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# If the value of $\int_1^k {(2x - 3)dx = 12}$, then find the value of k.$({\text{a}})$ $- 2$ and${\text{5}}$${\text{(b)}}$ ${\text{5}}$and ${\text{2}}$${\text{(c)}}$ ${\text{2}}$ and${\text{ - 5}}$${\text{(d)}}$None of these

Last updated date: 18th Mar 2023
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Hint: Evaluate the given integral carefully without missing any term in between.

We have the given integral as
$\int_1^k {(2k - 3)dx}$
After integrating the above equation, we get,
$= [{x^2} - 3x]_1^k$
$= ({k^2} - 3k) - (1 - 3)$
$= {k^2} - 3k + 2$
According to the question,
We are given that the value of the given integral is equal to $12$,
Therefore, we get
${k^2} - 3k + 2 = 12$
${k^2} - 3k - 10 = 0$
This equation can be re written in the form as
${k^2} - 5k + 2k - 10 = 0$
$k(k - 5) + 2(k - 5) = 0$
$(k + 2)(k - 5) = 0$
$\therefore k = - 2,5$
Therefore, the required solution is $({\text{a}})$ $- 2$ and${\text{5}}$.

Note: In these types of questions, the given integral is solved, then equated to the values given in the question, which on evaluation gives the value of the required variable.