Stuff

Question 1

Even though you have a turning point, that is not neccessarily the smallest points e.g. for cubics consider the end points

The minimum could be higher than the y intercept so check the stationary points, the intercepts and the end points of the domain

Answer 1

a

Question 2

Can do show thats from LHS or RHS e.g. show that A = B

Answer 2

a

Question 3

Write out properly, clearly, systematically

Answer 3

a

Question 4

0.5absinC = 0.5bcsinA = 0.5acsinB
/ by 0.5abc gives sine rule

Determine which one to use and where the height is drawn by rotating the traingle dependent on the asked variables

Answer 4

a

Question 5

Xn+1 = Xn + n+1

Answer 5

Xn+1 - Xn = n+1

Therefore the terms are increasing by n+1
And so the sum of these increases is (n+1)(n+2)/2

?

The sequence goes:
3, 6, 10, 15, 21
The difference of each term is: 3, 4, 5, 6
The difference of these terms are: 1, 1, 1
(Keep going until all 1's)
1's represent x^0 i.e. +c
3, 4, 5, 6 represent x^1 i.e. +bx
3, 6, 10, 15 represent x^2 i.e. +ax^2

Therefore the closed form of the formula is ax^2 + bx +c = an^2 + bn + c = Xn
By subbing in values for n and Xn i.e. n = 1 X(n=1) = 3 three times we can find answers for a, b and c via simultaneous equations

Or considering each term:
3, 6, 10, 15, 21
1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5,…, 1+2+3+…+(n+1)
Therefore the Xn = (n+1)(n+2)/2

Question 6

X^2 + Y^2 <=1 find the max of ax+by

Answer 6

X^2 + Y^2 = 1
Therefore sin^2(A) + cos^2(A) = 1
Therefore we can represent ax+by as asinA + bcosA which equals Rsin(A+c) where R^2 = (a^2 + b^2) therefore the max of ax+by = the max of asinA + bcosS = the max of Rsin(A+c) = the max of (a^2+b^2)^0.5

or just let a=b=1 !!!

Question 7

SYSTEMATICALLY, USUALLY NOT THAT LONG WINDED

LOOK AT OTHER QUESTIONS !!!

Answer 7

a

Question 8

2 letters, n-1 positions –>

Answer 8

2^n-1 combinations

Question 9

6 numbers 4 positions repeats allowed

Answer 9

6!/(6-4)! = 360

Question 10

Number of ways of arranging n things in a line is

Answer 10

n!

Question 11

Sin(pi/2 - x) =

Answer 11

cos(x)

Question 12

periods

Answer 12

2pi, p

Question 13

a^x = b

Answer 13

logab = x

Question 14

order matters, selecting r objects from n =

Answer 14

nPr = n!/(n-r)!

Question 15

order doesn’t matter, selecting r objects from n =

Answer 15

nCr = n!/r!(n-r)!

Question 16

f(x)/(x-c) = p(x) + r therefore

Answer 16

r = f(c)

Question 17

Am - Gm inequality

Answer 17

am > gm

Question 18

sinX + cosX / 2

Answer 18

> (sinXcosX)^0.5

Question 19

Talk about what you are thinking

Answer 19

if running out of time then write what you are thinking what you know and what you would have done

Question 20

Sum converges if

Answer 20

-1 < r < 1

|r| < 1

Question 21

function transformations

Answer 21

f(ax) f(x-a) af(x) f(x) + a

Question 22

Long division

Answer 22

replace missing terms with 0x^n
f(x)/p(x) = g(x) + r
p(x)|f(x) –> g(x) above (quotient), r at bottom (remainder)

Question 23

Circle theorem: the perpendicular from the centre of a circle to a chord bisects the chord (cuts in half)

Answer 23

a

Question 24

Point of inflection if second derivative = 0 and third does not

Answer 24

for stationary point check points above and below x using dy/dx to see if +ve or -ve either side

dy/dx > 0 increasing stationary point
< 0 decreasing stationary point

If d2y/dx2 = 0, then it could be a maximum, minimum or point of inflection, you must test the values of dy/dx either side of the stationary point, as before in the stationary points section.