Correct Answer: C. 4i - j

Explanation

\(\overrightarrow{OA} = (3, 4)\)

\(\overrightarrow{OB} = (5, -6)\)

\(\overrightarrow{OM} = (\frac{3 + 5}{2}, \frac{4 + (-6)}{2})\)

= \((4, -1) = 4i - j\)

Correct Answer: D. 0.976

Explanation

\(P(\text{at least one solves the problem}) = 1 - P(\text{none solving the problem})\)

= 1 - (0.4)(0.3)(0.2)

= 1 - 0.024

= 0.976

Explanation

Total number of arrangements:

(i)1. With replacement : Sample space (S) = {pp, qq, rr, ss}.

2. Without replacement : Sample space (S') = {pq, pr, ps, qr, qs, rs}

(ii)1. Total = 16; n(S) = 4 ;

\(\therefore p(S) = \frac{4}{16} = \frac{1}{4}\)

2. n(S') = 6;

\(\therefore p(S') = \frac{6}{16} = \frac{3}{8}\).

(b)(i) Numbers divisible by 4: 100, 104, 108, ..., 996.

\(T_{n} = a + (n - 1)d\)

\(996 = 100 + 4(n - 1) \implies 996 = 100 + 4n - 4\)

\(996 = 96 + 4n \implies 4n = 996 - 96 = 900\)

\(n = \frac{900}{4} = 225\)

(ii) Numbers divisible by both 3 and 4 are multiples of 12.

108, 120, 132, ..., 996.

\(996 = 108 + 12(n - 1)\)

\(996 = 108 + 12n - 12 \implies 996 = 96 + 12n\)

\(12n = 996 - 96 = 900\)

\(n = \frac{900}{12} = 75\)

Correct Answer: D. 3y + x + 14 = 0

Explanation

\(3x - y + 11 = 0 \implies y = 3x + 11\)

\(Gradient = 3\)

Gradient of perpendicular line = \(\frac{-1}{3}\)

\(\therefore \frac{y - (-5)}{x - 1} = \frac{-1}{3}\)

\(3(y + 5) = 1 - x\)

\(3y + x + 15 - 1 = 3y + x + 14 = 0\)

Correct Answer: B. GH¢ 5.66

Explanation

\(S.D = \sqrt{\frac{(x - \bar{x})^{2}}{n}}\)

\(S.D = \sqrt{\frac{160}{5}} = \sqrt{32}\)

= \(GH¢ 5.656 \approxeq GH¢ 5.66\)

Explanation

From Lami's theorem,

\(\frac{T_{1}}{\sin 150} = \frac{T_{2}}{\sin 140} = \frac{50}{\sin 70}\)

\(T_{1} = \frac{50 \sin 150}{\sin 70}\)

= \(\frac{50 \times 0.5}{0.9397} = 26.6N\)

\(T_{2} = \frac{50 \sin 140}{\sin 70}\)

= \(\frac{50 \times 0.6428}{0.9397} = 34.2N\)

The tensions are 26.6N and 34.2N respectively.

(b)(i) Given F = ?, u = 5 ms\(^{-1}\) ; v = 9 ms\(^{-1}\) ; t = 2s.

acceleration, \(a = \frac{v - u}{t}\)

= \(\frac{9 - 5}{2} = 2 ms^{-2}\)

Force, \(F = ma = 5 \times 2 \)

= \(10N\).

(ii) \(v = u + at\)

\(u = 9 ms^{-1}; a = 2 ms^{-2} ; t = 2s\)

\(v = 9 + 2(3)\)

= \(9 + 6 = 15 ms^{-1}\).

Correct Answer: A. \(\frac{2}{(1-x)^{2}}\)

Explanation

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

Using quotient rule, \(\frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^{2}}\), we have

\(\frac{dy}{dx} = \frac{(1-x)(1) - (1+x)(-1)}{(1-x)^{2}} = \frac{(1 - x +1 +x)}{(1-x)^{2}}\)

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

Correct Answer: D. 32/91

Explanation

Total balls = 8+4+2= 14

n(R) = 8, n (B) =4, n(G) = 2

Without replacement, it is p(RB) or p(BR)

= (8/14 x 4/13) + (4/14 + 8/13) = 16/91 + 16/91

= 32/91

Correct Answer: B. \(\frac{11}{6}\)

Explanation

\(h : x \to 2 - \frac{1}{2x - 3}\)

\(h(x) = \frac{2(2x - 3) - 1}{2x - 3} = \frac{4x - 7}{2x - 3}\)

Let x = h(y)

\(x = \frac{4y - 7}{2y - 3}\)

\(x(2y - 3) = 4y - 7 \implies 2xy - 4y = 3x - 7\)

\(y = \frac{3x - 7}{2x - 4}\)

\(h^{-1}(x) = \frac{3x - 7}{2x - 4}\)

\(\therefore h^{-1}(\frac{1}{2}) = \frac{3(\frac{1}{2}) - 7}{2(\frac{1}{2}) - 4}\)

= \(\frac{\frac{-11}{2}}{-3} = \frac{11}{6}\)

Correct Answer: C. \(\frac{1}{4}\)

Explanation

\(\log_{9} 3 = \log_{9} (9^{\frac{1}{2}}) = \frac{1}{2}\log_{9} 9 = \frac{1}{2}\)

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

\(x = \frac{1}{4}\)