Tamil Nadu Open University (TNOU) 2008 B.Sc Mathematics TRIGONOMETRY, ANALYTICAL GEOMETRY (3D) AND VECTOR CALCULUS UG 325 - Question Paper

UG-325 BMS-12/BMC-12

B.Sc. DEGREE EXAMINATION -JUNE 2008.

(BMS-12 : AY 2006-2007 onwards BMC-12 : AY 2007-2008 onwards)

First Year

Mathematics/Mathematics with Computer Application

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.

1.    If ^{sin x} = 86. Find an approximate value of x .

x 864

3. Find the angle between the planes 2x - y + z = 3 , x + y + 2z = 7.

4. Find k so that the lines ¹ = ² = ³ and

- 3 2k 2

x -1 y - 5 z - 6

may be perpendicular to each other.

3k 1 - 5

5.    Find the centre and radius of 2x² + 2y² + 2z² - 4x +16y + 8z + 20 = 0 .

6.    Find the value of a so that the vector F = (z + 3y) i + (x - 2z) j + (x + az) k is solenoidal.

7.    Find the maximum value of the directional derivative of the function $= 2x² + 3y² + 5z² at the point (1, 1, - 4).

8.    Prove that V x v(rⁿ ) = 0 .

PART B (5 x 10 = 50 marks)

Answer any FIVE questions.

Each question carries 10 marks.

9.    Prove that

32cos⁶ d = cos6 + 6cos40 + 15cos20 +10 .

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10.    Prove that

1 1 * 1³ o* 1³⁵ q*

1 cos * +--cos 2*--cos 3* +.....<x>

2    2 4    2 4 6

_ cos */ 4 V2 cos * / 2

11.    Find the equation of the plane which passes through the line of intersection of the planes

7x - 4y + 7z +16 _ 0 and 4x + 3y - 2z + 3 _ 0 and perpendicular to the plane x - y - 2z + 5 _ 0.

12.    Find the equation of the plane which passes through the point (1, 2, -1) and contains the line x +1 _ y -1 _ z + 2

2 _ _ -1 .

13.    Find the equation of the tangent plane to the sphere x² + y² + z² + 2x + 4y - 6z - 6 _ 0 at (1, 2, 3).

14.    Find the equation of the cone which passes through the coordinate axes as well as the lines

x _ y _ z _and x _ y _ z I_-2 _3 ^an 3_-!_ r

15.    A field F is of the form F _ (6xy + z³ )i + (x² - z) j + (3xz² - y) k . Show that F is conservative field and find a function such that

F _V$.

16. (a) Show that V² (e^r )= e^r + - e^r.

r

(b) Evaluate    JJJv - F dV    where

F = 2xyi + yz²j + xzk bounded by x = 0, y = 0, z = 0, x = 2, y = 1, z = 3.

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Attachment: tamil-nadu-open-university--tnou--2008-b-sc-mathematics-49090-1.pdf

Earning:   Approval pending.