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Consider a right angled triangle \( \boldsymbol{A B C} \) right angled at \( \boldsymbol{C} \) with integer sides. List-I gives inradius. List-II gives the number of triangles.
Column-I
(A) Find the sum of the series
\( 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{12}+\ldots . . . \infty \), where the
terms are the reciprocals of the positive integers whose only prime factors are twos and threes
(B) The length of the sides of \( \triangle A B C \) are \( a, b \) and \( c \) and \( A \) (Q) 10 is the angle opposite to side \( a \). If \( b^{2}+c^{2}=a^{2}+54 \) and \( b c=\frac{a^{3}}{\cos A} \) then the value of \( \left(\frac{b^{2}+c^{2}}{9}\right) \), is
(C) The equations of perpendicular bisectors of two (R) 13 sides \( A B \) and \( A C \) of a triangle \( A B C \) are \( x+y+1=0 \) and \( x-y+1=0 \) respectively. If circumradius of \( \triangle A B C \) is 2 units and the locus of vertex \( A \) is \( x^{2}+y^{2}+g x+c=0 \), then \( \left(g^{2}+c^{2}\right) \), is equal to
(D) Number of solutions of the equation (S) \( \cos \theta \sin \theta+6(\cos \theta-\sin \theta)+6=0 \) in \( [0,30] \), is equal to
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