[BA 1st Sem Question Papers, Dibrugarh University, 2013, Mathematics, General]
1. Answer the following questions: 1x3=3
- Define real sequence.
- State the condition for convergence of infinite geometric series
- If a polynomial
is divided by
where
is a constant, then state the remainder.
2. Answer the following questions: 2x5=10
- Prove that a convergent sequence cannot tend to two distinct limits.
- State the conditions of d’ Alembert’s ratio test for convergence of a series.
- Test for convergence of the series:
- If
are the roots of the equation
then show that
- Show that the equation
has a root between – 3 and – 2.
3. (a) Show that the sequence 
is convergent, if
is convergent, if 
3
(b) Use general principle of Cauchy’s criterion to show that 
is not convergent, where
is not convergent, where 
4
4. Test the convergence of any two of the following: 5x2=10
5. (a) If 
be the roots of the equation 
form the equation whose roots are
be the roots of the equation form the equation whose roots are
.
Or
Prove that every algebraic equation of degree n has exactly n real or imaginary roots.
(b) Solve the equation 
using Cardan’s method.
using Cardan’s method.
GROUP – B
(Trigonometry)
6. (a) What is the 2nd term in the expansion of 
in terms of 
?
in terms of ?
(b) State Gregory’s series. 1
(c) Prove that 2
7. State De Moivre’s theorem and prove it when n is any integer. 5
Or
If 
Prove that 
Prove that
8. (a) Express 
in the form 
2
in the form 2
(b) Prove that
where a and b are two real quantities. 3
(c) Prove that
3
Or
If 
prove that
prove that 
9. (a) If 
, prove that 
and 
are the roots of the equation 3
, prove that and are the roots of the equation 3
(b) Find the sum: 5
Or
Find the sum of the sines of a series of n angles which are in arithmetical progression.
GROUP – C
(Vector Calculus)
10. (a) Write the following expression correctly:
1
(b) Give the definition of vector differential operator
. 1
. 1
(c) If 
where 
prove that 
2
where prove that 2
11. What do you mean by directional derivative of 
at the point 
Find its values in the direction of coordinate axes.
at the point Find its values in the direction of coordinate axes.
Or
Find the directional derivative of 
at the point 
in the direction of the vector 
at the point in the direction of the vector
12. Answer any two of the following: 4x2=8
- Prove that
- If
determine
and
at
.
- Prove that:
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