Engineering Maths 1 - Jun 2014

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.

Choose the correct answer for the following :-

1 (a) (i) $$ If \ y_n=(\sqrt{17})^n e^{4x}\cos \left ( x-n \tan^{-1}\dfrac {1}{4} \right )then \ y= \\ (A) \ e^{4x}\cos x \\ (B) \ e^{2x}\sin 3x \\ (C) \ e^{x}\cos x \\ (D) \ None \ of \ these $$(1 marks) 1 (a) (ii) $$ \sin x=x-\dfrac {x^3}{3!}+\dfrac {x^5}{5!}-\dfrac {x^7}{7!}..... \ is, $$ (A) Taylor's series
(B) Exponential series
(C) Meclaurin's series
(D) None of these(1 marks) 1 (a) (iii) In the Rolle's theorem if F'(c) = then the tangent at the point x=c is,
(A) parallel to y-axis
(B) parallel to x-axis
(C) parallel to both axes
(D) None of these(1 marks) 1 (a) (iv) If y=3^x then y_r = _______
(A) (log x)3ⁿ
(B) 3(log x)ⁿ
(C) 3ⁿ log 3^x
(D) 3^x(log_e3)ⁿ(1 marks) 1 (b) If x=sin t, y=sin pt prove that, (1-x²)y_n+2-(2n+1)xy_n+1+(p²-n²)y_n=0(4 marks) 1 (c) State and prove Cauchy's mean value theorem in [0,16].(6 marks) 1 (d) $$ Expand \ \sqrt{1+\sin 2x} $$ by using Meclaurin's expansion.(6 marks)

Choose the correct answer for the following :-

2 (a) (i) The value of $$ \lim_{\lambda \rightarrow \infty}(1+x)^{1/x} \ is \\ (A) \ e \\ (B) 1\$$C) \ \dfrac {1}{e}\\ (D) \infty $$(1 marks) 2 (a) (ii) The angle between two curves r=ae⁰ and re^?=b is, $$ (A)\ \dfrac {\pi}{2}\$$B)\ \dfrac {\pi}{4} \\ (C) \ 0 \\ (D)\ \pi $$(1 marks) 2 (a) (iii) $$ \dfrac {ds}{dt}=\sqrt{\left ( \dfrac {dx}{dt} \right )^2-\left ( \dfrac {dy}{dt} \right )^2} $$ (A) Polar form
(B) Parametric form
(C) Cartesian form
(D) None of these(1 marks) 2 (a) (iv) $$ \lim_{x\rightarrow \infty}\dfrac {\log x}{\cot x}= _______ $$ (A) 1
(B) 0
(C) 2
(D) -2(1 marks) 2 (b) Find a & b, if $$ \lim_{x\rightarrow 0}\dfrac {x(1+a \cos x)-b\sin x}{x^3}=1 $$ (4 marks) 2 (c) Find the pedal equation of the curve r²=a² cos 2?(6 marks) 2 (d) Find the radius of curvature at any point t of the curve x=a(t+sin t) and y=a(1-cos t).(6 marks)

Choose the correct answer for the following :-

3 (a) (i) $$ If \ u=(x-y)^2+(y-z)^2+(z-x)^2 \ then \ \dfrac {\partial u}{\partial x}+ \dfrac {\partial u}{\partial y}+ \dfrac {\partial u}{\partial z} \ is, $$ (A) 1
(B) 24
(C) 2(x+y+z)
(D) 0(1 marks) 3 (a) (ii) $$ e^x \cos y=\dfrac {e}{\sqrt{2}}\left [ 1+ (x-1)- \left ( y-\dfrac {\pi}{4} \right )+\dfrac {(x-1)^2}{2}-(x-1)\left ( y- \dfrac {\pi}{4} \right )-\dfrac {1}{2}\left ( y-\dfrac {\pi}{4} \right )^2 \right ]+ ....... \\ (A) \ \left (1, \dfrac {\pi}{4}\right ) \\ (B) \ (0,0) \\ (C) \ (1,1) \\ (D) \ \left ( \dfrac {\pi}{4},1 \right ) $$(1 marks) 3 (a) (iii) $$ At \ (a,b) \ \dfrac {\partial^2 u}{\partial x^2}=A, \ \dfrac {\partial ^2u}{\partial y^2}=B \ and \ \dfrac {\partial^2 u}{\partial x \partial y}=H $$ and if AB-H² <0 then such a point is called,
(A) Maximum
(B) Minimum
(C) Saddle
(D) Extremum(1 marks) 3 (a) (iv) $$ If \ J=\dfrac {\partial (u,v)}{\partial (x,y)}, \ J=\dfrac {\partial (x,y)}{\partial (u,v)}, \ then \ JJ' \ is $$ (A) 0 (B) 2 (C) ? (D) 1(1 marks) 3 (b) If $u=f\left ( \dfrac {x}{y}, \dfrac {y}{z}, \dfrac {z}{x} \right ) $then prove that $x\dfrac {\partial u}{\partial x}+y\dfrac {\partial u}{\partial y}+z\dfrac {\partial u}{\partial z}=0$(4 marks) 3 (c) $$ If \ u=\dfrac {xy}{z}, \ v=\dfrac {yz}{x}, \ w=\dfrac {zx}{y} \ then \ show \ that \ J\left (\dfrac {u,v,w}{x,y,z} \right )=4 \ verify \ JJ'=1 $$(6 marks) 3 (d) For the kinetic energy $$ E=\dfrac {1}{2}mv^2 $$ find approximately the change in E as the mass m changes from 49 to 49.5 and the velocity 'y' change from 1600 to 1590.(6 marks)

Choose the correct answer for the following :-

4 (a) (i) The value of ? × ? ? is
(A) 0
$$ (B) \ \underset{R}{\rightarrow} $$
(C) ?
3(1 marks) 4 (a) (ii) Any motion in which the curl of the velocity is zero, then the vector $$ \underset{R}{\rightarrow} $$ is said to be,
(A) Constant
(B) Solenoidal
(C) Vector
(D) Irrotational(1 marks) 4 (a) (iii) In orthogonal curvilinear co-ordinates the jacobian $$ J=\dfrac {\partial (x,y,z)}{\partial (u,v,w)} \ is, \$$A) \ \dfrac {h_1}{h_2h_3} \\ (B) \ \dfrac {1}{h_1h_2h_3} \\ (C) \ h_1h_2h_3 \\ (D) \ \dfrac {h_3}{h_1h_2} $$(1 marks) 4 (a) (iv) A gradient of the scalar point function ?, ?? is,
(A) Scalar function
(B) Vector function
(C) ?
(D) zero(1 marks) 4 (b) Find the value of the constant a such that the vector field, $$ \underset{F}{\rightarrow}=(axy-z^3)i+(a-2)x^2j+(1-a)xz^2k $$ is irrotational and hence find a scalar function ? such that $$ \underset{F}{\rightarrow}=abla\phi $$(4 marks) 4 (c) Prove that curl $$ \left( curl \ \underset{A}{\rightarrow}\right )=abla\left (abla \cdot\underset{A}{\rightarrow} \right)-abla^2 \underset{A}{\rightarrow}. $$(6 marks) 4 (d) Express ?² ? in orthogonal curvilinear co-ordinates.(6 marks)

Choose the correct answer for the following :-

5 (a) (i) $$ The\ value \ of \ \int^{\pi/R}_0\cos^3 (4x)dx \ is, \\ (A)\ \dfrac {1}{3} \\ (B) \ \dfrac {1}{6} \\ (C) \ \dfrac {\pi}{3}\\ (D) \ \dfrac {1}{2} $$(1 marks) 5 (a) (ii) If the equation of the curve remains unchange after changing ? to -? the curve r=f(?) is symmetrical about.
(A) A line perpendicular to initial line through pole
(B) Radially symmetric about the point pole
(C) Symmetry does not exist.
(D) Initial line(1 marks) 5 (a) (iii) The volume of the curve r=a(1+cos ?) about the initial line is,
$$ (A) \ \dfrac {4\pi a^3}{3}\\ (B) \ \dfrac {2\pi a^3}{3}\\ (C) \ \dfrac {8\pi a^3}{3} \\ (D) \ \dfrac {\pi a^3}{3} $$(1 marks) 5 (a) (iv) The asymptote for the curve x³+y³=3axy is equal to,
(A) x+y+a=0
(B) x-y-a=0
(C) No Assymptote
(D) x+y-a=0(1 marks) 5 (b) $$ Evaluate \ \int^\pi_0 \dfrac {\log (1+\sin \alpha \cos x)}{\cos x}dx $$(4 marks) 5 (c) $$ Evaluate \ \int^{2a}_0 x^2 \sqrt{2ax-x^2}dx $$(6 marks) 5 (d) Find the area of surface of revolution about x-axis of the astroid x^2/3 + y^2/3 =a^2/3. (6 marks)

Choose the correct answer for the following :-

6 (a) (i) In the homogeneous differential equation $$ \dfrac {dy}{dx}=\dfrac {f_1(xy)}{f_2(xy)} $$ the degree of the function f₁(xy) and f₂(xy) are,
(A) Different
(B) Relatively prime
(C) Same
(D) None of these(1 marks) 6 (a) (ii) The integrating factor of the differential equation, $$ \dfrac {dy}{dx}+ \cot xy=\cos x \ is , $$
(A) cos x
(B) sin x
(C) -sin x
(D) cot x(1 marks) 6 (a) (iii) $$ Replacing \ \dfrac {dy}{dx}\ by \ \left ( -\dfrac {dy}{dx} \right ) \ in \ the \ differential \ equation \ f\left ( x,y,\dfrac {dy}{dx} \right )=0 $$ we get the differential equation of,
(A) Polar trajectory
(B) Orthogonal trajectory
(C) Parametric trajectory
(D) Parallel trajectory.(1 marks) 6 (a) (iv) Two families of curves are said to be orhogonal if every member of either family ctuts each member of the other family at, $$ (A) \ Zero \ angle \$$B) \ Right \ angle \\ (C) \ \dfrac {\pi}{6} \\ (D) \ \dfrac {2\pi}{3} $$(1 marks) 6 (b) $$ Solve \ (1+e^{x/y})dx+e^{x/y}\left ( 1-\dfrac {x}{y} \right)dy=0 $$(4 marks) 6 (c) $$ Solve \ \dfrac {dy}{dx}+x\sin 2y=x^3\cos^2y. $$(6 marks) 6 (d) Find the orthogonal trajectories of r²=a² cos² ?(6 marks)

Choose the correct answer for the following :-

7 (a) (i) $$ A=\begin {bmatrix}7&0&0\\0&7&0\\0&0&7\end{bmatrix} \ is \ called. $$ (A) Scalar matrix
(B) Diagonal matrix
(C) Identity matrix
(D) None of these(1 marks) 7 (a) (ii) If r=n and x=y=z=0. The equations have only ________ solution.
(A) Non trivial
(B) Trivial
(C) Unique
(D) Infinite(1 marks) 7 (a) (iii) In Gauss Jordan method, the coefficient matrix can be reduced to,
(A) Echelon form
(B) Unit matrix
(C) Triangular form
(D) Diagonal matrix(1 marks) 7 (a) (iv) The inverse square matrix A is given by, $$ (A) \ |A| \\ (B) \ \dfrac {adjA}{|A|}\\ (C) \ adjA \\ (D) \ \dfrac {|A|}{adjA} $$(1 marks) 7 (b) Find the Rank of the matrix $$ \begin {bmatrix}1&2&3&2\\2&3&5&1\\1&3&4&5 \end{bmatrix} $$(5 marks) 7 (c) Investigate the values of ? and ? such that the system of equations, x+y+z=6, x+2y+3z=10, x+2y-?z=? may be (i) Unique solution (ii) Infinite solution (iii) No solution(6 marks) 7 (d) Using Gauss elimination method solve,
2x₁ - x₂ + 3x₃=1, -3x₁ + 4x₂ - 5x₃=0, x₁ + 3x₂ - 6x₃=0(5 marks)

Choose the correct answer for the following :-

8 (a) (i) A square matrix A of order 3 has 3 linearly independent eigen vectors then a matrix P can be found such that P^-1 AP is a
(A) digonal matrix
(B) symmetric matrix
(C) unit matrix
(D) singular matrix(1 marks) 8 (a) (ii) The eigen values of matrix $$ \begin {bmatrix}2&\sqrt{2}\\ \sqrt{2}&2 \end{bmatrix} \ are, \\ (A)\ 2\pm\sqrt{6} \\ (B)\ 2\pm \sqrt{2}\\ (C) \ 1-\sqrt{6} \\ (D) \ None \ of \ these $$(1 marks) 8 (a) (iii) Solving the equation x+2y+3z=0, 3x+4y+4z=0, 7x+10y+12z=0, x,y and z values are,
(A) x=y=z=0
(B) x=y=z=1
(C) x?y?z?=1
(D) None of these(1 marks) 8 (a) (iv) The index and significance of the quadratic form, $$ x^2_1+2x^2_2-3x^2_3 $$ are respectively ______ and ______
(A) Index=1, Signature=1
(B) Index=1, Signature=2
(C) Index=2, Signature=1
(D) None of these(1 marks) 8 (b) Find all the eigen values and the corresponding eigen vector of the matrix, $$ A=\begin {bmatrix}8&-6&2\\-6&7&-4\\2&-4&3 \end{bmatrix} $$(4 marks) 8 (c) Reduce the matrix $$ A=\begin {bmatrix}11&-4&-7\\7&-2&-5\\10&-4&-6 \end{bmatrix} $$ into a diagonal matrix(6 marks) 8 (d) Reduce the quadratic form 3x²-5y²+3z²-2yz+2zx-2xy to the canonical form.(6 marks)