CBSE Class 12 Sample Paper for 2022 Boards (MCQ Based - for Term 1)

Question 34 - CBSE Class 12 Sample Paper for 2022 Boards (MCQ Based - for Term 1) - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at Sept. 4, 2021 by Teachoo

The area of a trapezium is defined by function ๐ and given by ๐(๐ฅ) = (10 + ๐ฅ) √(100 - x
^{
2
}
) , then the area when it is maximised is:

Question 34 The area of a trapezium is defined by function ๐ and given by ๐(๐ฅ) = (10 + ๐ฅ) โ("100 โ ๐ฅ2" ) , then the area when it is maximised is: (a) 75 cm2 (b) 7 โ3 cm2 (c) 75 โ3 cm2 (d) 5 cm2
๐(๐ฅ) = (๐+๐๐) (โ(๐๐๐โ๐๐))
Since A has a square root
It will be difficult to differentiate
Let Z = [๐(๐ฅ)]2
= (๐ฅ+10)^2 (100โ๐ฅ2)
Where f'(x) = 0, there Zโ(x) = 0
Differentiating Z
Z =(๐ฅ+10)^2 " " (100โ๐ฅ2)
Differentiating w.r.t. x
Zโ = ๐((๐ฅ + 10)^2 " " (100 โ ๐ฅ2))/๐๐
Zโ = [(๐ฅ + 10)^2 ]^โฒ (100 โ ๐ฅ^2 )+(๐ฅ + 10)^2 " " (100 โ ๐ฅ^2 )^โฒ
Zโ = 2(๐ฅ + 10)(100 โ ๐ฅ^2 )โ2๐ฅ(๐ฅ + 10)^2
Zโ = 2(๐ฅ + 10)[100 โ ๐ฅ^2โ๐ฅ(๐ฅ+10)]
Zโ = 2(๐ฅ + 10)[100 โ ๐ฅ^2โ๐ฅ^2โ10๐ฅ]
Zโ = 2(๐ฅ + 10)[โ2๐ฅ^2โ10๐ฅ+100]
Zโ = โ๐(๐ + ๐๐)[๐^๐+๐๐+๐๐]
Putting ๐ ๐/๐ ๐=๐
โ4(๐ฅ + 10)[๐ฅ^2+5๐ฅ+50] =0
(๐ฅ + 10)[๐ฅ^2+5๐ฅ+50] =0
(๐ฅ + 10) [๐ฅ2+10๐ฅโ5๐ฅโ50]=0
(๐ฅ + 10) [๐ฅ(๐ฅ+10)โ5(๐ฅ+10)]=0
(๐ + ๐๐)(๐โ๐)(๐+๐๐)=๐
So, ๐ฅ=๐ & ๐=โ๐๐
Since x is length, it cannot be negative
โด x = 5
Finding maximum area of trapezium
A = (๐ฅ+10) โ(100โ๐ฅ2)
= (5+10) โ(100โ(5)2)
= (15) โ(100โ25)
= 15 โ75
= 15 โ(25 ร 3)
= 15 ร โ๐๐ ร โ๐
= 15 ร 5 ร โ3
= 75โ๐ cm2
So, the correct answer is (C)

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CBSE Class 12 Sample Paper for 2022 Boards (MCQ Based - for Term 1)

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.