Explanation

(a) \(\frac{x^{2} - y^{2}}{3x + 3y}\)

\(\frac{(x + y)(x - y)}{3(x + y)}\) (Using difference of two squares)

= \(\frac{x - y}{3}\)

(b)(i)

\(\sin 37 = \frac{/PS/}{15}\)

\(/PS/ = 15 \times 0.6018\)

= \(9.03 cm\)

(ii) \(\cos 37 = \frac{/SK/}{15}\)

\(/SK/ = 15 \times 0.7986\)

= \(11.98 cm \)

\(\approxeq 12.0 cm\)

(iii) Area of the shaded portion = Area of rectangle PQRS - Area of triangle PSK.

/SR/ = 2(/SK/) = 2(11.98)

= 23.96 cm

Area of rectangle PQRS = \(23.96 \times 9.03 \)

= \(216.3588 cm^{2}\)

Area of triangle PKS = \(\frac{1}{2} \times 11.98 \times 9.03\)

= \(54.088 cm^{2}\)

Area of shaded portion : \((216.3588 - 54.088)cm^{2}\)

\(162.2708 cm^{2}\)

\(\approxeq 162 cm^{2}\)

Correct Answer: D. 87%

Correct Answer: A. 10! / (2!2!3!)

Correct Answer: D. 22

Explanation

The set is already in increasing order. We have five numbers in the set, however, we need to add another number to ensure the set will have a median of 15. This will make the set have six numbers. When there is an even number in the set, we need to take the two middle numbers by adding them then dividing by two. Let's say this number is x. Let's setup an equation. 8+x/2=15. The numerator represents the two middle numbers being added and divided by the denominator. If we multiply both sides by 2 and subtract 8 on both sides, x is 22. Make sure this answer doesn't violate the set. 22 is less than 25 but greater than 8, so therefore 22 is the correct answer.

Correct Answer: B. 84

Explanation

y 3x\(^2\) (2x +1) = 6x\(^3\) + 3x\(^2\)

dy/dx = 18x\(^2\) + 6x

At x = 2, 18(2)\(^2\) + 6(2)

= 72 + 12 = 84

Correct Answer: D. p = \(\frac{2qr - rs}{6}\)

Explanation

q = \(\frac{3p}{r} + \frac{s}{2}\)

q = \(\frac{6p + rs}{2r}\)

6p + rs = 2qr

6p = 2qr - rs

p = \(\frac{2qr - rs}{6}\)

Correct Answer: B. 10cm

Explanation

(a) The position vector of L, \(\overrightarrow{OL} = 5i + 6j\)

The coordinates of L = (5, 6)

The position vector of M, \(\overrightarrow{OM} = 13i + 4j\)

The coordinates of M = (13, 4).

Let \(\overrightarrow{OK} = xi + yj\) be the position vector of K.

\(\overrightarrow{LK} = (x - 5)i + (y - 6)j\), \(\overrightarrow{KM} = (13 - x)i + (4 - y)j\)

\(LK : KM = 2 : 3 ; \frac{LK}{KM} = \frac{2}{3}\)

\(3[(x - 5)i + (y - 6)j] = 2[(13 - x)i + (4 - y)j]\)

Equating by components,

\(i : 3x - 15 = 26 - 2x \implies 5x = 41\)

\(x = \frac{41}{5}\)

\(j : 3y - 18 = 8 - 2y \implies 5y = 26\)

\(y = \frac{26}{5}\)

The position vector of K is \(\overrightarrow{OK} = \frac{41}{5} i + \frac{26}{5} j\)

(b) \(\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}\)

= \(\begin{pmatrix} 8 \cos 60° \\ 8 \sin 60° \end{pmatrix} + \begin{pmatrix} 12 \cos 130° \\ 12 \sin 130° \end{pmatrix}\)

= \(\begin{pmatrix} 8 \times \frac{1}{2} \\ 8 \times \frac{\sqrt{3}}{2} \end{pmatrix} + \begin{pmatric} -12 \times 0.6428 \\ 12 \times 0.7660 \end{pmatrix}\)

= \(\begin{pmatrix} 4 \\ 4\sqrt{3} \end{pmatrix} + \begin{pmatrix} -7.7136 \\ 9.9192 \end{pmatrix} = \begin{pmatrix} -3.7136 \\ 16.12 \end{pmatrix}\)

\(|AC| = \sqrt{(-3.7136)^{2} + (16.12)^{2}} = \sqrt{13.794 + 259.9}\)

= \(\sqrt{273.694} = 16.544 \approxeq 16.5 km\)

\(AB = (8 \cos 60 i + 8 \sin 60 j) ; AC = (-3.7136 i + 16.12 j)\)

(ii)

Let the angle between \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) be \(\theta\).

Using sine rule,

\(\frac{\sin B}{AC} = \frac{\sin A}{BC}\)

\(\frac{\sin 110°}{16.5} = \frac{\sin \theta}{12}\)

\(\sin \theta = \frac{12 \sin 110°}{16.5}\)

\(\sin \theta = 0.6834\)

\(\theta = \sin^{-1} (0.6834) = 43.11°\)

Bearing of C from A = 60° + 43.11° = 103.11° \(\approxeq\) 103° (nearest degree)

Correct Answer: B. 15

Correct Answer: D. N8,268.75

Explanation

N75.00 x 110.25 = N8,268.75