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Paper Code: 41152
B-1502
M.Sc. Mathematics (Semester First) Examination, 2025
Real Analysis
Real Analysis Time: Three Hours ] [Maximum Marks: 70
Note: Attempt all the sections as per instructions.
Section-A
(Short Answer Questions)
Note: Attempt any four questions out of the following.
Each question carries 07 marks. Answer are required not excceding 300 words.
1.Show that the sequence <fn> where f_{n}(x) = nx * (1 – x) ^ n does not converge uniformly on [0,1].
2. State and prove Weierstrass’s M-test for uniform convergence
3. A continuous function defined on a measurable set is measurable.
4. The lower R-integral cannot exceed the upper R-integrable i.e. int a ^ b f <= integrate f df from a to – b
5. Let f be a continuous on [a, b] and let g be continuous and non-decreasing on [a, b] then f \in RS(g)
6. Let f(x) = x g(x) = x ^ 2 Does int b ^ 1 exist? If it exists, find its value.
7. Let f(x, y) = g(x) where g is nowhere differentiable. Show that f xy exist and is continuous and yet f_{x} does not exist.
8. If x + y + z = u y + z = uv z = uvw find the value of the Jacobian of x,y,z with regard to u,v,w.
M.Sc Maths 1st Year Question Paper 2025 Pdf
Section-B
(Long Answer Questions)
Note: Attempt any three questions out of following.
Each questions carries 14 marks.
9. Let f be a continuous function defined on [a,b]. Then there exists a sequence of polynomials which converges uniformly to f on [a,b]. 10. Show that the sequence {f} where f(x)=1+nx converges uniformly on R.
11. State and prove Beppo Levi’s theorem.
12. Let f be a bounded and g a non-decreasing function on [a,b]. Then feRS(g) if and only if for every €>0 there exists a partition p such that U(p,f,g)-L(p,f,g)<e.
13. If f(x,y) is differentiable of a point (a,b) then it is continuous at [a,b] but converse need not be true.
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