Explanation

(a)

\(\sum f = 16 + x + y = 30\)

\(\implies x + y = 14 ... (1)\)

\(\bar{x} = \frac{\sum fx}{\sum f}\)

\(52 = \frac{900 + 30x + 50y}{30}\)

\(1560 = 900 + 30x + 50y \implies 30x + 50y = 660\)

\(3x + 5y = 66 .... (2)\)

Solving equation (1) and (2),

From (1), x = 14 - y

\(3(14 - y) + 5y = 42 - 3y + 5y = 66\)

\(2y = 24 \implies y = 12\)

\(x = 14 - y = 14 - 12 = 2\)

(x, y) = (2, 12).

(b)

(c)

(d) From the histogram, Mode = 44.

Correct Answer: C. 4x² + 13x - 12 = 0

Explanation

Let x = \(\frac{3}{4}\) or x = -4

i.e. 4x = 3 or x = -4

(4x - 3)(x + 4) = 0

therefore, 4x² + 13x - 12 = 0

Correct Answer: D. q \(\alpha \frac{1}{p}\)

Explanation

p \(\alpha \frac{I}{Q}\); p \(\frac{k}{q}\) (where k is constant)

q = \(\frac{k}{p}\)

q \(\alpha \frac{1}{p}\)

Correct Answer: D. 3333km

Correct Answer: C. 6720

Explanation

The word has 8 letters with one letter repeated 3 times, therefore we have:

\(\frac{8!}{3!} = 6720\) ways.

Correct Answer: C. 240

Explanation

\(^{6}C_{4}(1)^{6-4}(-2x)^{4}\) = \(15\times1\times16x^{4} = 240x^{4}\)

The coefficient of \(x^{4}\)= 240

Explanation

\(F_{R} = F_{1} + F_{2} + F_{3}\)

\(F_{1} = (10 N, 090°) = \begin{pmatrix} 10 \cos 90 \\ 10 \sin 90 \end{pmatrix} = \begin{pmatrix} 0 \\ 10 \end{pmatrix}\)

\(F_{2} = (20 N, 210°) = \begin{pmatrix} 20 \cos 210 \\ 20 \sin 210 \end{pmatrix} = \begin{pmatrix} -10\sqrt{3} \\ - 10 \end{pmatrix}\)

\(F_{3} = (4 N, 330°) = \begin{pmatrix} 4 \cos 330 \\ 4 \sin 330 \end{pmatrix} = \begin{pmatrix} 2\sqrt{3} \\ -2 \end{pmatrix}\)

\(F_{R} = \begin{pmatrix} 0 \\ 10 \end{pmatrix} + \begin{pmatrix} -10\sqrt{3} \\ -10 \end{pmatrix} + \begin{pmatrix} 2\sqrt{3} \\ -2 \end{pmatrix}\)

= \(\begin{pmatrix} -8\sqrt{3} \\ -2 \end{pmatrix}\)

\(|F_{R}| = \sqrt{(8\sqrt{3})^{2} + (-2)^{2}}\)

= \(\sqrt{192 + 4} \)

= \(\sqrt{196} = 14 N\)

Correct Answer: B. 57\(^o\)

Explanation

lac = \(\frac{\theta}{360}\) x 2\(\pi\)r

7.5 = \(\frac{\theta}{360}\) x 2 x \(\frac{22}{7}\) x 7.5

7.5 = \(\frac{330\theta}{2520}\)

\(\theta\) = \(\frac{7.5}{0.1309}\)

\(\theta\) = 57.29

\(\theta\) = 57 \(^o\)

Correct Answer: B. 19

Explanation

The median is the middle number of the set, when it is listed in order from smallest to largest or vice versa. In this case we have an odd amount of numbers so we just count from each side until we find the number in the middle. 4,5,6,12,15,18,19,23,26,27,31,39,41 This gives us a final answer of 19 for the median.

Correct Answer: C. \(\frac{5}{12}\)

Explanation

\(p(P) = \frac{1}{2}, p(P') = \frac{1}{2}\)

\(p(Q) = \frac{1}{3}, p(Q') = \frac{2}{3}\)

\(p(R) = \frac{3}{4}, p(R') = \frac{1}{4}\)

p(exactly two hit the target) = p(P and Q and R') + p(P and R and Q') + p(Q and R and P')

= \((\frac{1}{2} \times \frac{1}{3} \times \frac{1}{4}) + (\frac{1}{2} \times \frac{3}{4} \times \frac{2}{3}) + (\frac{1}{3} \times \frac{3}{4} \times \frac{1}{2})\)

= \(\frac{1}{24} + \frac{6}{24} + \frac{3}{24}\)

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