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])

http://ascklee.org/CV/CV.pdf


1

Miscellaneous Exercise 3 Question 1

Sum of roots:

Product of roots:

Question 1(a) ( )

(

)

(

)

Question 1(b)

( ) ( )

(

)

(

)

Question 1(c)

√

√


2

Note: because

Question 1(Hence, …)

( )

( √

)

√

( )

(

)

The deduced equation:

√

Question 2(a)

( )

Real roots:

( ) ( )( )

Question 2(b)

√

√

( ) √

Distinct real roots:

( √ ) ( )( )

( )

( )( )


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:

( )( )


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:

( ) ( )( )


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:

( ) ( )( )


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:

( ) ( )


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


8

( )

( )

Product of roots:

( )

( )

( )

( )

Equating ( ) and ( ):

( )

( )

Multiply throughout by 3:

( )( )

(

)

(

)

Question 11(a)


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 equation ( ):


10

(

)

Question 11(b)

Sum of roots:

( )

Product of roots:

( )

( )

( )

( )

( )

( )

From ( ):

( )


( )

Squaring both sides of equation ( ):

( )

( )


( )


11

Question 12(a)

( )

( )

( )

( ) ( )

From (2) and (4):

( ) ( )

(

)

( )

(

)

( )

Question 12(b)

( )

( )

( )

( )


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


13

Question 14(a)

( ) ( )( )

( √ )( √ )

Question 14(b)

( )

( ) ( )( )

( )


( )

Divide throughout by :

( )

Question 15(a)

( )

[ ( )] ( )( )

[ ]


( )( )


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


15

( ) ( )

( ) ( )

( ) ( )( )

( )( )

Question 18

( )

No roots:

( )


( )( )

Question 19

( )( )

No roots, therefore is always positive!

Question 19(a)

( )( )

( )

Since is always positive,

Question 19(b)


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


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


18

The latest version of this file can be

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16 18 20 22 24 26t

74

75

76

77

78

79

80

h

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