1. No category

CONTENTS
1.

Pre‐requisite
 Complex Number and De-Moivre’s Theorem
1.
Complex Number
1-72
1.1 Root of a complex number
1-16
1.2 Hyperbolic functions, Inverse Hyperbolic functions.
17-36
1.3 Logarithmic of a Complex Number
37-50
1.4 Separation of real and imaginary parts
51-64
1.5 Expansion of
,
in terms of sines and cosines of
65-72
multiples of , Expansion of sinn , cosn in powers of sin , cos
Matrices and Numerical Methods
73-120
2.1 Types of Matrices
73-75
2.2 Theorems on matrices
76-78
2.3 Orthogonal and Unitary matrices
83-87
2.4 Rank of a Matrix, Reduction to normal form/Canonical form
88-94
2.5 Reduction to PAQ form
94-98
2.6 Inverse by elementary row transformation
99-99
2.7 Linear equations, Linear combination, dependence and independence 100-105
2.8 Echelon form, augmented matrices
105-106
2.9 Cconsistency and inconsistency of linear equations
107-120
2.
3.
4.
5.
i - xii
Solution of system of linear algebraic equations
3.1 Gauss Elimination Method
3.2 Gauss Jordan Method
3.3 Jacobi iteration Method
3.4 Gauss Seidal Method
3.5 Crout’s Method
Differential Calculus
4.1 Successive differentiation, nth derivative of standard functions
4.2 nth derivative of algebraic functions
4.3 nth derivative of trigonometric functions
4.4 nth derivative using De-Moivre’s theorem
4.5 Leibnitz’s Theorem
121-138
121-124
124-125
126-126
127-132
133-138
139-166
139-140
140-145
146-148
149-154
155-166
Partial Differentiation
5.1 Partial derivatives of first order and higher order
Differentiation of function of function, implicit function
5.2 Variables treated as constants
5.3 Differentiation of composite functions
5.4 Total differentials, Variables treated as constant
167-198
167-180
181-183
184-196
196-198
6.
Euler’s Theorem on Homogeneous functions
6.1 Homogeneous functions, Test of Homogeneous functions
6.2 Euler’s Theorem and examples
6.3 Deductions from Euler’s theorem
199-1214
199-200
200-205
206-214
7. Application of Partial differentiation (1)
7.1 Jacobean, properties of Jacobean
7.2 Jacobian of implicit function
215-226
215-224
224-226
8. Application of Partial differentiation (2)
8.1 Maxima and Minima of a function of two independent variables
9. Application of Partial differentiation (3)
9.1 Lagrange’s method of undetermined multipliers
227-244
227-244
245-254
245-254
10.
255-272
255-257
257-265
266-268
268-270
270-270
270-272
Expansion of functions
10.1 Expansions using Taylor’s Series
10.2 Expansions using Maclaurin’s Series
10.3 Expansion using standard expansions
10.4 Expansion using differentiation/Integration
10.5 Expansion using Method of inversion
10.6 Expansion using proper substitution
11.
Indeterminate forms
11.1 Indeterminate forms, L-Hospital Rule
11.2 Examples on direct evaluation of limits
11.3 Examples on finding constants
12. Curve fitting
12.1 Principle of Least squares, Method of Least squares
12.2 Straight line fitting
12.3 Parabolic curve fitting
12.4 Exponential curve fitting
List of Chapter wise formulas and relations
MU Question Papers from 2012 to 2015
Blue Print
Index
273-286
273-274
274-278
279-286
287-302
287-290
291-293
294-298
299-302
303-312
313-316
317-318
319-320
318
Blue Print (SEM- I/APM-I)
Blue Print of question paper AM I ( R- 2012 syllabus) Time: 3 Hours Total Marks : 80
Topic
Unit
01
1.1
1.2
1.3
1.4
2.1
2.2
3.1
3.2
3.3
4.1
4.2
4.3
Total
02
03
04
Q
1
2
3
4
5
6
S.Q
a
b
c
d
e
f
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
Unit Title
Complex Numbers- Powers &Roots
Complex Numbers- Circular Functions
Separation of real &imaginary parts
Expansion of sine and cosine, etc
Types of Matrices
Matrices-Linear algebraic equations
Successive Differentiation
Partial Differentiation
Euler’s Theorem
Partial Differentiation: Maxima and Minima
Taylor’s Theorem
Fitting of Curves
Chap
1.2
3.2
4.1
4.2
2.1
3.1
1.1
2.1
3.3
2.1
4.1
1.3
4.1
1.2
2.2
1.4
4.2
3.1
2.1
3.2
4.3
Total
Mks
3
3
3
3
4
4
6
6
8
6
6
8
6
6
8
6
6
8
6
6
8
120
Unitwise
Marks
06
09
08
06
16
14
12
09
08
15
09
08
Topicwise
Marks
120
120
29
30
29
32
Topic name
Relation between circular and hyperbolic function
Problems on basic partial derivatives
Jacobeans
Expansion standard series
Properties of matrices
Problems on std formula of successive derivatives
Powers and Roots of a complex number
Matrices PAQ/normal form
Euler’s theorem with deduction
Linear homo and non homogenous equations.
Maxima and minima/Lagrange’s method
Separation of real &imaginary parts
Jacobean of implicit fun/PD of implicit fun using Jacobean.
Logarithm of complex numbers
Matrices-Linear algebraic equations
Expansion of sine and cosine, etc
Expansion of series/indeterminate forms
Problems on Leibnitz’s theorem
Linear independent, dependent/from 2.2(from 2.1/2.2)
Composite/implicit functions
Fitting of curves/Regression
Blue Print (SEM- I/APM-I)
319
Blue Print of question paper AM I ( R- 2012 syllabus) Time: 3 Hours Total Marks : 80
Topic Unit Topic Name
01
1.1 Complex Numbers- Powers &Roots
1.2 Complex Numbers- Circular funs.
1.3 Separation of real &imaginary parts
1.4 Expansion of sine and cosine, etc
02
2.1 Types of Matrices
2.2 Matrices-Linear algebraic equations
03
3.1 Successive Differentiation
3.2 Partial Differentiation
3.3 Euler’s Theorem
04
4.1 Partial Diffrn:Maxima &Minima
4.2 Taylor’s Theorem
4.3 Fitting of Curves
Total
Wtge
06
09
08
06
16
14
12
09
08
15
09
08
120
Question Numbers
1
2
3
4
06
03
06
08
5
6
06
04
06
06
08
04
03
06
08
06
08
03
03
20
06
06
06
20
20
20
20
08
20
Note: (1) Each Question of 8 marks may be converted into two questions of 4 marks each
(2) No question on correlation coefficient is expected.
(2) Question number 1 is compulsory and 3 questions to be selected from the
remaining questions.
(REVISED COURSE)
(3 Hours)
Total Marks: 80
N.B. (1) Question No. 1 is compulsory.
(2) Attempt any three questions from question no. 2 to question no. 6.
(3) Figures to the right indicate full marks.
1. (a) If
, find the value of
and then
(P19) (3)
( ) find the value of
(b) If
(c) If
,
find
(P169) (3)
(
)
(
)
(P216) (3)
(d) Prove that
(P 264) (3)
(e) Show that every square matrix can be uniquely expressed as the sum a hermitian
matrix and a skew hermitian matrix.
(f) Find nth order derivative of
2. (a) Solve the equation
(b) Reduce the matrix A to the normal form and find its rank
[
(P. 77) (4)
(P. 148) (4)
(P. 12) (6)
(P. 90) (6)
]
(c) State and prove Euler’s theorem for homogeneous function of two variables and
(P202) (8)
√
hence verify the Euler’s theorem for
√
√
3. (a) Test the consistency of the following equations and solve them if they are consistent.
,
,
(b) Find all stationary values of
(c) Separate into real and imaginary parts of
( )
(P. 109) (6)
4. (a) If
(P 217) (6)
prove that
(b) Show that for real values of
and
(
,
)
(P. 46) (6)
(c) Solve the following equations by Gauss-Seidel method
27
,
,
5. (a) Expand
in a series of cosines of multiples of
(b) If
(c) If
√
prove that (
6. (a) Examine whether the vectors
are linearly dependent.
(b) If
(
, find
and
)
(
=[
],
(P130) (8)
(P 67) (6)
(P 280) (6)
)
(
],
=[
)
[
= .
(P 159) (8)
]
(P 105) (6)
), show that
(P195) (6)
(c) Fit a straight line for the following data
X
Y
1
49
2
54
(P 228) (6)
(P 54) (8)
3
60
(P 298) (8)
4
73
5
80
6
86
314
Applied Mathematics-I
(REVISED COURSE)
(3 Hours)
Total Marks: 80
N.B. (1) Question No. 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume suitable data if necessary.
(
1. (a) Prove that
)
(
)
(b) If
(P31) (3)
prove that
)
(P217) (3)
[
( )
]
(d) If
, prove that
(
) in powers of x. Hence prove that
(e) Find the series expansion of
(
)
(
)
(
)
(f) If A is a skew symmetric matrix of odd order then prove that it is singular
(
)
)
(
) = are given by
2. (a) Show that the roots of (
*
+ k = 1,2,3,4,5
(b) Find the non-singular matrices P and Q such that PAQ is normal form where
(P 147) (8)
(P 249) (4)
(c) If
,
find
[
(
(P169) (3)
(P 76) (4)
(P 7) (6)
(P 95) (6)
] Also find its rank of A.
(c) If
and
and u is a function of x and y
(P187) (8)
prove that
3. (a) . Find the value of for which the equations
,
,
=
have a solution and solve them completely for each value of .
(b) Divide 24 into three parts such that the product of the first, square of the second and
cube of the third is maximum.
(
(c) (i) If
*
(ii)
4. (a) S.t
, then prove that (
)
(
)
(
)
(
) = (
)
(P 112) (6)
(P247) (6)
(P 55) (4)
)+
(P 43) (4)
given that
and =
=
and
prove that
(
)
(
)
(
)
(c) Using Gauss-Siedel method solve the following system of equations
upto 3rd iterations.
,
,
(P218) (6)
(b) If
5. (a) Use De-Moivre’s Theorem to prove that
(b) Expand in powers of ,
=
, hence prove that
(P 71) (6)
(
)
)
(
)
(
(c) If
prove that (
√
6. (a) Examine the linear dependence or independence of vectors
],
[
] and
[
]
=[
(
(b) If
(c)
(PR P-x) (6)
(P128) (8)
+
)
=[
= . Hence find
],
), prove that
(c) (ii) Evaluate
1965
125
(
1966
140
)
( ) (P 160) (8)
(P 103) (6)
(P188) (6)
(i) Fit a straight line to the following data with x as independent variable:
X
Y
(P 257) (6)
1967
165
1968
195
(P 290) (4)
1969
200
(P 274) (4)
I.R. Khan, Theem College of Engineering, Boisar, Mob. +91 7666215501
University Question Papers
315
(REVISED COURSE)
(3 Hours)
N.B.
Total Marks: 80
(1) Question No. 1 is compulsory.
(2) Solve any three from the remaining.
(
1. (a) If
(b) If
Pages
(P. 55) (3)
) prove that
show that
(
)
(
(d) Prove that
(e) Express the relation in
)
(c) If
,
[
(P169) (3)
(
) show that
(
)
(
)
(P221) (3)
(P 253) (3)
(P. 86) (4)
for which
] is unitary
(f) Find nth derivative of
(
) , then show that
2. (a) If
where
(b) Find two non singular matrices P and Q such that PAQ is normal form.
Also find the rank of A where
[
(P. 152) (4)
(P. 15) (6)
(P 96) (6)
]
(c) State and prove Euler’s theorem for homogeneous function in two variables and
(P212) (8)
hence find the value of
(
)
for
3. (a) For what value of
the system of equations have a non trivial solution ?
Obtain the solution for for real values of
where
,
,
= .
(
)
(b) Find all stationary values of
(
)
(
)
(c) If
where
are real, then show that
4. (a) If
show that
( )
( )
)
)(
)
(b) If (
is real then one of the value of the principal values is (
(c) Solve by Crout’s Method the system of equations
,
5. (a) If
, the find
(b) By using Taylor’s Theorem, arrange in powers of x
(
)
(
)
(
)
(
) prove that
(c) If
(
)
(
)
(
)
6. (a) Solve correctly upto three iterations the following equations by Gauss-Siedel method
,
,
(b) If
(
) and
find
(c) Fit a curve
for the data:
x
1
2
3
4
5
6
y
2.51
5.82
9.93
14.84
20.55
27.06
(P. 115) (6)
(P241) (6)
(P. 26) (8)
(P182) (6)
(P. 47) (6)
(P. 134) (8)
(P 66) (6)
(P 246) (6)
(P. 156) (8)
(P. 129) (6)
(P186) (6)
(P 298) (8)
I.R. Khan, Theem College of Engineering, Boisar, Mob. +91 7666215501
316
Applied Mathematics-I
(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.
1. (a) If
(
prove that
(
(b) If
Pages
(P. 25) (3)
)
) prove that
(c) If
,
find
(P169) (3)
(
)
(
)
(P216) (3)
(
) in powers of upto
(d) Expand
(e) Show that every square matrix can be uniquely expressed as the sum a symmetric and a s
kew symmetric matrix.
(f) Find nth order derivative of
2. (a) Solve the equation
(b) Reduce the matrix A to the normal form and find its rank
[
hence find
√
where
3. (a) Determine the value of
,
[
the equations + + = , +
, prove that
(
)]
+
,
(P. 114) (6)
(
)
(
)
prove that
[
.
(
)
(
)
(P229) (6)
(P. 45) (8)
(P217) (6)
(
)]
where
(P. 43) (6)
(c) Using Gauss-Siedel iteration method to solve
(P. 127) (8)
,
5. (a) Expand
upto three iterations
in a series of cosines of multiples of
(P 68) (6)
(b) Evaluate
(P 278) (6)
prove that (
(c) If
)
(
)
(
)
(P. 161) (8)
6. (a) Examine the following vectors for linear dependence/independence
=[
(b) If
(P200) (8)
√
=
have a solution and solve them completely in each case.
(b) Find all stationary values of
,
(c) Separate into real and imaginary parts
( )
(b) If
(P. 146) (4)
(P. 4) (6)
(P. 88) (6)
]
(c) State and prove Euler’s theorem for homogeneous function of two variables and
4. (a) If
(P 253) (3)
(P. 76) (4)
],
=[
(
),
],
=[
(P. 100) (6)
] where
,
prove that
(P189) (6)
(c) Fit a straight to the following data:
(P 291) (8)
Year (X)
1951
Production(Y)
10
1961
1971
1981
1991
12
8
10
13
I.R. Khan, Theem College of Engineering, Boisar, Mob. +91 7666215501
University Question Papers
317
(REVISED COURSE)
(3 Hours)
Total Marks: 80
N.B. (1) Question No. 1 is compulsory.
(2) Attempt any three questions from the remaining five.
(3) Figures to the right indicate full marks.
Pages
1. (a) Prove that
(
(b) If
(c) If
) prove that
,
, find
(
(d) Prove that
(
)
(
)
)
(e) Show that every square matrix can be uniquely expressed as P+iQ
where P and Q are Hermitian matrices.
(f) Find the nth derivative of
(
2. (a) Show that the roots of (
) (
)
(
(
) = are given by
3. (a) Test the consistency and solve if consistent.
,
(b) Find all stationary values of
(c) If
(
)
show that
)
*
+ k = 1,2,3,4,5
√
find
,
by using Jacobean.
is real prove that (
)
in a series of cosines of multiples of
*
+
(
) , obtain ( )
(c) If
6. (a) Show that the vectors are linearly dependent and find the relation between them
], =[
],
[
] and
[
]
=[
prove that ( )
(b) If
( )
( )
(
1
2
2
6
3
7
4
8
(3)
5
10
(P 7) (6)
(P202)
(8)
(P. 107)
(6)
(P232)
(6)
(P 27)
(8)
(P219)
(6)
(P 47)
(P. 135)
(6)
(8)
(P 67)
(6)
(P 279)
(6)
6
11
7
11
(P. 164) (8)
(P. 102) (6)
)
(c) Fit a second degree parabolic curve to the following data:
X
Y
(P 255)
√
)
(b) Considering only the principal value, If (
(c) Solve the system of linear equation by Crout’s method
(b) Evaluate
(3)
,
,
5. (a) Expand
(P219)
(P 89) (6)
(iii)
4. (a) If
(3)
√
where
(ii)
(P172)
(P. 140) (4)
)
[
]
(c) State and prove Euler’s theorem for a homogeneous function of two variables.
(i)
(3)
(P 77) (4)
(b) Reduce the following matrix to normal form and find the rank
Hence find the value of
(P19)
(P180)
(6)
(P 294) (8)
8
10
9
9
I.R. Khan, Theem College of Engineering, Boisar, Mob. +91 7666215501