NIMCET 2022 — Mathematics PYQ

To solve this problem, we use the property that the volume of a parallelepiped formed by three vectors is equal to the magnitude of their Scalar Triple Product.1. Formula for Volume:The volume $V$ of a parallelepiped with adjacent edges $\vec{a}$, $\vec{b}$, and $\vec{c}$ is given by:$$V=|[\vec{a}\vec{b}\vec{c}]|$$This can be calculated using the determinant of the coefficients of the vectors:$$V=\left|\left|\begin{array}{ccc}{a}_1& amp;a_2& amp;a_3\\ b_1& amp;b_2& amp;b_3\\ c_1& amp;c_2& amp;c_3\end{array}\right|\right|$$2. Set up the Determinant:Given $V=15$ and vectors $\vec{a}=(2,3,4)$, $\vec{b}=(1,\alpha ,2)$, and $\vec{c}=(1,2,\alpha )$:$$\pm 15=\left|\begin{array}{ccc}2& amp;3& amp;4\\ 1& amp;\alpha & amp;2\\ 1& amp;2& amp;\alpha \end{array}\right|$$3. Expand the Determinant:Expand along the first row:$$15=2({\alpha }^2-4)-3(\alpha -2)+4(2-\alpha )$$$$15=2{\alpha }^2-8-3\alpha +6+8-4\alpha$$Simplify the terms:$$15=2{\alpha }^2-7\alpha +6$$4. Solve the Quadratic Equation:Rearrange the equation into standard form:$$2{\alpha }^2-7\alpha +6-15=0$$$$2{\alpha }^2-7\alpha -9=0$$Using the factorization method:$$2{\alpha }^2-9\alpha +2\alpha -9=0$$$$\alpha (2\alpha -9)+1(2\alpha -9)=0$$$$(2\alpha -9)(\alpha +1)=0$$This gives two possible values for $\alpha$:$2\alpha -9=0⟹\alpha =9/2$$\alpha +1=0⟹\alpha =-1$Conclusion:Comparing the results with the given options, the value of $\alpha$ is $9/2$.