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T\(_n\) = 29, a = 2, d = 3 and n = ? T\(_n\) = a + (n - 1 ) d 29 = 2 + (n - 1) x 3 29 = 2 + 3n - 3 3n = 29 + 3 - 2 = 30 n = \(\frac{30}{3}\) = 10

According to the laws of logarithms: log\(_a^x\) + log\(_a^y\) = 3 = log\(_a^{xy}\) =3 and this in index form = xy = a\(^3\)

(2p - q)\(^2\) - (p + q)\(^2\) Using the difference of two squares (2p - q)\(^2\) - (p + q)\(^2\) = ( 2p - q + p + q)(2p - q - (p + q)) = 3p(2p - q - p - q) = 3p(p - 2q)

(3 - 4\(\sqrt{2}\))(1 + 3\(\sqrt{2}\)) = a + b\(\sqrt{2}\), 3 + 9\(\sqrt{2} - 4\sqrt{2} - (4\sqrt{2} \times 3\sqrt{2}) = a + b\sqrt{2}\) 3 + 5\(\sqrt{2} - (4 \times 3\sqrt{2} \times \sqrt{2}) = a + b\sqrt{2}\) 3 + 5\(\sqrt{2} - (12 \times 2) = a + b\sqrt{2}\) 3 + 5\(\sqrt{2} - 24 = a + b\sqrt{2}\) -21 + 5\(\sqrt{2}\) = a + b\(\sqrt{2}\) Comparing both sides, we have b = 5

Simple interest = Amount - Principal = $(2,031.25 - 1,250) = $ 781.25 S.I = \(\frac{\text{PRT}}{100}\) 781.25 = \(\frac{1250 \times 12.5 \times T}{100}\) T = \(\frac{781.25 \times 100}{1250 \times 12.5}\) = 5 years.

log\(_3^{2x - 1}\) = 5 in index form becomes 3\(^5\) = 2x - 1 2x - 1 = 243 2x = 243 + 1 = 244 x = \(\frac{244}{2}\) = 122

Using P\(_n\) = P\(_o\)(1 + \frac{r}{100})^n\) P\(_n\) = 3000(1 + \frac{3}{100})^3\) = 3000(1.03)\(^3\) = 3278

% error = \(\frac{\text{Error}}{\text{Actual value}}\) x 100% Error = #570 - #540 = #30 % error = \(\frac{30}{570}\) x 100% = \(\frac{100}{19}\)% = 5\(\frac{5}{19}\)%

Nos of sides of a regular polygon is given by: \(\frac{(n - 2) \times 180}{\text{n}}\) = One interior angle of the polygon. where n = number of sides of the polygon 168 = \(\frac{(n - 2) \times 180}{\text{n}}\) Cross multiplying, we have: 168n = 180n - 360 168n - 180n = - 360 - 12n = - 360 n = \(\frac{-360}{-12}\) = 30.

Solving 3x - 2y = -5 and x + 2y = 9 simultaneously using elimination method 3x - 2y = - 5 ----------- (i) x + 2y = 9 ----------- (ii) 4x = 4 x = 1 put x = 1 into eqn ---(ii) x + 2y = 9 becomes 1 + 2y = 9 2y = 9 - 1 = 8 y = \(\frac{8}{2}\) = 4 x = 1 and y = 4 Therefore, \(\frac{x - y}{x + y}\) = \(\frac{1 - 4}{1 + 4}\) = \(\frac{- 3}{5}\)

W ∝ M and W ∝ \(\frac{1}{P}\) If a variable W varies partly as M and partly inversely as P, the relationship can be expressed as: W = k\(_1\)M + \(\frac{k_2}{P}\)

The total surface area A of a closed cylindrical barrel (closed at one end), is given by: A = \pir\(^2\) + 2\pirh = \pir(r + 2h) BUT, r = 0.245m, h = 2.5m A = \(\frac{22}{7}\) x 0.245( 0.245 + 2 x 2.5) A = 0.77(5.245) = 4.038650m\(^2\) ≈ 4.0m\(^2\)

V = \pil(R\(^2\) - r\(^2\)) Divide both sides by \pil \(\frac{V}{\pil}\) = (R\(^2\) - r\(^2\)) R\(^2\) = \(\frac{V}{\pil}\) + r\(^2\) Take the square root of both sides R = \(\sqrt{\frac{V}{\pil} + r^2}\)

Given: m: Edna is respectful, n: Edna is brilliant From the implication m ⇒ n: a) Contra-positive: If Edna is not brilliant (¬n), then she is not respectful (¬m): ¬n ⇒ ¬m b) Direct: If Edna is respectful (m), then she is brilliant (n) c) Invalid: If Edna is brilliant (n), we cannot conclude she is respectful (m) Thus, the valid conclusion is: ¬n ⇒ ¬m.

Let the number added to the fraction \(\frac{1}{8}\) be y So, \(\frac{1 + y }{8 + y}\) = \(\frac{1}{2}\) cross multiply 8 + y = 2 + 2y 2y - y = 8 - 2 y = 6.