Consider the sets$A = {(x$, y)$\ in $R $\times$ R :$x + y = 25}$,$B = {(x$, y)$\ in $R $\times$ R :$x + 9y = 144}$,$C = {(x$, y) 2 2 2 2$\ in $Z $\times$ Z : x$2 + y$2$\le$4} , and$D = A$$\cap$B. The total number of$one-one$functions from the set D to the set C is:
Step-by-Step Solution
Key Concept: Convert the condition into algebraic equations and simplify the resulting expression.
A : x$2 + y$$2 = 25...(i)$2 (3) 2 y B: x + = 1...(ii) 144 16 C: x +$y_{2}$2$\le$4...(iii) Solve (1) \& (2) 2 2$x + 9$$(25 - x$$) = 144$$2 - 8x = 144 - 225$= -81 9 x = $\pm$ 2$\sqrt{2}$2 By(1) $\Rightarrow$ y = $\pm$$\sqrt{25}$- x 81$\sqrt{119}$= $\pm$$\sqrt{25}$- = $\pm$ 8 2$\sqrt{2}$∴$D = A$$\cap$$B = 9$$\sqrt{119}$9$\sqrt{119}$-9$\sqrt{119}$-9 -$\sqrt{119}${( , ),( ,- ),( , ),( , )} 2$\sqrt{2}$2$\sqrt{2}$2$\sqrt{2}$2$\sqrt{2}$2$\sqrt{2}$2$\sqrt{2}$2$\sqrt{2}$2$\sqrt{2}$No. of elements in set$D = 4$2 2 ∵$C = {(x$, y)$\ in $Z $\times$ Z :$x + y$$\le$$4} = {(0$, 2), (2, 0), (0, -2),$(-2$, 0), (1, 1),$(-1$, -1), (1, -1),$(-1$, 1), (1, 0), (0, 1),$(-1$, 0), (0, -1), (0, 0)} No. of elements in set$C = 13$Total no. of$one-one$function from Set D to sec C $\Rightarrow$ 13 $\times$ 12 $\times$ 11 $\times$$10 = 17160$
Correct Answer: 3