Engineering Mathematics I- Engineering-Pune-November 2019
PUNE UNIVERSITY
F.E. (I Semester) EXAMINATION, 2019
ENGINEERING MATHEMATICS—I
(Phase–II)
(2019 PATTERN)
Time : 2½ Hours
Maximum Marks: 70
N.B.:— (i) Attempt Q. No. 1 or Q. No. 2, Q. No. 3 or Q. No. 4,
Q. No. 5 or Q. No. 6, Q. No. 7 or Q. No. 8.
(ii) Use of an electronic pocket calculator is allowed.
(iii) Assume suitable data, if necessary.
(iv) Neat diagrams must be drawn wherever necessary.
(v) Figures to the right indicate full marks.
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Q. 1 (a) If z = tan (y + ax) + (y – ax)3/2, find the value of
\frac{\partial ^{2}Z}{\partial X^{2}}-a^{2}\frac{\partial ^{2}z}{\partial y^{2}} .[6M]
(b) If T=\sin \left ( \frac{xy}{x^{2}y^{2}} \right )+\sqrt{x^{2}+y^{2}}
by using Euler’s theorem find x\frac{\partial T}{\partial x}+y\frac{\partial T}{\partial y} [6M]
(c) If u = x2 – y2, v= 2xy and z = f(u,v), then show that
x\frac{\partial T}{\partial x}+y\frac{\partial T}{\partial y} [6M]
OR
Q.2. (a) If x = u tan v, y =u sec v, prove that : [6M]
\left ( \frac{\partial u}{\partial x} \right )_{y}\left (\frac{\partial v}{\partial x}\right )_{y}=\left ( \frac{\partial u}{\partial y } \right )_{x}\left ( \frac{\partial v }{\partial y} \right )_{x}
(b) If \left ( \frac{\partial u}{\partial x} \right )_{y}\left (\frac{\partial v}{\partial x}\right )_{y}=\left ( \frac{\partial u}{\partial y } \right )_{x}\left ( \frac{\partial v }{\partial y} \right )_{x} , by using Euler’s theorem. prove that :
x^{2}\frac{\partial u^{2}}{\partial x^{2}}+2xy\frac{\partial u}{\partial x\partial y}+y^{2}\frac{\partial ^{2}u}{\partial y^{2}}=\frac{1}{4}\left ( tan^{3}u-tan\, u \right ) [6M]
(c) If x=\frac{cos\Theta }{u}, y=\frac{sin\Theta }{u} z= f\left ( x, y \right )
, then show that :
u\frac{\partial z}{\partial u}-\frac{\partial z}{\partial \Theta }=\left ( y-x \right )\frac{\partial z}{\partial x}-\left ( y+x \right )\frac{\partial z}{\partial y} [6M]
Q. 3 (a) If u = x + y + z, v = x2 + y2 + z2, w = xy + yz + zx
find \frac{\left ( u,v,w \right )}{\partial \left ( x,y,z \right )} [6M]
(b) Examine whether u=\frac{x-y}{1+xy} v= tan-1-tan-1y are functionally dependent, if so find the relation between them. [5M]
(c) Find the extreme values of x^{2}+x^{2}+\frac{2}{x}+\frac{2}{y} · [6M]
OR
Q. 4 (a) If u = x + y2, v = y + z2, w = z + x2, using Jacobian find \frac{\partial x}{\partial u} · [6M]
(b) A power dissipated in a resistor is given by P=\frac{\varepsilon ^{2}}{R}· If errors of 3% and 2% are found in
and R respectively, find the percentage error in P. [5M]
(c) Using Lagrange’s method find extreme value of xyz if x + y + z = a. [6M]
Q. 5 (a) Examine for consistency of the system of linear equations and solve if consistent :[6M]
x1 + x2 + x3 = 0
–2x1 + 5x2 + 2x3 = 1
8x1 + x2 + 4x3 = –1
(b) Examine for linear dependence or independence the vectors (1, 1, 1, 3), (1, 2, 3, 4), (2, 3, 4, 7). Find the relation between them if dependent. [6M]
(c) Determine the values of a, b, c when A is orthogonal where :[5M]
A= \begin{bmatrix}0& 2b& c\\a& b& -c\\a& -b& c\end{bmatrix}
OR
Q. 6 (a) Investigate for what values of a and b, the system of equations 2x – y + 3z = 2, x + y + 2z = 2, 5x – y + az = b have :
(1) No solution
(2) A unique solution
(3) An infinite number of solutions. [6M]
(b) Examine for linear dependence or independence the vectors
x1 = (2, 3, 4, –2), x2 = (1, 1, 2, –1), x3 =(–1/2, – 1, -1, 1/2 ) Find the relation between them if dependent. [6M]
(c) Determine the currents in the network given in figure below : [5M]
Q. 7 (a) Find the eigen values and the corresponding eigen vectors forthe following matrix : [6M]
A=\begin{bmatrix}4& 0& 1\\-2& 1& 0\\-2& 0& 1\end{bmatrix}(b) Verify Cayley-Hemilton theorem for A=\begin{bmatrix}1& -1& 0\\2& 3& -2\\-1& 0& 1\end{bmatrix} and use it to find A–1. [6M]
(c) Find a matrix P that diagonalizes the matrix
A=\begin{bmatrix}1 & 6 & 1\\1& 2& 0\\0& 0& 3\end{bmatrix} [6M]
OR
Q. 8 (a) Find the eigen values and the corresponding eigen vectors forthe following matrix : [6M]
A= \begin{bmatrix}5 & 0& 1\\0& -2 & 0\\1& 0& 5\end{bmatrix}
(b) Verify Cayley-Hamilton theorem for A=\begin{bmatrix}0 & 1& 0\\0 & 0& 1\\1& -3& 3\end{bmatrix} and useit to find A–1. [6M]
(c) Reduce the following quadratic form to the sum of the squares form : [6M]
Q = 2x2 + 9y2 + 6z2 + 8xy + 8yz + 6xz.