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Value of a parameter giving the maximum or minimum of a scalar triple product (volume)

This type of question, asked 8 times: 2 May 2023 S1, 4 May 2023 S1, 9 May 2024 S1, 10 May 2024 S2, 14 May 2024 S2, 11 Apr 2026 S2, 19 Apr 2026 S1, 19 Apr 2026 S2. All 8 questions below, oldest first.

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  1. Question 1

    2 May 2023 S1

    113
    The volume of parallelopiped formed by vectors i^+mj^+k^\hat{i} + m\hat{j} + \hat{k}, j^+mk^\hat{j} + m\hat{k} and mi^+k^m\hat{i} + \hat{k} becomes minimum when mm is

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  2. Question 2

    4 May 2023 S1

    120
    The value of aa for which the volume of parallelepiped formed by i^+aj^+k^\hat{i}+a\hat{j}+\hat{k}, j^+ak^\hat{j}+a\hat{k} and ai^+k^a\hat{i}+\hat{k} becomes minimum is

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  3. Question 3

    9 May 2024 S1

    107
    The value of α\alpha, so that the volume of the parallelepiped formed by i^+αj^+k^\hat{i}+\alpha\hat{j}+\hat{k}, j^+αk^\hat{j}+\alpha\hat{k} and αi^+k^\alpha\hat{i}+\hat{k} becomes maximum, is

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  4. Question 4

    10 May 2024 S2

    116
    The value of aa, so that the volume of parallelepiped formed by i^+aj^+k^\hat{i}+a\hat{j}+\hat{k}, j^+ak^\hat{j}+a\hat{k} and ai^+k^a\hat{i}+\hat{k} becomes minimum, is

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  5. Question 5

    14 May 2024 S2

    127
    Let x0x_0 be the point of local minima of f(x)=a⃗⋅(b⃗×c⃗)f(x) = \vec{a}\cdot(\vec{b}\times\vec{c}) where a⃗=xi^−2j^+3k^\vec{a} = x\hat{i}-2\hat{j}+3\hat{k}, b⃗=−2i^+xj^−k^\vec{b} = -2\hat{i}+x\hat{j}-\hat{k}, c⃗=7i^−2j^+xk^\vec{c} = 7\hat{i}-2\hat{j}+x\hat{k}, then value of a⃗⋅b⃗\vec{a}\cdot\vec{b} at x=x0x = x_0 is

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  6. Question 6

    11 Apr 2026 S2

    90
    Let x0x_0 be the point of local maxima of f(x)=aˉ⋅(bˉ×cˉ)f(x) = \bar{a}\cdot(\bar{b} \times \bar{c}) where aˉ=xi^−2j^+3k^,bˉ=−2i^+xj^−k^\bar{a} = x\hat{i} - 2\hat{j} + 3\hat{k}, \bar{b} = -2\hat{i} + x\hat{j} - \hat{k} and cˉ=7i^−2j^+xk^\bar{c} = 7\hat{i} - 2\hat{j} + x\hat{k} then the value of aˉ⋅cˉ\bar{a}\cdot\bar{c} at x=x0x = x_0 is

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  7. Question 7

    19 Apr 2026 S1

    90
    The value of xx so that the volume of the parallelopiped formed by the vectors i^+xj^+k^\hat{i} + x\hat{j} + \hat{k}, j^+xk^\hat{j} + x\hat{k} and xi^+k^x\hat{i} + \hat{k} is minimum, is

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  8. Question 8

    19 Apr 2026 S2

    89
    The maximum volume of a parallelopiped (in cubic units) with vectors (2ai^+k^),(aj^−ak^)(2a\hat{i} + \hat{k}), (a\hat{j} - a\hat{k}), and (3i^+aj^)(3\hat{i} + a\hat{j}), where a∈[0,1]a \in [0, 1], as its coterminous edges is...

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