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L Hospital's Rule

If Y = SX, ...

Question

If $Y = S X, Z = T X$ , all the variables being differentiable functions of x and lower suffixes denote the derivative w.r.t. to x and $∣ ∣ ∣ ∣ \begin{matrix} X & Y & Z X_{1} & Y_{1} & Z_{1} X_{2} & Y_{2} & Z_{2} \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ \begin{matrix} S_{1} & T_{1} S_{2} & T_{2} \end{matrix} ∣ ∣ ∣$ $X^{n}$ ,then $n =$

A

1

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B

2

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C

3

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D

4

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Solution

The correct option is C 3
By observation,
On the left hand side,
each of $Y, Z, Y_{1}, Y_{2}, Z_{1}, Z_{2}$ contain an X term. On multiply any two functions we get a function of type $X^{2}$
LHS is $X (Y_{1} Z_{2} - Z_{1} Y_{2}) - X_{1} (Y Z_{2} - Z Y_{2}) + X_{2} (Y Z_{1} - Z Y_{2})$
Since each term is different from other,
it is clear that the determinant will be a function of $X^{3}$
hence, $n = 3$

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