Consider the sets A= (x, y) R R: x^ 2 +y^ 2 =25 , B= (x, y) R R: … - Vidyadip

MCQ

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

A
15120
B
19320
✓
17160
D
18290

✓

Answer

Correct option: C.

17160

(C) 17160
$\mathrm{A}: \mathrm{x}^{2}+\mathrm{y}^{2}=25\qquad\ldots(1)$
$\text B : \frac{x^{2}}{144}+\frac{y^{2}}{16}=1\qquad\ldots(2)$
$\text C: x^{2}+y^{2} \leq 4\qquad\ldots(3)$
Solve (1) & (2)
$x^{2}+9\left(25-x^{2}\right)=144$
$-8 x^{2}=144-225=-81$
$x= \pm \frac{9}{2 \sqrt{2}}$
By $(1) \Rightarrow y= \pm \sqrt{25-x^{2}}$
$= \pm \sqrt{25-\frac{81}{8}}= \pm \frac{\sqrt{119}}{2 \sqrt{2}}$
$\therefore \mathrm{D}=\mathrm{A} \cap \mathrm{B}=$
$\left\{\left(\frac{9}{2 \sqrt{2}}, \frac{\sqrt{119}}{2 \sqrt{2}}\right),\left(\frac{9}{2 \sqrt{2}},-\frac{\sqrt{119}}{2 \sqrt{2}}\right),\left(\frac{-9}{2 \sqrt{2}}, \frac{\sqrt{119}}{2 \sqrt{2}}\right),\left(\frac{-9}{2 \sqrt{2}}, \frac{-\sqrt{119}}{2 \sqrt{2}}\right)\right\}$
No. of elements in set $\mathrm{D}=4$

$\because C=\left\{(\mathrm{x}, \mathrm{y}) \in \mathrm{Z} \times \mathrm{Z}: \mathrm{x}^{2}+\mathrm{y}^{2} \leq 4\right\}$
$=\{(0,2),(2,0),(0,-2),(-2,0),(1,1),(-1,-1),$ $(1,-1),(-1,1),(1,0),(0,1),(-1,0),(0,-1),$ $(0,0)\}$
No. of elements in set $\mathrm{C}=13$
Total no. of one-one function from
Set $D$ to $\sec C \Rightarrow 13 \times 12 \times 11 \times 10=17160$

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