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Success in SPM – Additional Mathematics – Form 4Chapter 2: Quadratic Equations1. Given i. ii. iii. and + + + are the roots of 3 − 2 + 1 = 0, find1Success in SPM – Additional Mathematics – Form 42Success in SPM – Additional Mathematics – Form 4 2. The equation 4 − 8 + 9 = 0 has roots i. + 3 and + 3 ii. and iii. and and , find3Success in SPM – Additional Mathematics – Form 44Success in SPM – Additional Mathematics – Form 4 3. Given + + = 0 has one root 4 times of another roo
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Success in SPM – Additional Mathematics – Form 4
1
Chapter 2: Quadratic Equations
1. Given � and � are the roots of 3� − 2� + 1 = 0, find
i. '
(+
'
)
ii. (
)+
)
(
iii. '
(*++
'
)*+
Success in SPM – Additional Mathematics – Form 4
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Success in SPM – Additional Mathematics – Form 4
3
2. The equation 4� − 8� + 9 = 0 has roots � and �, find
i. � + 3 and � + 3
ii. +
( and
+
)
iii. (
) and
)
(
Success in SPM – Additional Mathematics – Form 4
4
Success in SPM – Additional Mathematics – Form 4
5
3. Given 1� + 2� + 3 = 0 has one root 4 times of another root, find 3 in terms of 1 and 2.
Success in SPM – Additional Mathematics – Form 4
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4. 56 + 17� − 536 − 47� + 26 + 9 = 0 has a root negative of another root, find 6.
Success in SPM – Additional Mathematics – Form 4
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5. Given one root of 2:� + 55: + 37� − 5: + 57 = 0 is the reciprocal of another root, find
:.
Success in SPM – Additional Mathematics – Form 4
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6. The roots of the equation 2� + 6� + 3 = 0 where 6 > 0 are � and � whilst those of the
equation 3� − 2� + 3 are (
) and
)
(. Find 6.
Success in SPM – Additional Mathematics – Form 4
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7. Given � + 56 − 57� − 6 = 0 has roots which differ by 4. Find
i. The value of each root
ii. The value of 6.
Success in SPM – Additional Mathematics – Form 4
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8. Given that � is the root of each of the equations � − 5� + 6 = 0 and � − 6� + 36 = 0,
where 6 ≠ 0, find 6 and �.
Success in SPM – Additional Mathematics – Form 4
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9. Given that � and � are the roots of the equation � = � − 2,
i. Find the value of '
(A+
'
)A
ii. Prove that �B = 2 − 3�
Success in SPM – Additional Mathematics – Form 4
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Success in SPM – Additional Mathematics – Form 4
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10. Form the quadratic equation for which the sum of the roots is 2 and the sum of the
squares of the roots is 18.
Success in SPM – Additional Mathematics – Form 4
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11. Given the equation 53 + C7� − 58 + 4C7� + 53 + 4C7 = 0, find the value of C for
which
i. One root is the negative of the other
ii. One root is the reciprocal of the other
Success in SPM – Additional Mathematics – Form 4
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Success in SPM – Additional Mathematics – Form 4
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12. Write � + 66� + 144 in the form 5� + 17 + 2 and thus obtain expressions for 1 and 2
in terms of 6. Hence find the range of values of 6 such that � + 66� + 144 is positive
for all values of �, and deduce the corresponding range of values of 6.
Success in SPM – Additional Mathematics – Form 4
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Success in SPM – Additional Mathematics – Form 4
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13. The equation 3� + 1� + 120 = 0, where 1 > 0, has roots � and � where � − � = 3.
Evaluate � and � and hence, or otherwise, find the value of 1.
Success in SPM – Additional Mathematics – Form 4
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14. Obtain the quadratic equation whose roots are the squares of the roots of the equation
3� − 2� + 5 = 0.
Success in SPM – Additional Mathematics – Form 4
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15. Given that the equation 2� − 2� + 3 = 0 has the roots � and �, find the equation
whose roots are '
)A and
'
(A.
Success in SPM – Additional Mathematics – Form 4
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16. Given that � and � are the roots of the equation 2� = 3� − 4,
i. Form an equation whose roots are � − � and � − �.
ii. Show that 4�+ = � − 12
Success in SPM – Additional Mathematics – Form 4
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Success in SPM – Additional Mathematics – Form 4
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17. The quadratic equation � + 1� + 2 = 0 has roots −2 and 6. Find
i. The value of 1 and 2
ii. The range of values of 3 for which the equation � + 1� + 2 = 3 has no real roots
Success in SPM – Additional Mathematics – Form 4
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18. The equation � − 2� + 3 = 0 has roots � and � and the equation � − 4� + 1 = 0 has
roots I
( and
I
) . Find the value of 6 and of 1.
Success in SPM – Additional Mathematics – Form 4
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19. Find the values of 6 for which the line J = 2� + 6 is a tangent to the curve J =
'KL .
Success in SPM – Additional Mathematics – Form 4
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20. Given that �1 and �2 are the roots of :� + M� + N = 0, state in terms of some or all of :,
M and N,
i. The condition that �1= �2
ii. The value of �1+�2
Success in SPM – Additional Mathematics – Form 4
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21. Find the values of C for which the line J = C� is a tangent to the curve J = 3� − 1.
Success in SPM – Additional Mathematics – Form 4
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22. The line J = 2� mees the curve 3J = � − 10 at the points A5�1, J17 and B5�2 , J27
i. Obtain the quadratic equation whose roots are �1 and �2
ii. Without solving this equation, find the � co-ordinate of the midpoint of AB
