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\(\frac{6^{2n + 1} \times 9^n \times 4^{2n}}{18^n \times 2^n \times 12^{2n}}\) = \(\frac{(2 \times 3^{2n + 1} \times 3^{2n}) \times 2^{4n}}{(2 \times 9)^n \times 2^n \times (6 \times 2^{2n})}\) = \(\frac{(2^{2n + 1} \times 3^{2n + 1}) \times 3^{2n} \times 2^{4n}}{2^n \times 3^{2n} \times 2^n \times 2^{4n} \times 3^{2n}}\) = \(\frac{2^{2n} + 1 + 4^n \times 3^{2n} + 1 + 2^n}{2^{n + n + 4n} \times 3^{2n + 4n} \times 3^{2n + 2n}}\) = \(\frac{2^{6n + 1} \times 3^{4n + 1}}{2^{6n} x 3^{4n}}\) = 26n + 1 - 6n x 34n + 1 - 4n 2 x 3 = 6

\(\begin{array}{c|c} 2 & 25\\2 & 12 R 1\\2 & 6 R 0\\2 & 3 R 0\\2 & 1 R 1\\2&0 R 1\end{array}\) = (11001)

\(\frac{6.3 \times 10^5}{8.1 \times 10^3}\) \(\frac{7}{9} \times 10^{2}\) = \(0.77778 \times 10^{2}\) = \(77.778 \approxeq 77.80\)

\(4t^{2} + 7t - 1 = 0\) Using a = 4, b = 7, c = -1. \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) = \(\frac{-7 \pm \sqrt{7^{2} - 4(4)(-1)}}{2(4)}\) = \(\frac{-7 \pm \sqrt{49 + 16}}{8}\) = \(\frac{-7 \pm \sqrt{65}}{8}\) = \(\frac{-7 \pm 8.0623}{8}\) The positive answer = \(\frac{-7 + 8.0623}{8} = \frac{1.0623}{8}\) \(\approxeq 0.1328\) (4 decimal place)

Area = L x B L - B = 6 L = 6 + B area = 135 B(6 + B) = B\(^2\) + 6B - 135 = 0 (B - 9)(B + 15) = 0 B = 9 or - 15 L = 6 + 9 = 15

Surface area of a sphere = 4\(\pi\)r2 \(\frac{1}{16}\) of 4\(\pi\)r2 = \(\frac{\pi r^2}{4}\) P = \(\frac{\pi r^2}{4}\) r = 2\(\frac{P}{\pi}\)

\(\frac{\sin^{2} x}{1 + \cos x} + \frac{\sin^{2} x}{1 - \cos x}\) \(\frac{\sin^{2} x (1 - \cos x) + \sin^{2} x (1 + \cos x)}{1 - \cos^{2} x}\) = \(\frac{\sin^{2} x - \cos x \sin^{2} x + \sin^{2} x + \sin^{2} x \cos x}{\sin^{2} x}\) (Note: \(\sin^{2} x + \cos^{2} x = 1\)). = \(\frac{2 \sin^{2} x}{\sin^{2} x}\) = 2.

1st term a = 3, 5th term = 9, sum of n = 81 nth term = a + (n - 1)d, 5th term a + (5 - 1)d = 9 3 + 4d = 9 4d = 9 - 3 d = \(\frac{6}{4}\) = \(\frac{3}{2}\) = 6 Sn = \(\frac{n}{2}\)(6 + \(\frac{3}{4}\)n - \(\frac{3}{2}\)) 81 = \(\frac{12n + 3n^2}{4}\) - 3n = \(\frac{3n^2 + 9n}{4}\) 3n2 + 9n = 324 3n2 + 9n - 324 = 0 By almighty formula positive no. n = 9 = 3

T = \(\frac{4R_2}{R_1^{-1} + R_2^{-1} + 4R_3^{-1}}\) = \(\frac{4R_2}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{4}{R_3}}\) = \(\frac{4R_2}{\frac{R_2R_3 + R_1R_3 + 4R_1R_2}{R_1R_2R_3}}\) = \(\frac{4R_2 \times R_1 R_2 R_3}{R_2R_3 + R_1R_3 + 4R_1 R_2}\) = \(\frac{4R_1 \times R_2 R_3}{R_2R_3 + R_1R_3 + 4R_1R_2}\) T = \(\frac{4R_1 \times R_2 R_3}{R_2R_3 + R_1R_3 + 4R_1 R_2}\)

b = a + cp....(i) r = ab + \(\frac{1}{2}\)cp2.....(ii) expressing b2 in terms of a, c, r, we shall first eliminate p which should not appear in our answer from eqn, (i) b - a = cp = \(\frac{b - a}{c}\) sub. for p in eqn.(ii) r = ab + \(\frac{1}{2}\)c\(\frac{(b - a)^2}{\frac{ab + b^2 - 2ab + a^2}{2c}}\) 2cr = 2ab + b2 - 2ab + a2 b2 = 2cr - a2