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Engineering Maths 1 : Question Paper Jun 2014 - First Year Engineering (C Cycle) (Semester 1) | Visveswaraya Technological University (VTU)
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## 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=3x then yr = _______
(A) (log x)3n
(B) 3(log x)n
(C) 3n log 3x
(D) 3x(loge3)n
(1 marks)
1 (b) If x=sin t, y=sin pt prove that, (1-x2)yn+2-(2n+1)xyn+1+(p2-n2)yn=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
(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)

### 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)