VTU Engineering Maths 1 - June 2014 Exam Question Paper

VTU First Year Engineering (P Cycle) (Semester 1)
Engineering Maths 1
June 2014

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

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 M

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 M

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 M

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 M

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 M

1 (c) State and prove Cauchy's mean value theorem in [0,16].

6 M

1 (d)

$Expand \ \sqrt{1+\sin 2x}$ by using Meclaurin's expansion.

6 M

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 M

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 M

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 M

2 (a) (iv)

$\lim_{x\rightarrow \infty}\dfrac {\log x}{\cot x}= _______$ (A) 1
(B) 0
(C) 2
(D) -2

1 M

2 (b) Find a & b, if

$\lim_{x\rightarrow 0}\dfrac {x(1+a \cos x)-b\sin x}{x^3}=1$

4 M

2 (c) Find the pedal equation of the curve r²=a² cos 2?

6 M

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 M

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 M

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 M

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 M

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 M

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 M

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 M

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 M

Choose the correct answer for the following :-

4 (a) (i) The value of ? × ? ? is
(A) 0

$(B) \ \underset{R}{\rightarrow}$
(C) ?
3

1 M

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 M

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 M

4 (a) (iv) A gradient of the scalar point function ?, ?? is,
(A) Scalar function
(B) Vector function
(C) ?
(D) zero

1 M

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 M

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 M

4 (d) Express ?² ? in orthogonal curvilinear co-ordinates.

6 M

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 M

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 M

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 M

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 M

5 (b)

$Evaluate \ \int^\pi_0 \dfrac {\log (1+\sin \alpha \cos x)}{\cos x}dx$

4 M

5 (c)

$Evaluate \ \int^{2a}_0 x^2 \sqrt{2ax-x^2}dx$

6 M

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 M

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 M

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 M

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 M

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 M

6 (b)

$Solve \ (1+e^{x/y})dx+e^{x/y}\left ( 1-\dfrac {x}{y} \right)dy=0$

4 M

6 (c)

$Solve \ \dfrac {dy}{dx}+x\sin 2y=x^3\cos^2y.$

6 M

6 (d) Find the orthogonal trajectories of r²=a² cos² ?

6 M

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 M

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 M

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 M

7 (a) (iv) The inverse square matrix A is given by,

$(A) \ |A| \\ (B) \ \dfrac {adjA}{|A|}\\ (C) \ adjA \\ (D) \ \dfrac {|A|}{adjA}$

1 M

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 M

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 M

7 (d) Using Gauss elimination method solve,
2x₁ - x₂ + 3x₃=1, -3x₁ + 4x₂ - 5x₃=0, x₁ + 3x₂ - 6x₃=0

5 M

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 M

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 M

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 M

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 M

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 M

8 (c) Reduce the matrix

$A=\begin {bmatrix}11&-4&-7\\7&-2&-5\\10&-4&-6 \end{bmatrix}$ into a diagonal matrix

6 M

8 (d) Reduce the quadratic form 3x²-5y²+3z²-2yz+2zx-2xy to the canonical form.

6 M

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