Correct Answer: C. 3

Explanation

\(\sin \theta \leq 1\)

i.e Maximum value of \(\sin \theta \forall \theta = 1\).

Therefore, \(2 + \sin \theta \leq 2 + 1 = 3\)

Correct Answer: C. \(\frac{1}{13}(-5i + 12j)\)

Explanation

The unit vector \(\hat{n} = \frac{\overrightarrow{r}}{|r|}\)

\(\hat{n} = \frac{-5i + 12j}{\sqrt{(-5)^{2} + (12)^{2}} \)

= \(\frac{-5i + 12j}{13} \)

Correct Answer: D. -4

Explanation

√24 +√96 -√600 = y√6

convert all to basic surds

√24 = 2√6

√96 = 4√6

-√600 = -10√6

2√6 + 4√6 - 10√6 = -4√6

value of y = -4

Correct Answer: B. N200,000.00

Explanation

N1.00 = $0.075

N X

Explanation

(a)

(b)

(c) (i) Median (\(Q_{2}\)) = 49.5

(ii) Semi- Interquartile range = \(\frac{1}{2} (Q_{3} - Q_{1})\)

\(Q_{1} = 38.5\); \(Q_{3} = 59.5\).

Semi- Interquartile range = \(\frac{1}{2} (59.5 - 38.5)\)

= \(\frac{1}{2} (21)\)

= 10.5.

(iii) Pr(\(\geq 60kg = \frac{25}{100} = \frac{1}{4}\)

Correct Answer: C. 90°

Explanation

(a) \(f(x) = \int (4x - x^{2}) \mathrm {d} x\)

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

\(f(3) = 2(3^{2}) - \frac{3^{3}}{3} + c\)

\(21 = 18 - 9 + c\)

\(c = 21 - 9 = 12\)

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

(b) \(T_{n} = a + (n - 1) d\) (terms of an A.P)

\(T_{2} = a + d ; T_{4} = a + 3d ; T_{8} = a + 7d\)

These are consecutive terms of a G.P

\(i.e. \frac{T_{4}}{T_{2}} = \frac{T_{8}}{T_{4}}\)

\((T_{4})^{2} = T_{2} \times T_{8}\)

\((a + 3d)^{2} = (a + d)(a + 7d)\)

\(a^{2} + 6ad + 9d^{2} = a^{2} + 8ad + 7d^{2}\)

\(6ad + 9d^{2} = 8ad + 7d^{2} \implies 2ad = 2d^{2}\)

\(a = d\)

\(T_{3} + T_{5} = 20\)

\(a + 2d + a + 4d = 20\)

\(a + 2a + a + 4a = 20 \implies 8a = 20\)

\(a = d = \frac{5}{2}\)

(i) First four terms of A.P = \(\frac{5}{2}, 5, \frac{15}{2}, 10\)

(ii) \(S_{n} = \frac{n}{2} (2a + (n - 1)d)\)

\(S_{10} = \frac{10}{2} (2(\frac{5}{2}) + (10 - 1)(\frac{5}{2}))\)

= \(5(5 + \frac{45}{2})\)

= \(\frac{275}{2} = 137.5\)

Correct Answer: A. \(\begin{pmatrix} 3 & -3 \\ 8 & 2 \end{pmatrix}\)

Correct Answer: C. 60°

Explanation

\(2\sin^{2}\theta = 1 + \cos \theta \implies 2(1 - \cos^{2}\theta) = 1 + \cos \theta\)

\(2 - 2\cos^{2}\theta = 1 + \cos \theta\)

\(2 - 2\cos^{2}\theta - 1 - \cos \theta = 0\)

\(2\cos^{2}\theta + \cos \theta - 1 = 0\)

\(2\cos^{2}\theta + 2\cos\theta - \cos \theta - 1 = 0 \implies 2\cos \theta(\cos \theta + 1) - 1(\cos \theta + 1) = 0\)

\((2\cos \theta - 1)(\cos \theta + 1) = 0 \implies \cos \theta = \frac{1}{2} \)

\(\theta = \cos^{-1} \frac{1}{2} = 60°\)

Correct Answer: B. 2

Explanation

\(\frac{1}{5^{-y}} = 25(5^{4-2y})\)

\(\implies 5^{y} = (5^{2})(5^{4-2y})\)

\(5^{y} = 5^{2+4-2y}\)

Comparing bases, we have

\(y = 6 - 2y\)

\(3y = 6 \implies y = 2\)