Test Index

BITSAT 2018 Paper

Section: Mathematics

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Question : 117 of 150

Marks: +1, -0

Let

\vec{a}

\hat{i}-\hat{k}

\vec{b}

\widehat{xi}+\hat{j}+(1-x)\hat{k}

\vec{c}

\widehat{yi}+\widehat{xj}+(1+x-y)\hat{k}

\begin{bmatrix} \bar{a} & \bar{b} & \bar{c} \end{bmatrix}

depends on

only y
only x
both x and y
neither x nor y

Solution:

\vec{a}

=

\hat{i}-\hat{k}

,

\vec{b}

=

\widehat{xi}+\hat{j}+(1-x)\bar{k}

and

\vec{c}

=

\widehat{yi}+\widehat{xj}+(1+x-y)\hat{k}

\begin{bmatrix} \bar{a} & \bar{b} & \bar{c} \end{bmatrix}

=

\bar{a} \cdot \bar{b} \times \bar{c}

=

\begin{vmatrix} 1 & 0 & -1 \\ x & 1 & 1-x \\ y & x & 1+x-y \end{vmatrix}

= 1 [1 + x - y - x +

x^2

] - [

x^2

- y]

= 1 - y +

x^2-x^2

+ y]

= 1

Hence

\begin{bmatrix} \bar{a} & \bar{b} & \bar{c} \end{bmatrix}

is independent of x and y both

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