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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 1 variable, x which is the fifth data value and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1):
The fifth data value could be x = 9 or x = 12.
Since we don't have a unique solution, condition 1) is not sufficient.

Condition 2):
( x + 10 + 10 + 11 + 11 ) / 5 > 10
=> x + 10 + 10 + 11 + 11 > 50
=> x + 42 > 50
=> x > 8
Since we don't have a unique solution, condition 2) is not sufficient.

Conditions 1) & 2) :
Condition 1) tells us that x = 9 or x = 12, and condition 2) tells us that x > 8.
So, x = 9 or x = 12.
Since we don't have a unique solution, both conditions together are not sufficient.

Therefore, E is the answer.

Answer: E

If the original condition includes "1 variable", or "2 variables and 1 equation", or "3 variables and 2 equations" etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

Values = 10,10,11,11 (4 values)
let the fifth element = x
Target question = what is the value of x?
Statement 1 : the range of the data set = 2
Range = highest value - lowest value
2 = highest value - lowest value
This means that x= biggest value OR x= smallest value
first case (smallest value) :
x= 9 : the set = {9,10,10,11,11}
second case(biggest value)=
x = 12: the set = {10,10,11,11,12}
the answer to the target question is either 9 or 12 (it is not definite)
we cannot answer statement 1 with certainty. So it is NOT SUFFICIENT
STATEMENT 2 : The average of data values together than 10
There are so many values that satisfy statement 2
using two random values
x = 500 (case a)
x = 600 (case b)
1st case = average > 10
average = $$average=\ \frac{fx}{f}
average\ =\ \frac{\left(10+10+11+11+600\right)}{5}\ =\ \frac{642}{5}=128.4\ <greater\ than\ 10>$$ $$average=\ \frac{fx}{f}average\ =\ \frac{\left(10+10+11+11+600\right)}{5}\ =\ \frac{642}{5}
=128.4\ <greater\ than\ 10>$$
statement 2 cannot be answered with certainty so it is NOT SUFFICIENT
Statement 1 and 2 together are NOT SUFFICIENT
Answer is option E