Using Statement 1: There are two 3-digit numbers satisfying that condition which are 225 and 441.
Using Statement 2: we cannot determine X. Assuming both also we cannot uniquely determine X.
Hence, the question can't be answered even after using both the statements together.

Using Statement 1: If the card drawn is not a Ten probability= $$^4 C_2 / ^ {51} C_2=2/425$$.
If the card drawn is a Ten, probability is $$^3 C_2 / ^{51} C_2 = 1/425$$.
Hence the card drawn is a Ten.
Using Statement 2: Similarly as given above, we can say the card drawn is not a Queen.
But we cannot say if the card is a face card or not.
Hence, we can answer the question with one statement and not the other.

The question can not be answered using just one of the two statements.
Using both the statements together, (X,Y) can be either (252,36) or (180,108)
X-Y is different for both the answers.
Hence, the question can't be answered even after using both the statements together.

Using Statement 1: let the ages be a/r2, a/r, a, ar and ar2. Hence product of ages $$= a^5 = 1048576$$.
We can determine a but not r. So, the question cannot be answered by 1 alone.
Using Statement 2: The sum of the roots=-50/1= 50.
Hence the average=10.
So, the question can be answered using Statement 2, but not by using Statement 1

Let the time taken in man hours to construct the building be X.
Let M, W and C be the working speeds of one man, one woman and one child.
Using Statement 1: 1M+5W+9C=X/52.
Using Statement 2: 4M+8W+12C=X/28.
Answer cannot be deduced from either equation alone.
Combining both the statements and subtracting 1 from 2, we get
3(1M+1W+1C)=X/28-X/52=3X/182.
Hence, time taken is 182 days.
So, we can answer the question using both the Statements together, but not by using any of the two statements individually.

Using both the statements individually, we can't solve for the ages of A and B.
However, using both the statements together, we can conclude that A=B and both are perfect squares and cubes.
So $$A = B = x^6$$ for some x. Only 64 is a valid power of 6 in the given range.
So, we can answer the question using both the Statements together, but not by using any of the two statements individually

Let the cost of caramel, toffee and mango be x, y and z.
Using Statement 1: 3x+5y+7z=68.
Uding Statement 2: 2x+3y+4z=41.
We cannot determine answer from either of the two statements alone.
Using both the statements and multiplying the second equation by 2 and subtracting the first equation from it, we get
(4x+6y+8z)-(3x+5y+7z)=2*41-68=14.
or x+y+z=14
So, we can answer the question using both the Statements together, but not by using any of the two statements individually.

The original statement ensures that X<4 and X is odd.
So, X is either 1 or 3.
Using Statement 1: As 1 is not prime, $$X=3.$$
Using Statement 2: As $$1^{20} < 2^{30} , X=3$$
Hence, the question can be answered using either of the two statements alone.

Using either of the two statements alone, we can't calculate the value of X*Y.
Using both the statements together, we can conclude that the product of two numbers equals the product of its HCF and LCM = 108

The question can not be answered using just one of the two statements.
Using both the statements together, (X,Y) can be either (252,36) or (180,108)
X-Y is different for both the answers.
Hence, the question can't be answered even after using both the statements together.

Let the cost of caramel, toffee and mango be x, y and z.
Using Statement 1: 3x+5y+7z=68.
Uding Statement 2: 2x+3y+4z=41.
We cannot determine answer from either of the two statements alone.
Using both the statements and multiplying the second equation by 2 and subtracting the first equation from it, we get
(4x+6y+8z)-(3x+5y+7z)=2*41-68=14.
or x+y+z=14
So, we can answer the question using both the Statements together, but not by using any of the two statements individually.

Let the time taken in man hours to construct the building be X.
Let M, W and C be the working speeds of one man, one woman and one child.
Using Statement 1: 1M+5W+9C=X/52.
Using Statement 2: 4M+8W+12C=X/28.
Answer cannot be deduced from either equation alone.
Combining both the statements and subtracting 1 from 2, we get
3(1M+1W+1C)=X/28-X/52=3X/182.
Hence, time taken is 182 days.
So, we can answer the question using both the Statements together, but not by using any of the two statements individually.

$$x^{2}-x=12$$ => x = -3 or x = 4.
Now for both the values of x we have integer values of y which are valid and hence Statement 1 alone is not sufficient to determine the exact value of x.

$$y^{2}-y=2$$ => y = 2 or y = -1.
If y = 2, we get $$x^{2}\times(x-1) = 51$$ which doesn’t give any integral solution for x.
If y = -1, we get $$x^{2}\times(x-1) = 48$$ => x = 4. Hence x can be uniquely determined using Statement 2 and hence Statement 2 alone is sufficient to determine the value of x.