2012

(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)

1. (a) Find where the origin is to be shifted without changing the direction of the axes, so that the terms in x and y are removed from the equation

Or

If any angular displacement of the coordinate axes transforms the expression into the expressionthen show that

2. (a) Write the condition that the straight lines represented by are perpendicular to each other.

(b) Show that a homogeneous equation of the second degree always represents a pair of straight lines passing through the origin. 3

(c) Find the equations of the pair of straight lines represented by 3

Or

If the pairs of straight lines andbe such that each pair bisects the angle between the other pair, then prove that

(d) If the equation represents a pair of straight lines equidistant from the origin, show that

Or

Show that the area of the triangle formed by the lines represented by and is

3. (a)

(b)

(c)

Or

Find the condition that the lines be conjugate diameters of the conic

SECTION – II

(3-Dimension)

4. (a) Reduce the equation of the plane to the normal form.

(b) Write the equation of the plane parallel and at unit distance to plane.

(c) Express the equation of the in symmetrical form.

(d) Find the equation of the plane through the points and and perpendicular to the plane. 3

Or

Find the equation of the plane passing through the intersection of the planes and perpendicular to the plane.

(e) Find the equation of the two planes represented by the equation 3+1=4

Find the angle between them.

Or

Find the coordinates of the foot of the perpendicular drawn from the point to the plane . Find also the image of the point with respect to the plane.

5. (a) Find the shortest distance between the lines 3

and

and show that they are coplanar.

Or

A perpendicular is drawn from the origin to the line

Find the equation of the perpendicular and coordinates of its foot.

(b) Find the equation of the line of the shortest distance between the lines 4

and

Or

Find the surface generated by a straight line which meets the two linesat the same angle.

GROUP – B

(Analysis – I)

6. (a) Find the derivative of , where and .

(b) Find the curve whose curvature at any point on it is zero.

(c) Prove that the subnormal at any point of a parabola is of constant length.

(d) Evaluate:

Or

If, show that

7. (a) Verify Roll’s theorem for the function in 2

(b) Write the remainder after term of Taylor’s series in Lagrange’s Form. 1

(c) State and prove Lagrange’s mean value theorem. 4

Or

Using Maclaurin’s theorem, expand sin x in an infinite series in powers of x.

(d) State Darboux’s theorem.

(e) Show that function defined by

Is continuous at

8. (a) If , then show that

(b) If, prove that. 2

9. (a) Prove that , if 3

Or

Show that

(b) If, being a positive integer greater than 1, then deduces that, hence find the value of.

Or

Find the whole length of the curve.

(c) Show that, , m, n being positive integers greater than one, then show that .

Or

Using reduction formula of

Deduce

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