Tamil Nadu Open University (TNOU) 2009-1st Year B.Sc Mathematics " TRIGONOMETRY, ANALYTICAL GEOMETRY OF THREE DIMENSIONS AND VECTOR CALCULUS GU 476 BMS 12 / BMC 12 - Question Paper

wk 9

UG-476	BMS-12/
	BMC-12

B.Sc. DEGREE EXAMINATION JANUARY, 2009.

First Year

(AY 2006-07 batch onwards)

Mathematics

TRIGONOMETRY, ANALYTICAL GEOMETRY OF THREE DIMENSIONS 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.

1.    Prove that = 16 sin⁴ 6-20 sin² 0 + 5.

sin 6

2.    Prove that l-tanh²x = sech²x.

2 tan# _ 2524 $ approximately. 0 2523

4.    Find the equation to the plane through (2,-4,5) and is parallel to the plane 4x + 2j'-72 + 6 = 0 .

5.    Find the straight line through (3,2,-8) and perpendicular to - 3x + y + 2z - 2 = 0.

6.    Find the centre and radius of the sphere 7x² + ly² + Iz² + 28x - 42y + 56 z + 3 = 0 .

7.    Find unit vector normal to the surface x² +2y² +z² =7 at (1,-1,2)

8.    Prove that curl (grade/)) = 0 .

PART B (5 x 10 = 50 marks)

Answer any FIVE questions.

Each question carries 10 marks.

9.    Prove that

2⁵ cos⁶ 0 = cos60 + 6cos46> + 15cos26> +10.

10.    If cosc + z/y j=cosa + isina prove that cos 2x + cosh 2y = 2.

. . . sin2# sin30

11. Sum the series sm6H---1---K...o.

12.    Find the equation of the plane through (2,-3,1) and is perpendicular to the line joining the points (3,4,-1) and (2,-1,5).

13.    Show that the lines ^{x +} ₌ J + 5 _ 37_an(j

2 3-3

x + 1 ₌ j + 1 ₌ 3 + l coplanar and find the equation of

4 5 -1

the plane containing them.

14.    Find the equation of the sphere through the circle x² + y² + z² =9 , 2x + 3;y + 4z = 5 and the point (1,2,3).

15.    Show that F = yzi +3xj + xyk is irrotational. Find <f> so that F = V <f>.

16.    Show that Greens theorem in a plane can be deduced as a special case of Stokes theorem.

3    UG-476

Attachment: tamil-nadu-open-university--tnou--2009-b-sc-mathematics-49070-1.pdf

Earning:   Approval pending.