Higher Mathematics
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Transcript of Higher Mathematics
Higher Mathematics: HappySmile Higher Mathematics: HappySmile ProductionsProductions
Higher Higher MathematicsMathematics
Everything you wanted Everything you wanted to know about Higher to know about Higher
Maths but …Maths but …
or
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Pick a numberPick a number11 22 33 4.4. 55 6.6. 77 88 9.9. 1010 11.11. 12.12. 13.13. 14.14.
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Question 1Question 1
How do you draw How do you draw the graph of the graph of –f(x)–f(x) given the graph given the graph of of f(x)f(x) ? ?
answer
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Answer to Question 1Answer to Question 1
Reflect the graph Reflect the graph in the x-axisin the x-axis
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Question 2Question 2
For what values For what values is the functionis the functionf(x) = f(x) = √(x-a)√(x-a) undefined?undefined?
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Answer to Question 2Answer to Question 2
x < ax < aReturn to question
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Question 3Question 3
If If f(x) = xf(x) = xnn
then what isthen what isff’’(x)(x) ? ?
answer
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Answer to Question 3Answer to Question 3
nxnxn-1n-1
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Question 4Question 4What is theWhat is thegeneral equationgeneral equationof a straightof a straightline passingline passingthrough through (a,b)(a,b)with gradient with gradient mm ? ?
answer
x
y
(a,b)
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Answer to Question 4Answer to Question 4
y – b = m(x – a)y – b = m(x – a)
x
y
(a,b)
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Question 5Question 5
If If f(t)f(t) is the distance is the distance travelled in a certain travelled in a certain time t seconds, then time t seconds, then what does what does ff’’(t)(t) represent ?represent ?
answer
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Answer to Question 5Answer to Question 5
Speed (velocity)Speed (velocity)Return to question
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Question 6Question 6
What is an What is an altitudealtitude of a of a triangle ?triangle ?
answer
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Answer to Question 6Answer to Question 6A line from a A line from a vertexvertex, , perpendicularperpendicular to the to the opposite sideopposite side
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Question 7Question 7
What is What is loglogaaaaequal to ?equal to ?
answer
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Answer to Question 7Answer to Question 7
11Return to question
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Question 8Question 8
Name the steps Name the steps you take in order to you take in order to differentiate differentiate functions likefunctions likef(x) = f(x) = xx22+ 3x + 1+ 3x + 1
answer
√x
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Answer to Question 8Answer to Question 8(i)(i) Change roots to Change roots to
powerspowers(ii) (ii) split up into 3 split up into 3
fractionsfractions(iii) (iii) simplify each termsimplify each term(iv) (iv) differentiatedifferentiate
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Question 9Question 9
How do you find How do you find where a curve where a curve crosses the crosses the y-y-axisaxis ? ?
answer
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Answer to Question 9Answer to Question 9
Let x = 0 and Let x = 0 and solvesolve
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Question 10Question 10
How do you draw How do you draw the graph of the graph of 2f(x)2f(x) given the graph of given the graph of f(x)f(x) ? ?
answer
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Answer to Question 10Answer to Question 10
Double theDouble the y-coordinatesy-coordinates
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Question 11Question 11
How do you find the How do you find the gradient of a linegradient of a line, , given the given the angleangle the the line makes with the line makes with the positive directionpositive direction of of the the x-axisx-axis ? ?
answer
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Answer to Question 11Answer to Question 11
m = tan m = tan ΘΘ
x
y
ΘΘ
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Question 12Question 12
What is the What is the equation of a equation of a horizontal linehorizontal line passing through passing through the point the point (a,b)(a,b) ? ?
answer
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Answer to Question 12Answer to Question 12
y = by = b
x
y(a,b)
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Question 13Question 13
What do you know What do you know about the about the gradientgradient of of two perpendicular two perpendicular lineslines ? ?
answer
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Answer to Question 13Answer to Question 13They multiply They multiply together totogether togive -1give -1
i.e. i.e. mm11xxmm22 = -1 = -1x
y
m1
m2
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Question 14Question 14
What is a What is a medianmedian of a triangle ?of a triangle ?
answer
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Answer to Question 14Answer to Question 14
A line from a A line from a vertexvertex to the to the middlemiddle of the of the opposite sideopposite side
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Question 15Question 15
What is the What is the gradientgradient of a of a horizontal linehorizontal line ? ?
answer
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Answer to Question 15Answer to Question 15
00
x
y
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Question 16Question 16
How do you find How do you find the the rate of changerate of change of a function at a of a function at a particular point ?particular point ?
answer
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Answer to Question 16Answer to Question 16
(i) (i) differentiate differentiate (ii)(ii) fit in given x fit in given x value value
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Question 17Question 17
How do you find How do you find the the limitlimit of a of a recurrence relation recurrence relation fromfrom its its formulaformula ? ?
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Answer to Question 17Answer to Question 17(i) (i) let u let un+1 n+1 = u= unn = L = L(ii)(ii) solve equation solve equationRemember:Remember:For uFor un+1n+1==aauunn+b,+b,a limit exists if a limit exists if -1<a<1-1<a<1
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Question 18Question 18
How do you know How do you know when a recurrence when a recurrence relation of the form relation of the form uun+1n+1 = au = aunn + b + b convergesconverges ? ?
answer
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Answer to Question 18Answer to Question 18
-1 < a < 1-1 < a < 1Return to question
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Question 19Question 19Given a rec. relation in Given a rec. relation in the form the form uun+1n+1 = au = aunn + + bb and 3 consecutive and 3 consecutive terms, how do you find terms, how do you find the the values of a and bvalues of a and b??
answer
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Answer to Question 19Answer to Question 19(i) fit 1(i) fit 1stst term into u term into unn and and
2 2ndnd term into u term into un+1n+1(ii) fit 2(ii) fit 2ndnd term into u term into unn and and the 3 the 3rdrd term into u term into un+1n+1(iii) solve simultaneous (iii) solve simultaneous equationsequations
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Question 20Question 20
How doHow doyou showyou showthat a that a curvecurveis always is always decreasingdecreasing??
answer
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Answer to Question 20Answer to Question 20(i) differentiate(i) differentiate(ii) show that f’(x) is (ii) show that f’(x) is
negative and negative and a squarea square
e.g. –(x+a)e.g. –(x+a)22
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Question 21Question 21
How would you How would you find the maximum find the maximum or minimum value or minimum value of a function given of a function given its equation?its equation?
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Answer to Question 21Answer to Question 21(i) differentiate(i) differentiate(ii) let f’(x) = 0(ii) let f’(x) = 0(iii) solve to find the (iii) solve to find the stationary pointsstationary points(iv) draw the nature table(iv) draw the nature table(v) read off max or min(v) read off max or min
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Question 22Question 22
Which two points Which two points does the graph of does the graph of y = logy = logaaxx always always pass through ?pass through ?
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Answer to Question 22Answer to Question 22
(1,0) and (a,1)(1,0) and (a,1)Return to question
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Question 23Question 23
How do you find How do you find where a where a curvecurve is is decreasingdecreasing ? ?
answer
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Answer to Question 23Answer to Question 23(i)(i) differentiatedifferentiate
(ii)(ii) let flet f’’(x) = 0(x) = 0(iii)(iii)solve to find st ptssolve to find st pts(iv)(iv) draw nature tabledraw nature table(v)(v) read values where read values where graph is decreasinggraph is decreasing
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Question 24Question 24
If y is the equation If y is the equation of a curve, what is of a curve, what is represented by represented by dy/dxdy/dx ? ?
answer
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Answer to Question 24Answer to Question 24
The gradientThe gradientReturn to question
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Question 25Question 25
How do you find How do you find where a where a curvecurve crosses the crosses the x-x-axisaxis ? ?
answer
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Answer to Question 25Answer to Question 25
Let y = 0Let y = 0or f(x) = 0or f(x) = 0and and solve for xsolve for x
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Question 26Question 26
What is an What is an equivalent equivalent expression forexpression fory = ay = axx
using using logslogs ??answer
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Answer to Question 26Answer to Question 26
x = logx = logaay y Return to question
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Question 27Question 27
How do youHow do you show thatshow that a a curvecurve is is always always increasingincreasing ??
answer
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Answer to Question 27Answer to Question 27
(i)(i) DifferentiateDifferentiate(ii)(ii) show that fshow that f’’(x) is (x) is a a perfect perfect squaresquare
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Question 28Question 28
How do you find the How do you find the equation of a equation of a tangenttangent to a curve to a curve at the point when at the point when x x = a= a ? ?
answer
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Answer to Question 28Answer to Question 28(i)(i) DifferentiateDifferentiate(ii)(ii) fit fit aa into f into f’’(x) to get (x) to get the gradient (m)the gradient (m)
(iii)(iii) fit fit aa into f(x) to get into f(x) to get the tangent point (a,b)the tangent point (a,b)
(iv)(iv) use y-b=m(x-a)use y-b=m(x-a)
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Question 29Question 29
For what For what valuesvalues of of a function is the a function is the functionfunction said to be said to be undefinedundefined ? ?
answer
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Answer to Question 29Answer to Question 29
When you fit in a When you fit in a value of x and you value of x and you cannot get an answercannot get an answer
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Question 30Question 30
How do you draw How do you draw the graph of the graph of f(x-1)f(x-1) given the graph of given the graph of f(x) ?f(x) ?
answer
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Answer to Question 30Answer to Question 30
Move the graph Move the graph 1 unit1 unit to the to the rightright
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Question 31Question 31
How do you find How do you find f(f(g(x)g(x))) for given for given functions functions f(x)f(x) and and g(x)g(x) ? ?
answer
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Answer to Question 31Answer to Question 31
Fit Fit g(x)g(x) into into f(x)f(x)i.e. each x in i.e. each x in f(x)f(x) is is replaced by the replaced by the function function g(x)g(x)
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Question 32Question 32
What What two thingstwo things do do you require in order you require in order to find the equation to find the equation of a of a straight linestraight line ? ?
answer
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Answer to Question 32Answer to Question 32
The The gradientgradient of the of the line and line and a pointa point on on the linethe line
x
y
(a,b)m
1
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Question 33Question 33
How do you find How do you find the the midpointmidpoint of a of a lineline joining two joining two points ?points ?
answer
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Answer to Question 33Answer to Question 33
AddAdd the coordinates the coordinates and and divide by twodivide by two
xx11+ x+ x
22 , , yy11+ y+ y
22
22 22( )x
y(x2,y2)
(x1,y1)
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Question 34Question 34
What is the What is the gradientgradient of a of a verticalvertical line ? line ?
answer
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Answer to Question 34Answer to Question 34
undefinedundefined
x
y
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Question 35Question 35
How do you find How do you find the the medianmedian AM of AM of triangle ABC ?triangle ABC ?
answer
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Answer to Question 35Answer to Question 35(i)(i) find thefind the
mid pointmid pointof BC (M)of BC (M)
(ii)(ii) find thefind thegradient of AMgradient of AM
(iii)(iii)use use y-b = m(x-a)y-b = m(x-a)
A
BCM
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Question 36Question 36
Which Which two pointstwo points does the graphdoes the graphy = ay = axx always pass always pass through ?through ?
answer
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Answer to Question 36Answer to Question 36
(0,1) and (1,a)(0,1) and (1,a)Return to question
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Question 37Question 37
What is the What is the perpendicular perpendicular bisectorbisector of a line ? of a line ?
answer
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Answer to Question 37Answer to Question 37
A line which A line which cutscuts the the given line given line in halfin half at at 9090oo
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Question 38Question 38
How do you draw How do you draw the graph of the graph of f(x+1)f(x+1) given the given the graph of graph of f(x)f(x) ? ?
answer
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Answer to Question 38Answer to Question 38
Move the graph Move the graph 1 1 unitunit to the to the leftleft
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Question 39Question 39
How do you find How do you find the the equationequation of a of a perpendicular perpendicular bisectorbisector of a line ? of a line ?
answer
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Answer to Question 39Answer to Question 39 (i)(i) find the midpoint of the linefind the midpoint of the line
(ii)(ii) find the gradient of the linefind the gradient of the line(iii)(iii) find the gradient find the gradient perpendicular to the given perpendicular to the given lineline (iv)(iv) Use midpoint and gradient in Use midpoint and gradient in y-b = m(x-a)y-b = m(x-a)
M(a,b)
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Question 40Question 40
For what For what valuesvalues is is this function this function undefinedundefined ? ?f(x)f(x) = = xx
(x+2)(x-3)(x+2)(x-3)answer
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Answer to Question 40Answer to Question 40
-2 and 3-2 and 3Return to question
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Question 41Question 41
What are the two What are the two formulaeformulae used to used to find the find the areaarea of a of a triangletriangle ? ?
answer
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Answer to Question 41Answer to Question 41
A = A = ½base x height½base x heightA = ½bcsinAA = ½bcsinA
A
BC a
b c
B a s e
height
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Question 42Question 42
What What three three processesprocesses do you do you go through in go through in order to order to factorise a factorise a quadraticquadratic ? ?
answer
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Answer to Question 42Answer to Question 42
(i)(i) common factorcommon factor(ii)(ii) difference of two difference of two
squaressquares(iii)(iii) trinomialtrinomial
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Question 43Question 43
What is the What is the equationequation of a of a vertical linevertical line passing passing through through (a,b)(a,b) ? ?
answer
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Answer to Question 43Answer to Question 43
x = ax = a
x
y
(a,b)
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Question 44Question 44
What is the What is the Theorem of Theorem of Pythagoras ?Pythagoras ?
answer
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Answer to Question 44Answer to Question 44
For For ΔΔABC,ABC,right-angled at A,right-angled at A,aa22 = b = b2 2 + c+ c22
AB
C
a
c
b
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Question 45Question 45
What do you know What do you know about the about the gradients of two gradients of two parallelparallel lines? lines?
answer
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Answer to Question 45Answer to Question 45
They are the sameThey are the same Return to question
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Question 46Question 46
How do you draw How do you draw the graph of the graph of f’(x)f’(x) given the graph of given the graph of f(x)f(x) ? ?
answer
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Answer to Question 46Answer to Question 46 (i)(i) plot x coords of st. points on plot x coords of st. points on
x-axis (SPs become roots) x-axis (SPs become roots)(ii) look at each part of f(x) (ii) look at each part of f(x) separately: separately: if rising, graph of f’(x) is if rising, graph of f’(x) is above x-axisabove x-axis if falling, graphif falling, graph of f’(x) isof f’(x) is below x-axisbelow x-axis
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Question 47Question 47
How do you get the How do you get the gradientgradient of a line of a line with an equation likewith an equation like3x + 2y = 5 ?3x + 2y = 5 ?
answer
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Answer to Question 47Answer to Question 47
(i)(i) Rearrange into the Rearrange into the formformy = y = mmx + cx + c
(ii)(ii) read offread offgradient = gradient = mm
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Question 48Question 48
What is What is loglogaa11 equal to ?equal to ?
answer
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Answer to Question 48Answer to Question 48
00Return to question
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Question 49Question 49
How do you find How do you find the the length of a linelength of a line joining two joining two points ?points ?
answer
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Answer to Question 49Answer to Question 49
√√(x(x22 – x – x11))22 + (y + (y22 –y –y11))22
A(x1,y1)
B(x2,y2)
x
y
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Question 50Question 50
What is the What is the Converse of Converse of Pythagoras ?Pythagoras ?
answer
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Answer to Question 50Answer to Question 50
If aIf a22 = b = b22 + c + c22
then then ΔΔABC isABC isright-angled at A right-angled at A
AB
C
a
c
b
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Question 51Question 51
How do you find How do you find the the gradientgradient of a of a line joining line joining two two pointspoints ? ?
answer
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Answer to Question 51Answer to Question 51
m = ym = y22 – y – y11
xx22 – x – x11
A(x1,y1)
B(x2,y2)
x
y
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Question 52Question 52
How do you find How do you find the the altitudealtitude AN of AN of ΔΔABC ?ABC ?
answer
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Answer to Question 52Answer to Question 52(i)(i) find the gradient find the gradient
of BCof BC(ii)(ii) find the gradient find the gradient
of AN,of AN,perpendicularperpendicularto BCto BC
(iii)(iii)use y-b=m(x-a)use y-b=m(x-a)
A
NB
C
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Question 53Question 53
For a curve, how For a curve, how do you find the do you find the stationary points stationary points and their natureand their nature ? ?
answer
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Answer to Question 53Answer to Question 53(i)(i) differentiatedifferentiate(ii)(ii) let f’(x) = 0let f’(x) = 0(iii)(iii) solve to find solve to find
stationary pointsstationary points(iv)(iv) find y-coordinatesfind y-coordinates(v)(v) draw nature tabledraw nature table
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Question 54Question 54
How do you draw How do you draw the graph of the graph of 3+f(x)3+f(x) given the given the graph of graph of f(x)f(x) ? ?
answer
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Answer to Question 54Answer to Question 54
move graph move graph up 3up 3Return to question
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Question 55Question 55
How do you find How do you find where a where a curvecurve is is increasingincreasing ? ?
answer
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Exit Exit
Answer to Question 55Answer to Question 55(i)(i) differentiatedifferentiate
(ii)(ii) let f’(x) = 0let f’(x) = 0(iii)(iii)solve to find stationary solve to find stationary
pointspoints(iv)(iv) draw nature tabledraw nature table(v)(v) read values for which read values for which
graph is increasinggraph is increasing
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Question 56Question 56
How do you find How do you find where where two lines two lines intersectintersect ? ?
answer
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Answer to Question 56Answer to Question 56
Simultaneous Simultaneous equationsequations
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Question 57Question 57
How do you draw How do you draw the graph of the graph of 3-f(x)3-f(x) given the graph of given the graph of f(x)f(x) ? ?
answer
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Exit Exit
Answer to Question 57Answer to Question 57
ReflectReflect the graph in the graph in the the x-axisx-axis, , then move it then move it up 3up 3
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Question 58Question 58
How do you draw How do you draw the graph of the graph of f(-x)f(-x) given the graph of given the graph of f(x)f(x) ? ?
answer
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Answer to Question 58Answer to Question 58
ReflectReflect the graph in the graph in the the y-axisy-axis
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Question 59Question 59
How do you solve How do you solve equations likeequations like
100 = 0100 = 0 xx22
answer
?4 -
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Exit Exit
Answer to Question 59Answer to Question 59
(i)(i) multiply by the multiply by the denominator of the denominator of the
fraction (here xfraction (here x22))(ii)(ii) factorise and factorise and solvesolve
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Exit Exit
Question 60Question 60How do you find the How do you find the exact valuesexact values of ofsin(A+B)sin(A+B), , cos(A-B)cos(A-B) etc. etc.given thatgiven that
cosA = cosA = 33//55 and andsinB = sinB = 1212//1313 ? ? answe
r
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Exit Exit
Answer to Question 60Answer to Question 60 (i)(i) drawdraw
two two ΔΔss (ii)(ii) findfind
missing sidesmissing sides (iii)(iii) expandexpand
formulaformula (iv)(iv) fit in valuesfit in values
from from ΔΔss
A
3 5
B
12
13
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Exit Exit
Question 61Question 61
How do you solve How do you solve equations likeequations like
Cos2xCos2xoo - 5cosx - 5cosxoo = 2 = 2 ? ?(0 ≤ x ≤ 360)(0 ≤ x ≤ 360)
answer
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Exit Exit
Answer to Question 61Answer to Question 61
(i)(i) fit in 2cosfit in 2cos22xxoo-1 for -1 for cos2xcos2xoo
(ii)(ii) factorisefactorise(iii)(iii) solve the equationsolve the equation
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Question 62Question 62
What isWhat is sin xsin xcos xcos xequal to ?equal to ?
answer
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Answer to Question 62Answer to Question 62
tan xtan xReturn to question
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Question 63Question 63
How do you show How do you show that that x-1x-1 is a factor is a factor of the function of the function f(x)=xf(x)=x33-3x+2-3x+2 ? ?
answer
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Exit Exit
Answer to Question 63Answer to Question 63(i)(i) rewrite the function asrewrite the function asf(x)=xf(x)=x33++00xx22-3x+2-3x+2
(ii)(ii) use synthetic division use synthetic division with with 11 on the outside on the outside
(iii)(iii) show thatshow thatremainder = 0remainder = 0
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Question 64Question 64
What is theWhat is the sine rulesine rule ? ?
answer
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Exit Exit
Answer to Question 64Answer to Question 64
aa b b c c sinAsinA sinb sinC sinb sinC= =
A
B
C a
bc
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Exit Exit
Question 65Question 65
Given Given f’(x)f’(x) and a and a point on the curvepoint on the curve, , how do you findhow do you findf(x)f(x) ? ?
answer
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Exit Exit
Answer to Question 65Answer to Question 65
(i)(i) integrateintegrate(ii)(ii) fit in given point fit in given point
to to work outwork out value valueof of CC
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Exit Exit
Question 66Question 66
How do you solve How do you solve quadratic quadratic inequations likeinequations likexx2 2 - 5x + 6 ≤ 0 ?- 5x + 6 ≤ 0 ?
answer
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Exit Exit
Answer to Question 66Answer to Question 66
(i)(i) factorise factorise(ii)(ii) draw graph draw graph(iii) read values (iii) read values below x-axisbelow x-axis
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Question 67Question 67
How do you How do you changechange from from radiansradians to to degreesdegrees ? ?
answer
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Exit Exit
Answer to Question 67Answer to Question 67
Divide by Divide by ππ and and multiply by 180multiply by 180
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Question 68Question 68
What is the What is the condition for condition for real real rootsroots ? ?
answer
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Answer to Question 68Answer to Question 68
bb22 – 4ac ≥ 0 – 4ac ≥ 0Return to question
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Question 69Question 69How do you find the value How do you find the value of of aa in the polynomial in the polynomial xx33++aaxx22+4x+3+4x+3 given a given a factor of the polynomial or factor of the polynomial or the remainder when the the remainder when the polynomial is divided by a polynomial is divided by a number ?number ?
answer
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Exit Exit
Answer to Question 69Answer to Question 69 (i)(i) do synthetic divisiondo synthetic division(ii)(ii) let the expression let the expression = = 0 or the remainder0 or the remainder
(iii)(iii) solve the equationsolve the equation
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Exit Exit
Question 70Question 70How do you find How do you find f(x)f(x) if iff’(x) = 5-3xf’(x) = 5-3x22 and andthe curve passes through the curve passes through the point the point (1,9)(1,9) ? ?
answer
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Exit Exit
Answer to Question 70Answer to Question 70
(i)(i) f(x) = f(x) = ∫f'(x) dx∫f'(x) dx(ii)(ii) find C by replacing find C by replacing point (1,9) into f(x)point (1,9) into f(x)
(iii)(iii) write down write down completed completed formula for f(x)formula for f(x)
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Question 71Question 71
What isWhat issinsin22x + cosx + cos22xxequal to ?equal to ?
answer
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Answer to Question 71Answer to Question 71
11Return to question
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Question 72Question 72
How do you find the How do you find the equation of the equation of the tangenttangent to a circle at to a circle at a particular a particular point on point on the circumferencethe circumference ? ?
answer
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Exit Exit
Answer to Question 72Answer to Question 72 (i)(i) find thefind the
centrecentre(ii)(ii) find gradientfind gradientfrom centrefrom centreto pointto point(iii)(iii) find perpendicular find perpendicular gradientgradient(iv)(iv) use y-b=m(x-a)use y-b=m(x-a)
x
y
(a,b)
C
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Question 73Question 73
How do you findHow do you find xx22 + 1 + 1
√√xxanswer
∫ dx ?
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Exit Exit
Answer to Question 73Answer to Question 73 (i)(i) change root to change root to powerpower
(ii)(ii) split up into fractionssplit up into fractions(iii)(iii) simplify each termsimplify each term(iv)(iv) integrate each termintegrate each term(v)(v) REMEMBERREMEMBER +C+C
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Question 74Question 74
How do you show How do you show that the that the rootroot of a of a function lies function lies between two given between two given valuesvalues ? ?
answer
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Exit Exit
Answer to Question 74Answer to Question 74
fit in fit in two valuestwo values and and show one is show one is positivepositive andand one is one is negativenegative
x
+ve
-ve
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Exit Exit
Question 75Question 75
How do you find How do you find exact valuesexact values of of sin2xsin2x and and cos2xcos2x given given cosx =cosx =33//55 ? ?
answer
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Exit Exit
Answer to Question 75Answer to Question 75 (i)(i) draw adraw a
right-angledright-angledtriangletriangle
(ii)(ii) find the find the missing sidemissing side
(iii)(iii) expand the double expand the double angle formulaangle formula
(iv)(iv) fit in values from fit in values from ΔΔ
A3
5
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Exit Exit
Question 76Question 76
What is the What is the turning point ofturning point of
y=2(x-a)y=2(x-a)22+b+b ? ?MaxMax or or minmin ? ?
answer
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Answer to Question 76Answer to Question 76
(i)(i) (a,b)(a,b)minimumminimum
(a,b)
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Question 77Question 77
How do you How do you integrate xintegrate xnn ? ?
answer
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Answer to Question 77Answer to Question 77
xxn+1n+1
n+1n+1+ C
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Question 78Question 78
How do you How do you solve solve equations likeequations likecos2xcos2xoo-5sinx-5sinxoo = 0 = 0 ? ?
(0≤x≤360)(0≤x≤360)answer
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Exit Exit
Answer to Question 78Answer to Question 78
(i)(i) fit in 1-2sin fit in 1-2sin22xxoo for cos2xfor cos2xoo
(ii)(ii) factorise factorise(iii) solve equation(iii) solve equation
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Exit Exit
Question 79Question 79
How do you How do you complete the square complete the square for functions likefor functions like2x2x2 2 + 12x + 3 ?+ 12x + 3 ?
answer
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Exit Exit
Answer to Question 79Answer to Question 79
(i)(i) multiply out multiply out a(x+p)a(x+p)22+q+q
(ii)(ii) compare with compare with given functiongiven function
(iii) find a, p and q(iii) find a, p and q
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Exit Exit
Question 80Question 80
How do you How do you solve solve equations of the formequations of the formsin2xsin2xoo = 0.5 = 0.5 ? ?(0≤x≤360)(0≤x≤360)
answer
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Exit Exit
Answer to Question 80Answer to Question 80 (i) decide on the (i) decide on the 2 2
quadrants (sin is quadrants (sin is +ve+ve))(ii) press (ii) press INV sinINV sin to get to get
angleangle(iii) work out your 2 angles(iii) work out your 2 angles(iv) divide each by 2(iv) divide each by 2
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Exit Exit
Question 81Question 81
How do you How do you solve solve quadratic quadratic inequations likeinequations like
xx22+5x-6 ≥ 0 ?+5x-6 ≥ 0 ?answer
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Exit Exit
Answer to Question 81Answer to Question 81
(i)(i) factorisefactorise(ii)(ii) draw graphdraw graph(iii) read values (iii) read values above x-axisabove x-axis
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Exit Exit
Question 82Question 82
What is the What is the centrecentre and and radiusradius of a of a circle with equation circle with equation
xx2 2 + y+ y22 = r = r22 ? ?answer
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Exit Exit
Answer to Question 82Answer to Question 82
(i)(i) centre (0,0)centre (0,0)(ii)(ii) radius = rradius = r
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Question 83Question 83
How do you How do you calculate the area calculate the area under a curve ?under a curve ?
answer
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Exit Exit
Answer to Question 83Answer to Question 83
(i)(i) integrateintegrate(ii)(ii) fit in two limits fit in two limits
and subtract to and subtract to find areafind area
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Exit Exit
Question 84Question 84
How do you find How do you find the the rootroot of an of an equation between equation between two given values to two given values to 1 dp ?1 dp ?
answer
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Exit Exit
Answer to Question 84Answer to Question 84
iterationiterationReturn to question
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Question 85Question 85
How do you How do you solve solve equations of the formequations of the formsinsin22xxoo = 0.5 = 0.5 ? ?(0≤x≤360)(0≤x≤360)
answer
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Exit Exit
Answer to Question 85Answer to Question 85
(i)(i) rearrange to get rearrange to get sinxsinxoo = = ± ± ……
(ii)(ii) find answers in find answers in all 4 quadrantsall 4 quadrants
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Question 86Question 86
How do you name How do you name the the angle betweenangle between a a lineline and a and a planeplane ? ?
answer
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Exit Exit
Answer to Question 86Answer to Question 86(i) start at end of line (A)(i) start at end of line (A)(ii) go to where line meets(ii) go to where line meets
the plane (B)the plane (B)(iii) go to the point(iii) go to the point
on the planeon the plane directly underdirectly under the start of the line (C)the start of the line (C)
ABCABC
A
BC
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Question 87Question 87
What is the What is the condition for condition for equal equal rootsroots ? ?
answer
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Exit Exit
Answer to Question 87Answer to Question 87
bb2 2 – 4ac = 0– 4ac = 0Return to question
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Question 88Question 88
What is the turning What is the turning point of point of y = b-3(x-a)y = b-3(x-a)22 ? ?
max or min ?max or min ?answer
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Exit Exit
Answer to Question 88Answer to Question 88
(a,b)(a,b)
Maximum Maximum
(a,b)
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Question 89Question 89
What is the What is the quadratic formulaquadratic formula and and explainexplain when it when it is used ?is used ?
answer
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Exit Exit
Answer to Question 89Answer to Question 89x = x = -b-b±±√(√(bb22-4ac)-4ac)
2a2aIt is used to find roots It is used to find roots of a quadratic equation of a quadratic equation when it is difficult to when it is difficult to factorise.factorise.
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Question 90Question 90
How do you prove How do you prove that a line is a that a line is a tangent to a circletangent to a circle ? ?
answer
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Exit Exit
Answer to Question 90Answer to Question 90Rearrange line to makeRearrange line to makey =y = or or x =x =
Fit line into circleFit line into circleProve it has Prove it has equal equal rootsroots using using bb22-4ac = 0-4ac = 0 or or repeated rootsrepeated roots
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Exit Exit
Question 91Question 91
How do you find theHow do you find theexactexact value of value of
sin (sin (αα--ββ),),given that given that sinsinαα = =44//55
and and coscosββ = = 1212//1313 ? ? answer
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Exit Exit
Answer to Question 91Answer to Question 91 (i) draw triangles(i) draw triangles
for for αα and and ββ (ii) work out(ii) work out
coscosαα and sin and sinββ
(iii) expand(iii) expand formula for sin(formula for sin(αα--ββ))
(iv) insert exact values(iv) insert exact values
αα
4
5
12
13
ββ
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Exit Exit
Question 92Question 92
How do you How do you solve solve equations of the formequations of the formcosxcosxoo = - 0.8 = - 0.8 ? ?(0≤x≤360)(0≤x≤360)
answer
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Exit Exit
Answer to Question 92Answer to Question 92 (i) decide on the (i) decide on the 2 2 quadrants (cos is quadrants (cos is -ve-ve))
(ii) ignore the sign and (ii) ignore the sign and press press INV cosINV cos to get to get angle angle
(iii) work out your 2 angles(iii) work out your 2 angles
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Question 93Question 93
How do you How do you change change fromfrom degrees degrees toto radiansradians ? ?
answer
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Exit Exit
Answer to Question 93Answer to Question 93
Divide by 180Divide by 180 and and multiply by multiply by ππ
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Question 94Question 94
How do you find the How do you find the exact valuesexact values of of sin xsin x or or tan xtan x given given
cos x = cos x = aa ? ? bb
answer
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Exit Exit
Answer to Question 94Answer to Question 94(i) draw triangle(i) draw triangle
(ii) use Pythagoras (ii) use Pythagoras to fill to fill in missing sidein missing side
(iii) read values off (iii) read values off triangle triangle using using SOHSOHCAHCAHTOATOA
a
b
xx
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Exit Exit
Question 95Question 95
How do you How do you factorisefactorise a a cubiccubic expression likeexpression like
xx33-2x-2x22-x+2-x+2 ? ?answer
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Exit Exit
Answer to Question 95Answer to Question 95
Synthetic division Synthetic division using factors of using factors of last numberlast number
Remainder=0
factor 1 -2 -1 2
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Exit Exit
Question 96Question 96
What is the What is the centrecentre and and radiusradius of a circle of a circle of the formof the form
xx22+y+y22+2gx+2fy+c=+2gx+2fy+c=00 ? ?
answer
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Exit Exit
Answer to Question 96Answer to Question 96
Centre (-g,-f)Centre (-g,-f)Radius Radius √(√(gg22+f+f22-c)-c)
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Exit Exit
Question 97Question 97
How do you How do you remember the remember the exact exact valuesvalues of of 3030oo, 45, 45oo and 60and 60oo ? ?
answer
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Exit Exit
Answer to Question 97Answer to Question 97sin30sin30oo = ½ = ½Draw right-angledDraw right-angledtriangletriangle
Complete using PythagorasComplete using PythagorasDo similarDo similarfor tan 45for tan 45oo =1 =1
30o
60o
1 2
√3
45o
45o
1
1 √2
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Question 98Question 98
How do you How do you calculate the calculate the area area between two between two curvescurves ? ?
answer
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Exit Exit
Answer to Question 98Answer to Question 98(i) let equations equal (i) let equations equal each othereach other
(ii) solve to find limits(ii) solve to find limits(iii) integrate(iii) integrate
(upper - lower)(upper - lower) functions between limitsfunctions between limits
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Question 99Question 99
How do you solve an How do you solve an equation likeequation like
3sinx+1 = 03sinx+1 = 0 ? ?
answer
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Exit Exit
Answer to Question 99Answer to Question 99
(i) rearrange to (i) rearrange to sinx =sinx =(ii) decide on 2 quadrants(ii) decide on 2 quadrants(iii) ignore any –ve and (iii) ignore any –ve and press press INV sin to get INV sin to get angleangle
(iv) work out two answers(iv) work out two answers
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Question 100Question 100
What is the What is the condition for condition for no real no real rootsroots ? ?
answer
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Exit Exit
Answer to Question 100Answer to Question 100
bb22 – 4ac < 0 – 4ac < 0Return to question
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Exit Exit
Question 101Question 101
How do you findHow do you find
∫∫ xx33 dx ? dx ?
answer
aa
bb
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Exit Exit
Answer to Question 101Answer to Question 101
xx3+13+1
3+13+1
then then 11//44[[((bb44) - () - (aa44))]]
[[ ]] bb
aa
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Exit Exit
Question 102Question 102
How do you find How do you find where a where a lineline and a and a circle intersectcircle intersect ? ?
answer
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Exit Exit
Answer to Question 102Answer to Question 102
Rearrange line to getRearrange line to getx = …x = … or y = …or y = …
Fit into circle and solveFit into circle and solve
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Question 103Question 103
State the State the cosine rulecosine rule to find an to find an angleangle
answer
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Exit Exit
Answer to Question 103Answer to Question 103
cos A = cos A = bb2 2 + c+ c2 2 - a- a22
2bc2bcA
B
C a
bc
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Exit Exit
Question 104Question 104
What is the What is the centrecentre and and radiusradius of a circle of a circle of the formof the form
(x-a)(x-a)22+(y-b)+(y-b)2 2 = r= r22 ? ?answer
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Exit Exit
Answer to Question 104Answer to Question 104
Centre (a,b)Centre (a,b)Radius = rRadius = r
x
y
(a,b)Cr
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Question 105Question 105
State the cosine rule State the cosine rule to find a missing sideto find a missing side
answer
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Exit Exit
Answer to Question 105Answer to Question 105
aa22 = b = b22+c+c22-2bccosA-2bccosAA
B
C a
bc
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Question 106Question 106
How do you findHow do you find∫ ∫ (ax + b)(ax + b)nn dx dx ? ?
answer
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Exit Exit
Answer to Question 106Answer to Question 106 (i)(i) increase power by 1increase power by 1(ii)(ii) divide by new powerdivide by new power(iii)(iii) divide by the divide by the
derivative ofderivative ofthe bracketthe bracket
i.e. i.e. (ax+b)(ax+b)n+1n+1
a(n+1)a(n+1) + C+ C
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Exit Exit
Question 107Question 107
How do you findHow do you findthe the coordinatescoordinates of of a point which a point which divides a line in a divides a line in a ratioratio e.g. e.g. 3:23:2 ? ?
answer
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Exit Exit
Answer to Question 107Answer to Question 107 (i)(i) write in form write in form ABAB = = 33
BCBC 2 2(ii)(ii) cross-multiplycross-multiply(iii)(iii)write AB = (write AB = (bb--aa))(iv)(iv) solve to find missing solve to find missing
vectorvector(v)(v) rewrite as point (*,*)rewrite as point (*,*)
A
B
C
3
2
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Question 108Question 108
What is aWhat is a position vectorposition vector ? ?
answer
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Answer to Question 108Answer to Question 108
A vector which A vector which startsstarts atat the the originorigin
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Question 109Question 109
How do you express How do you express acosx+bsinx+cacosx+bsinx+cin the formin the formkcos(x-kcos(x-αα) etc?) etc?
answer
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Answer to Question 109Answer to Question 109(i) expand brackets (i) expand brackets and and
equate like termsequate like terms(ii) find k =(ii) find k =√√(a(a22+b+b22))(iii) identify quadrant (iii) identify quadrant αα is in is in
(iv) find (iv) find αα , tan , tanαα = = sinsinαα coscosαα
ATS
C
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Question 110Question 110
How do you How do you differentiate a differentiate a bracket without bracket without multiplying it out ?multiplying it out ?
answer
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Answer to Question 110Answer to Question 110
(i) multiply by old power(i) multiply by old power(ii) decrease power by 1(ii) decrease power by 1(iii) multiply by (iii) multiply by derivative of bracket derivative of bracket
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Question 111Question 111
What isWhat isLogLogaax – logx – logaayyequal to ?equal to ?
answer
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Answer to Question 111Answer to Question 111
xx
loglogaa yy
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Question 112Question 112
What do you get What do you get when you when you differentiatedifferentiate cosx ? cosx ?
answer
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Answer to Question 112Answer to Question 112
-sinx-sinx
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Question 113Question 113
How do you show How do you show that that two vectorstwo vectors are are perpendicularperpendicular ? ?
answer
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Answer to Question 113Answer to Question 113
Show that Show that aa..bb=0=0
a
b
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Question 114Question 114
How do you How do you integrate integrate sin axsin ax ? ?
answer
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Answer to Question 114Answer to Question 114
--11//aa cos ax cos ax + C+ C
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Question 115Question 115
How do you draw How do you draw a graph of the forma graph of the form
y = acosxy = acosxor or y = asinxy = asinx ? ?
answer
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Answer to Question 115Answer to Question 115
Draw y = cosxDraw y = cosxor y = sinx graphor y = sinx graphwith a with a maximum of amaximum of a and a and a minimum of -aminimum of -a
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Question 116Question 116
How do you find How do you find the maximum or the maximum or minimum values ofminimum values of
acosx + bsinx + cacosx + bsinx + c ? ?answer
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Answer to Question 116Answer to Question 116
(i) change (i) change acosx+bsinxacosx+bsinx into into Rcos(x-a)Rcos(x-a)
(ii) (ii) max is R+cmax is R+c
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Question 117Question 117
How do you find a How do you find a unit vectorunit vector parallelparallel to a given vectorto a given vector ? ?
answer
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Answer to Question 117Answer to Question 117
(i) find the length of (i) find the length of the given vector the given vector
(ii) (ii) divide all the divide all the components by components by
this length this length
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Question 118Question 118
How do you How do you integrate cos axintegrate cos ax ? ?
answer
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Answer to Question 118Answer to Question 118
11//aa sin ax sin ax + C+ C
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Question 119Question 119
How do you draw How do you draw a graph of the forma graph of the form
y = cos(x+a)y = cos(x+a) or or y = sin(x+a)y = sin(x+a) ? ?
answer
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Answer to Question 119Answer to Question 119
Move the graph of Move the graph of y=cosxy=cosx or or y=sinxy=sinx
aa units to the units to the LEFTLEFT
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Question 120Question 120
What is a unit What is a unit vector ?vector ?
answer
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Answer to Question 120Answer to Question 120
A vector of A vector of length 1 length 1 unitunit
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Question 121Question 121
How do you draw How do you draw a graph of the forma graph of the form
y = cos bxy = cos bx or or y = sin bx y = sin bx ??
answer
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Answer to Question 121Answer to Question 121
Draw the normal graph Draw the normal graph but fit in but fit in bb waveswaves between 0between 0oo and 360 and 360oo
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Question 122Question 122
What isWhat isloglogaa x + log x + logaa y y equal toequal to ??
answer
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Answer to Question 122Answer to Question 122
LogLogaa xy xy
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Question 123Question 123
What do you get What do you get when you when you differentiatedifferentiate sin xsin x ??
answer
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Answer to Question 123Answer to Question 123
cos xcos x
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Question 124Question 124
How do you find How do you find the the angleangle between between two vectorstwo vectors ??
answer
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Answer to Question 124Answer to Question 124
aa..bb aa bb
cos=
a
b
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Question 125Question 125
Given an equation Given an equation like like m = mm = mooee-3k-3k and and an amount by which an amount by which it has been decayed, it has been decayed, how do you how do you find k find k ??
answer
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Answer to Question 125Answer to Question 125
(i) fit in m and m(i) fit in m and moo
(ii) rearrange to get (ii) rearrange to get ee-3k-3k = =(iii) take logs(iii) take logs(iv) solve(iv) solve
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Question 126Question 126
If If uu = = aaii++bbjj++cckkthen what is then what is uu in in component formcomponent form ??
answer
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Answer to Question 126Answer to Question 126
U =abc
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Question 127Question 127
What do you get What do you get when you when you differentiatedifferentiate
coscosax ax ??answer
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Answer to Question 127Answer to Question 127
-asinaxReturn to question
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Question 128Question 128
How do you solve How do you solve an equation of the an equation of the form form aacosx + cosx + bbsinx sinx + + cc=0=0 ??
answer
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Answer to Question 128Answer to Question 128
Change acosx+bsinx Change acosx+bsinx into Rcos(x- into Rcos(x- ))
Rearrange and solveRearrange and solve
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Question 129Question 129
What is What is loglogaa x xnn equal toequal to ??
answer
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Answer to Question 129Answer to Question 129
nnloglogaa x x
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Question 130Question 130
How would you How would you differentiate a differentiate a function likefunction like
y = siny = sin33 x x ??answer
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Answer to Question 130Answer to Question 130 (i) write as (sin x)(i) write as (sin x)33
(ii) multiply by the power(ii) multiply by the power(iii) decrease power by one(iii) decrease power by one(iv) multiply by the derivative (iv) multiply by the derivative
of the bracketof the bracketi.e. i.e. 3cosx sin3cosx sin22xx
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Question 131Question 131
State the State the threethree rules of rules of logslogs ??
answer
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Answer to Question 131Answer to Question 131 (i) log(i) logaaxy = logxy = logaax + logx + logaayy
(ii) log(ii) logaa = log = logaax – logx – logaayy
(iii) log(iii) logaaxxnn = nlog = nlogaaxx
xy
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Question 132Question 132
How do you solve How do you solve equations of the equations of the formform
33xx = 0.155 = 0.155 ??answer
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Answer to Question 132Answer to Question 132
(i) take logs of both sides(i) take logs of both sides(ii) bring x down to front(ii) bring x down to front(iii) solve the equation(iii) solve the equation
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Question 133Question 133
Given experimental Given experimental data, how do you data, how do you find an equation in find an equation in the form the form y=aby=abxx or or y=axy=axbb ??
answer
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Exit Exit
Answer to Question 133Answer to Question 133 (i) take logs of both sides(i) take logs of both sides(ii) rearrange to get a (ii) rearrange to get a straight line equation straight line equation
(iii) determine type(iii) determine type(iv) find solution(iv) find solution
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Question 134Question 134
How would you How would you differentiate a differentiate a function likefunction like
y = sin y = sin aaxx ??answer
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Answer to Question 134Answer to Question 134
dydy//dxdx = = aacos cos aaxx
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Question 135Question 135
If If uu = =
then what is then what is uu ??answer
abc
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Answer to Question 135Answer to Question 135
work out lengthwork out length√√(a(a22+b+b22+c+c22))
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Question 136Question 136
How do you add or How do you add or subtract vectors subtract vectors ??
answer
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Answer to Question 136Answer to Question 136
add or subtract add or subtract matching matching componentscomponents
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Question 137Question 137
What doesWhat doesaa..aa equal ?equal ?
answer
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Answer to Question 137Answer to Question 137
aa22
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Question 138Question 138
How do you prove How do you prove that three 3-D that three 3-D points arepoints are
collinear ?collinear ?answer
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Answer to Question 138Answer to Question 138
Prove they are the Prove they are the same vector same vector multiplied by different multiplied by different or the same numbersor the same numbers
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Question 139Question 139
Express the equation Express the equation y=kxy=kxnn in the form of in the form of the equation of a the equation of a straight line, straight line, Y=nX+cY=nX+c..
answer
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Answer to Question 139Answer to Question 139
loglogyy = = nnloglogxx + log + logkk
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Question 140Question 140
Who loves Who loves maths ?maths ?
answer
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Answer to Question 140Answer to Question 140
ME !!!!!!ME !!!!!!
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Acknowledgements:Acknowledgements:
Presentation byPresentation byB. McKennaB. McKenna