Search any question & find its solution
Question:
Answered & Verified by Expert
If the volume of parallelopiped with conterminus edges $4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+\hat{\mathbf{k}},-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}+9 \hat{\mathbf{j}}+p \hat{\mathbf{k}}$ is 34 cubic units, then $p$ is equal to :
Options:
Solution:
1307 Upvotes
Verified Answer
The correct answer is:
-13
Coterminus edges of a parallelopiped are $4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+\hat{\mathbf{k}},-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}+9 \hat{\mathbf{j}}+p \hat{\mathbf{k}}$
Volume of parallelopiped $=34$
$\Rightarrow \quad\left|\begin{array}{rrr}4 & 5 & 1 \\ 0 & -1 & 1 \\ 3 & 9 & p\end{array}\right|=34$
$\Rightarrow \quad 4\left|\begin{array}{cc}-1 & 1 \\ 9 & p\end{array}\right|-5\left|\begin{array}{cc}0 & 1 \\ 3 & p\end{array}\right|+1\left|\begin{array}{cc}0 & -1 \\ 3 & 9\end{array}\right|=34$
$\Rightarrow \quad 4(-p-9)-5(-3)+1(3)=34$
$\Rightarrow \quad-4 p-36+15+3=34$
$\Rightarrow \quad 4 p=-36+18-34$
$\Rightarrow \quad p=-\frac{52}{4}=-13$
Volume of parallelopiped $=34$
$\Rightarrow \quad\left|\begin{array}{rrr}4 & 5 & 1 \\ 0 & -1 & 1 \\ 3 & 9 & p\end{array}\right|=34$
$\Rightarrow \quad 4\left|\begin{array}{cc}-1 & 1 \\ 9 & p\end{array}\right|-5\left|\begin{array}{cc}0 & 1 \\ 3 & p\end{array}\right|+1\left|\begin{array}{cc}0 & -1 \\ 3 & 9\end{array}\right|=34$
$\Rightarrow \quad 4(-p-9)-5(-3)+1(3)=34$
$\Rightarrow \quad-4 p-36+15+3=34$
$\Rightarrow \quad 4 p=-36+18-34$
$\Rightarrow \quad p=-\frac{52}{4}=-13$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.