Degree Sem II

CONTENTS
1.

Beta and Gamma functions
 1.1 Beta and Gamma functions and its properties
1.2 Relation between Beta and Gamma functions
1.3 Examples of Beta and Gamma functions
1.4 Duplication formula
1-30
2.
Differentiation under integral sign
2.1 Differentiation under the integral sign (DUIS)
2.2 Examples of with one parameter of DUIS
2.3 Examples of with two parameters of DUIS
31-50
3.
Rectification
3.1 Length of the arc of a curve given by
3.2 Length of a curve in parametric form
3.3 Length of a loop
3.4 Length of the arc of a curve given in polar form by
3.5 Length of the arc of a curve given in polar form by
51-72
4.
Exact differential equations (DE of first order and first degree)
4.1 Exact differential equations
4.2 IF
∫
where
4.3 IF
∫
where
4.4
(
4.5 Homogeneous differential equations (
73-100
)
)
5.
Linear differential equations
5.1 Linear differential equations and differential equations
5.2 Equations reducible to the linear form
5.3 Bernoulli’s Equation
101-120
6.
Linear differential equations with constant coefficients
121-152
6.1 Linear differential equations with constant coefficients, Complimentary
functions, particular integrals, solutions
6.2 D E of the type
where is (i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii) General method.
7.
Equation reducible to LDE with constant coefficient
7.1 Cauchy’s Homogeneous Linear Equation
7.2 Legendre’s Linear Equation
7.3 Method of Variation of Parameters
153-178
8.
Application of differential equation
8.1 Simple electrical circuits, examples
8.2 Examples on mechanical engineering
179-192
9.
Numerical solutions of Ordinary differential equations
9.1 Taylor’s series method
9.2 Euler’s method
9.3 Modified Euler’s method
9.4 Runge-Kutta method of 4th order
193-220
10. Double Integrals
10.1 Double Integrals
10.2 Evaluation of double integrals
10.3 Change of the order of integration
10.4 Evaluation over a given region
10.5 Changing to polar form
10.6 Applications of double integrals
i) Area by double integrations
ii) Mass of Lamina
221-306
11. Triple Integrals
11.1 Triple Integration- definition and evaluation
11.2. Application of Triple Integration
11.2.1. Region bounded by planes
11.2.2. Region bounded by spheres, hemispheres, ellipsoids
11.2.3. Region bounded by a cylinder or a cone or a Paraboloid
11.2.4. Region bounded by right circular cylinder/cone
307-342
12. Numerical integration and differentiation
12.1.Calculus of Finite differences
Finite Differences, Forward, Backward and Central differences,
Relations between the operators E, and ∇, Newton’s Forward and
Backward Difference Interpolation Formula, Lagrange’s
interpolation formula, Divided Difference, Newton’s divided
difference formula, Numerical Integration, Newton-Cote’s formula,
Trapezoidal Rule, Simpson’s One-third rule and three-eight Rule.
343-386
Chapter wise formulas and relations and MU Question Papers solution 2014(Nov)
Blue print and Index
387-408
409-412
Blue Print (SEM-II/APM-II)
409
Blue-print
Applied Maths –II
(Year 2013 onwards)
Weightage wise ‘Blue-print ‘Applied Maths -II (FEC201 Time: 3 Hours Total Marks : 80
Topic Unit Topic Name
01
1.1 Beta & Gamma functions & DUIS
1.2 Rectification
1.3 Exact Differential equation
02
2.1 Reducible to Linear DE
2.2 L.D.E. with constant co-efficient.
2.3 Cauchy’s/Legendre’s/varn of partr
2.4 Application of D.E
03
3.1 D.E. by 4 Numerical methods
3.2 Double Integration
04
4.1 Triple integration
4.2 Appln of Double & triple intgn
4.3 Numerical Integration
Total
Question Numbers
Wtge 1
2
3
4
11
03 08
06
06
10
04
06
06
09
03
06
08
08
06
14
08
19 7(3+4) 06
06
06
14
06
11
03
120
20 20 20 20
5
6
06
06
06
06
08
08
20
20
Note: (1) From subtopic 4.3-on Newton’s interpolation no examples are expected.
(2) Each Question of 8 marks may be converted into two questions of 4 marks each
410
Blue Print (SEM- II/APM-II)
Content wise ‘Blue-print’ Applied Maths -II (FEC201)
Topic Unit Unit Title**
No
No
01
1.1
Beta & Gamma functions & DUIS
1.2
Rectification
1.3
Exact Differential equation
02
2.1
Reducible to Linear Differential equation
2.2
Linear D.E. with constant co-efficient.
2.3
Cauchy’s/Legendre’s/variation of parameters
2.4
Application of D.E
03
3.1
Solving D.E. by Numerical methods
3.2
Double Integration
04
4.1
Triple integration
4.2 Application of Double and triple integration
4.3
Numerical Integration
Total
Unit wise
Marks
11
06
10
06
09
08
06
14
19
06
14
Topic wise
Marks
27
29
33
31
11
120
Blue print of question paper AM II (R 2012 syllabus
Q S.Q Chap Mks
Topic name
1 a
1.1 03 Beta & Gamma functions
b
2.2 03 Finding C.F or P.I
c
4.3 03 Relation between a, v , E
d
3.2 03 Change to polar co-ordinates and evaluate.
e
1.3 04 problems on exact equations
f
3.2 04 Evaluation of double integrals
2 a
2.1 06 Reducible to Linear differential equation
b
3.2 06 Evaluation by Change of order of Integration
c
1.1 08 Beta gamma function/DUIS
3 a
4.1 06 Evaluation of triple integration
b
4.2 06 Application of double integration
c
2.3 08 Cauchy’s/ Legender homogenous equations/variation of parameter
4 a
1.2 06 Rectification
b
2.2 06 Linear Differential Equation with constant co-efficient
c
3.1 08 Runga Kutta method
5 a
1.3 06 Reducible to exact differential equations.
b
3.1 06 Taylor’s/euler’s /euler’s modified method
c
4.3 08 Numerical integration
6 a
2.4 06 Application of differential equations
b
3.2 06 Double Integration over given region
c
4.2 08 Application of triple integrations
(REVISED COURSE)
(3 Hours)
N.B.
Total Marks: 80
(1) Question No. 1 is compulsory.
(2) Attempt any three Questions from remaining five.
(3) Figures to the right indicate full marks.
Pages Marks
1
(
(a). Evaluate ∫
)
(P.14) (3)
(b). Solve
(P.125) (3)
(c). Prove that
(d). Solve: * √
+
* √
(e). Change to polar coordinate and evaluate
(f). Evaluate
2
(P.346) (3)
(P.78) (3)
+
∫
∫
√
∫ ∫
(P.222) (4)
(a). Solve:
(P.108) (6)
(b). Change the order of integration and evaluate: ∫ ∫
(c) Prove that ∫
(
)
3. (a) Evaluate ∭
(
(
)
(P.238) (6)
)√
(
) using DUIS rule.
throughout the volumes of the sphere
(b).Find the area common to the two cardioids
(
(c). Apply the Method of Variation of Parameters to solve
4
(P.273) (4)
√
(a). Find the length of one arc of the cycloid
(
(P.320) (6)
) and
(
),
(
).
(P.288) (6)
(P.175) (8)
)
(P. 62) (6)
(b). Solve:
(c) Apply Runge -Kutta’s method of fourth order to find an approximation
value of y at
if
given that
when
choosing
5
(a). Solve: *
(P.140) (6)
(P.216) (8)
+
(P.85) (6)
,using Taylor’s
.
(b). If satisfies the equation
and with
when
Series Method for about
, find
when
and
(c). Compute the value of the definite integral ∫
by using
(i) Trapezoidal rule, (ii) Simpson’s ( )
6
(P.45) (8)
(P.194) (6)
(P.382) (8)
rule (iii) Simpson’s ( )
rule
Compare the result with exact values.
(a) . A radial displacement ‘u’ in rotating a disc at a distance r from the axis is given by
. Find the displacement given
when
(P.189) (6)
and
(b). Evaluate ∬
over the region bounded by
,
,
and
(c) Find the volume of the tetrahedron bounded by the coordinate planes and the plane
.
.
(P.246) (6)
(P.328) (8)
406
Applied Mathematics-II
(REVISED COURSE)
(3 Hours)
N.B.
(1)
(2)
(3)
(4)
Total Marks: 80
Question No. 1 is compulsory.
Attempt any three Questions from Question Nos. 2 to Questions No.6
Figures to the right indicate full marks.
Programing calculators are not allowed.
Pages Marks
1
(a). Evaluate ∫ √√
(b). Solve (
(c). Prove that (
)( – )
(d). Change to polar coordinate and evaluate
)
(
(e). Solve: (
2
(f). Evaluate
∫ ∫
(a). Solve:
(
(P.24) (3)
(P.125) (3)
(P.346) (3)
)
∫ ∫
√
(
)
(P.263) (3)
(P.97) (4)
)
√
(P.224) (4)
)
(P.118) (6)
(b). Change the order of integration and evaluate: ∫ ∫
(c) Evaluate ∫
3
and show that ∫
5
6
(
)
(
(a) Evaluate ∭
z
throughout the volumes bounded by the
planes
and
(b). Find the area bounded by parabola
and the line
(c). Use Method of Variation of Parameters to solve the equation
(
4
(P.244) (6)
(a). Find the length of the loop of the curve
)
(P.43) (8)
(P.314) (6)
(P.281) (6)
(P.176) (8)
)
(
)(
)
(P.58) (6)
(b). Solve:
(c) Apply Runge -Kutta’s method of fourth order to find an approximation
value of y at
if
given
when
in steps of
(P.137) (6)
(P.215) (8)
)
(a). Solve: (
(b) . Using Taylor’ Series Method solve
given that ( )
and hence find ( ) and ( )
(c). Compute the value of the definite integral ∫ (
) by
(i)Trapezoidal rulen, (ii) Simpson’s one third rule (iii)Simpson’s three-eight rule
Compare the result with exact values.
(P.91) (6)
(P.201) (6)
(a) . The motion of a particle is given by
(P.190) (6)
. Solve the equation
when
taking
at
. Show that the time of
maximum displacement is independent of the initial velocity.
)
(b). Evaluate ∬(
over the area of the triangle whose vertices are at
the point (0,1), (1,0), (1,2).
(c) Find the volume bounded by the cylinder
,
and the planes
and
(P.380) (8)
(P.255) (6)
(P.330) (8)
I.R. Khan, Theem College of Engineering, Boisar, Mob. +91 7666215501
407
Application of Differential Equation
(REVISED COURSE)
(3 Hours)
N.B.
Total Marks: 80
(1) Question No. 1 is compulsory.
(2) Attempt any three Questions from Question Nos. 2 to Questions No.6
(3) Figures to the right indicate full marks.
(4) Programing calculators are not allowed.
Pages Marks
1. (a) Evaluate ∫
(b) Solve: (
(c) Prove that
(P. 8) (3)
)
(d) Solve: (
(P. 114) (3)
(P. 346) (3)
)
(e) Evaluate ∬
(P. 88) (3)
, over the area included between the circle
(f) Evaluate ∫ ∫
and
. (P. 270) (4)
√
(
(P. 227) (4)
)√
2. (a) Solve: (
)
(
)
(P. 93) (6)
(b) Change the order of integration and evaluate: ∫ ∫
(c) Prove that ∫
∫
√
√
(P. 238) (6)
(P. 23) (8)
√
3. (a) Evaluate ∫ ∫ ∫
(P. 308) (6)
(b) Find the area of one the loop of the lemniscates
)
(c) Solve: (
4. (a) Show that the length of the arc of the parabola
(P.249) (6)
(P. 138) (8)
(P. 55) (6)
line
is
*
cut off by the
+
(b) Using the Method of Variation of Parameters solve
(
(c) Compute
, ( )
) given
(P. 171) (6)
by taking
using
(P. 212) (8)
Runge-Kutta method of fourth order correct to 4 decimal.
(
5. (a) Solve
)
(b) Solve
(
)
using Taylor’s series method. Find the
, y(0)
approximate values of y for
(c) Evaluate ∫
(P 112) (6)
(P.203) (6)
1.1.
by using
(i) Trapezoidal rule
(P. 383) (8)
(ii) Simpson’s ( )
rule
(iii) Simpson’s ( )
rule
Compare the result with exact values.
6. (a) The current in a circuit containing an inductance L, resistance R and voltage
is given by
=
, If
at
(b) Evaluate ∬
over the triangle bounded by
(c) (i). Find the volume of solid bounded by the surfaces
the plane
,
.
√
(c) (ii). Change to polar co-ordinates evaluate ∫ ∫√
√
(P. 183) (6)
find i.
,
, and
(P. 249) (6)
and
(P. 330) (4)
(P. 266) (4)
I.R. Khan, Theem College of Engineering, Boisar, Mob. +91 7666215501
408
Applied Mathematics-II
(REVISED COURSE)
(3 Hours)
N.B.
Total Marks: 80
(1) Question No. 1 is compulsory.
(2) Attempt any three Questions from Question Nos. 2 to Questions No.6
(3) Figures to the right indicate full marks.
(4) Programing calculators are not allowed.
1
(a). Prove that ∫ (
)(
)
(b). Solve (
(c). Prove that E = + 1 =
(d). Solve:
Pages Marks
(P.7) (3)
)
(
(P.124) (3)
(P.346) (3)

(P.74) (3)
)
∫ ∫
(e). Change to polar coordinate and evaluate
2
(f). Evaluate
∫ ∫
(a). Solve (
-
√
(
)
.
(P. 263) (4)
√
(P.224) (4)
)
(P.110) (6)
(b). Change the order of integration and evaluate: ∫ ∫
(
(c) Use DUIS to prove that ∫
3
(a) Evaluate ∫ ∫
∫
)
(
(P.237) (6)
√
(P.34) (8)
(P.313) (6)
)
(b). Find the area enclosed by rectangular hyperbola
(
(c). Solve:
4
(
.
for
to
is
(P.65) (6)
√ )].
(b). Solve:
(P.137) (6)
(c) Apply Runge -Kutta’s method of fourth order to find an approximation
(P.215) (8)
value of y at
(a). Solve: (
(b). Solve
if
given
)
when
(
in steps of
)
with initial condition
at
(P.90) (6)
by Taylor’s Series Method.
Obtain y as series in powers of x . Find approximation value of y for
Compare your result with exact value.
(c). Evaluate ∫
(P.382) (8)
rule (iii) Simpson’s ( )
rule
Compare the result with the exact values.
(a) . The current in a circuit containing an inductance L, resistance R and voltage
is given by
=
(P.200) (6)
.
by using
(i)Trapezoidal rule (ii) Simpson’s ( )
6
(P.282) (6)
(P.154) (8)
(a). Show that the length of arc of the parabola
[√
5
and the line
)
, If
at
find i.
(b). Evaluate ∬
over the region R given by
2
,
2 and
(c) Find the volume bounded by the cone
(P.183) (6)
(P.256) (6)
and the cylinder,
(P.335) (8)
I.R. Khan, Theem College of Engineering, Boisar, Mob. +91 7666215501

Download Report