Question
The number of elements in the set $$\left\{ {\left( {a,\,b} \right):2{a^2} + 3{b^2} = 35,\,a,\,b\, \in \,Z} \right\},$$ where $$Z$$ is the set of all integers, is :
A.
$$2$$
B.
$$4$$
C.
$$8$$
D.
$$12$$
Answer :
$$8$$
Solution :
$$\eqalign{
& {\text{Given set is}} \cr
& \left\{ {\left( {a,\,b} \right):2{a^2} + 3{b^2} = 35,\,a,\,b\, \in \,Z} \right\}, \cr
& {\text{We can see that,}} \cr
& 2{\left( { \pm 2} \right)^2} + 3{\left( { \pm 3} \right)^2} = 35{\text{ and }}2{\left( { \pm 4} \right)^2} + 3{\left( { \pm 1} \right)^2} = 35 \cr
& \therefore \,\left( {2,\,3} \right),\,\left( {2,\, - 3} \right),\,\left( { - 2,\, - 3} \right),\,\left( { - 2,\,3} \right),\,\left( {4,\,1} \right),\,\left( {4,\, - 1} \right),\,\left( { - 4,\, - 1} \right),\,\left( { - 4,\,1} \right) \cr
& {\text{are 8 elements of the set}}{\text{.}} \cr
& \therefore \,\,n = 8 \cr} $$