Question

Match the items of List-I with those of List-II (Here \( \Delta \) denotes the area of \( \triangle ABC \)).

Then the correct match is

• A. A-VI; B-I; C-II; D-III
• B. A-II; B-I; C-V; D-III
• C. A-II; B-VI; C-V; D-I
• D. A-VI; B-II; C-I; D-IV

Hint:

For matching-type problems in trigonometry and geometry, express each term in known formulas and find corresponding values systematically.

Answer 1

The Correct Option is B

A. ΣcotA

This represents the sum of the cotangents of the angles of a triangle. It is equivalent to (a² + b² + c²) / 4Δ where a, b, and c are the side lengths and Δ is the triangle's area. 
Therefore, A matches with II.

B. Σcot(A/2)

This represents the sum of the cotangents of the half-angles. This equals (a + b + c)² / 4Δ. 
Thus, B matches with I.

C. tanA : tanB : tanC = 1:2:3

The ratio of sines of the angles is proportional to the sides opposite to the respective angles. So, sin A : sin B : sin C = √5 : 2√2 : 3. 
C matches with V.

D. cot(A/2) : cot(B/2) : cot(C/2) = 3:7:9

The ratio of cotangents of half-angles is proportional to (s-a):(s-b):(s-c), where 's' is the semi-perimeter. Since a+b+c = 2s, the sides a:b:c are proportional to 12:5:13. 
Therefore, D matches with III.