2013

(November)

MATHEMATICS

(Major)

Course: 101

(Classical Algebra, Trigonometry and Vector Calculus)

Full Marks: 80

Pass Marks: 32

Time: 3 hours

The figures in the margin indicate full marks for the questions

GROUP – A

(Classical Algebra)

A sequence is bounded if and only if

- Its domain is bounded.
- Its range is bounded.
- It is bounded above.
- It is bounded below.

(b) Write the bounds of the sequence where

(c) Write the bounds of the sequence where

Is a bounded sequence

(d) Show that a sequence converges to one limit point only.

Or

Show that every bounded sequence with a unique limit point is convergent.

2. (a) Write when an infinite series diverges.

(b) Write the statement of Leibnitz test for convergence of an alternating series.

(c) Test the convergence of the infinite series.

(d) Test the convergence of the infinite series

(e) Show that if a series of positive monotonic decreasing terms converges, then as .

Or

Test the convergence of the infinite series.

3. (a) If be a root of an equation , then write a factor of . 1

(b) If and are two roots of the equation, then write the other two roots of the equation. 1

(c) Write the nature of the roots of the equation 2

(d) If are the roots of the equation then write the equation whose roots are 2

(e) If be the roots of the equation then find the value of

3

(f) Solve the equation by Cardan’s method. 6

Or

If be the roots of the equation then find the value of .

GROUP – B

(Trigonometry)

4. (a) Choose the correct answer:

is equal to

(b) Write the roots of the quadratic equation

(c) Express in powers of .

Or

If prove that

5. (a) Show that 2

(b) Show that 3

Or

Express in the form

6. (a) Write the interval of x for which

1

(b) Show that

3

Or

Expand in ascending powers of

7. (a) Choose the correct answer: 1

is equal to

(b) Write the period of 1

(c) Write the expansion of in terms of x. 2

(d) Find the sum of up to terms. 4

Or

Separate into real and imaginary parts.

GROUP – C

(Vector Calculus)

8. (a) Choose the correct answer: 1

A vector is a solenoidal vector of

(b) Find, where 2

(c) Show that, where is a scalar function and is a vector function. 5

Or

Show that where

(d) Show that is a vector perpendicular to the surface where is a constant. 3

(e) Evaluate 4

Or

Find the directional derivative of in the direction of

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