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-8, m, n, 19 = m + 8 = 19 - n m + n = 11 i.e. 1, 10

A regular has all sides and all angles equal. If each interior angle is 140^\circ each exterior angle must be 180^\circ - 140^\circ = 40^\circ The number of sides must be \(\frac{360^o}{40^o}\) = 9 sides hence 2k + 1 = 9 2k = 9 - 1 8 = 2k k = \(\frac{8}{2}\) = 4

PR\(^2\) + QS\(^2\) = kPQ\(^2\) SQ\(^2\) = SR\(^2\) + RQ\(^2\) PR\(^2\) + SQ\(^2\) = PQ\(^2\) + SR\(^2\) + 2RQ\(^2\) = 2PQ\(^2\) + 2RQ\(^2\) = 4PQ\(^2\) ∴ K = 4

Y \(\to\) 30^\circ X \(\to\) 120^\circ Z \(\to\) 30^\circ \(\frac{3}{\text{sin 30}}\) = \(\frac{YZ}{\text{sin 120}}\) YZ = \(\frac{3\times \sin 120}{\sin 30}\) \(3 \times \frac{\sqrt{3}}{2} \times 2 = 3\sqrt{3}\)

13 = 52 1 = \(\frac{52}{13}\) = 4 5 + 12 + 13 = 30 Total perimeter = 30 x 4 = 120cm

x = 10 cot35o - 10 cot55o = 10(cot35o - cot55o)

4 sin2x - 3 = 0 sin2x = \(\frac{3}{4}\) Sin x = \(\pm\) \(\sqrt{\frac{3}{4}}\) = \(\pm\) \(\sqrt{\frac{3}{4}}\) = 60o

Total floor area = 720 x 420 Area of one tile = 30 x 30 No. of tile required = \(\frac{720 \times 420}{30 \times 30}\) = 24 x 14 = 336 tiles

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\(\sqrt{3}\)

Rice = 75g, Margarine = 40g, Meat = 105g, Bread = 20g, Total = 240 Angle of sector represented by meat = \(\frac{105}{240}\) x \(\frac{360^\circ}{1}\) = 157.5^\circ

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

Arrange the numbers in order, 1.6, 2.1, 2.1| 2.4, 2.6| 2.6, 2.6, 3.7 n = median = \(\frac{2.4 + 2.6}{2}\) = 2.5 m = mode = 2.6 ∴ (n, m) = (2.5, 2.6)

The gradient of any curve is equal to zero at the turning points. i.e. maximum or minimum points. The points in the above curve are b, e, g, j, m

QMN = 60º MRQ = 60º(angle in the alternate segment are equal) MQN = 80º(angle sum of a \(\bigtriangleup\) = 180º) 60 + x = 80º(exterior angle = sum of opposite interior angles) x = 80º - 60º = 20º RMQ = 20º