JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

  • question_answer
    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 [IIT 1984]

    A) \[p=0\]

    B) \[p=1\] or \[-\frac{1}{3}\]

    C) \[p=-1\] or \[\frac{1}{3}\]

    D) \[p=1\] or \[-1\]

    Correct Answer: B

    Solution :

    • 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 \[4{{p}^{2}}+1={{(p+1)}^{2}}+1\]                   
    • \[\Rightarrow p+1=\pm \text{ }2p\Rightarrow p=1\] or \[-\frac{1}{3}.\]

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