miscellaneous exercise 19
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Solutions to Miscellaneous Exercise 19 (Ho Soo Thong & Khor Nyak Hiong’s Panpac Additional Mathematics)
Solved by: Dr Lee Chu Keong ([email protected])
Dr Lee Chu Keong (http://ascklee.org/)
1
Miscellaneous Exercise 19 Question 1(a)
∫
Question 1(b)
∫
∫
[
]
√
Note: Question 1 involves indefinite integrals1, and so, you must remember to add the
arbitrary constant, C.
Question 2(a)
∫ √
∫ ( )
[( )
( )
]
[( )
]
Question 2(b)
1 Ho Soo Thong and Khor Nyak Hiong (Section 19.1)
Dr Lee Chu Keong (http://ascklee.org/)
2
∫ ( )
∫
[
]
[
]
[
( )
( )
( )
] [
]
[
] [
]
Question 2(c)
∫( )
( )
∫
∫
∫
∫
∫( )
( )
∫
( )
∫ ( )
∫
( )
∫ ( )
∫ ( )
[( )
]
[( )
]
[( )
]
[( )
]
[( ) ]
[
]
Dr Lee Chu Keong (http://ascklee.org/)
3
[ ]
[
]
[ ]
[
]
(
)
Question 3(a)
∫ ( )
∫
[ ] [ ]
( ) ( ) ( )
( )( )
Question 3(b)
∫ ( )
∫ ( )
[( )
( )]
([( )
] [
( )
]) ( )
(( ) ( ) )
(( ) ( ))
(( ) )
(( ) )
( )
( )
Dr Lee Chu Keong (http://ascklee.org/)
4
Question 4(a)
At turning points,
.
Therefore, at ( ):
( )
Question 4(b)
This curve passes through ( ):
( ) ( )
At the point the curve meets the y-axis, the x-coordinate is zero:
The point the curve meets the y-axis is ( ).
Question 5(a)
∫
∫
∫
∫
At , the number of bacteria is A:
Dr Lee Chu Keong (http://ascklee.org/)
5
( )
Question 5(b)(i)
∫
∫
At , :
( )
Question 5(b)(ii)
Dr Lee Chu Keong (http://ascklee.org/)
6
Question 6
( )
[ ( )
] [
( )
( )]
( )
( )
√
√
√ √
√
( )
√
√
√
( )
√
∫ ( )
√
[ √ ]
[ √ ] [ √ ]
[ ] [ ]
Question 7(a)
∫ ( )
∫ ( )
∫ ( )
∫ ( )
∫ ( )
∫ ( )
Dr Lee Chu Keong (http://ascklee.org/)
7
Question 7(b)
∫ ( )
∫ ( )
∫ ( )
∫ ( )
∫ ( )
Question 7(c)
∫ ( )
∫ ( )
∫
∫ ( )
[ ]
( ) [ ]
[ ]
Question 8(a)
∫
[ ]
[(
) ( )]
(√
√
) ( ( ))
√
√
Question 8(b)
∫ (
)
[ (
)
]
Dr Lee Chu Keong (http://ascklee.org/)
8
[ (
)]
[
]
[
√
]
(√ )
Question 9
( ) ( )
( )
( )
( )
( )( )
∫
[
]
∫
[
]
[(
) (
)]
[(
) (
√
√
)]
Dr Lee Chu Keong (http://ascklee.org/)
9
√
√
√
√
√
√
√ √
√
√
√
Question 10(a)
∫( ) ∫
Question 10(b)
∫
∫
Work-in-Progress beyond this point!
Question 11
( )( )
( ) ( )
Let :
Let
:
Dr Lee Chu Keong (http://ascklee.org/)
10
( )( )
∫
( )( )
∫
∫
∫
∫
∫
( )
( )
(
)
Question 12
( )( )
( )
( ) ( )( ) ( )
When : 3
When
:
When : ( )
( )( )
( )
∫
( )( )
∫
∫
∫
( )
Question 13
( )( )
( ) ( )( )
Dr Lee Chu Keong (http://ascklee.org/)
11
When :
When : ( )
When : ( )
( )( )
∫
( )( ) ∫
∫
∫
( )( ) ∫
∫
Question 14
∫
∫
( )( )
( )( )
( ) ( )
Let :
Let :
( )( )
∫
( )( ) ∫
∫
∫
( ) ( )
(
)
Question 15(a)
( )
∫
[ ]
Dr Lee Chu Keong (http://ascklee.org/)
12
∫
[ ]
[
]
[
]
Question 15(b)
∫
[ ( )]
Question 16
[ ]
Question 16(a)
∫
∫
[ ]
[
]
[ ]
Question 16(b)
∫
∫
∫ ( )
Dr Lee Chu Keong (http://ascklee.org/)
13
∫
∫
∫
[
]
[
]
[ ]
Question 17(a)
∫
∫
( )
∫
[ ( )]
( ) ( )
(
)
(
)
Question 17(b)
[ ] ( ) (
)
∫
∫ ∫
Dr Lee Chu Keong (http://ascklee.org/)
14
∫
∫
∫
(
)
When :
∫
When :
∫
(
)
When :
∫
(
)
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