Anna University of Technology Tirunelveli 2009 B.E Aeronautical Engineering /B.Tech ,UARY ,MA 12-mathematics-I - Question Paper

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B.E/B.Tech. DEGREE EXAMINATION, APRIL/MAY 2009.

FIRST SEMESTER

MA 12 MATHEMATICS I

(Common to all B.E/B.Tech)

Time : Three hours    Maximum : 100 marks

Answer ALL questions.

PART A (10 x 2 = 20 marks)

-i f¹    5>l 1. Find the Eigen values of the matrix A if A =

0    4

v^u    y

cosd .^rsm0') .

-sin# cos0j

3.    Find the equation of the sphere which has its centre at (-1, 2, 3) and touches the plane 2x - y + 2z = 6.

4.    Find the equation to the cone with vertex at the origin and passes through the curve ax² + by² + cz² - 1, Ix + my + nz = p .

5.    Find the envelope of xcos a + y sin or = p where a is the parameter.

6. Find the radius of curvature at any point of y = cosh

7. If u = logfx² + xy + y²), prove that x + y = 2

ax ay

8. Find if u = x² +y², x = at²,y = 2at. dt

71

2 cos0

o o

u a

10. Change the order of integration of J J f(x,y) dxdy.

0 y

PART B (5 x 16 = 80 marks)

Find the Eigen values and Eigen vectors of the matrix

6-2 2 -2 3 -1 2-13

(ii) Using Cayley-Hamilton theorem find A ¹ if A

Or

(i) Reduce the Quadratic form 8x² +7y² +3z² ~ 12xy - 8yz + 4zx to a canonical form through an orthogonal transformation._______

If the matrices A and B are orthogonal prove that AB is orthogonal.

Find the tangent planes to the sphere x² +y² + z² -4x -2y -6z +5 = 0 which are parallel to the plane x + 4y +'82 = 0.

(ii) Find the equation to the right circular cylinder of radius 2 units and

(ii)

whose axis is the line

Or

Find the centre and radius of the circle in which the sphere

x² + y² + z² + 2x - 2y 4z 19 = 0 is cut by the plane x + 2y + 2z+7 = 0.

(ii) Find the equation of the right circular cone generated by the straight line drawn from the origin to cut the circle through the three points (l, 2, 2), (2,1, - 2) and (2, - 2, l).

13. (a) (i) Show that the radius of curvature p at any point (x, y) of the curve _x2/3 ₊y2/3 __a2/3 _sati_sfies p³ ~21axy.

(ii) Find the evolute of the parabola y² = 4ax .

Or

.(b) (i) Find the centre of curvature at

(ii) Considering the evolute of a curve as the envelope of its normals,

2 2 x y

find the evolute of + = 1.

a b

14. (a) (i) If u = and u = ^X - - , find    .

2x    2x    a (x,y)

(ii) Prove that the rectangular solid of maximum volume which can be inscribed in a sphere is a cube.

Or

(b) (i) Find the points on the surface z² = xy + 1 whose distance from the origin is minimum.

(ii) Expand at (l, l) in powers of x and y as far as the term of second degree.

(b) (i) Change the order of integration of | J .J*' rdxdy

o x *Jx²+y²

hence evaluate the same.

log 2 a- *+]ogy

(ii) Show that [ J J e^x+y+zdzdydx - -log²- .

ooo    2    Q

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Attachment: anna-university-of-technology-tirunelveli--2009-b-e-aeronautical-engineering-5407-1.pdf

Earning:   Approval pending.