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JEE MATHEMATICS MCQ: PROPERTIES OF TRIANGLES

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Analyzing +4 | -1
If a,  b,  ca,\;b,\;c be the sides of a Δ  ABC\Delta \;ABC and if roots of the equation a(b−c)x2+b(c−a)x+c(a−b)=0a\left( {b - c} \right){x^2} + b\left( {c - a} \right)x + c\left( {a - b} \right) = 0 are equal, then sin⁡2(A2),sin⁡2(B2),sin⁡2(C2){\sin ^2}\left( {\frac{A}{2}} \right),{\sin ^2}\left( {\frac{B}{2}} \right),{\sin ^2}\left( {\frac{C}{2}} \right) are in
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Correct option: C
HP
Option C (HP) is correct because if the roots of the given equation are equal, then a, b, and c are in HP, which leads to sin⁡2(A2),sin⁡2(B2),sin⁡2(C2){\sin ^2}\left( {\frac{A}{2}} \right),{\sin ^2}\left( {\frac{B}{2}} \right),{\sin ^2}\left( {\frac{C}{2}} \right) being in HP.

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If p1,p2,p3{p_1},{p_2},{p_3} are altitudes of a Δ  ABC\Delta \;ABC drawn from the vertices A,B,CA,B,C and Δ\Delta the area of the triangle, then p1−2+p2−2+p3−2p_1^{ - 2} + p_2^{ - 2} + p_3^{ - 2} is equal to If the sides of the triangles are 5k,  6k,  5k5k,\;6k,\;5k and radius of incircle is 6, the value of kk is equal to In a Δ ABC\Delta \,ABC, if a=(b−c)sec⁡θa = \left( {b - c} \right)\sec \theta, then 2 bcb−c  sin⁡  A2=\frac{{2\,\sqrt {bc} }}{{b - c}}\,\,\sin \,\,\frac{A}{2} = In Δ  ABC,b+ca\Delta \;ABC,\frac{{b + c}}{a} is equal to If Δ=a2−(b−c)2\Delta = {a^2} - {\left( {b - c} \right)^2}, where Δ\Delta is the area of Δ  ABC\Delta \;ABC, then tan⁡A\tan A is equal to If AA is the area and 2s2 s is the sum of the sides of a triangle, then
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