[BA 3rd Sem Question Papers, Dibrugarh University, 2015, Mathematics, Major, Analysis - I (Real Analysis)]
GROUP – A
(Differential Calculus)
(Marks: 35)
1. (a) If 
then write the value of
. 1
then write the value of. 1
(b) Evaluate: 2
(c) Find the length of the subnormal to the curve 
at the point 
3
at the point 3
(d) If
, then show that 
4
, then show that 4
Or
Find the radius of curvature at the point 
on the cycloid 
on the cycloid
2. (a) Write the statement of Darboux’s theorem. 1
(b) Show that
is continuous at 
2
2
(c) Show that 
4
4
Or
Expand 
in an infinite series using Maclaurin’s series.
in an infinite series using Maclaurin’s series.
(d) Verify Rolle’s Theorem for 
3
3
3. (a) State the Euler’s Theorem on homogeneous function of two variables. 1
(b) If
then show that
4
Or
If
then show that
4. (a) Write the statement of Schwartz’s theorem. 1
(b) if 
and 
, then show that
and , then show that 
4
(c) If z is a function of x and y and
, then prove that
, then prove that 
5
Or
Prove that
has a minimum value at 
.
GROUP – B
(Integral Calculus)
(Marks: 20)
5. (a) Write the value of
1
(b) Prove that 2
(c) Prove that 3
(d) Show that 4
Or
Using reduction formula, evaluate
6. (a) Write the formula for length of an arc between two points
and
when the curve is given in parametric form. 1
andwhen the curve is given in parametric form. 1
(b) Find the length of the arc of the curve
and
from 
to 
4
andfrom to 4
Or
Find the perimeter of the cardioids 
(c) Find the volume and surface area of the solid of revolution formed by rotation of the parabola 
about x-axis and bounded by 
5
about x-axis and bounded by 5
Or
Find the volume and surface of the solid of revolution of the ellipse
GROUP – C
(Riemann Integral)
(Marks: 25)
7. (a) Every bounded function defined on an interval [a, b] is Riemann integrable. State True or False. 1
(b) Prove that a constant function is always Riemann integrable. 3
(c) State and prove the necessary and sufficient condition for a function to be Riemann integrable. 4
Or
Prove that is a function is monotonic on [a, b], then it is Riemann integrable on [a, b].
8. (a) Define primitive of a function. 1
(b) If 
is bounded and integrable in
, and 
and 
are the bounds of in , then prove that
is bounded and integrable in, and and are the bounds of in , then prove that 
3
(c) If
Both exist and 
keeps the same sign throughout the interval, then prove that there exists a number 
between the bounds of such that
keeps the same sign throughout the interval, then prove that there exists a number between the bounds of such that 
3
9. (a) Give example of an improper integral of second kind. 1
(b) Test for convergence of 2
(c) Prove that
converges. 3
Also Read: Dibrugarh University Question Papers
10. Answer any one of the following: 4
- Prove that
Hence deduce
- Prove that
and
.
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