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9. straight line — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS9. straight line1 Mark
MCQ
A vector $a$ has components $ 2p$  and $1$  with respect to a rectangular cartesian system. The system is rotated through a certain angle about the origin in the anti-clockwise sense. If $a$ has components  $ p+1$  and  $1$  with respect to the new system, then
  • A
    $p = 0$
  • ✓
    $p = 1$ or $ - \frac{1}{3}$
  • C
    $p = - 1$ or $\frac{1}{3}$
  • D
    $p = 1or - 1$
✓

Answer

Correct option: B.
$p = 1$ or $ - \frac{1}{3}$
b
(b) If $x,\,\,y$ are the original components; $X,\,\,Y$the new components and $\alpha $ is the angle of rotation, then $x = X\cos \alpha - Y\sin \alpha $ and $y = X\sin \alpha + Y\cos \alpha $

$\therefore \,2p = (p + 1)\cos \alpha - \sin \alpha $ and $1 = (p + 1)\sin \alpha + \cos \alpha $

Squaring and adding, we get $4{p^2} + 1 = {(p + 1)^2} + 1$

$ \Rightarrow p + 1 = \pm {\rm{ }}2p \Rightarrow p = 1$ or $ - \frac{1}{3}.$

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