(a) Divide \(11111111_{two}\) by \(101_{two}\) (b) A sector of radius 6 cm has an angle of 105° at the centre. Calculate its: (i) perimeter ; (ii) area . [T...
Question 1 Report
(a) Divide \(11111111_{two}\) by \(101_{two}\)
(b) A sector of radius 6 cm has an angle of 105° at the centre. Calculate its:
(i) perimeter ; (ii) area . [Take \(\pi = \frac{22}{7}\)]
Answer Details
(a) Divide \(11111111_{two}\) by \(101_{two}\)
We use binary long division. Divisor is \(101_{two}\).
1 1 0 0 1 1
___________________
1 0 1 ) 1 1 1 1 1 1 1 1
1 0 1
-----
1 0 1
1 0 1
-----
0 0 1 1
1 1 1
1 0 1
-----
1 0 1
1 0 1
-----
0 0The quotient is \(11111111_{two} \div 101_{two} = 110011_{two}\), remainder \(0\).
Check in base ten: \(11111111_{two}=255\), \(101_{two}=5\), and \(255 \div 5 = 51 = 110011_{two}\). Correct.
(b) Sector: radius 6 cm, angle 105°, \(\pi=\tfrac{22}{7}\)
(i) Perimeter. First find the arc length.
\[\text{Arc} = \frac{\theta}{360}\times 2\pi r = \frac{105}{360}\times 2\times\frac{22}{7}\times 6 = \frac{7}{24}\times\frac{264}{7} = 11\text{ cm}\]
Perimeter = arc + the two straight radii:
\[P = 11 + 2(6) = 23\text{ cm}\]
(ii) Area.
\[A = \frac{\theta}{360}\times \pi r^{2} = \frac{105}{360}\times\frac{22}{7}\times 6^{2} = \frac{7}{24}\times\frac{792}{7} = 33\text{ cm}^{2}\]
Download The App On Google Playstore
Everything you need to excel in your exams
