[BA 3rd Sem Question Papers, Dibrugarh University, 2013, Mathematics, Major, Analysis - I (Real Analysis]
GROUP – A
(Differential Calculus)
(Marks: 35)
1. (a) If 
then write the value of 
is the positive integer. 1
then write the value of is the positive integer. 1
(b) Find the value of 2
(c) Show that the subnormal at any point of a parabola is of constant length. 3
Or
Find the curvature of the ellipse 
and 
at 
and at
(d) If 
then show that 4
then show that 4
Or
If 
is the angle between the radius vector and the tangent at any point of the curve 
then prove that 
is the angle between the radius vector and the tangent at any point of the curve then prove that
2. (a) State Darboux’s theorem in differential calculus. 1
(b) If a function 
is continuous in 
and differentiable in 
such that 
then prove that 
is strictly increasing on. 2
is continuous in and differentiable in such that then prove that is strictly increasing on. 2
(c) State and prove Cauchy’s mean value theorem. 3
Of
If 
is continuous on and differentiable in and 
then show that
is continuous on and differentiable in and then show that 
(d) Prove that 
4
4
Or
Expand 
in powers of 
for 
in powers of for
3. (a) State and Euler’s theorem on homogeneous function of two variables. 1
(b) If 
then prove that 
4
then prove that 4
Or
If 
, then prove that
, then prove that 
4. (a) State Schwarz’s theorem. 1
(b) If 
then show that 2
then show that 2
(c) If 
then prove that 
3
then prove that 3
(d) Prove that 
has neither maximum nor minimum at 
4
has neither maximum nor minimum at 4
Or
If 
, where 
, then prove that
, where , then prove that 
GROUP – B
(Integral Calculus)
(Marks: 20)
5. (a) If 
then write the relation between 
and 
1
then write the relation between and 1
(b) Show that 
2
2
(c) Evaluate: 
3
3
(d) Deduce the reduction formula for 
4
4
Or
Prove that 
6. (a) If 
then write the length of arc 
where A and B have abscissa a and b respectively. 1
then write the length of arc where A and B have abscissa a and b respectively. 1
(b) Find the whole length of the curve 
5
5
Or
Find the volume of the solid generated by revolving the ellipse 
about the major axis.
about the major axis.
(c) Find the surface of the solid formed by revolving the cardioids 
about the initial line. 4
about the initial line. 4
GROUP – C
(Riemann Integral)
(Marks: 25)
7. (a) What is the essential condition for a function 
to be Riemann integrable in [a, b]? 1
to be Riemann integrable in [a, b]? 1
(b) Prove that for any two partitions 
and 
of
. 2
and of. 2
(c) Show that every monotonic function on 
is Riemann integrable. 5
is Riemann integrable. 5
Or
If 
and 
are lower and upper bounds of 
on
, then show that
and are lower and upper bounds of on, then show that 
8. (a) State first mean value theorem of integral calculus. 1
(b) A function having primitive may not be continuous. Justify it. 3
(c) If 
and g are integrable on 
and
keeps the same sign over , then show that 
and 
where m, M are bounds of g on . 3
and g are integrable on andkeeps the same sign over , then show that and where m, M are bounds of g on . 3
9. (a) Define Abel’s test for convergence of integral of a product. 1
(b) Show that
Converges to 0 2
(c) Test for convergence (any one): 3
Also Read: Dibrugarh University Question Papers
10. (a) Prove that 4
(b) Prove that
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