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Probability Introduction :

Answers Of Sample Paper contain all the solutions of the question in sample paper .  It has answers with detail explanation. 

1Q)  If a, b , c are measures which form a triangle. For all n=2,3,4...etc, Prove  that \(\sqrt[n]{a}, \sqrt[n]{b}, \sqrt[n]{c}\)
also form a triangle .

Solution:

     If a, b , c  are the measures of triangle then sum of two sides is greater than the third side. 

   a+b > c 

  For all n=2,3,4,...etc, to prove \(\sqrt[n]{a}, \sqrt[n]{b}, \sqrt[n]{c}\) forms a triangle , it is enough to show that \(\sqrt[n]{a}+\sqrt[n]{b}> \sqrt[n]{c}\) .

Using Binomial Theorem we get 

\(( \sqrt[n]{a}+\sqrt[n]{b} )^{n}\) = \(a+\binom{n}{1}(\sqrt[n]{a})^{n-1}\sqrt[n]{b}+\binom{n}{2}(\sqrt[n]{a})^{n-2}(\sqrt[n]{b})^{2}+\cdots +b\).


.

\( (\sqrt[n]{a}+\sqrt[n]{b})^{n} > a+b > c= (\sqrt[n]{c})^{n}\)

\( (\sqrt[n]{a}+\sqrt[n]{b}) > \sqrt[n]{c} \).

Hence for all values of n=2, 3, 4 ....etc, \(\sqrt[n]{a}, \sqrt[n]{b}, \sqrt[n]{c}\) also form a triangle .

Answers Of Sample Paper

2Q) A square sheet of paper PQRS is so folded that the point Q fall on the mid point M of RS. Prove that the crease will divide QR in the ratio 5:3 .

Solution :    

      Let side of square = x .

   Q falls on the mid point M of RS. 

   Let crease be XY and  YR=a then QY=x-a

   We know that QY=YM as Q falls on M . 

    YR = a , YM=x-a  and triangle YMR is right angled triangle.

    Using Pythagoras Theorem we get

     \(MR^{2}+RY^{2}=YM^{2}\)

     \(\left ( \frac{x}{2} \right )^{2}\) +\( a^{2}\)=\( \left ( x-a \right )^{2} \)


     \(\frac{x^{2}}{4} + a^{2} =x^{2}-2ax + a^{2} \)

    \(2ax=\frac{3x^2}{4}\)

    \( a =\frac{3x}{8}\)

     \(\frac{x}{a}=\frac{8}{3}\)

      \(\frac{x-a}{a}=\frac{8-3}{3}\)

      \(\frac{x-a}{a} =\frac{5}{3}\)

      \(\frac{QY}{YR} =\frac{5}{3}\)

      QY:YR =5:3

  Hence it is proved .

Answers Of Sample Paper

 3Q) Prove that in any triangle ABC, if one angle is \(120^{\circ}\), the triangle formed by the feet of the angle bisectors is a right  angled.

Solution:


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