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B
right angled
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C
isosceles
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D
has at least one angle 600
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Solution
The correct options are A equilateral D has at least one angle 600 ∑sin3A=0 ⇒sin3A+sin3B+sin3C=0 by expanding the summation ⇒2sin(32(A+B))cos(32(A−B))+sin3C=0 using compound angle formula. ⇒2sin(32(A+B))cos(32(A−B))+2sin3C2cos3C2=0 using multiple angle formula. ⇒2sin(3π2−3C2)cos(32(A−B))+2sin3C2cos3C2=0 since A+B+C=π ⇒−2cos3C2cos(32(A−B))+2sin3C2cos3C2=0 ⇒−cos3C2[cos(32(A−B))−cos3C2]=0 by taking common terms ⇒cos3C2[cos(32(A−B))+cos(32(A+B))]=0 since cos(π−θ)=−cosθ ⇒cos3C2cos3A2cos3B2=0 by compound angle formula ⇒cos3C2=cos900 ⇒3C2=900 ⇒C=600 ∴C=600 or A=600 or B=600