## 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.

x_{1}=[1, -1, 2] x_{2}=[2 3 5] x_{3}=[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 n^{th} 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+42x^{2}+11x^{3}+x^{4} in powers of (x+2).(4 marks)
**4 (c)** If y=sin log (x^{2}+2x+1), then prove that

(x+1)^{2}y_{n+2}+(2n+1)(x+1)y_{n+1}+(n^{2}+4)y_{n=0}(4 marks)

### Solve any two of the following:

**5 (a)** If u=log (x^{3}+y^{3}-x^{2}y-xy^{2}) then prove that x^{2} u_{xx}+2_{xy} u_{xy}+ y^{2}u_{yy}=-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 xu_{x}+yu_{y} 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=(x^{2}-y^{2}) f(xy), then show that u_{xx}-u_{yy}=(x^{4}-y^{4}) 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)