# Ex 6.5,19 - Chapter 6 Class 12 Application of Derivatives (Term 1)

Last updated at April 15, 2021 by

Last updated at April 15, 2021 by

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Ex 6.5, 19 Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.Let radius be r of the circle & let π₯ be the length & π¦ be the breadth of the rectangle Now, Ξ ABC is right angle triangle (AB)2 + (BC)2 = (AC)2 π₯^2+π¦^2 = (2π)^2 π₯^2+π¦^2= 4π2 π¦2 = 4π2 β π₯2 (As AC is diameter of circle) π¦= β(4π"2 β " π₯"2" ) We need to maximize Area of rectangle Let A be the area rectangle Area of rectangle = Length Γ Breadth A = xy A = π₯ β(4π^2βπ₯^2 ) Since A has square root It will be difficult to differentiate So, we take Z = A2 Let Z = A2 Z = π₯^2Γ (β(4π^2βπ₯^2 ))^2 Z = π₯^2Γ(4π^2βπ₯^2 ) Z = 4π^2 π₯^2βπ₯^4 Since A is positive, A is maximum if A2 is maximum So, we maximize Z = A2 Diff. Z w.r.t π₯ πZ/ππ₯=π(4π^2 π₯^2βπ₯^4 )/ππ₯ πZ/ππ₯=4π^2Γ2π₯β4π₯^3 πZ/ππ₯=8π^2 π₯β4π₯^3 Putting πZ/ππ₯ = 0 8π^2 π₯β4π₯^3 = 0 4π₯^3β8π^2 π₯ = 0 π₯^3β2π^2 π₯ = 0 π₯ (π₯^2β2π^2 ) = 0 Therefore, π₯ = 0 Which is not possible Finding (π ^π π)/(ππ^π ) πZ/ππ₯=8π^2 π₯β4π₯^3 Diff w.r.t π₯ (π^2 Z)/(ππ₯^2 ) = π/ππ₯ [8π^2 π₯β4π₯^3 ] (π^2 Z)/(ππ₯^2 ) = 8π^2β4Γ3π₯^2 (π^2 Z)/(ππ₯^2 ) = 8π^2β12π₯^2 Putting π^π=ππ^π (π^2 Z)/(ππ₯^2 ) = 8π^2β12Γ2π^2 (π^2 Z)/(ππ₯^2 ) = 8π^2β24π^2 (π^2 Z)/(ππ₯^2 ) = β16π^2 < 0 Hence, (π^2 Z)/(ππ₯^2 ) < 0 at π₯^2=2π^2 Thus area is maximum when π₯^2=2π^2 Now, finding y π¦ = β(4π^2 βπ₯^2 ) Putting π₯^2=2π^2 π¦ = β(4π^2 βγ2πγ^2 ) π¦ = β(γ2πγ^2 ) π¦ = β2 π Therefore π₯ = π¦ = β2 π Hence area is maximum when π = π β΄ The rectangle is a square.

Ex 6.5

Ex 6.5, 1 (i)
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Ex 6.5, 1 (ii)

Ex 6.5, 1 (iii) Important

Ex 6.5, 1 (iv)

Ex 6.5, 2 (i)

Ex 6.5, 2 (ii) Important

Ex 6.5, 2 (iii)

Ex 6.5, 2 (iv) Important

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Ex 6.5, 3 (ii)

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Ex 6.5, 3 (iv) Important

Ex 6.5, 3 (v)

Ex 6.5, 3 (vi)

Ex 6.5, 3 (vii) Important

Ex 6.5, 3 (viii)

Ex 6.5, 4 (i)

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Ex 6.5, 5 (iv)

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Ex 6.5,19 Important You are here

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Ex 6.5, 27 (MCQ)

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Ex 6.5,29 (MCQ)

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.