MORE IN Engineering Maths 1
VTU First Year Engineering (C Cycle) (Semester 1)
Engineering Maths 1
January 2012
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) $If \ Y=\dfrac {x}{x-1}, \ then \ y_n \ is \\(A) \ \dfrac {(-1)^{n-1}n!a^n}{(x-1)^{n+1}}\\(B) \ \dfrac {(-1)^n n!}{(x-1)^{n+1}}\\(C) \ \dfrac {(-1)^n (n+1)!}{(x-1)^{n+1}}\\(D) \ \dfrac {(-1)^n n!}{(x-1)^n}\\$
1 M
1 (a) (ii) If y=log(ax+b), then yn is
$(A) \ \dfrac {(-1)^n n!a^n}{(ax+b)^n}\\(B) \ \dfrac {(-1)^{n-1}n!a^n}{(ax+b)^{n+1}}\\(C) \ \dfrac {(-1)^{n-1}(n-1)!a^{n}}{(ax+b)^{n}}\\(D) \ \dfrac {(-1)^n(n-1)!a^n}{(ax+b)^{n+1}}\\$
1 M
1 (a) (iii) If f(x)=sin x, x ?(0, ?), then by Rolle's theorem the value of 'x', where the Tangent is parallel to x-axis.
(A) 0
(B) ?/2
(C) ?/3
(D) ?/4
1 M
1 (a) (iv) Expansion of log (1+x) in powers of x is
$(A) \ x+\dfrac {x^2}{2}+\dfrac {x^3}{3}+....\\(B) \ x-\dfrac {x^2}{2}+\dfrac {x^3}{3}-\dfrac {x^4}{4}+....\\(C) \ 1-\dfrac {x}{1!}+\dfrac {x^2}{2!}-\dfrac {x^3}{3!}+ .... \\(D) \ \dfrac {x}{1!}-\dfrac {x^2}{2!}+\dfrac {x^3}{3!}-\dfrac {x^4}{4!}+....\\$
1 M
1 (b) If X=Tan(log y), show that (1+x2)yn+1+(2nx-1)yn+n(n-1)yn-1=0
4 M
1 (c) State and prove Cauchy's mean value theorem.
6 M
1 (d) Expand f(x)=sin (ex-1) in power's of 'x' upto the terms containing x4.
6 M

2 (a) (i) The indeterminate form of $\lim_{x\rightarrow}\left (\dfrac {x}{x-1}-\dfrac {(x-1)}{\log x} \right )\ is \\ \\(A) \ \infty - \infty \\(B) \ \dfrac {0}{0} \\(C) \ \dfrac {\infty}{\infty}\\(D) \ None \ of \ these$
1 M
2 (a) (ii) The angle between the radius vector and the tangent to the curve r=k e?Cot?, where K and ? are constant, is:
(A) K
(B) ?
(C) ?
(D) O
1 M
2 (a) (iii) The pedal equation of the curve r=a? is.
$(A) \ P^2=ar\\(B) \ \dfrac {1}{P^2}=\dfrac {a}{r^2}\\(C) \ \dfrac {1}{P^2}=\dfrac {1}{r^2}+a^2 \\(D) \ \dfrac {1}{P^2}=\dfrac {1}{r^2}+\dfrac {a^2}{r^4}\\$
1 M
2 (a) (iv) The radius of curvature at any point 't' on the curve defined by x=f(t), y=?(t) is given by
$(A) \ \dfrac {[(x)^2+(y)^2]^{3/2}}{x'y''-y'x''} \\(B) \ \dfrac {x'y''-y'x''}{[(x')^2+(y')^2]^{3/2}}\\(C) \ \dfrac {(x')^2+(y')^2}{(x'y''-y'x'')^{3/2}}\\(D) \ \dfrac {(x'y''-y'x'')^{3/2}}{(x')^2+(y')^2}\\$
1 M
2 (b) Find the angle of intersection between the curves rn cos(n?)=an and rnsin(n?)=bn.
4 M
2 (c) Show that the radius of curvature at any point '?' to the curve x=a(? + sin ?), y=a(1 - cos ?), is 4acos (?/2).
6 M
2 (d) Evaluate \ \lim_{x\rightarrow 0}\left ( \dfrac {a^x+b^x+c^x} {3}\right )^{1/2}
6 M

3 (a) (i) $if \ u=x^{y-1}, \ then \ \dfrac {\partial u}{\partial y} \ is \\ \\(A) \ x^{y-1} \log x \\(B) \ (y-1)x^{y-2} \\(C) \ x^{y-1} \log y \\(D) \ x^{y} \log x \\$
1 M
3 (a) (ii) If Z=f(u, v), where u=x+ct and v=x-ct, then $\dfrac {\partial z}{\partial t}$ is given by
$(A) \ \dfrac {\partial z}{\partial u}- \dfrac {\partial z}{\partial v}\\(B) \ \dfrac {\partial z}{\partial u}+ \dfrac {\partial z}{\partial v}\\(C) \ c \left (\dfrac {\partial z}{\partial u}- \dfrac {\partial z}{\partial v} \right )\\(D) \ c \left (\dfrac {\partial z}{\partial v}-\dfrac {\partial z}{\partial u} \right )\\$
1 M
3 (a) (iii) If x=u(1-v), y=uv, then $J\dfrac {x,y}{u,v}$ is equal to
(A) u
(B) 1/u
(C) uv
(D) u/v
1 M
3 (a) (iv) The necessary condition for the function f(x,y) to posses extreme values is
(A) fx=fy=0
(B) fxx-fyy=0
(c) (fxx) (fyy)-f2xy=0
(D) fx>0, fy>0
1 M
3 (b) $if \ u=f\left (\dfrac {y-x}{xy}, \dfrac {z-x}{xz} \right ), \ find \ x^2 \dfrac {\partial u}{\partial x}$
4 M
3 (c) If x+y+z=u, y+z=v and z=uvw, show that $j \left ( \dfrac {x,y,z}{u,v,w} \right )=uv.$
6 M
3 (d) The Horse power required to propel a steamer is proportional to the square of the distance and cube of the velocity. If the distance is increased by 4% and velocity increases by 3% find the percentage of increases in the Horse power.
6 M

4 (a) (i) $if \ \overrightarrow{R}=xi+yj+zk,|\overrightarrow{R}|=r, \ then \ abla r^{2} \ is \ equal \ to \\(A) \ \dfrac {\bar {R}}{r^2} \\(B) \ \dfrac {-\bar{R}}{2} \\(C) \ \dfrac {\bar{R}}{r}\\(D) \ 2\bar{R}\\$
1 M
4 (a) (ii) If F=3x2i-xyj+(a-3)x z k is solenoidal, then 'a' is equal to
(A) 0
(B) -2
(C) 2
(D) 3
1 M
4 (a) (iii) if A=x2i+y2j+z2 k, then curl A is given by
(A) 2xi+2yj+2zk
(B) 0
(C) $\dfrac {xi+yj+zk}{2}$
(D) 2x+2y+2z
1 M
4 (a) (iv) The scale factors for cylindrical coordinates system (? 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 M
4 (b) Prove \ that \ abla.\phi \bar{F}=abla\phi.\bar{F}+\phi(abla.\bar{F}).
4 M
4 (c) $if \ \bar{F}=2xy^3z^4i+3x^2y^2z^4j+4x^2y^3z^3k, find \\(i) \ (abla . \bar {F}) \\(ii) \ abla \times \bar{F}\\$
6 M
4 (d) Obtain the expression for ?.F in orthogonal curvilinear coordinate system (u1 u2 u3).
6 M

5 (a) (i) $Given \ \int^{1}_0 x^n dx=\dfrac {1}{x+1}, \ then \ \dfrac {d^2}{dx^3} \int^1_{0}x^n dx \ gives \\ (A)\ \int^1_0 (\log x)^2 x^n dx = \dfrac {2}{(1+n)^2} \\(B)\ \int^1_0 (\log x)^2 x^n dx= \dfrac {2}{(1+n)^3} \\(C)\ \int^1_0 (\log x)^n x^n dx= \dfrac {2}{(1+n)^2}\\(D)\ \int^1_0 (\log x)^2 x^n dx= \dfrac {-2}{(1+n)^3}\\$
1 M
5 (a) (ii) The value of the integral $\int^\pi_0 \sin^6 x \cos^5x \ dx \ is \\(A) \ 0 \\(B) \ \dfrac {8}{693}\\(C) \ \dfrac {8\pi}{693}\\(D)\ None \ of \ these$
1 M
5 (a) (iii) The volume of the solid generated by revolving the curve r=a(1+Cos ?) about the line ?=0 is given by
$(A)\ \dfrac {2\pi}{3}a^3 \int^\pi_0(1+ cos \theta)^3 \sin \theta \ d\theta \\(B)\ \dfrac {2\pi}{3}a^3 \int^\pi_0(1+\cos\theta)^3 \cos\theta \ d\theta \\(C)\ \dfrac {2\pi}{3}a^3 \int^{2\pi}_0(1+\cos\theta)^3 \sin\theta \ d\theta \\(D)\ \dfrac {4\pi a^3}{3}\\$
1 M
5 (a) (iv) The entire length of the asteroid x2/3+y2/3=a2/3 is
(A) 4a
(B) 8a
(C) 6a
(D) 3a
1 M
5 (b) Obtain the reduction formula of the integral ? Cosn x dx
4 M
5 (c) Using Leibnitz rule under differentiation under integral sing, evaluate $\int^{\pi}_0 \dfrac {\log (1+2 \cos x)}{\cos x}dx.$
6 M
5 (d) find the surface generated by revolving the cycloid x=a (? - \sin ?), y=a (1-\cos ?) about its base, (cosider one arc in the 1st quadrant).
6 M

6 (a) (i) The general solution of the differential equation $\dfrac {dy}{dx}=sec \left ( y/x \right )+y/x \ is$
(A) Tan y/x-logx=c
(B) Sin (y/x)-logx=c
(C) Cosec (y/x)-logx=c
(D) Cos (y/x)-logx=c
1 M
6 (a) (ii) Integrating factor for the differential equation $\dfrac {dx}{dy}+\dfrac {2x}{y}=y^2 \ is$
(A) y2
(B) ex2
(C) e2y
(D) ey2
1 M
6 (a) (iii) The general solution of the differential equation (x-y)dx+(y-x)dy=0 is
$(A)\ \dfrac {x^2}{2}-y -\dfrac {y^2}{2}=c\\(B)\ \dfrac {x^2}{2}-y+\dfrac {y^2}{2}=c \\(c)\ \dfrac {x^2}{2}-yx+\dfrac {y^2}{2}=c \\(D)\ None \ of \ these$
1 M
6 (a) (iv) Given the different equation of f(r, ? c)=0, we get differential equation of orthogonal trajectories by changing $\dfrac {d\theta}{dr} \ by$
$(A)\ \dfrac {1}{r}\dfrac {dr}{d\theta}\\(B)\ -r^2 \dfrac {dr}{d\theta}\\(C)\ \dfrac {-1}{r}\dfrac {dr}{d\theta}\\(D)\ r\dfrac {dr}{d\theta}\\$
1 M
6 (b) Solve (x2-4xy-2y2)dx+(y2-4xy-2x2)dy=0
4 M
6 (c) $Solve \ (x+2y^3)\dfrac {dy}{dx}=y$
6 M
6 (d) Find the orthogonal trajectories of the family of curves $\dfrac {x^2}{a^2}+\dfrac {y^2}{b^2+\lambda}=1$ ('?' being the parameter).
6 M

7 (a) (i) The rank of the matrix $\begin{bmatrix}6 &1 &3 &8 \\ 4&2 &6 &-1 \\ 10&3 &9 &7 \\ 16&4 &12 &15 \end{bmatrix}$ is equal to
(A) 2
(B) 3
(C) 4
(D) 1
1 M
7 (a) (ii) The exact solution of the system of equations 10x+y+z=12, x+10y+z=12, x+y+10z=12 by inspection is equal to
(A) [0 0 0]T
(B) [1 1 1]T
(C) [1 1 -1]T
(D) [-1 -1 -1]T
1 M
7 (a) (iii) If the given system of linear equation in 'n' variables is is consistant then number of linearly independent solution is given by
(A) n
(B) n-1
(C) r-n
(D) n-r
( Where 'r' stands for rank of co-efficient matrix).
1 M
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 M
7 (b) Using elementary row transformations find the rank of the matrix $\begin{bmatrix}0 &1 &-3 &-1 \\ 1&0 &1 &1 \\ 3&1 &0 &2 \\ 1&1 &-2 &0 \end{bmatrix}$
4 M
7 (c) Test for consistency and solve the system of equations x+4+3z=0, x-y+z=0, 2x-y+3z=0
6 M
7 (d) Applying Gauss Jordan method solve 2x+3y-z=5, 4x+4y-3z=3, 2x-3y+2z=2
6 M

8 (a) (i) The linear transformation y=Ax is regular if
(A) |A|=0
(B) |A|=1
(C) |A|=-1
(D) |A|?0
1 M
8 (a) (ii) The transformation ?=x Cos?-y Sin?, ?=x Sin?+y Cos? is orthogonal then the inverse of the transformation matrix is given by
$(A) \ \begin{pmatrix}\cos \alpha &\sin \alpha \\ -\sin \alpha&\cos \alpha \end{pmatrix}\\(B) \ \begin{pmatrix}\cos \alpha &-\sin \alpha \\ \sin \alpha&\cos \alpha \end{pmatrix}\\(C) \ \begin{pmatrix}\sin \alpha &\cos \alpha \\ \cos \alpha&-\sin \alpha \end{pmatrix}\\(D) \ \begin{pmatrix}-\sin \alpha &\cos \alpha \\ \cos \alpha& \sin \alpha \end{pmatrix}$
1 M
8 (a) (iii) The eigen vector 'x' of the matrix 'A' corresponding to eigen value '?' satisfy the equation
(A) AX=?X
(B) ?(A-X)=0
(C) XA-?A=0
(D) |A-?I|x=0
1 M
8 (a) (iv) Two square matrices A and B are similar if
(A) A=B
(B) B=P-1 AP
(C) A1=B1
(D) A-1=B-1
1 M
8 (b) Show that the transformation given below y1=2x1+x2+x3, y2=x1+x2+2x3, y3=x-2x3 is regular and find the inverse transformation.
4 M
8 (c) Find the matrix P which diagonalizes the matrix $A=\begin{bmatrix}-1 &1 &2 \\ 0&-2 &-1 \\ 0&0 &-3 \end{bmatrix}$
6 M
8 (d) Reduce the quadratic form $x^2_1+3x^2_2+3x^2_3-2x_2x_3$ in to canonical form by an appropriate orthogonal transformation which transforms x1 x2 x3 in terms of new variables y1 y2 y3
6 M

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