Question Paper: Engineering Mathematics-1 : Question Paper Dec 2014 - First Year Engineering (Semester 1) | Pune University (PU)
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## Engineering Mathematics-1 - Dec 2014

### First Year Engg (Semester 1)

TOTAL MARKS:
TOTAL TIME: HOURS

### Answer any one question from Q1 and Q2

1 (a) Examine for consistency the system of equations
x-y-z=2
x+2y+z=2
4x-7y-5z=2
and solve it consistent.
(4 marks)
1 (b) Examine whether the following vectors are linearly dependent or independent, find the relation between them.
x1=[1, -1, 2] x2=[2 3 5] x3=[3 2 1]
(4 marks)
1 (c) If cosec(x+iy)=u+iv, prove that $$i) \ \ \dfrac {u^2}{\sin^2 x} - \dfrac {v^2}{\cos^2 x}= (u^2 + v^2)^2 \\ ii) \ \dfrac {u^2}{\cosh^2 y}+ \dfrac {v^2}{\sinh^2 y} = (u^2 + v^2)^2$$(4 marks) 2 (a) A square lies above real axis in Argand diagram and two of its adjacent vertices are the origin and the point 2+3i. Find the complex numbers representing other two vertices.(4 marks) 2 (b) $$If \ arg(z+1) = \dfrac {\pi}{6} \ and \ arg(z-1) = \dfrac {2 \pi}{3} \ then \ find \ z.$$(4 marks) 2 (c) Find the Eigen values and Eigen vectors of following matrix. $$A= \begin{bmatrix} 1&1 &1 \\ 0&2 &1 \\0 &0 &3 \end{bmatrix}$$(4 marks)

### Answer any one question from Q3 and Q4

3 (a) Test convergence of the series (any one) $$i) \ \ i) \ \ \dfrac {1}{\sqrt{2}}+ \dfrac {1}{\sqrt{9}}+ \dfrac {1}{\sqrt{28}}+ \dfrac {1}{65}+ \cdots \ \cdots \\ ii) \ 1-\dfrac {1}{\sqrt{2}}+ \dfrac {1}{\sqrt{3}}- \dfrac {1}{\sqrt{4}}+ \cdots \cdots$$(4 marks) 3 (b) Prove that $$\log (1+ \sin x) = x - \dfrac {x^2}{2}+ \dfrac {x^3}{6}- \dfrac {x^4}{12}+ \cdots$$(4 marks) 3 (c) Find nth derivative of $$\dfrac {x^2}{(x-1)(x-2)}$$(4 marks)

### Solve any one:

4 (a) i) Evaluate $$\lim_{x\to 0} \ \log_{\tan x} \ \tan 4x$$ ii) Find the value of a and b if $$\lim_{n \to 0} \left [ x^{-3} \sin x + ax^{-2}+b \right ]=0$$(4 marks) 4 (b) Using Taylor's theorem expand 49-69x+42x2+11x3+x4 in powers of (x+2).(4 marks) 4 (c) If y=sin log (x2+2x+1), then prove that
(x+1)2yn+2+(2n+1)(x+1)yn+1+(n2+4)yn=0
(4 marks)

### Solve any two of the following:

5 (a) If u=log (x3+y3-x2y-xy2) then prove that x2 uxx+2xy uxy+ y2uyy=-3.(7 marks)

### Answer any one question from Q5 and Q6

5 (b) If x=u+v+w, y=uv+vw+wu, z=uvw and ϕ is a function of x,y,z then prove that $$u \dfrac {\partial \phi}{\partial u} + v \dfrac {\partial \phi}{\partial v}+ w \dfrac {\partial \phi}{\partial w} = x \dfrac {\partial \phi}{\partial x} + 2y \dfrac {\partial \phi}{\partial y}+ 3z \dfrac {\partial \phi}{\partial z}$$(6 marks) 5 (c) $$If \ ux+vy=0 \ and \dfrac {u}{v}+ \dfrac {v}{y}=1, \ then\ prove \ that \ \left ( \dfrac {\partial u}{\partial x} \right )y - \left (\dfrac {\partial v}{\partial y} \right )x = \dfrac {x^2 + y^2}{y^2 - x^2}$$(6 marks)

### Solve any two of the following:

6 (a) $$If \ u=\cos \left (\dfrac {xy}{x^2+y^2} \right )+ \sqrt{x^2 + y^2}+ \dfrac {xy^2}{x+y}$$ then find the value of xux+yuy at (3, 4).(7 marks) 6 (b) $$if \ x=\dfrac {\cos \theta}{u}, \ y=\dfrac {\sin \theta}{u},$$ Then prove that $$u \dfrac {\partial z}{\partial u}- \dfrac {\partial z}{\partial \theta} = (y-x) \dfrac {\partial z}{\partial x} - (y-x) \dfrac {\partial z}{\partial y}$$(6 marks) 6 (c) If u=(x2-y2) f(xy), then show that uxx-uyy=(x4-y4) f'(xy).(6 marks)

### Answer any one question from Q7 and Q8

7 (a) $$If \ x=r \sin \theta \cos \phi, \ y=r \sin \theta \sin \phi, \ z=r \cos \theta \ find \ \dfrac {\partial (x,y,z)}{\partial (r, \theta, \phi)}$$(4 marks) 7 (b) Examine for functional dependence $$u=\sin^{-1}x+\sin^{-1}y, \ v=x\sqrt{1-y^2}+y \sqrt{1-x^2}$$ if dependent find the relation between them.(4 marks) 7 (c) The area of a triangle ABC is calculated from the formula Δ=1/2 bc sin A. Errors of 1%, 2% and 3% respectively are made in measuring b,c,A. If the correct value of A is 30°, find the percentage error in the calculated value of area of triangle.(5 marks) 8 (a) $$If \ u^2 +xv^2 -uxy=0, \ v^2-xy^2+2uv+u^2=0, \ find \ \dfrac {\partial u}{\partial x}$$ by choosing u, v as dependent and x, y as independent variables.(4 marks) 8 (b) Show that $$u = \dfrac {x+y}{1-xy} , \ v=\tan^{-1} x+\tan^{-1}y$$ are functionally dependent and find the relation between them.(4 marks) 8 (c) Find all the stationary values of the function $$f(x,y)=x^3 +3 xy^3 - 15 x^2 - 15 y^2 + 72 x.$$ Find maximum value of f(x,y) at suitable point.(5 marks)