2017-2018 December

Separate into real and imaginary parts of cos`"^-1((3i)/4)`

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions

Show that the matrix A is unitary where A = `[[alpha+igamma,-beta+idel],[beta+idel,alpha-igamma]]` is unitary if `alpha^2+beta^2+gamma^2+del^2=1`

Concept: Inverse of a Matrix

If `z=tan(y-ax)+(y-ax)^(3/2)` then show that `(del^2z)/(delx^2)= a^2 (del^2z)/(dely^2)`

Concept: Partial Derivatives of First and Higher Order

`"If" x=uv & y=u/v "prove that" jj^1=1`

Concept: .Circular Functions of Complex Number

Find the n^th derivative of `x^3/((x+1)(x-2))`

Concept: nth derivative of standard functions

Using the matrix A = `[[-1,2],[-1,1]]`decode the message of matrix C= `[[4,11,12,-2],[-4,4,9,-2]]`

Concept: Rank of a Matrix Using Echelon Forms

`"If" sin^4θcos^3θ = acosθ + bcos3θ + ccos5θ + dcos7θ "then find" a,b,c,d.`

Concept: Expansion of π^π₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π ππβ1 (π₯),πππ β1 (π₯),π‘ππβ1 (π₯)

Using Newton Raphson method solve 3x – cosx – 1 = 0. Correct upto 3 decimal places.

Concept: Expansion of π^π₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π ππβ1 (π₯),πππ β1 (π₯),π‘ππβ1 (π₯)

Find the stationary points of the function x3+3xy2-3x2-3y2+4 & also find maximum and minimum values of the function.

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

Show that xcosecx = `1+x^2/6+(7x^4)/360+......`

Concept: Expansion of π^π₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π ππβ1 (π₯),πππ β1 (π₯),π‘ππβ1 (π₯)

Reduce matrix to PAQ normal form and find 2 non-Singular matrices P & Q.

`[[1,2,-1,2],[2,5,.2,3],[1,2,1,2]]`

Concept: Reduction to Normal Form

If y= cos (msin_1 x).Prove that `(1-x^2)y_n+2-(2n+1)xy_(n+1)+(m^2-n^2)y_n=0`

Concept: Expansion of π^π₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π ππβ1 (π₯),πππ β1 (π₯),π‘ππβ1 (π₯)

State and Prove Euler’s Theorem for three variables.

Concept: Eulerβs Theorem on Homogeneous functions with two and three independent variables (with proof)

Show that all roots of `(x+1)^6+(x-1)^6=0` are given by -icot`((2k+1)n)/12`where k=0,1,2,3,4,5.

Concept: Powers and Roots of Trigonometric Functions

Show that the following equations: -2x + y + z = a, x - 2y + z = b, x + y - 2z = c have no solutions unless a +b + c = 0 in which case they have infinitely many solutions. Find these solutions when a=1, b=1, c=-2.

Concept: Rank of a Matrix Using Echelon Forms

If Z=f(x.y). x=r cos θ, y=r sinθ. prove that `((delz)/(delx))^2+((delz)/(dely))^2=((delz)/(delr))^2+1/r^2((delz)/(delθ))^2`

Concept: Differentiation of Implicit Functions

If coshx = secθ prove that (i) x = log(secθ+tanθ). (ii) `θ=pi/2tan^-1(e^-x)`

Concept: Expansion of π^π₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π ππβ1 (π₯),πππ β1 (π₯),π‘ππβ1 (π₯)

Solve by Gauss Jacobi Iteration Method: 5x – y + z = 10, 2x + 4y = 12, x + y + 5z = -1.

Concept: Gauss Jacobi Iteration Method

Prove that `cos^-1tanh(log x)+ = π – 2(x-x^3/3+x^5/5.........)`

Concept: Expansion of π^π₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π ππβ1 (π₯),πππ β1 (π₯),π‘ππβ1 (π₯)

If` y= e^2x sin x/2 cos x/2 sin3x. "find" y_n`

Concept: Expansion of π^π₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π ππβ1 (π₯),πππ β1 (π₯),π‘ππβ1 (π₯)

Evaluate `Lim _(x→0) (cot x)^sinx.`

Concept: Expansion of π^π₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π ππβ1 (π₯),πππ β1 (π₯),π‘ππβ1 (π₯)

Prove that log `[sin(x+iy)/sin(x-iy)]=2tan^-1 (cot x tanhy)`

Concept: Expansion of π^π₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π ππβ1 (π₯),πππ β1 (π₯),π‘ππβ1 (π₯)

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## University of Mumbai previous year question papers Semester 1 (FE First Year) Applied Mathematics 1 with solutions 2017 - 2018

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