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Direction cosines satisfying a linear and a quadratic relation: angle between the two lines

This type of question, asked 4 times: 2021 Paper 3, 2021 Paper 21, 27 Jun 2022, 23 Jan 2026 S1. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 3

    62
    Let α\alpha be the angle between the lines whose direction cosines satisfy the equations l+m−n=0l + m - n = 0 and l2+m2−n2=0l^{2}+m^{2}-n^{2}= 0. Then the value of sin⁡4α+cos⁡4α\sin^{4}\alpha +\cos^{4}\alpha is:

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  2. Question 2

    2021 Paper 21

    61
    The angle between the straight lines, whose direction cosines are given by the equations 2l+2m−n=02l+ 2m-n= 0 and mn+nl+lm=0mn+nl+lm= 0, is :

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  3. Question 3

    27 Jun 2022

    74
    If two straight lines whose direction cosines are given by the relations l+m−n=0, 3l2+m2+cnl=0l + m - n = 0,\ 3l^{2}+m^{2}+ cnl = 0 are parallel, then the positive value of cc is:

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  4. Question 4

    23 Jan 2026 S1

    70
    Let the direction cosines of two lines satisfy the equations: 4l+m−n=04\mathcal{l +}m - n = 0 and 2mn+10nl+3lm=02mn + 10n\mathcal{l +}3\mathcal{l}m = 0. Then the cosine of the acute angle between these lines is :

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