Question
The following diagram shows a circle with centre O and radius 5 metres.
Points A and B lie on the circle and \(A\hat{O}B\) = 1.9 radians.
(a) Find the length of the chord [AB].
(b) Find the area of the shaded sector.
Answer/Explanation
Ans:
(a) EITHER
uses the cosine rule
AB2 = 52 + 52 -2 × 5 × 5 × cos1.9
OR
uses right-angled trigonometry
\(\frac{\frac{AB}{2}}{5}= sin 0.95\)
OR
uses the sine rule
\(\alpha \frac{1}{2}(\pi -1.9)(=0.6207…)\)
\(\frac{AB}{sin1.9}= \frac{5}{sin0.6207….}\)
THEN
AB = 8.1341…
AB = 8.13 (m)
(b) let the shaded area be A
METHOD 1
Attempt at finding reflex angle
\(A\hat{O}B = 2\pi -1.9 (=4.3831….)\)
substitution into area formula
\(A = \frac{1}{2}\times 5^{2}\times 4.3831…. OR \left ( \frac{2\pi -1.9}{2\pi } \right )\times \pi \left ( 5^{2} \right )\)
= 54.7898…
= 54.8 (m2)
METHOD 2
let the area of the circle be AC and the area of the unshaded sector be AU
Question
Consider the set of six-digit positive integers that can be formed from the digits 0 , 1 , 2, 3, 4 , 5, 6, 7, 8 and 9.
Find the total number of six-digit positive integers that can be formed such that
(a) the digits are distinct;
(b) the digits are distinct and are in increasing order.
Answer/Explanation
Ans:
(b) METHOD 1
EITHER
every unordered subset of 6 digits from the set of 9 non-zero digits can be arranged in exactly one way into a 6-digit number with the digits in increasing order.
OR
9C6 (×1)
THEN
= 84
METHOD 2
EITHER
removes 3 digits from the set of 9 non-zero digits and these 6 remaining digits can be arranged in exactly one way into a 6- digit number with the digits in increasing order.
OR
9C3 (×1)
THEN
= 84