Correct Answer: A. -1

Explanation

\(a * a^{-1} = e = -1\)

\(\therefore a^{-1} = -1\)

Explanation

(a) 5 female, 7 male = 12 in total.

(i) If no restriction, there are 12 teachers to fill up 4 vacancies.

Number of ways = \(^{12}C_{4}\)

= \(\frac{12!}{4! 8!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2}\)

= \(495\) ways.

(ii) No of ways if at least 2 of them are females

= No of ways [no restriction - 0 female - 1 female]

0 females implies 4 males

No of ways = \(^{7}C_{4} = \frac{7!}{4! 3!}\)

= \(\frac{7 \times 6 \times 5}{3 \times 2} = 35\) ways.

1 female = 1 female + 3 males

No of ways = \(^{5}C_{1} \times ^{7}C_{3}\)

= \(\frac{5!}{4! 1!} \times \frac{7!}{4! 3!}\)

= \(5 \times 35 = 175\) ways.

\(\therefore\) No of ways if at least 2 are females = 495 - (35 + 175) = 285 ways.

(b)

\(R_{S} = 1 - \frac{6 \sum d^{2}}{n(n^{2} - 1)}\)

= \(1 - \frac{6(49.5)}{7(7^{2} - 1)}\)

= \(1 - \frac{297}{336}\)

= \(\frac{13}{112} = 0.116\)

(ii) This implies that X and Y are fairly positively correlated. As X increases, Y increases as well.

Correct Answer: A. \(\frac{2x}{x - 3}, x \neq 3\)

Explanation

From the rules of binary operation, \(x * e = x\)

\(\implies x * e = 3x + 3e - xe = x\)

\(3e - xe = x - 3x = -2x\)

\(e = \frac{2x}{x - 3}, x \neq 3\)

Explanation

b i. x? = \(\frac{14+16+17+18+22+24+27+28+31+33}{10}\)

x? = \(\frac{230}{10}\)

y = \(\frac{22+19+15+13+10+12+3+5+3+2}{10}\)

= \(\frac{104}{10}\)

ii.

x?\(_1\) = \(\frac{14+16+17+18+22}{5}\) → \(\frac{79}{5}\) = 15.8

c.

d i. There is a negative correlation between X and Y. X increases as Y decreases

ii. The point labelled Y1 is the corresponding value on Y-axis when X = 20, Y1 = 12.6

Explanation

(a) \(h(x) = \frac{1}{x + 2}\) ;

Let y = h(x).

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

\(\frac{1}{y} = x + 2 \implies x = \frac{1}{y} - 2\)

\(\therefore h^{-1} (x) = \frac{1}{x} - 2\)

(b) \(g \circ h(x) = g(h(x))\)

\(g(h(-\frac{1}{2})) = g(\frac{1}{-\frac{1}{2} + 1})\)

= \(g(\frac{1}{\frac{3}{2}})\)

= \(g(\frac{2}{3})\)

= \((\frac{2}{3})^{2} - 2\)

= \(\frac{4}{9} - 2\)

= \(\frac{-14}{9}\)

Correct Answer: C. y? ~x ^ ~z

Correct Answer: B. \(\frac{-4\sqrt{3}}{3}\)

Explanation

\(a \Delta b\) = \(\frac{a+b}{\sqrt{ab}}\)

\(-3\Delta -1\) = \(\frac{-3 + -1}{\sqrt{-3\times -1}}\)

\(\frac{-4}{\sqrt{3}}\), rationalising, we have

\(\frac{-4 \times \sqrt{3}}{\sqrt{3}\times \sqrt{3}} = \frac{-4\sqrt{3}}{3}\)

Correct Answer: C. x = 0

Explanation

Substitute x by (x + h) to have f(x + h) = 3(x + h)\(^2\) + 2x + 2h - 1 - 3x\(^2\) - 2x + 1

= 6\(x\)h + 3h\(^2\) + 2h

Divided through by h and obtained \(\frac{f(x + h) - f(x)}{h} = \frac{6 \times h + 3h^2 + 2h}{h} = 6x + 3h + 2\)

Taking limit as h tends to zero, lim\(_{h \to o}\) \(\frac{f(x+h) -f(x)}{h}\)

= lim\(_{h \to o}\) (6x + 3h + 2) so that f\(^1\)(x) = 6x + 2

Correct Answer: A. 5

Explanation

\(\begin{pmatrix} 3 & 2 \\ 7 & x \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 12 \\ 29 \end{pmatrix}\)

\(\begin{pmatrix} 3 \times 2 + 2 \times 3 \\ 7 \times 2 + x \times 3 \end{pmatrix} = \begin{pmatrix} 12 \\ 29 \end{pmatrix}\)

\(\implies 14 + 3x = 29 \)

\(3x = 29 - 14 = 15\)

\(x = 5\)