## Engineering Maths 1 - Jun 2013

### First Year Engineering (P Cycle) (Semester 1)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.

### Choose your answer for the following :-

**1 (a) (i)** If y=3^{5x} then y_{n} is

(A) (3 log 5)^{n} e^{5x}

(B) (5 log 3)^{n} e^{5x}

(C) (5 log 3)^{-n} e^{5x}

(D) (5 log 3)^{n} e-5x(1 marks)
**1 (a) (ii)** if y=cos^{2} x then y_{n} is

(A) 2^{n+1} cos(n?/2+2x)

(B) 2^{n-1} cos(n?/2+2x)

(C) 2^{n-1} cos(n?/2-2x)

(D) 2^{2+1} cos(n?/2-2x)(1 marks)
**1 (a) (iii)** The Largrange's mean value theorem for the function f(x)=e^{x} in the interval [0, 1] is

(A) C=0.5413

(B) C=2.3

(C) 0.3

(D) none of these(1 marks)
**1 (a) (iv)** Expression of log (1+e^{x}) in powers of x is _____

$$ (A) \ \log 2-\dfrac {x}{2}+\dfrac {x^2}{8}+\dfrac {x^4}{192}+.... \$$B) \ \log 2+\dfrac {x}{2}+\dfrac {x^2}{8}-\dfrac {x^4}{192}+....\$$C) \ \log 2+\dfrac {x}{2}+\dfrac {x^2}{8}+\dfrac {x^4}{192}+....\$$D) \ \log 2-\dfrac {x}{2}-\dfrac {x^2}{8}-\dfrac {x^4}{192}+....$$(1 marks)
**1 (b)** if y^{1/m}+y^{-1/m}=2x prove that (x^{2}-1) y_{n+2} + (2n+1) xy_{n+1} +(n^{2}-m^{2}) y_{n}=0(6 marks)
**1 (c) ** Verify the Rolle's theorem for the functions: f(x)=e^{x} (sin x - cos x) in (?/4.5?/4).(6 marks)
**1 (d)** By using Maclaarin's theorem expand log sec x up to the term containing x^{6}(4 marks)

### Choose your answer for the following :-

**2 (a) (i)** The indeterminate form of $$ \lim_{x\rightarrow 0}\dfrac {a^x-b^x}{x} \ is \$$A) \ \log\left ( \dfrac {b}{a} \right ) \$$B) \ \log \left (\dfrac {a}{b} \right ) \$$C) \ 1\$$D) \ -1 $$(1 marks)
**2 (a) (ii)** The angle between radius vector and the tangent for the curves r=a(1-cos ?) is

(A) ?/2

(B) -?/2

(C) ?/2+?

(D) ?/2-?(1 marks)
**2 (a) (iii)** The polar form of a curve is _____

(A) r=f(?)

(B) ?=f(y)

(C) r=f(x)

(D) none of these(1 marks)
**2 (a) (iv)** The rate at which the curve is bending called _____

(A) Radius of curvature

(B) Curvature

(C) Circle of curvature

(D) Evaluate(1 marks)
**2 (b)** $$ Evaluate \ \lim_{x \rightarrow 0} \left ( \dfrac {\sin x}{x} \right )^{1/x^2} $$(6 marks)
**2 (c) ** Find the angles of intersection of the following pairs of curves, r=a?/(1+?); r=a/(1+?^{2})(6 marks)
**2 (d)** Find the radius curvature at (3a/2, 3a/2) on x^{3}+y^{3}=3axy(4 marks)

### Choose your answer for the following :-

**3 (a) (i)** If u=x^{2}+y^{2} then $$ \dfrac {(\partial^2u)}{(\partial x \partial y)}$$ is equal to

(A) 2

(B) 0

(C) 2x

(D) 2y(1 marks)
**3 (a) (ii)** If z=f(x,y) where x=u-v and y=uv then (u+v) $$ \left (\dfrac {\partial z}{\partial x} \right ) \ is $$

$$ (A) \ u\left (\dfrac {\partial z}{\partial u} \right )-v\left ( \dfrac {\partial z}{\partial v} \right )\$$B) \ u\left (\dfrac {\partial z}{\partial u} \right )+v \left (\ \dfrac {\partial z}{\partial v} \right )\$$C) \ \dfrac {\partial z}{\partial u}+\dfrac {\partial z}{\partial v} \$$D) \ \dfrac {\partial z}{\partial u}-\dfrac {\partial z}{\partial v} $$(1 marks)
**3 (a) (iii)** If x=r cos ?, y=r sin ? then $$ \dfrac {[\partial (r,\theta)]}{[\partial (x,y)]} \ is $$

(A) r

(B) 1/r

(C) 1

(D) -1(1 marks)
**3 (a) (iv)** In error and approximations $$ \dfrac {\partial x}{x}, \dfrac {\partial y}{y}, \dfrac {\partial f}{f} $$ are called

(A) relative error

(B) percentage error

(C) error in x,y and f

(D) none of these(1 marks)
**3 (b)** If x^{x} y^{y} z^{z}=c, show that \dfrac {\partial^2z}{\partial x \partial y}=-[x \log ex]^{-1}, when x=y=z(6 marks)
**3 (c) ** Obtain the Jacobian of $$ \dfrac {\partial (x.y.z)}{\partial (r.\theta . \phi)} $$ for change of coordinate from three dimensional Cartesian coordinates to spherical polar coordinates.(6 marks)
**3 (d)** In estimating the cost of a pile of bricks measured as 2m × 15m × 1.2m, the tape is stretched +1% beyond the standard length. If the count is 450 bricks to I cu.cm and bricks cost of 530 per 1000, find the approximate error in the cost(4 marks)

### Choose your answer for the following :-

**4 (a) (i)** If R=xi+yj+zk then div R

(A) 0

(B) 3

(C) -3

(D) 2(1 marks)
**4 (a) (ii)** If F=3x^{2}i-xyj+(a-3)xzk is solenoidal, then a is equal to

(A) 0

(B) -2

(C) 2

(D) 3(1 marks)
**4 (a) (iii)** If F=(x+y+1)i+j-(x+y)k then F. Curl F is _____

(A) 0

(B) x+y

(C) x+y+z

(D) x-y(1 marks)
**4 (a) (iv)** The scale factors for cylindrical coordinates system (? ? z) are given by

(A) (?, 1, 1)

(B) (1, ?, 1)

(C) (1, 1, ?)

(D) None of these(1 marks)
**4 (b)** Prove that curl A=g rad(div A)- ?^{2} A.(6 marks)
**4 (c) ** Find the constant a, b, c such that the vector F=(x+y+az)i+(bx+2y-z)j+(x+cy+2z)k is irrotational(6 marks)
**4 (d)** Derive an expression for ? ? A in orthogonal curvilinear coordinates. Deduce ? ? A is rectangular coordinates.(4 marks)

### Choose your answer for the following :-

**5 (a) (i) ** The value of $$ \int^{\infty}_0 e^{\alpha x}dx $$ is _____

(A) 1/e

(B) -1/e

(C) 1/?

(D) -1/?(1 marks)
**5 (a) (ii)** The value of the integral $$ \int^{\pi/2}_{0}\sin^{7} xdx \ is $$

(A) 35/16

(B) 16/35

(C) -16/35

(D) 18/35(1 marks)
**5 (a) (iii)** The volume generated by revolving the cardioid r=a(1+ cos ?) about the intial line is

$$ (A) \ \dfrac {(3\pi a^2)}{8} \$$B) \ \dfrac {(3\pi a^3)}{8} \$$C) \ \dfrac {(2\pi a^2)}{9} \$$D) \ None $$(1 marks)
**5 (a) (iv)** The area of the loop of the curve r=a sin 3? is _____

$$ (A) \ \dfrac {a^2}{12} \$$B) \ \dfrac {\pi}{12} \$$C) \ \dfrac {\pi a^2}{12} \$$D) \ none $$(1 marks)
**5 (b)** By applying differential under the integral sign evaluate $$ \int^{\pi/2}_0 \dfrac {\log (1+y \sin^2 x)}{\sin^2 x}dx $$(6 marks)
**5 (c) ** Evaluate $$ \int^{\pi/2}_0 \sin^n x \ dx $$ where n is any integer.(6 marks)
**5 (d)** Find the length of arch of the cycloid x=a (? - sin?); y=a (1- cos?); 0

(4 marks)

### Choose your answer for the following :-

**6 (a) (i)** The general solution of the differential equation (dy/dx)=(y/x)+tan(y/x) is

(A) sin (y/x)=c

(B) sin (y/x)=cx

(C) cos (y/x)=cx

(D) cos (y/x)=c (1 marks)
**6 (a) (ii)** An integrating factor for ydx-xdy=0 is

(A) x/y

(B) y/x

(C) 1(x^{2}y^{2})

(D) 1/(x^{2}+y^{2})(1 marks)
**6 (a) (iii)** The differential equation satisfying the relation x=A cos (mt-?) is

(A) (dx/dt)=1-x^{2}

(B) (d^{2}x/dt^{2})=-?^{2}x

(C) (d^{2}x/dt^{2})=-m^{2}x

(D) (dx/dt)=-m^{2}x(1 marks)
**6 (a) (iv)** The orthogonal trajectories of the system given by r=a? is

(A) r^{2}=ke^{?}

(B) r=ke^{?}

(C) r^{2} e^{-?2}= k

(D) r^{2}= k e^{-?2}(1 marks)
**6 (b)** Solve (x cos (y/x)+y sin (y/x)) y- (y sin (y/x) -x cos (y/x)) x(dy/dx)=0(6 marks)
**6 (c) ** Solve (1+y^{2})+(x-e^{tan-1y} )dy/dx=0(6 marks)
**6 (d) ** Prove that the system parabola y^{2}=4a(x+a) is self orthognal.(4 marks)

### Choose your answer for the following :-

**7 (a) (i) ** Find the rank of $$ \begin{bmatrix}3 &-1 &2 \\ -6&2 &4 \\ -3&1 &2 \end{bmatrix} $$

(A) 3

(B) 2

(C) 4

(D) 1(1 marks)
**7 (a) (ii)** The exact solution of the system of equation 10x+y+z=12, x+10y+z=12, x+y+10z=12 by inspection is equal to

(A) (-1, 1, 1)

(B) (1, 1, 1)

(C) (-1, -1, -1)

(D) None (1 marks)
**7 (a) (iii)** If the given system of linear equations in 'n' variables is consistent then the number of linearly independent-solution is given by

(A) n

(B) n-1

(C) r-n

(D) n-r(1 marks)
**7 (a) (iv)** The trivial solution for the given system of equations

qx-y+4z=0, 4x-2y+3z=0, 5x+y-6z=0 is

(A) (1, 2, 0)

(B) (0 4 1)

(C) (0 0 0)

(D) (1 -5 0)(1 marks)
**7 (b)** Using elementary transformation reduce each of following matrices to the normal form $$ \begin{bmatrix}1 &1 &1 &6 \\ 1&-1 &2 &5 \\ 3&1 &1 &8 \\ 2&-2 &3 &7 \end{bmatrix} $$(6 marks)
**7 (c) ** Test for consistency and solve the system, 2x+y+z=10, 3x+2y+3z=18, x+4y+9z=16(6 marks)
**7 (d)** Apply Gauss-Jordan method to solve the system of equations, 2x+5y+7z=52, 2x+y-z=0, x+y+z=9(4 marks)

### Choose your answer for the following :-

**8 (a) (i)** A square matrix A is called orthogonal if,

(A) A=A^{2}

(B) A=A^{-1}

(C) AA^{-1}=1

(D) None (1 marks)
**8 (a) (ii)** The eigen values of the matrix $$ \begin{bmatrix}6 &-2 &2 \\ -2&3 &-1 \\ 2&-1 &3 \end{bmatrix} \ are $$

(A) 2,3,8

(B) 2,3,9

(C) 2,2,8

(D) None (1 marks)
**8 (a) (iii)** The eigen vector X of the matrix A corresponding to eigen value ? and satisfy the equation.

(A) AX=?X

(B) ?(A-X)=0

(C) XA-A? =0

(D) |A-?|X=0(1 marks)
**8 (a) (iv)** Two square matrices A and B are similar if,

(A) A=B

(B) B=P^{-1}AP

(C) A'=B'

(D) A^{-1}=B^{-1}(1 marks)
**8 (b)** Show that the transformation, y_{1}=2x_{1}-2x_{2}-x_{3}, y_{2}= -4x_{1}+5x_{2}+3x_{3}, y_{3}=x_{1}-x_{2}-x_{3}, is regular and find the inverse transformations.(6 marks)
**8 (c) ** Diagonalize the matrix, $$ \begin{bmatrix}8 &-6 &2 \\ -6&7 &-4 \\ 2&-4 &3 \end{bmatrix} $$(6 marks)
**8 (d)** Reduce the quadratic form, $$ x^2_1 +2x^2_2 -7x^2_3-4x_1x_2+8x_2x_3 $$ into sum of squares.(4 marks)