Consider the sets , , and . The total number of one-one functions from the set D to the set C is: [2025]

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Solution

Q.
Consider the sets $A = {(x, y) \in R \times R : x^{2} + y^{2} = 25}$ , $B = {(x, y) \in R \times R : x^{2} + 9 y^{2} = 144}$ , $C = {(x, y) \in Z \times Z : x^{2} + y^{2} \leq 4}$ and $D = A \cap B$ . The total number of one-one functions from the set D to the set C is: [2025]

1 18290

2 15120

3 17160

4 19320

Ans.
(3)

We Have, $A : x^{2} + y^{2} = 25$ ... (i)

$B : \frac{x^{2}}{144} + \frac{y^{2}}{16} = 1$ ... (ii)

$C : x^{2} + y^{2} \leq 4$ ... (iii)

Solving (i) and (ii), we get

$x^{2} + 9 (25 - x^{2}) = 144 \Rightarrow - 8 x^{2} = - 81 \Rightarrow x = \pm \frac{9}{2 \sqrt{2}}$

From (i), $\frac{81}{8} + y^{2} = 25 \Rightarrow y^{2} = 25 - \frac{81}{8} \Rightarrow y = \pm \frac{\sqrt{119}}{2 \sqrt{2}}$

As, $D = A \cap B$

$= {(\frac{9}{2 \sqrt{2}}, \frac{\sqrt{119}}{2 \sqrt{2}}), (\frac{9}{2 \sqrt{2}}, - \frac{\sqrt{119}}{2 \sqrt{2}}), (\frac{- 9}{2 \sqrt{2}}, \frac{\sqrt{119}}{2 \sqrt{2}}), (- \frac{9}{2 \sqrt{2}}, - \frac{\sqrt{119}}{2 \sqrt{2}})}$

$\Rightarrow n (D) = 4$

Also, $C = {(x, y) \in Z \times Z : x^{2} + y^{2} \leq 4}$

={(0, 2), (0, –2), (2, 0), (–2, 0), (1, 1), (–1, –1), (–1, 1), (1, –1), (0, 1), (0, –1), (1, 0), (–1, 0), (0, 0)}.

$\Rightarrow n (C) = 13$

$∴$ Total number of one-one function from D to C = $^{13} P_{4}$ = 17160.

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