Engineering Mathematics I -Pune-November 2019 - Grad Plus

# Engineering Mathematics I -Pune-November 2019

## (2015 PATTERN)

Time : Two Hours
Maximum Marks : 50
N.B. :— (i) Attempt Q. 1 or Q. 2, Q. 3 or Q. 4, Q. 5 or Q. 6 and Q. 7 or Q. 8.
(ii) Neat diagram must be drawn wherever necessary.
(iii) Use of electronic pocket calculator is allowed.
(iv) Assume suitable data, if necessary.

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Q. 1. (a) Find the rank of the matrix : [4M]
\begin{bmatrix}1 & 2& -1& 3\\3& 4& 0& -1\\-1& 0& -2& 7\end{bmatrix}

(b) Find the eigen values and eigen vector corresponding to largest eigen value of a matrix : [4M]
A=\begin{bmatrix}4& 6& 6\\1& 3& 2\\-1& -4& -3\end{bmatrix}

(c) Solve x5 – 1 = 0 using DeMoivres Theorem. [4M]

OR

Q. 2. (a) Examine for linear dependence or independence of vectors : [4M]
x1 = (1, 1, –1), x2 = (2, 3, –5), x3 = (2, –1, 4).

(b) If cosec (x+iy) = u +iv, prove that : [4M]
\left ( u^{2} +v^{2}\right )^{2}=\frac{u^{3}}{sin^{2}x}-\frac{v^{2}}{cos^{2}x}

(c) Separate real and immaginary parts of (1+i )i . [4M]

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

(b) Expand: [4M]
sin^{-1}\left ( \frac{2x}{1+x^{2}} \right ) in ascending powers of x.

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

OR

Q.4. (a) Solve any one : [4M]
(i) Find the values of a and b if :
[;atex] \lim_{x\rightarrow 0}\left ( \frac{sinx}{x^{3}} +\frac{a}{x^{3}}+b\right )=0[/latex]

(ii) Evaluate :
\lim_{x\rightarrow \frac{\pi }{2}}\left ( sec\, x \right )^{cotx}

(b) Using Taylor’s theorem, expand 4x3+ 3x2 +2x+ 1 in ascending powers of (x + 1). [4M]

(c) If y  cos (m log x) , prove that :  [4M]
x^{2}y_{n+2}+\left ( 2n+1 \right )xy_{n+1}+\left ( m^{2}+n^{2}\right )y_{n}=0

Q.5. Solve any two : [6M]
(a) If u =4e-6x sin(pt-6x) satisfies the partial differential equation ut = uxx then find the value of Φ.

b) If T=s\left ( \frac{xy}{x^{2}+y^{2}} \right )+\sqrt{x^{3}+y^{2}}+\frac{x^{2y}}{x+y}
find the value of T T x y x y .[7M]

(c) If u  f (x – y, y -z, z – x) then find the value of : [6M]
\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y }+\frac{\partial u}{\partial z}

OR

Q.6. Solve any two :
(a) If x = u tan v, y = u sec v prove that : [6M]
(ux)y (vx)y=(uy)x (vy)x

(b) If u = f (r) where r = \sqrt{x^{2}+y^{2}} then prove that : [7M]
u_{xx}+u_{yy}=\frac{d^{2}f}{dr^{2}}+\frac{1}{r}\frac{df}{dr}

(c) If z = f (u,v) where u, v are homogeneous functions of degree 10 in x, y then prove that :
x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=10\left ( u\frac{\partial z}{\partial u}+\frac{\partial z}{\partial v} \right )[5M]

Q.7 (a) If u = x +3y2- z3 ,  v = 4x2 yz , w =2z2 – xy ,evaluate
[;atex] \frac{\partial \left ( u,v,w \right )}{\partial \left ( x,y,z \right )}[/latex] at (1,-1,0) [4M]

(b) Prove that the functions : [4M]
u = y+ z ,  v = x+ 2z2 ,  w = x – 4yz – 2y2 are functionally dependent.

(c) Find all the stationary points of the function :
x 3+3xy2-15x 2 -15y2 +72x .
Examine whether the function is maximum or minimum at those pionts. [5M]

OR

Q. 8. (a) If u + v= x2 + y2 , u- v =x +2y ,
find
\left ( \frac{\partial u}{\partial x} \right )_{y},  by using Jacobians.[4M]

(b) The focal length of a minor is found from the formula
\frac{1}{v}-\frac{1}{u}=\frac{2}{f} Find the percentage error in f given u and v are both of error 2% each. [4M]
(c) Find the stationary points of 2 T(x, y, z) = 8x2 +4yz -16z+ 600 if the condition  4x2 +y2 +4z 2 =16 satisfied. [5M]

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