Mathematics Questions and Answers

3(x – 1) ≤ 2 (x – 3)

= 3x – 3 ≤ 2x – 6

Add 3 to both sides

= 3x – 3+ 3 ≤ 2x – 6+ 3

= 3x ≤ 2x – 3

Subtract 2x from both the sides

= 3x – 2x ≤ 2x – 3 – 2x

∴ x ≤ -3

y = \(\frac {3x - 9c}{4x + 5d}\)

\(\frac {y}{1} = \frac {3x - 9c}{4x + 5d}\)

= y(4x + 5d) = 3x - 9c

= 4xy + 5dy = 3x - 9c

= 4xy - 3x = -9c - 5dy

= (4y - 3)x = -9c - 5dy

= x = \(\frac {-9c - 5dy}{4y - 3}\)

x = \(\frac {-(9c + 5dy)}{4y - 3}\)

Area of a circle (A) = \(\pi r^2\)

Given

\(\frac{dA}{dt} = 1.5cm^2s^{-1}\)

\(\frac{dr}{dt}\) = ?

A = 2cm\(^2\)

Now

2 = \(\pi r^2\)

= r\(^2 = \frac {2}{\pi}\)

r = \(\sqrt \frac {2}{\pi}\) cm = 0.798cm

\(\frac {dr}{dt} = \frac {dA}{dt} \times \frac {dr}{dt}\)

\(\frac {dA}{dr} = 2\pi r\) (differentiating A = \(\pi r^2)\)

\(\frac {dr}{dA} = \frac {1}{2\pi r}\)

\(\frac {dr}{dt} = 1.5 \times \frac {1}{2\times \pi \times 0.798} = 1.5 \times 0.199\)

\(\frac {dr}{dt} = 0.299cms^{-1}\) (to 3 s.f)

An interior angle of a regular polygon = \(\frac{(2n-4)\times 90}{n}\)

An exterior angle of a regular polygon = \(\frac{360}{n}\)

\(\frac{(2n-4)\times 90}{n}\) =5 \(\times\)  \(\frac{360}{n}\) (Given)
= (2n-4) x 90 = 5 x 360
= 180n - 360 = 1800
= 180n = 1800 + 360
= 180n = 2160
= n = \(\frac{2160}{180}\) = 12
The polygon has 12 sides which is dodecagon

Original population = 1,230
New population = 1,040
Decrease in population = 1,230 – 1,040 = 190
Percentage decrease in population = decrease in population   x 100%
                                                              original population         

= \(\frac {190}{1,230}\) x 100 = 15.45%