2015

(November)

MATHEMATICS

(Major)

Course: 301

[Analysis – I (Real Analysis)]

Full Marks: 80

Pass Marks: 32 (Backlog)/24 (2014 onwards)

Time: 3 hours

The figures in the margin indicate full marks for the questions

GROUP – A

(Differential Calculus)

(Marks: 35)

(b) Evaluate: 2

(c) Find the length of the subnormal to the curve at the point 3

(d) If, then show that 4

Or

Find the radius of curvature at the point on the cycloid

2. (a) Write the statement of Darboux’s theorem. 1

(b) Show that

is continuous at 2

(c) Show that 4

Or

Expand in an infinite series using Maclaurin’s series.

(d) Verify Rolle’s Theorem for 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

4

(c) If z is a function of x and y and, 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 pointsandwhen the curve is given in parametric form. 1

(b) Find the length of the arc of the curveandfrom 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

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

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

3

9. (a) Give example of an improper integral of second kind. 1

(b) Test for convergence of 2

(c) Prove that

converges. 3

10. Answer any one of the following: 4

- Prove that

Hence deduce

- Prove that

and.

***

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