Solution:

From a + b + c = 0 we get b= - (a + c)

So, in order to find root formula
x = (-b+/-sqrt(b^2-4ac))/2a --- (eq 1)

b^2 = (a^2+c^2+2ac)

b^2-4ac = (a-c)^2
Sqrt(b^2-4ac) = (a-c)

Substituting the value in eq 1
x=1 (or) x=c/a

Hence the roots are real.

Solution:

Let the age of Camilla = x
Then the age of Bella will be 2x
And the age of Anna will be 2x + 2

Now according to the question
(2x + 2) + 2x + x = 27
5x = 25
x = 5

Thus the age of Camilla is 5 years

Age of Bella = 2x = 2*5 = 10 years.

Solution:

Before moving forward with the solution do keep in mind that the answer is demanded in minutes.

Drops in the bucket = x
Time taken to leak z drops = t seconds
Time taken to leak one drop = t/z seconds

Time taken to drop x drops = (x.t)/z seconds
We must convert it into minutes for our final answer
So the minutes taken to leak x drops = (x.t) / (z.60)

Solution:

In this question, we have to find out the highest common factor
While dividing 1080 with 729, the remainder is 351
While dividing 729 by 351, the remainder is 27
While dividing 351 with 27, the remainder is 0.

Thus the required number is 27.

Solution:

No number can be formed using the given digits which is divisible by 9.

For any number to be divisible by 9, the sum of the digits must be divisible by 9. In the given problem, 6+4+8+1 = 19 which is not divisible by 9.

Thus no number under the given conditions can be formed with the given digits.

Solution:

The ant climbs 6 meters in one minute and then falls down 3 meters in the second minute, the distance that it covers in 2 minutes is 3 meters.

Thus for climbing 54 meters, the ant will require time = 52*2/3 = 36 minutes.

We have counted only for 54 meters as it will climb the remaining six meters in one minute.

Thus the total time required to climb the 60 meters wall = 36 + 1 = 37 minutes.

Solution:

The clock was 7 minutes late at 3pm on Wednesday and 8 minutes ahead at 4pm on Friday
Thus, the clock has gained 15 minutes in 9 + 24 + 16 = 49 hours
So 7 minutes will be gained in (7*49) 15 = 22.6 hours

Thus the clock will show the correct time at near about 1:30 pm on Thursday.

Solution:

Total age of x students = 30x (as per the question)
Total age of 12 new students that are merged in the group = 32*12 = 384
Total age of x+12 students = (x+12)*31 = 30x + 384

Thus according to the question,
31x + 372 = 30x + 384
x = 12

Solution:

20

The series is in the following pattern
2 = 1*2
6 = 2*3
12 = 3*4
20 = 4*5
30 = 5*6
42 = 6*7
56 = 7*8

Solution:

To solve this question, we will use a simple formula i.e. person*time/work

According to the question
(Person1*Time1)/Work = (Person2*Time2)/Work
Let us put in the values now
(100*60)/Work = (500*Time)/Work
Time = 12

Thus the same amount of food will last 12 days for 500 people.