VTU Engineering Maths 1 - January 2014 Exam Question Paper

VTU First Year Engineering (P Cycle) (Semester 1)
Engineering Maths 1
January 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)

$\if \ y=\dfrac {x+2}{x+}, \ then ] y_n \ is \\(A) \ \dfrac {(-1)^n(n+1)!}{(x-1)^{n-1}} \\(B) \ \dfrac {(-1)^n n!}{(x+1)^{n+1}}\\(C) \ \dfrac {(-1)^n n!}{(x+1)^n} \\(D) \ \dfrac {(-1)^{n-1}n!}{(x+1)^{n+1}}$

1 M

1 (a) (ii) If y=(ax+b)^m with m=n, then y_n is
(A) n! aⁿ
(B) 0
(C) n! bⁿ
(C) n!

1 M

1 (a) (iii) The geometrical intepretation of Lagrange's mean value theorem is

$(A)\ f'(C)=\dfrac {f(b)-f(a)}{b-a} \\(B)\ f'(C)=\dfrac {f(b)+f(a)}{b-a} \\(C)\ \dfrac {f'(C)}{g'(C)}= \dfrac {f(b)-f(a)}{g(b)-g(a)} \\(D)\ none \ of \ these$

1 M

1 (a) (iv) The Maclaurin's series expansion of e^-x is

$(A)\ 1+x+\dfrac {x^2}{2!}+\dfrac {x^3}{3!}-.....\\(B)\ 1-x+\dfrac {x^2}{2!}-\dfrac {x^3}{3!}+.....\\(C)\ x-\dfrac {x^2}{2!}+\dfrac {x^3}{3!}-.....\\(D)\ x+\dfrac {x^2}{2!}+\dfrac {x^3}{3!}+.....\\$

1 M

1 (b) If y= sin log (x²+2x+1), prove that (x+1)² y_n+2+(2n+1)(x+1)y_n+(n²+4)y_n=0

4 M

1 (c) If x is positive, show that

$x>\log (1+x)>x-\dfrac {1}{2}X^2$

6 M

1 (d) Using Maclourin's series, expand log (1+e^x) upto the terms containing x⁴.

6 M

Choose the correct answer for the following :-

2 (a) (i)

$\lim_{x \rightarrow \frac {\pi}{4}}\left (\dfrac {1-\tan x}{^\pi_4 -x } \right ) \ is \ equal \ to$
(A) 2
(B) -2
(C) 1
(D) -1

1 M

2 (a) (ii) If ? be the angle between the tangent and radius vector at any point on the curve r=f(?), then sin ? is equal to

$(A)\ dr/ds \\(B)\ r\dfrac{d\theta}{ds}\\(C)\ r\dfrac{d\theta}{dr}\\(D)\ ds/dr$

1 M

2 (a) (iii) The rate at which the curve is bending called
(A) radius of curvature
(B) curvature
(C) circle of curvature
(D) evolute

1 M

2 (a) (iv) The radius of curvature for polar curve r=f(?) is given by

$(A)\ \dfrac {(r^2+r^2_1)^{3/2}}{r^2+r^2_1-rr_2}\\(B)\ \dfrac {(r^2+r^2_1)^{3/2}}{r^2_1+2r^2-rr^2}\\(C)\ \dfrac {(r^2+r^2_1)^{3/2}}{r^2+2r^2_1-rr_2}\\(D)\ \dfrac {(r^2-r^2_1)^{3/2}}{r^2+2r^2_1-rr^2}$

1 M

2 (b) Find the Pedal equation of the curve r^m=a^m cos m?

4 M

2 (c) Find the radius of curvature for the curve

$y^2=\dfrac {a^2(a-x)}{x}$ where the curve meets the x-axis.

6 M

2 (d)

$Evaluate \ \lim_{x\rightarrow \infty}\left ( \dfrac {ax+1}{ax-1} \right )^x$

6 M

Choose the correct answer for the following :-

3 (a) (i)

$If\ u=\log (x^2+y^2+z^2),\ then \ \dfrac {\partial u}{\partial z} \ is \\(A)\ \dfrac {2x}{x^2+y^2+z^3} \\(B)\ \dfrac {2y}{x^2+y^2+z^2} \\(C)\ \dfrac {2z}{x^2+y^2+z^2} \\(D)\ \dfrac {2z}{x^2+y^2-z^2}$

1 M

3 (a) (ii) If u=f(x, y) and y is a function x, then

$(A)\ \dfrac {du}{dx}= \dfrac {\partial u}{\partial x}+ \dfrac {\partial u}{\partial y} \dfrac {dy}{dx}\\(B)\ \dfrac {\partial u}{\partial x}= \dfrac {du}{dx}+ \dfrac {\partial u}{\partial y}\dfrac {dy}{dx}\\(C)\ \dfrac {du}{dx}= \dfrac {\partial u}{\partial x}+ \dfrac {\partial u}{\partial y}\dfrac {\partial y}{\partial x}\\(D)\ \dfrac {\partial u}{\partial x}=\dfrac {du}{dx}+\dfrac {\partial u}{\partial y}\dfrac {\partial y }{\partial x}$

1 M

3 (a) (iii)

$if \ r=\dfrac {\partial^2 f}{\partial x^2},\ S=\dfrac {\partial^2f}{\partial x \partial y} \ and \ t=\dfrac {\partial^2 f}{\partial y^2}$ then the condition for the saddle point is
(A) rt-s²<0
(B) rt-s²=0
(C) rt-s²>0
(D) rt-s² ? 0

1 M

3 (a) (iv) If u=x+y+z, v=y+z, z=z, then

$J\left ( \dfrac {u,v,z}{x,y,z} \right )$ is equal to
(A) 1
(B) -1
(C) 0
(D) none of these

1 M

3 (b) The focal length of a mirror is given by the formula

$\dfrac {1}{v}-\dfrac {1}{u}=\dfrac {2}{f}$ if equal errors, 'e' are made in the determination of u and v. show that the resulting error in f is

$e \left ( \dfrac {1}{u}+\dfrac {1}{v} \right )$

4 M

3 (c) If u=f(2x-3y, 3y-4z, 4z-2x), prove that

$\dfrac {1}{2}\dfrac {\partial u}{\partial x}+\dfrac {1}{3}\dfrac {\partial u}{\partial y}+ \dfrac {1}{4}\dfrac {\partial u}{\partial z}=0$

6 M

3 (d) If x=u(1-v), y=uv, prove that JJ'=1

6 M

Choose the correct answer for the following :-

4 (a) (i) Directional derivative is maximum along
(A) tangent to the surface
(B) normal to the surface
(C) any unit vector
(D) coordinate axes

1 M

4 (a) (ii) If r=|x_i+y_j+2_k|, then ? rⁿ is
(A) nr^n-1
(B) r^n-1
(C) ?.? rⁿ
(D) none of these

1 M

4 (a) (iii) If f=3x²-3y²+4z², then curl (grad f) is
(A) 4x-6y+8z
(B) 4x_i-6y_j+8z k

$\vec{0}$
(D) 3

1 M

4 (a) (iv) If the base vectors e₁ and e₂ are orthogonal then |e₁ × e₂| is
(A) 0
(B) -1
(C) +1
(D) none of these

1 M

4 (b)

$If \ \vec{F}=(x+y+1)i+j-(x+y)k, \ show \ that \ \vec{F}\cdot curl \ \vec{F}=0$

4 M

4 (c) Find constant 'a' and 'b' such that

$\vec{F}=(axy+z^3)i+(3x^2-z)j+(bxz^2-y)k$ is irrotational. Also find a scalar function ? such that

$\vec{F}=abla \phi$

6 M

4 (d) Prove that a spherical coordinate system is orthogonal.

6 M

Choose the correct answer for the following :-

5 (a) (i)

$\int^\pi_0 \sin^7x \ dx \ is \ equal \ to \\(A)\ zero \\(B)\ \dfrac {32\pi}{35}\\(C)\ \dfrac {32}{35}\\(D)\ =\dfrac {35\pi}{32}$

1 M

5 (a) (ii) The asymptote of (2-x)y²=x³ is
(A) x=2
(B) y-axis
(C) x-axis
(D) none of these

1 M

5 (a) (iii) The area of the cordioid r=a(1- cos ?) is

$(A)\ \dfrac {3\pi a^2}{2}\\(B)\ \dfrac {3\pi}{2}\\(C)\ \dfrac {a^2}{2}\\(D)\ \dfrac {3a^2}{2}$

1 M

5 (a) (iv) The entire length of the asteroid x^2/3+y^2/3=a^2/3 is
(A) 6a
(B) 3a
(C) 2a
(D) a

1 M

5 (b)

$Evaluate \ \int^\pi_0 \log (1+ a \cos x)dx$ by differentiating under the integral sign.

4 M

5 (c) Evaluate

$\int^{2a}_0X^2\sqrt{2ax-x^2 }dx$ using reduction formula.

6 M

5 (d) Find the volume of generated by the revolution of the curve r=a(1+cos ?) about the initial line.

6 M

Choose the correct 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 M

6 (a) (ii) The family of straight lines passing through the origin is represented by the differential equation :
(A) ydx+xdy=0
(B) xdy-ydx=0
(C) xdx+ydy=0
(D) ydy-xdx=0

1 M

6 (a) (iii) The homogeneous differential equation Mdx+Ndy=0 can be reduced to a differential equation, in which the variables are seperated by the substitution
(A) y=vx
(B) x+y=v
(C) xy=v
(D) x-y=v

1 M

6 (a) (iv) The equation y-2x=c represents the orthogonal trajectories family
(A) y=ae^-2x
(B) x²+2y²=a
(C) xy=a
(D) x+2y=a

1 M

6 (b)

$Solve \ (x+1)\dfrac {dy}{dx}-y=e^{3x}(x+1)^2$

4 M

6 (c) Solve (1+xy) ydx+(1-xy)xdy=0

6 M

6 (d) Find the orthogonal trajectory of the cordioids r=a(1- cos ?)

6 M

Choose the correct answer for the following :-

7 (a) (i) If every minor of order 'r' of a matrix A is zero, then rank of A is
(A) greater than r
(B) equal r
(C) less than or equal to r
(D) less

1 M

7 (a) (ii) The trivial solution for the given system of equations x+2y+3z=0, 3x+4y+4z=0, 7x+10y+12z=0 is
(A) (1, 1, 1)
(B) (1, 0, 0)
(C) (0, 1, 0)
(D) (0, 0, 0)

1 M

7 (a) (iii) Matrix has a value. This statement
(A) is always true
(B) depends upon the matrices
(C) is false
(D) none of these

1 M

7 (a) (iv) If A is singular and ?(A)=?(A:B) then the system has
(A) unique solution
(B) infinitely many solution
(C) trivial solution
(D) no solution.

1 M

7 (b) Using elementary transformations, find the rank of the matrix

$\begin {bmatrix}1&2&0&-1 \\ 3&4&1&2 \\ -2&3&2&5 \end{bmatrix}$

4 M

7 (c) Show that the system x+y+z=4; 2x+y-z=1; x-y+2z=2 is consistent, solve the system.

6 M

7 (d) Apply Gauss-Jordan method to solve the system of equation: 2x+5y+7z=52; 2x+y-z=0; x+y+z=9

6 M

Choose the correct answer for the following :-

8 (a) (i) A square matrix A is called orthogonal if,
(A) A=A^L
(B) A^T=A^-1
(C) AA^-1=I
(D) none of these

1 M

8 (a) (ii) The eigen values of the matrix

$\begin{bmatrix}2 &\sqrt{2} \\ \sqrt{2}&2 \end{bmatrix} \ are \\(A)\ 1 \pm \sqrt{6}\\(B)\ 1 \pm \sqrt{5}\\(C)\ \sqrt{5}\\(D)\ 1$

1 M

8 (a) (iii) The index and signature of the quadratic form

$x^2_1+2X^2_2-3X^2_3$ are respectively
(A) 2,1
(B) 1,2
(C) 1,1
(D) none of these

1 M

8 (a) (iv) Two square matrices A and B are similar, if
(A) A=B
(B) B=P^-1AP
(C) A^T=B^T
(D) A^-1=B^-1

1 M

8 (b) Reduce the quadratic form 8x²+7y²+3z²-12yz+4zx-8xy to the canonical form

4 M

8 (c) Determine the characteristics roots and eigen vectors of

$A=\begin{bmatrix}8 &-6 &2 \\ -6&7 &-4 \\ 2&- 4&3 \end{bmatrix}$

6 M

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.

6 M

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