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List Match Sets
List - IList - IIIThe sides of a rectangle of greatestperimeter which is inscribed in a semicircle(P)108of radius 5 are λ1 and λ2 then 4(λ21+λ22) equalsIIIf the number of points of minima of f(x)=(Q)2x22x21 isλ then 18 λ isIIIIf f(x)=e2x2(a221)ex+8x+5(R)68is monotonically increasing for all x ϵ R,then the number of integers in range of 'a' areIVThe volume of a rectangular closed box is 72and the base sides are in the ratio 1 : 2.(S)54The least total surface area is 

Which of the following is the only CORRECT combination? 
  1. (III), (P)
  2. (IV), (P)
  3. (III), (Q)
  4. (II), (P)


Solution

The correct option is B (IV), (P)
I
P=2(5 sin θ+25 cos θ)
dPdQ=25(cos θ2 sin θ)=0
               tan θ=12
So λ1=5 sin θ=1
λ2=25 cos θ=4
λ21+λ22=17

II f(x)=x22x21

Points of minima are 2,0,2

III f(x)=2e2x2(a221)ex+80  
coefficient of the quadratic equation in ex is 1 so for e2x(a221)ex+40 to be true 
D 0  
(a221)2160
(a225)(a217)0
(a5)(a+5)(a17)(a+17)0

aϵ[5,17][17,5]
So, number of integral values of a=2 

IV
Let sides are x, 2x and h
x2h=36
S=2(2x2+hx+2hx)
S=4x2+6hx
S=4x2+216x
dsdx=8x216x2=0
x = 3 and h = 4
So S = 108.

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