imsc.res.in NBHM MA/MSc Scholarship Test Old Question Papers : Institute of Mathematical Sciences

Organisation : The Institute of Mathematical Sciences
Exam : NBHM MA/MSc Scholarship Test
Document Type : Old Question Papers
Location : India
Website : imsc.res.in

Download Model/Sample Question Papers:
1. October 2005 : https://www.pdfquestion.in/uploads/7251-nbhm_msc05.pdf
2. September 2006 : https://www.pdfquestion.in/uploads/7251-nbhm_msc06.pdf
3. September 2007 : https://www.pdfquestion.in/uploads/7251-nbhm_msc07.pdf
4. September 2008 : https://www.pdfquestion.in/uploads/7251-msc08.pdf
5. September 2009 : https://www.pdfquestion.in/uploads/7251-msc09.pdf
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9. September 2013 : https://www.pdfquestion.in/uploads/7251-msc13.pdf
10. September 2014 : https://www.pdfquestion.in/uploads/7251-msc14.pdf

IMSC NBHM MA/MSc Scholarship Tests Question Papers

1. Pick out the true statements:
(a) Every polynomial of odd degree with real coefficients has a real root.
(b) Every non-constant polynomial with real coefficients can be factorised such that every factor has real co effcients and is of degree at most two.

Related : Institute of Mathematical Sciences NBHM PhD Scholarship Test Model Question Paper 2017 : www.pdfquestion.in/10007.html

2. Let N > 1 be a positive integer. What is the arithmetic mean of all positive integers less than N and prime to it (including unity)?

3. Pick out the true statements:
(a) Every group of order 36 is abelian.
(b) A group in which every element is of order at most 2 is abelian.

4. How many abelian groups of order 8 are there (up to isomorphism)?

5. Let F and F’ be two finite fields with nine and four elements respectively. How many field homomorphisms are there from F to F’?

6. How many fields are there (up to isomorphism) with exactly 6 elements?

7. Let R be a ring and let p be a polynomial of degree n with coefficients in R. Then p has at most n roots in R. – True or False?

8. Pick out the true statements:
(a) A commutative integral domain is always a subring of a field.

(b) The ring of continuous real valued functions on the closed interval [0, 1] (with pointwise addition and pointwise multiplication as the ring operations) is an integral domain.

9. Let A be a 3×3 matrix with real entries which commutes with all 3×3 matrices with real entries. What is the maximum number of distinct roots that the characteristic polynomial of A can have?

10.Pick out the true statements:
(a) Let A be a hermitian N ×N positive definite matrix. Then, there exists a hermitian positive definite N × N matrix B such that B2 = A.

(b) Let B be a nonsingular N × N matrix with real entries. Let B’ be its transpose. Then B’B is a symmetric and positive definite matrix.

11. Let A be a singular N ×N matrix with real entires and let b be a N ×1 matrix (i.e. a column vector). Let A’ and b’ be their respective transposes. Consider the system of N linear equations in N unknowns written in matrix form as:

A’x = b.

Complete the following sentence: The above linear system has a solution, if, and only if, b’u = 0, for all column vectors u such that …

12. A point moves so that the sum of its distances from two fixed points is a constant. What is the path traced by this point?

13 . A point moves so that the line segments joining it to the points (2, 0) and (0, 2) are always perpendicular. What is the equation of its path?

14. Let ABCD be a parallellogram in the plane with area 1 and such that B and C lie on the x-axis and A and D lie on the line y = h. With BC as base, triangles BCE of area 1/4 are constructed. Write down the equation of the locus of E.

15. Do the lines x – 1 = y – 2 = z – 3 and 2(x + 1) = 3(y – 1) = 6(z – 4) intersect? If yes, give the point of intersection.

16. Write down the equation of the locus of a point which moves in the xy- plane so that it is equidistant from the straight lines y = x and y = -x.

17. What is the shape of the locus of a point which moves in the plane so that it is equidistant from a given point A and a given straight line ` (which does not contain the point A)?

18. What is the area of a quadrilateral in the xy-plane whose vertices are (0, 0), (1, 0), (2, 3) and (0, 1)?

19. What is the surface area of the sphere whose equation is given by x2 + y2 + z2 – 4x + 6y – 2z + 13 = 0?