MCQ
If $y = {a_0} + {a_1}x + {a_2}{x^2} + ..... + {a_n}{x^n},$ then ${y_n} = $
- A$n!$
- B$n!{a_n}x$
- ✓$n!{a_n}$
- DNone of these
${y_1} = {a_1} + 2{a_2}x + ...... + n{a_n}{x^{n - 1}}$
${y_2} = 2{a_2} + 6{a_3}x + ...... + n(n - 1){a_n}{x^{n - 2}}$
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${y_n} = n!{a_n}$.
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$\frac{{dy}}{{dx}} + \frac{1}{x}\sin 2y = {x^3}\,{\cos ^2}\,y$ represented by family of curves which is is givey by
$\overrightarrow{ a }=\hat{ i }+\hat{ j }+ n \hat{ k }, \quad \overrightarrow{ b }=2 \hat{ i }+4 \hat{ j }- n \hat{ k } \quad$ and $\overrightarrow{ c }=\hat{ i }+ n \hat{ j }+3 \hat{ k } \quad( n \geq 0),$ is $158 cu. Units$, then
$x-2 y=1, x-y+k z=-2, k y+4 z=6, k \in R$
consider the following statements :
$(A)$ The system has unique solution if $k \neq 2$, $k \neq-2$
$(B)$ The system has unique solution if $k =-2$.
$(C)$ The system has unique solution if $k =2$.
$(D)$ The system has no-solution if $k =2$.
$(E)$ The system has infinite number of solutions if $k \neq-2$
Which of the following statements are correct?