Match List-I with List-II and select the correct answer using the code given below the list. List - I List - II (P) The sides of a rectangle of greatest perimeter which is inscribed in a semicircle of

Question

Match List-I with List-II and select the correct answer using the code given below the list.

List - I	List - II
(P) The sides of a rectangle of greatest perimeter which is inscribed in a semicircle of radius $5$ are $λ_{1}$ and $λ_{2}$ then $4 (λ_{1}^{2} + λ_{2}^{2})$ equals	(1) 108
(Q) If the number of points of minima of $f (x) = \frac{x ^{2} - 2}{x ^{2} - 1}$ is $λ$ then $18 λ$ equals	(2) 11
(R) If $f (x) = e^{2 x} - 2 (a^{2} - 21) e^{x} + 8 x + 5$ is monotonically increasing for all $x \in R$ , then the number of integers in the range of ' a ' is	(3) 68
(S) The volume of a rectangular closed box is 72 and the base sides are in the ratio $1 : 2$ . The least total surface area is	(4) 54
	(5) 76

TestHub · Accepted Answer

(P) ​ P = 2 ( 5 ​ sin θ + 2 5 ​ cos θ ) ​ d θ d P ​ = 2 5 ​ ( cos θ − 2 sin θ ) = 0 ​ tan θ = 2 1 ​ ​ ​ ​ ​ ​ λ 1 ​ = 5 ​ sin θ = 1 λ 2 ​ = 2 5 ​ sin θ = 4 λ 1 2 ​ + λ 2 2 ​ = 17 f ( x ) = ​ x 2 − 1 x 2 − 2 ​ ​ (R) f ′ ( x ) = 2 e 2 x − 2 ( a 2 − 21 ) e x + 8 ≥ 0 2 e x + e x 8 ​ − 2 ( a 2 − 21 ) ⩾ 0 2 e x + e x 8 ​ ⩾ 8 ( A M ⩾ GM ) 8 − 2 a 2 + 42 ≥ 0 a ∈ [ − 5 , 5 ] (S) x , 2 x , h x 2 h = 36 S = 2 ( 2 x 2 + xh + 2 xh ) ⇒ S = 4 x 2 + 6 xh S = 4 x 2 + 6 x ( x 2 36 ​ ) ⇒ S = 4 x 2 + x 216 ​ dx dS ​ = 8 x − x 2 216 ​ x 3 = 8 216 ​ ⇒ x = 3 and h = 2 S = 108

Mathematics - Application of Derivative Question with Solution | TestHub

Options:

Doubts & Discussion

JEE Advanced (2028)