Let PQRS be quadrilateral. Two circle 01 and 02 are inscribed in triangles PQR and PSR respectively. Circle 01 touches PR at M and circle 02 touches PR at N. Find the length of MN.
I. A Circle is inscribed in the quadrilateral PQRS.
II. The radii of the circles 01 and 02 are 5 and 6 units respectively.

Given relationship is (PQ)(RQ) = XXX
Since X can take 9 values from 1 to 9 hence we have 9 possibilities
111 = 3 x 37 444 = 12 x 37 777 = 21 x 37
222 = 6 x 37 555 = 15 x 37 888 = 24 x 37
333 = 9 x 37 666 = 18 x 18 999 = 27 x 37
But out of these 9 cases only in 999, we get the unit's digit of two numbers the same. Since it is a unique value, hence we need neither statement I nor statement II to answer the question.

Given that Ram > Shyam, Vikram > Jay.
Hence from this we can conclude that neither Ram nor Vikram is the shortest. And we have to find the shortest. And we have to find the shortest among them:
Consider statement alone:
We know that that Ram is not the shortest, either Shyam or Jay is the shortest.
Hence (I) alone is not sufficient.
Consider statement I alone Shyam > Vikram.
From the given information and the information in (II), we get Ram > Shyam > Vikram > Jay.
Hence, (II) alone is sufficient.

Let x be the average height of the class and n be the number of students in the class.
Consider statements I alone
xn - 56 = (x + 1)(x -1)
? x + n = 57 .............(i)
Hence, the value of x cannot be found. So, I alone is not sufficient.
Consider statement I alone:
xn - 42 = (x + 1)(n - 1)
? x - n = 41 .............(ii)
Hence, the value of x cannot be found. So, II alone is not sufficient.
Both the statements together are suficient as the value of x can be found by solving (i) and (ii)

Given that their salaries are in the ratio of 3:4 and expenditure is in the ratio of 4:5 hence we can assume that salary of A and B are 3x and 4x and their expenditures are 4y and 5y.
Now we need to find the ratio of (3x - 4y)/(4x - 5y)
Consider statement I alone:
Giving of B is 25% of his salary hence his expenditure must be 75% so 3/4(4x) = 5y or 3x = 5y from this we can find the required ratio hence this statement is sufficient.
Consider statement II alone :
Given that 4x = 2000 or x = 500 but from this we can not find the value of y and hence we can not find the ratio of their savings.

Consider Statement I alone
Given that Area (?ABC) = Area(?PQR) since nothing about the sides or angles is mentioned, we can not say if they are congruent.Hence, I alone is not sufficient.
Consider Statement II alone
?ABC and ?PQR are right triangles. Nothing about the sides is given, Hence, II alone is not sufficient. Now using both I and II
Now we have two right angled triangle k with same area we may have different combination as only product of base and height is same. Hence even by using both the statement we can not find the answer.

From statement I,
E + B < A + D, we easily say that E is less than A, because B>D and as the statement suggest E + B < A + D.
? E < A.
? A is not the smallest integer.
Statement I is sufficient to answer.
From statement II, D < F
This statement is not sufficient to find the relation between A and E.

The data in Statement I, II and III together are necessary to answer the question .

The data given both the Statement I, II and II together are not sufficient to answer the question.

if the data in all the Statement I,II and III together are necessary to answer the question

From Statement I,
Total weight of Arun, Suraj and Vinay = 3 x Average weight = 3 x 68 kg
? Arun's weight = 3 x 68 - (78 + 46) = 80 kg
Now, total weight of Raju and Pradeep = 2 x 72 = 144 kg
? Pratap's weight = 144 - 68 = 76 kg
? Second highest weight = 78 kg
(Suraj's weight)
From statement II,
Total weight of Arun, Suraj, Vinay and Raju = 4 x 68 = 272 kg
? Arun's weight = 272 - (78 + 68 + 46)
? Arun's weight = 272 - 192 = 80 kg
? Second highest weight = 78 kg (Suraj's weight)