Ganit Prakash - Class-X - Quadratic Surd
Let us work out 9.1 Algebra
📘 Exercise 9.1 Solutions
Quadratic Surd(i) \( \sqrt{175} \) (ii) \( 2\sqrt{112} \) (iii) \( \sqrt{108} \) (iv) \( \sqrt{125} \) (v) \( 5\sqrt{119} \)
(i) \( \sqrt{175} \)
\[\begin{array}{l} = \sqrt{5 \times 5 \times 7} \\ = \sqrt{25 \times 7} \\ = 5\sqrt{7} \end{array}\](ii) \( 2\sqrt{112} \)
\[\begin{array}{l} = 2 \times \sqrt{16 \times 7} \\ = 2 \times 4\sqrt{7} \\ = 8\sqrt{7} \end{array}\](iii) \( \sqrt{108} \)
\[\begin{array}{l} = \sqrt{36 \times 3} \\ = \sqrt{6 \times 6 \times 3} \\ = 6\sqrt{3} \end{array}\](iv) \( \sqrt{125} \)
\[\begin{array}{l} = \sqrt{25 \times 5} \\ = 5\sqrt{5} \end{array}\](v) \( 5\sqrt{119} \)
\[\begin{array}{l} = 5 \times \sqrt{7 \times 17} \\ = 5\sqrt{119} \end{array}\](Since 119 has no perfect square factor, it remains in the same form.)
L.H.S:
\[\begin{array}{l} = \sqrt{108} - \sqrt{75} \\ = \sqrt{36 \times 3} - \sqrt{25 \times 3} \\ = 6\sqrt{3} - 5\sqrt{3} \\ = (6 - 5)\sqrt{3} \\ = 1\sqrt{3} \\ = \sqrt{3} \end{array}\]R.H.S: \( \sqrt{3} \)
\( \therefore \) L.H.S. = R.H.S. [Proved]
L.H.S:
\[\begin{array}{l} = \sqrt{98} + \sqrt{8} - 2\sqrt{32} \\ = \sqrt{49 \times 2} + \sqrt{4 \times 2} - 2\sqrt{16 \times 2} \\ = 7\sqrt{2} + 2\sqrt{2} - 2(4\sqrt{2}) \\ = 7\sqrt{2} + 2\sqrt{2} - 8\sqrt{2} \\ = (7 + 2 - 8)\sqrt{2} \\ = 1\sqrt{2} \\ = \sqrt{2} \end{array}\]R.H.S: \( \sqrt{2} \)
\( \therefore \) L.H.S. = R.H.S. [Proved]
L.H.S:
\[\begin{array}{l} = 3\sqrt{48} - 4\sqrt{75} + \sqrt{192} \\ = 3\sqrt{16 \times 3} - 4\sqrt{25 \times 3} + \sqrt{64 \times 3} \\ = 3(4\sqrt{3}) - 4(5\sqrt{3}) + 8\sqrt{3} \\ = 12\sqrt{3} - 20\sqrt{3} + 8\sqrt{3} \\ = (12 - 20 + 8)\sqrt{3} \\ = 0\sqrt{3} \\ = 0 \end{array}\]R.H.S: \( 0 \)
\( \therefore \) L.H.S. = R.H.S. [Proved]
\( \sqrt{12} + \sqrt{18} + \sqrt{27} - \sqrt{32} \)
Expression:
\[\begin{array}{l} = \sqrt{12} + \sqrt{18} + \sqrt{27} - \sqrt{32} \\ = \sqrt{4 \times 3} + \sqrt{9 \times 2} + \sqrt{9 \times 3} - \sqrt{16 \times 2} \\ = 2\sqrt{3} + 3\sqrt{2} + 3\sqrt{3} - 4\sqrt{2} \\ = (2\sqrt{3} + 3\sqrt{3}) + (3\sqrt{2} - 4\sqrt{2}) \\ = 5\sqrt{3} - \sqrt{2} \end{array}\]Answer: \( 5\sqrt{3} - \sqrt{2} \)
(a) Let us write what should be added with \( \sqrt{5} + \sqrt{3} \) to get the sum \( 2\sqrt{5} \).
(b) Let us write what should be subtracted from \( 7 - \sqrt{3} \) to get \( 3 + \sqrt{3} \).
(c) Let us write the sum of \( 2 + \sqrt{3} \), \( \sqrt{3} + \sqrt{5} \) and \( 2 + \sqrt{7} \).
(d) Let us subtract \( (-5 + 3\sqrt{11}) \) from \( (10 - \sqrt{11}) \) and let us write the value of subtraction.
(e) Let us subtract \( (5 + \sqrt{2} + \sqrt{7}) \) from the sum of \( (-5 + \sqrt{7}) \) and \( (\sqrt{7} + \sqrt{2}) \) and find value of subtraction.
(f) I write two quadratic surds whose sum is a rational number.
(a) Let the required number be \( x \).
\[\begin{array}{l} x + (\sqrt{5} + \sqrt{3}) = 2\sqrt{5} \\ \Rightarrow x = 2\sqrt{5} - (\sqrt{5} + \sqrt{3}) \\ \Rightarrow x = 2\sqrt{5} - \sqrt{5} - \sqrt{3} \\ \Rightarrow x = \sqrt{5} - \sqrt{3} \end{array}\]Answer: \( \sqrt{5} - \sqrt{3} \)
(b) Let the required number be \( x \).
\[\begin{array}{l} (7 - \sqrt{3}) - x = 3 + \sqrt{3} \\ \Rightarrow x = (7 - \sqrt{3}) - (3 + \sqrt{3}) \\ \Rightarrow x = 7 - \sqrt{3} - 3 - \sqrt{3} \\ \Rightarrow x = 4 - 2\sqrt{3} \end{array}\]Answer: \( 4 - 2\sqrt{3} \)
(c) Required sum:
\[\begin{array}{l} = (2 + \sqrt{3}) + (\sqrt{3} + \sqrt{5}) + (2 + \sqrt{7}) \\ = 2 + 2 + \sqrt{3} + \sqrt{3} + \sqrt{5} + \sqrt{7} \\ = 4 + 2\sqrt{3} + \sqrt{5} + \sqrt{7} \end{array}\]Answer: \( 4 + 2\sqrt{3} + \sqrt{5} + \sqrt{7} \)
(d) Value of subtraction:
\[\begin{array}{l} = (10 - \sqrt{11}) - (-5 + 3\sqrt{11}) \\ = 10 - \sqrt{11} + 5 - 3\sqrt{11} \\ = 15 - 4\sqrt{11} \end{array}\]Answer: \( 15 - 4\sqrt{11} \)
(e) First, finding the sum of \( (-5 + \sqrt{7}) \) and \( (\sqrt{7} + \sqrt{2}) \):
\[\begin{array}{l} \text{Sum } = (-5 + \sqrt{7}) + (\sqrt{7} + \sqrt{2}) \\ = -5 + 2\sqrt{7} + \sqrt{2} \end{array}\]Now, subtracting \( (5 + \sqrt{2} + \sqrt{7}) \) from the sum:
\[\begin{array}{l} = (-5 + 2\sqrt{7} + \sqrt{2}) - (5 + \sqrt{2} + \sqrt{7}) \\ = -5 + 2\sqrt{7} + \sqrt{2} - 5 - \sqrt{2} - \sqrt{7} \\ = (-5 - 5) + (\sqrt{2} - \sqrt{2}) + (2\sqrt{7} - \sqrt{7}) \\ = -10 + 0 + \sqrt{7} \\ = -10 + \sqrt{7} \end{array}\]Answer: \( -10 + \sqrt{7} \)
(f) Two quadratic surds whose sum is a rational number:
Let us consider two conjugate quadratic surds, for example: \( (2 + \sqrt{3}) \) and \( (2 - \sqrt{3}) \).
Their sum = \( 2 + \sqrt{3} + 2 - \sqrt{3} = 4 \), which is a rational number.
Answer: \( (2 + \sqrt{3}) \) and \( (2 - \sqrt{3}) \) (Other examples are also possible).
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