Descriptive type questions on Mathematics competitive examinations

Here we list some of the descriptive type questions asked in various competitive examinations on higher mathematics. We hope these questions will be helpful for the aspirants preparing for upcoming TN Arts and Science college recruitment exam.  

(The list given is not a final one. Visit the page frequently, as the questions will be updated often.)

You can type more questions(missed here) in comment box. They also will be updated.   

(Updated on 07 June 2024)

Algebra:

1. Prove that every group of order $p^2$ is abelian, where $p$ is a prime number.

2. P. T. any group of order 65 must be cyclic. 

3. S. T. group of order 30 has a subgroup of order 15. Is it necessarily normal?

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Linear Algebra:

1.  If T is an nxn matrix that commutes with all nxn matrices, then prove that T is a scalar matrix.

2. Obtain the characteristic polynomial and minimal polynomial for a given matrix.

3. Let A and B be two 3 × 3 complex matrices. Show that A and B are similar if and
only if χA = χB and μA = μB , where χA, χB are characteristic polynomials of A, B,
respectively and μA, μB are minimal polynomials of A, B, respectively.

4.  Find the spectrum of a nilpotent matrix.

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Real analysis:

1.  P. T. $f_n=n^2x^n(1-x)$ on [0,1] converges p.w. to zero but not uniformly.

2. P. T. $x+\sin x$ is a homeomorphism from $\mathbb{R}$ to $\mathbb{R}$.

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Measure theory:

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Complex analysis:

1. Determine the number of roots of the polynomial $3z^7+5z-1$ in the annular region $1<|z|<2.$ 

2. Prove that the function $u(x,y)=e^{2xy}\cos (x^2-y^2)$ is harmonic, and find the harmonic conjugate.

3. Compute $\int_{|z|=2}e^{e^{\frac{1}{z}}} dz$, where the circle is parametrized by  $t → 2e^{it}, 0 \le t \le 2π$.
 

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Topology:

1. Prove that every closed subspace of a compact space is compact. Also prove that an open subspace of a compact space need not be compact. 

2.Prove that if a topological space X is Hausdorff, then the diagonal $\Delta=\{(x,x):x\in X\}$ in XxX is closed. What about the converse? 

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Functional analysis:

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Ordinary Differential equations:

1. Using Greens function, find the solution of the ODE $y''+y'=1$, $y(0)=0$, $y(\pi/2)=0.$  

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Numerical methods on Ordinary Differential equations:

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Partial differential equations:

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Classical Mechanics:

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Calculus of variations

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Differential Geometry

Linear Integral equations

Operations Research

Probability and Statistics:

1. Obtain the CDF and desired probabilities from given PDF. 

2. Problems on Bayes theorem.

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