter 7Solve the following equations, giving
exact solutions72 xe
5)12ln( 2 x
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Starter 7Solve the following equations, giving
exact solutions72 xe
7ln2 x7ln2
1x
5)12ln( 2 x5)12ln(2 x
25)12ln( x
25
12 ex
212
5
ex
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Starter 8Show that can in be written in
the form Use the iteration starting with to generate Show that 5.5 is a root of the equation to
one decimal place.
0352 xx35 xx
351 nn xx
50 x 654321 ,,,,, xxxxxx
352 xx
35 xx
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Starter 8Use the iteration starting with to generate Show that 5.5 is a root of the equation to
one decimal place.
351 nn xx
50 x 654321 ,,,,, xxxxxx
2915.55
1
0
xx
Calculator:5 =5Ans+3====
537.5530.5518.5489.54275.52915.5
5
6
5
4
3
2
1
0
xxxxxxx
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Starter 8Show that 5.5 is a root of the equation to
one decimal place. 35)( 2 xxxf
525.0)55.5( f
547.0)45.5( f
Change of sign Root between 5.55 and 5.45Change of sign Root between 5.55 and 5.45Hence, x = 5.5 is a root to 1 decimal place.Back
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Starter 9Sketch the graph Hence, or otherwise, solve
52 xy
552 x
y=2x-5
52 xyy=5
x = 0 or 5 Back
when x = 3
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Starter 10 By differentiating find the coordinates of
the turning point on the curve State the nature of the turning point (i.e.
maximum or minimum).
xxy ln813
0813 2 x
xdxdyFor turning points
813 3 x3x22
2 816x
xdxyd
0816 22
2
x
xdxydwhen x =
3
Hence, minmum point at (3,-61.99)
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Starter 11Solve the following equations, giving
exact solutions422 xxx eee
12)35ln( 2 x
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Starter 11Solve the following equations, giving
exact solutions422 xxx eee
432 xx ee
432 xxTake logs base e
4x
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Starter 11Solve the following equations, giving
exact solutions12)35ln( 2 x12)35ln(2 x6)35ln( x
e to the power of 635 ex
536
ex Back
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Starter 12Complete the table:
ydxdy
x
x
n
a
e
xxxxx
kx
lnsectancossin
aa
e
xxx
xx
nkx
x
x
x
n
ln
tansecsec
sincos
1
2
1
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Starter 13Complete the table:
ydxdy
x
x
n
e
xxxx
xx
5
6
3
2
3
6ln7sec4tan
cos
2sin)13(
3ln)3(5
6
7tan7sec74sec4
sincos3
2cos2)13(6
5
6
1
2
2
12
x
x
x
n
e
xxx
xx
xxnx
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