UNIT I TRIGONOMETRY 10 hrs.Review of Complex numbers and De Moivre’s Theorem. Expansions of Sinnand Cosn; Sinand Cosin powers of , Sinnand Cosnin terms of multiples of . Hyperbolic functions – Inverse hyperbolic functions. Separation into real and imaginary parts of complex functions

I.Separate into real and imaginary parts of cos(x+iy) Find the real part of sin(x + iy).Separate into real and imaginary parts of cos(x+iy)Find the real part of Sin h (A+iB)Separate into real and imaginary parts of tan(x i y)

Separate sin (x + iy) into real and imaginary partsSeparate into real and imaginary parts of cos(x+iy) Separate real and imaginary parts of cosech (x + iy).II. Write down the expansion for tan n interms of power of tan. Write down the expansion for tan n interms of power of tan. Write the expansion of sin n.Write the expansion of sin n.Write the expansion of sin n.Write down the expansion for tan n interms of power of tan. Write the expansion of sin n.Expand cos4 in terms of cos.Expand sin5 in terms of sin.Expand cos4 in terms of cos.Expand cos in powers of cos and sin .Write cos4 in terms of a series of cosines of multiples of Expand Cos4 in a series of cosines of multiples of .Express in terms of cos

If sin (A + iB) = x + iy, prove that

If cos ( + i) = cos + isin prove that Sin2 = sin.

4. If x = cos + i sin, what is

Show that cos4 = 8cos4 - 8cos2 + 1 Show that Show that = 4 cos2 – 3.Show that

Show that

Prove that tan h-1 = log x for x>0.

Prove that tan h-1 = log x for x>0.

Prove that cosh2 x – sinh2 x = 1.

Prove that tan h-1 = log x for x>0.

Show that sinh 2x = 2sinhx coshx.

Prove that cosh2x – sinh2 x = 1.

PART-B1.Separate real and imaginary parts of cosech (x + iy)(b) Separate into real and imaginary parts of tanh(x + iy).(b) Separate tan-1(x + iy) into real and imaginary parts(b) Separate tanh-1 (x + iy) into real and imaginary parts.(a) Separate into real and imaginary part of tan-1 (x + iy).

(b) Separate tanh-1(x + iy) into real and imaginary parts. (b) Separate into real and imaginary parts of tanh(x + iy).(b) Separate tan-1(x + iy) into real and imaginary parts.(b) Separate tanh-1 (x + iy) into real and imaginary parts.

II.(a) Expand sin 6 in terms of sin .Find in terms of cosines powers of .(a) Expand sin 7 as a polynomial in sin , Hence show that

Sin /7 sin 2/7 sin 3/7 sin 4/7 sin 5/7 sin6/7 = -7/64(a) Expand sin 7 as a polynomial in sin , Hence show that

Sin /7 sin 2/7 sin 3/7 sin 4/7 sin 5/7 sin6/7 = -7/64(a) Expand sin 6 in terms of sin .(a) Expand sin 6 in terms of sinExpand Sin8 in a series of cosines of multiple of .. Expand Sin4Cos3 in a series of cosines of multiples of . (may-2012)Expand Sin4Cos3 in a series of cosines of multiples of .(a) Obtain the expansion of Sin7/Sin(a) Expand Sin3. Cos5 in a series of sines of multiples of .

(a) Expand sin 5 cos 4 in a series of sines of multiples of . (may-2013)(a) Prove that 64sin4 cos3 = cos7 - cos 5 = 3 cos 3 + 3cos.(a) Prove that cos7 sec = 64cos6 - 112cos4 + 56 cos2 - 7.

(a) Expand Sin3. Cos5 in a series of sines of multiples of .It cos(u+iv) = x+iy where u,v,x,y as real, prove that

(i) (1+x)2 + y2 = (Coshv + cos u)2

(ii) (1-x)2 + y2 = (Coshv – Cos u)2

(a) Prove that cos6 = [cos6 + 6cos4+15cos2+10].(a) Prove that sin6 =

(b) Prove that sin5 cos2= 1/26 [sin7 – 3sin5 + sin3+5sin]

(a) Prove that

(a) Find in powers of cos. (may-2012)

(a) Find in powers of cos.

Show that (a) Show that

[Cos 9 + cos 7 - 4cos 5 - 4cos 3 + 6cos]

(a) Show that

[Cos 9 + cos 7 - 4cos 5 - 4cos 3 + 6cos]

(a) Show that

[Cos 9 + cos 7 - 4cos 5 - 4cos 3 + 6cos]

Find in terms of cosines powers of

(a) Find in powers of cos.(a) Prove that

x2

x2

(a) Prove that (may-2012)(b) If Sin (

If x + iy = sin (A+iB) prove that

(b) If sin ( + i) = x + iy, prove that

If cos ( + i) = cos + isin prove that Sin2 = sin. (8

marks)

(b) If tan x/2 = tan h y/2, prove that sin hy = tanx and

y = log tan

(b) If tan = tan h prove that cos x cos hx = 1.12. It cos(u+iv) = x+iy where u,v,x,y as real, prove that

(i) (1+x)2 + y2 = (Coshv + cos u)2 (may-2012)(ii) (1-x)2 + y2 = (Coshv – Cos u)2

12. It cos(u+iv) = x+iy where u,v,x,y as real, prove that

(i) (1+x)2 + y2 = (Coshv + cos u)2

(ii) (1-x)2 + y2 = (Coshv – Cos u)2

Show that sinh 2x = 2sinhx coshx.Prove that cosh2 x – sinh2 x = 1.

(b) If tan (+ i) = tan + i sec,

Prove that

x2

x2

.(b) If tan x/2 = tan h y/2, prove that sin hy = tanx and

y = log tan

12. (b) Show that

(b) If x+iy = cos(A – iB), find the value of X2 +

(b) Show that

(or)12. (a) If show that is nearly equal to 3 ’

(b) If cos hu = sec , prove that u=log tan

(b) If sin( A i B) x i y , prove that

X2/Sin2 A –x2 /cos2 A

(b) Prove that tanh– 1(sin ) = cosh-1(sec ).

(b) If sin = tanh x prove that tan = sinh x.(b) If tan = tan h prove that cos x cos hx = 1.

12.

(b) If tan = tanh , prove that y = log tan

Cosh2B sinh2B

1a1

1a2

1an

Expand Sin4Cos3 in a series of cosines of multiples of .

(b) If tan (+ i) = tan + i sec,

Prove that

(a) Expand sin 7 as a polynomial in sin , Hence show thatSin /7 sin 2/7 sin 3/7 sin 4/7 sin 5/7 sin6/7 = -7/64(b) If tan x/2 = tan h y/2, prove that sin hy = tanx and

y = log tan

12. (b) Show that

UNIT II Characteristic equation of a square matrix - Eigen values and Eigen vectors of a real matrix- properties of Eigenvalues and Eigen vectors, Cayley-Hamilton theorem (without proof) verification – Finding inverse and power of a matrix.Diagonalisation of a matrix using similarity transformation (concept only) , Orthogonal transformation – Reduction ofquadratic form to canonical form by orthogonal transformation

1.Find the rank of the matrix

2.Find the sum and product of the eigen values of matrix.

3.Find the rank of the matrix 1 3 2 4 1 4 3 2 2 7 5 6 4 14 10 12

4.If a1, a2, … an are the eigen values of a square matrix A, prove that

6 –2 2–2 3 –1 2 –1 3

a.For what values of a and b the equations. , , … are the eigen values of A–1

5.The product of two eigen values of the matrix

is 16. Find the third eigen value.

6. Write down the matrix of the quadratic form x2 + y2 + z2 + xy + yz + zx.

7.Find the sum and product of the eigen values of the matrix

8. Prove the matrix is orthogonal.

9.State any two properties of eigen values of a matrix.

10. Use Cayley-Hamilton theorem to find the inverse of the matrix

11.Find the sum of the squares of the eigen values of A = .

12.. Determine the nature of the Quadrative form without reducing to the canonical form:

x2+3y2+6z2+2xy+2yz+4xz.Find the eigen value and eigen vectors of

13. Find the eigen value and eigen vectors of 2 -1 -8 4

14. The eigen values of the matrix A =

the third eigen value and the product of eigen values.

15.State cayly-hamilton theorem.

16..If = 3 and = -2 are twp eigen values of then find third eigen value

17.Find the rank of the matrix

18.. Find the sum and product of the eigen values of matrix.

19.State any two properties of eigen values of a matrix.

20.Use Cayley-Hamilton theorem to find the inverse of the matrix

21.If A= , find the eigen values of A-1 and A 3 .M12

22. Find the nature of the quadratic form M12

23. Find the sum and product of the eigen values of the matrix

24. State Cayley Hamilton theorem.

25.Find the rank of matrix.

26.. Find the sum and product of eigen values of the matrix

27.State Cayley-Hamilton theorem.-D11

28.Find the quadratic form corresponding to the matrix D11

27.State cayly-hamilton theorem.-D11

28.If = 3 and = -2 are twp eigen values of then find third eigen value. D11

29.If A = , then find the eigen values of A2.-M11

30.Write the matrix of the quadratic form.4x2 + 2y2 – 3z2 + 2xy + 4zx –M11

31. Define rank of a matrix.-M1132. Two eigen values of

2 2 133. A = 1 3 1 are equal to 1 each. Find the third eigen value.-M11

1 2 2

34.In the rank of A = is 2, find the value of k.-D10

35. Find the sum of the squares of eigenvalues of the matrix –D10

A =

36.Find sum and product of Eigen values of .-M10

37..Write the matrix of quadratic form (x12+3x2

2+6x32-2x1x2+6x1x3+5x2x3).-M10

38.State any one property of Eigen value of a matrix and verify it on the matrix .D09

39. Write down the quadratic form whose corresponding matrix –is . D09

a.1) x + y + z = 6x + 2y + 3z = 10x + 2y + az = b

have (i) No solution (ii) A unique solution (iii) Infinite number of solutions.(or)

A1. Reduce quadratic form to a canonical form through an orthogonal transformation.

a.2)If A and B are any two non-singular matrices of the same order. Prove that (AB) –1 = B–1 A–1.

(or)A2. If A is any square matrix, prove that ½ (A + AT) is a symmetric matrix and ½ (A –

AT) is a skew-symmetric matrix.(a3) Show that the equations 3x + y + 2z = 3, 2x – 3y – z = -3, x + 2y + z = 4 are consistent and

solve them.

(b) Find the eigen values and eigen vectors of the matrix.

(or)A3. Reduce the quadratic form 8x2 + 7y2 + 3z2 – 12xy – 8zy + 4xz to the canonical form

through an orthogonal transformation.a.4)Reduce the quadratic form in to its canonical form

by using orthogonal reduction.(or)

A4. Verify Cayley – Hamilton theorem for the matrix

Also find A – 1 and A4.

(a5) Find the Eigen values and Eigen vectors of the matrix

–1 2 3 8 1 –7–3 0 8

(b) Diagonalise the matrix A given above by similarity transformation. (or)

A5. (a) Find the inverse of the matrix by using Cay;ey-Hamilton theorem.

(b) Obtain an orthogonal transformation, which will transform the quadratic form 6x2 + 3y2 + 3z2 – 4xy – 2yz + 4zx into a canonical form.

a.6) Reduce the quadratic form 2x2 + 6y2 + 2z2 + 8xz to canonical form by orthogonal reduction. Find also the nature of the quadratic form.

(or)

A6. (a) Find the eigen values and eigen vectors of the matrix

(b) Verify Cayley Hamilton for the marix A =

a.7)Find the eigen values and eigen vectors of 2 2 0 2 1 1 –7 2 –3

A7. Using cayley-Hamilton theorem, find the inverse of the matrix

A =

a.8)Show that the quadratic form is positive semi definite.

(or)A8. Investigate for what values of a and b the simultaneous equations x + y + z = 6, x + 2y +

3z = 10, x + 2y + az = b. will have (a) no solution (b) unique solution (c) infinite solution

a.9)For what values of a and b the equations.x + y + z = 6x + 2y + 3z = 10x + 2y + az = b

have (i) No solution (ii) A unique solution (iii) Infinite number of solutions.(or)

A9. Reduce quadratic form to a canonical form through an orthogonal transformation.

(a.10) Find the Eigen values and Eigen vectors of the matrix

(b) Diagonalise the matrix A given above by similarity transformation. (or)

A10. (a) Find the inverse of the matrix by using Cay;ey-Hamilton theorem.

(b) Obtain an orthogonal transformation, which will transform the quadratic form 6x2 + 3y2 + 3z2 – 4xy – 2yz + 4zx into a canonical form.

a.11)Verify Cayley – Hamilton theorem for the matrix A= and hence

find A-1 and A4

M-12

(or)A11. Reduce the quadratic form 3x into a canonical form by

orthogonal reduction.-M12

a.12). Diagonalize the matrix by orthogonal transformation.

(or)

A12. (a) Show that the matrix satisfies its own

characteristic equation and hence find A-1.-M12

(b) Find the eigen values and eigen vectors of the matrix

M12

a.13)State Cayley–Hamilton theorem and find the inverse of the matrix A = using

Cayley – Hamilton theorem hence find A4.(or)

A13. Reduce 6x2 + 3y2 + 3z2 – 4xy – 2yz + 4xz into canonical form by an orthogonal transformation

a.14) Reduce the quadratic form into its canonical form using orthogonal reduction.-D11

(or)

A14.Using Cayley-Hamilton theorem find the inverse of the matrix D11

a.15)Show that the quadratic form is positive semi definite.

(or)A15. Investigate for what values of a and b the simultaneous equations x + y + z = 6, x + 2y +

3z = 10, x + 2y + az = b. will have (a) no solution (b) unique solution

(c) infinite solution

a.16). Find the eigen values and eigen vectors of

(a) the matrix

(b) Verify Cayley-Hamilton theorem for the matrix A = . Hence find its

inverse.(or)

A16. Reduce the quadratic form3x1

2 +5x22 +3x3

2 – 2x2x3 + 2x3x1 – 2x1x2

to a canonical form by orthogonal reduction. Find also index, signature and nature of the quadratic form.

(a17) Verify Cayley-Hamilton theorem for the matrix =M117 2 –2

A = –6 –1 26 2 –1

2 2 –7

(b) Find the eigen values and eigen vectors of 2 1 2 0 1 -3

(or)A17. Reduce 6x2 + 3y2 – 4xy – 2yz + 4xz + 3z2 into a canonical form by an orthogonal

reduction. Discuss the nature of quadratic form.-M11

a.18)Using cayley.Hamilton theorem find A-1 if ;

Also verify the theorem.-D10(or)

A18. Reduce the equation form 10x2 + 2y2 + 5z2 + 6yz – 10zx – 4xy to a canonical form.D10

a.19). Verify Cayley-Hamilton theorem for the matrix . Hence compute A-1.

–M10(or)

A19. Reduce the matrix to diagonal form by orthogonal transformation-M10

(a.20) Find the Eigen values and Eigen vectors of the matrix

(b) Diagonalise the matrix hence find A8.-D09

(or)

A20. (a) Find the inverse of the matrix using Cayley-Hamilton Therorem.

–D09

UNIT III GEOMETRICAL APPLICATIONS OF DIFFERENTIAL CALCULUS Curvature –centre, radius and circle of curvature in Cartesian co-ordinates only – Involutes and evolutes –envelope of family of curves with one and two parameters – properties of envelopes and evolutes – evolutes asenvelope of normal.UNIT IV FUNCTIONS OF SEVERAL VARIABLES 1Functions of two variables – partial derivatives – Euler’s theorem and problems - Total differential – Taylor’sexpansion – Maxima and minima – Constrained maxima and minima – Lagrange’s multiplier method – Jacobian –Differentiation under integral sign.UNIT V ORDINARY DIFFERENTIAL EQUATION Second order linear differential equation with constant coefficients – Particular Integrals for eax, sin ax, cos ax,xn, xneax, eax sinbx, eax cos bx. Equations reducible to Linear equations with constant co-efficient using x=et.

Simultaneous first order linear equations with constant coefficients - Method of Variations of Parameters.