May 2026

Simplify Surd Find the value \[ \sqrt{5 + 2\sqrt{6}} \] (a) \(\sqrt{3} – \sqrt{2}\) \quad (b) \(\sqrt{3} + \sqrt{2}\) \quad (c) \(\sqrt{5} + \sqrt{6}\) \quad (d) none of these Solution: \[ \sqrt{5 + 2\sqrt{6}} = \sqrt{a} + \sqrt{b} \] \[ (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} \] \[ a + b = […]

Simplify Surd Find the value \[ \sqrt{3 – 2\sqrt{2}} \] (a) \(\sqrt{2} – 1\) \quad (b) \(\sqrt{2} + 1\) \quad (c) \(\sqrt{3} – \sqrt{2}\) \quad (d) \(\sqrt{3} + \sqrt{2}\) Solution: \[ \sqrt{3 – 2\sqrt{2}} = \sqrt{a} – \sqrt{b} \] \[ (\sqrt{a} – \sqrt{b})^2 = a + b – 2\sqrt{ab} \] \[ a + b =

Find the Value Find the value If \[ x = \sqrt[3]{2 + \sqrt{3}} \] (a) 2 \quad (b) 4 \quad (c) 8 \quad (d) 9 Solution: \[ x^3 = 2 + \sqrt{3} \] \[ \frac{1}{x^3} = \frac{1}{2 + \sqrt{3}} \times \frac{2 – \sqrt{3}}{2 – \sqrt{3}} = 2 – \sqrt{3} \] \[ x^3 + \frac{1}{x^3} =

Find x and y Find the values of \(x\) and \(y\) If \[ \frac{5 – \sqrt{3}}{2 + \sqrt{3}} = x + y\sqrt{3} \] (a) \(x = 13, y = -7\) \quad (b) \(x = -13, y = 7\) \quad (c) \(x = -13, y = -7\) \quad (d) \(x = 13, y = 7\) Solution:

Simplify Expression Simplify If \[ \frac{1}{\sqrt{9} – \sqrt{8}} \text{ is equal to } \] (a) \(3 + 2\sqrt{2}\) \quad (b) \(\frac{1}{3 + 2\sqrt{2}}\) \quad (c) \(3 – 2\sqrt{2}\) \quad (d) \(\frac{3}{2} – \sqrt{2}\) Solution: \[ = \frac{1}{3 – 2\sqrt{2}} \times \frac{3 + 2\sqrt{2}}{3 + 2\sqrt{2}} \] \[ = \frac{3 + 2\sqrt{2}}{9 – 8} = 3

Find the Value Find the value If \[ x = \frac{\sqrt{3} – \sqrt{2}}{\sqrt{3} + \sqrt{2}}, \quad y = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} – \sqrt{2}}, \] then \[ x^2 + xy + y^2 = \ ? \] Solution: \[ x = \frac{(\sqrt{3} – \sqrt{2})^2}{3 – 2} = 5 – 2\sqrt{6} \] \[ y = \frac{(\sqrt{3} + \sqrt{2})^2}{3

Find the Value Find the value If \[ x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} – \sqrt{3}} \quad \text{and} \quad y = \frac{\sqrt{5} – \sqrt{3}}{\sqrt{5} + \sqrt{3}}, \] then \[ x + y + xy = \ ? \] Solution: \[ x = \frac{(\sqrt{5} + \sqrt{3})^2}{5 – 3} = \frac{5 + 3 + 2\sqrt{15}}{2} = 4 +

Find the Value Find the value \[ x + \sqrt{15} = 4 \] Solution: \[ x = 4 – \sqrt{15} \] \[ \frac{1}{x} = \frac{1}{4 – \sqrt{15}} \times \frac{4 + \sqrt{15}}{4 + \sqrt{15}} \] \[ = \frac{4 + \sqrt{15}}{16 – 15} = 4 + \sqrt{15} \] \[ x + \frac{1}{x} = (4 – \sqrt{15}) +

Find the Value Find the value \[ x = 7 + 4\sqrt{3}, \quad xy = 1 \] Solution: \[ y = \frac{1}{x} = 7 – 4\sqrt{3} \] \[ x + y = (7 + 4\sqrt{3}) + (7 – 4\sqrt{3}) = 14 \] \[ xy = 1 \] \[ x^2 + y^2 = (x + y)^2

Find the Value Find the value \[ x = \frac{2}{3 + \sqrt{7}} \] Solution: \[ x = \frac{2}{3 + \sqrt{7}} \times \frac{3 – \sqrt{7}}{3 – \sqrt{7}} = \frac{2(3 – \sqrt{7})}{9 – 7} \] \[ = 3 – \sqrt{7} \] \[ x – 3 = (3 – \sqrt{7}) – 3 = -\sqrt{7} \] \[ (x –