  Question

# 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)2∣∣x2−2x2−1∣∣ isλ then 18 λ isIIIIf f(x)=e2x−2(a2−21)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? (III), (P)(IV), (P)(III), (Q)(II), (P)

Solution

## The correct option is B (IV), (P)I P=2(√5 sin θ+2√5 cos θ) dPdQ=2√5(cos θ−2 sin θ)=0                tan θ=12 So λ1=√5 sin θ=1 λ2=2√5 cos θ=4 λ21+λ22=17 II f(x)=∣∣x2−2x2−1∣∣ Points of minima are −√2,0,√2 III f′(x)=2e2x−2(a2−21)ex+8⩾0   coefficient of the quadratic equation in ex is 1 so for e2x−(a2−21)ex+4⩾0 to be true  D ≤0   (a2−21)2−16≤0 (a2−25)(a2−17)≤0 (a−5)(a+5)(a−√17)(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=8x−216x2=0 ⇒ x = 3 and h = 4 So S = 108.  Suggest corrections   