Engineering Mathematics I-Engineering-Pune- May 2019 - Grad Plus

# Engineering Mathematics I-Engineering-Pune- May 2019

## (2015 PATTERN)

Time : Two Hours
Maximum Marks : 50
N.B. (i) Neat diagrams must be drawn wherever necessary.
(ii) Figures to the right indicate full marks.
(iii) Use of logarithmic tables, slide rule, Mollier charts, electronic
pocket calculator and steam tables is allowed.
(iv) Assume suitable data, if necessary.
Attempt Q. No. 1 or 2, Q. No. 3 or 4, Q. No. 5 or 6, Q. No. 7 or 8.
____________________________________________________________________________________________________________________________

Q.1.a) Reduce the following matrix to its normal form and hence find its rank:[4M]
A=\begin{bmatrix}-2& -8& -12\\1& 4& 4\\0& 0& 1 \end{bmatrix}

b) Find eigen values and eigen vector corresponding to highest eigen value of the following matrix :[4M]

A=\begin{bmatrix}-2& -8& -12\\1& 4& 4\\0& 0& 1\end{bmatrix}

c) Solve the equation x3-1=0 by applying Demoivre’s theorem. [4M]

OR

Q. 2.a) Investigate for what values of k the system of equations x + y + z = 1, 2x + y + 4z = k, 4x + y + 10z =k2have infinite number of solutions. [4M]

b) Find locus of z such that: [4M]

|z+1| = |z-i|.

c) If sin(x + iy) = μ + iν prove that μ2 cosec2 x – ν2 sec2  x = 1 and μ2 sec h2y + ν2 cosec h2y = 1. [4M]

Q. 3.a)Solve any one: [4M]

(i) Test for convergence the
\sum_{n=1}^{\infty }\frac{2n-1}{n(n+1)(n+2)}

(ii) Test the convergence of the series :
1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}+...........

b)Expand log (1 + x + x2 + x3) upto the term in x8. [4M]

(c) Find the nth derivative of. [4M]
y=\frac{x}{\left ( x-1 \right )\left ( x-2 \right )\left ( x-3 \right )}\cdot

OR

Q.4.a) Solve any one: [4M]

(i) Evaluate:
\lim_{x\rightarrow 0}\, \frac{e^{x}-1-x}{log\left ( 1+x \right )-x}\cdot

(ii) Evaluate :
\lim_{x\rightarrow 0}\left ( sin x \right )^{tanx}

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

c) If : [4M]
y=e^{tan^{-1}x}
prove that :
(1 + x2)yn+2+ [2(n + 1) x – 1]yn+1 + n(n + 1)yn = 0.

Q.5.Solve any two: [6M]
(a) If z = tan(y + ax) + (y – ax)3/2, find the value of
\frac{\partial ^{2}z}{\partial x^{2} }-\alpha ^{2}\frac{\partial ^{2}z}{\partial y^{2}}\cdot.
b) If x2 = aμ+ bν and y2 = aμ – bν, then prove that : [6M]
\left ( \frac{\partial \mu }{\partial x} \right )_{y}\cdot\, \left ( \frac{\partial x}{\partial \mu }\right )_{\nu }=\left ( \frac{\partial \nu }{\partial y} \right )_{x}\cdot \left ( \frac{\partial y}{\partial \nu } \right )_{\mu }

c) If u = sin-1(x2 + y2)1/5, then prove      [7M]
x^{2}\mu _{xx}+2xy\, \mu _{xy}+y^{2}\mu _{yy}=\frac{2}{5}tan\mu \left [ \frac{2}{5}tan^{2}\mu -\frac{3}{5} \right ]\cdot

Q. 6 Solve any two : [7M]
a) If f\left ( x,y \right )=\frac{1}{x^{2}}+\frac{1}{xy}+\frac{log x -log y}{x^{2}y^{2}}[,latex] </span><span style="font-weight: 400;">then prove that . [latex] x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}+2f=0

b) If u = f(r) where r=\sqrt{x^{2}+y^{2}} ,
then prove that :
\mu _{xx}+\mu _{yy}={f}''\left ( r \right )+\frac{1}{r}{f}'(r)\cdot [6M]

c) If z = f(x, y) where x=eμ cos ν and y=eμ sin v, then prove that: [6M]
y\frac{\partial z}{\partial \mu }+\frac{\partial z}{\partial \nu }=e^{2n}\frac{\partial z}{\partial y}

Q. 7 a) If : [6M]
x=\mu \nu ,y=\frac{\mu +\nu }{\mu -\nu }
Find : \frac{\partial \left ( \mu ,\nu \right )}{\partial \left ( x,y \right )}

b) If : [4M]
μ= x+y+z, ν= x2 + y2 +z2,ω= x3 + y3+z3
find \frac{\partial x}{\partial \mu }\cdot

(c) Divide the number 120 into three parts so that the sum of their products taken two at a time shall be maximum. [5M]

OR

Q. 8.(a) Examine for functional dependence and independence μ = x + y + z , ν = x2 + y2 + z2, ω = xy + yz + xz.  [4M]

b) Find the percentage error in the area of an ellipse with an error of 1% is made in measuring its major and minor axis. [4M]

c) Find the extreme values of f(x, y) = x3 + y3 - 3axy, a > 0.[5M]

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