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Q) Two tangents pq and qr are drawn from an external point to a circle with centre 'o'.balu said the quadrilateral ORP is a circle do you agree ? Give reason?

sol) what is cyclic quadrilateral?
         A) The sum of either pairs( /_A  + /_C = 180 deg    or /_B + /_D 180 deg) of opposite angles of cyclic quadrilateral is 180 degrees

Given:- PQ and PR are  two tangents drawn at points "Q" and "R" from an external point "P" with center "O"

Required to prove(RTP) :- QORP is a cyclic Quadrilateral

Proof :- OR _|_ PR and OQ _|_ PQ [ tangent at a point on the circle is perpendicular to the radius through point of contact]

∠ORP = 90°

∠OQP = 90°

∠ORP + ∠OQP = 180°

Hence balu said correct QOPR is a cyclic quadrilateral. 

As the sum of the opposite pairs of angle is 180°

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Problem:-

Sol) cos 15 can be written as

= cos (45-30)   [cos (a-b) form]

= cos45 cos30+sin45 sin30   

= _/2/2 . _/3/2 +  _/2/2 . 1/2

= _/6 + _/2
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         4


Explanation :- you may be having question why we took cos( 45-30)  why not  cos (35-20)?

Because if you see trigonometric ratios table you have value only for  {0,30,45,60, and 90}.So we

Add and subtract numbers depending upon the question Asked.As in here for cos 15 we cant

 take for cos 15=/= cos(60-45) == cos (25) right?


2) point :- You may also have doubt when we have cos ( 45) value is  1/ root 2  how we got _/2/2?  in this we multiply and divide numerator and denominator by _/2   

_/2 / _/2 * 1 / _/2 = _/2/2


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Problem :-   

Sol Least Common Multiple (L . C .M) of 15,25,75

in order to check whether the LCM is correct or not just divide it with the given numbers( 15,25,75)  in this case 75/15=5 (divisble)  , 75/25 =3 ,  75/75=1