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Determinants

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 26 times: 1 in 2021, 5 in 2022, 5 in 2023, 7 in 2024, 4 in 2025, 4 in 2026. This one: 8 Apr 2026 S2 See all 26 of this type

    53
    If the system of linear equations : x+y+z=6x+y+z= 6, x+2y+5z=10x+ 2y+ 5z= 10, 2x+3y+λz=μ2x + 3y +\lambda z =\mu. has infinitely many solutions, then the value of λ+μ\lambda+\mu equals.

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  2. Question 2

    This type asked 19 times: 2021 Paper 4, 2021 Paper 19, 26 Jun 2022, 28 Jun 2022, 27 Jul 2022, 24 Jan 2023, 25 Jan 2023, 8 Apr 2023, 10 Apr 2023, 13 Apr 2023, 15 Apr 2023, 31 Jan 2024, 5 Apr 2024, 6 Apr 2024, 9 Apr 2024, 22 Jan 2025, 2 Apr 2025, 3 Apr 2025, 7 Apr 2025. This one: 7 Apr 2025 See all 19 of this type

    65
    Let AA be a 3×33 \times 3 matrix such that ∣adj(adj(adjA))∣=81|adj(adj(adjA))| = 81. If
    S={n∈Z:(∣adj(adjA)∣)(n−1)22=∣A∣(3n2−5n−4)}S =\left\{ n \in\mathbb{Z}:(|adj(adjA)|)^{\frac{(n- 1)^{2}}{2}}= |A|^{\left( 3n^{2}- 5n- 4 \right)} \right\}
    then ∑n∈S∣A(n2+n)∣\sum_{n\in S}^{} \left| A^{\left( n^{2}+n \right)} \right| is equal to.

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  3. Question 3

    This type asked 14 times: 2021 Paper 1, 2021 Paper 21, 2021 Paper 22, 2021 Paper 24, 14 Jun 2022, 25 Jun 2022, 26 Jun 2022, 27 Jun 2022, 28 Jun 2022, 25 Jul 2022, 10 Apr 2023, 23 Jan 2025, 21 Jan 2026 S2, 2 Apr 2026 S1. This one: 2 Apr 2026 S1 See all 14 of this type

    53
    Let α,β∈R\alpha,\beta\in R be such that the system of linear equations
    x+2y+z=5x+ 2y+z= 5
    2x+y+αz=52x+y+\alpha z= 5
    8x+4y+βz=188x+ 4y+\beta z= 18
    has no solution. Then βα\frac{\beta}{\alpha} is equal to :

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  4. Question 4

    This type asked 9 times: 2021 Paper 2, 2021 Paper 26, 25 Jan 2023, 29 Jan 2023, 31 Jan 2023, 6 Apr 2023, 10 Apr 2023, 13 Apr 2023, 30 Jan 2024. This one: 30 Jan 2024 See all 9 of this type

    73
    Consider the system of linear equations x+y+z=4μ, x+2y+2λz=10μ, x+3y+4λ2z=μ2+15x+y+z=4\mu,\ x+2y+2\lambda z=10\mu,\ x+3y+4\lambda^{2}z=\mu^{2}+15, where λ,μ∈R\lambda,\mu\in R. Which one of the following statements is NOT correct?

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  5. Question 5

    This type asked 8 times: 2021 Paper 10, 2021 Paper 23, 27 Jun 2022, 8 Apr 2023, 4 Apr 2024, 6 Apr 2024, 5 Apr 2026 S1, 6 Apr 2026 S2. This one: 6 Apr 2026 S2 See all 8 of this type

    54
    The sum of all possible values of θ∈[0,2π]\theta\in \lbrack 0,2\pi\rbrack, for which the system of equations :
    xcos3θ−8y−12z=0xcos3\theta- 8y- 12z= 0
    xcos2θ+3y+3z=0xcos2\theta+ 3y+ 3z= 0
    x+y+3z=0x+y+ 3z= 0
    has a non-trivial solution, is equal to:

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  6. Question 6

    This type asked 6 times: 2021 Paper 6, 2021 Paper 9, 2021 Paper 16, 2021 Paper 18, 1 Feb 2023, 29 Jan 2025. This one: 29 Jan 2025 See all 6 of this type

    52
    Let M and m respectively be the maximum and the minimum values of
    f(x)=∣1+sin⁡2xcos⁡2x4sin⁡4xsin⁡2x1+cos⁡2x4sin⁡4xsin⁡2xcos⁡2x1+4sin⁡4x∣,x∈Rf(x) =\left| \begin{matrix} 1 +\sin^{2}x & \cos^{2}x & 4\sin4x \\ \sin^{2}x & 1 +\cos^{2}x & 4\sin4x \\ \sin^{2}x & \cos^{2}x & 1 + 4\sin4x \end{matrix} \right|,x \in R
    Then M4−m4M^{4}-m^{4} is equal to :

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  7. Question 7

    This type asked 5 times: 2021 Paper 3, 2021 Paper 12, 30 Jan 2023, 23 Jan 2025, 29 Jan 2025. This one: 29 Jan 2025 See all 5 of this type

    136
    Let α,β(α≠β)\alpha,\beta(\alpha \neq \beta) be the values of m , for which the equations x+y+z=1;x+2y+4z=mx + y + z = 1;x + 2y + 4z = m and x+4y+10z=m2x + 4y + 10z =m^{2} have infinitely many solutions. Then the value of ∑n=110(nα+nβ)\sum_{n = 1}^{10} \left( n^{\alpha}+n^{\beta} \right) is equal to :

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  8. Question 8

    This type asked 4 times: 27 Jun 2022, 30 Jan 2024, 31 Jan 2024, 3 Apr 2025. This one: 3 Apr 2025 See all 4 of this type

    61
    If y(x)=∣sin⁡xcos⁡xsin⁡x+cos⁡x+1272827111∣,x∈Ry(x) =\left| \begin{matrix} \sin x & \cos x & \sin x+ \cos x+ 1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{matrix} \right|,x\in\mathbb{R}, then d2ydx2+y\frac{d^{2}y}{dx^{2}}+ y is equal to.

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  9. Question 9

    This type asked 3 times: 12 Apr 2023, 6 Apr 2024, 5 Apr 2026 S2. This one: 5 Apr 2026 S2 See all 3 of this type

    53
    If f:N→Zf:N \rightarrow Z is defined by f(n)=∣n−1−5−2n23(2k+1)2k+1−3n33k(2k+1)3k(k+2)+1∣,k∈Nf(n) =\left| \begin{matrix} n & - 1 & - 5 \\ - 2n^{2} & 3(2k+ 1) & 2k+ 1 \\ - 3n^{3} & 3k(2k+ 1) & 3k(k+ 2) + 1 \end{matrix} \right|,k\in N, and ∑n=1kf(n)=98\sum_{n= 1}^{k} f(n) = 98, then kk is equal to:

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  10. Question 10

    This type asked 3 times: 2021 Paper 19, 24 Jan 2023, 22 Jan 2026 S2. This one: 22 Jan 2026 S2 See all 3 of this type

    51
    Let n be the number obtained on rolling a fair die. If the probability that the system x−ny+z=6x - ny + z = 6, x+(n−2)y+(n+1)z=8x + (n - 2)y + (n + 1)z = 8, (n−1)y+z=1(n - 1)y + z = 1 has a unique solution is k6\frac{k}{6}, then the sum of k and all possible values of nn is :

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