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Values of a parameter for which the angle between two vectors is obtuse

This type of question, asked 4 times: 2 May 2023 S1, 21 Apr 2025 S2, 25 Apr 2025 S2, 26 Apr 2025 S2. All 4 questions below, oldest first.

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  1. Question 1

    2 May 2023 S1

    149
    For all real xx, the vectors Cxi^−6j^−3k^Cx\hat{i} - 6\hat{j} - 3\hat{k} and xi^+2j^+2Cxk^x\hat{i} + 2\hat{j} + 2Cx\hat{k} make an obtuse angle with each other, then the value of CC can be in

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  2. Question 2

    21 Apr 2025 S2

    124
    The value of m∈Rm\in\mathbb{R}, when angle between the vectors pˉ=myi^−6j^+3k^\bar{p}=my\widehat{i}- 6\widehat{j}+ 3\widehat{k} and qˉ=yi^+2j^+2myk^\bar{q}=y\widehat{i}+ 2\widehat{j}+ 2my\widehat{k} is obtuse angle, is

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  3. Question 3

    25 Apr 2025 S2

    127
    If the vectors a‾=c(log⁡7x)i^+2j^+3k^\overline{a}= c\left( \log_{7}x \right)\widehat{i}+ 2\widehat{j}+ 3\widehat{k} and b‾=(log⁡7x)i^+3c(log⁡7x)j^−4k^\overline{b}=\left( \log_{7}x \right)\widehat{i}+ 3c\left( \log_{7}x \right)\widehat{j}- 4\widehat{k} make obtuse angle for any x>0x> 0, then c belongs to

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  4. Question 4

    26 Apr 2025 S2

    134
    The values of xx for which the angle between the vectors a‾=2x2i^+4xj^+k^\overline{a}= 2x^{2}\widehat{i}+ 4x\widehat{j}+\widehat{k} and b‾=7i^−2j^+xk^\overline{b}= 7\widehat{i}- 2\widehat{j}+x\widehat{k} is obtuse, are

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