Question : 24 The value of dr is

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$If D_{r} = ⎛ ⎜ ⎝ \begin{matrix} 2^{r - 1} & 2 . 3^{r - 1} & 4 . 5^{r - 1} α & β & γ 2^{n} - 1 & 3^{n} - 1 & 5^{n} - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣, then the value of \sum_{r = 1}^{n} D_{r} is$

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D_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2^{r - 1} & {2.3}^{r - 1} & {4.5}^{r - 1} x & y & z 2^{n} - 1 & 3^{n} - 1 & 4^{n} - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

, then the value of

\sum_{r = 1}^{n} D_{r}

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Question 44.Dr. S. Radhakrishnan has-

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Question 3
If 24x is a multiple of 3, where x is a digit, what is the value of x?

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The value of

L i m x ⟶ \infty \frac{d}{d x} \sqrt{3} \int - \sqrt{3} \frac{r^{3}}{(r + 1) (r - 1)}

dr,is

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