,.......

Jlints for Challenge Yourself

I Gradients, Derivatives and Differentiation Techniques
Chapter I
What do you obtain when you solve dy = 41
I' d.x '

chapter 12 Applications of Differentiation

2, Let S 1 and 5l represent the distance travelled in air and in water respectively. Then find an expression for t, the
time taken.

Chapter 13 Differentiation of Trigonometric, Exponential and Logarithmic Functions and their Applications

1. Take In on both sides.

3. Recall the Double Angle Formulae for cosine.

Chapter 14 Integration

1. (ii) Use the fact that integration is the reverse of differentiation.

2. Factorise the denominator.
3. (i) Use Product Rule.
(iii) Integrate the given equation with respect to x.

Chapter 15 Applications of Integration

I. Consider the shape of the respective curve.

2. Rewrite sin,x in terms of tan x and sec' x.

COS X

3. Write down a definite integral to find the area. What do you observe about the terms within the integral?

Chapter 16 Kinematics

I. Form a pair of simultaneous equations to find the values of a and b.

Chapter 17 Proofs in Plane Geometry

I, (iii)
Let M be a point on the tangent at A such that angle PAM= y. What can we say about angles ACP
andBDP?
2· What
can we say about triangles ABS and AQX? How about triangles AQX and SQX?

99
.........__
Hin b fo r ChaUcnge Yourself }
E
 ...
Answer Keys
Chapter If c;raJic-nh, _ (a) 6 x~ +6x =+ 6 12. (o) /1 = - 1. k = - 6
2 (d) 6(x+S-6~ )' ( 1---l_)
Oct'ivatn1..~ anti (b)f:i111=- l. 11 = I
(3x+ l )' I[;,
Ditkrent,atiun r\.'1.:hn1qu"~!i. 13. (i) (2. 19). (-2. -25)
Sx- 16
)-j (1-f)
(b)
PractisC' N o w 1 (ii) O (iii) 7 (c) ½( x +f
3~( 2x- s )'
I
(a) -7 I
6x ' + 3x+ l4
I
l./x
I
(b) 3ef;' (c) _
Jcix' - 7)'( 1+ 4x )
14. (a) (b) 3 l/x' (0 ¥J<Jx-1)'
I
(c) 0 (d) -74 15.x < l orx > J 8. e ,, = -2x +3
3
3. .2;..
37>7 9. (2. 2)
(c) -2-Jx' (0 o E.~r.rci sc 11 B "
4 3
Practi se Now 8 I. (a) 14 (2 < + S) , 10. (i) --+-
2x -5 x+l
(b) -32(3 - 4.,·i
Practise Now 2
I. (i) 15., 0 - t\x; 30x - 6 (ii) _ _ _s_ __--2_
.. 2 (c) m x - 4) ( 2x- s )· (x +l )'
I. (a) -2,. + _l_ (u) x >
5 5
x· 3ef;' I (d) 27(x + 4)' I I. a=
4 .b= - 3
2. -
3
or I 4
(b) - ~ + a (e) -15 (5
. ., - 3)' IJ. O < x < l6
3x"
Practise Now 9 4 '
2.
-R+ 2Jx
4 3
3 - 3x : : x < -l o r x > l (0 i(i- 1) Exercise I IC
I. (a) -9., 0 - 4x + 6
(x ' +1 )' 3
2. (a) - (3x+2 )' (b) (Sx + 1)' (40x + I )
Practise Now 3
Practise Now 10 (c) 2(x - 3)'(5x + 3)
I. 28 30
Q ::!S. X <2 (b) ( 2-Sx )' (d) 3( I + Sx)'(S0.,0 + 4x - 5)
2 I
. 6 2. IS
Exercise 11 A 72 x
I
(c) - (2+ 3x ' )' 3. - 3, 5 or -
Practise Now 4 I. (a) IOx'
(b) 4l/x' 11
I. a= 8. b = 3 6x 7 __1_
4.
2. a = 5. b = - 6 (d) _2_ (d) - (x' + 2)' 2Jx
(c) O
x• 24-105x J
60 5.
Practise Now 5 2 (a)
(c) - -- (0 0 (c) (3 -4x )'
I. (a) 84x'( 4x' - I)' 31./x'
9x (b) 12x ' -2x+9
4 ( 2x+3) 2. (a) IOOx ' +~ (O 2(5- 3x ' )' .J2x' +3
(b) .Jx' +3x-4 X

5 I (c) (2lx-2 )J (J x -1 )'

(b) 4nx ' - 2 , . - -- 3.
x · 3ef;' (a) .J2x+3
(c)~ (d) 3x ' +4x
( 2-Sx 'J' 20 Sx
(c) 14x ' - .Jx' +4
3x ' (b)- 8
2. 18 6. - - -. ;lor3
3. (a) 2rr, (b) amr · ' (2 -x )·
9x '
Practise Now 6 (c) rrny" · ' -4
(c) 5
4. (a) -3 (b) 30.!2_
7. - 5 or l2
I. (a) 8(2x + 7)' (7x - I) 4x
16 1 I
(b) 15-16x-2Sx '
(d)~
3 (2x · +S ) 8. 8 11 = 4 and b =
2
(c) 31 (d) 4~
2.Jsx+4 27
3b<
5
2x
2 _ (a) 3a +
5. (a) 8 (b) 4~ (cl-.~)' 9. (i) --y !'S; X !'S; 2
27 q (27-x")
(a-bx ') (c) 2 (d) I
(b) 2(5ax-3x')' (l0a'-18a'x' I E.,:erdse 11 D
6. (a) -5 (b) 7.!2_
-1Sax'+2lx') 16 (O - J (x-3 )'
7. (4. 3) 1. ( a ) -3- .
4. (2-x )·
Practise Now 7 5
8. 61 + 5, I < - 5. -18
I. (a) ___ 6_ _
6 (b)--s-.
(1+4x )·
(4x-3 )· 9. (a) 9 I - I+
3
2 6. -9 )' ( 3 ) 6x : +48x+-l
( b ) --7 -
7 _ (a) s(x ·, +-;3 -')x---r
x (c)
(3x ' - d
(b) Sr' - 31' + 41
(7-3x )'
4x (d)~4
(c) 41 ' +4-~
1' (b) - 3J(2x ' - s)' (1-2x' )'
IO. (i) a = - I, c = 2
(e) 2x ' -2x
(ii) (-1 , 2)
I I. a = 3, b = - 8
(c) ----:r;--
5(4-Jx )'
( 2x-l )'

(0~
4 (x ' +J )'

A ns wer Kc-rs
200
1111111111
 ExC"rdsc 11 F 9. a = - 16.5, b = 54 Practise Now 12
.\ - ,\
J. (R)
i.<• l ~
X

(c) x < - I
> -5 (b) X > 2 10. Increasing
(ii)H.7)
6.\' 1 + IO
(d) x < 3 or x > 5
(b) - ~ Practise Now 13
(e) x < - 3 or x > 4 Ch.1ptcr 12 Applic,1tions of
x-6 (iii) r = 2.8,
5 Differentiation
(0 - I < x< I minimum A = 65. t cml
(d ~
(x-3 )' Practise Now 1
7
2. (a) X < I (b) x > - Practi se Now 14
d) Jx' +4x 4 1 y= -Ix+~· y = Ix+!.!.
. 2 4 ' 3 2
(2 ~
(x+ ll (C) - 4 < X < 2
5 13 9 49
= 7 -lr (Ii)
(I) £! cm'
16
. I (ii)!.
(d) - 1 <X< 2 2· y = 9x-9 ;y = -sx•rr r

3, (1) 9 25
(e)
3
-I < x < 4 Practise Now I 5
l , 8/1 :2, k •- I Practise Now 2
3 (i) .J251 ' -2601+680 m
(ii) Yes (0 x < - l or x > I
s. (i) 7 I. (i) l;y=-x-4
(ii) I = 5.2, m inimum distance=
6. 6.32 units 3. 0 < X < )
4
(ii)( I, -5) (Iii) ¾ 2m
7, (i) (.,0 + 5)(x - I)
4. x>3
2. (i) y= -¾x+2 Exercise 12A
( i xi·)+S
~ .x-
~1 5, (i) -3 < X < 2

3,o).
(a) y= 5x- 7
(ii) -8 1 or44 (ii) A( 2 8(0, 2); ( 1}, I) I.
(b)y=-16x
{iii)~-~
J(x' , s)' 3(x -1 )' 6. e -2x' - I Ox (iii)No (c) y= 2
7. (i) 0 < X < 7
1 I
9. (I) -H!::!!:,
k- X 8. 0 < x < 3 or x > 6 Practise Now J (d)y= Ix-I
9. (a) B (b) C 2; (-0.451, -0.823), (-2.22, J.82)
2 (a) y = __!..,x+~
E.xrrdsr tl E (c)A ' 10 10
Practise Now 4
1. (a) 12.,0 ; 36.," (b)y= -9x+~
(b) S; O (c) 6x + 5; 6 Review Exercise 11 I. 4 3

I 2 67 (c)y=x+2
I
(d)-,; ' I. (a) - (b) 18 2. 2
X X 4 (d)x = .!_
(c) 30(3x + 2)'; 8l0(3x + 2)' 3
I I
(c) 2.
16
(d) 0 Practise Now 5
7 4 79
<o--
20 ,--=-
4J(x-4 )' (e) -5 (f) -13, 13 (i) 9.6rr cm' /s (ii) 57.6rrcm' /s 3. y =
4 x+l;y= -
7 x+IT
. 23 3 446
I 2 2, (a) 6(ax' + bx)' (2ax + b) Practise Now 6 4 · (,) 3 ;y = -23x+23
l. (a) l-7 ; ?° (b) 10(2ax + b)'(5x' - ab)'
(i) ~rr cm/s (ii) 5 cm'/s ") 23
2 I 2 6 (l 7ax' + 6bx - a'b) 3 (II -82
(b) 7 -7 ; 7-7 2abx~ +2b x +2acx 2 3 2

(c)!l,.Jx' __1_,
(c) - ,
(bx' -ex)·
Practise Now 7
(-1, 10), maximum point;
5. (½,4)and (-½,-4)
2 2,JX
(d) acx' +2bcx+a' (3, -22), minimum point 6. er= 5x
~.fx+-1- (cx' -a ')' 7. y =X 5; (0, -5)
4 4-Jx' Practise Now 8
-
7
3. (a) 25(5x - 4)'; 500(5x - 4)' 8. y= 2x- 7; 4
1 _ _ ; _ 2_ ,
(d) _ _ (0, 0), point of inflexion;
(x -1 )' (x -1 ) b ) ~ · 3x'+30x 9. a = 2, b = -3, c = 5
( ,.---,-; ' 3 27) . . .
v2 x +5 v C2x' +5l ( 4.- 256 , mm1mum pomt 10. (i) y= 3x-5
(c)-~ ; ~
(x-4 ) (x -4 ) 150x 2 +45
(c) 15xv,x- +,; .J5x "+3
, Practise Now 9 (ii) (½ , -1¾)
(O x' +2x . _ 2_
(x+I )' , (x+I)' (d) 6x ; 12-18x
2
(i) (½,-~)and (-4, 159)
12. (i) h = 2, k = l

3. (I)
. (3
4' 3sI) ' (-4•-ls
I 5) 1(2+9x' )' 1(2+9x ' )'
") (I3•- 327
( 11 20) ,mm1mumpomt;
. . .
(ii) (¾,2¾)
(e) _ _2_7_.
13. Yes
(ii) 152 (3x-5l'' (3x-5)' (-4, 159), maximum point
14. (29.3, 8.58)
S. • • 16,b, 226
(O 3x+34 ; - ~
Practise Now 10
7. (i) ~.fx +.2_ . J <2x+9)' j (2x+9)' Exe rcise 128
2 Jx' (3, 1), point of inflexion;
I. 0.268
3 I
4. (½ , -1½), (-½,1½) 3 11 ) . . . 2. 6x' - l 2x + 4; x .;; 0 or x ;;,, 2
4.fx-2,R ( 4 ,-16 128 , mm,mum pomt
4
(ii) No (iii)No 5. a = V, b = -l 3. 4rr cm'/s
s. e-x' 6. I 2r' - 6r - I 8; -I < r <
2
Practise Now 11 4. 120 cm'/s
1
Minimum gradient= -31.4, 5.
7
·
5
18' 5
8
x= 1.6 12 cmls
8. (i) Increasing 6. (i) 3.6rr cm'/s

(ii) Decreasing (ii)No

Answer Keys
201
 1111111:

15. (I) " =·' · /, =4 14. 9.27 km. 16.1 kn,

7
. T
1.:to
cm ),s \ z -fq •,t ;;
(ii) (0. 5). poi nt of inllcxion
(I) ( , __, )' 15. (ll)x = 15;S24 00Q
8.. -45 umls '/s
16. (I) ( I. - 2). minimum point ; 16. 70.71 km/h
L' • (il) l l. 0). muimum poin l : (4. 5). m u..x imum point;
9. -
20 um v s (5. $ ). minimum point t6. 2), minimum point
17.llnoon
10• .xxht>t ..:m '/ s S. (a) (.\. l 71l. minimum('\'int;
11. No
I l. 4 c-m ' rs;, l.o ,._,,_1:/s (-21.-m½)· Exercise 12 D
I. Chapter 13 Differentiation
minimum poin t: of Trigonometric,
I.\. (i) 2. emfs (ii) No 2. (I) It =
x·
(ii) 2.52
Exponential and
Sff.
1-t. (i) + l l' (2.!.2. 171~)·
S
nuu:imum
(Ill) 19.0m'
l.ogarithmic Functions
and I heir Applications
(ii) 0.0l m/ s point O= 80- 2r
3 _ (i) Practise Now I
(b) ( 1, 0). min imum point; r
15. ,;;,._ mis "' l - l. -4 ). maximum point
(11)20 (HI) Ma.,imum
1. (a) 3 sec1 x- 2 cosx
4. (i) so ,
16.. ~ a emfs. (,) (2 . .l). minimum JX.-'int, (b) 3,'(5 sin x + x cos x)
-4. n (0. - 1). muimum poi nt 5. (i) nr ·, + ( -12-nri·
- ) 2(5x+l)sinx+l0cosx
2
17. ( i) 10 fI cm
-.J; 6. (ii) (0, 0). minimum point (ii) No (c (5x+I )'
7. (a) (O. 0 ). point o i in flexio n 6. 44.6 Ill (d) 6(3x + 5 cos x)'
ntt ¼-II c:rn/ s (b) (0. 5). point ofintlexion, '0
7. (I) ,, = .:, 2. II= 1,b =- 1
( I. 4 ). min imum poi nt
x·
1 (iii) x = 2.61. minimum
18. (i) '. =' cm/s (c) (- 1.0l. point ofinilexion; Practise Now 2
_,t, n: 8. No
(ii) 1.59 .:mis
( I 3)
- - .6- . maximum . 9. x = 100.y=
400
3 I. (a) -IOsin( 2x+%)
4 4

1x')
19. (i) °"'-.-rea.-., (ii) No
(iii) -6-lSk crn/s
J>'"""'lint
(d) (-2, -6). point of inflexion;
10. (ii) (15x-
(iii) 937.5 cm'
cm' (b) 12xsec '(3x -
1 )+

El.~rci.~ tl:C
(2. -6). point of intle.xion;
(0. -70). minimum point
11. (ii)224 cm 4tan(3x-})
13. 6. 16 unit'
I. (a ) {-1}.-1{).minimum 9. (i) 6.\ - .lOx + 36: (l. 35).
0
14. (i) S275
(c) l0sin-l xcos5x - 8sin5xcos4x
sin 1 4x
(3. 34) 15. (i) 8km
point 2. sec n.x (nx tan nx + 1)
(ii) 1lx - 30; (2. 35),
(b) ( I. -2). m:i.-cimum point ma.timum point; (3. 3-t). Re,icw Exercise 12 Practise Now 3
(c) {1½ - - 3¼) . minimum
minimum point 1. 24.\.! + 22x - )0, -35 < X < 43
10. (i) a= I. b = 16 (a) - 20 cos ' (5x) sm
. (5x)
3 9 9
-3, 89I)'
1 27)
(ii) Minimum point 2. (i) (I, 0), ( 4•-128
. ( l
pomr~
11. (i) p = -4, q = -1 (ii) (I, 0), point of inflexion;
(b) ~sin'(~),o{;)-
maximum point
(ii) (-+-34¾) I 27)
( 4,- 128 , minimum 20tan( 2x+} )s~ 2x+%)
1

(}.s). minimumpoint: (

(... ) (-12,1 -34:i°3);

(d)
point
H--3).maximum
lU
3. 1681 Practise Now 4
minimum point; (2. 51), 5. k=8,p= 13,q=-2,c = 12 I. 0.207
point maximum point
(e) (2, -64 ). minimumpoint; 6.
.
(1) e)' =40x
1
2. -3
(M -I <x < 2
( -2, 64), maximum point l . dx dy
(1i)e dt = I, dt= -40 Practise No\\' 5
(f) ( 3, 0 ). minimum point; 12. (i) a = 16, b = 1
(a) 15e" (b) -4e'· ~
(- 3. -1 2).maximum point (ii)(a) D<creasing k
7.
16 cm /s
3

(g) (0 , 4 ), point of infktion (b) Increasing (c) (6x -5)e'•'·"

(h) ( I. -6), point of inflexion (iii) Minimum point 8. (i) ½(18-3x+l2x ' -2x '); (d) 2e'' '' (cos 2x - sin 2x)
1. (al ( I. 6). maximum point; (iv)96 e' '' '(5x-l )
(2~ S). minimum point ·i 33 _ .!.(-3+24x-6x ')
I 3 . (1 p = -T . q = 54 2 (e) 3x'
(b) (i-, ·27-'' ). minimum
_/
(ii)(9, -120.5) (ii)..!..unil/s
I05 (f) _.!_,,,, .. ,
2·
poin t: (iii) (2. 51), 9. (ii) Greater

(-1.4½).ma.timum ma."Cimu.m point:

(9, -120.5),
10. (ii) r = 1.97, It= 3.28
23 4
Practise Now 6
96
point minimum point 11. (ii) 120x- ~ ;
x·
(c) (5. 150), min imum point
(d) (0. 0 ). ma.-cimum point;
(h•) Minimum gradient =
-36.75, X = 5.5 120 +
4 8
6? Practise Now 7

(4, S). minimum point 7

X
(a) _i_ (bH cot 4x
14. (i) - - - -, (iii) x ;: 2.68, minimum Sx+2
3. (-3. 34 ), maximum point; (lx -1 )" 20-ISn
12. r;:8; - -cm' (c)~
t l . 2). minimum point 3
(ii) No 3(x'+2x )
13. (ii) 26.5 (iii) Maximum

\ o,...,n Kt"n
202
' ..........
 6. (a) (2ax+b )e u'••• 6. (i) -3 sin x - 5 cos x
prscti" Nows 7. (a) ~ , x ' sin ( -}x ) +
(b) -e•'-• (ii) 2. 11 (iii) Minimum

Je, -
1- i .b4
2 nx cos
).. 0. 198 (c) ~ er.;:; 7. (a) ( 2le)
2 X
Je(ln(e' +6))' (b) 4 cos 3x sec' 4x-
(d) ne ' (b-1)
(¼, -¾)
3- ~ 3 tan 4x sin 3x
(c) 10 , 8. (b)
practis< Now 9
11
(sinS x +cos5x )· 9. (ii) I
I. )
·=Jx +-+-
12 2 '
8. (i) 4 (ii) y =-.!.x+2
4
-4 sin .!. x- (9-x lcos.!. x
(d) 4 4 Exercise 13C 9. 2+48./3
y = -~ x+ ~+ . ' Ix
3 36 2 4 sm I. (a) _I_ 10. (i) -4. 12 units/ s
4 x-5
(b)~
x' -3
).. 5 (ii) Decreasing
8. (a) sin!:_!:cos!:
J. A(!, l ),y: - lx+ 3 X X X (c) 2x-2 (d) 1 I. (i) 28.3 g (ii) 29.2 days
x 2 -2x 9x-l
(b) 3a tan' (ax+ b) sec' (iii) 0.593 g/day
practise Now IO (ax+ b) 6 I
23 (e) 2x+ 7 (O 2(1+x) 12. -0.0540 thousand units/
1· (i) (3x-4 )(2x +S) (c) a cos ax cos bx - month; -0.00 I I 7 thousand
b sin ax sin bx (g) _.!. 2
(h) 5-2x
units/month
X
(ii)~units/s (d) acosax cos bx +b sinax sinbx 13. 4 watts/s

(i,-./2) ,
23 cos 2 bx 2. (a) -tan x
9. No (b) sec ' x- 7 cosx 14. (i) (f,-1) ,
l.
./3 units/s IO. 8P = 8, q = l or p = -16, tanx- 7 sm x
(3411 ' -./2)
5
q =-2 (c) _ _ 3___ 1_
3. 3(1 - lx)e·" - !Se·" ,
11. -4.99 2(3x-l ) 2x+l
-10.0 units/s

Practise Now 11
12. -1.73 2
(d) 3x - 3(5x+4 )
5 (ii) ( f, -1) , maximum;
13. a= 5, b = -10, c = -10

I. 3+3 ln .,; (i, -¾) 14. (i) a cos ax - b sin bx,

-a' sin ax - b' cos bx
3. 3.10
I
(i ,-./2), minimum;
4.
-./2),
(a) xlnlO
l. 1.02, maximum (ii)S 3
2 ( 1t , minimum
15.a=-l,b=O 4
(b) (2x+3 )ln5
Practise Now 12
15. (i) 2e·' COS X
(ii) 1.1 3 (iii) Maximum Exercise 13B (c) cotx
Ina
(ii)!!.
I. (a) Se' (b) 3e'"' ' 2
Exercise 13A (iii) Maximum point
(d)~
I. (a) Scosx + 3 sec' x (c) -2xe''•' (d) ~e"· 1 (x' +!)Ina
9 3./3 3./3
(b) 9x(2 cos x - x sin x) 17. 0, 16 , -16
sinx+ (7 -x )cosx
2. (a) \-::-e
2-vx
5 (b) 5 .5.
- x 2 e.r (e) 6x +7
3

I
2x 1 +7 x 18. l <X< I
(c) 6sin 2 X
I
I_ (d) e.r - ~
(c) _ _ (0 -2 tan 2x
(d) 5(4 tan x + nx)' 19. (i) - 33 (ii) _73
(4 sec' x + n)
2,R e 5. 11 x+l
3. (a) e",,, cos x 6. (a) !Ox'(! + 4 In 5x) (iii) 0
2. (a) 28 cos 7x - I
(b) -2e='-' sin 2x 20. i-+,fj
(b) (I+x ')' +6x(l+x' ) 2
(b) 24sec '(3x-i) (C) 4e• 1ua SeCl X x+e 21. (ii) 1.07 (iii) Maximum
(d) 3(cosx + sin x)e''"'•'M' ln (x+e) 22. (i) 50(28 + sin 20) cm'
(c) Ssi~Sx
cos· sx 4. 5 (ii) 15 cm'/s
(c)-1-
(d) _ 9sec,' 0.9x 5. (a) x(x + 2)e' x-.fx 4kr'
23. - -- cm' /s
tan · 0.9x (b) e'(cos x - sin x) 5
(d) 1-l~x
3. (a) -4 cos' 3x sin 3x (c) e'(2 cos 2x + sin 2x) sx·
7. (a) -308 (b) 2.12 Review Exercise 13
(b) 14tan'(2x--i) (d) -2e' sin x
8. 3.41 ; 2.72 I. (a) 211 - 2n sin nx
(e) xe""' (½xcosx+l) (b)2 cos 2x
sec '(2x-'i) 10. -0.009 00

(c) 24 sin' Bx cos Bx - (0 e·•(-

2-Jx
1
Exercise 13D
(c) 3 sin (6x - 2) - 4 cot 4x
(d) 6e'-' + e' "
~._s_
2 sin 0.Sx
(g) 2xe·'' (I-x ')
Sn s./3 (e)
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3. 2;y = 2x- 2
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9
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5
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e" 8. 0.340 cm/s
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9. AB = 40 m, BC= I 20 m
5 3 . I I , 3 ,,---.
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6 4
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5
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9. -JO uni t/s Practise Now 8

10. Sx 2 _
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; -
15 .
umts/s
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X
4 8 3. a= 5, b = -2 I
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4. (i) 1-~ (ii)~ 3 4 2x
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minimum 4
(iii) J 6. (a) fs(3x- 7 )' +c
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2.59 7. 6 Practise Now 9
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9. (ii) 19.0

min value = - -4 -
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45
1
16. Min concentration = (c) x- - -+ c (d) 1o~(2x+7 )'
Chapter 14 Integration )e h
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5
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20. (ii) -i; Maximum 5 7. m = + 6994

21. (i) V = 3nh'

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9 In (6 - 9x )+ c
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2
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(ii)h = 12, minimum 8
Practise Now 11
(b) +4,/x +c (b) !!...ef? +2.. i.W+<
I. (i) - 2<' sin x
Challenge Yourself 46 16
(c) ...!...x ' +Ix -...2....+ c
5
3. a= -2, b =4, c =2 12 4 16x (ii) -
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64x 16· 20

An.S'"''C.1" Ktrs

E
204
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) l~•7
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5
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20 J ( ~3 I . I
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3
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21 5 (ii) 3x'+5
e'
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X 15
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l. (a)y; x+!cos2x+J 1
13, (i) y= 2tan~+ ~ (ii) ftx+sinxcosxl+c
2 2 .,_,3 II. -lncosx+c
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4 4 andq=-4 I
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5 14, I sm
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4
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.
I .!..sin2x+~x-.!..x 2 +c 9
cosx+c 2 4
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.
18. 6 sec• x - 6 sec' x; ( 11
I 2 4
Exercise 14C
(d) 3tan(4+3xl+c 5 l (iii)x+ C
tan x+5tanx+c
I, (ii) xJ3x+I +c 3 1
14.(i) - -
3x+8 x 1nx
2, (i)
3 I
(ii)
J(3x+4 )
4 tnOnx)+c
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J3x+4 +c

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 8 Practise Now 9 Exercise I SB
15. (i) 5+-- 8 - (i) 2sec ' (2x-~)
2x-5
I. (i) 3 cos x - 3x sin x I. (i) 60x(5.r - 3)'
(ii) Increasing (ii)0.571 units' 4
(li)2 (ii) -665
(iii) Decreasing 9. (I) 0.457 (ii) 3
2. (ii)-'3+1 2. (ii)~
(iv) Sx + 4 In (2x _ 5) _ 9 9 IO. (ii) 0.5 (iii) 0.5 rn'

Challenge You rself

Practise Now I 0
I. 51 units 2
3. (ii) i 1 I. (a) 68
I
3 units'
Cl I ,.r;.
I. (i)~ 2. (../2-I) units'
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3
8x ' -27
I 2 (ii) 68.!.units'
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x-1 x-3 3

4 I. 14~units 2 (ii)2.7 1 12. (i) Sn

2 24 3 (ii) 3.25 units'
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6. (ii)~+.!. (ii) 2
4 2
3. (i) = x d 'y + dy 7. (i)4x(9-5x')' (27x-95x' -20) (iii) 7.75 units'

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dx
I ,
dx '
5
dx
Practise Now 12
4.5 units2 (ii) - IO 679 760.25 14. eh = -3, k= 32
(m y= x -lnx+ 15. (ii) 2+-3-.
3 3 8. (ii)4.47 x-2' 0 ·704 units
. '
Practise Now 13 9. (i) 4xe" (ii) 3e' + I 17. 1.25 units'
16 IO. (i) x + 2x In x - 3 18. (i) P(0, 4), Q(3, 3)
C hapter 15 Applications of (ii)7.34
Integration Practise Now 14 (ii) 6¼ units'
II. (i) 5x' + ISx' in x
Practise Now 1 I. 1.!. units2

¥
3 (ii)48.99 19· (.!.e• +1-.!.) units' ·
(a) (b) 10 2. (i) P(0, 17), Q(8.5, 0), R(3, 11) 3 a a '
12. (i) 4+.!. _ ( _)-
(ii) 45.25 units' x 2 x+3
(c) 158 3 [ ( I-; le"+;] units'
81 (iii)~
108 2(x-3)
Practise Now 2 (ii)4x+Jnx- 20. 17
(a) 78 (b) 37.8 Practise Now 15 3
1n[(x+3)(x-3)]+c;
2 Review Exercise 15
(i) P(I , I) (ii) ¾units' 2.98 64
Practise Now 3 I. (a) (b) 66.0
Exercise ISA 13. (iii) JJ-1-% 3
-ff
(a)
2 I (c) - (d) 34.1
(b) .!.
1. (a) 6
6 14. 4n+--
3"3
(b) +2-~ 2 (e) 0.566 (0 1.06
6

Practise Now 4
2 (c) 1905
7
(d)
3
I
15. (iii) 1 2. 4 or6
2 n 3n
(e) 3 (0 -;; 16. 4 cos 4x- 2 sin 4x 3.
1. (a) 19.09 (bl 20.09 2' T
123 17. (i) I
2. 1.108 (g) -~ (h) 4. (i) 2--s- (ii) 0.562
7 2 x+3
Exercise I 5C
Practise Now 5 Cil _I3 (j) -I I .
5. (i) 4xe2.ir
1
(ii) 22.3
I. (a) 21 units' 6. (i) I+ lnx (ii) 2.55
(a) 2.92 (b) 16.6
2. (a) -IO (b) _!_g 3
3 (b) 5.71 units' 7. (ii)0.143
Practise Now 6 (c) (d) 8 (c) 2.72 units' 3x 3 +4x 2 -4x+l
55 8. (i)
(a) -IO (bl 8 n' (d) 6.91 units' x'(l-2x)'
3. (a) (b) +1
4 2. (a) 5.21 units' (ii).!..!.
Practise Now 7
(c) I J3 n (b) 1.51 units'
8
1. (ii) 13.76 2 (d) 3+4-12 9. 2
(c) 0.386 units'
2. (i) 2xe'-'+ e'-' 4. (a) 1.19 (b) 0.781 10. 0.305
(d) 2.51 units' II. (i) P(4,8),Q(6,0)
(ii) .!.(e' +I ) (c) 39.4 (d) 86.9
2 3. 9 units'
(e) 11.7 (0 7.00 (ii)8: 19
3. (i) 3x + 6x In x
(g) 73.9 (h) -0.0754
4. (i) P(0, 15), Q(3, 9),
12. (i) PH, 4½ ), Q(2, 8),
(ii) 14.72
5. (a) 8.05 (b) 0.462 +½,o) R(4, 2)
(d) 5.16
Practise Now 8 (c) 2.08
(e) 0.472 (0 21.5
(ii) 12: 13
5. (a) (2-../2) units'
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ll SuI units~
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(i) ~ - - l -
2x+3 x+I
6. (a) 50 (b) -2 (b) units' 14. (i) ~+ln(2+xl

i
(ii) 6.85 2+x
7. W= .!.k(x'
2
- x') 6. units'
2 I
(ii) l--2-
8. 0.733 2+x
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10. (i) -tan x (ii) ½ln2
15. (ii) 13.9 units'
16. 62

Answe,. Keys

E
206
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8. (i) 0.405 minute 9. (i) 55 (ii) 33 s
y 0 urself
a,:all••&• (b) O
(ii) - 178 m/ min (Iii) 0.523 km
_ (a) 0 (iii) 56.1 m (Iv) a = 500e 20'
1
(c) O 9. (i) 1.6 1 s (ii) N o JO. (i) v = kn sec' nl,
a = 2kn' tan nl sec' nl
2- ~units' 10. 8" s
1 I. (i) 5.85 mis; -0.0685 m is'
J. 4-4.9 11. (i) No (ii) m/s (iii) 14.5 m
12. a p =2, q =-4, r = 2
_,!M1Hiil¥ Exerci se 16B
Challenge Yourself
I. (ii)39 m
practist Now I
(ii) 42 mis' I. (i) s = -t' +13 t' +I01 + 4
(i) J6rn/S
(iv) 8.5 m (ii) 9.04 (iii) 254 m
(iil) I or 1.5 Chapter 17 Proofs in Plane
(iv) 250 m Geomctq
M sm
practist Now 2
2. (i) T m (ii) 6 s Practise Now 3
3. (i) 8.5 m is (ii) 0 2. (ii) 6 ADR, 6 CQD
(i) 6.5J cm .
4. (i) 4 or 6 (ii) 36 m (iii) 6 QCP
(ii) v=3 + Scos 2t;a = -16 sin 21

(iii)6.64m (iii) m (iv) m Practise Now 10

3 3
5. (i) a = 41 + I - I Ok (i) 33°
practist Now 3
(ii)2
(i) 0_405 s (ii) 58.8 m
(iii) 0 < I < 2.5 or I > 7 Exercise 17 A

(iv) 2.5 or 7 11. (i) 6CDA (ii) 6 QKC

Practist Now4

(i) s = ~t ' -1 2 -21+9 (v) s = !.1 ' _.!.2.1 +351 2

Exercise 17B
3 2
(il)I 6. (i) -4 ,;;; V ,;;; 12 8. (ii) 6PAQ and 6PBR
• I
(1v) 243 m (ii) a = 3, b = 3, c = -4
(iii)30m 3 Exercise l 7C
7. (i) \" (ii) 9.02 m
Practise Now 5 9. (iii) 6 CDA
(i) 3or 5 (ii) 36cm 8. (i) 4.47 (ii) 36.6 m 17. (i) LQAD

(iii)36cm (iv) 2i cm
9. (i) 0.408
(iii) 24.4 m/s
(ii) 17.9m

Revision Exercise Fl
Practise Now 6 10. h =-2, k = 12
I. (a) 0.183 (b) 4.49
(il)2.70m 11 (ii) 860 m
. 3 2. (ii) _g
12. (i) 7 m / s (iii) 8 3
Exercise 16A 5
(iv) 31.5 m 3. ( i i ) - --
I. (i) 133 emfs (ii) 2 em fs' 2(x+2)2
(iii) 11 (iv) 452 cm 4. (i) a = 9, b = 16
Review Exercise 16
(v) 452 cm
I. (i) 4 (ii) I.SI m (ii)y= 9x+~-10
2. (i) v=6t' - 6t - 12;a= 121-6 X
10
(ii)2 2. (i)
27
mis (iii) 5m 5. (i) A(3, O) (ii) ~o units'
(iii) -13.5 m/s 3. (i) -2.94 mis' (iii) No
3. (i) 24 s, -576 cm/s (iii) 5.42 m 2 0
6. (i) ~ m
(ii)2048cm 4. (i) 1 h
(ii) 15 s, -37.5 mis
4. (i) -5 mis (ii) 4 s, -45 mis 5. (i) V= 312 - 201 + 25;
(iii) 100m a= 61- 20
Revision Exercise F2
5. (i) 3.29m (ii)5m (iii) 5
1000 1. (a) 53 (b) 0.428
(ii)v = ( 2-
3
°;2) mis, (iv) } or 5 (v) n m
2. y = -x' + 2x' + X - 7

9.,fi_
6. (i) b = -24, c = ½ 3. (ii) a = -¾, b ¼, c
= = ¾
a= --2- mis' (ii)-22.5 mis; 3 mis'
(iii) 0.243 minutes 4. (i) -1 (ii) 2
6. (i) 4m 7. (i) t, = ~.l, =7 ···> 628 umts
(ii) 3 mis (m 81
. 2

(iii) -6,/s mis' (ii) 925 m (""") 827

m 12m 5. (i) 37 (ii) 20 s
(iv) 4.72 m 24
(iv) 9 mis' (iii) 0.218 km
7.
(i) v = k sin 21, a = 2k cos 21 (iv) SOOe'"
8. (i) 29.4 m/s (ii) 44.1 m
(ii)O' 2:
2 , rr {""")
1u y es

C A nswer Ke )'S
207 E