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Question 1

area of sector

Answer 1

0.5*r(^2)θ

Question 2

area of segment

Answer 2

0.5r^2 (θ − sin(θ))

Question 3

arc length

Answer 3

rθ

Question 4

area of trapezium

Answer 4

(a+b)/2 *h

Question 5

cos2A

Answer 5

cos(^2)A-sin(^2)A
2cos(^2)A-1
1-2sin(^2)A

Question 6

sin2A

Answer 6

2sinAcosA

Question 7

tan2A

Answer 7

(2tanA)/(1-tan(^2)A)

Question 8

derivative of a^(kx)

Answer 8

a(^kx) k ln a

Question 9

integral of f(AX+B)

Answer 9

1/a f(ax+b)

Question 10

prove the sum of the first n terms of an arithmetic series is 1/2n(2a+(n-1)d)

Answer 10

1. Sn= a, a+d, a+2d+…….+ a+(n-2)d + a+(n-1)d
2. reverse sequence(Sn): a+(n-1)d + a+(n-2)d +………+ a+2d + a+d + a
3. add the two sequences: 2Sn= n(2a+(n-1)d)
4. divide by 2: Sn= n/2 (2a+(n-1)d)

Question 11

Proof of geometric series

Answer 11

(1) Sn= a+ ar +ar2+ ar3+…..+ar(n-2)+ ar(n-1)
(2) rSn= ar+ ar2+ ar3 +…..+ ar(n-1) + ar(n)
(1)-(2) Sn-rSn = a-ar(n)
Sn(1-r)= a(1-r(n))
Sn= a(1-r(n))/1-r

Question 12

Volume of sphere

Answer 12

4/3pi r^2

Question 13

prove the first principles, that the derivative of sin x is cos x

Answer 13

f’(x)= (f(x+h)-f(x))/h
(sin(x+h)-sin(x))/h
(sinx cosh + cosx sinh - sin x)/ h
(cos h -1)/h)sinx + (sin h)/h)cosx

cos h-1/ h ——> 0
sin h/h——>1
0sin x +1 cosx
=cos x

Question 14

how to convert degrees to radians

Answer 14

multiply value of degrees by pi/180

Question 15

proof of infinite primes

Answer 15

Assumption: there are a finite number of prime numbers,
p1, p2, p3, up to pn
. Let X= (p1* p2* p3……Pn)+1
None of the prime numbers are a factor of X as they all leave a remainder of 1 when X is divided by them.
But X must have at least one prime factor.
This is a contradiction

Question 16

area of parrellogram

Answer 16

side a* side b* sin((angle between A and B))

Question 17

when a vector has travelled north east what does that mean

Answer 17

the I components are equal to the j components