Calculate Mr. Gatua's PAYE (Pay As You Earn) given his salary of Sh. 80,000 and the tax brackets and reliefs provided.

Mr. Gatua's annual salary is Sh. 80,000. This is his taxable income.

• First bracket (Sh. 1 - 1500): Taxable amount = Sh. 1500 Tax = $1500 \times 10\% = \text{Sh. } 150$

• Second bracket (Sh. 1501 - 3000): Taxable amount = $3000 - 1500 = \text{Sh. } 1500$ Tax = $1500 \times 15\% = \text{Sh. } 225$

• Third bracket (Sh. 3001 - 4500): Taxable amount = $4500 - 3000 = \text{Sh. } 1500$ Tax = $1500 \times 25\% = \text{Sh. } 375$

• Fourth bracket (Above Sh. 4500): Taxable amount = $80,000 - 4500 = \text{Sh. } 75,500$ Tax = $75,500 \times 35\% = \text{Sh. } 26,425$

Total Gross Tax = $150 + 225 + 375 + 26,425 = \text{Sh. } 27,175$

Monthly personal relief = Sh. 1056
Annual personal relief = $1056 \times 12 = \text{Sh. } 12,672$

PAYE = Total Gross Tax - Annual Personal Relief
PAYE = $27,175 - 12,672 = \text{Sh. } 14,503$

Mr. Gatua's PAYE is $\boxed{\text{Sh. } 14,503}$.

The third term ($T_3$) is $3 \frac{1}{3} = \frac{10}{3}$.

The sixth term ($T_6$) is $11 \frac{1}{4} = \frac{45}{4}$.

Simplify the fraction:

Take the cube root of both sides:

The common ratio is $\boxed{\frac{3}{2}}$ and the first term is $\boxed{\frac{40}{27}}$.

Given triangle RSQ:
Angle RSQ (angle S) = $50^\circ$
Side SQ = 11.83 cm
Side QR = 12 cm
We need to find the radius ($R$) of the circumcircle.
The sine rule states that for a triangle with sides $a, b, c$ and opposite angles $A, B, C$, the radius of the circumcircle $R$ is given by:

In this case, we have side QR (opposite angle S) and angle S.
So, $a = QR = 12 \text{ cm}$ and $A = \text{angle RSQ} = 50^\circ$.

Rounding to two decimal places:

The radius of the circle is $\boxed{7.83 \text{ cm}}$.