Miscellaneous Exercise 3
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Transcript of Miscellaneous Exercise 3
Dr Lee Chu Keong (http://ascklee.org/)
0
Solutions to Miscellaneous Exercise 3 (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 3 Question 1
Sum of roots:
Product of roots:
Question 1(a) ( )
(
)
(
)
Question 1(b)
( ) ( )
(
)
(
)
Question 1(c)
√
√
Dr Lee Chu Keong (http://ascklee.org/)
2
Note: because
Question 1(Hence, …)
( )
( √
)
√
( )
(
)
The deduced equation:
√
Question 2(a)
( )
Real roots:
( ) ( )( )
Question 2(b)
√
√
( ) √
Distinct real roots:
( √ ) ( )( )
( )
( )( )
Dr Lee Chu Keong (http://ascklee.org/)
3
Question 3(a)
( )( )
Question 3(b)
( √ )( √ )
√ √
Question 4(a)
( )( )
Question 4(b)
( )
Therefore, the equation has no roots:
( ) ( )( )
Question 5
( )
Question 5(a)
Coincident roots:
( )( )
Dr Lee Chu Keong (http://ascklee.org/)
4
( )( )
Question 5(b)
Distinct real roots:
( )( )
( )( )
Question 5(c)
One positive root and one negative root, which means that the product of the roots must
be negative!
Question 6(a)
( )
( )
( )
Tangent to the curve ⟹ two equal real roots:
( ) ( )( )
Dr Lee Chu Keong (http://ascklee.org/)
5
Question 6(b)
Real roots:
( ) ( )( )
( √ )( √ )
√ √
Largest negative integer is
Question 7(a)
( ) ( )
Equal roots:
( ) ( )( )
( )( )
Question 7(b)
Multiply throughout by :
Two distinct real roots:
( ) ( )( )
Dr Lee Chu Keong (http://ascklee.org/)
6
( )( )
Question 8(a)
√
√
√
√
√
Question 9(a)
One root is twice the other: and
Sum of roots:
( )
( )
Product of roots:
( )
Substitute ( ) into ( ):
(
)
( )
Multiply both sides by 9:
( ) ( )
Dr Lee Chu Keong (http://ascklee.org/)
7
( )( )
is discarded as it will result in roots that are zero.
Question 9(b)
( )( )
Question 10(a)
( )( )
Multiply by 2:
Comparing coefficients, we get:
Question 10(b)
The roots are: and ( )
Sum of roots:
3 2 1 1x
2
1
1
2
3
4
y
Dr Lee Chu Keong (http://ascklee.org/)
8
( )
( )
Product of roots:
( )
( )
( )
( )
Equating ( ) and ( ):
( )
( )
Multiply throughout by 3:
( )( )
(
)
(
)
Question 11(a)
Dr Lee Chu Keong (http://ascklee.org/)
9
Question 11(a)(i)
Two equal roots:
( ) ( )
Divide throughout by 4:
( )
Question 11(a)(ii)
Two real roots, one being trice the other: and
Sum of roots:
( )
Product of roots:
( )
Substitute ( ) into ( ):
( )
Substitute
into equation ( ):
Dr Lee Chu Keong (http://ascklee.org/)
10
(
)
Question 11(b)
Sum of roots:
( )
Product of roots:
( )
( )
( )
( )
( )
( )
From ( ):
( )
Substitute ( ) into ( ):
( )
Squaring both sides of equation ( ):
( )
( )
Substitute ( ) into ( ):
( )
Dr Lee Chu Keong (http://ascklee.org/)
11
Question 12(a)
( )
( )
( )
( ) ( )
From (2) and (4):
( ) ( )
(
)
( )
(
)
( )
Question 12(b)
( )
( )
( )
( )
Dr Lee Chu Keong (http://ascklee.org/)
12
Substituting into ( ):
Question 13(a)
( )( )
Blue curve:
Red line:
Question 13(b)
( )
( )
( ) ( )
1.5 1.0 0.5 0.5 1.0 1.5 2.0x
1
2
3
4
5
6
y
Dr Lee Chu Keong (http://ascklee.org/)
13
Question 14(a)
( ) ( )( )
( √ )( √ )
Question 14(b)
( )
( ) ( )( )
( )
Substitute ( ) into ( ):
( )
Divide throughout by :
( )
Question 15(a)
( )
[ ( )] ( )( )
[ ]
Divide throughout by 4:
( )( )
Dr Lee Chu Keong (http://ascklee.org/)
14
Question 15(b)
Blue curve: ( )( )
Red line:
Question 16(a)
( ) ( )
( ) ( )( )
( )
Question 16(b)
( )
(
)
Multiply throughout by ( ):
( ) ( )
1 1 2 3x
4
3
2
1
1
2
y
Dr Lee Chu Keong (http://ascklee.org/)
15
( ) ( )
( ) ( )
( ) ( )( )
( )( )
Question 18
( )
No roots:
( )
Divide throughout by 4:
( )( )
Question 19
( )( )
No roots, therefore is always positive!
Question 19(a)
( )( )
( )
Since is always positive,
Question 19(b)
Dr Lee Chu Keong (http://ascklee.org/)
16
( )( )
is always positive, and so:
( )
This can be seen from the graph of below:
Question 20(a)
( ) ( )( )
1.0 0.5 0.5x
1
1
2
3
4
y
Dr Lee Chu Keong (http://ascklee.org/)
17
Question 20(a)
( )
Multiply throughout by 5
( )( )
0.5 1.0 1.5 2.0 2.5 3.0x
5
10
y
0 5 10 15 20 25 30t
20
40
60
80
h
Dr Lee Chu Keong (http://ascklee.org/)
18
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