Engineering Maths 1 - Jun 2013

First Year Engineering (C 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.

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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=cos² x then y_n is
(A) 2ⁿ⁺¹ cos(n?/2+2x)
(B) 2^n-1 cos(n?/2+2x)
(C) 2^n-1 cos(n?/2-2x)
(D) 2²⁺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 (x²-1) y_n+2 + (2n+1) xy_n+1 +(n²-m²) 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 x⁶(4 marks)

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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 x³+y³=3axy(4 marks)

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Choose your answer for the following :-

3 (a) (i) If u=x²+y² 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)

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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=3x²i-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)

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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)

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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(x²y²)
(D) 1/(x²+y²)(1 marks) 6 (a) (iii) The differential equation satisfying the relation x=A cos (mt-?) is
(A) (dx/dt)=1-x²
(B) (d²x/dt²)=-?²x
(C) (d²x/dt²)=-m²x
(D) (dx/dt)=-m²x(1 marks) 6 (a) (iv) The orthogonal trajectories of the system given by r=a? is
(A) r²=ke^?
(B) r=ke^?
(C) r² e^-?2= k
(D) r²= 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+y²)+(x-e^tan-1y )dy/dx=0(6 marks) 6 (d) Prove that the system parabola y²=4a(x+a) is self orthognal.(4 marks)

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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)

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Choose your answer for the following :-

8 (a) (i) A square matrix A is called orthogonal if,
(A) A=A²
(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^-1AP
(C) A'=B'
(D) A^-1=B^-1(1 marks) 8 (b) Show that the transformation, y₁=2x₁-2x₂-x₃, y₂= -4x₁+5x₂+3x₃, y₃=x₁-x₂-x₃, 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)