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Value of an unknown making three given vectors coplanar (linearly dependent)

This type of question, asked 7 times: 4 May 2023 S1, 15 May 2023 S2, 11 May 2024 S2, 22 Apr 2025 S2, 11 Apr 2026 S2, 13 Apr 2026 S2, 19 Apr 2026 S1. All 7 questions below, oldest first.

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  1. Question 1

    4 May 2023 S1

    132
    The number of distinct real values of λ\lambda, for which the vectors −λ2i^+j^+k^-\lambda^2\hat{i}+\hat{j}+\hat{k}, i^−λ2j^+k^\hat{i}-\lambda^2\hat{j}+\hat{k} and i^+j^−λ2k^\hat{i}+\hat{j}-\lambda^2\hat{k} are coplanar, is

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  2. Question 2

    15 May 2023 S2

    139
    If the vector c⃗\vec{c} lies in the plane of a⃗\vec{a} and b⃗\vec{b}, where a⃗=i^−j^+2k^, b⃗=i^+j^+k^\vec{a}=\hat{i}-\hat{j}+2\hat{k},\, \vec{b}=\hat{i}+\hat{j}+\hat{k} and c⃗=xi^−(2−x)j^−k^\vec{c}=x\hat{i}-(2-x)\hat{j}-\hat{k}, then the value of xx is

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  3. Question 3

    11 May 2024 S2

    114
    If a⃗=i^+j^+k^\vec{a} = \hat{i}+\hat{j}+\hat{k}, b⃗=4i^+3j^+4k^\vec{b} = 4\hat{i}+3\hat{j}+4\hat{k} and c⃗=i^+αj^+βk^\vec{c} = \hat{i}+\alpha\hat{j}+\beta\hat{k} are linearly dependent vectors and ∣c⃗∣=3|\vec{c}|=\sqrt{3}, then the value of α\alpha and β\beta are respectively.

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  4. Question 4

    22 Apr 2025 S2

    118
    The number of integral values of pp for which the vectors (p+1)i^−3j^+pk^,pi^+(p+1)j^−3k^(p + 1)\widehat{i}- 3\widehat{j}+ p\widehat{k},p\widehat{i}+ (p + 1)\widehat{j}- 3\widehat{k} and −3i^+pj^+(p+1)k^- 3\widehat{i}+ p\widehat{j}+ (p + 1)\widehat{k} are linearly dependent vectors, are

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  5. Question 5

    11 Apr 2026 S2

    87
    If aˉ=i^+j^+k^,bˉ=i^−j^+2k^\bar{a} = \hat{i} + \hat{j} + \hat{k}, \bar{b} = \hat{i} - \hat{j} + 2\hat{k} and cˉ=xi^+(x−2)j^−k^\bar{c} = x\hat{i} + (x - 2)\hat{j} - \hat{k} are three vectors in which cˉ\bar{c} lies in the plane of aˉ\bar{a} and bˉ\bar{b}, then x=⋯x = \cdots

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  6. Question 6

    13 Apr 2026 S2

    90
    If aˉ=i^+j^+k^,bˉ=i^,cˉ=c1i^+c2j^+c3k^\bar{a} = \hat{i} + \hat{j} + \hat{k}, \bar{b} = \hat{i}, \bar{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k} with c1=1,c2=2c_1 = 1, c_2 = 2, then value of c3c_3 such that aˉ,bˉ,cˉ\bar{a}, \bar{b}, \bar{c} are coplanar is ____

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  7. Question 7

    19 Apr 2026 S1

    91
    The sum of all real values of λ\lambda for which the vectors a⃗=λi^+j^+k^,b⃗=i^+λj^+2k^,c⃗=2i^+3j^+λk^\vec{a} = \lambda\hat{i} + \hat{j} + \hat{k}, \vec{b} = \hat{i} + \lambda\hat{j} + 2\hat{k}, \vec{c} = 2\hat{i} + 3\hat{j} + \lambda\hat{k} are coplanar is...

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