RGPV Engineering Mathematics -I - December 2015 Exam Question Paper

RGPV First Year Engineering (Set A) (Semester 1)
Engineering Mathematics -I
December 2015

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

1(a) Find the percentage error in the area of an ellipse if 1% error is made in measuring the major and minor axes.

2 M

1(b) Discuss the maxima and minima of the function x³+y³-3axy.

2 M

1(c) Find the radius of curvature at any point of the curve x=acost, y=bsint

3 M

Solve any one question from Q.1(d) &Q.1(e)

1(d) Use Taylor's theorem to prove that

$\tan^{-1}(x+h)=\tan^{-1}x+(h\sin\theta).\dfrac{\sin\theta}{1}-(h\sin\theta)^2\frac{\sin 2\theta}{2}\\ +(h\sin\theta)^3\dfrac{\sin 3\theta}{3}-\cdots+(-1)^{n-1}(h\sin\theta)^n\dfrac{\sin n\theta}{n}+\cdots$
Where θ=cot^-1x.

7 M

1(e) Expand e^rcosx by Maclaurin's theorem.

7 M

2(a) Find the limit when n → ∞ of the series

$\sum ^n_{r=1}\dfrac{n^2}{(n^2+r^2)^{3/2}}$

2 M

2(b) Evaluate

$\int ^1_0\left ( \log \dfrac{1}{y} \right )^{n-1}dy$ .

2 M

2(c) Change the order of integration in

$\int^q_0 \int ^a_y \dfrac{xdxdy}{x^2+y^2}$ and hence evaluate the same.

3 M

Solve any one question from Q.2(d) &Q.2(e)

2(d) Prove that

$\displaystyle \beta (m,n)=\dfrac{\Gamma m \Gamma n}{\Gamma (m+n)}, m>0,n>0$ .

7 M

2(e) Find the volume bounded by the paraboloid x²+4y²+z=4 and the xy-plane. Also sketch the curve.

7 M

3(a) Solve (ycosx+siny+y)dx+(sinx+xcosy+x)dy=0

2 M

3(b) Solve p²+2pycotx-y²=0

2 M

3(c) Solve (D²+2D+1)y=xcosx.

3 M

Solve any one question from Q.3(d) &Q.3(e)

3(d) Solve

$x^2\dfrac{d^2y}{dx^2}+5x\dfrac{dy}{dx}+4y=x \log x$

7 M

3(e) Solve by the method of variation of parameters

$\dfrac{d^2y}{dx^2}+y=cosec x$ .

7 M

4(a) Find the rank of the matrix A, where

$A=\begin{pmatrix} 2 & 1 & -1\\ 0 & 3 & -2\\ 2 & 4 & -3 \end{pmatrix}$ . If λ be an eigen value of a non singular matrix A.

2 M

4(b) Show that λ^-1 is an eigen value of A^-1.

2 M

4(c) Solve the equations:
x₁+3x₂+2x₃=0, 2x₁-x₂+3x₃=0, 3x₁-5x₂+4x₃=0, x₁+17₂+4x₃=0.

3 M

Solve any one question from Q.4(d) &Q.4(e)

4(d) Find the eigen values and eigen vectors of the matrix

$A=\begin{pmatrix} -2 & 2 & -3\\ 2 & 1 & -6\\ -1 & -2 & 0 \end{pmatrix}$

7 M

4(e) Verify Cayley - Hamilton theorem for the matrix.

$A=\begin{pmatrix} 2 & -1 & 1\\ -1 & 2 & -1\\ 1 & -1 & 2 \end{pmatrix}$ . Hence compute A^-1.

7 M

5(a) Define the following with examples:
i) Union of two fuzzy set
ii) Intersection of two fuzzy set

2 M

5(b) Define the following :
i) Simple graph
ii) Multigraph
iii) Degree of vertex
iv) Isolated vertex.

2 M

5(c) Prove that the number of vertices of odd degree in a graph is always even.

3 M

Solve any one question from Q.5(d) &Q.5(e)

5(d) If (B, +,•, ') be a Boolean algebra and a,b be any two elements of B. Then show that (a+b)'=a'.b' ∀ a,b ∈ B.

7 M

5(e) Write the function f(x, y, z) = x.y'+x.z+x.y into conjunctive normal form in three variables.

7 M

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