Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
3A_C0001
Find the value of 6! . (1 ar" Solution:
6 $ 4 # 2 1
6! %20
1A (1"
3A_C0002
Find the value of 4! #! . (1 ar" Solution: 4! #!
4 # 2 1 # 2 1
24 6 1&
1A (1"
3A_C0003
Find the value of ($! " 2 . (1 ar" Solution:
($! "
($ 4 # 2 1" 2
2
(120 " 2
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3.1 3.1
14 400
1A (1"
3A_C0004
Find the value of
4! $! #!
. (1 ar"
Solution:
4! $!
#!
4 #! $ 4 #! #!
(4 $ 4" #! #!
4 $ 4 24
1A (1"
3A_C0005 '
Find the value of C # . (2 ars" Solution:
C #'
'! (' #"! #!
1
' & % 6! 6! #!
'& % # 2 1
&4
1A (2"
3A_C0006 & & Find the value of C 2 C 6 .
(2 ars" Solution:
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3.2 3.2
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
& & C 2 C 6
&! (& 2"! 2!
&! (& 6"! 6!
1
&! 6! 2!
&! 2! 6!
0
1A (2"
3A_C0007
Find the value of
C $% C #%
. (2 ars"
Solution:
C
%$
C #%
%! ( % $ "! $! %! ( % #"! #!
1 %! 2! $!
%! 4! #!
4! #! 2! $! 4! (# 2!" 2! ($ 4! "
# $
1A (2"
3A_C0008 6
Find the value of #!C # . (2 ars" Solution:
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3.3 3.3
#!
#!C #6
6! (6 #"! #!
1
#!
6 $ 4 #! #! #!
6 $ 4 120
1A (2"
3A_C0009
)i*lif+
( n 2"! . ( n 1"!
(2 ars" Solution:
( n 2"! ( n 1"! 1
( n 2"(n 1" n ( n 1"! ( n 1"!
( n 2"(n 1" n
1A (2"
3A_C0010
)i*lif+
(2 016 n"! . ( 2 01& n"!
(2 ars" Solution:
(2 016 n "! ( 2 01& n "!
1
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3.4 3.4
(2 016 n"! ( 2 01& n"(2 01% n" (2 016 n"!
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
1
1A
(2 01& n"(2 01% n"
(2"
3A_C0011
)i*lif+
n! (n 1"! (n 1"! (n 2"!
. (2 ars"
Solution:
n! (n 1"!
(n 1"! (n 2"!
n! (n 1" n! (n 1" n! (n 2"(n 1" n!
1
-1 ( n 1", n! -1 ( n 2",(n 1" n! n2
1A
( n #"(n 1"
(2" 3A_C0012 n 1
Find C n
in ters of n. (2 ars"
Solution:
C nn 1
( n 1"! -(n 1" n, ! n!
1
( n 1" n ! 1! n !
n 1
1A (2"
3A_C0013 n '&
Find C n '$ in ters of n. (2 ars" Solution:
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3.5 3.5
'& C nn '$
(n '&"! -(n '&" (n '$",! (n '$"!
1
(n '&"(n '%"(n '6" (n '$"! #! ( n '$"! ( n '&"( n '% "( n '6"
1A
6
(2" 3A_C0014
Find
C #n C nn #1
in ters of n. (2 ars"
Solution:
C #n
C nn #1
n! ( n #"! #! ( n 1"! -( n 1" ( n #",! ( n #"!
1
n! 2! (n 1"! #!
n ( n 1"! 2! ( n 1"! (# 2! "
n #
1A (2"
3A_C0015
)olve C 2n 1 $n . (2 ars" Solution: n 1
C 2 ( n 1"!
-( n 1" 2,! 2! ( n 1" n ( n 1"! ( n 1"! 2!
$n $n
1
$n
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3.6 3.6
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
(n 1"n 2
$n
( n 1" n 10n -(n 1" 10,n 0 ( n '" n 0 n '
or 0 (reected"
1A (2"
3A_C0016
Prove that C r n11 C r n 1 C r n 1 C r n 1 . (# ars" Proof:
/..). C r n11 C r n 1
( n 1"! -( n 1" ( r 1",! ( r 1"!
( n 1"! ( n r "! (r 1"!
( n 1"! -(n 1" r ,! r !
1
( n 1"! ( n 1 r "! r !
(n 1"!-(n 1 r " (r 1", (n 1 r "! (r 1"!
( n 1"! ( n 2r " ( n 1 r "! (r 1"!
..). C r n 1 C r n 1
n! n! - n ( r 1",! ( r 1"! - n ( r 1",! ( r 1"!
n! n! ( n 1 r "! ( r 1"! ( n 1 r "! ( r 1"!
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3.7 3.7
1
n!-(n 1 r "(n r " ( r 1"r ,
( n 1 r "! (r 1"! 2 2 2 n!(n n nr nr r r r r "
(n 1 r "! (r 1"!
n! ( n 2 n 2nr 2r " (n 1 r "! (r 1"!
n!- n( n 1" 2r (n 1", (n 1 r "! ( r 1"! n! ( n 2r "(n 1" ( n 1 r "! (r 1"! ( n 1"! ( n 2r " ( n 1 r "! (r 1"!
/..). ..).
C r 1 C r
n 1
n 1
C r n 1 C r n1
1 (#"
3B_C0017
t is given that T 1 1 3 T 2 0 3 T # 2 3 T 4 1 and T $ 6 . #
(a) Find the value of
T i 1
2i 1
.
4
(b) Find the value of
(T T i
i 1
".
i 1
(4 ars" Solution:
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3.8 3.8
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
#
(a)
T
T 1 T # T $
2i 1
i 1
1 1 2 6 '
1A (2"
4
(b)
(T T
i 1
i
(T 1 T 2 " (T 2 T # " (T # T 4 " (T 4 T $ "
"
i 1
1 T 1 2T 2 2T # 2T 4 T $ 1 2 0 2 2 2 (1" 6 '
1A (2"
3B_C0018 $
n each of the folloing3 find the value of
a
i
.
i 1
(a) ai $ 1 (b) ai i (c) ai i!
(# ars" Solution: $
(a)
a
i
$
$ i 1
i 1
$$$$$ 2$
1A (1" $
$
(b)
a
i
i 1
i 1
1
1
1
2
1 #
1 4
1 $
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3.9 3.9
1 i
1#% 60
1A (1"
$
(c)
a
i
$
i! i 1
i 1
1! 2! #! 4! $! 1 2 6 24 120 1$#
1A (1"
3B_C0019 &
Find the value of
(2'm #" . m#
(2 ars" Solution: &
&
&
m#
m#
2'm (#"
(2'm #" m# &
&
m #
m#
2' m ( #"
1
2'(# 4 $ 6 % &" (& # 1"(#"
2' ## 6 (#" '#'
1A (2"
3B_C0020 6
Find the value of
(2
r
1" .
r #
(1 ar" Solution: 6
(2
r
1"
(2# 1" (24 1" (2$ 1" (26 1"
r #
' 1% ## 6$
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.10 3.10
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
124
1A (1"
3B_C0021 4
Find the value of
C
2n n
.
n2
(2 ars" Solution: 4
C
2n n
C 24 C #6 C 4&
n2
4! 6! &! ( 4 2"! 2! ( 6 #"! #! (& 4"! 4!
4 # 2! 6 $ 4 #! & % 6 $ 4! 2! ( 2 1" #! (# 2 1" 4! ( 4 # 2 1"
1
6 20 %0 '6
1A (2"
3B_C0022 #
Find the value of
log(i 1 2 " . i
i 1
(2 ars" Solution: #
log(i 1 2 " i
log(1 1 21 " log(2 1 2 2 " log(# 1 2# "
i 1
log 2 log $ log 10
log(2 $ 10"
1
log 100 2
1A (2"
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.11 3.11
3B_C0023
Find the value of
#
k 2
k 1
k !
$
(1" k . k 4
(1 ar" Solution: #
k 1
2
k
k !
$
(1" k
k 4
1 2
# 2
12 1!
22 2!
#2 #!
( 1" 4 ( 1" $
1 (1"
'
1A
2
(1"
3B_C0024 4
Find the value of
(1
2
2 2 #2 n 2 " .
n 1
(2 ars" Solution: 4
(1
2
2 2 #2 n2 " 12 (12 22 " (12 22 #2 " (12 22 #2 42 "
1
n 1
1 $ 14 #0 $0
1A (2"
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.12 3.12
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
3B_C0025 #
E5*ress
#
#
#
#
#
#
in suation notation . (1 ar"
Solution:
# # # # # # #
%
#
1A
r 1
(1"
3B_C0026 2 #
E5*ress 1 # $
1' #%
in suation notation . (2 ars"
Solution:
1
2 #
# $
1' #%
1 2 1 1 1'
r 1
2 2 2 1
# 2 # 1
1' 2 1' 1
r
1 1A
2r 1
(2" 3B_C0027 & & & & & & & & E5*ress C 0 C & C 1 C % C 2 C 6 C & C 0 in suation notation .
(2 ars" Solution:
C 0&C && C 1&C %& C 2&C 6& C &&C 0& C 0&C && 0 C 1&C && 1 C 2&C && 2 C &&C && &
&
C C & r
r 0
& & r
1 1A (2"
3B_C0028
E5*ress x 6 2C 16 x $ 2 2 C 26 x 4 2 6 in suation notation . (2 ars" Solution: 0 6 60 21 C 16 x 6 1 22 C 26 x6 2 26 C 66 x 6 6 x 6 2C 16 x $ 2 2 C 26 x 4 2 6 2 C 0 x
6
2 C x r
6 r
6 r
r 0
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3.13 3.13
1 1A
(2" 3B_C0029 $
$
a
f
i
4 3 ai & and ai # 3 find the value of each of the folloing.
$
(a
2
2ai 1"
2
1"(2ai #"
i
#
i 1
i 1
i 1
(a)
$
2
i 1
$
(a
(b)
i
i 1
($ ars" Solution: $
(a
(a)
2
i
$
$
$
i 1
i 1
ai 2 ai 1
2ai 1"
2
i 1
i 1
1
& 2 4 ($ 1 1"(1" &&$ 21
1A (2"
$
(a
(b)
i
2
$
(2ai #ai 2ai #"
1"(2ai #"
i 1
#
2
i 1
1 $
$
$
$
i 1
i 1
2 ai # ai 2 ai (#" #
i 1
2
i 1
1
2 ( #" # & 2 4 ($ 1 1"(#"
6 24 & 1$ #%
1A (#"
3B_C0030 &
a
f
i
&
&
&
i 1
i 1
i 1
4 3 bi 6 and ai bi 12 3 find the value of (#ai 2"(2bi $" .
i 1
(# ars" Solution:
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3.14 3.14
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
&
(#a
i
&
(6ai bi 1$ai 4bi 10"
2"(2bi $"
i 1
i 1
1 &
&
&
&
i 1
i 1
i 1
i 1
6 ai bi 1$ ai 4 bi (10"
1
6 12 1$ ( 4" 4 6 (& 1 1"(10" %2 60 24 &0
%6
1A (#"
3B_C0031 '
t is given that
a
n
12 . Find the values of the folloing.
n 1
'
(a)
(1 6a " i
i 1 $
'
(2 a " ($ #a "
(b) #
n
n
n 1
n 6
(6 ars" Solution: '
(a)
(1 6a "
'
'
i 1
i 1
1 6 ai
i
i 1
1 '
'
1 6 a n i 1
1
n 1
(' 1 1"(1" 6 12 &1
1A (#"
$
' ' # 2 # an $ # an n 1 n 1 n6 n 6 $
'
(2 a " ($ #a "
(b) #
n 1
n
n
n 6
$
'
$
'
n 1
n6
n 1
n6
# 2 $ # an # an
$
1 '
#($ 1 1"(2" (' 6 1"($" # an n 1
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.15 3.15
1
#0 20 # 12 26
1A (#"
3B_C0032 &
x
f
2
i
&
10# and ( xi 2"2 6# 3 find the ean of x1 3 x2 3 x# 3 3 x% and x& .
i 1
i 1
(# ars" Solution: &
( x
2"2 6#
i
i 1 &
( x
2
i
4 xi 4" 6#
i 1
&
x
2
i
i 1 &
x
10# 4
i
&
&
i 1
i 1
4 xi 4 6#
1
(& 1 1"(4" 6#
i 1 &
x
i
1&
1A
i 1
euired ean
x x x 1
2
#
x
&
&
&
xi
i 1
& 1& & '
1A
4
(#" 3B_C0033
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.16 3.16
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
$
Find the value of
n 1
n i . i 1
n
- int7
i 1 2 # n , i 1
(2 ars" Solution: $
n 1
n i i 1
$
(1 2 # n" n 1
1 1 (1 2" (1 2 #" (1 2 # 4" (1 2 # 4 $"
1 # 6 10 1$ #$
1A (2"
3C_C0034
E5*and ( x 6" 4 using Pascal8s triangle. (2 ars" Solution:
1( x 4 " 4( x # "(6" 6( x 2 "(6 2 " 4( x "(6# " 1(6 4 "
( x 6" 4
1 x 4 24 x # 216 x 2 &64 x 1 2'6
1A (2"
3C_C0035
E5*and ( a b " 6 using Pascal8s triangle. (2 ars" Solution:
( a b " 6
1( a 6 " 6( a $ "( b " 1$( a 4 "( b" 2 20( a # "( b" # 1$( a 2 "( b " 4 6( a "( b " $ 1( b" 6
1 a 6 6a $b 1$a 4b 2 20a #b# 1$a 2b 4 6ab$ b 6
1A (2"
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.17 3.17
3C_C0036 4
$ E5*and 2 x using Pascal8s triangle. x
(2 ars" Solution:
$ 2 x x
4
2
#
$ $ $ $ 1(2 x " 4( 2 x " 6( 2 x " 2 4( 2 x " 1 x x x x 4
4
#
1 16 x 4 160 x 2 600
1 000 x 2
62$ x 4
1A (2"
3C_C0037
E5*and (# x 1" 4 using the 9inoial Theore. (2 ars" Solution:
(# x 1"
4
C 04 (# x" 4 C 14 (# x "# (1" C 24 (# x" 2 (12 " C #4 (# x"(1# " C 44 (14 "
1 &1 x 4 10& x # $4 x 2 12x 1
1A (2"
3C_C0038
E5*and ( 4 x 2 " 6 using the 9inoial Theore. (2 ars" Solution:
(4 x 2 " 6
C 06 ( 4 6 " C 16 ( 4$ "( x 2 " C 26 ( 4 4 "( x 2 " 2 C #6 ( 4# "( x 2 " # C 46 ( 4 2 "( x 2 " 4 C $6 ( 4"( x 2 " $ C 66 ( x 2 " 6 4 0'6 6 14 4 x 2 # &4 0 x 4 1 2&0 2& 0 x 6 24 0 x & 24 x10 x12
1 1A (2"
3C_C0039
E5*and (# p 4q " 4 using the 9inoial Theore. (2 ars" Solution:
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.18 3.18
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
(# p 4q "
C 04 (# p " 4 C 14 (# p"# ( 4q" C 24 (# p" 2 ( 4q" 2 C #4 (# p "( 4q"# C 44 ( 4q" 4
4
1 &1 p 4 4#2 p # q &64 p 2 q 2 %6& pq # 2$6q 4
1A (2"
3C_C0040 $
1 E5*and % x 2 using the 9inoial Theore. x
(2 ars" Solution: $
2
2 1 $ 2 $ $ 2 4 1 $ 2 # 1 $ 2 2 1 % x C 0 (% x " C 1 (% x " C 2 (% x " C # (% x " x x x x 1
4
1 C 4$ (% x 2 " C $$ x x
#
1
$
16 &0% x10 12 00$ x % # 4#0 x 4 4'0 x
#$ x
2
1
1A
x $
(2" 3C_C0041 $
1 (a) E5*and 4 x 2 . # $
1 (b) 9+ using the result of (a)3 e5*and 4 x . # x
(4 ars" Solution:
1 (a) 4 x 2 #
$
2
1 1 1 C ( 4 x " C ( 4 x " C 2$ ( 4 x 2 "# C #$ (4 x 2 " 2 # # # 4 $ 1 1 1 $ 2 $ C 4 ( 4 x " C $ # # $ 0
1 024 02 4 x10
2 $
$ 1
2
#
4
1 2&0 2& 0 x & 640 x 6 160 x 4 20 x 2 1 # ' 2% &1 24#
1A (2"
1 (b) 4 x # x
$
1 1 4 x 2 # x
$
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.19 3.19
1 1 $ 4x2 x #
$
1
1 1 2&0 x & 640 x 6 160 x 4 20 x 2 1 10 - Fro the result of (a) , $ 1 024 x x # ' 2% &1 24#
1 024 x $
1 2&0 x # #
640 x '
160 2% x
20 &1 x
#
1
1A
24# x $
(2"
3C_C0042
E5*and (10 x 1"1# in ascending *oers of x u* to the ter in x # . (2 ars" Solution:
C 01# (11# " C 11# (10 x"(112 " C 21# (10 x" 2 (111 " C #1# (10 x"# (110 "
(10 x 1"
1#
1 1 1#0 x % &00 x 2 2&6 000 x #
1A (2"
3C_C0043 10
1 E5*and 2 x # in descending *oers of x u* to the ter in x10 . 2 2 x
(2 ars" Solution: 10
1 # 1 1 C 010 ( 2 x # "10 C 110 ( 2 x # " ' C 210 ( 2 x # "& 2 2 x 2 2 2 x 2 x 2 x #
2
4
1 1 C ( 2 x " 2 C 410 ( 2 x # " 6 2 2 x 2 x 10 #
#
1
%
1 024 x #0 2 $60 x 2$ 2 &&0 x 20 1 '20 x1$ &40 x10
1A (2"
3C_C0044
E5*ress the :inoial e5*ansion of ( x 1#"12 in ascending *oers of x in suation notation. (1 ar" Solution:
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.20 3.20
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
( x 1#"
12
12
C
12 r
r 0
1#12 r x r
1A (1"
3C_C0045 $
4 E5*ress the :inoial e5*ansion of x 2 in descending *oers of x in suation notation. x (1 ar" Solution: $
2 4 x x
$
C r $ ( x 2 " $ r
r 0
4 x
r
$
C
$
r
r
4
r
10 #r
x
1A
0
(1" 3C_C0046
Find the coefficient of x # in the e5*ansion of (1 $ x " % . (2 ars" Solution:
The ( r 1" th ter
C r % ($ x " r C r % $r x r
1A
For the ter in x # 3 r # .
% # Coefficient of the ter in x # C # $
4 #%$
1A (2"
3C_C0047
Find the coefficient of x & in the e5*ansion of ($ 2 x"10 . (2 ars" Solution:
The (r 1" th ter C r 10 ($10 r "( 2 x " r
C r 10 $10 r 2 r (1" r x r For the ter in x & 3 r & . 10 10 & & 2 ( 1" & Coefficient of the ter in x & C & $ 2&& 000
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
1A
1A
3.21 3.21
(2" 3C_C0048 &
2 Find the constant ter in the e5*ansion of x # . x
(# ars" Solution:
The (r 1" th ter C
&
r
( x # "& r
2
r
x
C r & (1" r 2 r x 24 4 r
1A
For the constant ter3 24 4r 0 r 6
1A
& 6 6 Constant ter C 6 ( 1" 2
1 %'2
1A (#"
3C_C0049
Find the coefficient of the 4th ter in the e5*ansion of (# x "10 in ascending *oers of x. (# ars" Solution:
The ( r 1" th ter C r 10 (#10 r "( x " r
C r 10 #10 r ( 1" r x r
1A
For the 4th ter3 r 1 4 r #
1A
10 10 # ( 1" # Coefficient of the 4th ter C # #
262 440
1A (#"
3C_C0050 1#
1 Find the coefficient of the 'th ter in the e5*ansion of 2 x 2 in ascending *oers of x. 4 x (# ars" Solution: 1# r
1 The (r 1" th ter C 4 x 1# r
( 2 x 2 " r
C r 1# 2#r 26 x #r 1#
1A
For the 'th ter3 r 1 ' r &
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
1A
3.22 3.22
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
1# # ( & " 26 Coefficient of the 'th ter C & 2
1 2&%
1A
4
(#" 3C_C0051 %
a' a Find the coefficient of $ in the e5*ansion of #a 2 . #b b (# ars" Solution:
a The ( r 1" th ter C (#a " #b C r % #% 2 r a14 r b r % r
2
r
% r
1A
a' For the ter in $ 3 r $ . b
1A
a' % % 2 ($" Coefficient of the ter in $ C $ # b
%
1A
'
(#"
3C_C0052 '
1 x # . Find the constant ter in the e5*ansion of 2 x (# ars" Solution: '
1 x# 1 2 2 x x x
'
1 The ( r 1" th ter C 2 x
' r
' r
( x " r
C r ' ( 1" r x # r 1&
1A
For the constant ter3 #r 1& 0 r 6
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
1A
3.23 3.23
Constant ter C 6 ( 1" '
6
&4
1A (#"
3C_C0053
1 ( x # x 2 " % Find the coefficient of 2 in the e5*ansion of . 12 x x (# ars" Solution:
The ( r 1" th ter
1 12
x
C r % x % r ( # x 2 " r
C r % (1" r #r x r $
1A
1
For the ter in x 2 3 r $ 2 r #
Coefficient of the ter in
1A 1
x
2
C #% ( 1" # ## '4$
1A (#"
3C_C0054 6
x Find the coefficient of x in the e5*ansion of (1 x " 1 . # 4
2 $
($ ars" Solution:
n the e5*ansion of (1 x 2 " $ 3 the ( r 1" th ter C r $ ( x 2 " r
C r $ x 2 r
1A
For the ter in x 4 3 2r 4
r 2
$ Coefficient of the ter in x 4 C 2
10
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
1A
3.24 3.24
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
6
x n the e5*ansion of 1 3 #
x the ( r 1" th ter C #
r
6 r
C r 6 # r x r
1A
For the ter in x 4 3 r 4 .
6 4 Coefficient of the ter in x 4 C 4 #
$ 2%
1A 6
x n the e5*ansion of (1 x " 1 3 # 2 $
the coefficient of the ter in x 4 10
$ 2%
2%$
1A
2%
($" 3C_C0055
x 2 2 consists of each of the folloing ters. Find each ter ;eterine hether the e5*ansion of 2 x if it e5ists. (a) A constant ter 1 (b) A ter in 2
x
($ ars" Solution:
x 2 ( 1 " r C The th ter 2
& r
& r
2 x
r
C r & ( 1" r 2 2 r & x16 #r (a) For the constant ter3 16 #r 0 r
16
1A
# &
x 2 2 does not )ince r is is not an integer3 the e5*ansion of 2 x
consist of a constant ter.
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
1A
3.25 3.25
(2" 1 (b) For the ter in x 2 3 16 #r 2 r 6
1A &
1 x 2 2 consists of a ter in 2 . The e5*ansion of x 2 x
1A
1
& 6 2 ( 6 " & 2 x The ter in x 2 C 6 (1" 2
44&
1A
x 2
(#"
3C_C0056
/et r :e the coefficient of x r in the e5*ansion of (# % x " ' . Find
# . $ (4 ars"
Solution:
The ( r 1" th ter C r ' (#' r "( % x " r
C r ' #'r % r ( 1" r x r
1A
For the ter in x # 3 r # . #
C #' #' #%# (1"#
C #' #6 % #
1A
For the ter in x $ 3 r $ .
C $' #' $%$ (1"$
$
C $' #4 % $
1A
C #' #6%# C $' #4%$
# $
&4 #2 126 %
2
6 4'
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
1A
3.26 3.26
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
(4"
3C_C0057
/et T r :e the coefficient of x r in the e5*ansion of (1 2 x" 20 . f T r 2 7 T r 26 7 # 3 find the value of r . (4 ars" Solution:
The (r 1" th ter C r 20 ( 2 x" r
C r 20 (1" r 2 r x r T r C r ( 1" 2 20
r
r
T r 2 C r 2 ( 1" 20
1A
r 2
2
r 2
T r 2 7 T r 26 7 #
C r 20 2 ( 1" r 2 2r 2
C r ( 1" 2 20
r
r
26
1
#
6C r 20 2 1#C r 20 6
20! 20! 1# - 20 ( r 2",! ( r 2"! ( 20 r "! r ! 6 ( 20 r "! (1& r "!
1# ( r 2"!
r !
6( 20 r "(1' r " 1#( r 2"( r 1" 6( r 2 #' r #&0 " 1#( r 2 #r 2" % r 2 2%#r 2 2$4 0 2 r #'r #22 0
r
#'
#' 2 4(1"( #22 "
1
2(1"
#' $#
%
2
or 46 (reected"
1A (4"
3C_C0058 A r
/et $6 A11
# x 2
k
r 3 here k is :e the coefficient of x in the e5*ansion of is a constant. f 2 x 1$ A$ 3 find the values of k .
(# ars" Solution:
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.27 3.27
10 r
# x 2 r ( 1 " C The th ter 2 10 r
k x
r
C r 10 #10 r 2 r 10 k r x 20 #r
1A
For the ter in x11 3 20 #r 11 r #
C #10 #10 # 2# 10 k #
A11
C #10 #% 2 % k # For the ter in x $ 3 20 #r $ r $
C $10 #10 $ 2 $ 10 k $
A$
C $10 #$ 2 $ k $ $6 A11 1$ A$
$6 C #10 #% 2 % k # 1$ C $10 #$2 $ k $
k 2
10
2
10
2
1
$6C # # 1$C $ 2
k 4 2
k 2
or
1A
2
(#"
3C_C0059 '
2 1 ax is 6%23 here a is a non<=ero real nu:er3 find the f the constant ter in the e5*ansion of x value of a. (# ars" Solution:
The ( r 1" th ter C ( ax " ' r
2
' r
1 x
r
C r ' a ' r x1& #r
1A
For the constant ter3 1& #r 0 r 6
1A
Constant ter 6%2
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.28 3.28
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
'
C 6 a
'6
a a
#
6%2 &
1A
2
(#" 3C_C0060
f the coefficient of x 2 in the e5*ansion of ( value of n.
2 x 1"
n
is '03 here n is a *ositive integer3 find the (# ars"
Solution:
The (r 1" th ter C r n ( 2 x" r r
1A
C 2 2 x r n r
For the ter in x 2 3 r 2 .
Coefficient of the ter in x 2 '0
2 2
C 2 '0 n 2
2
n(n 1" 2
'0
2 n n '0 0
1
( n 10"(n '" 0 n 10
or ' (reected"
1A (#"
3C_C0061
f n is a *ositive integer and the coefficients of x # and x 4 in the e5*ansion of ( x #" n are eual3 find the value of n. (# ars" Solution:
The (r 1" th ter C r n (#n r " x r
1A
For the ter in x # 3 r # .
# n n# Coefficient of the ter in x C # (# "
For the ter in x 4 3 r 4 .
n4
Coefficient of the ter in x C 4 (# 4
n
"
C #n (#n # " C 4n (#n 4 "
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
1A
3.29 3.29
#
n! (n #"!#! 4!
#
#!
n! ( n 4"!4! ( n #"! ( n 4"!
12 n # n
1A
1$
(#" 3C_C0062 n
$ 1 f the $th ter in the e5*ansion of ax 2 3 here a is a non< in descending *oers of x is 2% # x =ero real nu:er and n is a *ositive integer3 find the values of a and n. (# ars" Solution:
The (r 1" th ter C ( ax " 2
n r
n r
1 # x
r
C r n a n r # r x 2 n #r
1A
For the $th ter3 ter 3 r 4 .
The $th ter is a constant ter. 2n #(4" 0 n 6
The $th ter 6
C 4 a
64
4
#
1$
1A
$ 2%
$ 2%
a
2
a
2
&1
a
$ 2%
1 1
or
1A
1
(#" 3C_C0063 &
2 1 & 2 x ( x 2" is 20&3 here k is f the coefficient of x in the e5*ansion of is a non<=ero real kx nu:er3 find the value of k . ($ ars" %
Solution:
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.30 3.30
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
&
1 n the e5*ansion of 2 x 2 3 kx
the ( r 1" th ter C r & ( 2 x 2 " & r
1 kx
r
C r & 2& r k r x16 #r
1A
For the ter in x % 3 16 #r % r #
& & # # Coefficient of the ter in x % C # 2 k
1 %'2 k #
1A
n the e5*ansion of ( x 2" & 3 the ( r 1" th ter C r & (2& r " x r
1A
For the ter in x % 3 r % .
& &% Coefficient of the ter in x % C % 2
16
1A
&
1 n the e5*ansion of 2 x 2 ( x 2"& 3 kx the coefficient of the ter in x % 1 %'2k # 16 .
1 %'2 k # 16 20& k # & k 2
1A ($"
3C_C0064 6 (a) E5*and ( 2 x " .
(b) t is given that (2 # " 6 a b # 3 here a and b are integers. >sing the result of (a)3 find the values of a and b. (# ars" Solution: 6 (a) ( 2 x "
C 06 (26 " C 16 (2$ "( x" C 26 (2 4 "( x" 2 C #6 (2# "( x"# C 46 (2 2 "( x" 4 C $6 (2"( x"$ C 66 ( x"6 24 0 x 2 160 x # 60 x 4 12 x $ x 6 64 1'2 x 240
1A (1"
(b) ( 2
# "6
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3.31 3.31
64 1'2 ( # " 240 ( # " 2 160 ( # " # 60 ( # " 4 12 ( # "$ ( # " 6
1
- Fro the result of (a) ,
64 1'2 # %20 4&0 # $40 10& # 2% 1 #$1 %&0 #
a
1 #$13
b %&0
1A (2"
3C_C0065 4
4
2 2 t is given that x # x # a bxc 3 here a3 b and c are integers. x x
(a) Find the values of a3 b and c. 4
4
1 1 10 $ 10 $ . (b) ence find the value of $ $
($ ars" Solution:
2 (a) x # x
4
2
x 4 & 2 x # x
4
4 1
4
4
#
24 #2 16 x 4 x & x12
1A 2
#
2 2 2 2 C 04 ( x 4 " C 14 ( x # " # C 24 ( x 2 " # C #4 ( x " # C 44 # x x x x
x4 & 4
#
2 2 2 2 C ( x " C ( x " # C 24 ( x 2 " # C #4 ( x" # C 44 # x x x x 4 0
24 x
4
#2 &
x
16
4
1A
12
x
4
64 2 2 x # x # 16 & x x x
16 64 x &
a
16 3
b
64 3
c &
1A (#" 4
1 1 2 2 1 1 10 $ 10 $ (b) 1 # 1 # $ $ $ ( $ " $ ( $ " 4
4
1 16 64 $
&
16 64 62$
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
4
3.32 3.32
-Fro the result of (a) ,
1
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
40 016
1A (2"
3C_C0066
t is given that that (k x " n x n 2nx n1 1&0 x n 2 ters involving loer *oers of x. f k is is a non<=ero real nu:er and n is a *ositive integer3 find the values of k and and n. (4 ars" Solution:
x n C 1n ( x n1 "(k " C 2n ( x n2 "(k 2 "
(k x " n
x n nkxn1
n(n 1" 2
2
k x
n 2
1A
9+ co*aring the coefficients of x n 1 and x n 2 3 e have
nk 2n ....................... (1" n(n 1" 2 2 k 1&0 ........ (2"
1
Fro (1"3 k 2 )u:stitute k 2 into (2"3 n(n 1" (2" 2 1&0
1A
2
n n '0 0 2
(n 10"(n '" 0 n
10
or ' (reected"
1A (4"
3C_C0067
t is given that that (1 kx" n 1 ## x 4'$x 2 ters involving higher *oers of x. f k is is a non<=ero real nu:er and n is a *ositive integer3 find the values of k and and n. (4 ars"
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.33 3.33
Solution:
(1 kx" kx " n
1 C 1n (kx" C 2n (kx"2
1 nkx
n(n 1" 2
k 2 x 2
1A
9+ co*aring the coefficients of x and x 2 3 e have
nk ## ................................... (1" n(n 1" 2 2 k 4'$..................... (2" ##
Fro(1"3 Fro (1"3 k
n
1
....................... (#"
)u:stitute (#" into (2"3 2
n ( n 1" ## 4'$ 2 n n 1 10 n 11 11n 11 10 n n 11
1A
)u:stitute n 11 into (#"3 k
## 11
#
1A (4"
3C_C0068
t is given that (2 x# 1"n a bx# cx6 dx' ters involving higher *oers of x. f d 2&b and n is a *ositive integer3 find the values of c and n. (4 ars" Solution:
1 C 1n (2 x# " C 2n (2 x# "2 C #n (2 x# "#
( 2 x # 1" n
1 n(2 x # "
n(n 1" 2
(4 x "
1 2nx# 2n(n 1" x 6
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
6
n(n 1"(n 2" 6
4n(n 1"(n 2" #
3.34 3.34
(& x "
' x
'
1A
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
9+ co*aring the coefficients of x # and x ' 3 e have
b 2n ...................................... (1" 4n(n 1"(n 2" ................ (2" d #
1
)u:stitute (1" and (2" into d 2&b 3 4 n ( n 1"( n 2" 2&( 2 n " # ( n 1"( n 2" 42
n 2 #n 40 0 ( n &"( n $" 0 n &
or $ (reected"
1A
9+ co*aring the coefficient of x 6 3 e have c 2 n( n 1"
2(&"(& 1" 112
1A (4"
3C_C0069
t is given that (1 2 x" m (1 % x"n 2 22 x kx 2 ters involving higher *oers of x. f k is a constant3 and m and n are *ositive integers3 find the values of k 3 m and n. (4 ars" Solution:
(1 2 x "
1 C 1m (2 x" C 2m (2 x"2
m
1 2mx 2m( m 1" x 2 (1 % x "
1 %nx
1 C 1n (% x" C 2n (% x"2
n
4'n(n 1" 2
2 x
(1 2 x " m (1 % x " n
4'n(n 1" 2 2 (2m %n" x 2m(m 1" x 2
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.35 3.35
1A
9+ co*aring the coefficients of x and x 2 3 e have
(2m %n" 22 ................................. (1" 4'n(n 1" k .................. (2" 2m(m 1" 2
1
Fro (1"3 2m %n 22
m and n are m
43
n
*ositive integers.
2
1A
)u:stitute m 4 and n 2 into (2"3 2(4"( 4 1"
4'( 2"( 2 1" k 2 k 24 4' %#
1A (4"
3C_C0070
(a) E5*and ( 2 x "( 2 x " . 6 6 (b) ence e5*and (2 x " ( 2 x " .
(# ars" Solution:
(a) (2 x"(2 x"
22 x 2 4 x2
1A (1"
(b) ( 2 x "6 ( 2 x " 6
-( 2 x "(2 x ",6
1
(4 x 2 " 6
C 06 (46 " C 16 (4$ "( x 2 " C 26 ( 44 "( x 2 " 2 C #6 (4# "( x 2 "# C 46 (4 2 "( x 2 " 4 C $6 (4"( x 2 "$ C 66 ( x 2 " 6 4 0'6 6 14 4 x 2 # &40 &4 0 x 4 1 2& 0 x 6 24 0 x & 24 x10 x12
1A (2"
3C_C0071
(a) E5*and ( x 1"( x 2 x 1" .
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3.36 3.36
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
% 2 % (b) ence e5*and ( x 1" ( x x 1" .
(# ars" Solution:
(a) ( x 1"( x 2 x 1"
x# x 2 x2 x x 1
x# 1
1A (1"
-( x 1"( x 2 x 1",%
(b) ( x 1" % ( x 2 x 1" % 1
( x # 1" %
C 0% ( x # " % C 1% ( x # " 6 (1" C 2% ( x # " $ ( 1" 2 C #% ( x # " 4 ( 1" # C 4% ( x # "# ( 1" 4 C $% ( x # " 2 ( 1" $ C 6% ( x # "(1" 6 C %% (1" % x 21 % x1& 21 x1$ #$ x12 #$ x ' 21 x 6 % x # 1
1A (2"
3C_C0072
(a) E5*and (1 x " % in ascending *oers of x u* to the ter in x 2 . % (b) The coefficient of x 2 in the e5*ansion of (1 kx "(1 x " is %. >sing the result of (a)3 find the value of k . (# ars"
Solution:
1 C 1% x C 2% x 2
(a) (1 x " % 1 % x 21x 2
1A (1"
(b) (1 kx"(1 x" (1 kx"(1 % x 21x " -Fro the result of (a) , %
2
Coefficient of the ter in x 2 % (1"( 21" ( k "( % " %
1 1A
k 2
(2" 3C_C0073
(a) E5*and (# x 2 " $ in ascending *oers of x u* to the ter in x 4 . # 2 $ (b) ence e5*and (# x x "(# x " in ascending *oers of x u* to the ter in x 4 .
(# ars" Solution:
(a) (# x 2 " $
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#$ C 1$ (#4 "( x 2 " C 2$ (## "( x 2 "2
3.37 3.37
24# 40$ x 2 2%0 x 4
1A (1"
# 2 $ (b) (# x x "(# x " (# x x # "(24# 40$ x 2 2%0x 4 "
(#"( 24 #" ( 1"( 24 #" x (#"( 40 $" x 2 -( 1"( 40 $" (1"( 24 #", x # (#"( 2% 0" x 4 1 1A
%2' %2 ' 24 # x 1 21 $ x 64& 64 & x &10 &1 0 x 2
#
4
(2" 3C_C0074
(a) E5*and ( x 2 4" 6 in descending *oers of x u* to the ter in x & . 4 2 2 6 (b) ence find the coefficient of x12 in the e5*ansion of ( x # x 4"( x 4" .
(# ars" Solution:
(a) ( x 2 4" 6
( x 2 "6 C 16 ( x 2 "$ (4" C 26 ( x 2 "4 (42 " x12 24 x10 240 x &
1A (1"
(b) ( x 4 # x 2 4"( x 2 4"6 ( x 4 # x 2 4"( x12 24 x10 240 x& "
Coefficient of the ter in x12 (1"( 240 " (#"( 24 " (4"(1"
#16
1 1A (2"
3C_C0075
/et x
1 x
# (a) x
# 3 find the values of each of the folloing. 1 x
$ (b) x
#
1 x
$
(6 ars" Solution: #
2
1 1 1 1 (a) x C 0# x # C 1# ( x 2 " C 2# ( x" C ## x x x x
## x # # x ## x #
#
1
# 1 x x #
1 1 # x x # x
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
1
3.38 3.38
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
# x #
x #
1 x
1
#
#
x
#
#(#"
## #(#" 1&
1A (#"
$
2
#
4
1 1 1 1 1 1 (b) x C 0$ x$ C 1$ ( x 4 " C 2$ ( x# " C #$ ( x 2 " C 4$ ( x" C $$ x x x x x x
#$ x $ $ x # 10 x #$ x $ $
x $
1 x
$
$
1 x
$
1
10 $ 1 # $ x x x
1 # 1 1 $ 10 x x # x $ x x
# x
$
1
$(1&" 10(#"
#$ $(1&" 10(#" 12#
1A (#"
3C_C0076 1 1 3 find the values of each of the folloing. ! 2 1 1 # $ (a) ! # (b) ! $ ! !
/et !
(6 ars" Solution: #
2
1 1 1 1 (a) ! C 0# ( ! # " C 1# ( ! 2 " C 2# ( ! " C ## ! ! ! !
#
1
#
# 1 1 # ! # ! # ! ! 2 #
1 1 1 # ! # # ! ! ! 2
1
#
1 1 1 # ! # # ! 2 2 #
1 1 ! # # ! 2 2 #
1
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.39 3.39
1#
1A
&
(#" $
2
1 1 1 1 (b) ! C 0$ ( ! $ " C 1$ ( ! 4 " C 2$ ( ! # " C #$ ( ! 2 " ! ! ! ! 4
1 1 C ( ! " C $$ ! !
#
$
$ 4
1
$
10 $ 1 1 $ # # $ ! $ ! 10 ! ! ! ! 2 $
# 1 1 1 1 $ ! $ $ ! # 10 ! ! ! ! 2
1
$
1 1 1# 1 $ ! $ $ 10 ! 2 & 2 $
1 1# 1 ! $ $ 10 ! 2 & 2 $
1
101
1A
#2
(#" 3C_C0077 &
(1 2 x "% ( x 2" 6 x Find the coefficient of x in the e5*ansion of 1 . x 2 x 4 2
(4 ars" Solution: &
x n the e5*ansion of 1 3 2 r
x the (r 1"th ter C 2 C r & (1" r 2 r x r & r
For the ter in x3 r 1 .
& 1 Coefficient of the ter in x C 1 ( 1" 2
1
4 n the e5*ansion of the (r 1"th ter
(1 2 x" % x 2 1
x
2
1A
3
C r % (2 x" r
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.40 3.40
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
C r % 2r x r 2 For the ter in x3 r 2 1 r #
Coefficient of the ter in x C # 2 %
#
2&0 n the e5*ansion of
( x 2" x
1
the (r 1"th ter
4
1A
6
3
4
C r 6 x 6r ( 2" r
x C r 6 (1" r 2 r x 2 r
For the ter in x3 2 r 1 r 1
6 1 1 Coefficient of the ter in x C 1 ( 1" 2
12
1A
&
% 6 (1 2 x " ( x 2" x n the e5*ansion of 1 3 x 2 x 4 2
coefficient of the ter in x 4 2&0 12 264
1A (4"
3C_C0078 n
1 x 2 in ascending ascending *oers *oers of x in suation notation3 (a) E5*ress the :inoial e5*ansion of k x
here k is is a *ositive real nu:er and n is a *ositive integer. (b) f the #rd ter of the e5*ansion is
%6 x#
3 find the values of k and and n.
(c) ence find the coefficient of the ter ith the highest *oer.
($ ars" Solution: n
n 1 x 2 n 1 C (a) r k x r 0 x
n
C k n r
n r
r
x 2 k
r
x #r n
1A
r 0
(1" (b) For the #rd ter3 r 2 .
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.41 3.41
The #rd ter C 2n k 2 x #( 2"
n
C 2n k 2 x 6 n
The #rd ter is a ter in
6 n #
1 . x #
n '
1A
Coefficient of the ter in
1 $%6 x #
C 2 k $%6 '
2
k 2 k
1
1 16 1 1 or 4 (reected" 4
1A (#"
1 4
'
(c) The coefficient of the ter ith the highest *oer C ''
262 144
1A (1"
3C_C0079
The coefficients coefficients of x and x 2 in the e5*ansion of (1 2 x" # (1 ax 2 " 6 are b and 0 res*ectivel+3 here is a non<=ero real nu:er and b is a real nu:er.
a
(a) Find the values of a and b. (b) Find the coefficient of x # .
($ ars" Solution: # # 2 # # (a) (1 2 x" 1 C 1 (2 x" C 2 (2 x" (2 x"
1 6 x 12 x 2 & x# (1 ax 2 " 6 1 C 16 (ax2 "
1 6ax2
(1 2 x " # (1 ax 2 " 6
(1 6 x 12 x 2 & x # "(1 6ax 2 "
1 6 x 12 x 2 & x# 6ax2 #6ax# 1 6 x (12 6a " x 2 (& #6a" x #
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.42 3.42
1A
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
9+ co*aring the coefficients of x and x 2 3 e have
b 6 .......................... (1" 12 6a 0 ................ (2"
1
Fro (1"3 b 6 Fro (2"3 12 6a 0
1A
a 2
1A (4"
(b) Coefficient of the ter in x # & #6a
& #6 ( 2" 64
1A (1"
3C_C0080
t is given that (1 x" m (1 2 x" n 1 ' x 2 ters involving higher *oers of x3 here here *ositive integers. Find the values of m and n.
m and n
are
(4 ars" Solution:
1 C 1m x C 2m x 2
(1 x" x " m
1 mx (1 2 x "
m(m 1" 2
x2
1 C 1n ( 2 x" C 2n ( 2 x" 2
n
1 2nx 2n(n 1" x 2 (1 x " m (1 2 x " n
m(m 1" 2 2 1 mx x -1 2nx 2n(n 1" x , 2
m(m 1" 1 (m 2n" x 2mn 2n(n 1" x 2 2
1A
9+ co*aring the coefficients of x and x 2 3 e have
m 2n 0 ............................................ (1" m(m 1" 2 2mn 2n(n 1" ' ......... (2"
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.43 3.43
1
Fro (1"3 m
2n ................................................... (#"
)u:stitute (#" into (2"3 2 n ( 2 n 1" 2( 2 n " n 2 n ( n 1" ' 2
2n 2 n 4 n 2 2 n 2 2 n '
#n ' n #
1A
)u:stitute n # into (#"3 m 2(#" 6
1A (4"
3C_C0081
(a) E5*and (1 # x"(1 #x" . (b) ence e5*and (1 # x"# (1 #x"# .
2 (c) The coefficient of x 4 in the e5*ansion of x
k
2%
# # (1 # x" (1 #x" is 3 here k is is a non< 2 x 2
=ero real nu:er. nu:er. Find the value of k . ($ ars" Solution:
(a) (1 # x"(1 #x"
1 (#x" 2
1 'x2
1A (1"
(b) (1 # x " # (1 # x " # 1 (1 ' x 2 " #
-(1 # x"(1 #x",#
C 0# C 1# ( ' x 2 " C 2# ( ' x 2 " 2 C ## ( ' x 2 "# 1 2% x 2 24# x 4 %2'x 6
1A (2"
2 (c) x
k 2 k # # 2 4 6 ( 1 # x " ( 1 # x " 24# x %2' %2' x " x 2 (1 2% x 24# 2 x x
Coefficient of the ter in x 4 2% (1"( 2% " k ( %2' " 2
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
2% 2
1
3.44 3.44
Effective Effe ctive Learn ing Mathe M athemati matics cs Module M odule 2: Algebra Al gebra and Calcu lus Question Bank | Volume olum e 1
%2'k
2% 2 1
k
1A
$4
(2"
3C_C0082 x " n and ( 1 x" x " n in ascend (a) E5*ress E5*ress the e5*ansi e5*ansions ons of (# x" ascendin ing g *oe *oers rs of x in suation suation notation3 here n is a *ositive integer.
(b) Prove that (# x" (1 x" n
n
n
C (1" -# n r
n r
r
(1" n , x r .
r 0
100
(c) ence find
C
( 2" r (#100 r 1" .
100 r
r 0
($ ars" Solution:
x " (a) (# x" n
n
C (# n r
r 0 n
C
n r
n r
n r
#
"( x " r
(1" x r
r
1A
r 0
( 1 x " n
n
C (1" n r
n r
x r
1A
r 0
(2" (b) (# x " ( 1 x " n
n
n
C # n r
n r
(1" x r
r
r 0
n
C (1" n r
n r
x r
r 0
n
C r n (1" r -#n r x r (1" n 2r x r , r 0 n
C r n (1" r -#n r (1" n ,x r
1
r 0
(1" (c) )u:stitute n 100 and x 2 into (# x" (1 x" n
n
n
C (1" -# n r
r 0
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.45 3.45
r
n r
(1" n , x r 3
(# 2"
100
100
(1 2"
100
C r 100 (1"r -#100 r (1"100 ,2r
1
r 0
1100 1100
100
C
100 r
( 2" r (#100 r 1"
r 0 100
C
100 r
( 2" r (#100 r 1" 2
1A
r 0
(2" 3C_C0083
n . (a) f n is a *ositive integer and r is is a non
2 2 2 2 (b) >sing (a) or otherise3 sho that (C 0 " (C 1 " (C 2 " (C n " n
n
n
n
( 2 n"! . ( n! " 2 (4 ars"
Proof: 2n 2n 2n 2n 2 2n 2n (a) (1 x" C 0 C 1 x C 2 x C 2n x
(1 x" C 0 C 1 x C 2 x C n x n
n
n
n
2
n
(1 x" 2 n (1 x " n (1 x" n
C 0 C 1 x C 2 x C 2 n x 2n
2n
2n
2
2n
n
2n
(C 0n C 1n x C 2n x2 C nn xn "(C 0n C 1n x C 2n x 2 C nn x n "
1
9+ co*aring the coefficient of x r 3 e have C r C 0 C r C 1 C r 1 C 2 C r 2 C r C 0 2n
n
n
n
n
n
n
n
n
1 (2"
n into C 0nC r n C 1nC r n1 C 2nC r n 2 C r nC 0n C r 2n 3 (b) )u:stitute r C 0 C n C 1 C n 1 C 2 C n 2 C n C 0 C n n
n
n
n
n
n
n
n
2n
C 0nC 0n C 1nC 1n C 2nC 2n C nnC nn
(C 0n " 2 (C 1n " 2 (C 2n " 2 (C nn " 2
1 (2n"!
(2n n"!n!
( 2 n "! ( n! " 2
1 (2"
2014 Chung Tai Educational Press. Press. All rights reserved. reserved.
3.46 3.46
