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Relation among letter parameters when three vectors such as ai + j + k, i + bj + k, i + j + ck are coplanar

This type of question, asked 5 times: 9 May 2023 S1, 13 May 2024 S2, 19 Apr 2025 S1, 22 Apr 2025 S1, 11 Apr 2026 S1. All 5 questions below, oldest first.

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  1. Question 1

    9 May 2023 S1

    113
    If the vectors ai^+j^+k^a\hat{i}+\hat{j}+\hat{k}, i^+bj^+k^\hat{i}+b\hat{j}+\hat{k}, i^+j^+ck^\hat{i}+\hat{j}+c\hat{k} (a≠b,c≠1a\neq b, c\neq1) are coplanar, then 11−a+11−b+11−c\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c} has the value

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  2. Question 2

    13 May 2024 S2

    119
    If the vectors pi^+j^+k^,i^+qj^+k^p\hat{i}+\hat{j}+\hat{k},\hat{i}+q\hat{j}+\hat{k} and i^+j^+rk^ (p≠q≠r≠1)\hat{i}+\hat{j}+r\hat{k}\ (p\neq q\neq r\neq1) are coplanar, then the value of pqr−(p+q+r)pqr-(p+q+r) is

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  3. Question 3

    19 Apr 2025 S1

    129
    If the points A(2−x,2,2),B(2,2−y,2)A(2 -x,2,2),B(2,2 -y,2), C(2,2,2−z)C(2,2,2 - z) and D(1,1,1)D(1,1,1) are coplanar, then the locus of point P(x,y,z)P(x,y,z) is

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  4. Question 4

    22 Apr 2025 S1

    124
    If the vectors mi^+mj^+nk^,i^+k^,ni^+nj^+pk^m\widehat{i}+m\widehat{j}+n\widehat{k},\widehat{i}+\widehat{k},n\widehat{i}+n\widehat{j}+p\widehat{k} lie in a plane then...

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  5. Question 5

    11 Apr 2026 S1

    90
    If a,b,ca, b, c are distinct non-negative numbers and the vectors ai^+aj^+ck^,i^+k^,ci^+cj^+bk^a\hat{i} + a\hat{j} + c\hat{k}, \hat{i} + \hat{k}, c\hat{i} + c\hat{j} + b\hat{k} lie in the same plane, then the value of cc is...

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