Let Najib, Adil, and Eoin be able to do the work alone in 1/N, 1/A, 1/E days.

Since Najib and Adil together do one-fourth work in a day. In three days, they do three-fourth of the work. So, Eoin does one-fourth of the work in three days. Hence Eoin does the whole work alone in 12 days. 1/E =12.

Hence

N+ A = 0.25.......... (1)

A+ E =1/6............ (2)

E = 1/12................. (3)

Solving the three equations we have

1/N= 6(1/A)=12

Hence amount of work done by the three of them together in a day = 1/12+1/6+1/12=1/3

Time taken by three of them together to complete the whole work = 3 days.

Given, Mistry can fill the tank in 60 minutes

So Saha can fill the tank in (60 X 100)/125 minutes = 48 minutes.

So Datta can empty (or fill) the tank in (48 X 100)/120 = 40 minutes.

Now, amount of the tank filled in 1 minute = (1/60+ 1/48 - 1/40) part of the tank = 1/80 part of the tank.

Therefore, the tank will be filled in 80 minutes.

Let log_ca,=x and log_bc, log_ab,=y and z respectively

So xyz=1 and 2y=x+z(AP)

Also a,b,c are in GP so b²=ac

Or, log_bb²= log_ba+log_bc

Or, 2=(1/z)+y

From above we have 2(2-1/z)=x+z

Or, x=4-(2/z)-z

Putting the value of x

We get (4-2/z-z)(2-1/z)z=1

Or, 2z²-7z+2=0 (i)

Now,

Common difference = z-y = z-(2-1/z)=(z²-2z+1)/z

Using (i) we get (2z²-4z+2)/2z= 3z/2z = 3/2

Total no. of possible outcomes = ¹²C₃ = 220. Number of ways of selecting three defective bulbs from 6 defective bulbs = ⁶C₃=20. Probability that the room is not lighted = 20/220=1/11. Probability that the room is lighted= 1-1/11=10/11.

Money borrowed that becomes 441 in 1 year = 441 / (1 + 5/100) = 441 x 20 /21 = 420

Money borrowed that becomes 441 in 2 years = 441 / (1 + 5/100)² = 441 x (20/21) x (20/21) = 400

Total Money = 420 + 400 = 820

By Appollonius theorem,

6² + 8² = 2 * (5)² + 2 * (QM)²

Therefore, QM = 5 units

M is the midpoint of QR.

So, length of QR = 10 units

Let the list price be P

Merchant bought it at a discount of 20% i.e. cost price = 0.8P

Profit = 20% of selling price

Selling price = cost price + profit

Therefore, Selling price = cost price + 20% of selling price

SP = CP + 0.2 SP

0.8 SP = CP

Cost price = 0.8 × selling price

Hence,

Selling price = P

He gives 20% on marked price

Therefore,

Selling price = 0.8 × marked price

Marked price = P/0.8 = 1.25 P = 1.25 list price

Hence, option 3

As we know that orthocenter is the point of the intersection of the perpendiculars drawn from the vertices to the opposite sides of the triangle.

In the above figure AD is perpendicular to BC and BE is perpendicular to AC.

O is the orthocenter of the triangle ABC.

Now we have to find out the equation of AD and BE.

Slope of BC =-(6/4) = -(3/2)

So the slope of AD will be (2/3) and it is passing through (0,0) hence the equation of AD is , 2x - 3y = 0 ....(1)

Similarly slope of BE is - (2/3) and it is passing through (8,0), hence the equation of BE is, 3y = -2( x - 8)

⇒ 3y = -2x + 16

⇒ 2x + 3y = 16 .....(2)

Now the intersection of (1) and (2) will give us the co-ordinates of orthocenter.

On solving (1) and (2) we will get (x, y) = (4, 8/3)