Match the following. List-I List-II (a) If twice the square on the diameter of a circle is equal to the sum of the squares on the sides of the inscribed triangle ABC, then sin 2 A + sin 2 B + sin 2 C

Question

Match the following. List-I List-II (a) If twice the square on the diameter of a circle is equal to the sum of the squares on the sides of the inscribed triangle ABC, then sin 2 A + sin 2 B + sin 2 C = (p) 27 (b) If in a triangle ABC, 4 sin A ​ = 5 sin B ​ = 6 sin C ​ , then the value of cos A + cos B + cos C is (q) 10 (c) If a , b , c , d are the sides of a quadrilateral, the minimum value of d 2 a 2 + b 2 + c 2 ​ is (r) 2 (d) If in any triangle ABC, ∏ ( sin A sin 2 A + sin A + 1 ​ ) is always greater than (s) 16 23 ​ (t) 3 1 ​ (One or more entries can match more than one option.)

TestHub · Accepted Answer

(a) 2 ( 2 R ) 2 = a 2 + b 2 + c 2 = 4 R 2 ( sin 2 A + sin 2 B + sin 2 C ) ⇒ sin 2 A + sin 2 B + sin 2 C = 2 (r) (b) a : b : c = 4 : 5 : 6 , so cos A = 60 25 + 36 − 16 ​ = 4 3 ​ , cos B = 48 16 + 36 − 25 ​ = 16 9 ​ , cos C = 40 16 + 25 − 36 ​ = 8 1 ​ ; sum = 16 23 ​ (s). (c) 3 ( a 2 + b 2 + c 2 ) ≥ ( a + b + c ) 2 > d 2 (since d < a + b + c ), so d 2 a 2 + b 2 + c 2 ​ > 3 1 ​ ; the greatest lower bound is 3 1 ​ ( t ) . (d) ∏ ( sin A + s i n A 1 ​ + 1 ) = ∏ { ( sin A ​ − s i n A ​ 1 ​ ) 2 + 3 } > 3 ⋅ 3 ⋅ 3 = 27 ; being always greater than 27, it is also always greater than 10 , 2 , 16 23 ​ and 3 1 ​ ( p , q , r , s , t ) .

List-I	List-II
(a) If twice the square on the diameter of a circle is equal to the sum of the squares on the sides of the inscribed triangle ABC, then $sin^{2} A + sin^{2} B + sin^{2} C =$	(p) 27
(b) If in a triangle ABC, $\frac{sin A}{4} = \frac{sin B}{5} = \frac{sin C}{6}$ , then the value of $cos A + cos B + cos C$ is	(q) 10
(c) If $a, b, c, d$ are the sides of a quadrilateral, the minimum value of $\frac{a ^{2} + b ^{2} + c ^{2}}{d ^{2}}$ is	(r) 2
(d) If in any triangle ABC, $\prod (\frac{sin ^{2} A + sin A + 1}{sin A})$ is always greater than	(s) $\frac{23}{16}$
	(t) $\frac{1}{3}$

Mathematics - Properties of Triangle Question with Solution | TestHub

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