Engineering Mathematics I- Engineering-Pune-November 2018 - Grad Plus

# Engineering Mathematics I- Engineering-Pune-November 2018

## (2015 Pattern) (Credit System)

Time: 2 Hours
Mar. Marks: 50 Instructions to the candidates:
1) Solve Q.1 or 0.2, 0.3 or 0.4.0.3 or 0.6, Q.7 or 0.8.
2) Near diagrams must be drawn wherever necessary
3) Figures to the right indicate full marks.
4) Use of an electronic pocket calculator is allowed.
5) Assume suitable data, if necessary.
____________________________________________________________________________________________________________________________

Q.1) a) Examine for consistency of system of equations [4M]
x+y-32=-1
4x – 2y + 6z=8
15x – 3y + 92 = 21
if consistent solve it.

b) Find eigen values of the matrix. [4M]
\begin{bmatrix}2& 0& -1\\0& 2& 0\\-1& 0& 2\end{bmatrix}
Also find eigen vector corresponding to smallest eigen value,

c) Two opposite vertices of a square are represented by complex numbers
9 + 12i and -5 + 10i. Find the complex number representing the other two vertices of the square. [4M]

OR

Q.2) a) Examine for Linear dependence or independence of vectors X1=(3, 1, -4), x2=(2, 2, -3).x3 = (0, -4,1). If dependent find the relation between them. [4M]

b) Solve x4+ x3 + x2 + x + 1 = 0, by using DeMoivre’s theorem. [4M]

c) If sinh(Θ+ iΦ) = cos α + i sinα, prove that sin4Θ= costΦ. [4M]

Q.3) a) Solve any one: [4M]
i) Test the convergence of the series : \sum_{n=1}^{\infty }\frac{(n+1)^{n}}{n!}
ii) Test the convergence of the series : \frac{1}{1+2^{-1}}+\frac{2}{1+2^{-2}}+\frac{3}{1+2^{-3}}+..........

b) Prove that : [4M]
log(1+x+x^{2}+x^{3}+x^{4})= x+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\frac{x^{4}}{4}-\frac{4}{5}x^{5}+.......

c) Find the derivative of [4M]
y=\frac{1}{(x-1)^{2}(x-2)}\cdot

OR

Q. 4) a) Solve any one : [4M]
1) Find a &b, if
\lim_{x\rightarrow 0}\frac{x(-a\, cos x +1)+b\, sinx}{x^{3}}=\frac{1}{3}
ii) Prove that
\lim_{x\rightarrow \infty }\left ( \frac{ax+1}{ax-1} \right )^{x}=e^{\frac{2}{a}}

b) Expand 2x3 + 7x2 + x- 6 in powers of (x-3). [4M]

c) If y = a cos (m log x) + b sin(m log x), show that

x2yn+2+(2n+1)xyn+1+(n+m2)yn=0 [4M]

Q.5) Solve any two: [6M]
a) If μ=log(x3+y3-x2y-xy2) prove that
\left ( \frac{\partial }{\partial x}+\frac{\partial }{\partial y} \right )^{2}\mu =\frac{-4}{\left ( x+y \right )^{2}}

b) If x=eμ tan ν, y=eμ sec ν, find the value of
\left [ x\frac{\partial \mu }{\partial x}+y\frac{\partial \mu }{\partial y} \right ]\cdot \left [ x\frac{\partial \nu }{\partial x} +y\frac{\partial \nu }{\partial y}\right ] [7M]

C) If ν = f (ex-y,ey-z,ez-x) then show that
\frac{\partial \nu }{\partial x}+\frac{\partial \nu }{\partial y}+\frac{\partial \nu }{\partial z}=0 [M]

Q. 6 Solve any two : [6M]
a) Find \frac{\partial \mu }{\partial x} If μ =x.log(xy) and x3 + y3+3xy=0

b) If \mu =sin^{-1}\left ( \frac{x+y}{\sqrt{x}\sqrt{y}} \right ) prove that x^{2}\frac{\partial ^{2}\mu }{\partial x^{2}}+2xy\frac{\partial ^{2}\mu }{\partial x\partial y}+y^{2}\frac{\partial ^{2}\mu }{\partial y^{2}}=-\frac{sin \mu\, cos2 \mu }{4\, cos^{3}\mu } [7M]

c) If x2 = aμ + bν, y2 = aμ – bν prove that (μx)y (xu)v = (νy)x (yν)μ where a, b are
constants. [6M]

Q. 7 a) If μx=yz, νy=zx, ωz=xy find \frac{\partial(\mu ,\nu ,\omega) }{\partial(x,y,z) }. [4M]

b) Examine for functional dependence u=y+z ν=x+2z2, w=x-4yz-zy2. [4M]

c) Find the extreme values of f(x, y) = 3x2 – y2 +x3.[5M]

OR

Q.8) a) If μ = x+y2, ν=y+z2 , w=z+xFind \left ( \frac{\partial x}{\partial \mu } \right )_{\nu \cdot \omega } by using Jacobians. [4M]

b) The area of a triangle ABC, is calculated from the formula Δ = 1/2 bc sinA. Errors of 1%, 2% & 3% respectively are made in measuring b, c, A. If the correct values of A is 45°. Find the % error in the calculated values of Δ. [4M]

c) Find stationary values of a3x2 + b3y2 + c3z2. where \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1 [5M]

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