NIMCET 2020 — Mathematics PYQ

For two vectors $\vec{A}$ and $\vec{B}$ at an angle $\theta$ to each other:

- Dot Product is defined as: $\vec{A}\cdot \vec{B}=|\vec{A}||\vec{B}|\cos \theta$.
- Cross Product is defined as: $\vec{A}\times \vec{B}=\vec{n}|\vec{A}||\vec{B}|\sin \theta$ where $\vec{n}$ is the unit vector perpendicular to the plane containing $\vec{A}$ and $\vec{B}$.

For three vectors $\vec{A}$, $\vec{B}$ and $\vec{C}$:

- Triple Cross Product: is defined as: $\vec{A}\times (\vec{B}\times \vec{C})=(\vec{A}\cdot \vec{C})\vec{B}-(\vec{A}\cdot \vec{B})\vec{C}$.
- Triple Scalar Product (Box Product): is defined as: $[\vec{A}\vec{B}\vec{C}]=\vec{A}\cdot (\vec{B}\times \vec{C})=\left|\begin{array}{ccc}{a}_1& amp;a_2& amp;a_3\\ b_1& amp;b_2& amp;b_3\\ c_1& amp;c_2& amp;c_3\end{array}\right|$.

Volume of a parallelepiped, with vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ as its sides, is given by the box product of the three vectors.
- Volume = $[\vec{a}\vec{b}\vec{c}]$.

Calculation:
The sides of the parallelepiped are:
$\vec{a}=2\hat{i}+3\hat{j}+4\hat{k}$
$\vec{b}=\hat{i}+\alpha \hat{j}+2\hat{k}$
$\vec{c}=\hat{i}+2\hat{j}+\alpha \hat{k}$

$\therefore$ Volume $=[\vec{a}\vec{b}\vec{c}]=\left|\begin{array}{ccc}2& amp;3& amp;4\\ 1& amp;\alpha & amp;2\\ 1& amp;2& amp;\alpha \end{array}\right|=15$

$\Rightarrow 2({\alpha }^2-4)+3(2-\alpha )+4(2-\alpha )=15$

$\Rightarrow 2{\alpha }^2-8+6-3\alpha +8-4\alpha =15$

$\Rightarrow 2{\alpha }^2-7\alpha -9=0$

$\Rightarrow 2{\alpha }^2-9\alpha +2\alpha -9=0$

$\Rightarrow \alpha (2\alpha -9)+(2\alpha -9)=0$

$\Rightarrow (2\alpha -9)(\alpha +1)=0$

$\Rightarrow 2\alpha -9=0\text{ OR }\alpha +1=0$

$\Rightarrow \alpha =\frac{9}{2}\text{ OR }\alpha =-1$.