BE Biotechnology

BE IT (Information Technology)

BE Computer Engineering

BE Electrical Engineering

BE Marine Engineering

BE Electronics and Telecommunication Engineering

BE Biomedical Engineering

BE Electronics Engineering

BE Instrumentation Engineering

BE Construction Engineering

BE Production Engineering

BE Civil Engineering

BE Printing and Packaging Technology

BE Mechanical Engineering

BE Chemical Engineering

BE Automobile Engineering

Academic Year: 2018-2019

Date: December 2018

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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` ).

Chapter: [6.02] Logarithm of Complex Numbers

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

Chapter: [7] Matrices

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

Chapter: [6.01] Successive Differentiation

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

Chapter: [6.02] Logarithm of Complex Numbers

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

Chapter: [7] Matrices

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

Chapter: [7] Matrices

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

Chapter: [5] Complex Numbers

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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)]`

Chapter: [7] Matrices

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

Chapter: [9] Applications of Partial Differentiation , Expansion of Functions

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)`

Chapter: [6.01] Successive Differentiation

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

Chapter: [10] Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations

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.

Chapter: [6.01] Successive Differentiation

State and prove Euler’s Theorem for three variables.

Chapter: [8] Partial Differentiation

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`.

Chapter: [5] Complex Numbers

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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.

Chapter: [7] Matrices

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

Chapter: [6.01] Successive Differentiation

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).`

Chapter: [8] Partial Differentiation

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

Chapter: [10] Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations

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

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

II. cos hy cos x = 1

Chapter: [6.02] Logarithm of Complex Numbers

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].`

Chapter: [8] Partial Differentiation

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

Chapter: [9] Applications of Partial Differentiation , Expansion of Functions

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

Chapter: [9] Applications of Partial Differentiation , Expansion of Functions

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