Q1 if a = 1/3-root2 find 1) a square- 1 / a square 2) a square + 1 / a - Maths -

Question

Q1 if a = 1/3-root2 find 1) a square- 1 / a square 2) a square +
1 / a square
Q2 if x square + 1/ x square = 83 then find the value of x cube
- 1/ x cube
Q3 if xx square is a factor of ax quadra + bx cube + cx square +
dx + e then prove that a+c+e = b+d = 0
Q4 if a line perpendicular to any one of the two ll lines show
that it is perpendicular to the other
Q5 find the value of k if 6/3root2-2root3 = 3roo2-kroot3
Q6 Fill in the blanks
i) In Euclid's geometary there are --------- undified terms
namely-----------
ii) a line seperates a plane into --------- parts namely
-------------
Q7 if x + 1 / x = 1 / 2 - root3 then x-1 / x =??

Ankush Jain · Accepted Answer

1)

$a = \frac{1}{3 - \sqrt{2}} \Rightarrow \frac{1}{a} = 3 - \sqrt{2} Also, a = \frac{1}{3 - \sqrt{2}} \times \frac{3 + \sqrt{2}}{3 + \sqrt{2}} = \frac{3 + \sqrt{2}}{9 - 2} = \frac{3 + \sqrt{2}}{7} ∴ a^{2} - \frac{1}{a^{2}} = (\frac{3 + \sqrt{2}}{7})^{2} - (3 - \sqrt{2})^{2} = \frac{9 + 2 + 6 \sqrt{2}}{49} - (9 + 2 - 6 \sqrt{2}) = 11 + 6 \sqrt{2} - 539 + 294 \sqrt{2} = - 528 + 300 \sqrt{2} = 300 \sqrt{2} - 528$

5) $\frac{6}{3 \sqrt{2} - 2 \sqrt{3}} = 3 \sqrt{2} - k \sqrt{3}$

$\Rightarrow 6 = (3 \sqrt{2})^{2} - 3 k \sqrt{6} - 6 \sqrt{6} + 6 k \Rightarrow 36 - 3 k \sqrt{6} - 6 \sqrt{6} + 6 k = 6 \Rightarrow k (6 - 3 \sqrt{6}) = - 30 + 6 \sqrt{6} \Rightarrow k = \frac{- 10 + 2 \sqrt{6}}{2 - \sqrt{6}} \Rightarrow k = \frac{- 10 + 2 \sqrt{6}}{(2 - \sqrt{6})} \times \frac{(2 + \sqrt{6})}{(2 + \sqrt{6})} \Rightarrow k = \frac{- 20 + 4 \sqrt{6} - 10 \sqrt{6} + 12}{4 - 6} \Rightarrow k = \frac{- 8 - 6 \sqrt{6}}{- 2} = 4 + 3 \sqrt{6}$

7) $x + \frac{1}{x} = \frac{1}{2 - \sqrt{3}}$

$\Rightarrow x + \frac{1}{x} = \frac{2 + \sqrt{3}}{(2 - \sqrt{3}) (2 + \sqrt{3})} \Rightarrow x + \frac{1}{x} = \frac{2 + \sqrt{3}}{4 - 3} \Rightarrow x + \frac{1}{x} = 2 + \sqrt{3} \Rightarrow (x + \frac{1}{x})^{2} = (2 + \sqrt{3})^{2} \Rightarrow x^{2} + \frac{1}{x^{2}} + 2 = 4 + 3 + 4 \sqrt{3} \Rightarrow x^{2} + \frac{1}{x^{2}} - 2 = 3 + 4 \sqrt{3} \Rightarrow (x - \frac{1}{x})^{2} = 3 + 4 \sqrt{3} \Rightarrow x - \frac{1}{x} = \sqrt{3 + 4 \sqrt{3}}$

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