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Everything you wanted Everything you wanted to know about Higher to know about Higher

Maths but …Maths but …

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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.

15.15. 1616 1717 1818 1919 2020 2121 2222 2323 2424 2525 2626 2727 2828

2929 3030 3131 32.32. 33.33. 34.34. 35.35. 3636 37.37. 3838 39.39. 4040 4141 4242

43.43. 4444 45.45. 4646 47.47. 4848 49.49. 5050 51.51. 52.52. 5353 5454 5555 56.56.

5757 5858 5959 6060 6161 6262 6363 6464 6565 6666 6767 6868 6969 7070

7171 7272 7373 7474 7575 7676 7777 7878 7979 8080 8181 8282 8383 8484

8585 8686 8787 8888 8989 9090 9191 9292 9393 9494 9595 9696 9797 9898

9999 100100 101101 102102 103103 104104 105105 106106 107107 108108 109109 110110 111111 112112

113113 114114 115115 116116 117117 118118 119119 120120 121121 122122 123123 124124 125125 126126

127127 128128 129129 130130 131131 132132 133133 134134 135135 136136 137137 138138 139139 140140


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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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For what values For what values is the functionis the functionf(x) = f(x) = √(x-a)√(x-a) undefined?undefined?

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x < ax < aReturn to question


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If If f(x) = xf(x) = xnn

then what isthen what isff’’(x)(x) ? ?

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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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y – b = m(x – a)y – b = m(x – a)

x

y

(a,b)

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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 ?

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Speed (velocity)Speed (velocity)Return to question


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What is an What is an altitudealtitude of a of a triangle ?triangle ?

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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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What is What is loglogaaaaequal to ?equal to ?

answer


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11Return to question


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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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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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Let x = 0 and Let x = 0 and solvesolve

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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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Double theDouble the y-coordinatesy-coordinates

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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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m = tan m = tan ΘΘ

x

y

ΘΘ

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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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y = by = b

x

y(a,b)

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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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What is a What is a medianmedian of a triangle ?of a triangle ?

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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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What is the What is the gradientgradient of a of a horizontal linehorizontal line ? ?

answer


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00

x

y

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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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(i) (i) differentiate differentiate (ii)(ii) fit in given x fit in given x value value

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How do you find How do you find the the limitlimit of a of a recurrence relation recurrence relation fromfrom its its formulaformula ? ?

answer


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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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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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-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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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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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?

answer


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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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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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(1,0) and (a,1)(1,0) and (a,1)Return to question


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How do you find How do you find where a where a curvecurve is is decreasingdecreasing ? ?

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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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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 ? ?

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The gradientThe gradientReturn to question


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How do you find How do you find where a where a curvecurve crosses the crosses the x-x-axisaxis ? ?

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Let y = 0Let y = 0or f(x) = 0or f(x) = 0and and solve for xsolve for x

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What is an What is an equivalent equivalent expression forexpression fory = ay = axx

using using logslogs ??answer


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x = logx = logaay y Return to question


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How do youHow do you show thatshow that a a curvecurve is is always always increasingincreasing ??

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(i)(i) DifferentiateDifferentiate(ii)(ii) show that fshow that f’’(x) is (x) is a a perfect perfect squaresquare

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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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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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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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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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Move the graph Move the graph 1 unit1 unit to the to the rightright

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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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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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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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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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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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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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What is the What is the gradientgradient of a of a verticalvertical line ? line ?

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undefinedundefined

x

y

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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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Which Which two pointstwo points does the graphdoes the graphy = ay = axx always pass always pass through ?through ?

answer


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(0,1) and (1,a)(0,1) and (1,a)Return to question


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What is the What is the perpendicular perpendicular bisectorbisector of a line ? of a line ?

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A line which A line which cutscuts the the given line given line in halfin half at at 9090oo

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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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Move the graph Move the graph 1 1 unitunit to the to the leftleft

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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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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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-2 and 3-2 and 3Return to question


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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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A = A = ½base x height½base x heightA = ½bcsinAA = ½bcsinA

A

BC a

b c

B a s e

height

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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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(i)(i) common factorcommon factor(ii)(ii) difference of two difference of two

squaressquares(iii)(iii) trinomialtrinomial

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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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x = ax = a

x

y

(a,b)

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What is the What is the Theorem of Theorem of Pythagoras ?Pythagoras ?

answer


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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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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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They are the sameThey are the same Return to question


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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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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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(i)(i) Rearrange into the Rearrange into the formformy = y = mmx + cx + c

(ii)(ii) read offread offgradient = gradient = mm

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What is What is loglogaa11 equal to ?equal to ?

answer


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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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√√(x(x22 – x – x11))22 + (y + (y22 –y –y11))22

A(x1,y1)

B(x2,y2)

x

y

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What is the What is the Converse of Converse of Pythagoras ?Pythagoras ?

answer


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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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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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m = ym = y22 – y – y11

xx22 – x – x11

A(x1,y1)

B(x2,y2)

x

y

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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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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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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) ? ?

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move graph move graph up 3up 3Return to question


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How do you find How do you find where a where a curvecurve is is increasingincreasing ? ?

answer


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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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How do you find How do you find where where two lines two lines intersectintersect ? ?

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Simultaneous Simultaneous equationsequations

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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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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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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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ReflectReflect the graph in the graph in the the y-axisy-axis

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How do you solve How do you solve equations likeequations like

100 = 0100 = 0 xx22

answer

?4 -


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(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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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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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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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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(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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What isWhat is sin xsin xcos xcos xequal to ?equal to ?

answer


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tan xtan xReturn to question


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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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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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What is theWhat is the sine rulesine rule ? ?

answer


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aa b b c c sinAsinA sinb sinC sinb sinC= =

A

B

C a

bc

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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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(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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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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(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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How do you How do you changechange from from radiansradians to to degreesdegrees ? ?

answer


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Divide by Divide by ππ and and multiply by 180multiply by 180

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What is the What is the condition for condition for real real rootsroots ? ?

answer


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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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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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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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(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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What isWhat issinsin22x + cosx + cos22xxequal to ?equal to ?

answer


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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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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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How do you findHow do you find xx22 + 1 + 1

√√xxanswer

∫ dx ?


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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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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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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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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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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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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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(i)(i) (a,b)(a,b)minimumminimum

(a,b)

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How do you How do you integrate xintegrate xnn ? ?

answer


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xxn+1n+1

n+1n+1+ C

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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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(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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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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(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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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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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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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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(i)(i) factorisefactorise(ii)(ii) draw graphdraw graph(iii) read values (iii) read values above x-axisabove x-axis

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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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(i)(i) centre (0,0)centre (0,0)(ii)(ii) radius = rradius = r

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Exit Exit


How do you How do you calculate the area calculate the area under a curve ?under a curve ?

answer


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(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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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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iterationiterationReturn to question


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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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(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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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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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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What is the What is the condition for condition for equal equal rootsroots ? ?

answer


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bb2 2 – 4ac = 0– 4ac = 0Return to question


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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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(a,b)(a,b)

Maximum Maximum

(a,b)

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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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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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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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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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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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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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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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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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How do you How do you change change fromfrom degrees degrees toto radiansradians ? ?

answer


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Divide by 180Divide by 180 and and multiply by multiply by ππ

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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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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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How do you How do you factorisefactorise a a cubiccubic expression likeexpression like

xx33-2x-2x22-x+2-x+2 ? ?answer


Exit Exit


Synthetic division Synthetic division using factors of using factors of last numberlast number

Remainder=0

factor 1 -2 -1 2

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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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Centre (-g,-f)Centre (-g,-f)Radius Radius √(√(gg22+f+f22-c)-c)

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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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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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How do you How do you calculate the calculate the area area between two between two curvescurves ? ?

answer


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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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How do you solve an How do you solve an equation likeequation like

3sinx+1 = 03sinx+1 = 0 ? ?

answer


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(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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What is the What is the condition for condition for no real no real rootsroots ? ?

answer


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bb22 – 4ac < 0 – 4ac < 0Return to question


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How do you findHow do you find

∫∫ xx33 dx ? dx ?

answer

aa

bb


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xx3+13+1

3+13+1

then then 11//44[[((bb44) - () - (aa44))]]

[[ ]] bb

aa

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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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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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State the State the cosine rulecosine rule to find an to find an angleangle

answer


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cos A = cos A = bb2 2 + c+ c2 2 - a- a22

2bc2bcA

B

C a

bc

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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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Centre (a,b)Centre (a,b)Radius = rRadius = r

x

y

(a,b)Cr

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State the cosine rule State the cosine rule to find a missing sideto find a missing side

answer


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aa22 = b = b22+c+c22-2bccosA-2bccosAA

B

C a

bc

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How do you findHow do you find∫ ∫ (ax + b)(ax + b)nn dx dx ? ?

answer


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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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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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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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What is aWhat is a position vectorposition vector ? ?

answer


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A vector which A vector which startsstarts atat the the originorigin

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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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Exit Exit


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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(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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What isWhat isLogLogaax – logx – logaayyequal to ?equal to ?

answer


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xx

loglogaa yy

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What do you get What do you get when you when you differentiatedifferentiate cosx ? cosx ?

answer


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-sinx-sinx

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How do you show How do you show that that two vectorstwo vectors are are perpendicularperpendicular ? ?

answer


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Show that Show that aa..bb=0=0

a

b

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How do you How do you integrate integrate sin axsin ax ? ?

answer


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--11//aa cos ax cos ax + C+ C

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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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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

Return to question


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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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(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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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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(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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How do you How do you integrate cos axintegrate cos ax ? ?

answer


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11//aa sin ax sin ax + C+ C

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y = cos(x+a)y = cos(x+a) or or y = sin(x+a)y = sin(x+a) ? ?

answer


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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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What is a unit What is a unit vector ?vector ?

answer


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A vector of A vector of length 1 length 1 unitunit

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y = cos bxy = cos bx or or y = sin bx y = sin bx ??

answer


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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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What isWhat isloglogaa x + log x + logaa y y equal toequal to ??

answer


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LogLogaa xy xy

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What do you get What do you get when you when you differentiatedifferentiate sin xsin x ??

answer


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cos xcos x

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How do you find How do you find the the angleangle between between two vectorstwo vectors ??

answer


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aa..bb aa bb

cos=

a

b

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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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(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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If If uu = = aaii++bbjj++cckkthen what is then what is uu in in component formcomponent form ??

answer


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U =abc

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What do you get What do you get when you when you differentiatedifferentiate

coscosax ax ??answer


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-asinaxReturn to question


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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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Change acosx+bsinx Change acosx+bsinx into Rcos(x- into Rcos(x- ))

Rearrange and solveRearrange and solve

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What is What is loglogaa x xnn equal toequal to ??

answer


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nnloglogaa x x

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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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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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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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(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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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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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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How would you How would you differentiate a differentiate a function likefunction like

y = sin y = sin aaxx ??answer


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dydy//dxdx = = aacos cos aaxx

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If If uu = =

then what is then what is uu ??answer

abc


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work out lengthwork out length√√(a(a22+b+b22+c+c22))

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How do you add or How do you add or subtract vectors subtract vectors ??

answer


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add or subtract add or subtract matching matching componentscomponents

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What doesWhat doesaa..aa equal ?equal ?

answer


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aa22

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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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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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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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loglogyy = = nnloglogxx + log + logkk

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Who loves Who loves maths ?maths ?

answer


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ME !!!!!!ME !!!!!!

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Acknowledgements:Acknowledgements:

Presentation byPresentation byB. McKennaB. McKenna

higher mathematics

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