2015

(November)

MATHEMATICS

(General)

Course: 301

[Group – A: Coordinate Geometry and

Group – B: 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

(Coordinate Geometry)

SECTION – I

(2 –Dimension)

(b) Find the new coordinates of the point, if the frame of reference is rotated through an angle, without changing the origin. 2

(c) Find the angle through which it is to be rotated to remove term from. 2

2. (a) represents a pair of straight lines. Determine whether they are parallel or perpendicular to each other. 1

(b) Find the angle between the pair of lines. 2

(c) Find the pair of lines represented by. 3

(d) Show that the lines andform an equilateral triangle. 6

Or

Discuss the nature of the conic represented by the equation

and reduce it to canonical form.

3. (a) Find the equation of tangent at to the conic . 2

(b) Find the centre of the conic. 3

(c) Find the pole of a given line with respect to a conic. 5

Or

Determine whether the conic represented by

has a single centre, infinitely many centres or no centre.

SECTION – II

(3-Dimension)

4. (a) Plane is described by an equation. Write the degree of the equation. 1

(b) The plane passes through a particular point. Write the coordinates of that point. 1

(c) Find the intercepts on the axes made by the plane. 2

(d) Find the equation of the plane passing through the points (3, 1, 1) and (1, -2, 3), and parallel to x-axis. 3

Or

Find the angle between the planes and.

(e) Find the equation of the line in symmetric form passing through (2, 0, 4) and (1, -2, 3). 3

5. (a) Find the length of the shortest distance between the following lines: 4

(b) Find the equation of the perpendicular line to the lines;and z-axis. 4

GROUP – B

(Analysis – I)

6. (a) If , then find the value of. 1

(b) If, then find the value of. 2

(c) Find the length of subtangent to at. 3

Or

Find the radius of curvature of at any point.

(d) Evaluate any one of the following: 4

7. (a) Write Maclaurin’s theorem with Lagrange’s form of remainder. 1

(b) Write the geometrical meaning of Lagrange’s mean value theorem. 2

(c) Show that a function, which is derivable at a point, is continous at that point. 2

(d) State and prove Rolle’s Theorem. 5

Or

If a function is derivable on a closed interval and, are of opposite signs, then there exists at least one point between and such that.

8. (a) If , then find . 1

(b) Verify Euler’s theorem for. 4

Or

If , then show that . 1

9. (a) Write the condition when .

(b) Evaluate any one of the following: 4

(c) Obtain the reduction formula for. 5

Or

Find the perimeter of the asteroid.

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