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(x – 3) ∝ y2 X-3 = Ky2 K = X-3 / y2 = 5-2/22 = 2/4 = 1/2 ∴X-3 = 1/2y2 X-3 = 1/2 (6)2 X-3 = 1/2 x 36/1 X-3 = 18 X = 21

p ∝ \(\frac{1}{q}\) p = \(\frac{k}{q}\) K = q\(^2\)p = 4\(^2\)(8) ∴ p = \(\frac{128}{\text{q}}\) 32 = \(\frac{128}{\text{q}}\) q\(^2\) = \(\frac{128}{32}\) q\(^2\) = 4 q = \sqrt4 = \(\pm\) 2

4X - 7 \(\leq\) 3X and 3X - 4 \(\leq\) 4X 4X - 3X \(\leq\) 7 and 3X - 4X \(\leq\) 4 X \(\leq\) 7 and -X \(\leq\) 4 = X \(\geq\) -4 Range -4 \(\leq\) x \(\leq\) 7

x\(^2\) - 5x + 6 = 0 (X-2)(X-3) = 0 X-2 = 0 implies X = 2 X-3 = 0 implies X = 3 ∴ x \le 2, x \ge 3

U5 = 24, n = 5 and U11 = 96, n = 11 Un = a + (n-1)d 24 = a + (5-1)d imply 24 = a+4d .....eqn1 96 = a + (11-1)d imply 96 = a+10d ...eqn2 eqn1 - eqn2 -72 = -6d d = 72/6 = 12 but 24 = a+4d 24 = a + 4(12) 24 = a + 48 a = 24-48 a = -24

X * Y = 2X - 3Y + 2 2*3 = 2(2) - 3(3) + 2 =4-9+2 = -3 But -3 does not belong to positive integer

mΔn = m+n+mn Let e be the identity element ∴mΔe = eΔm = m m+e+me = m e+me = m-m e+me = 0 e(1+m) = 0 e = 0 / (1+m) e = 0

Exterior angle = sum of two interior opposite angles y = 50^\circ (vert. opp ∠s are equal) X = y + 30 X = 50 + 30 X = 80^\circ

Exterior angle = 360 / n = 360 / 12 = 30o

a = a(base ∠s of Iss \Delta) ∴ a+a+74 = 180 2a + 74 = 180 2a = 180-74 2a = 106 a = 53 ∴∠OPQ = 53o

Area of the figure = Area of the rectangle + area of the semicircle \(=L \times h + \frac{1}{2}\pi r^2\) 5 \(\times 15 + \frac{1}{2} \times \frac{22}{7} \times \left(\frac{5}{2}\right)^2\) =75+\(\frac{(22\times 25)}{8\times7}\) =75 + \(\frac{550}{56}\) =84.8cm\(^2\)

If a length of a chord is equal to the length of a radius in the same circle, the triangle formed y the chord and the radii is an equilateral triangle ∴each angle = 60o

V = \pir2h 1m = 100cm 14cm = 1400cm \(∴V = \frac{22}{7} \times \frac{100x100x1400}{1000}\\ = 44, 000 liters\)

X = \(\frac{6+4}{2}\) = \(\frac{10}{2}\) = 5 Y = \(\frac{2+2}{2}\) = \(\frac{4}{2}\) =2 ∴ X, Y = (5,2)

3X + 2Y + 1 = 0 2Y = -3X - 1 \(\frac{-3}{2}X - \frac{1}{2}\) Gradient of 3X + 2Y +1 = 0 is -3/2 Gradient of a line perpendicular to 3X + 2Y + 1 = 0 \(=-1 \div \frac{3}{2}\) =-1 \(\times \frac{-2}{3} = \frac{2}{3}\)