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Vector Algebra

JEE Mains Maths. Try each one first, then open its answer.

Top 10 JEE Mains PYQs to practise first

The questions and kinds of question asked most often come first. Each one says how often and in which papers.

  1. Question 1

    This type asked 18 times: 2021 Paper 3, 2021 Paper 6, 2021 Paper 7, 24 Jan 2023, 29 Jan 2023, 31 Jan 2023, 1 Feb 2023, 8 Apr 2023, 13 Apr 2023, 31 Jan 2024, 1 Feb 2024, 5 Apr 2024, 8 Apr 2024, 29 Jan 2025, 3 Apr 2025, 24 Jan 2026 S2, 5 Apr 2026 S1, 8 Apr 2026 S2. This one: 8 Apr 2026 S2 See all 18 of this type

    64
    Let a→=4i^−j^+3k^,b→=10i^+2j^−k^\overrightarrow{a}= 4\widehat{i}-\widehat{j}+ 3\widehat{k},\overrightarrow{b}= 10\widehat{i}+ 2\widehat{j}-\widehat{k} and a vector c→\overrightarrow{c} be such that 2(a→×b→)+3(b→×c→)=0→2(\overrightarrow{a}\times\overrightarrow{b}) + 3(\overrightarrow{b}\times\overrightarrow{c}) =\overrightarrow{0}. If a→⋅c→=15\overrightarrow{a}\cdot\overrightarrow{c}= 15, then c→⋅(i^+j^−3k^)\overrightarrow{c}\cdot (\widehat{i}+\widehat{j}- 3\widehat{k}) is equal to :

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  2. Question 2

    This type asked 13 times: 2021 Paper 18, 25 Jun 2022, 28 Jun 2022, 29 Jun 2022, 25 Jul 2022, 24 Jan 2023, 25 Jan 2023, 27 Jan 2024, 31 Jan 2024, 6 Apr 2024, 2 Apr 2025, 21 Jan 2026 S1, 6 Apr 2026 S1. This one: 6 Apr 2026 S1 See all 13 of this type

    73
    If a→=i^+j^+k^,b→=j^−k^\overrightarrow{a} = \widehat{i} + \widehat{j} + \widehat{k},\overrightarrow{b} = \widehat{j} - \widehat{k} and c→\overrightarrow{c} be three vectors such that a→×c→=b→\overrightarrow{a} \times \overrightarrow{c} = \overrightarrow{b} and a→⋅c→=3\overrightarrow{a} \cdot \overrightarrow{c} = 3, then c→⋅(a→−2b→)\overrightarrow{c} \cdot (\overrightarrow{a} - 2\overrightarrow{b}) is equal to ____\_\_\_\_ .

  3. Question 3

    This type asked 10 times: 2021 Paper 17, 2021 Paper 23, 2021 Paper 25, 27 Jul 2022, 29 Jul 2022, 25 Jan 2023, 30 Jan 2023, 1 Feb 2024, 6 Apr 2024, 23 Jan 2026 S1. This one: 23 Jan 2026 S1 See all 10 of this type

    53
    Let a→=−i^+j^+2k^,b→=i^−j^−3k^,c→=a→×b→\overrightarrow{a}= -\widehat{i}+\widehat{j}+ 2\widehat{k},\overrightarrow{b}=\widehat{i}-\widehat{j}- 3\widehat{k},\overrightarrow{c}=\overrightarrow{a}\times\overrightarrow{b} and d→=c→×a→\overrightarrow{d}=\overrightarrow{c}\times\overrightarrow{a}. Then (a→−b→)⋅d→(\overrightarrow{a}-\overrightarrow{b}) \cdot\overrightarrow{d} is equal to :

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  4. Question 4

    This type asked 8 times: 2021 Paper 10, 27 Jun 2022, 29 Jul 2022, 30 Jan 2023, 30 Jan 2024, 31 Jan 2024, 7 Apr 2025, 4 Apr 2026 S2. This one: 4 Apr 2026 S2 See all 8 of this type

    65
    Let u^\widehat{u} and v^\widehat{v} be unit vectors inclined at an acute angle such that ∣u^×v^∣=32|\widehat{u}\times\widehat{v}| =\frac{\sqrt{3}}{2}. If A→=λu^+v^+(u^×v^)\overrightarrow{A}=\lambda\widehat{u}+\widehat{v}+ (\widehat{u}\times\widehat{v}). Then λ\lambda is equal to :

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  5. Question 5

    This type asked 8 times: 2021 Paper 1, 2021 Paper 8, 2021 Paper 24, 26 Jun 2022, 28 Jun 2022, 24 Jan 2025, 24 Jan 2026 S2, 2 Apr 2026 S2. This one: 2 Apr 2026 S2 See all 8 of this type

    64
    Let the vectors a→=−i^+j^+3k^\overrightarrow{a}= -\widehat{i}+\widehat{j}+ 3\widehat{k} and b→=i^+3j^+k^\overrightarrow{b}=\widehat{i}+ 3\widehat{j}+\widehat{k}. For some λ,μ∈R\lambda,\mu\in R, let c→=λa→+μb→\overrightarrow{c}=\lambda\overrightarrow{a}+\mu\overrightarrow{b}. If c→⋅(3i^−6j^+2k^)=10\overrightarrow{c}\cdot (3\widehat{i}- 6\widehat{j}+ 2\widehat{k}) = 10 and c→⋅(i^+j^+k^)=−2\overrightarrow{c}\cdot (\widehat{i}+\widehat{j}+\widehat{k}) = - 2, then ∣c→∣2|\overrightarrow{c}|^{2} is equal to :

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  6. Question 6

    This type asked 7 times: 2021 Paper 4, 2021 Paper 20, 28 Jun 2022, 13 Apr 2023, 4 Apr 2024, 9 Apr 2024, 22 Jan 2026 S2. This one: 22 Jan 2026 S2 See all 7 of this type

    58
    Let a→=2i^−j^+k^\overrightarrow{a}= 2\widehat{i}-\widehat{j}+\widehat{k} and b→=λj^+2k^,λ∈Z\overrightarrow{b}= \lambda\widehat{j}+ 2\widehat{k},\lambda \in Z be two vectors, Let c→=a→×b→\overrightarrow{c}=\overrightarrow{a}\times\overrightarrow{b} and d→\overrightarrow{d} be a vector of magnitude 2 in yz-plane. If ∣c→∣=53|\overrightarrow{c}| =\sqrt{53}, then the maximum possible value of (c→⋅d→)2(\overrightarrow{c}\cdot\overrightarrow{d})^{2} is equal to :

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  7. Question 7

    This type asked 6 times: 1 Feb 2023, 8 Apr 2023, 30 Jan 2024, 5 Apr 2024, 6 Apr 2024, 9 Apr 2024. This one: 9 Apr 2024 See all 6 of this type

    170
    Let a→=2i^+αj^+k^, b→=−i^+k^, c→=βj^−k^\overrightarrow{a}= 2\widehat{i}+ \alpha\widehat{j}+\widehat{k},\ \overrightarrow{b}= -\widehat{i}+\widehat{k},\ \overrightarrow{c}= \beta\widehat{j}-\widehat{k}, where α\alpha and β\beta are integers and αβ=−6\alpha\beta = - 6. Let the values of the ordered pair (α,β)(\alpha,\beta) for which the area of the parallelogram of diagonals a→+b→\overrightarrow{a}+\overrightarrow{b} and b→+c→\overrightarrow{b}+\overrightarrow{c} is 212\frac{\sqrt{21}}{2}, be (α1,β1)\left( \alpha_{1},\beta_{1} \right) and (α2,β2)\left( \alpha_{2},\beta_{2} \right). Then α12+β12−α2β2\alpha_{1}^{2}+\beta_{1}^{2}-\alpha_{2}\beta_{2} is equal to

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  8. Question 8

    This type asked 6 times: 2021 Paper 8, 28 Jul 2022, 29 Jan 2023, 8 Apr 2023, 12 Apr 2023, 15 Apr 2023. This one: 15 Apr 2023 See all 6 of this type

    79
    Let SS be the set of all (λ,μ)(\lambda,\mu) for which the vectors λi^−j^+k^,i^+2j^+μk^\lambda\widehat{i}-\widehat{j}+\widehat{k},\widehat{i}+ 2\widehat{j}+ \mu\widehat{k} and 3i^−4j^+5k^3\widehat{i}- 4\widehat{j}+ 5\widehat{k}, where λ−μ=5\lambda - \mu = 5, are coplanar, then ∑(λ,μ)∈S80(λ2+μ2)\sum_{(\lambda,\mu) \in S} 80\left( \lambda^{2}+\mu^{2} \right) is equal to :

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  9. Question 9

    This type asked 5 times: 27 Jan 2024, 29 Jan 2024, 28 Jan 2025, 28 Jan 2026 S1, 28 Jan 2026 S2. This one: 28 Jan 2026 S2 See all 5 of this type

    67
    Let P be a point in the plane of the vector AB→=3i^+j^−k^\overrightarrow{AB}= 3\widehat{i}+\widehat{j}-\widehat{k} and AC→=i^−j^+3k^\overrightarrow{AC}=\widehat{i}-\widehat{j}+ 3\widehat{k} such that PP is equidistant from the lines AB and AC . If ∣AP→∣=52|\overrightarrow{AP}| =\frac{\sqrt{5}}{2}, then the area of the triangle ABPABP is :

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  10. Question 10

    This type asked 5 times: 2021 Paper 22, 4 Apr 2024, 22 Jan 2025, 7 Apr 2025, 28 Jan 2026 S1. This one: 28 Jan 2026 S1 See all 5 of this type

    69
    For three-unit vectors a→,b→,c→\overrightarrow{a},\overrightarrow{b},\overrightarrow{c} satisfying ∣a→−b→∣2+∣b→−c→∣2+∣c→−a→∣2=9 and |\overrightarrow{a}-\overrightarrow{b}|^{2}+ |\overrightarrow{b}-\overrightarrow{c}|^{2}+ |\overrightarrow{c}-\overrightarrow{a}|^{2}= 9\text{~and~} ∣2a→+kb→+kc→∣=3|2\overrightarrow{a}+k\overrightarrow{b}+k\overrightarrow{c}| = 3 the positive value of kk is :

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