[BA 1st Sem Question Papers, Dibrugarh University, 2012, Mathematics, Major]
1. Answer the following questions: 1x4=4
- Define a null sequence.
- Give an example of an oscillatory series.
- Write the necessary condition for convergence of an infinite series.
- Write the equation whose roots are the roots of the following equation with opposite signs:
2. Answer the following questions:
- Write the limit point(s) and the range set of the following sequence: 2
Or
Show that the series
does not converge.- Prove that every convergent sequence is bounded. Is the converse true? 2+1=3
- Using comparison test, find whether the series
is convergent. 3
Or
Using Leibnitz test for the convergence of alternating series, show that the series
Converges for 
3. (a) State and prove the factor theorem for polynomial equations. 4
(b) Show that the sequence
, where 
is convergent. 4
, where is convergent. 4
(c) Test for convergence of the series 5
(d) Test the convergence of the series
5
4. (a) From the equation whose roots are the squares of the differences of the roots of the cubic equation
5
Or
If 
are the roots of the equation 
then find the equation whose roots are
are the roots of the equation then find the equation whose roots are
(b) Discuss the Cardan’s method of solving a general cubic equation. 5
Or
Solve the equation 
by Cardan’s method.
by Cardan’s method.
GROUP – B
(Trigonometry)
5. (a) How many different values can be obtained for the following expression? 1
(b) Distinguish between 
and 
where 
is a complex quantity. 1
and where is a complex quantity. 1
6. Answer (any two): 2x2=4
- Express
in the form
.
- Find the real part of
.
- Find all the values of
.
7. (a) Using De Moivre’s theorem, prove that 
are the roots of the cubic equation 
5
are the roots of the cubic equation 5
(b) Sum to n terms the series
4
(c) Expand 
in ascending powers of 
in ascending powers of
(d) If
then show that
5
GROUP – C
(Vector Calculus)
8. (a) What is the physical interpretation of directional derivative of 
in the direction
in the direction
(b) What do you mean by a solenoidal vector?
9. If 
has constant magnitude, then show that 
and 
are perpendicular to each other provided 
has constant magnitude, then show that and are perpendicular to each other provided
Or
Show that 
Where 
is the magnitude of
. 2
is the magnitude of. 2
10. (a) Show that 
is a vector perpendicular to the surface 
, where K is a constant. 3
is a vector perpendicular to the surface , where K is a constant. 3
(b) Prove that 
4
4
(c) If 
and 
then find curl
at the point 
. 4
and then find curlat the point . 4
Or
Prove that
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