RAJALAKSHMI ENGINEERING COLLEGE

DEPARTMENT OF MATHEMATICS MA 2111 ENGINEERING MATHEMATICS -1 MATRICES Part A

1. Find the eigen values of the matrix A =

2145

2. Find the constant a and b such that the matrix

b

a1

4 has 3 and -2 as its

eigen values.

3. Show that if is a characteristic root of the matrix A, then + k is a characteristic root of the matrix A + k I.

4. If be an eigen value of a non-singular matrix A, show that || A is an

eigen value of adjA.

5. If the characteristic equation of a matrix A is 2 - 4 +3 = 0, find the

characteristic equation of IA 43 1

6. Are

26

and

16

eigen vectors of

4132

.

7. Let A =

113151311

If ( 1, 0, -1)T is an eigen vector corresponding to some

eigen value of the matrix A. find .

8. Discuss the nature of QF 2xy + 2yz + 2zx. 9. Write down the matrices of the following QF 2x2 +3y2+6xy. 10. Write down the QF corresponding to the following matrices

23003602

00410211

RAJALAKSHMI ENGINEERING COLLEGE DEPARTMENT OF MATHEMATICS

MA 2111 ENGINEERING MATHEMATICS -1 ANALYTICAL GEOMETRY

PART A

1. Find the equation of the sphere whose centre is ( 2, -3, 4) and radius 5? 2. Show that the spheres x2+y2+z2=25, x2+y2+ z2-18x-24y-40z + 225=0 touch

externally.

3. Find the equation of the sphere on this join ( 2,-3,1) and ( 1,-2,-1) as diameter.

4. Find the equation of sphere having its centre on the plane 4x-5y-z=3 and passing through the circle x2+y2+z2 -2x-3y+4z + 8=0,

x2+y2+z2+4x+5y-6z+2 = 0.

5. Prove that the plane x + 2y z = 4 cuts the sphere x2+y2+z2 x + z -2 =0 in a circle of radius unity.

6. Find the equation of the tangent plane to the sphere 3(x2+y2+z2) -2x -3y -4z -22 = 0 at the point ( 1,2,3).

7. Find the equation to the cone whose vertex is the origin and base the circle x =a; y2 + z2 = b2.

8. Find the equation of the right circular cone whose vertex is at the origin, whose axis is the line x = y / 2 =z / 3 and which has semi vertical angle of

30o

9. Find the equation of the quadric cylinder with generators parallel to x axis and passing through the curve ax2 + by2 + cz2 = 1, lx+ my+ nz = p.

10. Find the equation of the quadric cylinder whose generators intersect the curve ax2 + by2 = 2z, lx+ my+ nz = p and are parallel to Z axis.

RAJALAKSHMI ENGINEERING COLLEGE DEPARTMENT OF MATHEMATICS

MA 2111 ENGINEERING MATHEMATICS -1 FUNCTIONS OF SEVERAL VARIABLES

PART A

1. Find the first order partial derivatives of the function u = yx

2. If Z = log( x2 + xy + y2), prove that .2=+

yzy

xzx

3. If U = f( y-z, z-x, x-y), prove that 0=+

+

zu

yu

xu

4. Given U = sin(x/y), x = et and y = t2 find dtdu

5. If U = x2 + y2 + z2 and x = e2t, y = e2t cos3t, z = e2tsin3t find dtdu

6. If yx

zwxz

yVzy

xU === ,, show that .0),,(),,( =

zyxwvu

7. If U = x (1-y); V = xy prove that JJ = 1.

8. Show that 2

)1log(2xxyxxey +=+ ( approximately)

9. Find the stationary points of xy ( a x-y). 10. State necessary conditions for a maximum or a minimum.

RAJALAKSHMI ENGINEERING COLLEGE DEPARTMENT OF MATHEMATICS

MA 2111 ENGINEERING MATHEMATICS -1 DIFFERENTIAL CALCULUS

PART-A

1. Find the radius of curvature for a circle and straight line. 2. Find the radius of curvature at any point of the catenary y = c cosh (x / c). 3. Find the radius of curvature at any point of y = logsinx. 4. Find at on x = 3a cos - a cos3 , y = 3a sin - a sin3 . 5. Find at ( -2,0) on y2 = x3 + 8. 6. Find the circle of curvature at (0,0) on x + y=x2+y2+x3. 7. Write the method to find the evolute of a given curve y = f(x). 8. State two important properties of the evolute. 9. Find the envelope of the family of straight lines y = mx +a/m.

10. Find the envelope of 1=+by

ax subject to a + b = c, where c is known

constant.

RAJALAKSHMI ENGINEERING COLLEGE DEPARTMENT OF MATHEMATICS

MA 2111 ENGINEERING MATHEMATICS -1 MULTIPLE INTEGRALS

PART A

1. Evaluate 21

5

2

dxdyxy

2. Evaluate 20

2

0

rdrd

3. Evaluate dxdyyx +34

2

1

2)(

4. Evaluate 0

cos

0

rdrd

5. Shade the region of integration

a xa

xax

dxdy0

22

2

6. Transform the integral

0 0

y

dxdy to polar coordinates.

7. Change the order of integration 1

0

2

2

),(x

x

dydxyxf

8. Express the region x > 0, y > 0, z > 0, x2 + y2 + z2 < 1 by triple integration. 9. Find the area enclosed by the circle x2 + y2 = a2. 10. Find the area of the region bounded by y2 = 4x and x2 = 4y.

RAJALAKSHMI ENGINEERING COLLEGE MA 2111

1. Find the eigen values and eigen vectors of the matrix

2 2 02 5 00 0 3

andhence diagonalize it through orthogonal reduction.

2. Verify Cayley-Hamilton theorem for A =

2 0 10 2 01 0 2

and hencefind A1 and A4.

3. Find the eigen values and eigen vectors of the matrixA =

2 2 21 1 11 3 1

.

4. Find the inverse of the matrix A =

7 2 26 1 26 2 1

by using Cayley-Hamilton theorem.

5. show that

0 1 01 0 00 0 1

is orthogonal. Find its inverse. Verify that itseigen values are of unit modulus.

6. Find the characteristic equation of A =

2 1 10 1 01 1 2

and hence expressthe matrix A5 in terms of A2, A and I.

7. If (0, 1, 1)T , (2,1, 1)T , (1, 1,1)T be the eigen vectors of matrix Acorresponding to the eigen values 1, 1, 4 then find the matrix A.

8. Reduce the quadratic form 8x21 + 7x22 + 3x

23 12x1x2 + 8x2x3 + 4x1x3

to the canonical form through orthogonal transformation. Hence showthat it is positive semi-definite.

9. Reduce the quadratic form 6x21 + 3x22 + 3x

23 4x1x2 2x2x3 + 4x1x3

into sum of squares by orthogonal transformation. Write also rank,index and signature.

10. Reduce the quadratic form 17x2 30xy + 17y2 to a canonical formand find the nature of conic 17x2 30xy + 17y2 = 128. Find also thelengths and directions of the principal axes.

QUESTION BANK 1 MATHEMATICS I

RAJALAKSHMI ENGINEERING COLLEGE MA 2111

11. Find the equation of the sphere which passes through the points (2, 0, 0),(0, 2, 0) and (0, 0, 2) and which has its radius as small as possible.

12. Find the equations of the tangent planes to the sphere x2+y2+z2+2x4y+6z7 = 0 which passes through the line 6x3y23 = 0 = 3z+2.

13. Find the equations of sphere which pass through the circle x+2y+3z =8, x2 + y2 + z2 2x 4y = 0 and touch the plane 4x+ 3y = 25.

14. Find the center and radius of the circle x2+y2+z28x+4y+8z45 = 0and x 2y + z 3 = 0.

15. Show that the spheres x2 + y2 + z2 + 6y + 2z + 8 = 0 and x2 + y2 +z2 + 6x + 8y + 4z + 20 = 0 cut orthogonally. Find their plane ofintersection. Also prove that this plane is perpendicular to the linejoining the center.

16. Find the equation of the sphere passing through the points (0, 3, 0),(2,1,4) and cutting orthogonally the two spheres x2 + y2 + z2 +x 3z 2 = 0 and 2x2 + 2y2 + 2z2 + x+ 3y + 4 = 0.

17. Find the equation of the cone whose vertex is (1, 2, 3) and guidingcurve is the circle x2 + y2 + z2 = 4, x+ y + z = 1.

18. Prove that 9x2 + 9y2 4z2 + 12yz 6zx+ 54z 81 = 0 represents acone. Find also its vertex.

19. Find the equation of the cylinder whose axis is x1 =y2 =

z3 and whose

guiding curve is the ellipse x2 + 2y2 = 1, z = 3.

20. Find the equation of the right circular cylinder passing throughA(3, 0, 0)and having the axis x 2 = z, y = 0.

21. Find , at (a, 0) on y2 = a3x3x .

22. Show that the curves y = a2 (exa + e

xa ) and y = a2 (2 +

x2

a2) have the

same curvature at their crossing with the Y-axis.

23. Show that at on x = 3a cos a cos 3, x = 3a sin a sin 3 is3a sin .

24. Show that the line joining any point on x = a( + sin ), y = a(1cos ) to its center of the curvature is bisected by the line y = 2a.

QUESTION BANK 2 MATHEMATICS I

RAJALAKSHMI ENGINEERING COLLEGE MA 2111

25. For the curve rn = an cosn, prove

=anr1n

n+ 1.

26. Find the circle of curvature at (0, 0) on x+ y = x2 + y2 + x3.

27. Obtain the equation of the evolute of the ellipse x2

a2+ y

2

b2= 1.

28. Show that the evolute of the tractrix x = a(cos t+log tan t2), y = a sin tis the catenary y = a coshxa.

29. Given that x23 + y

23 = c

23 is the envelope of xa +

yb = 1. Find the

necessary relation between a and b.

30. Find the evolute of x2

a2+ y

2

b2= 1 considering it as the envelope of

normals.

31. Prove that if f(x, y) = 1ye (xa)2

4y , then fxy = fyx.

32. If u = (1 2xy + y2) 12 prove that

x

[(1 x2)u

x

]+

y

[y2

u

y

]= 0.

33. Verify Eulers Theorem for the function

u = sin1x

y+ tan1

x

y.

34. If u = log x4+y4

x+y , show that xux + y

uy = 3.

35. By changing the independent variables u and v to x and y by meansof the relations x = u cos v sin, y = u sin + v cos, show that2zu2

+ 2z

v2transforms into

2zx2

+ 2z

y2.

36. If u = xyz, v = xy + yz + zx,w = x + y + z, show that (u,v,w)(x,y,z) =(x y)(y z)(z x).

37. Expand tan1( yx) in the neighborhood of (1, 1), using Taylors series.

38. Discuss the maxima and minima of x3y2(1 x y).

QUESTION BANK 3 MATHEMATICS I

RAJALAKSHMI ENGINEERING COLLEGE MA 2111

39. Show that the rectangular solid of maximum volume that can be in-scribed in a given sphere is a cube.

40. Find the maximum and minimum distances of the point (3, 4, 12) fromthe sphere x2 + y2 + z2 = 1.

41. Evaluate

(x+y)2dxdy over the area bounded by the ellipse x2

a2+ y

2

b2=

1.

42. Change the order of integration in

I = 10

2xx2

xydydx

and hence evaluate the same.

43. Evaluate 21

4x20 (x+ y)dydx by changing the order of integration.

44. Evaluate 10

2x2x

xx2+y2

dydx by changing the order of integration.

45. Evaluate 10

2xx

x

ydxdy

by changing the order of integration.

46. Evaluate 10

1x0

1xy0

xyzdzdydx.

47. Find the volume of the solid in the positive octant bounded by theparabolic z = 36 4x2 9y2.

48. Change a2

0

a2y2y

log(x2 y2)dydx,

a > 0 into polar co-ordinates and hence evaluate it.

49. Evaluate a0

a2x2axx2

dxdya2 x2 y2

using polar co-ordinates.

50. Evaluate

R xydxdy where R is the region in the first quadrant thatlies between the circles x2 + y2 = 4 and x2 + y2 = 25.

QUESTION BANK 4 MATHEMATICS I

QP - MA2111 _PART - A_.pdfQP - MA2111 (PART - B).pdf