2013

(November)

MATHEMATICS

(General)

Course: 301

[Group – A: Coordinate Geometry and

Group – B: Analysis – I (Real Analysis)]

Full Marks: 80

Pass Marks: 32

Time: 3 hours

The figures in the margin indicate full marks for the questions

GROUP – A

(Coordinate Geometry)

SECTION – I

(2- Dimension)

(b) What will be the transformation of the equation when the two axes are turned through an angle

Or

To which point the origin will be shifted, so that the first degree terms in the equationmay be missed, when referred to the parallel displacement of the axes?

2. (a) Find the angle between the two lines represented by the equation

(b) Determine the distance between the two parallel lines represented by the equation 3

Or

For what value of k the equation

(c) Show that the two straight lines represented byare equally inclined to the straight line. 3

(d) Prove that the product of the lengths of perpendiculars drawn from the pointto the line is

Or

Find the area of the triangle formed by the lines represented by

and axis of .

3. (a) Reduce the equation 2

into standard form.

(b) Find the equation of the tangent drawn at the point to the conic 4

Or

Show that the line will be a tangent to the conic if

(c) Define Chord of contact, diameter and conjugate diameter. 2+1=3

Or

Determine the pole of the line with respect to the conic .

SECTION – II

(3-Dimension)

4. (a) Write the direction cosines of any normal to the plane .

(b) Find the points of intersection of the plane with the coordinate axes.

(c) What is the equation of the plane parallel to the planeand passing through the origin?

(d) Find the equation of the plane passing through the point and perpendicular to each of the planes and. 3

Or

Find the locus of the point whose distance from the origin is three times its distance from the plane.

(e) Find the bisector of that angle between the planes and which contains the origin. 4

Or

Find the condition of representing two planes by the equation.

5. (a) Show that the two lines

are coplanar. Find the equation of the plane through the two lines.

Or

Prove that the plane through the line and perpendicular to the plane through the lines and

(b) Determine the shortest distance between the lines and the axis. 4

Or

Find the equation of the line of shortest distance between the lines

GROUP – B

(Analysis – I)

6. (a) If , where n is a positive integer, show that . 2

(b) Find the radius of curvature of the parabola at the vertex. 2

(c) Show that in any curve 2

(d) If show that. 4

Or

Evaluate: 2+2=4

7. (a) State Lagrange’s mean value theorem. 1

(b) Find the value of in Rolle’s Theorem where in 2

(c) In the mean value theorem find c if and give a geometrical interpretation of the result. 3+1=4

Or

If find where and where. 4

(d) Show that the function denied by

Is continuous at

8. (a) State and prove Euler’s theorem on homogeneous function of two variables. 3

Or

If then show that

(b) Show that if . 2

9. (a) Prove that if 3

Or

Show that

(b) If (m, n being positive integers), then deduce that 5

Or

Find the length of the arc of the parabola cut-off by its latus rectum.

(c) Establish the reduction formula 2

where n is an integer.

***

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