Page 1 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
Multiple Choice Test 1 1 D 2 B 3 A 4 A 5 B 6 D 7 A 8 A 9 B 10 B 11 A 12 C 13 C 14 B 15 A 16 D 17 A 18 C 19 B 20 D 21 B 22 B 23 A 24 C 25 A 26 B 27 B 28 B 29 A 30 B 31 B 32 C 33 B 34 C 35 A 36 A 37 B 38 B 39 C 40 B 41 B 42 C 43 B 44 C 45 B Multiple Choice Test 1 worked answers
1 1 3arg ( 3 3 ) tan3
i − − π −5π− − = − π = − π = − 6 6
Page 2 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
D 2 (x + iy)2 = –5 + 12i
x2 – y2 + i (2xy) = –5 + 12i x2 – y2 = –5
62 = 12xy yx
⇒ =
22
36 5∴ − = −xx
x4 + 5x2 – 36 = 0 (x2 + 9) (x2 – 4) = 0 x2 = 4, x = ± 2
When x = 2, = =6 32
y
62, 32
x y= − = = −−
∴ 5 12i− + = 2 + 3i or –2 – 3i B
3 (1 – i)5 − =1 2i
1arg (1 ) tan ( 1)4
i − −π− = − =
55 5 5(1 ) 2 cos sin 4 2 cos sin
4 4 4i i i
−π −π − π − π − = + = + 4
1 14 22 2
i = − + −
= –4 + 4i A
4 1 2 2 (1 2 ) 2z i z i− + = ⇒ − − = Circle centre (1, –2) radius 2 A
5 y = (x + 1) ln (x + 2) d 1ln ( 2)d 2y xxx x
+= + +
+
when = = +d 10, ln2d 2yxx
B
6 + += +sin (2 3) sin (2 3)d [ ] 2 cos (2 3)d
x xe x ex
D 7 x = t2 + t
y = 2t – 1 y = 2x – 3 2t – 1 = 2 (t2 + t) – 3 2t – 1 = 2t2 + 2t – 3 2t2 = 2
Page 3 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
t2 = 1, t = ± 1 A
8 x3y + 2xy = 4y Differentiate wrt x:
3 2d d d3 2 2 4d d dy y yx x y x yx x x+ + + =
When x = 0, y = 0 d 0dyx
⇒ =
A 9 y = x lnx
d 1 ln 1 lndy x x xx x
= + = +
B 10 f(x) = cos 2x
( )f x′ = –2 sin 2x ( )f x′′ = –4 cos 2x
B
11 2
2 24 3 ( 1) ( 3)x x x x
≡+ + + +
2( 1) ( 3) 1 3
A Bx x x x
≡ ++ + + +
2 ( 3) ( 1)A x B x⇒ = + + + When x = –1, 2 = 2A ⟹ A = 1 When x = –3, 2 = –2B ⟹ B = –1
2 1 1( 1) ( 3) 1 3x x x x
∴ ≡ −+ + + +
A
12 =+ − + −∫ ∫2 2 2
1 1d d4 ( 1) 2 ( 1)
x xx x
11 1tan2 2
x c− − = +
C
13 1 1
2 2 20 0
1 1d d1 1 1
x xx xx x x+
= ++ + +∫ ∫
11 2
0
1tan ( ) ln ( 1)2
x x− = + +
1 ln 24 2π
= +
C
14 1 21 1(ln ) d (ln )2
x x x cx
= +∫
B
Page 4 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
15 2 21d
2x xxe x e c= +∫
A
16 =
−∑50
1
(3 2)r
r
=
= −∑50
1
3 2 (50)r
r
= −3 (50) (51) 2 (50)
2
= 3725 D
17 (1 – 2x2)9 4th term = 9C3 (–2x2)3 = –672x6 A
18 ++
+ +! ( 1)!
( 1)! ( 2)!n n
n n
+ + +=
+( 2) ( !) ( 1)!
( 2)!n n n
n
+ + +=
+( 2) ! ( 1) !
( 2)!n n n n
n
+ + +=
+![ 2 1]
( 2)!n n n
n
=!n +
+ +(2 3)
( 2) ( 1) ( !)n
n n n
2 3( 2) ( 1)
nn n
+=
+ +
C 19 (2 + 3x)n
coefficient of x3: nC3 2n – 3 (3)3 coefficient of x4: nC4 2n – 4 (3)4
3 33
4 44
C (2 ) (3 ) 8C (2 ) (3 ) 15
n n
n n
−
− =
!2 8( 3)!3!
!3 15( 4)!4!
nn
nn
− = −
( 4)!n − 4!( 3) ( 4)!n n− −
453!
=
= −20 34
n
n = 8 B
Page 5 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
20 1 2( 1) ( 2)(1 ) 1 ( 1) ( ) ( )2!
x x x− − −+ = + − − + −
= 1 + x + x2 D
21 −
+
432 12
x
−31 < < 12
x
2 2< <3 3
x
B
22 = + + + +2 3
2 (2 ) (2 )1 22! 3!
x x xe x
∞
=
= ∑0
2!
n n
n
xn
B 23 f(x) = e4x
f1(x) = 4e4x when x = 1, f(1) = e4 f1(1)= 4e4 f(x) = f(1) + (x – 1) f1(1) e4x = e4 + e4 (x – 1) A
24 ex = 25x – 10 ex – 25x + 10 = 0 f(x) = ex – 25x + 10 f(0) = 10 f(1) = e – 25 + 10 = –ve Root in [0, 1] C
25 f(x) = 2 sinx – x f(1.5) = 0.49499 f(2) = –0.18141
+=
+1(1.5) (0.18141) 2 (0.49499)
0.49499 0.18141x
= 1.8659 A
26 B 27 2 – 6 + 18 – 54 + …
2 (–3)r-1 B
28 f(x) = x3 + 3x2 + 5x + 9 f'(x) = 3x2 + 6x + 5
− + − + − += − −
− + − +
3 2
2 2
( 2.5) 3 ( 2.5) 5 ( 2.5) 92.53 ( 2.5) 6 ( 2.5) 5
x
= –2.457
Page 6 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
B 29 a1 = 1, an + 1 = 2an + 4
a2 = 2a1 + 4 = 2(1) + 4 = 6 a3 = 2a2 + 4 = 2(6) + 4 = 16 1, 6, 16 A
30 an = 5 (2n – 1) – 4 B
31 P(exactly two heads) 38
=
B
32 126
52
125
2
×51
29
17
×
8
502
172
=
C 33 7! – 6! × 2!
= 5 × 6! = 3600 B
34 6! × 7P3 = 151 200 C
35 × = =6 93 2 15
5
720 240C C 720,1001C
A
36 532
A
37 =3 0.3
10
B 38 P(A) = 0.2, P(B) = 0.5
∩ = × = × =P (A B) P(A) P(B) 0.5 0.2 0.1 P (A B) 0.2 0.5 0.1 0.6∪ = + − =
( ) P(A) P(B)P (A B)P A BP(B)∩
= =P(B)
0.2=
( ) P(A B )P A B 0.2P(B )
′∩′ = =′
B 39 No. in committee = 5
6 seniors, 4 juniors 4C1 × 6C4 + 4C2 × 6C3 + 4C3 × 6C2 + 4C4 ×6C1 = 60 + 120 + 60 + 6 = 246 C
Page 7 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
40 1 1 21 1 3
0 5a
−
Cofactor of 3 = –a B
41 1 3 1 3 1 1
20 5 5 0a a
+ +
= 5 + 5 – 3a – 2a = 10 – 5a B
42 10 – 5a ≠ 0 a ≠ 2 C
43 PI is y = a cos 2x + b sin 2x B
44 − + =2
2
d d4 4 0dd
y y yxx
m2 – 4m + 4 = 0 (m – 2)2 = 0 m = 2 y = (Ax + B) e2x C
45 5d 2dyx y xx− =
− = 4d 2dy y xx x
22 d 2ln ln
2
1x x xxe e ex
−− −∫ = = =
= ∫ 22
1 dy x xx
32
1 13
y x cx
= +
5 213
y x cx= +
B Multiple Choice Test 2 1 C 2 A 3 D 4 D 5 A 6 C
Page 8 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
7 C 8 A 9 A 10 C 11 C 12 C 13 B 14 A 15 C 16 B 17 B 18 A 19 B 20 B 21 C 22 B 23 D 24 D 25 A 26 D 27 C 28 D 29 A 30 C 31 B 32 D 33 B 34 B 35 B 36 A 37 D 38 A 39 C 40 D 41 B 42 A 43 C 44 A 45 C Multiple Choice Test 2 worked answers 1 i49 = i48 i = (i2)24 (i) = (–1)24 i = i
C
2 + + + + + += × = = = +
− − +
23 3 2 6 5 5 5 12 2 2 5 5
i i i i i i ii i i
A 3 − − = =2 2 8 2 2i
arg 1 2( 2 2 ) tan2
i − − −3π − − = − π = − 4
Page 9 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
π−4− − =
3
2 2 2 2i
i e D
4 (2 + i) (2 + i) = 4 + 4i – 1 = 3 + 4i (3 + 4i) (3 + 4i) = 9 + 24i + 16i2 = –7 + 24i D
5 –x2 + y2 – 2xy – 2 = 0 Differentiate wrt x:
d d2 2 2 2 0d dy yx y x yx x
− + − − =
d (2 2 ) 2 2dy y x y xx
− = +
d 2 2d 2 2y y x x yx y x y x
+ += =
− −
A 6 y = ln (3x2 + 5)
2
d 6d 3 5y xx x=
+
C 7 y = sin–1 (3x)
2
d 3d 1 9yx x=
−
=− 2
91 9x
=− 2
119
x
C 8 y = tan–1 (x2 + 2)
=+ +2 2
d 2d 1 ( 2)y xx x
A 9
+ +
+ +
− −
2 2
2
13 2
3 2
3 2
x x xx x
x
∴ 2
2
3 213 2 ( 1) ( 2)x x
x x x x+
≡ −+ + + +
11 2
B Cx x
≡ − + + +
Page 10 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
3 2 ( 2) ( 1)x B x C x− − ≡ + + + When 1, 1x B= − =
2 1 41( 1)( 2) 1 2
xx x x x
∴ = + −+ + + +
A = 1, B = 1, C = 4− A
10 2
1 d9 4
xx−
∫
11 2sin2 3
x c− = +
C 11 sin 4 cos4 dθ θ θ∫
1 sin8 d2
θ θ= ∫
1 cos816
cθ= − +
C 12 y = 2t2 + 5
x = t + 3
= =d d4 , 1d dy xtt t
= ÷d d dd d dy y xx t t
= 4t C
13 2 cos , 2 2 sinx yθ θ= = + d d2sin , 2 cosd d
x yθ θθ θ= − =
d 2 cos cotd 2 sinyx
θ θθ
= = −−
2 23
2
d d d cosec 1[ cot ] cosecd d d 2 sin 2
y xx
θθ θθ θ θ
+= − ÷ = = −
−
B
14 ∫ cossin dxx e x = – e cos x + c A
15 Given that = + 1 d2 ,d
x yyx
is
ln y = (x + 1) ln 2 1 d dln 2 ln 2
d d= ⇒ =
y y yy x x
= 2x + 1 ln 2 C
Page 11 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
16 = + +2
12
x xe x
B 17 x = 0, y = 2
2 d d2 4 0, 1d dy yx x
+ = = −
22
2
d d d2 2 2 0d d dy y yy yx x x
+ + =
22
2
d2 (2) ( 1) + 4 2 ( 1) 0d
yx
− + − =
2
2
d 1d 2
yx
=
− +
2 1= 2 + ( ) ( 1)2! 2xy x +…
21= 2 + ...4
x x− +
B
18 −∑= 1
(5 4 )n
r
r
=
= −∑ ∑= 1 1
5 4n n
r r
r
−
( + 1)= 5 42
n nn
= 5n – 2n (n + 1) = n [5 – 2n – 2] = n (3 – 2n) A
19 =
−∑50
1
(5 4 )r
r
= =
= −∑ ∑50 50
1 1
5 4r r
r
= −
50 (51)5 (50) 42
= –4850 B
20 when x = 0, y = 2 2d 2 (2) 1 9
dyx= + =
2
2
d d4 4 2 9 72d d
y yyx x
= = × × =
y = y(0) + (x – 0) 2( 0)(0) (0) ...
2!xy y−′ ′′+ +
y = 2 + 9x + 36x2 +…= 5.24
Page 12 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
x = 0.3, y = 2 + 9(0.3) = 4.700 B
21 Third iteration C
22 u1 = 2, u n + 1 = 2 u n, n ≥ 1 u 2 = 2 u 1 = 4 u 3 = 2 u 2 = 2(4) = 8 u 4 = 16 2, 4, 8, 16 B
23 Divergent D 24 u n = 2n D
25 − − −+ = + − +3 2( 3) ( 4)(1 2 ) 1 ( 3) (2 ) (2 )
2!x x x
= 1 – 6x + 24x2 A
26 −− < < < <
1 11 2 1,2 2
x x
D 27 Sn = pn + qn2
S3 = 6 ⇒ 3p + 9q = 6 p + 3q = 2 [1] S5 = 11 ⇒ 5p + 25q = 11
1155
p q+ = [2]
[2] – [1] ⇒ = =1 12 ,5 10
q q
3 210
p + =
1710
p =
17 1,10 10
p q= =
C
28 = + 217 1S10 10n n n
− = − + − 21
17 1S ( 1) ( 1)10 10n n n
117S S10n n nu n−= − = 21
10n+
1710
−1710
n + − 2110
n + −1 15 10
n
= +16 110 5
n
= +1 (2 16)
10n
Page 13 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
D 29 2 ln x + x – 2 = 0
f(x) = 2 ln x + x – 2 f(1) = 1 – 2 = –1 f(2) = 2 ln 2 + 2 – 2 = 2 ln 2 [1, 2] A
30 = − + −2 4 6(2 ) (2 ) (2 )cos 2 1
2! 4! 6!x x xx
Coefficient of −= − =
66 2 4
6! 45x
C
31 − + = +2
2
d d3 2 1dd
y y y xxx
AQE: m2 – 3m + 2 = 0 (m – 1) (m – 2) = 0 m = 1 or 2 y = Aex + Be 2x B
32 2
2
d d, , 0d dy yy ax b ax x
= + = =
Substituting into the differential equation: –3a + 2 [ax + b] = x + 1 2a = 1
12
a =
–3a + 2b = 1 3 2 1
2b−
+ =
=522
b
54
b =
1 52 4
y x= +
D
33 y = Aex + Be2x 1 52 4
x+ +
When x = 0, y = 1
⇒ = + + ⇒ + = −5 11 A B A B4 4
[1]
2d 1A 2Bd 2
x xy e ex= + +
When x = 0, d 2dyx=
Page 14 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
12 A 2B2
⇒ = + +
+ =3A 2B2
[2]
[2] – [1] ⇒ 3 1 7B2 4 4
= + =
A = –2 27 1 52
4 2 4x xy e e x= − + + +
B
34 4d 2 2d
xy y ex− =
IF = 2 2dx xe e− −∫ = 2 4 22 dx x xye e e x− −= ∫ 2 22 dx xye e x− = ∫
ye–2x = e2x + c y = 1 + ce2x B
35 when x = 0, y = 4, 4 = 1 + c ⇒ c = 3 y = 1 + 3e2x B
36 8! = 40 320 A 37 6! × 3! = 4320 D 38 5! × 6P3 = 14 400
A 39 10C2 × 10C3 × 2 = 10 800 C
40 × × =3 2 1 16 5 4 20
D
41 1 52 4 14 6 7
x pay qz r
− =
− −
− +4 1 2 1 2 4
56 7 4 7 4 6
a
= –34 – 5 (10) + a (28) = –84 + 28a B
42 a ≠ 3 A
Page 15 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
43 − − −
1 5 3 182 4 1 74 6 7 29
R2 → R2 – 2R1 R3 → R3 – 4R1
181 5 30 14 5 430 14 5 43
− − −
− − +
3 3 2R R R→ +
181 5 30 14 5 430 0 0 0
− −
Infinite set of solutions. C
44 = −
1 3 4M 2 5 1
3 8 4
− −= − +
5 1 2 1 2 5M 3 4
8 4 3 4 3 8
= 28 – 33 + 4 = –1 A
45 Matrix of cofactors −
= + − + − + −
28 11 120 8 123 9 1
1
28 20 23M 11 8 9
1 1 1
−
− = − − − −
− − = − − −
28 20 2311 8 9
1 1 1
C Multiple Choice Test 3 1 B 2 C 3 A 4 C 5 B 6 B 7 B
Page 16 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
8 A 9 D 10 D 11 C 12 C 13 B 14 D 15 B 16 C 17 A 18 B 19 C 20 A 21 A 22 B 23 A 24 C 25 D 26 B 27 B 28 D 29 A 30 C 31 D 32 A 33 C 34 D 35 D 36 C 37 D 38 A 39 C 40 C 41 D 42 A 43 A 44 B 45 B Multiple Choice Test 3 worked answers
1 2 2 33 3 3
i i ii i i
+ + += ×
− − +
+ +=
26 510i i
+=
5 510
i
1 12 2
i= +
B
Page 17 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
2 − −
− − = − π − 1 3arg ( 1 3 ) tan
1i
− π=
32
C
3 2 22 cos sin 22 2
i i π π + = + 4 4
= +2 2i A
4 π π + 6 6
6
cos sini
= π + πcos sini π= ie
C 5 − + =1 4 3z i
(1 4 ) 3z i⇒ − − = Circle centre (1, – 4) radius 3 B
6 y = ecos(2x) cos 2d 2 sin (2 )
d= − xy x e
x
B
7 = + =+
3 12,2
x t yt
−= = − +2 2d d3 , ( 2)d dx yt tt t
= ÷d d dd d dy y xx t t
−= ×
+ 2 2
1 1( 2) 3t t
2= −+2
13 ( 2)t t
B
8 −= +
− + − +3 4 A B
( 3) ( 2) 3 2x
x x x x
1 23 2x x
= +− +
A
9 2
1 d4 4 5
xx x+ +∫
Page 18 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
= + + −
∫ 2
1 d14 5 12
xx
2
1 d14 42
xx
= + +
∫
2
1 1 d4 1 1
2
xx
= + +
∫
11 1tan4 2
x c− = + +
11 2 1tan4 2
x c− + = +
D
10 +
= + 1ln2
xyx
= ln (x + 1) – ln (x + 2) d 1 1d 1 2yx x x= −
+ +
+ − +=
+ +2 ( 1)
( 1) ( 2)x xx x
=+ +
1( 1) ( 2)x x
D
11 ∫ 2sin (2 ) dx x
= −∫1 1 cos 4 ) d2
x x
1 1 sin 42 8
x x c= − +
C
12 ∫ 3 dxxe x
3 31 13 9
x xxe e c= − +
C 13 xy + 2y2 + 3x = 3
d d4 3 0d dy yx y yx x+ + + =
When x = 1, y = 0 ⇒ d 3dyx= −
B
14 + +=
+ + + +∫ ∫2 2
2 1 2 4d d24 5 4 5
x xx xx x x x
Page 19 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
21 ln 4 52
x x c= + + +
D 15 y = tan–1(x2)
4
d 2d 1y xx x=
+
B 16 Divergent C 17 un = (–1)n (n + 1)2
u6 = (–1)6 (6 + 1)2 = 49 A
18 = =
−∑ ∑ 2
1 1
1n n
r r
r
+ += −
( 1) (2 1)6
n n nn
[ ]= − + +6 ( 1) (2 1)6n n n
= − − −2[6 2 3 1]6n n n
= − + −2(2 3 5)6n n n
B
19 + − −+
=−
( 1) ( ) ( 1) ( 2)!( 1)!( 2)!
n n n nnn −( 2)!n
= (n2 – 1) n = n3 – n C
20 9
2 2xx
−
− − 9 2 9 2C ( )
rr
r xx
18 – 3r = 0 ⇒ r = 6 Term independent of x: 9C6 (–2)6 = 5376 A
21 finite A (22) (3r + 1)
B
(23) +∑= 1
(3 1)n
r
r
A
Page 20 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
24 −
+12(1 2 )x
2 3
1 3 1 3 51 2 2 2 2 21 (2 ) (2 ) (2 )2 2! 3!
x x x
− − − − − = + − + + +
Fourth term: − 352
x
C
25 = + + +2 3
2 (2 ) (2 )1 22! 3!
x x xe x
= + + +2 341 2 23
x x x
D 26 f(x) = ex, f '(x) = ex, f ''(x) = ex
21 1 1( 1)( 1)
2!x xe e x e e−= + − + +…
2( 1)1 ( 1)2!
xe x −
= + − +
B 27 ln (1 + x) is valid for
–1 < x ≤ 1 B
28 f(x) = x3 + 10x2 + 10x – 4 f(0) = – 4 f(1) = 1 + 10 + 10 – 4 = 17 Root lies in the interval [0, 1] D
29 = − +2 4
cos 12! 4!x xx
= − +2 4(0.1) (0.1)cos (0.1) 1
2 24
= 0.9950041667 0.995 A
30 2S 13
n
n = −
= − =12 1S 13 3
24 5S 19 9
= − =
25 1 29 3 9
u = − =
C 31 5!
D
Page 21 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
32 OE P W R 4! × 2! = 4! × 2 A
33 =5 4 3 2 1 4 80
3!
C 34 18C9 × 2C2 = 18C9
D 35 5C4 + 5C3
+ 5C2 = 5 + 10 + 10 = 25 D
36 0.75 × 0.40 = 0.30 C
37 1P (A) =3
1P (A B )6
′ ′∩ =
1P (A B)6
′∪ =
1 5P (A B) 16 6
∪ = − =
D
38 = − −
1 2 1 1 0 4AB 1 0 1 1 1 1
3 2 2 2 1 0
1 3 61 1 45 4 14
= −
A
39 −1 1 01 2 1
1 3 2
− −= − +
2 1 1 1 1 20
3 2 1 2 1 3
= 1 – (–3) = 4 C
40-42 2
2
d 4d
y y xx
− =
CF: 2
2
d 0d
y yx
− =
AQE: m2 – 1 = 0 m = ± 1 ∴ y = Aex + Be– x
Page 22 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
PI: Let y = ax + b
=ddy ax
2d 0d
yx=
Substituting into the differential equation: –(ax + b) = 4x a = –4 b = 0 ∴ y = –4x General solution is y = Aex + Be–x – 4x
40 y = Aex + Be–x C 41 y = –4x D 42 When x = 0, y = 2 ⇒ A + B = 2 [1]
xd A Be 4d
xy ex
−= − −
When d0, 0dyxx
= =
A B 4⇒ − = [2] Adding [1] and [2]: 2A = 6, A = 3 A = 3, B = –1 ∴ y = 3ex – e–x – 4x A
43 1 2 1
A 1 1 13 3 a
= −
1 2 11 1 1
3 3 a−
1 1 1 1 1 12
3 3 3 3a a− −
= − +
= (a – 3) – 2 (–a – 3) + (–3 – 3) = a – 3 + 2a + 6 – 6 = 3a + 3 A 0 1a= ⇒ =
A
44 2d 1dy y xx x− =
11 d ln ln 1x x xxe e e
x−− −∫ = = =
Page 23 of 23
Unit 2 Answers: Multiple Choice Tests © Macmillan Publishers Limited 2013
= ∫ 21 1 dy x xx x
1 dy x xx
= ∫
21 12
y x cx
= +
312
y x cx= +
B
45 when x =1, y = 2 ⇒ 122
c= +
122
c = −
=32
31 32 2
y x x= +
B
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