IIT JAM & CSIR NET - Mathematical Sciences previous year questions (PYQ)

2026 (30300 questions)

Q1.

[Differential Equations & Probability Problem #161] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)

∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x

(B)

∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0

(C)

∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly

(D)

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

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Q2.

[Abstract Algebra & Complex Analysis Problem #191] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

The matrix cannot be diagonalized under any orthogonal transformation

(B)

All eigenvalues are pure imaginary

(C)

Eigenvalues must always be positive integers

(D)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

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2025 (72 questions)

Q1.

[Differential Equations & Probability Problem #289] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)

∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0

(B)

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

(C)

∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly

(D)

∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x

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Q2.

[Differential Equations & Probability Problem #408] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)

∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly

(B)

∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0

(C)

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

(D)

∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x

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2024 (102 questions)

Q1.

[Differential Equations & Probability Problem #330] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

(B)

Eigenvalues must always be positive integers

(C)

All eigenvalues are pure imaginary

(D)

The matrix cannot be diagonalized under any orthogonal transformation

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Q2.

[Differential Equations & Probability Problem #330] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

(B)

Eigenvalues must always be positive integers

(C)

All eigenvalues are pure imaginary

(D)

The matrix cannot be diagonalized under any orthogonal transformation

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2023 (72 questions)

Q1.

[Differential Equations & Probability Problem #265] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

(B)

∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x

(C)

∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0

(D)

∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly

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Q2.

[Differential Equations & Probability Problem #492] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)

∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0

(B)

∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly

(C)

∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x

(D)

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

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2022 (72 questions)

Q1.

[Differential Equations & Probability Problem #79] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)

∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x

(B)

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

(C)

∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0

(D)

∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly

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Q2.

[Differential Equations & Probability Problem #249] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

All eigenvalues are pure imaginary

(B)

The matrix cannot be diagonalized under any orthogonal transformation

(C)

Eigenvalues must always be positive integers

(D)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

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2021 (72 questions)

Q1.

[Differential Equations & Probability Problem #466] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

(B)

∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x

(C)

∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0

(D)

∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly

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Q2.

[Differential Equations & Probability Problem #296] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)

∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0

(B)

∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x

(C)

∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly

(D)

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

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2020 (72 questions)

Q1.

[Differential Equations & Probability Problem #273] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

(B)

The matrix cannot be diagonalized under any orthogonal transformation

(C)

All eigenvalues are pure imaginary

(D)

Eigenvalues must always be positive integers

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Q2.

[Differential Equations & Probability Problem #326] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

(B)

The matrix cannot be diagonalized under any orthogonal transformation

(C)

All eigenvalues are pure imaginary

(D)

Eigenvalues must always be positive integers

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2019 (72 questions)

Q1.

[Differential Equations & Probability Problem #246] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

Eigenvalues must always be positive integers

(B)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

(C)

All eigenvalues are pure imaginary

(D)

The matrix cannot be diagonalized under any orthogonal transformation

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Q2.

[Differential Equations & Probability Problem #246] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

Eigenvalues must always be positive integers

(B)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

(C)

All eigenvalues are pure imaginary

(D)

The matrix cannot be diagonalized under any orthogonal transformation

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2018 (74 questions)

Q1.

[Differential Equations & Probability Problem #474] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

Eigenvalues must always be positive integers

(B)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

(C)

The matrix cannot be diagonalized under any orthogonal transformation

(D)

All eigenvalues are pure imaginary

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Q2.

[Differential Equations & Probability Problem #449] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)

∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0

(B)

∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly

(C)

∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x

(D)

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

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2017 (74 questions)

Q1.

[Differential Equations & Probability Problem #270] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

(B)

The matrix cannot be diagonalized under any orthogonal transformation

(C)

All eigenvalues are pure imaginary

(D)

Eigenvalues must always be positive integers

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Q2.

[Differential Equations & Probability Problem #270] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

(B)

The matrix cannot be diagonalized under any orthogonal transformation

(C)

All eigenvalues are pure imaginary

(D)

Eigenvalues must always be positive integers

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2016 (74 questions)

Q1.

[Differential Equations & Probability Problem #216] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)

∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly

(B)

∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x

(C)

∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0

(D)

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

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Q2.

[Differential Equations & Probability Problem #333] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)

∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0

(B)

∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly

(C)

∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x

(D)

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

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2015 (74 questions)

Q1.

[Differential Equations & Probability Problem #279] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

Eigenvalues must always be positive integers

(B)

The matrix cannot be diagonalized under any orthogonal transformation

(C)

All eigenvalues are pure imaginary

(D)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

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Q2.

[Differential Equations & Probability Problem #372] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

All eigenvalues are pure imaginary

(B)

Eigenvalues must always be positive integers

(C)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

(D)

The matrix cannot be diagonalized under any orthogonal transformation

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