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Tn = a + (n-1)d 10th term = 32 32 = 3/2 + 9d 9d = 32 - 3/2 9d = 61/2 d = 61/18 : T4 = 3/2 + 3(61/18) T4 = 3/2 + 61/6 → 35/3

The common difference is 6. Just add 6 to get the next term 8, 14, 20 and 26

3 * 5 = 3 - 5 + (3 x 5)\(^2\) -2 + (15)\(^2\) -2 + 225 = 223

Volume of a rectangular pyramid = \(\frac{length * width * height}{3}\) or \(\frac{area * height}{3}\) = \(\frac{24 * 7.5}{3}\) → \(\frac{180}{3}\) Volume of the rectangular pyramid = 60cm\(^3\)

Definition of a circle

Condition for Parallelism means that their gradient value is same line 2x - y = 5 is rearranged As y = 2x - 5 from y = mx + c : line 2x - y = 5 has the gradient of 2 A parallel line with gradient of 2 and intercept of 5 → 2x - y = -5 Rearranged as y = 2x + 5

Given Data: x\(_1\) = -3, x\(_2\) = 2, y\(_1\) = 5, y\(_2\) = 10 coordinates of the mid-point of the line = (\(\frac{x_1 + x_2}{2}\) , \(\frac{y_1 + y_2}{2}\)) (\(\frac{-3 +2}{2}\) , \(\frac{5 + 10}{2}\)) = \(\frac{-1}{2}\)) , \(\frac{15}{2}\))

\(\frac{x^3 + 6x - 7}{x-1}\): When numerator is differentiated → 3x\(^2\) + 6 When denominator is differentiated → 1 : \(\frac{3x^2 + 6}{1}\) substitute x for 1 \(\frac{3 * 1^2 + 6}{1}\) = \(\frac{3 + 6}{1}\) = \(\frac{9}{1}\) = 9

The initial velocity (v) is obtained by differentiating the displacement (S) with respect to time, which gives: v = \(\frac{dS}{dt}\) = \(\frac{d}{dt}\)(\(3t^2\) + 5t - 10) = 6t + 5 The initial velocity is the velocity of the particle at time t = 0. Substituting t = 0 into the velocity equation: v(0) = 6(0) + 5 = 5 Therefore, the initial velocity of the particle is 5 m/s.

Recall chain rule: u = 2x +1; du = 2dx → dx = \(\frac{du}{2}\) u\(^3\) = ∫ u\(^3\) \(\frac{du}{2}\) → \(\frac{1}{2}\) ∫ u\(^3\) = \(\frac{1*u^4}{2*4}\) = \(\frac{u^4}{8}\) → \(\frac{{2x+1}^4}{8}\) + C

Firstly; solving for x 6 = \(\frac{2 + 5 + x+1 + x+2 + 7 + 9}{6}\) cross multiply to have: 6 * 6 = 2 + 5 + x+1 + x+2 + 7 + 9 36 = 2x + 26 36 - 26 = 2x 10 = 2x x = 5 Median = \(\frac{7+6}{2}\) → 6.5

When rearranged: 10, 10, 13, 14, 17, 18 median = \(\frac{13+14}{2}\) = \(\frac{27}{2}\) or 13.5

mean (4) = \(\frac{2x + 4x + 2x - 13 + 6x}{4}\) 16 = 14x - 13 16 + 13 = 14x x = 29/14 or 2.07

Zoology has 7 letters in total, with O repeated thrice \(\frac{7!}{3!}\) → \(\frac{7*6*5*4*3*2*1}{3*2*1}\) = 840ways

\(^{3^2 + 1}\)C\(_{3+5}\) \(^{9 + 1}\)C\(_{3+5}\) \(^{10}\)\(C_8\) = \(\frac{10!}{8! 2!}\) \(\frac{10 * 9 * 8!}{8! 2!}\) = \(\frac{10 * 9}{2}\) = 45