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S = \(\sqrt{\frac{2R + T}{2RT}}\) Squaring both sides, \(S^{2} = \frac{2R + T}{2RT}\) \(S^{2} (2RT) = 2R + T\) \(2S^{2} RT - 2R = T\) \(R = \frac{T}{2TS^{2} - 2}\) = \(\frac{T}{2(TS^2 - 1)}\)

\(f(x) = 2x^2 - 5x + 3\) \(f(x + 1) = 2(x + 1)^2 - 5(x + 1) + 3\) = \(2(x^2 + 2x + 1) - 5x - 5 + 3\) = \(2x^2 + 4x + 2 - 5x - 2\) = \(2x^2 - x\)

\(2^{(x^3 - x^2 - 2x)} = 1\) \(x^3 - x^2 - 2x = 0\) \(x(x^2 - x - 2) = 0\) \(x^2 - x - 2 = 0\) \((x + 1)(x - 2) = 0\) x = 2 is the positive answer.

\(\frac{324 - 4x^2}{2x + 18}\) = \(\frac{18^2 - (2x)^2}{2x + 18}\) = \(\frac{(18 - 2x)(18 + 2x)}{(2x + 18)}\) 18 - 2x = 2(9 - x) or -2(x - 9)

\(y^3 -4xy + xy^3 - 4y \) = \(y^3(1 + x) - 4y(1 + x) \((y^3 - 4y)(1 + x) = (y^3(1 + x) - 4y(1 + x))\) ∴ = \(y(1 + x)(y + 2)(y - 2)\)

4a2 - 12ab - C2 + 9b2 rearranges: (4a2 - 12ab + 9b2) - c2 (2a - 3b)(2a - 3b) - c2 = (2a - 3b)2 - c2 = (2a - 3b + c)(2a - 3b - c)

\(\frac{1}{2}\)(3y - 4x)2 = (8x2 + kxy + Ly2) \(\frac{1}{2}\)(9y2 - 24xy + 16x2) = 8x2 + kxy + Ly2 \(\frac{9}{2}\)y2 - 12xy) = kxy + Ly2 k = -12 ∴ L = \(\frac{9}{2}\)

4x2 + 5x + Q To make a complete square, the coefficient of x2 must be 1 = x2 + \(\frac{5x}{4}\) + \(\frac{Q}{4}\) Then (half the coefficient of x2) should be added i.e. x2 + \(\frac{5x}{4}\) + \(\frac{25}{64}\) ∴ \(\frac{Q}{4}\) = \(\frac{25}{64}\) Q = \(\frac{4 \times 25}{64}\) = \(\frac{25}{16}\)

\(\frac{1}{x + 1}\) - \(\frac{1}{x - 2}\) = \(\frac{x - 2 - x - 1}{(x + 1)(x - 2)}\) = \(\frac{-3}{(x + 1)(2 - x)}\)

\[\frac{x(x + 1)^{-\frac{1}{2}} - (x + 1)^{\frac{1}{2}}}{(x + 1)^{\frac{1}{2}}}\] The numerator can be rewritten as: \[x(x + 1)^{-\frac{1}{2}} - (x + 1)^{\frac{1}{2}} = \frac{x}{(x + 1)^{\frac{1}{2}}} - (x + 1)^{\frac{1}{2}}\] Combining the terms in the numerator over a common denominator: \[= \frac{x - (x + 1)}{(x + 1)^{\frac{1}{2}}}\] \[= \frac{x - x - 1}{(x + 1)^{\frac{1}{2}}} = \frac{-1}{(x + 1)^{\frac{1}{2}}}\] Substituting this back into the original expression gives: \[\frac{\frac{-1}{(x + 1)^{\frac{1}{2}}}}{(x + 1)^{\frac{1}{2}}}\] This simplifies to: \[\frac{-1}{(x + 1)^{\frac{1}{2}} \cdot (x + 1)^{\frac{1}{2}}} = \frac{-1}{(x + 1)}\] = \[{\frac{-1}{x + 1}}\]

Sum of nth term of a G.P = Sn = \(\frac{ar^n - 1}{r - 1}\) sum of the first two terms = \(\frac{ar^2 - 1}{r - 1}\) x = a(r + 1) sum of the last two terms = Sn - Sn - 2 = \(\frac{ar^n - 1}{r - 1}\) - \(\frac{(ar^{n - 1})}{r - 1}\) = \(\frac{a(r^n - 1 - r^{n - 2} + 1)}{r - 1}\) (r2 - 1) ∴ \(\frac{ar^{n - 2}(r + 1)(r - 1)}{1}\)= arn - 2(r + 1) = y = a(r + 1)r^n - 2 y = xrn - 2 = yrn - 2 \(\frac{y}{x}\) = r = (\(\frac{y}{x}\))\(\frac{1}{n - 2}\)

To find the volume of metal used for the cylindrical pipe, we calculate the volume of the outer cylinder and subtract the volume of the inner cylinder. Convert dimensions to consistent units - Outer diameter = \(7.2 \, \text{cm}\), so outer radius \( R = \frac{7.2}{2} = 3.6 \, \text{cm} \) - Inner diameter = \(2.8 \, \text{cm}\), so inner radius \( r = \frac{2.8}{2} = 1.4 \, \text{cm} \) - Height \( h = 1 \, \text{m} = 100 \, \text{cm} \) Calculate the volume of the outer cylinder The formula for the volume of a cylinder is: \(V = \pi R^2 h\) For the outer cylinder: \(V_{\text{outer}} = \pi (3.6^2)(100) = \pi (12.96)(100) = 1296\pi \, \text{cm}^3\) Calculate the volume of the inner cylinder For the inner cylinder: \(V_{\text{inner}} = \pi (1.4^2)(100) = \pi (1.96)(100) = 196\pi \, \text{cm}^3\) Calculate the volume of metal used The volume of metal is given by: \(V_{\text{metal}} = V_{\text{outer}} - V_{\text{inner}} = 1296\pi - 196\pi = 1100\pi \, \text{cm}^3\) Thus, the volume of metal used for the cylinder is: 1100\(\pi\)cm\(^3\)

xz\(^2\) = 6\(^2\) + 6\(^2\) 36 + 36 = 72 xz = \(\sqrt{72}\) 6\(\sqrt{2}\) = XT \(\frac{6\sqrt{2}}{2}\) = \(\frac{3}{\sqrt{2}}\) OT\(^2\) = 5\(^2\) + (3\(\sqrt{2}\))2 = 25 + 18 OT = 4\(\sqrt{3}\)

Total age of the whole family = 21 x 14 = 294 Total age without the grandfather = 20 x 12 = 240 Age of grandfather = 294 - 240 = 54 years

In \(\bigtriangleup\)PHK and \(\bigtriangleup\)PAR = \(\frac{PH}{PQ} = \frac{PK}{PR}\) \(\frac{4}{7} \times \frac{PR-KR}{PR}\) 4PR = 7(PR - KR) = 7PR - 7KR \(\frac{KR}{PR} = \frac{3}{7}\) KR:PR = 3:7