When the piece of iron is lowered into the water, the water level in the bucket increases. And so the pressure at the bottom of bucket also increases. Thus the total thrust at bottom also increases. Therefore the spring balance reading increases. Thus (a) is correct.

The weight of the man mg is acting on the floor. As the lift is moving up with an acceleration, say a, the man too in it is having this acceleration. The force producing this acceleration is ma. By Newton's third law, this force will have an equal reaction pressing the floor equal to ma. Therefore the total resultant reaction on the floor is mg+ma = m(g+a) which is mg, the weight of the man. So (a) is correct.

Combined efficiency of all the three boats = 60 passenger/trip 

Now, consider option(a)

15 trips and 150 passengers means efficiency  of B1 =  10 passenger/trip 

which means in carrying 50 passengers B1 must has taken 5 trips. So the rest trips equal to 5 (10-5 = 5) in which B2 and B3 together carried remaining 250 (300 - 50 = 250) Passengers.

Therefore the efficiency of B2 and B3 = 250/5 = 50 passenger/trip  

Since, the combined efficiency of B1, B2 and B3 is 60. Which is same as given in the first statement hence option(a) is correct.

Let the length of the train be L metres and speeds of the train Arun and Sriram be R, A and S respectively, then 

       LR-A=36 ---------- (i) 

and LR+K=24 ---------(ii)

From eq.(i) and (ii)

              3(R - A ) = 2 (R + K) 

⇒  R = 3A + 2K 

In 30 minutes (i.e 1800 seconds), the train covers 1800R (distance) but the Arun also covers 1800 A (distance) in the same time. Therefore distance between Arun and Sriram, when the train has just crossed Sriram

               = 1800 ( R - A) - 24 ( A + K)

 Time required = 1800(R-A)-24(A+K)(A+K)

= (3600 - 24) = 3576 s                            

s1s2=t2t1

⇒120s2=916=34

⇒s2=160km/h

Time taken by Akash = 4 h

Time taken by Prakash = 3.5 h

For your convenience take the product of times taken by both as a distance.

Then  the distance = 14km

Since, Akash covers half of the distance  in 2 hours(i.e at 8 am)

Now, the rest half (i.e 7 km) will be coverd by both prakash and akash

Time taken by them = 7/7.5 = 56 min

Thus , they will cross each other at  8 : 56am.

Let the normal speed be x km/h, then 

80x-80(x+4)=1

⇒x2+4x-320=0

⇒x (x + 20) - 16 (x + 20) = 0

(x + 20 ) (x - 16) =0

x = 16 km/h

 Therefore (x + 4) = 20 km/h

 Therefore increased speed = 20 km/h

kl>k+1l+1 for (k,l) > 0 and  k > l 

Let     k = x+1    and   l = x

Therefore, x+1x>(x+1)+1(x)+1

 (x + 1) > x

Therefore, log(x+1)log(x)>log(x+2)log(x+1)

⇒logxx+1 >logx+1x+2