Tamil Nadu Open University (TNOU) 2008 B.Sc Mathematics Trigonometry Analytical Geometry and Vector Calculus UG 446 / UG 449 - Question Paper

UG-446/UG-449	BMS-12/
	BMC-12

B.Sc. DEGREE EXAMINATION -JULY 2008.

Mathematics/Mathematics with Computer Application

First Year

TRIGONOMETRY, ANALYTICAL GEOMETRY (3D) AND VECTOR CALCULUS

Time : 3 hours    Maximum marks : 75

PART A (5 x 5 = 25 marks)

Answer any FIVE questions.

Each question carries 5 marks.

2 tanh x

1. Prove that sinh 2x =

1 -tanh² x

₀ J ,. tan 2x - 2 tan x

2. Evaluate lim---

x 0    x

3.    Find the equation of the plane through the point (1, 2, 4) and parallel to the plane 2x + 6y - 8z + 9 = 0.

4.    Find the angle between the straight lines

x - 1 y + 1 z - 1 ! x - 2 y + 4 z - 4

-= --=- and-= --=-

4 3 1    3 15

5.    Find the equation of the sphere having centre at (7, 4, -3) and radius 6.

6.    Show that

F = (y² - z² + 3yz - 2x) i + (3xz + 2xy) j + (3xy - 2xz + 2z) k is irrotational and solenoidal.

7. If F = 3xyi - y²j, evaluate JF dr where C is

C

the curve on the xy plane y = 2x² from (0, 0) to (1, 2).

8. Prove that div\ | = ², if r = xi + yj + zk .

I r ) r

PART B (5 x 10 = 50 marks)

2    UG-446/UG-449

Answer any FIVE questions. Each question carries 10 marks.

If tan = tanh prove that

2 2

(a) sinh y = tan x and

(b) y = log tan |-4 + 2 |.

10.    Sum to n terms the series

sin² a + sin² 2a + sin² 3a + ...

11.    Find the equation of the plane through the points (2, 2, 1) and (9, 3, 6) and perpendicular to the plane 2x + 6 y + 6 z = 9.

12.    Prove that the lines ¹ = - = ³ and

2 3 4

^x + ¹ = ^{y 2} = ² are coplanar, find the equation of the plane containing them.

13.    Find the equation of the tangent plane to the sphere x² + y² + z² - 4x + 2y - 6z + 5 = 0 at the point (2, 2, -1).

14.    Show that the equation

2x¹ + 2y² + 7z² - 10yz - 10zx + 2x + 2y + 26z - 17 = 0

represents a cone. Find the co-ordinates of the vertex.

15.    Find the directional derivative of xyz - xy²z³ at the point (1, 2, -1) in the direction of the vector

i - j - 3k .

16.    If F = 4xzi - y² j + yzk, evaluate jjF ndS where

S is the surface of the cube bounded by x = 0, x = 1,

y = 0, y = 1, z = 0, z = 1.

4    UG-446/UG-449

1

Attachment: tamil-nadu-open-university--tnou--2008-b-sc-mathematics-49091-1.pdf

Earning:   Approval pending.