Mathematics Questions and Answers

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%

$\int_0^1 4x - 6^3\sqrt x^2$ dx
=$\int_0^1 4x - 6x^{2/3}$ dx
=$(2x^2 - \frac{8x^{5/3}}{5})_0^1$
=$(2(1)^2 - \frac{18(1)^{5/3}}{5}) - (2(0)^2 - \frac{18(0)^{5/3}}{5}$
= - $\frac{8}{5}$ - 0
    - $\frac{8}{5}$