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JEEVAN ENGINEERS ACADEMY Engineering Mathematics
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INDEX
1. Solutions of Equations
2. Matrices
3. Geometry
4. Differentiation
5. Partial Differentiation
6. Integration
7. Vector Calculus
8. Infinite series
9. Fourier Series
10. Differential equations of I order
11. Linear Differential equations
12. Partial Differential equations
13. Complex variables
14. Laplace Transforms
15. Fourier Transforms
16. Z-Transforms
17. Probability & Statistics
18. Numerical methods
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1. SOLUTIONS OF EQUATIONS
* f(x) = a0 xn + a1 x
n-1 +…….. + an-1 x+an (a0≠ 0)
Polynomial in x of degree n
* f (x) = 0 algebraic equation of degree „n‟
* If f (x) has function such as trigonometric, logarithmic exponential etc. then f (x) = 0 is called
transcendental equation
* value of „x‟ which satisfies f (x) = 0 is called root.
* process of finding roots of equation solution of that equation.
Properties
1. if α is a root of equation f(x) = 0 then polynomial f (x) is exactly divisible by (x- α) and conversely.
2. Every equation of nth degree has „n‟ roots (real or imaginary)
3. If f(a) and f(b) have different signs, then the equation f (x) = 0 has atleast one root between x = a & x = b.
4. In an equation with real coefficients, imaginary roots occur in conjugate pairs as α + iβ & α - iβ
Similarly we have a+ √ b & a- √ b as roots
* Every equation of odd degree has atleast one real root.
5. Descarte’s rule
The equation f(x) = 0 cannot have more +ve roots than the changes of signs in f(x) : and more -ve roots than the
changes of signs in f (-x)
* If an equation of nth degree has at the most „p‟ +ve roots & „q‟ –ve roots, then it follows that the equation has atleast
n- (p+q) imaginary roots.
If α1, α2, α3….. αn be the roots of equation
a0xn +a1x
n-1 +a2x
n-2+ ……..+ an-1x+an = 0
then
Quadratic equation
ax2+bx+c=0 has roots
Roots are equal if b
2-4ac = 0
Roots are real & distinct if b2-4ac >0
Roots are img if b2-4ac < 0
Progressions
1. Numbers a, a+d, a+2d….. are said to be in Arithmetic progression (A.P)
nth term Tn = a + (n-1) d
Sum Sn = n/2 (2a+(n-1)d)
2. Numbers a, ar, ar2…….. are said to be in Geometric progression (G.P)
nth term Tn = a rn-1
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Sum
3. Numbers 1/a, 1/a+d, 1/a+2d…… are said to be in Harmonic progression (H.P) reciprocal of
A.P.
nth term
4.If a and b are two numbers, then their
Arithmetic mean =
Geometric mean =
Harmonic mean =
5. Natural numbers are 1, 2, 3, ……. n then
Transformation of equations
1. To find an equation whose roots are m times the roots of the given equation, multiply second
term by „m‟ third term by m2 and so on (all missing terms supplied with zero coefficients).
2. To find an equation whose roots are reciprocal of the roots of the given equation, change x to
1/x.
3. To diminish the roots of an equation f(x) = 0 by h, divide f (x) by (x-h) successively.
* To increase, h taken as - ve (or) divide by (x+h)
Reciprocal Equations
Equation is unaltered on changing x to 1/x
1. A reciprocal equation of an odd degree having coefficients of terms equidistant from beginning
& ending equal, it has root = -1
2. A reciprocal equation of an terms equidistant from the beginning & end equation but oppose in
sign, it has root = 1.
3. A reciprocal equation of even degree having coefficient of terms equidistant from beginning &
end equal but opposite in sign, it has roots = 1 and -1.
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2.Linear Algebra: Determinants,Matrices
* Linear Algebra comprises of theory & application of linear system of equations, linear
transformations and eigen value problems.
* In linear algebra we make a systematic use of matrices and to a lesser extent determinants and
their properties.
Matrix
→ A system of „mn‟ numbers arranged in a rectangular array of „m‟ rows and „n‟ columns is
called a matrix of order m x n.
→ if m = n, it is called a square matrix of order „n‟.
Determinants
→ expression is called det of II order
det = a1 b2 –a2 b1
is called det of III order
→ det of nth order is
a1, b2, c3 ----- ln is called leading or principal diagonal
Minor
Minor of an element in a determinant is the determinant obtained by deleting the row and
the column which intersect in that element.
Eg: if
Co factor
Co factor of any element in a det is its minor with proper sign. The sign of an element in the „ith‟
row & „jth‟ column is (-1)i+j
Cofactor of b3 is B3 = (-1)3+2
x minor of b3
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C2
Laplace’s expansion
A determinant can be expanded in terms of any row (or column) as follows: “Multiply
each element of the row (or column) in terms of which we intend expanding the det, by its
cofactor and then add up all these terms”.
i.e. ∆ = a1 A1 +b1B1 +c1C1
= a1 (b2 c3- b3 c2)- b1(a2 c3-a3 c2) + c1 (a2 b3-a3b2)
ai Aj+bi Bj+ci cj = ∆ when i = j
= 0 when i ≠ j
Properties of Determinants
1. A determinant remains unaltered by changing its rows into columns and columns into rows
2. If two parallel lines of a det are interchanged, the det retains its numerical value but changes in
sign.
* if any line of a det be passed over „m‟ parallel lines resulting det ∆1 = (-1)
m ∆.
3. A determinant vanishes if two parallel lines are identical.
4. If each element of a line be multiplied by the same factor, the whole determinant is multiplied
by that factor.
5. If each element of a line consists of m terms the det can be expressed as sum of m dets.
6. If to each elements of a line be added equi multiples of the corresponding elements of one or
more parallel lines, the determinants remains unaltered.
7. If the elements of a determinant ∆ are functions of x and two parallel lines become identical
when x = a then (x-a) is a factor of ∆.
Multiplication of determinants
The product of two determinants of the same order is itself a determinant of that order.
then
Special Matrices
Row and Column matrices
* A matrix having a single row is called row matrix
[ 1 3 5 7]
* A matrix having a single column is called column matrix
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Square matrix
* A matrix having „n‟ rows & „n‟ columns is called a square matrix of order „n‟
leading or principal diagonal = 1, 3, 5
sum of diagonal elements is called trace of „A‟.
if |A| = 0 then matrix is said to be singular otherwise non singular.
Diagonal matrix
A square matrix all of whose elements except those in leading diagonal are zero is called
diagonal matrix.
A diagonal matrix whose all the leading diagonal elements are equal is called a scalar matrix.
Ex:
Diagonal Scalar
Unit matrix
A diagonal matrix of order „n‟ which has unity for all its diagonal elements is called a
unit matrix or an identity matrix of order „n‟ & denoted by „In‟.
Eg:
Null Matrix
If all the elements of a matrix are zero, it is called 0 null or zero matrix & is denoted by „0‟.
Symmetric
A square matrix A = [aii] is said to be symmetric when aij = aji i & j
Skew symmetric
If aij = - aji i & j so that all the leading diagonal elements are zero
Triangular Matrix
A square matrix all of whose elements below the leading diagonal are zero is called upper
triangular matrix.
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A square matrix all of whose elements above the leading diagonal are zero, is called a lower
triangular matrix.
Matrix Operations
1. Equality of matrices
Two matrices A and B are said to equal if and only if
(i) they are of same order and
(ii) each element of A is equal to corresponding element of B.
2. Addition and Subtraction of matrices
If A, B be two matrices of the same order, then their sum A+B is defined as the matrix each
element of which is the sum of the corresponding elements of A and B
Similarly A-B is defined as a matrix whose elements are obtained by subtracting the elements
of B from the corresponding element of A.
* Only matrices of same order can be added or subtracted
* Addition of matrices is commutative
i.e. A+B = B+A.
* Addition & subtraction of matrices is associative
i.e. (A+B) –C = A+ (B-C) = B +(A-C)
3. Multiplication of matrix by a scalar
The product of a matrix A by a scalar k is a matrix whose each element is „K‟ times
corresponding elements of A.
* Distribution law holds for such products
K(A+B) = KA + KB
* If AB = 0 it does not necessarily imply that A or B is a null matrix
4. Multiplication of matrices
Two matrices can be multiplied only when the number of column in the first is equal to
the no. of rows in the second.
* such matrices are said to be comformable
* Multiplication of matrices is associative
(AB) C = A (BC)
* Multiplication of matrices is distributive
A(B+C) = AB+AC.
* If A be a square matrix then product AA is defined as A2.
* If A2 = A then the matrix A is called idempotent
* If A2 = 0, then the matrix A is called Nilpotent.
* If A2= I, then the matrix A is called involuntary
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5. Transpose of a matrix
The matrix obtained from any given matrix A by interchanging rows and columns
denoted by A1 or A
T.
* transpose of m x n matrix is n x m matrix
* transpose of the transpose of a matrix coincides with itself i.e. (A1)
1 = A
* For a symmetric matrix A1 = A
* For a skew symmetric matrix A1 = A
* (AB)1 = B
1A
1
* every square matrix can be uniquely expressed as a sum of symmetric and skew symmetric
matrix.
6. Adjoint of a square matrix
1. Adjoint of A is the transposed matrix of cofactors of A.
7. Inverse of a matrix
If A be any matrix then a matrix B if it exists, such that AB=BA= I, is called the inverse of A.
i.e.
* both matrix and its inverse must be non singular
* Inverse of a matrix is unique
* (AB)-1
= B-1
A-1
Rank of a Matrix
A matrix is said to be of rank „r‟ when
(i) It has at least one non-zero minor of order r, and
(ii) Every minor of order higher than „r‟ vanishes
i.e.
Rank of a matrix is the largest order of any non-vanishing minor of the matrix.
If a matrix has a non-zero minor of order r, its rank is ≥ r.
If all minors of a matrix of order (r+1) are zero, its is rank is ≤ r.
Rank is denoted by ρ (A)
Elementary transformation of matrix
1. The interchange of any two rows (columns) [Rij] /[Cij]
2. The multiplication of any row (column) by a non-zero number [KRi]/[K ci]
3. The addition of a constant multiple of the elements of any row (column) to the corresponding
elements of any other row (column) [Ri+PRj]/[Ci +Pci]
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* Elementary transformation do not change either the order or rank of a matrix.
* The value of minors may get changed by the transformation (1) & (2), their zero or non zero
character remains unaffected.
Equivalent matrix
Two matrices A and B are said to be equivalent if one can be obtained from the other by a
sequence of elementary transformations.
Two equivalent matrices have
same rank.
Gauss Jordan Method of finding inverse
“These elementary row transformations which reduce a given square matrix. A to the unit matrix
when applied to unit matrix I gives the inverse of A”.
Working
Write two matrices A&I side by side. Then perform the same row transformation on both.
As soon as A is reduced to I, the other matrix represents A-1
.
Normal form
Every non zero matrix A of rank „r‟ can be reduced by a sequence of elementary transformations
to the form
is called normal form.
* Rank of matrix A is r if and only if it can be reduced to normal form.
* Any quantity having n-components is called a vector of order‟n‟
Linear dependence
The vector are said to be linearly dependent, if these exists r numbers λ1, λ2--- λr not
all zero such that
λ1 + λ2 +… + λr = 0
* If no such numbers other than zero exists, the vectors are said to be linearly independent
Consistency of Linear System of Equations
Find the ranks of the coefficient matrix [A] and the augmented matrix [AK] by reducing A to the
triangular form by elementary row operations. Let the rank of [A] be „r‟ and that of [AK] be „r‟
[AX] =K
(i) If r ± r‟ the equation are in consistent, i.e. no solution.
(ii) If r = r‟ = n, the equations are consistent and there in unique solution
(iii) If r= r‟ < n, the equations are consistent and there are infinite number of solutions.
System of linear homogeneous equations
Consider the homogeneous linear equations [ AX] = 0
a11 x1+ a12 x2 +….. +a1nxn = 0
a21 x1 +a22 x2+ … +a2nxn = 0
…………………………….
am1x1 +am2 x2 +…..+amn xn = 0
let rank „r‟ of the coefficient matrix A by reducing it to the triangular form then
1. If r = n, the equations have only a trivial zero solution
x1 = x2 ----- xn = 0
* If r < n the eqns have (n-r) linearly independent solutions.
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2. When m < n the solution is always other than x1 = x2 = ----- xn = 0, The no. of solutions is
infinite.
3. When m = n, the necessary & sufficient condition for solutions other than x1 = x2 = ----- xn = 0
is that the determinant of the coefficient matrix is zero.
* Equations are consistent and a solution is non-trivial solution.
Orthogonal transformation
The linear transformation y = Ax is said to be orthogonal if it transforms
Y12 +y
22 + …… +yn
2 into x
21 +x
22 +…..+xn
2
The matrix of an orthogonal transformation is called an orthogonal matrix.
* Square matrix A is said to be orthogonal if AAT =A
T A = I
* If A is orthogonal, AT and A
-1 are also orthogonal
* If A is orthogonal then |A| = ± 1
Characteristic equation
is called
the characteristics equation.
* The roots of the equation
(-1)n λ
n +K1 λ
n-1 + K2 λ
n-2 + ……Kn = 0 are called the characteristic roots or latent roots or
eigen values.
Eigen vectors
If then the
Linear transformation y = Ax carries the column
Vector X into column vector y by means of square matrix A.
X= [ x1 x2….. xn]1 is known eigen vector or latent vector.
Properties of Eigen values
1. Any square matrix A and its transpose AT and its transpose A
T have same eigen values.
2. Eigen values of a diagonal matrix are just the diagonal elements of the matrix.
3. The eigen value of an idempotent matrix are either zero or unity.
4. The sum of the eigen value of matrix is the sum of the elements of the principal diagonal.
5. The product of eigen value of a matrix A is equal to its determinant.
6. If λ is an eigen value of a matrix A, then 1/ λ is the eigen value of A-1
.
7. If λ is an eigen value of an orthogonal matrix, then 1/ λ is also its eigen value.
8. If λ1, λ2 ….. λn are eigen value of matrix A, then Am has the eigen values λ1
m, λ2
m……. λ
mn
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Cayley- Hamilton Theorem
Every square matrix satisfies its own characteristic equation.
Reduction to Diagonal form
If a square matrix A of order n has „n‟ linearly independent eigen vectors, then a matrix P can be
found such that P-1
AP is a diagonal matrix.
* The matrix P which diagonalises A is called the modal matrix of A.
* Resulting diagonal matrix D is known as Spectral Matrix of A.
* The diagonal matrix has the eigen values of A as its diagonal elements.
* The matrix P, which diagonalise A, constitutes the eigen vectors of A
* A square matrix of order n is called similar to a square matrix A of order n if
= p-1
AP.
Reduction of Quadratic form to Canonical form
* A homogeneous expression of the second degree in any no. of variables is called a quadratic
form.
Procedure
Let λ1, λ2, λ3 be eigen values of matrix A and
Then where
Here the quadratic form is reduced to a canonical for (or sum of squares form or principal axes
form)
i.e. λ1 x2 + λ2 y
2 + λ3 z
2
* The number of positive terms in canonical form of a quadratic form is known as Index of the
form.
* Signature of the quadratic form is the difference of +ve terms and –ve terms in its canonical
form.
Nature of a quadratic form
A real quadratic form x1Ax in „n‟ variables is said to be
i. Positive definite if all the eigen values of A > 0
ii. negative definite if all the eigen values of A < 0
iii. Positive semidefinite if all the eigen values of A≥0 and atleast one eigen value = 0.
iv. negative semidefinite if all the eigen values of A ≤0 and atleast one eigen value = 0
v, indefinite if some of the eigen values of A are +ve and others – Ve
Complex Matrices
1. Conjugate of a matrix
If the elements of a matrix A= [ars] are complex numbers αrs + iβrs & αrs &βrs are real then
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is called conjugate matrix
2. Hermitian matrix
A square matrix A such that AT= is said to be Hermitian matrix
3. Skew Hermitian Matrix
A square matrix A such that AT = is said to be a skew Hermitian matrix.
Properties
1. Any square matrix A can be written as the sum of a Hermitian and skew Hermitian matrices.
2. If A is a Hermitian matrix then (iA) is a skew Hermitian matrix.
3. The eigen values of a Hermitian matrix are real
4. The eigen value of a skew Hermitian matrix are purely imaginary or zero
4. Unitary matrix
A square matrix U such that is called a unitary matrix.
Properties
1. Inverse of a unitary matrix is unitary.
* Inverse of an orthogonal matrix is orthogonal
2. Transpose of a unitary matrix is unitary
* Transpose of an orthogonal matrix is orthogonal
3. Product of two unitary matrices is a unitary matrix
* product of two orthogonal matrices is an orthogonal
4. The eigen value of a unitary matrix has absolute value I.
* The eigen value of a orthogonal matrix has absolute value I.
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3. GEOMETRY
Co-ordinate of a point
y
0
P
y
x
r
x
Cartesian (x, y) polar (r, θ)
x = r cos θ, y = r sin θ
r = √ (x2+y
2), θ = tan
-1 (y/x)
* Distance between two points
(x1y1) & (x2,y2)=
* Point of division of the line joining (x1,y1) & (x2,y2) in the ratio m1:m2 is
Triangle
A (x1y1) B (x2y2), C (x3, y3)
* Area
* Centroid (point of intersection of medians) is
* Incentre (point of intersection of internal bisectors of angle)
a,b,c, are length of sides of triangle
Circumcentre is the pt of intersection of right bisectors of sides of triangle.
Orthocentre is the pt of intersection of perpendicular drawn from vertices to the opposite sides
of triangle.
Straight line
Slope of line joining points (x1y1) & (x2 y2) =
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* Slope of the line ax+by+c = 0 is –a/b i.e. -
Equations of a line
o Having slope m and cutting an intercept „C‟ on y-axis is y = mx+C.
o Cutting intercepts a & b from the axes is
o Passing through (x1y1) & having slope m is y-y1= m (x-x1)
o Passing through (x1y1) & making an <θ with x-axis is
o Through the point of intersection of lines ax+b1y+c1=0 & a2x+b2y+c2= 0 is
a1x+b1y+c1+k(a2x+b2y+c1+K(a2x+b2y+c2) = 0
Angle between two lines having slopes m1 & m2 is
o Two lines are parallel if m1 = m0
o Two lines are perpendicular if m1m2 = 1
o Any line parallel to line ax+by+c= 0 is ax+by+k = 0
o Any line perpendicular to line ax+by+ c= 0 is bx-ay+ k = 0
Length of the perpendicular from (x1y1) to line ax+by+c=0 is
Circle
Equation of the circle having centre (h, k) & radius r is (x-h)2 + (y-k)
2 = r
2
Equation x2+y
2+2yx+2fy+c = 0 represents a circle having centre (-g1 f) and radius = √ (g
2+f
2-
c)
Equation of the tangent at the point (x1y1) to the circle x2+y
2=a
2 is xx1+yy1 = a
2
Condition for the line y = mx+c to touch the circle x2+y
2= a
2 is c= √(1+m
2)
Length of tangent from the point (x1y1) to the circle x2+y
2+2gx+2fy+c = 0
is
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Parabola
Standard equation of parabola is y2= 4 ax
o Parametric equations are x = at2, y = 2 at
o Latus rectum L L‟ = 4 a Focus S (a, 0)
o Directrix ZM is x+a = 0
Focal distance of any point P (x1, y1) on parabola y2= 4ax is Sp= x1+a
Equation of tangent at (x1y1) to parabola y2 = 4ax is yy1 = 2a (x+x1)
Condition for the line y= mx+c to touch parabola y2=4ax is c = a/m
Equation of normal to parabola y2= 4ax in terms of its slope m is y =mx-2am-am
3.
Ellipse
Standard equation is
o Parametric equations are x = a cos θ, y = b sin θ
o Eccentricity =
o Latus rectum LSL1 = 2b
2/a, Foci S (-ae, 0) & S
1 (a e, 0)
o Directrices ZM (x = a/e) & Z1M
1 (x = a/e)
Sum of focal distances of any point on the ellipse is equal to major axis i.e.
SP+S1P = 2a
Equation of the tangent at the point (x1 y1) to the ellipse
is
Condition for the line y = mx+c to touch ellipse
is
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Hyperbola
↑
→M M
y
x
L
L
SC
Z Z
1
11
Standard equation is
o Parametric equations are
x= a sec θ, y = b tan θ
Eccentricity e =
Latus rectum
Directrices ZM (x= a/e) & z1m
1(x= -a/e)
Equation of tangent at the point (x1,y1) to hyperbola
is
Condition for the line y = mx+c to touch hyperbola
is
Asymptotes of hyperbola are &
Equation of rectangular hyperbola with asymptotes as axes is xy= c2
Its parametric equations are x = ct, y = c/t
Nature of a Conic
The equation ax2+2hxy+by
2+2gx+2fy+c = 0 represents
A pair of lines, if
A circle, if a=b, h =0, ∆≠ 0
A parabola, if ab-h2 = 0,∆≠ 0
An ellipse, if ab-h2> 0, ∆≠ 0
A hyperbola, if ab-h2 < 0, ∆≠ 0
A rectangular hyperbola if
ab-h2< 0, ∆≠ 0, a+b=0.
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Volumes & Surface areas
Solid volume Curved surface area Total surface area
1. Cube (side „a‟) a3 4a
2 6a
2
2. Cuboid (length l, breadth
b, height h) lbh 2(l+b) h 2(lb+bh+hl)
3. Sphere (radius r) (4/3) πr 3 - 4πr
2
4. Cylinder
(base radius r, height h π r2h 2πrh 2πr(r+h)
5. Cone (base radius r, height
h) [ l= √ r2+h
2] (1/3) πr
2h πrl π r (r+l)
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CALCULUS
[Differentiation, Integration]
4. Differentiation
Some standard Derivatives
,
,
,
,
,
,
,
,
,
,
,
D
n (a
mx)= m
n (loga)
n . a m
x
Dn (e
mx) = m
n e
mx
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Dn Sin (ax+b) = a
n Sin (ax+b+nπ/2)
Dn Cos (ax+b) = a
n Cos (ax+b+ nπ/2)
Dn [e
ax Sin (bx+c)] = (a
2+b
2)
n/2 e
ax sin (bx+c+ntan
-1(b/a)
Dn [ e
ax Cos (bx+c)] = (a
2+b
2)
n/2 e
ax cos (bx+c+ntan
-1 (b/a))
Leibnitz’s theorem
If u, v be two functions of x possessing derivatives of the nth order then
(uv)n = unV +nc1 un-1 V1 + nc2 un-2 v2 +…….+ncnuVn
Rolle’s theorem
If (i) f(x) is continuous in [a,b]
(ii) f(x) exists for every value of c of x in (a,b) such that f(c) = 0
(iii) f(a) = f(b)
then there exists at least one value C of x in (a, b) such that f1 (c) = 0
Lagrange’s Mean-value Theorem
If (i) f(x) is continuous in [a,b] and
(ii) f1 (x) exists in (a, b) then there is atleast one value c of x in (a, b) such that
Cauchy’s Mean-value theorem
If (i) f(x) and G(x) be continuous in [a,b]
(ii) f1(x) and g
1(x) exist in (a,b)
(iii) g1(x) ≠ o for any value of x in (a,b) then
There is atleast one value c of x in (a,b) such that
Taylor’s Theorem
If (i) f(x) and its first (n-1) derivatives be continuous in [a, a+h] and
(ii) fn (x) exists for every value of x in (a, a+b)
Then there is atleast one number θ (0 < θ<1), such that
f(a+h) = f (a) + h f1(a) + h
2/2 f
11 (a)+…. + h
n/n! f
n (a+ θh).
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Maclaurin’s series
If f (x) can be expanded as an infinite series, then
f (x)= f(0) + xf1(0) +
Indeterminate forms
To evaluate Lt [ (f(x)/φ(x)] in 0/0 form, differentiate the numerator & denominator separately as
many times as would be necessary to arrive a determinate form.
Form ∞/∞
Applying L‟ Hospital‟s rule.
i.e. Differentiating numerator & Denominator separately as many times as would be necessary.
Increasing & Decreasing Functions
In the function y = f(x), if y increases as x increase it is called an increasing function of x.
In the function y = f(x), if y decreases as x increases it is called a decreasing function of x.
Maxima & Minima
Procedure to find maxima & minima
1. Put the given function = f(x)
2. Find f1(x) and equate it to zero. Solve this equation and let its roots be a, b, c….
3. Find f11
(x) and substitute in it by turns x= a, b, c
* If f11
(a) is –ve, f (x) is maximum at x = a
* If f11
(a) is +ve, f(x) is minimum at x = a.
4. Sometime f11
(x) may be difficult to find out or f11
(x) may be zero at x= a. Then
* If f‟(x) changes sign from +ve to –ve as x passes through a,f(x) is minimum at x = a.
* If f1 (x) changes sign from – ve to +ve as x passes through a,f(x) is maximum at x = a.
* If f1 (x) does not change sign while passing through x = a, f(x) is neither maximum nor
minimum at x =a.
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5. PARTIAL DIFFERENTIATION
If z= f(x, y) a function of two variables x & y then & partial derivative of z with respect to x
is denoted by
(or) (or) fx (x,y)
or Dxf and is given by
* Partial derivating of x with respect to y is given by
* Sometimes we may use the notation
Any function f (x, y) which can be expressed in the form xn φ (y/x) is called a homogenous
function of degree n in x and y
Euler‟s theorem
If u be a homogeneous function of degree n in x and y then.
If u f(x, y) where x = φ(f) & y = ψ (t) then total derivative is given by
If t = x
If f (x, y) = c be an implicit relation between x and y then
Taylor‟s expansion of f (x, y) in powers of (x-a) & (y-b) is given by
f(x, y)= f (a, b) + [(x-a) fx (a, b)+ (y-b) fy (a, b)]
+ ½! [ (x-a)2 fxx (a, b) +2(x-a) (y-b) fxy (a, b)+ (y-b)
2 fyy (a, b) ]+…..
Maclaurin‟s expansion of f (x, y) is given by f (x, y) = f (0, 0)+ [ x fx (0, 0)+ y fy (0, 0)]+ ….
f (a, b) is said to be stationary value of f(x, y) if fx (a, b) = 0 and fy (a, b) = 0 i.e., the function is stationary at
(a, b) but converse is not true.
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Procedure to find maxima & minimum values of f (x, y)
1. Find and equate each to zero. Solve these as simultaneous equation in x & y.
Let (a, b) (c, d)… be pair of values.
2. Calculate the value of
for
each pair of values
3. (i) If rt-s2 > 0 & r< 0 at (a, b), f (a, b) is a max value.
(ii) If rt-s2> 0 & r> 0 at (a,b), f (a, b) is a min value.
(iii) If rt-s2 < 0 at (a, b) f (a, b) is not an extreme value. i.e. (a, b) is a saddle point.
(iv) If rt-s2= 0 at (a, b) the case is doubtful & needs further investigation.
Larange‟s method
1. Write F= f (x, y, z) + λ φ (x, y, z)
2. Obtain the equations
3. Solve the above equations together with φ (x, y, z) = 0. The values of x, y, z so obtained will give the stationary
value of f (x, y, z)
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6. INTEGRATION
Some Standard Integration Formulae
Sinx dx = - Cos x, cosx dx = sin x
tanx dx = - log (cos x) cotx dx = log (sinx)
sec x dx = log (sec x+tan x), cosec xdx= log (cosecx-cot x)
,
,
,
, cos hx dx = sin hx
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tan hx dx = log (cos hx) cot hx dx = log (sin hx)
sech2xdx = tan hx , cosech
2x dx = - cot hx.
Reduction Formulae
( π/2, only if
both m & n are even)
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Definite Integrals
Properties
1.
2.
3.
4.
5. if f (x) is an even function
= 0 if f (x) is an odd function
6.
= 0 if f (2a-x) = - f (x)
7. f(x) g(x) dx = f (x) g(x) dx - f1(x) [ g(x) dx] dx.
Important Integrals
if n is even
= 0 if n is odd.
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Improper Integrals
is said to be an improper integral of first kind if a = - ∞ or b = ∞ or both.
* is said to be an improper integral of second kind if f (x) is infinite for one or
more values of x in [a, b]
* is said to be convergent if the values of integral is finite.
7. VECTOR CALCULUS
A quantity which is completely specified by its magnitude & direction is called vector.
P
A
Q
A vector of unit magnitude is called a unit vector ( )
A vector of zero magnitude (which have no direction) is called a zero vector (0).
The vector represents the negative of i.e -
Two vectors having the same magnitude & the same (or parallel) directions are said to
be equal ( ).
Addition of vectors
Subtraction of vectors
Any three position vectors are collinear, if ,where
In vector algebra, the division of a vector by another vector is not defined.
Scalar or Dot Product
The scalar or dot product of two vectors is defined as the scalar ab cos θ, where θ is
the angle b/n
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Properties
1. Scalar product of two vectors is commutative
2. The necessary & sufficient condition for two vectors to be perpendicular is that their scalar
product should be zero.
3. The square of a vector is a scalar which stands for the square of its magnitude.
4. For the mutually perpendicular unit vectors
5. Scalar product of two vectors is distributive
6. Schwarz inequality
7. Scalar product of two vectors is equal to the sum of the products of their corresponding
components.
8. Angle between two lines whose direction cosines are l, m, n & , , is
9. Angle between two lines whose direction ratios are a, b, c & , , is
10. Projection of the line joining two points (x1, y1, z1) & (x2, y2, z2) on a line whose direction
cosines are l, m, n is
l (x2- x1) + m (y2 – y1)+ n (z2 – z1)
Vector or Cross Product
The vector or cross product of two vectors
is defined as a vector such that
(i) its magnitude is ab sin θ, θ being angle between
(ii) its direction is perpendicular to the plane of
(iii) it forms with a right handed system.
Properties
1. Vector product of two vectors is not commutative
2. The necessary and sufficient condition for two non-zero vectors to be parallel is that their
vector product should be zero.
3. For the orthonormal vector triad
4. Relation b/n Scalar & vector products is
5. Vector product of two vectors is distributive
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6. If then
Scalar Product of three vectors
If be any three vectors then the scalar or dot product with is called the scalar
product of three vectors and is written as
Properties
1. The condition for three vectors to be coplanar is that their scalar triple product should vanish.
i.e.
2. If any two vectors of a scalar triple product are equal the product vanishes i.e.
3. Every scalar triple product
(i) is independent of the position of the dot or cross and
(ii) depends upon the cyclic order of vectors
4. Scalar triple product is distributive
i.e.
5. If ,
then
Vector Product of three vectors
If be any three vectors, then the vector or cross product of with is called
the vector product of these vectors and is written as
*
*
If be any four vectors then
*
*
Vector differential operator Del, written as is defined by
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The vector function f is defined as the gradient of the scalar point function f and is written
as grad f.
* grad f =
The divergence of a continuously differentiable vector point function is denoted by div .
*
The curl of a continuously differentiable vector point function is given by
is solenoidal.
is called Laplace‟s equation. is Laplacian operator
is called irrotational.
Green’s theorem
If φ (x, y) , ψ (x, y) φy & ψx be continuous in a region E of the xy-plane bounded by a
closed curve C, then
Stoke’s theorem
If S be an open surface bounded by a closed curve C and be any
continuously differentiable vector- point function, then
Where is a unit external normal at any point
of s.
Gauss divergence theorem
If is a continuously differentiable vector function in the region E bounded by the closed
surface S, then
Where is the unit external normal vector.
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8. INFINITE SERIES
An ordered set of real numbers a1, a2, a3…..an is called a sequence and is denoted by (an).
If the number of terms is unlimited, then the sequence is said to be an infinite sequence
Limit
A sequence is said to tend to a limit l, if for every ε > 0, a value N of n can be found such
that for n ≥ N.
Convergence
If a sequence (an) has a finite limit, it is called a convergent sequence.
* If (an) is not convergent it is said to be divergent.
Bounded Sequence
A sequence (an ) is said to be bounded, if there exists a number K such that an < k for every
n.
Monotonic sequence
The sequence (an) is said to increase steadily or to decrease steadily according as an+1 ≥ an or
an+1 ≤ an for all values of n. Both increasing and decreasing sequences are called monotonic
sequences.
A monotonic sequence always tends to a limit, finite or infinite.
A sequence which is monotonic and bounded is convergent
If u1, u2, u3… un…. be an infinite sequence of real numbers, then u1+u2+u3+…. +un+…….∞ &
is called an infinite series.
* Denoted by ∑un & sum its first n terms by Sn
If ∑un = u1+u2+u3 + …. +un + ……∞ &
Sn = u1+u2+u3+ …. +un then
1. If „Sn‟ tends to a finite limit as n → ∞ , the series ∑un is said to be convergent.
2. If „Sn‟ tends to ± ∞ as n→∞, the series ∑un is said to be divergent
3. If „Sn‟ does not tend to a unique limit as n → ∞ then the series ∑un is said to be oscillatory or
non-convergent
Properties of series
1. The convergence or divergence of an infinite series remains unaffected by the addition or
removal of a finite number of its terms.
2. If a series in which all the terms are positive is convergent, the series remains convergent even
when some or all of its terms are negative.
3. The convergence or divergence of an infinite series remains unaffected by multiplying each
term by a finite number.
An infinite series in which all the terms after some particular term are positive term series.
* A series of positive terms either converges or diverges to +∞.
* If , the series ∑un is convergent.
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* If series ∑un is divergent
Tests for convergence of a series
Comparison tests
1. If two positive term series ∑un & ∑un be such that
a. ∑un converges.
b. un ≤ vn for all values of n, then ∑un also converges.
2. If two positive term series ∑un & ∑vn be such that
a. ∑vn diverges
b. un ≥ vn for all values of n, then ∑un also diverges.
3. Limit form
If two positive term series ∑un & ∑vn be such that = finite quantity (≠ 0) then
∑un & ∑vn converge or diverge together.
Integral test
A positive term series f(1)+ f(2) + ……..+ f(n) + ….. where f (n) decreases as n increases,
converges or diverges according as the integral,
is finite or in finite
Comparison of ratios
If ∑un & ∑vn be two positive term series, the ∑un converges if (i) ∑vn converges and (ii) from
and after some particular term
D‟Alembert‟s ratio test
In a positive term series ∑un, if , then the series converges for λ < 1
And divergence for λ > 1
Cauchy‟s root test
In a positive series ∑un, if , then the series converges for λ < 1
And divergence for λ > 1
A series in which the terms are alternately +ve or – ve is called alternating series.
An alternating series u1 – u2 +u3- u4 +…..converges if
(i) Each term is numerically less than its preceding term and
(ii)
* If the given series is oscillatory
If the series of arbitrary terms u1+u2+u3+…… un+….. be such that the series |u1|+ |u2|+ |u3| +
…..+ |un| + …. is convergent, then the series ∑un is said to be absolutely convergent.
* If ∑|un| is divergent but ∑un is convergent, then ∑un is said to be conditionally convergent.
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9. FOURIER SERIES
The fourier series for the function f (x) in the interval α < x< α+ 2π is given by
Where
f(x) cos nx dx
f (x) sin nx dx
a0, an, bn values are known as Euler‟s formulae
Dirichlet’s conditions
Any function f (x) can be developed as a Fourier series
where a0,an,bn constants provided:
1. f(x) is periodic, single valued and finite
2. f(x) has a finite number of discontinuities in any one period.
3. f (x) has at the most a finite number of maxima and minima.
A function f (x) is said to be even if f(-x) = f(x)
A function f (x) is said to be odd if f(-x) = - f(x)
Fourier series for the even function f(x) in the interval (-c, c) is
where
bn = 0
* If a periodic function f (x) is even, its Fourier expansion contains only cosine terms.
Fourier series for the odd function f (x) in the interval (-c, c) is
Where a0 = 0, an = 0
If a periodic function f (x) is odd, its Fourier expansion contains only sine terms.
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10. DIFFERENTIAL EQUATIONS OF FIRST ORDER
A differential equation is an equation which involves differential coefficients or differentials.
An ordinary differential equation is that in which all the differential coefficients have reference
to a single independent variable.
A partial differential equation is that in which there are two or more independent variables and
partial differential coefficients with respect to any of them.
The order of a differential equation is the order of the highest derivative appearing in it.
The degree of a differential equation is the degree of the highest derivative occurring in it, after
the equation has been expressed in a form free from radicals and fractions as far as the derivatives
are concerned.
A solution (or integral) of a differential equation is a relation between the variables which
satisfies the given differential equation.
The general (or complete) solution of a differential equation is that in which the number of
arbitrary constants is equal to the order of the differential equation.
A particular solution is that which can be obtained from the general solution by giving
particular values to the arbitrary constants.
Solutions of I order and first degree differential equations
1. Variables- Separable Method:
If in an equation it is possible to collect all functions of x and dx on one side and all the functions
of y and dy on the other side, then the variables are said to be separable. Thus the general form of
such an equation is f (y) dy = φ (x) dx.
Integrating both sides, we get ∫ f (y) dy = ∫φ (x) dx+c as its solution.
2. Homogeneous Equations
Are of the form
Where f (x, y) and φ (x, y) are homogenous functions of the same degree in x and y.
To solve a homogenous equation (i) put y = vx then
(iii) Separate the variables v and x and integrate
3. Equations reducible to homogenous form
The equations of the form
Can be reduced to the homogenous form as follows:
Case (i) when
Putting x = X+h, y = Y+k (h, k being constants)
Where
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solved by putting Y = Ux.
Case (ii) when
Let
Put ax+by= t and solve by variable separable method
4. Linear equations
A differential equation is said to be linear if the dependent variable and its differential
coefficients occur only in the first order degree and not multiplied together.
The standard form of a linear equation of the first order commonly known as Leibnitz‟s linear
equation is
where P, Q are functions of x,
Solution is
Integrating Factor (IF) =
Solution is y (I.F) =
5. Bernoulli‟s Equations
The equation where P, Q are functions of x, is reducible to the Leibniqz‟s
linear equation and is usually called the Bernoulli‟s equation.
To solve, divide both side by yn so that
Which is Leibnitz‟s equation in z & can be
solved easily.
6. Exact Differential equations
A differential equation of the form M (x, y) dx+ N (x, y) dy = 0 is said to be exact if its left
hand member is the exact differential of some function u (x, y) i.e. du = Mdx+Ndy = 0
Solution is u (x, y) = C
The necessary and sufficient condition for the differential equation Mdx+Ndy = 0 is
The solution of Mdx+Ndy = 0 is
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(terms of N not containing x) dy = C
(y constant)
7. Equations reducible to exact equations
Sometimes a differential equation which is not exact, can be made so on multiplication by a
suitable factor called an integrating factor (I.F)
(i) I.F. Found by inspection:-
The I.F. can be found after regrouping the terms of the equation and recognizing each
group as being a part of an exact differential
xdy+ydx = d (xy)
(ii) I.F. of a homogenous Equation
If Mdx+Ndy = 0 be a homogeneous equation in x & y,
Then is an I.F. (Mx+Ny≠ 0)
(iii) I.F. for an equation of type f1 (x y) ydx + f2 (xy ) xdy = 0
If the equation Mdx+ Ndy = 0 be of this form, then is an I.F. (Mx-Ny ≠ 0)
iv. In the equation Mdx+Ndy = 0
a. If be a function of x only = f (x) say, then is an integrating factor.
b. If be a function of y only = F (y) say, then is an integrating
factor.
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11. LINEAR DIFFERENTIAL EQUATIONS
Linear Differential equations are those in which the dependent variable and its derivatives
occur only in the first degree and are not multiplied together
General form
Where P1 P2….. Pn and x are functions of x only.
Linear differential equations with constant coefficients are of the form
Where K1, K2…… Kn are constant
Procedure to solve the equation
Of which the symbolic form is
(Dn+ K1D
n-1 +……+ Kn-1 D+Kn) y = x
Step – I to find the complementary function (CF)
(i) Write the Auxiliary equation
i.e. Dn+ K1 D
n-1 +……+ Kn-1 D+ Kn = 0 and
solve it for D.
(ii) Write the C.F. as follows:
Roots of A.E. C.F
1. m1, m2, m3 … (real & different roots) c1em1x
+ c2 em2x
+ c3 em3x
+ ….
2. m1, m1, m3….(two real & equal roots) (c1+c2 x) em1x
+ C3 em3x
+….
3. m1, m1, m1, m4… (three real & equal roots) (c1+c2x+c3x2)e
m1x +c4 em
4x +….
4. α+iβ, α- iβ, m3 … (a pair of imaginary roots) eax
(c1cosβx+c2 sinβx) +c3 em3x
+…
5. α±β, α±β, m5 … (2 pairs of equal imaginary roots) eax
[(c1+c2x)cos βx+(c3+c4x)sin βx] +c5
em5x
+ ….
Step – II To find the particular Integral (P.I)
From Symbolic from
(i) When X = eax
Put D = a [ f(a)≠ 0]
Put D = a [f (a) = 0; f1(a) ≠ 0]
Put D= a [ f1 (a) = 0, f
11 (a) ≠ 0]
And so on.
(ii) when x = sin (ax+b) or cos (ax+b)
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Sin (ax+b) [or cos (ax+b)], Put D2 = -a
2 [ φ (-a
2) ≠ 0]
Sin (ax+b) [ or cos (ax+b)] Put D2 = -a
2
[φ (-a2)= 0, φ
1 (-a
2) ≠ 0]
And so on.
(iii) When X = xm, m being a +ve integer
P.I.
(iv) When X = eax
V, where V is a function of x
P.I.
(V) when x is any function of x
P.I.
* Resolve into partial fractions & operate each partial fraction on x remembering that
Step III To find complete solution (C.S.)
C.S. is y = C.F. + P.I.
Method of Variation of Parameters
This method is quite general & applies to equations of the form
y11
+Py1+qy = X
where P, q & X are functions of x.
It gives P.I. =
Where y1 and y2 are the solutions of y
11+Py
1+qy = 0 &
is called Wronskian of y1, y2.
Cauchy‟s homogenous Linear equation
Where X is function of x.
Such equations can be reduced to linear differential equations with constant coefficient by
putting
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X = et or t = log x, then D= d/dt
Legendre‟s Linear equation
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12. PARTIAL DIFFERENTIAL EQUATIONS
x and y are independent variable and z is dependent variable
Z = f (x, y) then
A linear partial differential equation of the first order, commonly known as Lagrange‟s linear
equation is of form
Pp+Qq = R
When P, Q, R are functions of x,y, z. This equation is called quasi-linear equation.
To solve equation PP +Qq = R
(i) form the subsidiary equations
(ii) solve these simultaneous equations, giving u = a & v= b as its solutions.
(iii) write the complete solution as φ (u, v) = 0 or u = f (v)
Procedure to solve the equation
Its symbolic form is (D
n+ K1 D
n-1 D
1+ …,+Kn D
n) z = F (x, y)
Or f (D, D1) z = F (x, y)
Step- I to find C.F.
(i) Write the A.E.
mn +K1 m
n-1 + ….+ Kn = 0 & solve it for „m‟.
(ii) Write the C.F. as follows
Roots of A.E. C.F.
1. m1, m2, m3…. (distinct roots) f1(y+m1 x) +f2(y+m2 x) + f3 (y+m3 x)+ ….
2. m1, m2, m3… (two equal roots) f1(y+m1 x)+ x f2 (y+m2 x)+f3 (y+m3 x)+….
3. m1,m1, m1… (three equal roots) f1 (y+m1 x)+ x f2 (y+m1x)+x2 f3 (y+m3 x) + ….
Step- II To find P.I.
From the symbolic form
(i) when F (x, y) =
(ii) [Put D = a & D1 =b]
(iii) when F (x, y) = sin (mx+ny) or cos (mx+ny)
Sin or cos (mx+ny) [ Put D2 = - m
2, DD
1= - mn, D
‟2 = - n
2]
(iii) when F (x, y) = xm y
n
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(iv) when F (x, y) is any function of x & y
Step- III
C.S. = C.F. + P.I. = Z
Non-linear equations of the first order
Form I. f (p, q)= 0 i.e. equations containing p & q only its complete solution is z = ax+by+c
where a & b are connected by relation f (a, b) = 0.
z = ax+ φ (a) y+c, a & c are arbitrary constants.
From- II f (z, p, q) = 0, i.e. equations not containing x & y.
(i) assume u = x+ay & substitute P = dz/du, q= a dz/du in equation.
(ii) solve the resulting ordinary differential equation in z & u.
(iii) replace u by x+ay.
Form- III f (x, p) = F (y, q) i.e. equations in which z is absent and the terms containing x & p can
be separated from those containing y & q.
Solution is z = ∫ φ (x) dx+ ∫ψ (y) dy +b.
Form-IV Z = Px+qy+f (P, q)
Its complete solution is z = ax+by+f (a, b) which is obtained by writing a for P & b for q in the
given equation.
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13. COMPLEX VARIABLES
A number of the form x+iy, where x and y are real numbers and i=√(-1) is called a complex
number.
X is called the real part of x+iy and is written as R (x+iy) and y is called the imaginary part
and is written as I (x+iy)
A pair of complex numbers x+iy and x-iy are said to be conjugate of each other.
Properties
1.If x1+iy1= x2+iy2 then x1-iy1 = x2- iy2
2. Two complex numbers x1+iy1 and x2 +iy2 are said to be equal when
R (x1+iy1)= R (x2+ iy2) i.e. x1 = x2
And I (x1+ iy1) = I (x2 + iy2) i.e. y1 = y2
3. Sum, difference product and quotient of any two complex numbers is itself a complex number.
If x1+iy1 and x2+iy2 be two given complex number.
Then (i) their sum = (x1 +x2) + i (y1+y2)
(ii) Their difference = (x1- x2) + i(y1- y2)
(iv) Their product = x1x2- y1 y2 + i (x1y2+x2y1)
(v) Their quotient
4. Every complex number x+iy can always be expanded in the form r (cos θ + i sin θ)
The number r = √ x2+y
2 is called the modulus of x+iy and is written as mod (x+iy) or |x+ iy|
The angle θ is called the amplitude or argument of x+iy and is written as amp (x+iy) or arg
(x+iy). θ = tan-1
(y/x)
Cos θ + i sin θ is briefly written as c is θ
De Moivre‟s Theorem
If n be (i) an integer, +ve or - ve
(cos θ + i sin θ)n = cos n θ + i sin n θ
(ii) a fraction + ve or – ve one of the values of (cos θ+ i sin θ)n is cos n
θ + isin n θ.
Cis θ1 .. Cis θ2 … Cis θn = C is (θ1+ θ2+ …..+ θn)
(Cos θ- i Sin θ)n = Cosnθ- isin n θ = (cos θ + i sin θ)
-n
(Cis m θ)n = Cis mn θ = (Cis n θ)
m
Complex Function
If for each value of the complex variable z (= x+iy) in a given region R, we have one or more
values of w (= u+iv), then w is said to be a complex function of z and we write w = u (x, y) + I
v(x, y) = f (z) where u, v are real functions of x & y.
If to each value of z, there corresponds one & only one values of w, then w is said to be a
single valued function of z otherwise a multivalued function
Eg: w = 1/z is a single valued function
W = √ z is a multivalued function
Exponential function of a complex variable
The exponential function of the complex variable z = x+iy is
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Properties
1. Exponential form of Z = reiθ
2. ez is periodic function having imaginary period 2 πi,
3. ez is not zero for any value of z
4.
Circular function of a complex variable
The circular functions of the complex variable z is given as
Properties
1. Sin z, cos z are periodic with period 2 π, tan z, cot z are periodic with π.
2. cos z, sec z are even functions while sinz, cosec z are odd functions
3. Zeros of Sin z are given by z = ± 2 n π & zeroes of cos z are given by z = ± ½ (2n+1) π, n = 0,
1, 2…..
Euler‟s theorem
eiθ = cos θ + i sin θ, where θ is real or complex
Hyperbolic functions
If x be real or complex
(i) is defined as hyperbolic sine of x [ sin hx]
(ii) is defined as hyperbolic cosine of x [ cos hx ]
Properties
1.Sin hz & cos hz are periodic with period 2 π i
2. cos hz is an even function & sin hz is an odd function
3. Sin ho= 0 , cos ho = 1, tan ho = 0.
4. Relations b/n hyperbolic & circular functions
Sin i x = i sin hx
Cos i x = cos h x
Tanix = i tan hx
5. Formulae of hyperbolic functions
cosh2 x- sin h
2 x = 1
Sec h2 x+ tan h
2 x = 1
Cot h2 x- Cosech
2 x = 1
Sin h (x±y) Sin hx cos hy ± cos hx sin hy
Cos h ( x ±y) = cos hx cos hy± sin hx sin hy
Tan h (x ±y) =
Sin h2x = 2 Sin hx cos hx
Cos h2x = cos h2 x + sin h
2 x = 2 cos h
2 x -1 = 1+2 sin h
2 x
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Sin h 3x = 3Sin hx+ 4 Sin h3 x
Cos h3x = 4 cosh3 x- 3 Cos hx
Tan h 3x =
Sin hx + Sin hy =
Sin hx- sin hy
Cos h x+ coshy
Cos hx- Cos hy
Inverse hyperbolic functions
Sin h-1
z = log [ Z+ √z2+1]
Cos h-1
z = log [ z+ √ z2-1]
Tan h-1
z = ½ log [ (1+z)/(1-z)]
Logarithmic function of a complex variable
If z ( = x + I y) and w ( = u + I v) be so related that ew = z, then w is said to be a logarithm of z to
the base „e‟ and is written as w = logez.
Logarithm of a complex number has an infinite no. of values and is therefore a multi-valued
function.
Log Z = log (x+ iy) = 2 in π + log (x+ iy )
Log (x+ iy) = log ( √ x2 + y
2) + i [ 2n π + tan
-1 (y/x)].
Exponential series ex =
Sine, Cosine, Sin h or Cosh series
Logarithmic series
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Gregory‟s series
Binomial Series
Geometric series
a+ar+ar2 + ….to n terms =
a+ar+ar2+ ….. ∞ =
Derivative of f(z)
The derivative of w = f (z) is defined to be
provided the limit exists and has the same value for all the different ways in which δz approaches
zero.
The necessary and sufficient conditions for the derivative of the function w = u(x, y)+ i
v(x, y) = f (z) to exist for all values of z in a region R, are
(i) are continuous functions of x & y in R.
(ii) Cauchy- Riemann equations
Analytic functions
A function f (z) which is single valued & possess a unique derivative with respect to z at all
points of a region R, is called an analytic function of z in that region.
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A function which is analytic everywhere in the complex plane is known as an entire function.
A point at which an analytic ceases to possess a derivative is called a singular point of the
function.
If a complex function is one known to be analytic it can be differentiated just in the ordinary
way.
The real & imaginary parts of an analytic function are called conjugate functions.
Cauchy‟s theorem
If f (z) is an analytic function and f1 (z) is continuous at each point within and on a closed curve C,
then ∫c f (z) d z= 0
Cauchy‟s integral formula
If f (z) is analytic within and on a closed curve and if a is any point within C, then
Taylor‟s series
If f (z) is analytic inside a circle C with centre at a, then for z inside C,
f (z) = f (a) + f1 (a) (z-a) +
Laurent‟s Series
If f (z) is analytic in the ring- shaped region R bounded by two concentric circles C and C1 of
radii r & r1 ( r > r1) and with centre at a, then for all z in R
f (z) = a0 + a1 (z-a) + a2 (z-a)2 + ….+ a-1 (z-a)
-1 + a-2 (z-a)
-2 + ….
Where
Γ being any curve in R, encircling C1.
A zero of an analytic function f (z) is that value of z for which f (z) = 0
Singularities of analytic functions
(i) Isolated singularity
If z = a is a singularity of f (z) such that f (z) is analytic at each point in its
neighbourhood, then z = a is called isolated singularity.
(ii) Removable singularity
If exists finitely, then z = a is a removable singularity.
(iii) Poles
If all the negative powers of (z-a) in f (z) after the nth are missing then the singularity at z
= a is called a pole of order n
f (z) = a0+ a1(z-a) + a2 (z-a)2+ …. + a-1 (z-1)
-1 + ….
A pole of first order is called a simple pole.
(iv) Essential Singularity
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If the number of negative powers of (z-a) in f (z) is infinite then z = a is called an
essential singularity
In this case, does not exist.
Residues
The coefficient of (z-a)-1
in the expansion of f (z) around an isolated singularity is called the
residue of f(z) at that point
Residue theorem
If f (z) is analytic in a closed curve C except at a finite number of singular points within C, then
x (sum of the residues at the singular points within C).
Calculation of residues
1. If f (z) has a simple pole at z = a then
Res f (a) =
2. Res f (a )
Where ψ (z) = (z-a) F(z), F(a) ≠ 0.
3. If f (z) has a pole of order n at z = a then
Res f (a) =
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14. LAPLACE TRANSFORMS
Let f (t) be a function of t defined for all positive values of t. Then the Laplace transforms of f
(t) denoted by L { f (t)} is defined by
L { f (t)} =
Provided that the integral exists.
If f (t) = L-1
{ f (s)} then f (t) is called the inverse Laplace transform of f (s).
Important Laplace transforms
L (1) =
L (tn) =
L (eat) =
L (Sin at ) =
L (cos at) =
L (sin h at)=
L (cos h at ) =
L (e at t
n) =
L (eat Sin bt) =
L (eat Cos bt) =
L (eat Sinh bt) =
L (eat Cosh bt) =
Properties
1. Linearity property
If a, b, c be any constants and f, g, h any functions of t then
L [ a f (t) + b g (t) – c h (t)] = aL { f (t)} + bL { g (t)} – c L [ h (+1)}
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2. First shifting property
If L { f (t)} = then
L { eat f (t)} =
3. Change of scale property
If L {f (t)} = then
L { f(at)} =
If f (t) is continuous and is finite, then the Laplace transform of f(t) i.e.
exists for s > a
If f1 (t) be continuous and L {f (t)}= f(s) then L {f
1(t)}
If f1 (t) and its first (n-1) derivatives be continuous then L {f
n (t)}
If L { f (t)} =
If L {f (t)} = then
L { tn f (t) } where n = 1, 2, 3, ….
If L { f (t)} = then
ds provided integral exists.
Inverse transforms
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Convolution theorem
If L-1
{ } = f (t) and L-1
{ } = g (t)
Then L-1
{ } =
If f (t) is a periodic function with period T, then
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15. FOURIER TRANSFORMS
The fourier transform of f(x) is given by
F(s) =
The inverse fourier transform of F (s) is given by
The fourier sine transform of f(x) is 0 < x < ∞ is
o Inverse Fourier sine transform of Fs(s) is
The Fourier cosine transform of f (x) is 0 < x < ∞ is
Fc(s)=
o Inverse Fourier cosine transform of Fc(s) is
The finite fouirier sine transform of f(x) in 0 <x < c, is defined as
o The inverse finite fourier sine transform of Fs(n) is given by
The finite fourier cosine transform of f(x) in 0< x< c is defined as
o The inverse finite fourier cosine transform of Fc(n) is
Properties
1. Linear property
If F(s) & G(s) are fourier transforms of f(x) & g(x) respectively then
F[ a f(x) + b g(x)] = a F (s) + b G(s)
Where a & b are constants
2. Change of scale property
If F(s) is the complex Fourier transform of f(x), then
F { f (ax)}
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3. Shifting property
If F(s) is the complex fourier transform of f(x), then
F { f(x-a)} = eisa
F(s)
4. Modulation theorem
If f(s) is the complex fourier transform of f(x) then
F{f(x) cos ax} =1/2 [ F(s+a) + F (s-a)]
Fs{ f(x) cos ax} = ½ [ Fs (s+a) + Fs (s-a)]
Fc{ f(x)sin ax} = ½ [ Fs (s+a) - Fs (s-a)]
Fs{ f(x) sin ax} = ½ [ Fs (s-a) – Fc (s+a)]
Convolution
The convolution of two functions f(x) and g(x) over the interval (- ∞, ∞) is defined as f * g =
The fourier transform of the convolution f(x) & g(x) is the product of their Fourier transforms
i.e. F {f(x) * g (x)} = F {f(x)} . F {g (x)}
Parseval’s Identity
If the fourier transforms of f(x) & g(x) are F(s) & G(s) then
i)
ii) Where bar implies the complex conjugate
Parseval‟s identities for fourier sine & cosine transforms are
(i)
(ii)
(iii)
(iv)
Relation between Fourier & Laplace transforms
If f (t) = e-xt
g (t), t > 0 then
= 0, t < 0
F {f (t)} = L { g (t)}
The Fourier transform of the nth derivative of f(x) is
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Inverse Laplace transforms by method of Residues
= Sum of the residues of at the poles of f(s)
6. Z-TRANSFORMS
If the function un is defined for discrete values (n = 0, 1, 2, ….) and un = 0 for n < 0, then its z-
transform is defined to be
whenever the infinite series converges
The inverse Z-transform is written as
Z-1
[ U(z)] = un
Standard Z- transforms
Sequence un (n≥0) Z- transform U(z)
1
-1
K
n
n2
np
δ (n) = 1, n = 0 1
= 0, n ≠ 0
u (n)= 0, n <0 Z/ Z-1
= 1 n ≥ 0
an
n an
n2 a
n
Sin n θ
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Cos nθ
an sin nθ
an cos n θ
Sin hnθ
Cos hnθ
an sinhnθ
an cos hnθ
an un U (Z/a)
un+1 z [ U (z) – u0]
un+2 Z2 [ U (z)- u0- u1z
-1]
un+3 z3 [ U(z) – u0-u1 z
-1-u2 z
-2]
un-k z-k
U (z)
n un - z d/dz [ U (z)]
u0
Properties
1. Linearity property
If a, b, c be any constants and un, vn, wn be any discrete functions, then
Z(a un+bv n-cwn) = aZ(un) + bZ (vn) – cZ(wn)
2. Damping rule
If z(un) = U(z) then z(a-n
un) = U (az)
Z (an un)= U (z/a)
3. Shifting un to the right
If z (un) = U (z) then Z (un-k) = z-k
U(z), (k > 0]
4. Shifting un to the left
If z(un) = U(z) then
Z (un+k) = zk [ U(z)– u0-u1 z
-1- u2 z
-2 …. -uk-1 z
-(k-1)]
5. Multiplication by n
If z(un) = U(z) then Z (n un) = - z dU(z)/dz.
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Initial value theorem:
If z(un) = U(z), then u0 =
Final value theorem:-
If z(un) = U(z) then
Some standard inverse Z transforms
U(z) Inverse z- transform un
an u (n)
(n+1)an u (n)
½! (n+1) (n+2) an u (n)
an-1
u (n-1)
(n-1) an-2
u (n-2)
½ (n-1) (n-2) an-3
u (n-3)
Convolution theorem
If Z-1
[ U (z)] = un and z-1
[ V(z)] = vn then
= un *vn
The region of the z-plane in which U(z) converges absolutely is known as region of
convergence (ROC) of U(z).
For a right sided sequence the ROC is |z| > |a|
.0 z =a
→
For a left handed sequence the ROC is |z| < |b|
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b
0 →
For a two-sided sequence ROC is |b| < |z| < |c|
0
b c
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17. PROBABILITY & STATISTICS
Permutations
The number of permutations of n different things taken r at a time is
n (n-1) (n-2)… (n-r+1)= nPr =
The no. of circular permutations formed with „n‟ objects is (n-1)!
If the direction is not specified or considered then no. of circular permutations is
The no. of permutations of n objects of which n1 are alike, n2 are alike and n3 are alike is
Combinations (or) Selections
The number of combinations of n different objects taken r at a time is
ncn-r = ncr
2n objects can be divided into two equal groups in
(m+n+p) objects can be divided into three groups of m objects, n objects, p objects in
ways.
The number of straight lines drawn through „n‟ points on a circle is nc2.
The no. of diagonal of a polygon with n vertices is
The no. of triangles formed by joining vertices of a polygon with n vertices is nc3 =
Basic Terminology
Exhaustive events;
A set of events is said to be exhaustive if it includes all the possible events.
Mutually Exclusive events:
If the occurrence of one of the events procludes the occurrence of all others, then such a set of
events is said to be mutually exclusive
Equally Likely events:
If one of the events cannot be expected to happen in preference to another then such events are
said to be equally likely.
Odds in favour of an event:
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If the number of ways favourable to an event A is m and the no. of ways not favourbale to A is n
then odds in favour of A = m/n
Odds against A = n/m
Probability
If there are n exhaustive, mutually exclusive & equally likely cases of which m are favourable to
an event A, then probability (P) of happening of A is
P (A) = m/n
Chance of A not happening is q or P (A1)
P (A) + P (A
1) = 1 always
If an event is certain to happen then its probability is unity.
If an event is certain not to happen then its probability is zero.
Statistical (or empirical) definition
If in n trials, an event A happens m times, then the probability (p) of happening of A is given by
P = p (A) =
Random experiment:
Experiments which are performed essentially under the same conditions & whose results cannot be
predicted are known as random experiments.
Sample Space:
The set of all possible outcomes of a random experiment is called sample space for that
experiment(s).
The elements of the sample space S are called the sample points.
Event
The outcome of a random experiment is called an event.
Every subset of a sample space S is an event.
The null set φ is also an event & is called an impossible event.
Probability of an impossible event is zero i.e. P (φ) = 0.
Axioms
(i) The numerical value of probability lies between 0 & 1.
i.e. for any event A of S, 0≤ p(A) ≤1.
(ii) The sum of probabilities of all sample events is unity i.e. p (s) = 1
(iii) Probability of an event made of two or more sample events is the sum of their probabilities
Notations
(i) Probability of happening of events A or B is written as P (A+B) or P (AUB)
(ii) Probability of happening of both the events A & B is written as P (AB) or P (A∩B).
(iii) „Event A implies ( ) event B‟ is expressed as A B.
(iv) „Event A & B are mutually exclusive‟ is expressed as A∩B = φ
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For any two events A & B
P (A∩B1) = P(A)- P (A ∩B)
P (A1∩B) = P (B) – P (A ∩B)
Addition Law of probability (Theorem of Total Probability)
If the probability of an event A happening as a result of a trial is P (A) and the probability of a
mutually exclusive event B happening is P (B), then the probability of either of the events
happening as a result of the trial is P(A+B) or P ( A U B) = P (A) + P (B)
If A, B are any two events (not mutually exclusive) then
P (A+B) = P (A) + P (B)- P(AB)
i.e. P(A UB) = P(A) + P(B) – P(A∩B)
Independent events
Two events are said to be independent, if happening or failure of one does not affect the
happening or failure of the other.
Otherwise the events are said to be dependent.
Conditional probability
For two dependent events A & B, the symbol P (B/A) denotes the probability of occurrence of B,
when A has already occurred. It is known as the conditional probability and is read as a probability
of B given A`.
Multiplication Law of Probability (Theorem of compound probability)
If the probability of an event A happening as a result of trial is P (A) and after A has happened the
probability of an event B happening as a result of another trial (i.e. conditional probability of B
given A) is P (B/A) then the probability of both the events A & B happening as a result of two
trials is P (AB) or P(A ∩B) = P (A). P (B/A)
The conditional probability of A given B is P (A/B) then
P (A ∩B) = P (B). P (A/B)
If the events A& B are independent i.e. if the happening of B does not depend on whether A
has happened or not, then P (B/A) = P (B) & P(A/B)= P (A).
Therefore P (AB) or P (A∩B) = P (A) . P (B)
If P1, P2 be the probabilities of happening of two independent events, then
(i) The probability that the first event happens & the second fails is P1 (1-P2)
(ii) The probability that both events fail to happen is (1-P1) (1-P2)
(iii) The probability that atleast one of the events happens is 1- (1-P1) (1-P2). This is commonly
known as their cumulative probability
If P1, P2, P3… Pn be the chances of happening of n independent events, then their cumulative
probability is
1- (1-p1) (1-p2) (1-p3)… (1-pn)
Baye’s theorem
An event A corresponds to a number of exhaustive events B1, B2….Bn If P (Bi) and P (A/Bi) are
given then
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Random Variable
If a real variable x be associated with the outcome of a random experiment, then since the values
which x takes depend on chance, it is called a random variable or stochastic variable or simply a
variate.
If a random variables takes a finite set of values, it is called a discrete variate.
If it assumes an infinite number of uncountable values, it is called a continuous variate.
Discrete Probability Distribution
If the probability that X takes the values xi, is pi then
P (x= xi)= Pi or p (xi) for i = 1, 2
Where (i) p (xi) ≥ 0 for all values of i,
(ii) ∑ p(xi) = 1
The set of values xi with their probability Pi constitutes a discrete probability distribution of
the discrete variable X.
The distribution function F(x) of the discrete variate X is defined by
F(x) = P (X ≤x) = P (xi) where x is any integer.
The distribution function is also sometimes called cumulative distribution function.
Continuous Probability Distribution
The probability distribution of a continuous variate x is defined by a function f(x) such that the
probability of the variate x falling in the small interval x-1/2 dx to x+1/2 dx is f(x) dx
i.e. P (x- ½ dx ≤ x ≤ x+1/2 dx) = f (x) dx
* f (x) is called the probability density function
* the continuous curve y = f (x) is called probability curve
* the density function f (x) is always positive &
If F(x) = P (X ≤x) = then F(x) is defined as the cumulative distribution
function or distribution function of continuous variate X.
F1(x) = f(x) ≥ 0, F(x) is a non-decreasing function.
F(- ∞) = 0 ;
F (∞) = 1
= F(b) – F(a)
Expectation
The mean value (μ) of the probability distribution of a variable X is known as its expectation
, discrete distribution
, continuous distribution
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Variance
=
ζ is the standard deviation of the distribution
The rth moment about mean (denoted by μr) is defined by
μr = ∑ (xi- μ)r f (xi), discrete
=
Mean deviation from the mean is given by
∑ |xi- μ| f(xi), discrete
The probability of r success is ncr P
r q
n-r
The probability of atleast „r‟ successes in n trials
= ncr Pr q
n-r + ncr+1 P
r+1 q
n-r-1 + …. + ncn P
n
Binomial Distribution
If we perform a series of independent trials such that for each trial P is the probability of a
success and q that of a failure, then the probability of r successes in a series of n trials is given by P (x
=r) =ncr Pr q
n-r
The probability of the number of successes obtained is called the binomial distribution
Mean = np
Standard deviation = √(npq)
Variance = npq
Skewness =
Poisson Distribution
It is a distribution related to the probabilities of events which are extremely rare, but
which have a large number of independent opportunities for occurrence.
Mean = λ
Standard deviation = √ λ
Variance = λ
Skewness = 1/ λ
Normal Distribution
The normal curve is of form
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where
Mean deviation from the mean (μ)
Moments about mean
μ2n+1 = 0
μ2n = (2n-1) (2n-3) … 3.1ζ2n
Coefficient skewness is zero
Mean
If x1, x2, x3…. Xn are a set of n values of a variate, then the arithmetic mean (or simply mean) is
given by
In a frequency distribution if x1, x2, …. xn be the mid values of the class intervals having
frequencies f1, f2….. fn respectively we have
Median
If the values of a variable are arranged in the ascending order of magnitude, the median is the
middle item if the number is odd and is the mean of the two middle items if the number is even
Median =
Where L = Lower limit of the median class.
N= total frequency
f= frequency of the median class
h= width of the median class
C= cumulative frequency upto the class preceding the median class
Mode
The mode is defined as that value of the variable which occurs most frequently, i.e, the value of
the maximum frequency
Mode =
Where L = Lower limit of class containing mode.
∆1 = excess of modal frequency over freq of the preceding class.
∆2 = excess of modal freq over following class.
h= size of modal class.
Curves having a single mode are termed as unimodal, those having two modes as bi-modal
and those having more than two modes as multi-modal.
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Mean-Mode = 3 (Mean-Median)
Geometric mean
If x1 x2, …. xn are a set of n observations then the geometric mean is given by
G.M= (x1 x2… xn) 1/n
Harmonic mean
If x1, x2, …. Xn be a set of n observations, then the harmonic mean is defined as the reciprocal of
the (arithmetic) mean of the reciprocal of the quantities
Standard Deviation (ζ)
Correlation
When the changes is one variable are associated or followed by changes in the other, is called
correlation.
Data connecting such two variables is called bivariate population.
If an increase (or decrease) in the values of one variable corresponds to an increase (or
decrease) in the other, the correlation is said to be positive.
If the increase (or decrease) in one corresponds to the decrease (or increase) in the other, the
correlation is said to be negative.
If there is no relationship indicated between two variables they are said to be independent or
uncorrelated.
The numerical measure of correlation is called the coefficient of correlation and is defined by
the relation
Where x = deviation from the mean
Y = deviation from the mean
ζx = S.D. of x. series
ζy = S.D. of y series
n = no. of values of two variables
Lines of Regression
A line of best fit for the given distribution of dots is called the line of regression.
If there are two lines, such that one giving the best possible mean values of y for each
specified value of x and the other giving the best possible values of x for given values of y.
The former is known as the line of regression of y on x and the latter as the line of regression
of x on y.
The line of regression of y on x is
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o Slope is called regression coefficient = r ζy/ζx
The line of regression of x on y is
o Regression coefficient = r ζx/ζy
The correlation coefficient r is the geometric mean between the two regression co-efficients
i.e. we have
* The acute angle between two regression lines is
When (1) r= 0, θ = π/2, the two lines of regression are perpendicular to each other.
(2) r = ±1, θ= 0 or π, the lines of regression coincide and there is perfect correlation between
variables x and y.
Rank Correlation
A group of n individuals may be arranged in order of merit with respect to some characteristic.
The same group would give different orders for different characteristics. Considering the orders
corresponding to two characteristics A and B, the correlation between these n pairs of ranks is
called the rank correlation in the characteristics A and B for that group of individuals
Rank correlation coefficient
Where di = xi-yi
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18. NUMERICAL METHODS
Solution of Algebraic and Transcendental equations
To find the roots of an equation f (x) = 0, we start with a known approximate solution and apply
any of the following methods.
1. Bisection method:
If f(x) is continuous between a & b, and f(a) & f(b) are of opposite signs then there is a root
between a & b.
Let f (a) be-ve & f(b) be +ve, then the first approximation to the root is x1 = ½ (a+b)
If f(x1)= 0, then x1 is a root of f(x) = 0.
Otherwise, the root lies between a & x1 or x1 & b according as f (x1) is positive or negative.
Then we bisect the interval as before and continue the process until the root is found to desired
accuracy.
2. Method of false position (or) Regular-falsi method (i)
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Here we choose two points x0 & x1 such that f(x0) and f(x1) are of opposite of signs i.e., the graph
of y = f(x) crosses the x-axis between these points.
This indicates that a root lies between x0 & x1 consequently f (x0) f (x1) < 0
Equation of the chord joining the points A[x0, f(x0)] & B[ x1, f(x1)] is
The method consisting in replacing the curve AB by means of the chord AB and taking the
point of intersection of the chord with the x-axis as an approximation to the root.
3. Newton Raphson Method
(Order of convergence = 2)
Secant method (1.62)
Iterative formula to find 1/N is xn+1 = xn (2- Nxn)
Iterative formula to find√ N is xn+1 = ½ (xn + N/xn)
Iterative formula to find 1/√ N is xn+1 = ½ (xn + 1/N xn)
Iterative formula to find is
Solution of Non linear simultaneous equations = Newton Raphson method
Consider the equations f(x,y) = 0, g (x, y) = 0. If an initial approximation (x0 y0) to a solution has
been found by graphical method or otherwise, then a better approximation (x1, y1) as
X1 = x0+h, y1 = y0+K so that
F(x0 + h, y0+k) = 0 & g(x0 + h, y0 +K)= 0
Finite differences
Suppose we are given the following values of y = f(x) for a set of values of x:
X: x0 x1 x2 … xn
Y: y0 y1 y2 … yn
Then the process of finding the values of y corresponding to any value of x = xi between x0 & xn
is called interpolation
Interpolation is the technique of estimating the value of a function for any intermediate value
of the independent variable.
The process of computing the value of the function outside the given range is called extra
polation.
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1. Forward differences
The difference y1-y0, y2-y1, …yn-yn-1 when denoted by ∆y0, ∆y1, … ∆yn-1 respectively are
called the first forward differences where ∆ is the forward difference operator.
i.e. first forward differences are
∆yr = yr+1 - yr
∆2 yr = ∆yr+1 - ∆yr
∆p yr = ∆
P-1 yr+1 - ∆
P-1 yr defined p
th forward difference.
Forward Difference Table
Value of x Value of y 1st diff 2
nd diff 3
rd diff
X0 y0
∆y0
X0+h y1 ∆2y0
∆y1 ∆3y0
X0+2h y2 ∆y2 ∆2y1
X0+3h y3 ∆3y1
∆2 y2
∆y3
X0 +4h y4
2. Backward differences
The differences y1-y0, y2- y1, ….. yn – yn-1
When denoted by y1, y2, … yn respectively, are called the first backward difference
where is the backward difference operator
yr = yr – yr-1
2 yr = yr - yr-1,
3 yr =
2 yr-
2 yr-1 etc.
Backward difference table
Value of x Value of y 1st diff 2
nd diff 3
rd diff
X0 y0
y1
2y2
X0+h y1 3y3
y2
2y3
3y4
X0+2h y2
y3
2y4
X0+3h y3
y4
X0+ 4h y4
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3. Central differences
The central difference operator δ is defined by y1-y0= δy1/2 , y2-y1 = δy3/2 …. yn- yn-1
= δyn-1/2
δy3/2 – δy1/2 = δ2 y1, δy5/2 – δy3/2 = δ
2 y2, …..
δ2y2 – δ
2y1 = δ
3y3/2 and so on.
Central difference table
Value of x Value of y 1st diff 2
nd diff 3
rd diff
X0 y0
δy1/2
δ2y1
X0+h y1 δ3y3/2
δy3/2
X0+2h y2 δ2y2
δy5/2 δ3y5/2
δ2y3
X0+3h y3
δy7/2
X0+ 4h y4
Newton‟s forward interpolation formula
Where
Newton‟s backward interpolation formula
Where
Central difference Interpolation Formulae
x y 1st diff 2
nd diff 3
rd diff
x0-2h y-2
∆y-2 (=δy-3/2)
∆2y-2(=δ
2y-1)
∆3y-2 (=δ
3y-1/2)
X0-h y-1
∆y-1 (=δy-1/2)
X0 y0 ∆2y-1 (=δ
2y0)
∆y0 (=δy1/2) ∆3y-1(=δ
3y1/2)
X0+h y1 ∆2y0(= δ
2y1)
∆y1 (=δy3/2)
X0 +2h y2
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1. Gauss‟s forward interpolation formula
….
In the central difference notation
2. Gauss‟s backward interpolation formula
+…..
3. Stirling‟s formula
4. Bessel‟s Formula
5. Everett‟s formula
Where P=1-q
Interpolation with Unequal intervals
1. Lagrange‟s formula
If y = f(x) takes the values y0, y1, …. Yn corresponding to x = x0, x1, … xn then
F(x)= (x-x1) (x-x2)… (x-xn) y0 + (x-x0) (x-x2)… (x-xn)
(x0-x1) (x0-x2).. (x0 –xn) (x1-x0) (x1-x2)… (x1-xn)
y1+ …+ (x-x0) (x-x1)… (x-xn-1) yn
(xn-x0) (xn-x1)… (xn-xn-1)
2. Divided difference
If (x0, y0), (x1, y1), (x2, y2)… be given points, then the first divided differences for the arguments
x0, x1 is defined by the relation
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3. Newton‟s divided difference formula
Y = f(x) = y0 + (x-x0) [ x0, x1] + (x-x0) (x-x1) [ x0, x1, x2] +….
Where [x0, x1] = and so on
Numerical Integration
The process of evaluating a definite integral from a set of tabulated values of the integral f(x) is
called numerical integration
This process when applied to a function of a single variable is known as quadrature.
1. Newton- Cote‟s quadrature formula
n = 1, 2, 3,
2. Trapezoidal rule
Putting n = 1 is Newton-cote‟s quadrature formula
[ (y0+yn)+2 (y1+y2+…+yn-1)]
3. Simpson‟s one-third rule
Putting n = 2
[ (y0+yn)+4 (y1+y3+ …+yn-1) +2 (y2+y4+…+yn-2)]
4. Simpson‟s three eighth rule
Putting n = 3
[ (y0+yn)+3 (y1+y2+y4+y3+…..+yn-1)+2 (y3+y6+…+yn-3)]
5. Weddle‟s rule
Putting n = 6
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(y0+5y1+y2+6y3+y4+5y5+2y6+5y7+y8+….)
Numerical Solution of Ordinary Differential equations
Methods for finding the solution of first order differential equations of form
dy/dx = f (x, y), given y (x0) = y0 are as follows
1. Picard‟s Method
First approximation y1 to the solution is
y=y0 +
y1= y0 +
Second approximation y2 = y0 +
Third approximation y3= y0 + and so on.
2. Taylor series Method
y(x) = y0 + (x-x0) (y1)0 +
3. Euler‟s Method
dy/dx = f (x, y), y (x0) = y0
y1 = y0 + h f (x0, y0)
y2 = y1 + hf (x0+h y1)
:
:
yn = yn-1 +h f (x0 + h, yn-1)
4. Modified Euler‟s Method
Y1(1)
= y0 + h/2 [ f (x0, y0)+ f (x0 +h, y1)]
Y1(2)
= y0 + h/2 [ f (x0, y0) + f (x0+h, y1(1)
)]
Y2(1)
= y1+h/2[ f (x0+h, y1)+ f (x0+2h, y2)]
5. Runge‟s method
Procedure
Calculate successively
K1 = h f (x0, y0)
K2 = h f (x0+ ½ h, y0+ ½ K1)
K1 = h f (x0 + h, y0+K1)
K3 = h f (x0 + h, y0 + K1)
Finally K = 1/6 (K1+4K2+ K3)
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y = y0 +K is the solution
6. Runge-Kutta method
Procedure
Calculate successively
K1 = h f (x0,y0)
K2 = h f (x0+ ½ h, y0+ ½ K1)
K3 = h f (x0 +1/2 h, y0+1/2 K2)
K4 = h f (x0 + h, y0 + K3)
Finally K = 1/6 (K1+2K2+ 2K3+K4)
y1 = y0 +K is the solution
7. Predictor-corrector methods
(i) Milne‟s method
Given dy/dx = f (x,y) and y = y0 when x = x0; to find an approximate value of y for x =
x0+ nh by Milne‟s method, proceed as follows
y0 = y(x0) being given, we compute
y1 = y (x0+h), y2 =(x0+2h), y3 = y (x0+3h) by picard‟s or Taylor‟s series method
Next we calculate
f0 = f(x0 y0), f1 = f (x0+h, y1), f2 = f (x0 + 2h, y2), f3 = f (x0 +3h, y3),
then y4 = y0 + 4h/3 (2f1- f2+2f3) called predictor
y4 = y2 + h/3 (f2+4f3+f4) called corrector
y5 = y (x0 +5h) = y1+ 4h/3 (2f2-f3+2f4) Predictor
y5 = y3+ h/3 (f3+4f4+ f5) and so on
(ii) Adams- Bashforth method:
Given dy/dx = f (x, y) & y0 = y (x0) we compute
y-1= y (x0-h), y-2 = y (x0- 2h), y-3= y (x0-3h)
by Taylor‟s series or Euler‟s method or Runge- Kutta method
Next we calculate
f-1 = f (x0-h, y-1), f-2 = f (x0-2h, y-2) f-3 f(x0-3h, y-3)
y1 = y0 + h/24 (55f0- 59f-1 +37f-2 -9f-3)
Adams- Bashforth predictor formula
y1= y0+ h/24 [ 9f1+19f0-5f-1+f-2)
Correctors‟ formula.
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GATE OLD QUESTION PAPERS
GATE-2005 One Mark Questions
1. The following differential equation has
a. degree = 2, order = 1 b. degree= 1, order = 2
c. degree= 4, order = 3 d. degree= 2, order = 3
2. A fair dice is rolled twice. The probability that an odd number will follow an even number is
a. b.
c. d.
3. A solution of the following differential equation is given by
a. y = e2x
+ e-3x
b. y = e2x
+ e3x
c. y = e-2x
+ e3x
d. y = e-2x
+ e-3x
GATE- 2005 Two Marks Questions
4. In what range should Re(s) remain so that the Laplace transform of the function e(a+2)t+5
exits.
a. Re (s) > a+2 b. Re (s) > a+7
c. Re (s) < 2 d. Re (s) > a + 5
5. Given the matrix , the eigen vector is
a. b.
c. d.
6. Let and Then (a+b) =
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a. b.
c. d.
7. The value of the integral is
a. 1 b. π
c. 2 d. 2 π
8. The derivative of the symmetric function drawn in given figure will look like
→
↑
a. →
↑
b. →
↑
c.
→
↑
d.
→
↑
9. Match the following and choose the correct combination
Group-I Group-II
E. Newton-Raphson method 1. Solving nonlinear equations
F. Rung-kutta method 2. Solving linear simultaneous equations
G. Simpson‟s Rule 3. Solving ordinary differential equations
H. Gauss elimination 4. Numerical integration
5. Interpolation
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6. Calculation of Eigen values
a. E-6, F-1, G-5, H-3 b. E-1, F-6, G-4, H-3
c. E-1, F-3, G-4, H-2 d. E-5, F-3, G-4, H-1
10. Given an orthogonal matrix
a. b.
c. d.
GATE-2006 One Mark Questions
11. The rank of the matrix is
a. 0 b. 1
c. 2 d. 3
12. x x P, where P is a vector is equal to
a. P x x P - 2 P b.
2 P + ( x P)
c. 2 P + x P d. ( .P)-
2 P
13. ,where P is a vector, is equal to
a. P. d b. x xP. d
c. x P. d d. ∫∫∫ .P dv
14. A probability density function is of the form
p(x) = Ke-α|x|
, x (-∞,∞)
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The value of K is
a. 0.5 b. 1
c, 0.5α d. α
15. A solution for the differential equation (t) + 2x(t) = δ (t) with initial condition x(0-) = 0 is
a. e-2t
u(t) b. e2tu(t)
c. e-t u(t) d.e
t u(t)
16. A low-pass filter having a frequency response H(jω) = A (ω) ejφ(ω)
does not produce any phase
distortion if
a. A (ω) = Cω2, φ (ω) = kω
3
b. A (ω) = Cω2, φ (ω) = kω
3
c. A (ω) = Cω, φ (ω) = kω2
d. A (ω) = C, φ (ω) = kω-1
GATE-2006 Two Marks Questions
17. The eigen values and the corresponding eigen vectors of a 2 x 2 matrix are given by
Eigen value Eigen vector
λ1 = 8
λ2 = 4
The matrix is
a. b.
c. d.
18. For the function of a complex variable W = In z (where, W = u + jv and Z = x + jy, the u =
constant lines get mapped in Z-plane as
a. set of radial straight lines b. set of concentric circles
c. set of confocal hyperbolas d. set of confocal ellipses
19. The value of the contour integral |z-j|=2 dz in positive sense is
a. j π/2 b. - π/2
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c. -jπ/2 d. π/2
20. The integral sin3 θ d θ is given by
a. 1/2 b. 2/3
c. 4/3 d. 8/3
21. Three companies X, Y and Z supply computers to a university. The percentage of computers
supplied by them and the probability of those being defective are tabulated below
Company % of computers Probability of being
Supplied defective
X 60% 0 .01
Y 30% 0.02
Z 10% 0.03
Given that a computer is defective, the probability that it was supplied by Y is
a. 0.1 b. 0.2
c. 0.3 d. 0.4
22. For the matrix the eigen value corresponding to the eigen vector
a. 2 b. 4
c. 6 d. 8
23. For the differential equation the boundary conditions are
(i) y = 0 for x = 0 and (ii) y = 0 for x = a
The form of non-zero solutions of y (where m varies over all integers) are
a. b.
c. d.
24. As x increased from - ∞ to ∞, the function f (x) =
a. monotonically increases
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b. monotonically decreases
c. increases to a maximum value and then decreases
d. decreases to a minimum value and then increases
GATE-2007 One Mark Questions
25. The following plot shows a function y which varies linearly with x. The value of the integral I
= y dx is
X
Y
1 2 3
1
2
3
-1
a. 1.0 b. 2.5
c. 4.0 d. 5.0
26. For |x| << 1, coth (x) can be approximated as
a. x b. x2
c. 1/x d. 1/x2
27.
a. 0.5 b. 1
c. 2 d. not defined
28. Which one of the following is stricity bounded?
a. 1/x2 b. e
x
c. x2 d. e
-x2
29. For the function e-x
the linear approximation around x = 2 is
a. (3-x) e-2
b. 1-x
c. [3+2√ 2- (1+ √ 2) x] e-2
d. e-2
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GATE-2007 Two Marks Questions
30. It is given that X1, X2…. XM are M non-zero orthogonal vectors. The dimension of the vector
space spanned by the 2M vectors X1, X2 … XM, – X1, - X2,… - XM is
a. 2M
b. M+1
c. M
d. dependent on the choice of X1, X2… XM
31. Consider the function f(x) = x2 – x-2. The maximum value of f(x) in the closed interval [-4, 4]
is
a. 18 b. 10
c. -2.25 d. indeterminate
32. An examination consists of two papers, Paper 1 and Paper 2. The probability of failing in
Paper 1 is 0.3 and that in Paper 2 is 0.2. Given that a student has failed in Paper 2, the probability
of failing in Paper 1 is 0.6. The probability of a student failing in both the papers is
a. 0.5 b. 0.18
c. 0.12 d. 0.06
33. The solution of the differential equation under the boundary conditions
(i) y = y1 at x= 0 and
(ii) y = y2 at x = ∞, where k, y1 and y2 are constant is
a. y = (y1 – y2) exp(-x/k2) + y2
b. y = (y2 – y1) exp (-x/k) + y1
c. y = (y1 – y2) sin h(x/k) + y1
d. y = (y1-y2) exp (-x/k) + y2
34. The equation x3- x
2 + 4x -4 = 0 is to be solved using the Newton- Raphson method. If x = 2 is
taken as the initial approximation of the solution, then the next approximation using this method
will be
a. 2/3 b. 4/3
c. 1 d. 3/2
35. Three functions f1 (t), f2(t) and f3(t) which are zero outside the interval [0, T] are shown in the
figure. Which of the following statements is correct?
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a. f1(t) and f2 (t) are orthogonal
b. f1(t) and f3(t) are orthogonal
c. f2(t) and f3(t) are orthogonal
d. f1(t) and f2(t) are orthonormal
36. If the semi-circular contour D of radius 2 is as shown in the figure. Then the value of the
integral is
→
↑
0
→
↑↓
↓
D
j
02
-j2
a. j π b. – j π
c. – π d. π
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GATE-2008 One Mark Questions
37. All the four entries of the 2 x 2 matrix are nonzero, and one of its eigen
values is zero. Which of the following statements is true?
a. p11 p22 – p12 p21 = 1 b. p11 p22 – p12 p21 = -1
c. p11 p22 – p12 p21 = 0 d. p11 p22 + p12 p21 = 0
38. The system of linear equations
4x+ 2y = 7
2x+y = 6
has
a. a unique solution
b. no solution
c. an infinite number of solutions
d. exactly two distinct solutions
39. The equation sin (z) = 10 has
a. no real or complex solution
b. exactly two distinct complex solutions
c. a unique solution
d. an infinite number of complex solutions
40. For real values of x, the minimum value of the function f(x)= exp (x) + exp (-x) is
a. 2 b. 1
c. 0.5 d. 0
41. Which of the following functions would have only odd powers of x in its Taylor series
expansion about the point x = 0?
a. sin (x3) b. sin (x
2)
c. cos (x3) d. cos (x
2)
42. Which of the following is a solution to the differential equation
a. x(t) = 3e-t b. x(t) = 2e
-3t
c. x(t) = (-3/2) t2 d. x(t) = 3t
2
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GATE-2008 Two Marks Questions
43. The recursion relation to solve x = e-x
using Newton Raphson method is
a. b. c.
d.
44. The residue of the function f(z) =
a. b.
c. d.
45. Consider the matrix The value of eP is
a.
b.
c.
d.
46. In the Taylor series expansion of exp(x) + sin (x) about the point x = π, the coefficient of (x-
π)2 is
a. exp (π) b. 0.5 exp (π)
c, exp (π) + 1 d. exp (π) -1
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47. The value of the integral of the function g(x,y) = 4x3 + 10y
4 along the straight line segment
from the point (0,0) to the point (1,2) in the x-y plane is
a. 33 b. 35
c. 40 d. 56
48. Consider points P and Q in the x-y plane, with P = (1, 0) and Q= (0,1). The line integral
along the semicircle with the line segment PQ as its diameter
a. is-1
b. is 0
c. is 1
d depends on the direction (clockwise or anticlockwise) of the semicircle
GATE-2009 One Mark questions
49. The order of the differential equation + y 4 = e
-t is
a. 1 b. 2
c. 3 d. 4
50. A fair coin is tossed 10 times. What is the probability that Only the first two tosses will yield
heads?
a. b.
c. d.
51. If f(z) = c0 + c1z-1
, then dz is given by
a. 2 πC1 b. 2π (1+C0)
c. 2 πjC1 d. 2 πj (1+C0)
GATE-2009 Two Marks Questions
52. The Taylor series expansion of at x = π is given by
a. b.
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c. d.
53. Match each differential equation in Group I to its family of solution curves from Group II
Group I Group II
a. 1. Circles
b. 2. Straight lines
c. 3. Hyperbolas
d.
a. A-2, B-3, C-3, D-1 b. A-1, B-3, C-2, D-1
c. A-2, B-1, C-3, D-3 d. A-3, B-2, C-1, D-2
54. The eigen values of the following matrix are
a. 3, 3 + 5j, 6-j b. -6, + 5j, 3 + j, 3-j
c. 3 + j, 3-j, 5 + j d. 3, -1 + 3j, -1-3j
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ANSWERS & EXPLANTIONS
1. (b)
Order is highest derivative term. Degree is power of highest derivative term.
2. (d)
Since both events are independent of each other.
P(odd/even) =
3. (b)
A.E. D2 – 5D + 6 = 0
(D-2) (D-3) = 0
D = 2, 3
Therefore y = e2x
+ e3x
4. (a)
f(t)= e(a+2) t+5
= e5.e
(a+2)t
Therefore for L.T. to exists, Re (s) > a + 2
5. (c)
A- λI = 0
(-4- λ) (3- λ) -8 = 0
-12 + 4 λ-3 λ + λ2- 8 = 0
λ2 + λ – 20 = 0
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(λ +5) (λ-4) = 0
Therefore λ = -5, 4
Let eigen vector be
Putting λ1 = - 5
m1 + 2m2 = 0
Taking the value m1 = 2 & m2 = -1
Thus the eigen vector correspondence to eigen value λ1 = -5 is
Again, eigen vector X2 corresponding to eigen value λ2 = 4 is
-8m1 + 2m2 = 0
4m1 – m2 = 0
- 8m1 = - 2m2
m1 = - 4; m2 = - 1 m1 = -2, m2 = - 8
When λ = -5,
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6. (a)
[AA-1
]= I
- 2a-0.1b=0, 3b = 1 b = 1/3
7. (a)
Comparing with
Here µ = 0
8. (c)
Given function has negative slope in +ve half and +ve slope in –ve half. So its differentiation
curve is satisfied by (c).
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9. (C)
10. (c)
[AAT]
-1 = I
For orthogonal matrix AAT = I i.e. unity matrix
Inverse of I = I
11. (c)
R3 → R1- R3
Therefore Rank = 2
12. (d)
From vector triple product.
A x (B x C) = B (A.C)- C(A. B)
A = , B = , C=P
x x P = ( .P) – P ( . )= ( .P) - 2 P
13. (a)
∫∫ (∆ X P) ds = P.d (strokes Theorem)
14. (c)
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2K= α
K = 0.5 α
15. (a)
(t) + 2x(t) = δ (t)
Taking L.T. on both sides
sX(s) –x(0) + 2X(s) = 1
X(s) [s+2] = 1
X(t) = e-2t
u(t)
16. (b)
For distortion less transmission
Phase response should be linear φ (ω) = Kω
17. (a)
[λI-A] = 0
(λ-6)2 – 4 = 0
λ = 8, 4
* By property of eigen matrix sum of diagonal elements should be equal to sum of values of λ.
18. (b)
W= nZ = loge z
u + jv= loge (x+jy) = ½ log (x2 + y
2) + itan
-1 (y/x)
u is constant
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½ log (x2 + y
2) = C
X2 + Y
2 = C [equation of circle having same centre (0, 0)]
19. (d)
Polo (0,2) lies enside the circle |z-j| = 2
By Cauchy‟s integral formula.
| z-j| = 2
20. (c)
, Let cos θ = t
-sin θ d θ = dt, θ = 0, cos 0 = 1 = t
θ = π, cos π=-1 = t
=
I = (2/3) + (2/3) = 4/3
21. (d)
S→ supply by y d → defective
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Probability that the computer was supplied by y, if the product is defective
P(s ∩d) = 0.3 x 0.02 = 0.006
P (d) = 0.6 x .01 + 0.3 x 0.02 + 0.1 x 0.03 = 0.015
22. (c)
Given eigen vector
(4- λ) (101) + 2(101) = 0 4- λ + 2 = 0
λ = 6
23. (a)
D2 + K
2 = 0
D = ± jK
Y = A cos Kx + jB sin kx
At x= 0, y = 0 A = 0, y = jB sin kx
x=a, y =0 0 = B sinka
B ≠ 0 else y = 0 always
Sinka = 0
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24. (a)
At x → ∞, functions value increases.
25. (b)
Y = X + 1
= ½ (9-4) = 2.5
26. (c)
27. (a)
28.(d)
→
↑
x
y
→
2y= 1/x x
y= e
→x
y
→
↑
y=
↑y
→x→
↑
→
y
x →
y= 1/x2
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→
↑
x
y
→
y= x -x2y= e
→x
y
→
↑
y=
↑y
→x→
↑
→
y
x →
y= 2
29. (a)
30. (c)
31. (a)
f (x) = x2 – x – 2
= (x+1) (x-2)
f(-4) = 18
f (+4) = 10
f(x) is maximum in interval [-4, 4] at x = -4
32. (c)
P (A) = 0.2 A→ failing in paper 1
P (B) = 0.3 B → failing in paper 2
P (A/B) = 0.6
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Prop. of failing in both P (A∩B) = P(A/B) x P(B)
= 0.6 x 0.2 = 0.12
33. (d)
Complementary function
y = C1ex/k
+ C2 –x/k
……. (1)
X = 0; y = y1 = C1 + C2… (2)
X = ∞; y = y2
34. (b)
f (x) = x3 – x
2 + 4x-4
= 3x2 – 2x+4
f(2) = 8
f(2) = 12
35. (c)
Two functions f(x) & g(x) are said to be orthogonal if
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=
= 0
36. (a)
= 2πj x (sum of residne)
Residue at pole s = -1 is 0
Residue at pole s = 1 is
37. (c)
Eigen values are the roots of the determinant formed by matrix [si-P]
[sI-P] = 0 (s-p11) (s-p22) – p12p21 = 0
S2 – (p11 + p22) s+p11p22 – p11 p22 – p12p21 = 0
Since, one of the its eigen values is zero, therefore, putting s = 0
P11p22 – p12p21 = 0
Which is the desired condition
38. (b)
The system can be written in matrix from as
The Augmented matrix [A | B] is given by
2
2 1
4
6
7
Performing Gauss elimination on this [A|B] as follows:
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2
2 1
4
6
7
_R
2
_
_R2
2
2_4
R1
1R1=
2
0 0
4
5/2
7
Now Rank [A|B]= 2
(The number of non-zero rows in [A|B]
Rank [A] =1
(The number of non-zero rows in [A])
Since Rank [A|B] ≠ Rank [A]
The system has no solution
39. (a)
Sin (z) = 10
Since maximum value of sin (z) = 1,
Therefore, the above equation has no real or complex solutions
40. (a)
f(x) = ex + e
-x
Arithmetic mean of ex and 1/e
x is
Geometric mean of ex and 1/e
x is
It is known that A.M. ≥ G.M.
Therefore, (ex + e
-x) min = 2
41. (a)
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42. (b)
(D+3) X(t) = 0
D = - 3
So, x(t) = CeDt
= Ce-3t
43. (c)
The given equation to be solved is x = e-x
Which can be rewritten as
f (x) = x- e-x
= 0
= 1 + e-x
The Newton-raphson iterative formula is
44. (a)
Since is finite and non-zero, f(z) has a pole of order two at z = 2
The residue at z = a is given for a pole of order n as
Res f(a) =
Here n = 2 (pole of order 2) & a = 2
Res f (2)
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= [-2 (z+2)-3
] z=2
46. (b)
f (x) = ex + sin x
we wish to expand about x = π
Taylor‟s series expansion about X = a is
Now about x = π
f (x) = f(π) + (x- π) (π) +
The coefficient of (x- π)2 is
Here f(x) = ex + sinx
(x) = ex + cosx
(x) = ex _sinx
(π) = eπ – sin π
= e
π – 0 = e
π
The coefficient of (x- π)2 is
47. (a)
Equation of straight line from point (0,0) to (1,2) is
Or
Y = 2x
G (x,y) = 4x3 + 10 y
4
= 4x3 + 10 (2x)
4
= 4x3 + 160x
4
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= 1 + 32 = 33
48. (b)
=
=
49. (b)
Hieghest derivative of differential equation is 2
51. (d)
f(z)= c0 + c1 z-1
Unit circle
It has one pole at origine
So
52. (b)
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Now
f‟ (π) = 0
(π) = -1/6
so the expansion is
f(x) = -1 + (-1/6) (x- π)2 +…..
53. (a)
A.
log y = log x + log c
= log cx
Y = cx….. Equation of straight line
B.
log y = -log x + log c
log yx = log c
yx = c
y = c/x….. Equation of hyperbola.
C.
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y2 – x
2= c
2
D.
x2 + y
2 = c
2…. Equation afcircle
54. (d)
Sum of the eigen values are the sum of the principle diagonal element of the matrix
Sum of the diagonal current
= 3-1-1
=1
Sum of the eigen values
= 3-1 + 3j-1-3j
= 3-1-1
=1
Hence (d) option is correct.