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Model Question Paper with effect from 2017-18
USN 17MAT11
First Semester B.E.(CBCS) Examination
Engineering Mathematics-I
(Common to all Branches)
Time: 3 Hrs Max.Marks: 100
Note: Answer any FIVE full questions, choosing at least ONE question from each module.
Module-I
1. Find the th n derivative of

x
(06 Marks)
Find the angle between the curves: r and r (07 Marks)
Find the radius of curvature for the curve: x y 3axy 3 3 at 2,3a (07 Marks)
OR
2. If x sint, y cosmt prove that 0 2 2
2 1
2 n x y n xy m n y (06 Marks)
With usual notation, prove that
dr
d
r

(07 Marks)
Find the radius of curvature for the cycloid x y (07 Marks)
Module-II
3. Find the Taylors series of x sin in powers of



2

x up to fourth degree terms. (06 Marks)

If








x y
x y
u
3 3
1 tan prove that u
y
u
y
x
u
x 2





(07 Marks)
If u yz x v zx w xy z show that 4. (07 Marks)
OR
4. Evaluate
x
x x x x
x
a b c d

0 4
lim






(06 Marks)
Using Maclaurin's series, prove that ...
2 3 24
1 sin 2 1
2 3 4

x x x
x x (07 Marks)
Ifu f y z prove that








z
u
y
u
x
u
(07 Marks)
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Module-III
5. A particle moves along a curve whose parametric equations are x e y 2cos3t, z 2sin 3t, t
where t is the time. Find the velocity and acceleration in the direction of i 3 j 2k

at t 0. (06 Marks)
Find divF

and curlF

at the point where F grad y z 3xyz 3 3 3

(07 Marks)
Show that vector field 2 2 F xi yj x y

is both solenoidal irrotational (07 Marks)
OR
6. For any scalar field and vector field, prove that DivF F

(06 Marks)

Find the angle between the surfaces 9 2 2 2 x y z and 3 2 2 z x y at (07 Marks)
If F y az)i (bx 2y j cy 2z)k

find such that F

is irrotational. (07 Marks)
Module-IV
7. Obtain reduction formula for sin
2
0


xdx n n (06 Marks)
Solve the differential equation: sin cos . 2 r
d
dr
r

(07 Marks)
Find the orthogonal trajectory of the family of curves 1.
2
2
2
2



a
y
a
x
-parameter) (07 Marks)
OR
8. Evaluate:
2
0
2 2
8
5
2

x x x dx (06 Marks)

Solve the differential equation xdy 0. (07 Marks)
A bottle of mineral water at a room temperature of F 0 72 is kept in a refrigerator where the
temperature is 44 . 0 F After half an hour, water cooled to 61 . 0 F What is the temperature of
the mineral water in another half an hour? (07 Marks)
Module-V
9. Solve the system of equations 83x 4z 95; 7x 52y 29z 71, (06 Marks)
using Gauss-Seidel method.
Using Rayleigh's power method, find largest eigen value and eigen vector of the matrix:












6 3 5
4 4 1
1 3 2
by taking X as initial eigen vector. (07 Marks)
Reduce 8x 7y 3z 12xy 4xz 8yz 2 2 2 into canonical form using orthogonal transformation.
Also indicate the nature, index, rank, and signature of the quadratic form. (07 Marks)
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OR
10. Find the rank of the matrix














2 2 3 7
3 1 1 8
1 1 2 5
1 1 1 6
(06 Marks)
Reduce the matrix




1 2
5 4
A into the diagonal form (07 Marks)
Show that the transformation 1 1 2 3 y x 2x 5x 2 1 2 3 y 2x 4x 3 2 3 y 2x (07 Marks)
is regular. Write down the inverse transformation.