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Additional Mathematics Revision 1. Simplify the following: a) −8 + 5 − 18 b) 12 ! !! ×3 !! ! 2. Factorize the following: a) 3 ! + 7 + 2 b) 2 ! − 7 + 6 3. a) Find the quotient and the remainder for each of the following division of polynomial. (6 ! − 3 ! + 2 ! + 11 − 6) ÷ (2 − 1) b) Put in form ≡ . + () 4. When ! − 5 ! − 3 + 8 is divided by − 2, the remainder is 13. Find the value of the constant . 5. Determine which of the following linear divisors are factors of 3 ! − 2 ! − 13 ! + 8 + 4 a) 3 + 1 b) 2 − 1 c) − 1 6. If ! + ! + + 6 is divisible by ! + 3 + 2, evaluate the values of the constant p and q. 7. Use the quadratic formula to solve a) 2 ! − 3 − 2 = 0 b) 3 ! + 10 − 8 = 0
Transcript of Additional Mathematics Revision - mathsville.weebly.com · Additional Mathematics Revision 13....
Additional Mathematics Revision
1. Simplifythefollowing:a) −8𝑥 + 5𝑦 − 18𝑥b) 12𝑥!𝑦!!×3𝑥!!𝑦!
2. Factorizethefollowing:a) 3𝑥! + 7𝑥 + 2b) 2𝑥! − 7𝑥 + 6
3. a)Findthequotientandtheremainderforeachofthefollowingdivisionofpolynomial.(6𝑥! − 3𝑥! + 2𝑥! + 11𝑥 − 6)÷ (2𝑥 − 1)b)Putinform𝑝 𝑥 ≡ 𝑔 𝑥 . 𝑞 𝑥 + 𝑟 (𝑥)
4. When𝑘𝑥! − 5𝑥! − 3𝑥 + 8isdividedby𝑥 − 2,theremainderis13.Findthevalueoftheconstant𝑘.
5. Determinewhichofthefollowinglineardivisorsarefactorsof3𝑥! − 2𝑥! − 13𝑥! + 8𝑥 + 4
a) 3𝑥 + 1b) 2𝑥 − 1c) 𝑥 − 1
6. If𝑥! + 𝑝𝑥! + 𝑞𝑥 + 6isdivisibleby𝑥! + 3𝑥 + 2,evaluatethevaluesofthe
constantpandq.
7. Usethequadraticformulatosolve
a) 2𝑥! − 3𝑥 − 2 = 0b) 3𝑥! + 10𝑥 − 8 = 0
Additional Mathematics Revision
8. Expressthequadraticfunction𝑓 𝑥 = 4𝑥! − 16𝑥 − 6intheform𝑎(𝑥 + ℎ)! + 𝑘.Hence:
a) Statetheturningpointandstatewhetheritisamaximumor
minimumb) Statethemaximumorminimumvalueof𝑥c) Statetheaxisofsymmetryd) Statetherangeof𝑓forthegivendomaine) Statethe𝑦interceptf) Solve𝑎(𝑥 + ℎ)! + 𝑘 = 0g) Statethe𝑥interceptsh) Sketchagraph𝑓 𝑥 =𝑎(𝑥 + ℎ)! + 𝑘
9. Determinetherootsofeachofthefollowingquadraticequation.
a) 6(𝑥 − 4)! + 3 = 0b) 3(𝑥 − 1)! − 4 = 0c) 5(𝑥 − 3)! = 0
10. Solveeachofthefollowingequations.a) 5!!!! − 26(5!)+ 5 = 0b) 6𝑥 − 17 𝑥 + 5 = 0c) 2𝑥! − 7𝑥! + 3 = 0
11. If𝛼,𝛽aretherootsoftheequation3𝑥! − 6𝑥 − 4 = 0,findthevalueofa) (i)𝛼!𝛽 + 𝛽!𝛼(ii)(𝛼!𝛽)( 𝛽!𝛼)(iii)(𝛼 + 1)(𝛽 + 1)b)(iv)(𝛼 + 1)+ (𝛽 + 1)(v) 𝛼 + !
! + ( 𝛽 + !
! )
12. Solvethefollowingpairofsimultaneousequation𝑦 + 3 = 5𝑥! + 2𝑥𝑦 + 6𝑥 = 1
Additional Mathematics Revision
13. Solveeachofthefollowingquadraticinequalities,usinganalgebraicmethodandagraphicalmethod.
a) (i)2𝑥! − 𝑥 − 15 < 0(ii)2𝑥! − 𝑥 − 15 ≤ 0b) (i)2𝑥! − 𝑥 − 15 > 0
(ii)2𝑥! − 𝑥 − 15 ≥ 0
d) !!!!!!!!
≥ 0
14. (a)Sketchthegraphofeachofthefollowinglinearfunctionsforthegivendomain.(b)Hence,determinetherangeofeachfunction.(i) 𝑓 𝑥 = 3𝑥 + 3; Domain:{−3,−2,−1, 0, 1, 2}(ii) 𝑔 𝑥 = 3𝑥 + 3; Domain:−3 ≤ 𝑥 ≤ 2(iii) ℎ 𝑥 = 3𝑥 + 3; Domain:𝑥 ∈ 𝑅
15. Afunction𝑓isdefinedby𝑓:𝑥 → !!!!!!!
.
a) Statethevalueof𝑥forwhich𝑓isnotdefined,andstatethedomainof𝑓.
b) Findtheinversefunction𝑓!!,andstatethedomainof𝑓!!.
16. Given𝑓: 𝑥 → 1𝑥+2 , 𝑥 ≠ 2and𝑔 𝑥 = 2𝑥 − 5,find
a) 𝑔𝑓 b) 𝑓𝑔c)𝑓!d)𝑔!e) 𝑔!!𝑓!!
17. Thefunction𝑓isdefinedby𝑓: 𝑥 → 2𝑥𝑥−5,for𝑥 ≠ 5.
a)Findanexpressionfor(i)𝑓!(ii)𝑓!!b)Hence,findthenonzerovalueof𝑥forwhich𝑓! = 𝑓!!.
Additional Mathematics Revision
18. Differentiate the following: a) 𝑦 = 𝑥!
b) 𝑦 = (𝑥 − 15)!"
c) 𝑦 = 6
d) 𝑦 = (2𝑥! − 5)!"
e) 𝑥!" sin 𝑥
f) sin 𝑥 cos 𝑥
g) 𝑦 = 𝑥3𝑥5
h) 𝑦 = !"# !!!"# !!
i) 𝑦 = 2𝑥! − 2sin 3𝑥! + 5cos 5𝑥!
19. Differentiate the following:
a) 𝑦 = (3𝑥! − 5𝑥)!
b) 𝑝 = 15(12𝑡! − !!𝑡)
!!
c) 𝑦 = 𝑡𝑎𝑛𝑥
d) y = 2 tan!𝑥
e) 𝑦 = (tan 8𝑥!)!!
f) 𝑦 = ln(𝑥! − 5𝑥)
g) 𝑦 = 𝑒!!!
h) 𝑦 = 𝑒!"# !!!
i) 𝑦 = 4𝑥! − 3
j) 𝑦 = 𝑥!𝑐𝑜𝑠!2𝑥
k) 𝑦 = 𝑥 2𝑥 − 5
Additional Mathematics Revision
l) 𝑦 = !!!!!!
!"#!!!
m) 𝑦 = !
!!!
20. If 𝑥!𝑦 + 𝑦!𝑥 = 12𝑦. Find !"!"
at (0,1).
21. Find the equation of the tangent in a curve 𝑦 = 𝑥! − 6𝑥 + 25 at (0,5). Given that 𝑦 = !!!!
!!!! show that !"
!" is always positive.
22. Find the gradient of the tangent to the curve 𝑦 = 𝑥𝑐𝑜𝑠𝑥 at the point where
𝑥 = 0.5 giving your answer to (2dp).
23. A curve of equation 𝑦! + 3𝑦 ln 𝑥 = 10. Find !"!"
at (𝑒, 2). Note: 𝒍𝒏 𝒆 = 𝟏.
24. Integratethefollowing
a) 𝑒!!!
b) ln(𝑥! − 5𝑥)
c) 𝑥!
d) 𝑒!"# !!!
e) 𝑥3𝑥5
f) (6+ 2𝑥2 − 2sin3𝑥9 + 5cos5𝑥3)
g) tan!𝑥
h) 15(12𝑡! − !
!𝑡)
!!
i) (2𝑥! − 5)!"!"
!
