Correct Answer: D. 76^o

Explanation

(3y - 450 + (y + 25) + t = 360^o(angles at a point)

3y - 45 + y + 25 + y = 360

5y = 360 + 20

5y = 380

y = \(\frac{380}{5}\)

= 76^o

Correct Answer: A. dodecagon

Explanation

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

Explanation

(a) Area of the shaded segment = Area of sector - Area of triangle

= \(\frac{\theta}{360°} \times \pi r^{2} - \frac{1}{2} ab \sin \theta\)

Area of sector = \(\frac{63}{360} \times \frac{22}{7} \times 10^{2}\)

= \(55 cm^{2}\)

Area of triangle = \(\frac{1}{2} \times 10 \times 10 \times \sin 63°\)

= \(50 \times 0.891 = 44.55 cm^{2}\)

\(\therefore \text{Area of the shaded segment} = 55 - 44.55 = 10.45 cm^{2}\)

(b) (i) Total surface area = \(2\pi r (r + h)\)

= \(2 \times \frac{22}{7} \times 3 (3 + 6)\)

= \(\frac{132}{7} \times 9 = \frac{1188}{7}\)

= \(169.71 cm^{2}\)

(ii) Volume of tin = \(\pi r^{2} h\)

= \(\frac{22}{7} \times 3 \times 3 \times 6 = \frac{1188}{7}\)

\(\approxeq 169.71 cm^{3}\)

Correct Answer: B. 9

Correct Answer: D. a = 5 or a = -2

Explanation

3a + 10 = a\(^2\)

a\(^2\) - 3a - 10 = 0

a\(^2\) - 5a + 2a - 10 = 0

a(a - 5) + 2(a - 5) = 0

(a - 5)(a + 2) = 0

a = 5 or a = -2.

Correct Answer: C. 135°

Explanation

Let \(\mu\) = 40 n(M) = 23, n(M \(\cup\) P) = 5, n(M \(\cup\) C) = 8, n(P \(\cup\) C) = 5, n(M \(\cup\) P \(\cup\) C) = 3, n(C) only = x and n(P) only = 2x

(a)(i) We have 13 + 2 + 3 + 5 + x + 2 + 2x = 40 and solving for x will yield x = 5, so that the number of students that studied only Physics = 2(5) = 10

(a)(ii) The number of students that studied only one subject is 13 + 10 + 5 = 28

(b) The probability that a student selected at random studies exactly two subjects = \(\frac{5 + 2 + 2}{40} = \frac{9}{48}\) or 0.225

Explanation

p(have valid license) = p = 0.8

p(no valid license) = q = 0.2.

The binomial probability distribution is

\((p + q)^{6} = p^{6} + 6p^{5} q + 15p^{4} q^{2} + 20p^{3} q^{3} + 15p^{2} q^{4} + 6p q^{5} + q^{6}\)

(a) p(2 have valid licenses) = \(15p^{2} q^{4}\)

= \(15 (0.8)^{2} (0.2)^{4} = 15(0.64)(0.0016)\)

= \(0.01536 \approxeq 0.015\) (3 d.p)

(b) p(more than 3 have valid licenses) = \(p^{6} + 6p^{5} q + 15p^{4} q^{2}\)

= \((0.8)^{6} + 6(0.8)^{5} (0.2) + 15(0.8)^{4} (0.2)^{2}\)

= \(0.262144 + 0.393216 + 0.24578\)

= \(0.90112 \approxeq 0.901\) (3 d.p)

(c) p(at least 5 have valid licenses) = \(p^{6} + 6p^{5} q\)

= \((0.8)^{6} + 6(0.8)^{5} (0.2)\)

= \(0.262144 + 0.393216\)

= \(0.65536 \approxeq 0.655\).

Explanation

Line : \(2y = x + 3\)

\(\therefore y = \frac{1}{2} (x + 3)\)

\(x^{2} + y^{2} - 2x + 6y - 15 = 0\)

\(x^{2} + (\frac{1}{2} (x + 3))^{2} - 2x + 6(\frac{1}{2} (x + 3)) - 15 = 0\)

\(x^{2} + \frac{x^{2}}{4} + \frac{6x}{4} + \frac{9}{4} + 3x + 9 - 15 = 0\)

\(\frac{5x^{2}}{4} + \frac{10x}{4} - \frac{15}{4} = 0\)

\(\equiv 5x^{2} + 10x - 15 = 0\)

\(5x^{2} - 5x + 15x - 15 = 0 \implies 5x(x - 1) + 15(x - 1) = 0\)

\((5x + 15)(x - 1) = 0 \implies \text{x = -3 or 1}\)

When \(x = -3\), \(y = \frac{1}{2} (-3 + 3) = 0\)

When \(x = 1\), \(y = \frac{1}{2} (1 + 3) = 2\)

The coordinates are M(-3, 0) and N(1, 2).

Correct Answer: A. N 70° W

Explanation

The bearing S 20° W is equivalent to a bearing of 200°.

Moving clockwise 90° ⇒ 200° + 90°

= 290°

This is equivalent to N 70° W.