Solved by Mahatab Hossain Rezon

a. LCM of 5,8 & 20 = 40
Total Capacity = 40 units
Efficiency, P1 = 40/5 = 8
P2 = 40/8 = 5
Leakage = 40/20 = 2
Req. Time be T hours
ATQ,
40 = T(8+5-2)
=> T = 40/11 i.e. 3(7/11) hours [Ans.]

b. Outer Radius be R m
ATQ, 2πR-2πr = 88
=> (R-r) = 88/2π = 44/π = 44×(7/22) = 14 m [Ans.]

c. Let, Passenger be = x
ATQ,
5500/x : 5500/(x-5) = 1:2
=> 2x-10 = x
=> x = 10
Availed passengers = 10-5 = 5 [Ans.]

d. (a – 1/a) = 3
=> (a – 1/a)² = 3²
=> a² + 1/a² – 2 a 1/a = 9
=> a² + 1/a² = 9+2
=> (a² + 1/a²)² = 11²
=> a⁴ + 1/a⁴ + 2 a²(1/a²) = 121
=> a⁴ + 1/a⁴ = 121-2 = 119 [proved]

e. Let, daughter present age be x years
Mother present age = 60-x
ATQ,
(60-x-5)/(x-5) = 4/1
=> 55-x = 4x-20
=> x = 15
10 years after age ratio,
Daughter : Mother = 15+10 : 65-15+10 = 25:55 = 5:11 [Ans.]

f. Let, bench be x
ATQ,
5(x-4) = 3x+20
=> 5x-20 = 3x+20
=> x = 20 [Ans.]

g. Let, smaller number be x
Larger be y
ATQ,
3x+4y = 71 —-(1)
Again,
2y-5x = 3—–(2)
Now, (2)*2 -(1) =>
3x+10x = 71-6
=> x = 65/13 = 5
So, 3*5+4y = 71
=> y = (71-15)/4 = 14
Smaller = 5
Larger = 14 [Ans.]
Thank You. 