OCR A-LEVEL MATHS REVISED
QUESTIONS WITH GUARANTEED A+
GRADE ANSWERS
- Sn = n/2 (2a + (n-1)d)
- Sn = n/2 (a + l) where a is the first term and l is the last term - Correct Answers -formula of an
arithmetic series
the sum of the terms of an arithmetic sequence - Correct Answers -what is an arithmetic series
- Un = a + (n-1)d
- a = the first term
- d = the common difference - Correct Answers -nth term of an arithmetic sequence
- Un = ar^(n-1)
- a = first term
- r = common ratio - Correct Answers -nth term of a geometric sequence
- Sn = a(1-r^n) / 1-r
- Sn = a(r^n - 1) / r-1
where r does not equal 1 - Correct Answers -formula of first n terms of a geometric sequence
the sum of the values tend towards infinity - Correct Answers -divergent sequence
- the sum of the values tend towards a specific number
- it is only convergent if |r|<1 - Correct Answers -convergent sequence
,a / 1-r - Correct Answers -sum to infinity of a geometric series
- Correct Answers -series can be shown using sigma notation
- defines each term of a sequence as a function of the previous term
- to find the members of the sequence substitute in n=1, n=2 ... using the previous terms given -
Correct Answers -recurrence relation of form Un+1 = f(Un)
it is decreasing - Correct Answers -if Un+1 < Un for all n ∈ ℕ, what is true of the sequence
- it is periodic
- means that the terms repeat in a cycle
- k = the order of the sequence (how often the terms repeat) - Correct Answers -if Un+k = Un for
all n ∈ ℕ, what is true of the sequence
(x+y)(x-y) - Correct Answers -x^2-y^2
* (a-sqrt(b) / a-sqrt(b)) - Correct Answers -rationalising the denominator of e.g. 1/sqrt(b)+a
b^2 - 4ac > 0 has 2 distant real roots
B^2 -4ac = 0 has on real repeated root
b^2 - 4ac < 0 has no real roots - Correct Answers -using the discriminant to find number of roots
if f(x) = a(x+p)^2 + q, then the turning point is (-p,q) - Correct Answers -completing the square to
find the turning point
< is dotted line
,≤ is solid line - Correct Answers -using lines to represent < and ≤
x=0 and y=0 - Correct Answers -where are the asymptotes of y = k/x
translationBupBbyBaBunitsB-BCorrectBAnswersB-yB=Bf(x)B+Ba
translationBleftBbyBaBunitsB-BCorrectBAnswersB-yB=Bf(x+a)
stretchBverticallyBbyBscaleBfactorBaB-BCorrectBAnswersB-yB=Baf(x)
stretchBbyBscaleBfactorB1/aBhorizontallyB-BCorrectBAnswersB-yB=Bf(ax)
reflectionBinBx-axisB-BCorrectBAnswersB-yB=B-f(x)
reflectionBinBy-axisB-BCorrectBAnswersB-yB=Bf(-x)
mB=B(y2B-By1)/(x2B-Bx1)B-BCorrectBAnswersB-calculatingBtheBgradientBwithB2Bpoints
y-y1=m(x-x1)B-BCorrectBAnswersB-anotherBwayBtoBcalculateBequationBofBaBline
y=B-(1/m)xB-BCorrectBAnswersB-equationBofBlineBperpendicularBtoByB=Bmx
SqrtB((x2B-Bx1)^2B+B(y2B-By1)^2B)B-BCorrectBAnswersB-distanceBbetweenB(x1,y1)BandB(x2,y2)
x^2B+By^2B=Br^2B-BCorrectBAnswersB-equationBofBcircleBcentreB(0,0)
, (x-a)^2B+B(y-b)^2B=Br^2B-BCorrectBAnswersB-equationBofBcircleBcentreB(a,b)
centre:B(-f,-g)
radius:BsqrtB(f^2B+Bg^2B-c)B-BCorrectBAnswersB-centreBandBradiusBofBx^2B+By^2B+B2fxB+B2gyB+BcB=B0
perpendicularB-BCorrectBAnswersB-
aBtangentBtoBaBcircleBisB......BtoBtheBradiusBofBtheBcircleBatBtheBpointBofBintersection
theBcentreBofBaBcircleB-BCorrectBAnswersB-theBperpendicularBbisectorBofBaBchordBwillBgoBthrough.....
aBrightBangleB-BCorrectBAnswersB-theBangleBinBaBsemicircleBisBalways
- CorrectBAnswersB-ifB∠PRQB=B90°BthenBRBliesBonBtheBcircleBwithBdiameterBPQ
B B
-findBtheBequationsBofBtheBperpendicularBbisectorsBofB2BdifferentBchords
-findBtheBcoordinatesBofBtheBintersectionBofBtheBperpendicularBbisectorsB-BCorrectBAnswersB-
findBtheBcentreBofBaBcircleBgivenBanyB3Bpoints
ifBf(p)B=B0BthenB(x-p)BisBaBfactorBofBf(x)B-BCorrectBAnswersB-factorBtheorem
startingBfromBknownBfactsBorBdefinitionsBthenBusingBlogicalBstepsBtoBreachBtheBdesiredBconclusionB
-BCorrectBAnswersB-proofBbyBdeduction
breakingBtheBstatementBintoBsmallerBcasesBandBprovingBeachBcaseBseparatelyB-BCorrectBAnswersB-
proofBbyBexhaustion
anBexampleBthatBdoesBnotBworkBforBtheBstatementB-BCorrectBAnswersB-proofBbyBcounter-example
QUESTIONS WITH GUARANTEED A+
GRADE ANSWERS
- Sn = n/2 (2a + (n-1)d)
- Sn = n/2 (a + l) where a is the first term and l is the last term - Correct Answers -formula of an
arithmetic series
the sum of the terms of an arithmetic sequence - Correct Answers -what is an arithmetic series
- Un = a + (n-1)d
- a = the first term
- d = the common difference - Correct Answers -nth term of an arithmetic sequence
- Un = ar^(n-1)
- a = first term
- r = common ratio - Correct Answers -nth term of a geometric sequence
- Sn = a(1-r^n) / 1-r
- Sn = a(r^n - 1) / r-1
where r does not equal 1 - Correct Answers -formula of first n terms of a geometric sequence
the sum of the values tend towards infinity - Correct Answers -divergent sequence
- the sum of the values tend towards a specific number
- it is only convergent if |r|<1 - Correct Answers -convergent sequence
,a / 1-r - Correct Answers -sum to infinity of a geometric series
- Correct Answers -series can be shown using sigma notation
- defines each term of a sequence as a function of the previous term
- to find the members of the sequence substitute in n=1, n=2 ... using the previous terms given -
Correct Answers -recurrence relation of form Un+1 = f(Un)
it is decreasing - Correct Answers -if Un+1 < Un for all n ∈ ℕ, what is true of the sequence
- it is periodic
- means that the terms repeat in a cycle
- k = the order of the sequence (how often the terms repeat) - Correct Answers -if Un+k = Un for
all n ∈ ℕ, what is true of the sequence
(x+y)(x-y) - Correct Answers -x^2-y^2
* (a-sqrt(b) / a-sqrt(b)) - Correct Answers -rationalising the denominator of e.g. 1/sqrt(b)+a
b^2 - 4ac > 0 has 2 distant real roots
B^2 -4ac = 0 has on real repeated root
b^2 - 4ac < 0 has no real roots - Correct Answers -using the discriminant to find number of roots
if f(x) = a(x+p)^2 + q, then the turning point is (-p,q) - Correct Answers -completing the square to
find the turning point
< is dotted line
,≤ is solid line - Correct Answers -using lines to represent < and ≤
x=0 and y=0 - Correct Answers -where are the asymptotes of y = k/x
translationBupBbyBaBunitsB-BCorrectBAnswersB-yB=Bf(x)B+Ba
translationBleftBbyBaBunitsB-BCorrectBAnswersB-yB=Bf(x+a)
stretchBverticallyBbyBscaleBfactorBaB-BCorrectBAnswersB-yB=Baf(x)
stretchBbyBscaleBfactorB1/aBhorizontallyB-BCorrectBAnswersB-yB=Bf(ax)
reflectionBinBx-axisB-BCorrectBAnswersB-yB=B-f(x)
reflectionBinBy-axisB-BCorrectBAnswersB-yB=Bf(-x)
mB=B(y2B-By1)/(x2B-Bx1)B-BCorrectBAnswersB-calculatingBtheBgradientBwithB2Bpoints
y-y1=m(x-x1)B-BCorrectBAnswersB-anotherBwayBtoBcalculateBequationBofBaBline
y=B-(1/m)xB-BCorrectBAnswersB-equationBofBlineBperpendicularBtoByB=Bmx
SqrtB((x2B-Bx1)^2B+B(y2B-By1)^2B)B-BCorrectBAnswersB-distanceBbetweenB(x1,y1)BandB(x2,y2)
x^2B+By^2B=Br^2B-BCorrectBAnswersB-equationBofBcircleBcentreB(0,0)
, (x-a)^2B+B(y-b)^2B=Br^2B-BCorrectBAnswersB-equationBofBcircleBcentreB(a,b)
centre:B(-f,-g)
radius:BsqrtB(f^2B+Bg^2B-c)B-BCorrectBAnswersB-centreBandBradiusBofBx^2B+By^2B+B2fxB+B2gyB+BcB=B0
perpendicularB-BCorrectBAnswersB-
aBtangentBtoBaBcircleBisB......BtoBtheBradiusBofBtheBcircleBatBtheBpointBofBintersection
theBcentreBofBaBcircleB-BCorrectBAnswersB-theBperpendicularBbisectorBofBaBchordBwillBgoBthrough.....
aBrightBangleB-BCorrectBAnswersB-theBangleBinBaBsemicircleBisBalways
- CorrectBAnswersB-ifB∠PRQB=B90°BthenBRBliesBonBtheBcircleBwithBdiameterBPQ
B B
-findBtheBequationsBofBtheBperpendicularBbisectorsBofB2BdifferentBchords
-findBtheBcoordinatesBofBtheBintersectionBofBtheBperpendicularBbisectorsB-BCorrectBAnswersB-
findBtheBcentreBofBaBcircleBgivenBanyB3Bpoints
ifBf(p)B=B0BthenB(x-p)BisBaBfactorBofBf(x)B-BCorrectBAnswersB-factorBtheorem
startingBfromBknownBfactsBorBdefinitionsBthenBusingBlogicalBstepsBtoBreachBtheBdesiredBconclusionB
-BCorrectBAnswersB-proofBbyBdeduction
breakingBtheBstatementBintoBsmallerBcasesBandBprovingBeachBcaseBseparatelyB-BCorrectBAnswersB-
proofBbyBexhaustion
anBexampleBthatBdoesBnotBworkBforBtheBstatementB-BCorrectBAnswersB-proofBbyBcounter-example
