Explanation

(a) \(\sin x = \frac{5}{13}\)

Using SOHCAHTOA, then Opp = 5 and Hyp = 13

\(13^{2} = 5^{2} + Adj^{2}\)

\(169 = 25 + Adj^{2}\)

\(Adj = \sqrt{169 - 25} = \sqrt{144} = 12\)

\(\cos x = \frac{12}{13}\)

\(\tan x = \frac{5}{12}\)

\(\frac{\cos x - 2\sin x}{2\tan x} = \frac{\frac{12}{13} - 2(\frac{5}{13})}{2(\frac{5}{12})}\)

= \(\frac{\frac{2}{13}}{\frac{5}{6}}\)

= \(\frac{12}{65}\)

(b)

Taking \(\Delta LPB\),

Using Pythagoras theorem, \(/BP/^{2} = /BL/^{2} - /LP/^{2}\)

\(i.e. /BP/ = \sqrt{12^{2} - 9.6^{2}}\)

= \(\sqrt{144 - 92.16} = \sqrt{51.84} = 7.2m\)

\(/PA/ = 10m - 7.2m = 2.8m\)

\(/LA/^{2} = x^{2}\)

\(x^{2} = /LP/^{2} + /PA/^{2}\)

\(x^{2} = 9.6^{2} + 2.8^{2}\)

= \(92.16 + 7.84\)

\(x^{2} = 100 \implies x = 10m\)

(ii) \(\tan \theta_{y} = \frac{9.6}{2.8} = 3.4286\)

\(\theta_{y} = \tan^{-1} (3.4286) \approxeq 74°\)

Correct Answer: D. 1000

Explanation

¹/₃log₁₀P = 1

log₁₀P^1/3 = log₁₀10

P^1/3 = 10

P = 1000

Correct Answer: C. x - sin x + k

Correct Answer: C. 13cm

Explanation

given that diameter = 140cm ⇒radius = 70cm, volume of cylinder = 200litres = \(200,000cm^3\)

volume of cylinder = base area times height = \(\pi r^2h\)

\( 200,000cm^3 = \frac{22}{7} \times 70 \times 70 \times h\)

\(200,000 = \frac{107,800h}{7}\)

cross multiply

1,400,000 = 107,800h

\( h = \frac{1,400,000}{107,800}\)

= 13cm (to the nearest cm).

Correct Answer: D. 229

Explanation

Find the median of the following number set: 14,15,76,345,222,976,300,117,236,426 Recall that the median is the middle value of a number set when it is arranged in ascending order. So we must begin by arranging the set in ascending order. 14,15,76,117,222,236,300,345,426,976 In this case, because we have an even number of terms, we do not have a single middle number. This means we need to take the average of the middle two terms to find our median. Median=222+236/2=229 So our answer is 229.

Correct Answer: D. 36°

Explanation

Angle at the centre O is twice the angle at the circumference.

i.e O = 72° * 2 = 144°

∠OPS = 180 - 144 = 36°

Correct Answer: C. 15

Explanation

\(2u m/s\) \(0 m/s\) : \(\frac{1}{2} u m/s\) \(u m/s\)

P Q P Q

Before collision : After collision

Momentum before collision = \(M_{P} (2u) kg m/s\)

Momentum after collision = \(-\frac{1}{2}u (M_{P}) + M_{Q} (u)\)

\(2u (M_{P}) + \frac{1}{2} u (M_{P}) = M_{Q} (u)\)

\(\frac{5}{2} M_{P} = M_{Q}\)

\(\frac{M_{P}}{M_{Q}} = \frac{2}{5}\)

The ratio = 2:5

(b)

From cosine rule, \(5^{2} = 3^{2} + 7^{2} - 2 \times 3 \times 7 \cos (180 - \alpha)\)

\(25 = 9 + 49 - 42(-\cos \alpha)\)

\(25 = 58 + 42\cos \alpha\)

\(\cos \alpha = \frac{-33}{42} = -0.7857\)

\(\alpha = \cos^{-1} (-0.7857) = 141.79°\)

(c)

\(\overrightarrow{VB} = \frac{1}{2}\overrightarrow{AB} = \frac{1}{2} \begin{pmatrix} -4 \\ 6 \end{pmatrix} = \begin{pmatrix} -2 \\ 3 \end{pmatrix}\)

\(\overrightarrow{CV} + \overrightarrow{VB} = \overrightarrow{CB}\)

\(\overrightarrow{CV} = \overrightarrow{CB} - \overrightarrow{VB}\)

= \(\begin{pmatrix} 2 \\ -3 \end{pmatrix} - \begin{pmatrix} -2 \\ 3 \end{pmatrix}\)

= \(\begin{pmatrix} 2 + 2 \\ -3 - 3 \end{pmatrix}\)

= \(\begin{pmatrix} 4 \\ -6 \end{pmatrix}\)

Correct Answer: C. x?2

Explanation

x ? 2 is represented by an arrow with an filled-in dot above the 2.

The arrow points in the direction of all the numbers that are greater than 2.

Correct Answer: B. 5