Further Mathematics Questions and Answers

\(3^{2x-2} - 28(3^{x-2}) + 3 = 0\)

\(\frac{3^{2x}}{3^2} - \frac{28.3^x}{3^2} + 3 = 0\)

\(\frac{3^{2x}}{9} - \frac{28.3^x}{9} + 3 = 0\)

let p = 3\(^x\)

\(\frac{p^{2}}{9} - \frac{{28p}}{9} + 3 = 0\)

multiply through by 9
p\(^2\) - 28p + 27 = 0
p\(^2\) - p - 27p + 27 = 0
p (p - 1) - 27(p - 1) = 0
(p-1)(p-27) = 0
p = 1 or 27
when p = 1
p = 3\(^x\)
3\(^x\) = 1
3\(^x\) = 3\(^0\)
x = 0
when p = 27
3x = 27
3x = 33
x = 3

6C\(_2 = \frac{6!}{[6-2]![2!]}\)

\(\frac{6*5*4!}{4!*2!}\)

= \(\frac{6*5}{2}\)

= 15

Un = kn\(^2\) + pn,

U\(_1\) = -1,

U\(_5\) = 15,

when n = 1
U1 = k(1)\(^2\)+ p(1) = -1
k + p = -1 --------eqn1
when n = 5
U\(_5\) = k(5)\(^2\)+ p(5) = 15
25k + 5p = 15 --------eqn2
multiply eqn1 by 5 and eqn2 by 1
5k + 5p = -5 -------eqn3
25k + 5p = 15 -------eqn4
eqn4 - eqn3
20k = 20
k = 1
sub for k in eqn1
1 + p = -1
p = -1 -1 = -2

from the equation of motion

u = 5m/s, a = 2m/s\(^2\), t = 4s 

s = ut + \(\frac{1}{2} at^2\)

s = 5*4 + \(\frac{1}{2} 2*4^2\)

s =  20 + 16

s = 36m

P = { 20, 25, 30, 35}, Q = {21, 24, 27, 30, 33}, R = {21, 23, 25, 27, 29, 31, 33, 35}

(P⋃Q)∩R = {20, 21, 24, 25, 27, 30, 33, 35} ∩ {21, 23, 25, 27, 29, 31, 33, 35}

= {21, 25, 27, 33, 35}