MCQ
The coefficient of $x ^7$ in $\left(1-x+2 x^3\right)^{10}$ is $........$.
- A$960$
- B$950$
- C$940$
- ✓$960$
where $r_1+r_2+r_3=10$ and $r_2+3 r_3=7$
$\begin{array}{lll}r_1 & r_2 & r_3 \\ 3 & 7 & 0 \\ 5 & 4 & 1 \\ 7 & 1 & 2\end{array}$
Required coefficient
$=\frac{10 !}{3 ! .7 !}(-1)^7+\frac{10 !}{5 ! .4 !}(-1)^4(2)+\frac{10 !}{7 ! \cdot 2 !}(-1)^1(2)^2$
$=-120+2520-1440=960$
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$\mathrm{A}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}:(\mathrm{x}-2)^{2}+\mathrm{y}^{2} \leq 4\right\}$
$\mathrm{B}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}: \mathrm{x}^{2}+\mathrm{y}^{2} \leq 4\right\} \text { and }$
$\mathrm{C}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}:(\mathrm{x}-2)^{2}+(\mathrm{y}-2)^{2} \leq 4\right\}$
If the total number of relation from $\mathrm{A} \cap \mathrm{B}$ to $\mathrm{A} \cap \mathrm{C}$ is $2^{\mathrm{p}}$, then the value of $\mathrm{p}$ is :
$x^2+y^2+5 x-3 y+4=0 .$
Then which of the following statements is (are) TRUE?
$(A)$ $\alpha=-1$ $(B)$ $\alpha \beta=4$ $(C)$ $\alpha \beta=-4$ $(D)$ $\beta=4$