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COMBINED FIRST AND SECOND SEMESTER B.TECH (ENGINEERING)
EXAMINATION; EN 14 101 – ENGINEERING MATHEMATICS I
Time: 3 Hours
Max: 100 marks
PART A: Analytical/problem solving SHORT questions
Answer Eight questions out of Ten (Each Question carries 5 marks)
1. Evaluate lim
−
x−
− −l g
2 . If x=u(1+v),y=v(1+u) Find
∂ ,
∂
,
3. Discuss the convergence of∑∞= sin
4. Ify=xn logx, Show that yn+1=
!
5. Using Cayley Hamilton find A2 if A=
6. Reduce the matrix A=
−
into Echelon form
−
7. If f(x)= x sinx is expanded as Fourier series in(-2,2),find a0
8. Find the half rangecosine series of f(x)={{π
9 State Raabe’s test
10 find eigen values of the matrix
−
π
<�<
π
<�<�
}
−
PART B: Analytical/Problem solving DESCRIPTIVE questions
Answer all questions (Each Question carries 15 marks)
11 a (i)If the centre of curvature of
the eccentricity is
√
+ =1 at one end of minor axis lies at the other end, prove
a (ii) Examinethe maxima and minima of f(x,y)=y2+4xy+3x2+x3
OR
12 b (i) If u = sin (
−
−
, then show that x
∂
∂
+y
∂
= 2 tanu
∂
b (ii) Show that the rectangular solid of maximum volume that can be inscribed in a sphere is a
cube.
13.a (i)Test the convergence of the seriesx −
√
+
√
- +………..
√
a (ii)Find the power series expansion of log(cos x) as far as and including the term containing x 4
OR
14 b (i)If y=⌈x + √x − ⌉m Prove that x −
yn+2+(2n+1)xyn+1+(n2-m2)yn= 0
2n−2
b (ii)Examine the convergence of the series ∑∞=
15a (i) Diagonalise the matrix
−
16b (i) For what values of and
+
−
√
OR
do the system of equations x+y+z=6,x+2y+ z= , x +2y+3z=10
have i no solution, ii unique solution iii more than one solution
b (ii)Given that A=
−
equal to the Trace A and
−
Verify that the sum and product of
the eigen values of A are
lAl respectively
17a (i) Obtain the first three coefficients in the Fourier cosine series for y, where y is given in the
following table
X 0 1 2 3 4 5
Y 3 9 6 8 2 7
OR
18 b (i) Obtain the Fourier cosine series of f(x)=e
x∈ [ , π]
b (ii) Express the function f(x)=x when -1<x<1 as a Fourier series with interval 2
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