Find the profit or loss of the following:CP=Rs 300, SP= Rs 832
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If there is some scalar function 
such that
then we want to find
such that
So the vector field
is conservative, which means the fundamental theorem applies; the line integral of 
along any path 
parameterized by some vector-valued function 
over 
is given by
In this case,
then we want to find
So the vector field
In this case,
3. If and
, then
Then
(only one solution)
4. If , then
. So we're told that
but this is impossible, because that would that for any
, we have -5 = -2. This is false, so this system has no solution.
7. "y is 6 less than 3 times x" means . "x is 2 more than one-third of y" means
. Solving the second equation for
gives
, which would mean
. But for the same reason as in (4), there is no solution. For any value of
, this equation gives the false conclusion that -6 = -2.