Correct Answer: D. -(6x - 2)sin(3x² - 2x)

Explanation

(a) 5, 9, 13, ... is an A.P with the first term = 5 and common difference = 9 - 5 = 13 - 9 = 4.

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

\(T_{25} = 5 + (25 - 1) \times 4\)

= \(5 + 24 \times 4\)

= \( 5 + 96 = 101\)

(b) \(\frac{8}{9}, \frac{-4}{3}, 2, ...\) is a G.P with first term = \(\frac{8}{9}\)

\(r = \frac{T_{2}}{T_{1}} = -\frac{4}{3} \div \frac{8}{9} = \frac{-4}{3} \times \frac{9}{8} = -\frac{3}{2}\)

\(T_{n} = ar^{n - 1}\) (terms of a G.P)

\(T_{5} = (\frac{8}{9})(-\frac{3}{2})^{5 - 1}\)

= \(\frac{8}{9} \times \frac{81}{16} = \frac{9}{2}\)

(c) \(T_{n} = ar^{n - 1}\) (terms of a G.P)

\(T_{3} = ar^{2} = 48 ... (1)\)

\(T_{6} = ar^{5} = 14\frac{2}{9} ... (2)\)

\((2) \div (1) : r^{3} = \frac{128}{9} \div 48 = \frac{128}{9} \times \frac{1}{48}\)

\(r^{3} = \frac{8}{27}\)

\(r = \sqrt[3]{\frac{8}{27}} = \frac{2}{3}\)

From (1), \(ar^{2} = 48 \implies a \times (\frac{2}{3})^{2} = 48\)

\(4a = 48 \times 9 \implies a = \frac{48 \times 9}{4} = 108\)

\(T_{2} = 108 \times \frac{2}{3} = 72\)

\(T_{4} = 48 \times \frac{2}{3} = 32\)

\(\therefore\) The first four terms of the sequence are \(108, 72, 48, 32\).

Correct Answer: D. 157.5°

Correct Answer: D. 97.2N

Explanation

Horizontal component = 230 cos 64 = 97.2N

Correct Answer: A. 0.9180

Correct Answer: C. \(\frac{79}{156}\)

Explanation

Sin x = \(\frac{5}{13}\)

0^o \(\leq\) x \(\leq\) 90^o, (cos x - tan x)

AC² = AB² + BC²

13² = 5² + BC²

169 - 25 + BC²

169 - 25 = BC²

144 = BC²

Cos x = \(\frac{Adj}{Hyp}\) = \(\frac{12}{13}\)

BC = \(\sqrt{144}\)

BC = 12

tan x = \(\frac{opp}{adj} = \frac{5}{12}\)

BC = 12

cos x - tan x = \(\frac{12}{13} - \frac{5}{12}\)

\(\frac{144 - 65}{156} = \frac{79}{156}\)

Correct Answer: B. y = \(\frac{k}{x}\)

Correct Answer: D. r(35 + 2q)

Correct Answer: B. 9

Explanation

\(\frac{3^{n - 1} \times 27^{n + 1}}{81^{n}}\)

= \(\frac{3^{n - 1} \times 3^{3(n + 1)}}{3^{4n}}\)

= 3\(^{n - 1 + 3n + 3 - 4n}\)

= 3\(^{4n - 4n - 1 + 3}\)

= 3²

= 9

Correct Answer: C. -23

Explanation

\(\begin{bmatrix}2& -1&3\\4&1&2\\1&-3&1\\\end{bmatrix}\)

= 2[(1 x 1) - (2 x -3)] - (-1)[(4 x 1) - (2 x 1)] + 3[(4 x -3) - (1 x 1)]

= 2(1 - (-6)) + 1(4 - 2) + 3(-12 - 1)

= 2(1 + 6) + 1(2) + 3(-13)

= 2(7) + 1(2) + 3(-13)

= 14 + 2 - 39

∴ |D| = -23