Correct Answer: D. -1

Explanation

Midpoint between \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) = \((\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})\).

\((-k, k) = (\frac{2 + (1 + k)}{2}, \frac{-4 + (k + 1)}{2})\)

\(-k = \frac{k + 3}{2} \implies -2k = k + 3\)

\(-3k = 3 \implies k = -1\)

Correct Answer: B. \(\frac{1}{(x + 1)^{2}}\)

Explanation

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

Using quotient rule because the function is of the form \(\frac{u(x)}{v(x)}\)

\(\frac{\mathrm d y}{\mathrm d x} = \frac{v\frac{\mathrm d u}{\mathrm d x} - u\frac{\mathrm d v}{\mathrm d x}}{v^{2}}\)

\(\frac{\mathrm d y}{\mathrm d x} = \frac{(x + 1) . 1 - x . 1}{(x + 1)^{2}}\)

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

Correct Answer: A. -30

Explanation

\(\int (3x^{2} - 2x - 12) \mathrm {d} x = \frac{3x^{2 + 1}}{2 + 1} - \frac{2x^{1 + 1}}{2} - 12x\)

= \(x^{3} - x^{2} - 12x\)

\((x^{3} - x^{2} - 12x)|_{-2}^{3} = ((3^{3}) - (3^{2}) - 12(3)) - ((-2^{3}) - (-2^{2}) - 12(-2))\)

= \((27 - 9 - 36) - (-8 - 4 + 24) = -18 - 12 = -30\)

Correct Answer: A. 8.25

Explanation

Mean \(\bar{x}\) = \(\frac{3 + 7 + 6 + 2 + 8 + 5 + 9 + 1 + 4 + 10}{10} \)

= \(\frac{55}{10} = 5.50\)

Variance = \(\frac{\sum (x - \bar{x})^{2}}{n}\)

= \(\frac{82.5}{10}\)

= 8.25

Explanation

(a) Resultant of forces, \(R = -63j + (32.14i + 38.3j) + (14i - 24.25j)\)

= \(46.14i - 48.95j\)

Magnitude |R| = \(\sqrt{(46.14)^{2} + (-48.95)^{2}}\)

= \(\sqrt{2128.9 + 2396.1}\)

= \(\sqrt{4525}\)

= \(67.268 \approxeq 67.3N\) (to 1 d.p)

(b) \(a = \frac{F}{m}\)

= \(\frac{67.268}{5} = 13.454 ms^{-2}\)

\(\approxeq 13.5 ms^{-2}\)

Correct Answer: C. \(1, -\frac{1}{3}\)

Explanation

At turning point, \(\frac{\mathrm d y}{\mathrm d x} = 0\).

Given \(x^{3} - x^{2} - x + 6 \)

\(\frac{\mathrm d y}{\mathrm d x} = 3x^{2} - 2x - 1 = 0 \)

\(3x^{2} - 3x + x - 1 = 0 \implies (3x + 1)(x - 1) = 0\)

\(x = \frac{-1}{3}, 1\)

Explanation

\(3x^{2} - 6x + 10\)

\(3(x^{2} - 2x) + 10\)

\(3[x^{2} - 2x + (\frac{1}{2}(-2))^{2}] + 10 - 3\)

= \(3(x - 1)^{2} + 7\)

The minimum value is 7 and this occurs at x - 1 = 0 ==> x = 1.

Correct Answer: D. -4

Explanation

\(a \Delta b = a + b + 4\)

\(a \Delta e = a + e + 4 = a\)

\(e = a - a - 4 = -4\)

Correct Answer: C. \(3 ms^{-2}\)

Explanation

The two time differentiation of distance with respect to time will give the acceleration.

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

\(\frac{\mathrm d s}{\mathrm d t} = v = 3t - 3\)

\(\frac{\mathrm d v}{\mathrm d t} = a = 3\)

Correct Answer: B. 0

Explanation

\(\log_{10} (\frac{1}{3} + \frac{1}{4}) + 2\log_{10} 2 + \log_{10} (\frac{3}{7})\)

\(\frac{1}{3} + \frac{1}{4} = \frac{7}{12}\)

= \(\log_{10} (\frac{7}{12} \times 2^{2} \times \frac{3}{7})\)

= \(\log_{10} 1 = 0\)