Skip to main content

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 10 Nov 2022)

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?

********************

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.

********************

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 $\mathbb{R}$.

********************
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  $t → 2e^{it}, 0 \le t \le 2Ï€$.
 

********************

Differential equations:

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

********************

Probability and Statistics:

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

2. Problems on Bayes theorem.

********************



Comments

Post a Comment

Popular posts from this blog

Countable and Uncountable

Here we collect the various examples given in books and options asked in CSIR, GATE, TRB, JAM examinations about countable sets.  Find which of the following sets are countable?  1. The set of rational numbers $\mathbb{Q}$  2. The set of natural numbers $\mathbb{N}$ 3. The set of real numbers $\mathbb{R}$ 4. The set of prime numbers $\mathbb{P}$ 5. The set of complex numbers with unit modulus. 6. The set of algebraic numbers. 7. The set of transcendental numbers. 8. The set of all polynomials with integer coefficients. 9. The set of all polynomials with rational coefficients. 10. The set of all polynomials over $\mathbb{R}$ with rational roots. 11. The set of all monic polynomials over $\mathbb{R}$ with rational roots. 12. The product set $\mathbb{N} \times \mathbb{N}$ 13. The product set $\mathbb{N} \times \mathbb{R}$ 14. $\wp(\mathbb{N})$, The set of all subsets of $\mathbb{N}$  15. The set of all finite subsets of $\mathbb{N}$  16. The set of all functions from   $\mathbb{N}$ to $\
Post a Comment
Read more

NBHM Problem...

Let T  be a nilpotent linear operator on the vector space $\mathbb{ R}^5$  (i.e., $ T^k=0$  for some k). Let $d_i$  denote the dimension of the kernel of $ T^i$ . Which of the following can possibly occur as a value of $ (d_1,d_2,d_3)$ ?  1. (1,2,3) 2. (2,3,5) , 3. (2,2,4) , 4. (2,4,5) For the excellent proof using Jordan Canonical form visit  https://nbhmcsirgate.theindianmathematician.com/2020/04/nbhm-2020-part-c-question-26-solution.html#.XqCM3QR0R10.whatsapp Here we present a proof (incomplete in preciousness) on the fact "A linear transformation is completely determined by its behaviour on a basis."
Post a Comment
Read more