Linear Algebra and Tensors - Mathematical Physics - Physics Previous Year Questions

1

2007 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2007

The eigenvalues of a matrix are \( i, -2i \) and \( 3i \). The matrix is

A

unitary

B

anti-unitary

C

Hermitian

D

anti-Hermitian

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2

2007 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2007

The eigenvalues of a matrix are i, -2i and 3i. The matrix is

A

unitary.

B

anti-unitary.

C

Hermitian.

D

anti-Hermitian.

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3

2007 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2007

The eigenvalues and eigenvectors of the matrix \(\begin{bmatrix} 5 & 4 \\ 1 & 2 \end{bmatrix}\) are

A

\(6, 1\) and \(\begin{bmatrix} 4 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \end{bmatrix}\)

B

\(2, 5\) and \(\begin{bmatrix} 4 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \end{bmatrix}\)

C

\(6, 1\) and \(\begin{bmatrix} 1 \\ 4 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \end{bmatrix}\)

D

\(2, 5\) and \(\begin{bmatrix} 1 \\ 4 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \end{bmatrix}\)

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4

2007 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2007

If \(\vec{r} = x \hat{i} + y \hat{j}\), then

A

\(\vec{\nabla} \cdot \vec{r} = 0\) and \(|\vec{\nabla} \times \vec{r}| = r\)

B

\(\vec{\nabla} \cdot \vec{r} = 2\) and \(|\vec{\nabla} \times \vec{r}| = r\)

C

\(\vec{\nabla} \cdot \vec{r} = 2\) and \(|\vec{\nabla} \times \vec{r}| = \frac{r}{2}\)

D

\(\vec{\nabla} \cdot \vec{r} = 3\) and \(|\vec{\nabla} \times \vec{r}| = \frac{r}{2}\)

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5

2007 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2007

Consider a vector \(\vec{p} = 2\hat{i} + 3\hat{j} + 2\hat{k}\) in the coordinate system \((\hat{i}, \hat{j}, \hat{k})\). The axes are rotated anti-clockwise about the Y axis by an angle of \(60^0\). The vector \(\vec{p}\) in the rotated coordinate system \((\hat{i}', \hat{j}', \hat{k}')\) is

A

\((1-\sqrt{3})\hat{i}' + 3\hat{j}' + (1+\sqrt{3})\hat{k}'\)

B

\((1+\sqrt{3})\hat{i}' + 3\hat{j}' + (1-\sqrt{3})\hat{k}'\)

C

\((1-\sqrt{3})\hat{i}' + (3+\sqrt{3})\hat{j}' + 2\hat{k}'\)

D

\((1-\sqrt{3})\hat{i}' + (3-\sqrt{3})\hat{j}' + 2\hat{k}'\)

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6

2008 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2008

For arbitrary matrices \(E, F, G\) and \(H\), if \(EF – FE = 0\) then \(Trace(EFGH)\) is equal to

A

\(Trace(HGFE)\)

B

\(Trace(E)Trace(F)Trace(G)Trace(H)\)

C

\(Trace(GFEH)\)

D

\(Trace(EGHF)\)

Check answer

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7

2008 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2008

An unitary matrix \(\begin{pmatrix} ae^{i\alpha} & b \\ ce^{i\beta} & d \end{pmatrix}\) is given, where \(a, b, c, d, \alpha\) and \(\beta\) are real. The inverse of the matrix is

A

\(\begin{pmatrix} ae^{-i\alpha} & -ce^{-i\beta} \\ b & d \end{pmatrix}\)

B

\(\begin{pmatrix} ae^{i\alpha} & ce^{i\beta} \\ b & d \end{pmatrix}\)

C

\(\begin{pmatrix} ae^{-i\alpha} & b \\ ce^{-i\beta} & d \end{pmatrix}\)

D

\(\begin{pmatrix} ae^{-i\alpha} & ce^{-i\beta} \\ b & d \end{pmatrix}\)

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8

2008 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2008

The eigenvalues of the matrix \(\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}\) are

A

\(\frac{1}{2}(\sqrt{3} \pm i)\) when \(\theta = 45^\circ\)

B

\(\frac{1}{2}(\sqrt{3} \pm i)\) when \(\theta = 30^\circ\)

C

\(\pm 1\) since the matrix is unitary

D

\(\frac{1}{\sqrt{2}}(1 \pm i)\) when \(\theta = 30^\circ\)

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9

2008 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2008

Under a certain rotation of coordinate axes, a rank-1 tensor \(v_a\) (\(a=1,2,3\)) transforms according to the orthogonal transformation defined by the relations \(v'_1 = \frac{1}{\sqrt{2}}(v_1 + v_2)\), \(v'_2 = \frac{1}{\sqrt{2}}(-v_1 + v_2)\), \(v'_3 = v_3\). Under the same rotation a rank-2 tensor \(T_{ab}\) would transform such that

A

\(T'_{11} = T_{11} T_{12}\)

B

\(T'_{11} = T_{11}\)

C

\(T'_{11} = T_{11} + 2T_{12} - T_{21}\)

D

\(T'_{11} = \frac{1}{2}(T_{11} + T_{22} + T_{12} + T_{21})\)

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10

2008 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2008

For \( f(x) = a_0 + a_1 x + a_2 x^2 \in V \) and \( g(x) = b_0 + b_1 x + b_2 x^2 \in V \), which one of the following constitutes an acceptable scalar product?

A

\((f,g) = a_0^2 b_0 + a_1^2 b_1 + a_2^2 b_2 \)

B

\((f,g) = a_0^2 b_0^2 + a_1^2 b_1^2 + a_2^2 b_2^2 \)

C

\((f,g) = a_0 b_0 - a_1 b_1 + a_2 b_2 \)

D

\((f,g) = a_0 b_0 + \frac{a_1 b_1}{2} + \frac{a_2 b_2}{3} \)

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11

2008 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2008

Using the scalar product obtained in the above question, identify the subspace of \( V \) that is orthogonal to \( (1 + x) \).

A

\(\{ f(x) : b(1 - x) + cx^2 ; b, c \in R \} \)

B

\(\{ f(x) = b(1 - 2x) + cx^2 ; b, c \in R \} \)

C

\(\{ f(x) : b + cx^2 ; b, c \in R \} \)

D

\(\{ f(x) : bx + cx^2 ; b, c \in R \} \)

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12

2009 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2009

The value of the contour integral, \(\left| \oint_C \vec{r} \times d\vec{\theta} \right|\), for a circle C of radius r with center at the origin is

A

\(2\pi r\)

B

\(\frac{r^2}{2}\)

C

\(\pi r^2\)

D

\(r\)

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13

2009 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2009

The eigenvalues of the matrix \( A = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} \) are

A

real and distinct

B

complex and distinct

C

complex and coinciding

D

real and coinciding

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14

2009 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2009

Let \( T_{ij} = \sum_k \varepsilon_{ijk} a_k \) and \( \beta_i = \sum_{l,j} \varepsilon_{ijl} T_{ij} \), where \( \varepsilon_{ijk} \) is the Levi-Civita density, defined to be zero if two of the indices coincide and +1 and −1 depending on whether \( jik \) is even or odd permutation of 1,2,3. Then \( \beta_i \) is equal to

A

\(2a_i \)

B

\(-2a_i \)

C

\(a_i \)

D

\(-a_i \)

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15

2009 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2009

Consider the set of vectors in three-dimensional real vector space \( \mathbb{R}^3 \), \( S = \{(1,1,1), (1,-1,1), (1,1,-1)\} \). Which one of the following statements is true?

A

\(S \) is not a linearly independent set.

B

\(S \) is a basis for \(\mathbb{R}^3 \).

C

The vectors in \(S \) are orthogonal.

D

An orthogonal set of vectors cannot be generated from \(S \).

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16

2010 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2010

Consider an anti-symmetric tensor \( P_{ij} \) with the indices \( i \) and \( j \) running from 1 to 5. The number of independent components of the tensor is

A

3

B

10

C

9

D

6

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17

2010 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2010

The eigenvalues of the matrix \( \begin{pmatrix} 2 & 3 & 0 \\ 3 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} \) are

A

5, 2, -2

B

-5, -1, -1

C

5, 1, -1

D

-5, 1, 1

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18

2010 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2010

25 persons are in a room. 15 of them play hockey, 17 of them play football and 10 of them play both hockey and football. Then the number of persons playing neither hockey nor football is:

A

2

B

17

C

13

D

3

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19

2010 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2010

The question below consists of a pair of related words followed by four pairs of words. Select the pair that best expresses the relation in the original pair.
Unemployed : Worker

A

fallow : land

B

unaware : sleeper

C

wit : jester

D

renovated : house

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20

2011 · Physics · Mathematical Physics · Linear Algebra and Tensors

Physics (PH) 2011

Two matrices A and B are said to be similar if \( B = P^{-1}AP \) for some invertible matrix P. Which of the following statements is NOT TRUE?

A

Det A = Det B

B

Trace of A = Trace of B

C

A and B have the same eigenvectors

D

A and B have the same eigenvalues

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