(a) Divide \(11111111_{two}\) by \(101_{two}\) (b) A sector of radius 6 cm has an angle

Explanation

(a) This can either be done using the long division method (easier) or convert everything to base ten and divide, then re-convert to base two.

I'll be using the latter method.

\(11111111_{two} = 1 \times 2^{7} + 1 \times 2^{6} + 1 \times 2^{5} + 1 \times 2^{4} + 1 \times 2^{3} + 1 \times 2^{2} + 1 \times 2^{1} + 1 \times 2^{0}\)

= \(128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255_{ten}\)

\(101_{two} = 1 \times 2^{2} + 0 \times 2^{1} + 1 \times 2^{0} \)

= \(4 + 0 + 1 = 5_{ten}\)

\(\frac{255}{5} = 51_{ten}\)

= \(110011_{two}\)

(b)

(i) Length of arc AXB = \(\frac{105}{360} \times 2 \times \frac{22}{7} \times 6cm\)

= \(11cm\)

\(\therefore Perimeter = 6cm + 11cm + 6cm = 23cm\)

(ii) Area of sector = \(\frac{105}{360} \times \frac{22}{7} \times 6cm \times 6cm\)

= \(33 cm^{2}\)