Correct Answer: B. 128

Explanation

\(S_{\infty} = \frac{a}{1 - r}\) (for an exponential series)

\(r = \frac{24}{96} = \frac{6}{24} = \frac{1}{4}\)

\(S_{\infty} = \frac{96}{1 - \frac{1}{4}} = \frac{96}{\frac{3}{4}}\)

= \(\frac{96 \times 4}{3} = 128\)

Correct Answer: B. 3.0 s

Explanation

\(F = ma \)

\(10 = 2.5a \implies a = 4 ms^{-2}\)

Since it is a retarding movement, then \(a = -4 ms^{-2}\).

\(v = u + at; v = 0 ms^{-1}, u = 12 ms^{-1}\)

\(0 = 12 + (-4t) \implies 0 = 12 - 4t\)

\(4t = 12 \implies t = 3 s\)

Correct Answer: D. \(-\frac{3}{2}\)

Explanation

\(\log_{0.25} 8 = x\)

\(8 = 0.25^{x}\)

\(2^{3} = (2^{-2})^{x} \implies 3 = -2x\)

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

Correct Answer: A. \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

Explanation

(a) Resultant of the forces acting on the plane = (6i + 2j) + (5i + 4j) + (4i - j) = 15i + 5j

\(F = \frac{m \Delta v}{t}\)

Resultant force = \(\frac{mv}{t} \)

\(15i + 5j = \frac{5v}{t}\)

\(v = \frac{(15i + 5j). t}{5} = 3t i + t j\)

(b) velocity after 4 seconds = \(3(4) i + 4 j = 12 i + 4 j\)

\(\therefore |12i + 4j| = \sqrt{12^{2} + 4^{2}} = \sqrt{144 + 16}\)

= \(\sqrt{160} \)

= \(4\sqrt{10} m/s\)

Correct Answer: A. 1

Explanation

\(\lim \limits_{x \to 1} \frac{1 - x}{x^{2} - 3x + 2}\)

\(\frac{1 - x}{x^{2} - 3x + 2} = \frac{-(x - 1)}{(x - 1)(x - 2)}\)

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

\(\lim \limits_{x \to 1} \frac{1 - x}{x^{2} - 3x + 2} = \lim \limits_{x \to 1} \frac{-1}{x - 2}\)

= \(\frac{-1}{1 - 2} = \frac{-1}{-1} = 1\)

Explanation

When f(x) is divided by (x - a), the remainder is f(a). When (x - a) is a factor, then f(a) = 0.

\(f(x) = 6x^{4} + mx^{3} - 13x^{2} + nx + 14\)

(a) When divided by (x + 2), \(f(-2) = 6(-2^{4}) + m(-2^{3}) - 13(-2^{2}) + n(-2) + 14 = 0\)

= \(96 - 8m - 52 - 2n + 14 = 0\)

\(58 = 8m + 2n .... (1)\)

When divided by (x - 1), \(f(1) = 6(1^{4}) + m(1^{3}) - 13(1^{2}) + n(1) + 14 = 0\)

\(6 + m - 13 + n + 14 = 0\)

\(m + n = -7 ... (2)\)

\(m = -7 - n \) (from equation 2)

\(\therefore 58 = 8(-7 - n) + 2n \)

\(-56 - 8n + 2n = 58\)

\(-6n = 58 + 56 = 114 \implies n = -19\)

\(m = -7 - (-19) = -7 + 19 = 12\)

\(\therefore \text{m and n = 12 and -19}\)

\(\therefore f(x) = 6x^{4} + 12x^{3} - 13x^{2} - 19x + 14\)

(b) When divided by (x + 1)

\(f(-1) = 6(-1^{4}) + 12(-1^{3}) - 13(-1^{2}) - 19(-1) + 14\)

= \(6 - 12 - 13 + 19 + 14 = 14\).

Correct Answer: D. 175

Explanation

From the five(5) men viable for the position of the chairman, one of them must be selected.

This implies; 4 men and 3 women are open for committee membership.

7 C\(_3\) = \(\frac{7!}{3!(7 - 3)!}\) =

\(\frac{7⋅6⋅5⋅4⋅3⋅2⋅1}{3.2.1 \times 4.3.2.1}\)

= 35 \(\times\) 5 = 175

Correct Answer: A. 32

Explanation

Reacting force - Weight = Net force

R = W + ma = mg + ma

480 = 10p + 5p

480 = 15p

p = 32 kg

Correct Answer: B. -4

Explanation

\((\sqrt{3} - 5\sqrt{2})(\sqrt{3} + \sqrt{2}) = 3 + \sqrt{6} - 5\sqrt{6} - 10\)

= \(-7 - 4\sqrt{6} = p + q\sqrt{6}\)

\(\therefore q = -4\)