Correct Answer: A. 3x + 1

Explanation

Using the remainder theorem, if (x - a) is a factor of f(x), then f(a) = 0.

Check the options and get the answer.

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

Explanation

Gradient = \(\frac{-4 - 4}{-3 - 3}\)

= \(\frac{-8}{6}\)

= \(\frac{4}{3}\)

Normal = -(\(\frac{1}{\frac{1}{4}}\))

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

Explanation

F\(_1\) = (10N, 090\(^o\)), F\(_2\) = (20N, 210\(^o\))

F\(_3\) = (4N, 33\(^o\))

F\(_3\) component x - axis

= - 4 cos 60\(^o\) = - 2N

F\(_2\) along y - axis

= 20 x cos 30\(^o\) = -1.7320

\(\frac{8N}{-1.732N}\)

Resultant force

= \(\sqrt{8^2 + (-1.732)^2}\)

= \(\sqrt{65.999}\)

= 8.185N

= 8.2N

Explanation

(a & c)

(b) \(\bar{x} = \frac{10 + 20 + 30 + 40 + 50 + 60 + 70 + 80}{8} = \frac{360}{8} = 45\)

\(\bar{y} = \frac{1.0 + 1.1 +1.2 + 1.4 + 1.6 + 1.8 + 2.0 + 2.3}{8} = \frac{12.4}{8} = 1.55\)

\(\therefore (\bar{x}, \bar{y}) = 45, 1.55\).

(d)(i) Equation of line of best fit :

\(\frac{y - 1.8}{x - 60} = \frac{1.8 - 1.55}{60 - 45}\)

\(\frac{y - 1.8}{x - 60} = \frac{0.25}{15} = 0.017\)

\(y - 1.8 = 0.017(x - 60) \implies y - 1.8 = 0.017x - 1.02\)

\(y = 0.017x - 1.02 + 1.8 = 0.017x + 0.78\)

(ii) When x = 75 days, \(y = 0.017(75) + 0.78\)

= \(2.055\)

Explanation

(a) \(p(six) = p = \frac{1}{6} ; p(\text{not six}) = q = \frac{5}{6}\)

\((p + q)^{6} = p^{6} + 6p^{5}q + 15p^{4}q^{2} + 20p^{3}q^{3} + 15p^{2}q^{4} + 6pq^{5} + q^{6}\)

(i) p(exactly 3 sixes) = \(20p^{3}q^{3}\)

= \(20(\frac{1}{6})^{3} (\frac{5}{6})^{3}\)

= \(20(\frac{1}{216})(\frac{125}{216})\)

= \(\frac{2500}{46656}\)

= 0.0536 \(\approxeq\) 0.054.

(ii) p(at most 3 sixes) = p(0 six) + p(1 six) + p(2 sixes) + p(3 sixes)

= \(q^{6} + 6pq^{5} + 15p^{2}q^{4} + 20p^{3}q^{3}\)

= \((\frac{5}{6})^{6} + 6(\frac{1}{6})(\frac{5}{6})^{5} + 15(\frac{1}{6})^{2} (\frac{5}{6})^{4} + 20(\frac{1}{6})^{3} (\frac{5}{6})^{3}\)

= \(\frac{46250}{46656}\)

= \(0.991\) (to 3 d.p)

(b) p(a defective screw) = 0.08 ; p(non- defective screw) = 1 - 0.08 = 0.92.

The binomial distribution = \((p + q)^{10}\).

(i) p(exactly 2 defective) = \(^{10}C_{2} (0.08)^{2} (0.92)^{8}\)

= \(45(0.0064)(0.5132)\)

= \(0.1478 \approxeq 0.148\)

(ii) p(not more than 2 defective) = p(0 defective or 1 defective or 2 defective)

= \(^{10}C_{0} p^{0} q^{10} + ^{10}C_{1} p^{1} q^{9} + ^{10}C_{2} p^{2} q^{8}\)

= \(1(0.92)^{10} + 10(0.08)(0.92)^{9} + 0.1478\)

= \(0.4344 + 0.3777 + 0.1478\)

= \(0.9599 \approxeq 0.960\) (3 d.p)

Correct Answer: C. 11.25m

Explanation

Maximum height (H) = \(\frac{u^{2}}{2g}\)

= \(\frac{15^{2}}{2 \times 10} = \frac{225}{20}\)

= \(11.25m\)

Correct Answer: B. 2.16

Explanation

\(\bar{x} = \frac{24}{6} = 4\)

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

= \(\sqrt{\frac{28}{6}} = \sqrt{4.667} = 2.16\)

Correct Answer: B. 4.0

Explanation

\(y = ax^{2} + bx + c\)

Gradient = \(\frac{\mathrm d y}{\mathrm d x} = 2ax + b = 8x + 4\)

Equating, we have

\(2a = 8 \implies a = 4\)

\(b = 4\)

For minimum value, gradient = 0

\(8x + 4 = 0 \implies x = -\frac{1}{2}\)

At \(x = -\frac{1}{2}, y = 1\)

\(1 = 4(-\frac{1}{2})^{2} + 4(-\frac{1}{2}) + c\)

\(1 = 1 - 2 + c\)

\(1 = c - 1 \implies c = 2\)

\(a, b, c = 4, 4, 2\).

Correct Answer: C. m + n - 2mn

Explanation

P(only one alive) = P(husband and not wife) + P(wife and not husband)

= m (1 - n) + n ( 1 - m)

= m - mn + n - mn

= m + n - 2mn