2018-2019 December

1) Question No. 1 is compulsory

2) Attempt any 3 questions from remainging five questions.

Show that sec h^{-1}(sin θ) =log cot (`theta/2` ).

Concept: Logarithmic Functions

Show that a matrix A = `1/2[(sqrt2,-isqrt2,0),(isqrt2,-sqrt2,0),(0,0,2)]` is unitary.

Concept: Types of Matrices

Evaluate `lim_(x->0) sinx log x.`

Concept: nth derivative of standard functions

Find the n^{th} derivative of y=e^{ax} cos2 x sin x.

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions

If 𝒙 = r cos θ and y= r sin θ prove that JJ-1=1.

Concept: Types of Matrices

Using coding matrix A=`[(2,1),(3,1)]` encode the message THE CROW FLIES AT MIDNIGHT.

Concept: Types of Matrices

Find all values of `(1 + i)^(1/3` and show that their continued product is (1+ 𝒊 ).

Concept: Powers and Roots of Trigonometric Functions

Find the non-singular matrices P & Q such that PAQ is in normal form where`[(1,2,3,4),(2,1,4,3),(3,0,5,-10)]`

Concept: Types of Matrices

Find maximum and minimum values of x^{3} +3xy^{2} -15x^{2}-15y^{2}+72x.

Concept: Maxima and Minima of a Function of Two Independent Variables

If U = `e^(xyz) f((xy)/z)` prove that `x(delu)/(delx)+z(delu)/(delx)2xyzu` and `y(delu)/(delx)+z(delu)/(delz)=2xyzu` and hence show that `x(del^2u)/(delzdelx)=y(del^2u)/(delzdely)`

Concept: Successive Differentiation

By using Regular falsi method solve 2x – 3sin x – 5 = 0.

Concept: Regula – Falsi Equation

If y=sin[log(x^{2}+2x+1)] then prove that (x+1)^{2}y_{n+2} +(2n +1)(x+ 1)y_{n+1} + (n^{2}+4)y_{n}=0.

Concept: Leibnitz’S Theorem (Without Proof) and Problems

State and prove Euler’s Theorem for three variables.

Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)

By using De Moivre's Theorem obtain tan 5θ in terms of tan θ and show that `1-10 tan^2(pi/10)+5tan^4(pi/10)=0`.

Concept: D’Moivre’S Theorem

Investigate for what values of λ and μ the equations

2x + 3y + 5z = 9

7x + 3y - 2z = 8

2x + 3y + λz = μ have

A. No solutions

B. Unique solutions

C. An infinite number of solutions.

Concept: Types of Matrices

Find nth derivative of `1/(x^2+a^2.`

Concept: nth derivative of standard functions

If z = f (x, y) where x = e^{u} +e^{-v}, y = e^{-u} - e^{v} then prove that `(delz)/(delu)-(delz)/(delv)=x(delz)/(delx)-y(delz)/(dely).`

Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)

Solve using Gauss Jacobi Iteration method

2𝒙 + 12y + z – 4w = 13

13𝒙 + 5y - 3z + w = 18

2𝒙 + y – 3z + 9w = 31

3𝒙 - 4y + 10z + w = 29

Concept: Gauss Jacobi Iteration Method

If y = log `[tan(pi/4+x/2)]`Prove that

I. tan h`y/2 = tan pi/2`

II. cos hy cos x = 1

Concept: Logarithmic Functions

If U `=sin^(-1)[(x^(1/3)+y^(1/3))/(x^(1/2)+y^(1/2))]`prove that `x^2(del^2u)/(del^2x)+2xy(del^2u)/(delxdely)+y^2(del^2u)/(del^2y)=(tanu)/144[tan^2"U"+13].`

Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)

Expand 2 𝒙^{3} + 7 𝒙^{2} + 𝒙 – 6 in power of (𝒙 – 2) by using Taylors Theorem.

Concept: Taylor’S Series Method

Expand sec x by McLaurin’s theorem considering up to x^{4} term.

Concept: Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)

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