Manual for Krash Manual for Krash
Why Krash? Why Krash?
Towards the end of preparation, a student has lost the time to revise all the chapters from Towards the end of preparation, a student has lost the time to revise all the chapters from his / her class notes / standard text books. This is the reason why K-Notes is specifically his / her class notes / standard text books. This is the reason why K-Notes is specifically intended for Quick Revision and should not be considered as comprehensive study material. intended for Quick Revision and should not be considered as comprehensive study material. It’s very overwhelming for a student to even think about finishing 300
It’s very overwhelming for a student to even think about finishing 300-400 questions per-400 questions per subject when the clock is ticking at the last moment. This is the reason why Kuestion serves subject when the clock is ticking at the last moment. This is the reason why Kuestion serves the purpose of being the bare minimum set of questions to be solved from each chapter the purpose of being the bare minimum set of questions to be solved from each chapter during revision.
during revision.
What
What is is Krash?Krash?
Krash is a combination of K-Notes and Kuestion which effectively becomes a great tool for Krash is a combination of K-Notes and Kuestion which effectively becomes a great tool for revision. A 50 page or less notebook for each subject which contains all concepts covered in revision. A 50 page or less notebook for each subject which contains all concepts covered in GATE Curriculum in a concise manner to aid a student in final stages of his/her preparation. GATE Curriculum in a concise manner to aid a student in final stages of his/her preparation. A set of 100 questions or less for each subject covering almost every type which has been A set of 100 questions or less for each subject covering almost every type which has been previously asked in GATE. Along with the Solved examples to refer from, a student can try previously asked in GATE. Along with the Solved examples to refer from, a student can try similar unsolved questions to improve his/her problem solving skills.
similar unsolved questions to improve his/her problem solving skills. When do I start using Krash?
When do I start using Krash?
It is highly recommended to use Krash in the last 2-3 months before GATE Exam. It is highly recommended to use Krash in the last 2-3 months before GATE Exam.
How do I use Krash? How do I use Krash? Once you finish the enti
Once you finish the entire K-Notes for a particular subject, you should practice the respectivere K-Notes for a particular subject, you should practice the respective Subject Test / Mixed Question Bag containing questions from all the Chapters to make best Subject Test / Mixed Question Bag containing questions from all the Chapters to make best use of it. Kuestion should be used as a tool to improve your speed and accuracy use of it. Kuestion should be used as a tool to improve your speed and accuracy subject-wise. It should be treated as a supplement to our K-Notes and should be attempted once wise. It should be treated as a supplement to our K-Notes and should be attempted once you are comfortable with the concepts and basic problem solving ability of the subject. You you are comfortable with the concepts and basic problem solving ability of the subject. You should refer
K-should refer K-Notes before solving any “Type” problems from Kuestion.Notes before solving any “Type” problems from Kuestion.
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Krash - Sample
Krash - Sample
Engineering Maths
Engineering Maths
(Linear Algebra &
(Linear Algebra &
Differential Equation)
Differential Equation)
LINEAR ALGEBRA
LINEAR ALGEBRA
MATRICES MATRICES
A matrix is a rectangular
A matrix is a rectangular array of numbers (or functions) array of numbers (or functions) enclosed in brackets. Theseenclosed in brackets. These numbers (or function) are called entries of elements of the matrix.
numbers (or function) are called entries of elements of the matrix. Example: Example: 2 0.4 8 2 0.4 8 5
5 -32 -32 00 order order = = 2 2 x x 3, 3, 2 2 = = number number of of rows, rows, 3 3 = = number number of of columnscolumns
Special Type of Matrices Special Type of Matrices 1.
1. Square Square MatrixMatrix
A m x n matrix is called as a square matrix if m = n i.e. number
A m x n matrix is called as a square matrix if m = n i.e. number of rows = number of columnsof rows = number of columns The elements
The elements aa when i = jijij when i = j
a a a 11 11 2a 222...
are called diagonal are called diagonal elementselementsExample: Example: 1 1 22 4 4 55 2.
2. Diagonal Diagonal MatrixMatrix
A square matrix in which all non-diagonal elements are zero and diagonal elements may or A square matrix in which all non-diagonal elements are zero and diagonal elements may or may not be zero.
may not be zero. Example: Example:
1 1 00 0 0 55 Properties Properties a.a. diag [x, y, z] + diag [p, q, r] = diag [x + p, y + q, z + r]diag [x, y, z] + diag [p, q, r] = diag [x + p, y + q, z + r] b.
b. diag [x, y, z]diag [x, y, z] ×× diag [p, q, r] = diag [xp, yq, zr] diag [p, q, r] = diag [xp, yq, zr] c. c.
1 1 1 1 11 11 ddiiaag g xx, , yy, , zz ddiiaag g ,, ,, xx yy zz
d.
d.
diag diag x, x, y, y, zz
tt = diag [x, y, z] = diag [x, y, z] e.e.
diiadag g xx, , yy, , zz
nn diiadag g xx ,, yy ,, zz nn nn nn f.f. Eigen value of diag [x, y, z] = x, y & zEigen value of diag [x, y, z] = x, y & z g.
g. Determinant of diag [x, y, z] = xyzDeterminant of diag [x, y, z] = xyz
3.
3. Scalar Scalar MatrixMatrix
A diagonal matrix in which all diagonal elements are equal. A diagonal matrix in which all diagonal elements are equal.
4.
4. Identity Identity MatrixMatrix
A diagonal matrix whose all diagonal elements are 1. Denoted by I A diagonal matrix whose all diagonal elements are 1. Denoted by I Properties: Properties: a. a. AI = IA = AAI = IA = A b. b. n n I I II c. c. 1 1 I I II d. d. det(I) = 1det(I) = 1 5.
5. Null Null matrixmatrix
An m x n matrix whose all elements are zero. Denoted by O. An m x n matrix whose all elements are zero. Denoted by O. Properties: Properties: a. a. A + O = O + A = AA + O = O + A = A b. b. A + (- A) = OA + (- A) = O 6.
6. Upper Upper Triangular Triangular MatrixMatrix A square matrix whose lower off d
A square matrix whose lower off diagonal elements are zero.iagonal elements are zero. Example: Example: 3 3 4 4 55 0 0 6 6 77 0 0 0 0 99 7.
7. Lower Lower Triangular Triangular MatrixMatrix A square matrix whose upper off
A square matrix whose upper off diagonal elements are zero.diagonal elements are zero. Example: Example: 3 3 0 0 00 4 4 6 6 00 5 5 7 7 99 8.
8. Idempotent Idempotent MatrixMatrix
A matrix is called Idempotent if
A matrix is called Idempotent if A A 22 AA
Example: Example: 1 1 00 0 0 11 9.
9. Involutary Involutary MatrixMatrix
A matrix is called Involutary if
Matrix Equality Matrix Equality Two matrices
Two matrices
AA m m nn and and
BB p p qq are equal if are equal if mm = = p p ; ; n n = = q q i.e., i.e., both both have have same same sizesize ij
ij a
a = = bb for all values of i & j.ijij for all values of i & j.
Addition of Matrices Addition of Matrices
For addition to be performed, the size of both matrices should be same. For addition to be performed, the size of both matrices should be same. If [C] = [A] + [B]
If [C] = [A] + [B] Then
Then c c iij j a a iij j bbiijj
i.e., elements in same position in the two matrices are added. i.e., elements in same position in the two matrices are added. Subtraction of Matrices
Subtraction of Matrices [C] = [A]
[C] = [A] –– [B] = [A] + [ [B] = [A] + [––B]B]
Difference is obtained by subtraction of all elements of B from elements of A. Difference is obtained by subtraction of all elements of B from elements of A. Hence here also, same size matrices should be
Hence here also, same size matrices should be there.there. Scalar Multiplication
Scalar Multiplication The product of any m
The product of any m×× n matrix A n matrix A
aa jk jk and any scalar c, written as cA, is the m and any scalar c, written as cA, is the m ×× n matrix n matrix cA =
cA =
caca jk jk obtained by multiplying each entry in A by c. obtained by multiplying each entry in A by c.
Multiplication of two matrices Multiplication of two matrices Let
Let
AAm m nn and and
BB p p qq be two matrices and C = AB, then for multiplication, [n = p] should be two matrices and C = AB, then for multiplication, [n = p] should hold. Then, hold. Then, n n jk jk j j 11 iikk iijj cc a a bb
Properties: Properties: If AB exists then BA If AB exists then BA does not necessarily exists.does not necessarily exists.
Example:
Example:
AA 3 3 44 , ,
BB 4 4 55 , then AB exits but BA does not exists as 5 , then AB exits but BA does not exists as 5 ≠≠ 3 3So, matrix multiplication is not commutative. So, matrix multiplication is not commutative.
Matrix multiplication is not associative.Matrix multiplication is not associative.
A(BC)
A(BC) ≠≠ (AB)C . (AB)C .
Matrix Multiplication is distributive with respect to matrix additionMatrix Multiplication is distributive with respect to matrix addition
A(B + C) = AB +AC A(B + C) = AB +AC
If If AB AB = = ACAC B B = = C C (if (if A A is is non-singular)non-singular)
BA = CA
BA = CA
B B = = C C (if A (if A is is non-singular)non-singular)Transpose of a matrix Transpose of a matrix
If we interchange the rows by columns of a matrix and vice versa we obtain transpose of a If we interchange the rows by columns of a matrix and vice versa we obtain transpose of a matrix. matrix. eg., A = eg., A = 1 1 33 2 2 44 6 6 55 ;; T T 1 1 2 2 66 A A 3 3 4 4 55 Conjugate of a matrix Conjugate of a matrix
The matrix obtained by replacing each element of matrix by its complex conjugate. The matrix obtained by replacing each element of matrix by its complex conjugate. Properties Properties a. a.
A A AA b. b.
A A B B A
A BB c. c.
KKA A KK AA d. d.
AAB B AA BBTransposed conjugate of a matrix Transposed conjugate of a matrix
The transpose of conjugate of a matrix is called transposed conjugate. It is represented by The transpose of conjugate of a matrix is called transposed conjugate. It is represented by
A A .. a. a.
A A AA b. b.
A B A B
A BA B c. c.
KKA A KKAA d. d.
AAB B B B A A Trace of matrix Trace of matrixTrace of a matrix is sum of all diagonal elements of the matrix. Trace of a matrix is sum of all diagonal elements of the matrix. Classification of Real Matrix
Classification of Real Matrix a.
a. Symmetric Matrix :Symmetric Matrix :
A A TT AA b.b. Skew symmetric matrix :Skew symmetric matrix :
A A TT AA c.c. Orthogonal Matrix :Orthogonal Matrix :
AATT AA11; ; AAA =A =IITT
Note: Note: a.
a. If A & B are symmetric, then (A + B) & (AIf A & B are symmetric, then (A + B) & (A –– B) are also symmetric B) are also symmetric b.
b. For any matrixFor any matrix AAAATT is always symmetric. is always symmetric. c.
c. For any matrix,For any matrix, T T A + A A + A 2
2 is symmetric & is symmetric &
T T A A AA 2
2 is is skew skew symmetric.symmetric. d.
d. For orthogonal matrices,For orthogonal matrices, A A 11
Classification of complex Matrices Classification of complex Matrices a.
a. Hermitian matrix :Hermitian matrix :
A A AA
b.
b. SkewSkew –– Hermitian matrix : Hermitian matrix : A A AA c.
c. Unitary Matrix :Unitary Matrix :
A A A A ;; A 11 AA A 11
DeterminantsDeterminants
Determinants are only defined for
Determinants are only defined for square matrices.square matrices. For a 2
For a 2 ×× 2 matrix 2 matrix
1 11 1 1122 2 21 1 2222 a a aa a a aa
== 1111 2222 1212 2211 a a aa aa aaMinors & co-factor Minors & co-factor
If If 1 11 1 112 2 1133 2 21 1 222 2 2233 3 31 1 332 2 3333 a a a a aa a a a a aa a a a a aa Minor of element Minor of element 221 1 2211 112 2 1133 3 32 2 3333 a a aa a a :: MM a a aa Co-factor of an element
Co-factor of an element
a
a
iijj
1
1
i i jjM
M
iijjTo design cofactor matrix, we replace each element by
To design cofactor matrix, we replace each element by its co-factorits co-factor Determinant
Determinant
Suppose, we need to calculate a 3 × 3 determinant Suppose, we need to calculate a 3 × 3 determinant
3 3 3 3 33 1 1 j j 11 j j 22 j j 22 j j 33j j 33jj j j 1 1 j j 1 1 j j 11a
a cco
of
f a
a
a
a cco
of
f a
a
a
a cco
of
f a
a
We can calculate determinant along any row or column of the matrix. We can calculate determinant along any row or column of the matrix.
Properties Properties
Value of determinant is invariant under row & column interchange i.e.,Value of determinant is invariant under row & column interchange i.e., A A TT AA
If any row or column is completely zero, thenIf any row or column is completely zero, then
A
A 0
0
If two rows or columns are interchanged, then value of determinant is multiplied by -1.If two rows or columns are interchanged, then value of determinant is multiplied by -1.
If one row or column of a matrix is multiplied by ‘k’, then determinant also becomes kIf one row or column of a matrix is multiplied by ‘k’, then determinant also becomes k
times. times.
If A is a matrix of order n × n , thenIf A is a matrix of order n × n , then n
n
K
KA A K AK A
Value of determinant is invariant under row or column transformationValue of determinant is invariant under row or column transformation
A
AB
B
A
A ** B
B
A A nn AAnn AA 11 11 A A Adjoint of a Square Matrix Adjoint of a Square Matrix Adj(A) =
Adj(A) =
cof Acof A
TTInverse of a matrix Inverse of a matrix Inverse of a matrix only
Inverse of a matrix only exists for square matricesexists for square matrices
1 1 AdAdj j AA A A A A Properties Properties a. a. AAA A 1 1 A A A 11A II b. b.
A
AB
B
11
B
B A
1 1A
11 c. c.
AABBC C
11 C B C 1 1 B A 1 1 A11 d. d.
T T 1 1 T T 11 A A
AA e.e. The inverse of a 2 × 2 matrix should be rememberedThe inverse of a 2 × 2 matrix should be remembered
1 1 a a bb 11 dd bb c c dd aad d bbcc cc aa I.I. Divide by determinant.Divide by determinant. II.II. Interchange diagonal element.Interchange diagonal element. III.
Rank of a Matrix Rank of a Matrix a.
a. Rank is defined for Rank is defined for all matrices, not necessarily a square matrix.all matrices, not necessarily a square matrix. b.
b. If A is a matrix of order m × n,If A is a matrix of order m × n, then Rank (A) ≤ min (m, n)
then Rank (A) ≤ min (m, n) c.
c. A number r is said to be rank of matrix A, if and only ifA number r is said to be rank of matrix A, if and only if
There is at least one square sub-There is at least one square sub-matrix of A of order ‘r’ whose determinant is nonmatrix of A of order ‘r’ whose determinant is non-zero.-zero.
If there is a sub-matrix of order (r + 1), then determinant of such sub-matrix should be 0.If there is a sub-matrix of order (r + 1), then determinant of such sub-matrix should be 0. Linearly Independent and Dependent
Linearly Independent and Dependent Let X
Let X11 and Xand X22 be the non-zero vectors be the non-zero vectors
If XIf X11=kX=kX22 or X or X22=kX=kX11 then X then X11,X,X22are said to be are said to be Linearly Dependent vectors.Linearly Dependent vectors.
If XIf X 11 kXkX22 or X or X22 kXkX11 then X then X11,X,X22are said to be Linearly Independent vectors.are said to be Linearly Independent vectors.
Note: Note: Let X
Let X11,X,X22………. X………. Xnn be n vectors of matrix A be n vectors of matrix A
If rank(A)=no of vectors then vector XIf rank(A)=no of vectors then vector X11,X,X22………. X………. Xnn are Linearly Independent are Linearly Independent
If rank(A)<no of vectors then vector XIf rank(A)<no of vectors then vector X11,X,X22………. X………. Xnn are Linearly Dependent are Linearly Dependent
System of Linear
System of Linear EquationsEquations
There are two type of linear equations There are two type of linear equations
Homogenous equationHomogenous equation n n 1 11 1 1 1 112 2 2 2 11nn a x a x a x a x ... . a x a x 00 n n 2 211 11 2222 22 22nn
a
a xx
a
a xx
... a
a xx
0
0
---mn mn nn m m1 1 1 1 mm2 2 22 a a x x a a x x ... a . a x x 00This is a system of ‘m’ homogenous equations in ‘n’ variables This is a system of ‘m’ homogenous equations in ‘n’ variables
Let Let 11 11 1122 1133 11nn 11 2 211 2222 22nn 22 mn mn nn m m11 mm22 m m nn n n 11 m m 11 a a aa aa aa xx 00 a a aa aa xx 00 --A A ;; xx ;; 00== --a a aa aa xx 00
This system can be represented as This system can be represented as AX = 0 AX = 0
Important Facts Important Facts
Inconsistent systemInconsistent system
Not possible for homogenous system as the trivial solution Not possible for homogenous system as the trivial solution
T T TT n n 1 1 22 x x , x , x ..., x , x 00, 0, 0, ., ..., 0, 0
always exists.always exists.
Consistent unique solutionConsistent unique solution
If rank of A = r and r = n
If rank of A = r and r = n |A| |A| ≠ 0≠ 0, so A is non-singular. Thus trivial solution exists., so A is non-singular. Thus trivial solution exists.
Consistent infinite solutionConsistent infinite solution
If r < n, number of independent equation < (number of variables) so, value of (n If r < n, number of independent equation < (number of variables) so, value of (n –– r) r) variables can be assumed to compute rest
variables can be assumed to compute rest of r variables.of r variables. This solution set has the following additional properties: This solution set has the following additional properties: 1.
1. If u and v are twoIf u and v are two vectors vectors representing solutions to a homogeneous representing solutions to a homogeneous system, then thesystem, then the vector sum u + v is also a solution to the system.
vector sum u + v is also a solution to the system.
2.
2. If u is a vector representing a solution to a homogeneous system, andIf u is a vector representing a solution to a homogeneous system, andr r is any is any scalar, scalar,
then
thenr r u is also a solution to the system.u is also a solution to the system.
Non-Homogenous Equation Non-Homogenous Equation n n 1 11 1 1 1 112 2 2 2 11n n 11 a x a x a x a x ... . a x a x bb n n 2 21 1 1 1 222 2 2 2 22n n 22 a x a x a x a x ... . a x a x bb ---m mn n n n nn m m1 1 1 1 mm2 2 22 a a x x a a x x ... a . a x x bb This is a system of ‘m’ non
This is a system of ‘m’ non-homogenous equation for n variables.-homogenous equation for n variables.
11 11 2 2 1 122 11nn 11 11 2 211 2222 22nn 22 mn mn nn mm m m11 mm22 m m nn m m 11 a a aa aa xx bb a a aa aa xx bb A A ;; XX ;; BB== --a a aa aa xx bb Augmented matrix = [A | B] = Augmented matrix = [A | B] = 11 11 12 12 nn 11 2 211 2222 22 mn mn mm m m1 1 mm22 a a aa aa bb a a a a bb a a aa aa bb
Conditions Conditions
InconsistencyInconsistency
If r(A) ≠
If r(A) ≠ r(A | B), r(A | B), system is system is inconsistentinconsistent
Consistent unique solutionConsistent unique solution
If r(A) = r(A | B) = n, we have consistent unique solution. If r(A) = r(A | B) = n, we have consistent unique solution.
Consistent Infinite solutionConsistent Infinite solution
If r(A) = r (A | B) = r & r < n, we have infinite solution If r(A) = r (A | B) = r & r < n, we have infinite solution
Specifically, if p is any solution of the system Ax=B, then entire solution set can be described
Specifically, if p is any solution of the system Ax=B, then entire solution set can be described
as {p+v: v is any solution to Ax=O}.
as {p+v: v is any solution to Ax=O}.
The solution of system of equations can be obtained by using Gauss elimination Method. The solution of system of equations can be obtained by using Gauss elimination Method. (Not required for GATE)
(Not required for GATE) Note:
Note:
Let ALet Anxnnxn and rank(A)=r, then the number of Linearly Independent and rank(A)=r, then the number of Linearly Independent solutions of Ax = 0 is “nsolutions of Ax = 0 is
“n--r” r”..
Cramer’s Rule Cramer’s Rule
Suppose we have the following system of
Suppose we have the following system of equations,equations,
1 1 1 1 11 11 2 2 2 2 22 22 3 3 3 3 33 33
a
a x
x b
b y
y c
c z
z d
d
a
a x
x b
b y
y c
c z
z d
d
a
a x
x b
b y
y c
c z
z d
d
We define the following matrices, We define the following matrices,
1 1 11 11 11 11 11 1 1 22 22 22 22 22 22 22 3 3 33 33 33 33 33 1 1 11 11 11 11 11 3 3 22 22 22 22 22 22 3 3 33 33 33 33 33 d d bb cc aa dd cc d d bb cc aa dd cc d d bb cc aa dd cc a a bb dd aa bb cc a a bb dd aa bb cc a a bb dd aa bb cc
Then, the solution is given by, Then, the solution is given by,
3 3 1 1 22 x x y y zz
This rule can only be applied in case of unique solution. This rule can only be applied in case of unique solution.
Eigen values & Eigen Vectors Eigen values & Eigen Vectors If A is n ×
If A is n × n square matrix, n square matrix, then then the equationthe equation Ax
Ax = λ x= λ x is called Eigen value problem.is called Eigen value problem. Where λ is called as Eigen value of A. Where λ is called as Eigen value of A. x is called as Eigen vector of A.
x is called as Eigen vector of A.
Characteristic polynomial Characteristic polynomial 1 111 1122 11nn 2 211 2222 22nn mn mn m m1 1 mm22 a a a a aa a a a a aa A A II a a a a aa
Characteristic equation Characteristic equation AA II 00The roots of characteristic equation are
The roots of characteristic equation are called as characteristic roots or the Eigen called as characteristic roots or the Eigen values.values. To find the Eigen vector, we need to solve
To find the Eigen vector, we need to solve
A
A I
I x
x 0
0
This is a system of homogenous linear equation. This is a system of homogenous linear equation. We substitute each value of λ one by one & cal
We substitute each value of λ one by one & calculate Eigen vector corresponding to eachculate Eigen vector corresponding to each Eigen value.
Eigen value. Important Facts Important Facts a.
a. If x is an eigenvector of A corresponding to λ, the KX is also an Eigenvector where K is aIf x is an eigenvector of A corresponding to λ, the KX is also an Eigenvector where K is a constant.
constant. b.
b. If a n × n matrix has ‘n’ distinct Eigen values, we have ‘n’ linearly independIf a n × n matrix has ‘n’ distinct Eigen values, we have ‘n’ linearly independent Eigenent Eigen vectors.
vectors. c.
c. Eigen Value of Eigen Value of Hermitian/Symmetric matrix are real.Hermitian/Symmetric matrix are real. d.
d. Eigen value of Skew - Eigen value of Skew - Hermitian / SkewHermitian / Skew –– Symmetric matrix are purely imaginary or zero. Symmetric matrix are purely imaginary or zero. e.
e. Eigen Value of unitary or orthogonal matrix are such that | λ | = 1.Eigen Value of unitary or orthogonal matrix are such that | λ | = 1. f.
f. IfIf 1 1 , , 22...,,nn are Eigen value of are Eigen value of A,A, k k ,k 1 1 ,k ...,k 22 ,knn are Eigen are Eigen values values of kA.of kA.
g.
g. Eigen Value ofEigen Value of
A
A
11 are reciprocal of Eigen value of A. are reciprocal of Eigen value of A. h.h. IfIf
n n 1
1 , , ...,,22 are Eigen are Eigen values of values of A,A,
n n 1 1 22
A
A
A
A
A
A
,
,
,
, ...
...,
...,
are Eigen values of Adj(A).are Eigen values of Adj(A). i.i. Sum of Eigen values = Trace (A)Sum of Eigen values = Trace (A)j.
j. Product of Eigen values = |A|Product of Eigen values = |A| k.
Cayley - Hamiltonian Theorem Cayley - Hamiltonian Theorem
Every matrix satisfies its own Characteristic equation. Every matrix satisfies its own Characteristic equation. e.g., If characteristic equation is
e.g., If characteristic equation is
n n n n 11 n n 1 1 22 C C CC ...CC 00 Then, Then, C AC11Ann CC A22An n 11... CC IInn OO
Where I is identity matrix Where I is identity matrix
O is null matrix O is null matrix
DIFFERENTIAL EQUATION
DIFFERENTIAL EQUATION
The order of a deferential equation is the order of highest derivative appearing in it.The order of a deferential equation is the order of highest derivative appearing in it.
The degree of a differential equation is the degree of the highest derivative occurring in it,The degree of a differential equation is the degree of the highest derivative occurring in it, after the differential equation is expressed in a form free from radicals & fractions.
after the differential equation is expressed in a form free from radicals & fractions.
For equations of first order & first degree For equations of first order & first degree
Variable Separation methodVariable Separation method
Collect all function of x & dx on one side. Collect all function of x & dx on one side. Collect all function of y & dy on other side. Collect all function of y & dy on other side. like f(x) dx = g(y) dy like f(x) dx = g(y) dy Solution: Solution:
f f x
x ddx x
g g y
y ddy y cc Example: Example: dzdz 1 1 x tx taannz z 11 dx dx dz
dz
xtanz
xtanz
dx
dx
dz dz xdx xdx tanz tanz cco
ott zzd
dzz
xxd
dxx
xx
22llo
og
g ssiin
nz
z
cc
2
2
An equation is linear if it can be written as: An equation is linear if it can be written as:
y y' P x ' P x y y r r xx If r(x) = 0 ; equation is homogenous If r(x) = 0 ; equation is homogenous else r(x)else r(x)≠≠ 0 ; equation is non-homogeneous 0 ; equation is non-homogeneous
y(x) =
y(x) = cece
p x dxp x dx is the solution for is the solution for homogenous formhomogenous form for non-homogenous form, h =for non-homogenous form, h =
P x d xP x d x
hh hh y y x x e e
e rrde dx cx c Example: Example: t t dxdx x x tt dt dt dx
dx
xx
Divide
Divide by
by t,
t,
1
1
dt
dt
tt
1 1//tt llooggttIIF
F
e d
e dtt
e
e
tt
Solution is Solution is 2 2 tt x x tt 11 tt ddtt cc 2 2
Exact differential equationExact differential equation
An equation of the form An equation of the form
M(x, y) dx + N (x, y) dy = 0 M(x, y) dx + N (x, y) dy = 0 For equation to be exact.
For equation to be exact. M M NN y y xx
; then only ; then only this method cthis method can be applied.an be applied.
The solution can be given by any one of the following two equations, The solution can be given by any one of the following two equations,
M
Mdx
dx
te
term
rms o
s of N
f N n
not
ot co
con
nta
tain
inin
ing
g 'x
'x' d
' dy =
y = cc
(y
(y co
con
nssta
tan
nt)
t)
N
Ndy
dy
te
term
rms o
s of M
f M n
not
ot co
cont
ntai
ain
nin
ing '
g 'y'
y' dx
dx = c
= c
(treat x as constant)(treat x as constant)Else, some special terms can be rearranged as
Else, some special terms can be rearranged as given below,given below,
xxd
dy
y yyd
dxx
d log xy
d log xy
xy
xy
22x
x
1
1 yyd
dx
x xxd
dy
y
yyd
dx
x xxd
dyy
d log
d log
y
y x
x
/
/
y
y
yy
xxyy
y
y
xxd
dy
y y
yd
dxx
d log
d log
x
x
xy
xy
1 1 2 2 22xx
yyd
dxx xxd
dyy
d
d tta
an
n
yy
xx
yy
1 1 2 2 22
y
y
xxd
dy
y y
yd
dxx
d tan
d tan
xx
x
x
y
y
2 2 22 2 2 222 x
2 xd
dx
x yyd
dyy
d
d llo
og
g x
x
yy
x
x
yy
x x x x xx 2 2e
e
y
ye
e d
dx
x e
e d
dy
y
d
d
y
y
y
y
y y y y yy 2 2e
e
xxe
e d
dy
y e
e d
dxx
d
d
xx
xx
2 2 22 2 2x
x
2
2xxy
yd
dx x
x x d
dy
y
d
d
y
y
y
y
2 2 22 2 2 y y 22xxyyddy y y y ddxx d d xx xx
2 2 2 2 22 2 2 44x
x
2
2xxy
y d
dx
x 2
2x
x y
yd
dy
y
d
d
y
y
y
y
2 2 2 2 22 2 2 44y
y
2
2x
x y
yd
dy
y 2
2xxy
y d
dxx
d
d
x
x
xx
Example:Example:
y c
y co
ossy
y
1
1
dx
d
x x x
x xssiin
ny
y 1
1 d
dy
y 0
0
2
2 xx
yyd
dx x
x xd
dy
y
cco
os y
s yd
dx
x xx ssiin
nyyd
dy
y
1
1
d
dx
x d
dy 0
y 0
2
2 xx
1
1
d
d xxy
y d
d xx cco
oss y
y
d
dx
x d
dy
y 0
0
2
2 xx
xxy xy x cco oss y y x x y y 0 0
Integrating factorsIntegrating factors
An equation of the form An equation of the form
P(x, y) dx + Q (x, y) dy = 0 P(x, y) dx + Q (x, y) dy = 0
This can be reduced to exact form by multiplying both sides by IF. This can be reduced to exact form by multiplying both sides by IF.
Rule I Rule I
If the given equation Pdx + Qdy = 0
If the given equation Pdx + Qdy = 0 (1) (1) isn’t an isn’t an exact exact equation but equation but iit is a homogenoust is a homogenous
equation and Px + Qy
equation and Px + Qy 0, then 0, then 11 (2)(2)
P
Px x QQyy
is an integrating factor of (1). is an integrating factor of (1).
Rule II Rule II
In the given equation
In the given equation
Pd
P
dx
x Q
Qd
dy
y 0
0
((1
1))
is not exact equation but it is in the form is not exact equation but it is in the form
f x
f xy y
y yd
dx
x g
g xxy x
y xd
dy
y 0
0
andand
Px Q
P
x Qy
y 0
0
.. ThenThen IIF F 11 ...((22)) P Px x QQyy If If 1 1 P P Q Q Q
Q y y xx is a function of x, then is a function of x, then R(x) =R(x) = 1 1 PP Q Q Q Q yy xx
. Then, Integrating Factor . Then, Integrating FactorIF = exp IF = exp
R x d xR x d x
Rule IV Rule IV Otherwise, if Otherwise, if 11PP Q Q PP x x yy is a function of y then, S(y) = is a function of y then, S(y) =
Q Q PP 1 1 P P x x yy
..Then, IntegratingThen, Integrating Factor, IF = exp
Factor, IF = exp
S y d yS y d y
Bernoulli’s equationBernoulli’s equation
The equation
The equation dydy PPy y QQyynn
dx dx
Where P & Q are function of x Where P & Q are function of x
Divide both sides of the equation by Divide both sides of the equation by nn
y
y & put & put yy 1 1 nn zz
dz dz P P 1 1 n n z z Q Q 1 1 nn dx dxThis is a linear equation & can be solved easily. This is a linear equation & can be solved easily.
Clairaut’s equationClairaut’s equation
An equation of the form y = Px + f (P), is known as Clairaut’s equation where P =
An equation of the form y = Px + f (P), is known as Clairaut’s equation where P =
dydydxdx The solution of this equation is y = cx + f (c) where c = constantThe solution of this equation is y = cx + f (c) where c = constant
Linear Differential Equation of Higher Order Linear Differential Equation of Higher Order Constant coefficient differential equation
Constant coefficient differential equation
n n n n n n 11 n n 11 n n 1 1 d d y y d d yy k k ... . k k y Xy X dx dx dxdx
Where X is a function of x only Where X is a function of x only a.
a. IfIf y y ,, y 1 1 y ,...22,...,y.,ynn are n independent solution, then are n independent solution, then
n n nn 1
1 11 22 22
c y
c y c yc y ... c yc y xx is complete solution is complete solution where
where c c ,c 1 1 ,c ,....2 2 ,...,c.,cnn are are arbitrary arbitrary constants.constants.
b.
b. The procedure of finding solution of nThe procedure of finding solution of nthth order differential equation involves computing order differential equation involves computing complementary function (C. F) and particular Integral (P. I).
c.
c. Complementary function is solution ofComplementary function is solution of
n n n n n n 11 n n 11 n n 1 1 d d y y d d yy k k ... . k k y 0y 0 dx dx dxdx d.
d. Particular integral is particular solution ofParticular integral is particular solution of
n n n n n n 11 n n 11 n n 1 1 d d y y d d yy k k ... . k k y xy x dx dx dxdx e.
e. y = CF + PI is complete solutiony = CF + PI is complete solution
Finding complementary function Finding complementary function Method of differential operator Method of differential operator Replace Replace dd dx dx by D by D→→ dy dy Dy Dy dx dx Similarly, Similarly, n n n n d d dx dx byby n n D D →→ n n n n n n d d yy D D yy dx dx n n n n n n 11 n n 11 n n 1 1 d d y y d d yy k k ... . k k y 0y 0 dx dx dxdx becomes becomes
nn nn 11
n n 1 1 D D k Dk D ... kk yy 00 LetLet mm ,m11,m ,..22,...,m,m be nn be roots roots ofof
n n 1 1 n n 11 n n D D k k D D ... . K K 00 ………….(i)………….(i) Case I:
Case I: All roots are real & distinct All roots are real & distinct
D m D m D 1 1
D m m ... 22
. D
D m m nn
00 is is equivalent equivalent to to (i)(i)y = y = m m x11x m m x2 2 x m m xnnx n n 1 1 22 c
c e e c c e e ... . c c ee is solution is solution of differential of differential equationequation
Case II:
Case II: If two roots are real & equal i.e., If two roots are real & equal i.e., m m 1 1 m m 22 mm
y = y =
22
mxmx m m xx33 m m xxnn n n 1 1 33 c c
c x c x e e
c c e e
...
. c c ee Case III:Case III: If two roots are complex conjugate If two roots are complex conjugate
1 1 m m
jj ;; m m 22
jj y = y =
1 1
22
m m xxnn n n xx e e c c ' c' coos x s x c c ' s' siin x n x ... . c c eeFinding particular integral Finding particular integral Suppose differential equation is Suppose differential equation is
n n n n 11 n n n n 11 n n 11 d d y y d d yy k k ... . k k y Xy X dx dx dxdx
PI = PI =
n n 1 1 22 n n 1 1 22 W W xx WW xx WW xx yy XXddxx yy XXddxx ... yy XXddxx W W xx WW xx WW xx
WhereWhere y y ,, y 1 1 y ,...22,...y.y are solutions of Homogenous from of differential equations.nn are solutions of Homogenous from of differential equations.
n n 1 1 22 n n 1 1 22 n n nn nn n n 1 1 22 y y y y yy y y ' y ' y ' ' y y '' W W xx y y y y yy
n n 1 1 22 i i 11 n n 1 1 22 i i 11 ii n n n n nn nn n n 1 1 22 i i 11 y y y y yy 0 0 yy y y ' ' y y ' ' y 'y ' 0 0 y y '' W W xx 0 0 y y y y yy 1 1 yy
ii WW xx is obtained from W(x) by replacing i is obtained from W(x) by replacing ithth column by all zeroes & last 1. column by all zeroes & last 1.
Example: Example:
3x 3x 2 2 2 2 e e D D 66D D 9 9 yy xx
22 3x3x22 e e D D 3 3 yy xx Finding Complimentary Function, Finding Complimentary Function,
D D 33,,33
1 1 22 3 3x x 33x x 33xx 1 1 22 11 22 yy yy y y c c c c x x e e cc e e c xxec e Wronskian is given by,Wronskian is given by,
3 3xx 33xx 6x 6x 3 3xx 33xx 33xx e e xxee W W ee 3 3ee 33xxee ee 3x 3x 3x 3x 1 1 33xx 33xx 0 0 xxee W W xexe 1 1 33xxee ee
3x 3x 3x 3x 2 2 3x3x e e 00 W W ee 3e 3e 11 Coefficients for Particular integral can be computed as, Coefficients for Particular integral can be computed as,
3 3x x 33xx 2 2 6x6x
e
e
xe
xe
A
A
d
dx
x llo
og
gxx
x
x
e
e
3 3x x 33xx 2 2 6x6xe
e
e
e
B
B
dx
dx
x
x
e
e
1
1
logx
logx
xx
P P 1 1 22y
y A
Ay
y B
Byy
Euler-Cauchy Equation Euler-Cauchy Equation An equation of the form An equation of the formn n n n 11 n n n n 11 n n n n 11 n n 11 d d y y d d yy x x k k x x ... . k k y y 00 dx dx dxdx
Substitute y = x Substitute y = xmm
The equation becomes The equation becomes
m m m m 1 1 ... . m m n n k mk 11m((m m 11))... . m m n n 1 1 ... . k k x nnx mm 00 The roots of equation areThe roots of equation are Case I:
Case I: All roots are real & distinct All roots are real & distinct
1 1 2 2 nn m m mm mm n n 1 1 22 y y c c x x c c x x ... . c c xx Case II:
Case II: Two roots are real & equal Two roots are real & equal
1 1 22 m m m m mm
mm mm33 mm n n 1 1 2 2 33 nn y y c c c n c n x x x x c x c x ... c . c xx Case III:Case III: Two roots are cTwo roots are complex conjugate of eaomplex conjugate of each otherch other
1 1 m m jj ; ; m m 22 jj y = y =
mm33
mmnn n n 3 3 x x AA ccoos s nnx x BB ssiin n nnx x c c x x ... c . c xxType 1: Linear Algebra
Type 1: Linear Algebra
For Concepts, please refer to Engineering Mathematics K-Notes, Linear Algebra For Concepts, please refer to Engineering Mathematics K-Notes, Linear Algebra
Problems: Problems:
01.
01. If for a matrix, rank equals both the number of rows and number of columns, then theIf for a matrix, rank equals both the number of rows and number of columns, then the matrix is called
matrix is called (A)
(A) Non-singular Non-singular (B) (B) singularsingular (C)
(C) transpose transpose (D) (D) minorminor
02.
02.The equationThe equation
2 2 2 2 1 1 11 1 1 1 1 1 1 00 y y x x xx
represents a parabola passing through the points. represents a parabola passing through the points.
(A) (A) (0, (0, 1), 1), (0, (0, 2), 2), (0, (0, -1) -1) (B) (B) (0, (0, 0), 0), (-1, (-1, 1), 1), (1, (1, 2)2) (C) (C) (1, (1, 1), 1), (0, (0, 0), 0), (2, (2, 2) 2) (D) (D) (1, (1, 2), 2), (2, (2, 1), 1), (0, (0, 0)0) 03. 03. If matrixIf matrix XX 22 a a 11 a a a a 1 1 1 1 aa and and 2 2 X
X X X I I 00 then the inverse of X is then the inverse of X is
(A) (A) 1 1 a 22a 11 a a aa
(B)(B) 22 1 1 a a 11 a a a a 1 a1 a (C) (C) 22 a a 11 a a a a 1 1 a a 11 (D)(D) 2 2 a a a a 1 1 aa 1 1 1 1 aa 04.04.Consider the matricesConsider the matrices X , X 4 4 3 3 4 , Y Y 4 3 3 aanndd PP2 32 3. The order of. The order of
T T 1 1 T T TT P P X X Y Y P P will be will be (A) (A) 2x2 2x2 (B) (B) 3x33x3 (C) (C) 4x3 4x3 (D) (D) 3x43x4 05.
05.The rank of the matrixThe rank of the matrix
1 1 1 1 11 1 1 1 1 00 1 1 1 1 11 is is (A) (A) 0 0 (B) (B) 11 (C) (C) 2 2 (D) (D) 33
06.
06. The matrixThe matrix
2 2 2 2 33 M M 2 2 1 1 66 1 1 2 2 00
has given values -3, -3, 5. An eigen vector corresponding has given values -3, -3, 5. An eigen vector corresponding
to the eigen value 5 is
to the eigen value 5 is
1 21 2
11
TT. One of the eigen vector of the matrix. One of the eigen vector of the matrix MM33 is is(A) (A)
1 1 8 8
11
TT (B)(B)
1 1 2 2 11
TT (C) (C) 33 TT 1 1 2 2 11 (D)(D)
T T 1 1 1 1
11 07.07.The system of equations x + y + z = 6, x + 4y + 6z = 20, x + 4y +The system of equations x + y + z = 6, x + 4y + 6z = 20, x + 4y + zz has no solutions has no solutions
for values of
for values of and and given by given by
(A)
(A) 6 6, , 2200 (B)(B) 6 6, , 2200 (C)
(C) 6 6, , 2200 (D)(D) 6 6, , 2200
08.
08. Let P be 2x2 real orthogonal matrix andLet P be 2x2 real orthogonal matrix and xx is a real vector is a real vector
T T 1
1 22
x
x xx with length with length
1 1 22 2 2 22 1 1 22 xx x x xx . Then which one of the following statement is correct?. Then which one of the following statement is correct?
(A)
(A) px px xx where at least one where at least one vector satisfiesvector satisfies px px xx
(B)
(B) px px xx for all vectors for all vectors xx
(C)
(C) px px xx where at least one vector satisfies where at least one vector satisfies pxpx xx
(D)
(D) No No relationship relationship can can be be established established betweenbetween x and pxx and px 09.
09.The The characteristics characteristics equation equation of of a a 3 3 x x 3 3 matrix matrix P P is is defined defined asas
3 3 22I
I P P 22 11 00
. If I denotes identity matrix then the inverse of P will be. If I denotes identity matrix then the inverse of P will be
(A)
(A) P P P 222 P 2II (B)(B) P P 22 P P II
(C)
(C)
P P 22
P P II
(D)(D)
P P P 222 P 2II
10.
10. A set of linear equations is represented by the matrix equations Ax = b. The necessaryA set of linear equations is represented by the matrix equations Ax = b. The necessary condition for the existence of a solution for the system is
condition for the existence of a solution for the system is (A)
(A) A A must must be be invertibleinvertible (B)
(B) b must b must be linearlbe linearly dey dependent on pendent on the cthe columns of olumns of AA (C)
(C) b must b must be linearly be linearly independent on independent on the columns the columns of Aof A (D) None.
11.
11. Consider a non-homogeneous system of linear equations represents mathematically anConsider a non-homogeneous system of linear equations represents mathematically an over determined system. Such a system will
over determined system. Such a system will bebe (A)
(A) Consistent Consistent having having a a unique unique solution.solution. (B)
(B) Consistent Consistent having having many many solution.solution. (C)
(C) Inconsistent Inconsistent having having a a unique unique solution.solution. (D)
(D) Inconsistent Inconsistent having having no no solution.solution.
12.
12. The trace and determinant of a 2x2 matrix are shown to be -2 and -35 respectively. Its The trace and determinant of a 2x2 matrix are shown to be -2 and -35 respectively. Its eigen values are
eigen values are (A)
(A) -30, -30, -5 -5 (B) (B) -37, -37, -1-1 (C)
