Download Visvesvaraya Technological University (VTU) BE/B.Tech First And Second Semester (1st sem and 2nd sem) 20192020 Jan ( Bachelor of Engineering) 17 Scheme 17MAT11 Engineering Mathematics I Previous Question Paper
USN
First Semester B.F. Degree Examination, Dec.2019/Jan.2020
Engineering Mathematics ? I
Time: 3 hrs.
Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
ModuleI
I a. Find the n
th
derivative of sin 2x Cos X. (06 Marks)
b. Prove that the following curves cuts orthogonally r = a(1+ sin 0) and r = a(1? sin 0) .
(07 Marks)
c. Find the radius of the curvature of the curve r = a sin nO at the pole. (07 Marks)
2 a.
b.
c.
c.
OR
If, tan y = x , prove that (1 + x
2
+ 2(n +1)xy
ri+1
+ n(n + 1)y
r
, = 0 .
'WO
With usual notations, prove that tan(I) = .
dr
Find the radius of curvature for the curve n

y = a(x
2
+ y
2
) at (2a, 2a).
x + y eli ill
, prove that x +
y
= 
1
sin 2U U.
Ax +
3)
( x.
4
a
y
4
Find the Jacobian of u = x
2
+ y

+ z

, v = xy + yz +zx , w = x + y + z .
b. If U = cot

OR
r,
4 a. Evaluate lirn
tan x
x
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
Module2
3 a. Using Maclaurin's series prove that VI + sin 2x = +x x ++....
6 24
b. Find the Taylor's sense of log(cos x) about the point x = ?
jr
upto the third degree.
(x y
z
y z x
(07 Marks)
cu
prove that x ?+ y +z? Ou = . (07 Marks)
ax ay Oz
c. If u = f
5
a.
Module3
If x = t
2
+ 1, y = 4t ? 3, z = 2t
2
6t represents the parametric equation of a curve then, find
velocity and acceleration at t = 1. (06 Marks)
b. Find the constants a and b such that F = (axy + z
3
)i+(3x
2
? z)j+(bxz
2
? y )k is irrotational.
Also find a scalar function
4
such that F = V4. (07 Marks)
c. Prove that div(curl A) = 0 . (07 Marks)
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17MAT11
USN
First Semester B.F. Degree Examination, Dec.2019/Jan.2020
Engineering Mathematics ? I
Time: 3 hrs.
Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
ModuleI
I a. Find the n
th
derivative of sin 2x Cos X. (06 Marks)
b. Prove that the following curves cuts orthogonally r = a(1+ sin 0) and r = a(1? sin 0) .
(07 Marks)
c. Find the radius of the curvature of the curve r = a sin nO at the pole. (07 Marks)
2 a.
b.
c.
c.
OR
If, tan y = x , prove that (1 + x
2
+ 2(n +1)xy
ri+1
+ n(n + 1)y
r
, = 0 .
'WO
With usual notations, prove that tan(I) = .
dr
Find the radius of curvature for the curve n

y = a(x
2
+ y
2
) at (2a, 2a).
x + y eli ill
, prove that x +
y
= 
1
sin 2U U.
Ax +
3)
( x.
4
a
y
4
Find the Jacobian of u = x
2
+ y

+ z

, v = xy + yz +zx , w = x + y + z .
b. If U = cot

OR
r,
4 a. Evaluate lirn
tan x
x
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
Module2
3 a. Using Maclaurin's series prove that VI + sin 2x = +x x ++....
6 24
b. Find the Taylor's sense of log(cos x) about the point x = ?
jr
upto the third degree.
(x y
z
y z x
(07 Marks)
cu
prove that x ?+ y +z? Ou = . (07 Marks)
ax ay Oz
c. If u = f
5
a.
Module3
If x = t
2
+ 1, y = 4t ? 3, z = 2t
2
6t represents the parametric equation of a curve then, find
velocity and acceleration at t = 1. (06 Marks)
b. Find the constants a and b such that F = (axy + z
3
)i+(3x
2
? z)j+(bxz
2
? y )k is irrotational.
Also find a scalar function
4
such that F = V4. (07 Marks)
c. Prove that div(curl A) = 0 . (07 Marks)
1 of 2
17MAT11
OR
6 a. Find the component of velocity and acceleration for the curve r = 2t

i + t
2
 4t)j+(3t 5)k
at the points t = 1 in the direction of i 3 j+ 2k . (06 Marks)
b. If t = V(xy
3
z
2
) , find div t and curl t at the point (1, I, 1). (07 Marks)
c. Prove that curl(grad 4)) = 0. (07 Marks)
Module4
y
7 a. Prove that  dx = 3m using reduction formula. (06 Marks)
0  x
b.
Solve (x
2
+ y + x)dx + xydy = 0 . (07 Marks)
c. Find the orthogonal trajectory of rn = a sin nO . (07 Marks)
OR
8
a. Find the reduction formula for icosn xdy and hence evaluate cos" xdx (06 Marks.
0
b. Solve ye" dx +(w" + 2y)dy = 0 . (07 Marks)
c. A body in air at 25?C cools from 100?C to 75?C in 1 minute. Find the temperature of the
body at the end of 3 minutes. (07 Marks)
Module5
2
1
3 1
9 a. Find the rank of the matrix A =
I 2
I 0
0 1
by reducing to row echelon form.
4 1 1
2 3 1 b. Find the largest eigen and the corresponding eigen vector for
(06 Marks)
by taking the
2 1
5
initial approximation as [1, 0.8, 0.81
1
by using power method. Carry out four iterations.
(07 Marks)
c. Show that the transformation y
1
= 2x
1
2x
2
x. y, = 4x
1
+ 5x, + 3x? y
1
= x,  x, x
3
is regular. Find the inverse transformation. (07 Marks)
OR
10 a. Solve the equations 5x + 2y + z =12 , x + 4y + 2z =15 , x + 2y + 5z = 20 by using Gauss
Seidal method. Carryout three iterations taking the initial approximation to the solution as
(1, 0, 3). (06 Marks)
3

Diagonalize the matrix A =
b.
4
(07 Marks)
c. Reduce the quadratic form 8x
2
+ 7y
2
+3z
2
12xy + 4xz 8yz into canonical form by
orthogonal transformation. (07 Marks)
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This post was last modified on 02 March 2020