2016-2017 December

(1) Question no. 1 is compulsory.

(2) Attempt any 3 questions from remaining five questions.

If `cos alpha cos beta=x/2, sinalpha sinbeta=y/2`, prove that:

`sec(alpha -ibeta)+sec(alpha-ibeta)=(4x)/(x^2-y^2)`

Concept: Review of Complex Numbers‐Algebra of Complex Number

If `z =log(e^x+e^y) "show that rt" - s^2 = 0 "where r"= (del^2z)/(delx^2),t=(del^2z)/(dely^2)"s"=(del^2z)/(delx dely)`

Concept: Review of Complex Numbers‐Algebra of Complex Number

If x = uv, y `=(u+v)/(u-v).`find `(del(u,v))/(del(x,y))`.

Concept: Review of Complex Numbers‐Algebra of Complex Number

If `y=2^xsin^2x cosx` find `y_n`

Concept: Review of Complex Numbers‐Algebra of Complex Number

Express the matrix as the sum of symmetric and skew symmetric matrices.

Concept: Addition of a Matrix

Evaluat `lim_(x->0) (e^(2x)-(1+x)^2)/(xlog(1+x)`

Concept: L‐ Hospital Rule

Show that the roots of x^{5 }=1 can be written as 1, `alpha^1,alpha^2,alpha^3,alpha^4` .hence show that `(1-alpha^1) (1-alpha^2) (1-alpha^3)(1-alpha^4)=5.`

Concept: Powers and Roots of Trigonometric Functions

Reduce the following matrix to its normal form and hence find its rank.

Concept: Reduction to Normal Form

Solve the following equation by Gauss-Seidel method upto four iterations

4x-2y-z=40, x-6y+2y=-28, x-2y+12z=-86.

Concept: Gauss Seidal Iteration Method

Investigate for what values of μ and λ the equations x+y+z=6, x+2y+3z=10, x+2y+λz=μ has

1) No solution

2) A unique solution

3) Infinite number of solutions.

Concept: consistency and solutions of homogeneous and non – homogeneous equations

If `u=x^2+y^2+z^2` where `x=e^t, y=e^tsint,z=e^tcost`

Prove that `(du)/(dt)=4e^(2t)`

Concept: Review of Complex Numbers‐Algebra of Complex Number

Show that `sin(e^x-1)=x^1+x^2/2-(5x^4)/24+`...................

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)

Expand `2x^3+7x^2+x-6` in powers of (x-2)

Concept: Expansion of sinnθ, cosnθ in powers of sinθ, cosθ

If x = u+v+w, y = uv+vw+uw, z = uvw and φ is a function of x, y and z

Prove that

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

If tan(θ+iφ)=tanα+isecα

Prove that

1)`e^(2varphi)=cot(varphi/2)`

2) `2theta=npi+pi/2+alpha`

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

Find the roots of the equation `x^4+x^3 -7x^2-x+5 = 0` which lies between 2 and 2.1 correct to 3 places of decimals using Regula Falsi method.

Concept: Regula – Falsi Equation

If y=(x+√x^{2}-1 ,Prove that

`(x^2-1)y_(n+2)+(2n+1)xy_(n+1)+(n^2-m^2)y_n=0`

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

Using the encoding matrix `[(1,1),(0,1)]` encode and decode the messag I*LOVE*MUMBAI.

Concept: Application of Inverse of a Matrix to Coding Theory

Considering only principal values separate into real and imaginary parts

`i^((log)(i+1))`

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions

Show that `ilog((x-i)/(x+i))=pi-2tan6-1x`

Concept: Logarithmic Functions

Using De Moivre’s theorem prove that]

`cos^6theta-sin^6theta=1/16(cos6theta+15cos2theta)`

Concept: D’Moivre’S Theorem

If u `=sin^(-1)((x^(1/3)+y^(1/3))/(x^(1/2)-y^(1/2)))`, Prove that

`x^2(del^2u)/(delx^2)+2xy(del^2u)/(delxdely)+y^2(del^2u)/(dely^2)=tanu/144(tan^2u+13)`

Concept: System of Homogeneous and Non – Homogeneous Equations

Find the maxima and minima of `x^3 y^2(1-x-y)`

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

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