1

Contents

Manual for K-Notes ................................................................................. 2

LINEAR ALGEBRA ..................................................................................... 3

CALCULUS ............................................................................................. 14

VECTOR CALCULUS ................................................................................ 19

DIFFERENTIAL EQUATIONS ................................................................... 22

COMPLEX FUNCTIONS ........................................................................... 27

PROBABILITY AND STATISTICS ............................................................... 30

NUMERICAL METHODS ......................................................................... 35

© 2014 Kreatryx. All Rights Reserved.

3

LINEAR ALGEBRA

MATRICES

A matrix is a rectangular array of numbers (or functions) enclosed in brackets. These numbers

(or function) are called entries of elements of the matrix.

Example:

2 0.4 8

5 -32 0 order = 2 x 3, 2 = no. of rows, 3 = no. of columns

Special Type of Matrices

1. Square Matrix

A m x n matrix is called as a square matrix if m = n i.e, no of rows = no. of columns

The elements ij

a when i = j 11 22a a ......... are called diagonal elements

Example:

1 2

4 5

2. Diagonal Matrix

A square matrix in which all non-diagonal elements are zero and diagonal elements may or

may not be zero.

Example:

1 0

0 5

Properties

a. diag [x, y, z] + diag [p, q, r] = diag [x + p, y + q, z + r]

b. diag [x, y, z] × diag [p, q, r] = diag [xp, yq, zr]

c.

11 1 1diag x, y, z diag , ,

x y z

d. t

diag x, y, z = diag [x, y, z]

e. n n n ndiag x, y, z diag x , y , z

f. Eigen value of diag [x, y, z] = x, y & z

g. Determinant of diag [x, y, z] = xyz

3. Scalar Matrix

A diagonal matrix in which all diagonal elements are equal.

4

4. Identity Matrix

A diagonal matrix whose all diagonal elements are 1. Denoted by I

Properties

a. AI = IA = A

b. nI I

c. 1I I

d. det(I) = 1

5. Null matrix

An m x n matrix whose all elements are zero. Denoted by O.

Properties:

a. A + O = O + A = A

b. A + (- A) = O

6. Upper Triangular Matrix

A square matrix whose lower off diagonal elements are zero.

Example:

3 4 5

0 6 7

0 0 9

7. Lower Triangular Matrix

A square matrix whose upper off diagonal elements are zero.

Example:

3 0 0

4 6 0

5 7 9

8. Idempotent Matrix

A matrix is called Idempotent if 2A A

Example:

1 0

0 1

9. Involutary Matrix

A matrix is called Involutary if 2A I .

5

Matrix Equality

Two matrices m nA and p qB are equal if

m = p ; n = q i.e., both have same size

ij

a = ij

b for all values of i & j.

Addition of Matrices

For addition to be performed, the size of both matrices should be same.

If [C] = [A] + [B]

Then ij ij ij

c a b

i.e., elements in same position in the two matrices are added.

Subtraction of Matrices

[C] = [A] – [B]

= [A] + [–B]

Difference is obtained by subtraction of all elements of B from elements of A.

Hence here also, same size matrices should be there.

Scalar Multiplication

The product of any m × n matrix A jka and any scalar c, written as cA, is the m × n

matrix cA = jkca obtained by multiplying each entry in A by c.

Multiplication of two matrices

Let m nA and p qB be two matrices and C = AB, then for multiplication, [n = p]

should hold. Then,

n

jkj 1

ik ijC a b

Properties

If AB exists then BA does not necessarily exists.

Example: 3 4A , 4 5B , then AB exits but BA does not exists as 5 ≠ 3

So, matrix multiplication is not commutative.

6

Matrix multiplication is not associative.

A(BC) ≠ (AB)C .

Matrix Multiplication is distributive with respect to matrix addition

A(B + C) = AB +AC

If AB = AC B = C (if A is non-singular)

BA = CA B = C (if A is non-singular)

Transpose of a matrix

If we interchange the rows by columns of a matrix and vice versa we obtain transpose of a

matrix.

eg., A =

1 3

2 4

6 5

;

T 1 2 6A3 4 5

Conjugate of a matrix

The matrix obtained by replacing each element of matrix by its complex conjugate.

Properties

a. A A

b. A B A B

c. KA K A

d. AB AB

Transposed conjugate of a matrix

The transpose of conjugate of a matrix is called transposed conjugate. It is represented by A .

a.

A A

b. A B A B

c. KA KA

d. AB B A

7

Trace of matrix

Trace of a matrix is sum of all diagonal elements of the matrix.

Classification of real Matrix

a. Symmetric Matrix : T

A A

b. Skew symmetric matrix : T

A A

c. Orthogonal Matrix : T 1 TA A ; AA =I

Note:

a. If A & B are symmetric, then (A + B) & (A – B) are also symmetric

b. For any matrix TAA is always symmetric.

c. For any matrix,

TA + A

2 is symmetric &

TA A

2 is skew symmetric.

d. For orthogonal matrices, A 1

Classification of complex Matrices

a. Hermitian matrix : A A

b. Skew – Hermitian matrix : A A

c. Unitary Matrix : 1A A ;AA 1

Determinants

Determinants are only defined for square matrices.

For a 2 × 2 matrix

11 12

21 22

a a

a a

= 11 22 12 21a a a a

Minors & co-factor

If

11 12 13

21 22 23

31 32 33

a a a

a a a

a a a

8

Minor of element 12 13

21 2132 33

a aa : M

a a

Co-factor of an element i j

ij ija 1 M

To design cofactor matrix, we replace each element by its co-factor

Determinant

Suppose, we need to calculate a 3 × 3 determinant

3 3 3

1 j 1 j 2 j 2 j 3 j 3 jj 1 j 1 j 1

a cof a a cof a a cof a

We can calculate determinant along any row of the matrix.

Properties

Value of determinant is invariant under row & column interchange i.e., TA A

If any row or column is completely zero, then A 0

If two rows or columns are interchanged, then value of determinant is multiplied by -1.

If one row or column of a matrix is multiplied by ‘k’, then determinant also becomes k times.

If A is a matrix of order n × n , then

nKA K A

Value of determinant is invariant under row or column transformation

AB A * B

nnA A

1 1A

A

Adjoint of a Square Matrix

Adj(A) = T

cof A

9

Inverse of a matrix

Inverse of a matrix only exists for square matrices

1

Adj AA

A

Properties

a. 1 1AA A A I

b. 1 1 1AB B A

c. 1 1 1 1ABC C B A

d. T1

T 1A A

e. The inverse of a 2 × 2 matrix should be remembered

1a b d b1

c d c aad bc

I. Divide by determinant.

II. Interchange diagonal element.

III. Take negative of off-diagonal element.

Rank of a Matrix

a. Rank is defined for all matrices, not necessarily a square matrix.

b. If A is a matrix of order m × n,

then Rank (A) ≤ min (m, n)

c. A number r is said to be rank of matrix A, if and only if

There is at least one square sub-matrix of A of order ‘r’ whose determinant is

non-zero.

If there is a sub-matrix of order (r + 1), then determinant of such sub-matrix

should be 0.

10

Linearly Independent and Dependent

Let X1 and X2 be the non-zero vectors

If X1=kX2 or X2=kX1 then X1,X2 are said to be L.D. vectors.

If X 1 kX2 or X2 kX1 then X1,X2 are said to be L.I. vectors.

Note

Let X1,X2 ……………. Xn be n vectors of matrix A

if rank(A)=no of vectors then vector X1,X2………. Xn are L.I.

if rank(A)

11

Consistent infinite solution

If r < n, no of independent equation < (no. of variables) so, value of (n – r) variables can be

assumed to compute rest of r variables.

Non-Homogenous Equation

n11 1 12 2 1n 1a x a x .......... a x b

n21 1 22 2 2n 2a x a x .......... a x b

-------------------------------------

-------------------------------------

mn n nm1 1 m2 2a x a x .......... a x b

This is a system of ‘m’ non-homogenous equation for n variables.

11

2

12 1n 1 1

21 22 2n 2

mn n mm1 m2 m 1m n

a a a x b

a a a x b

A ; X ; B= -

-

a a a x b

Augmented matrix = [A | B] =

11 n12 1

21 22 2

mn mm1 m2

a a a b

a a b

a a a b

Conditions

Inconsistency

If r(A) ≠ r(A | B), system is inconsistent

Consistent unique solution

If r(A) = r(A | B) = n, we have consistent unique solution.

Consistent Infinite solution

If r(A) = r (A | B) = r & r < n, we have infinite solution

12

The solution of system of equations can be obtained by using Gauss elimination Method.

(Not required for GATE)

Note

Let An n and rank(A)=r, then the no of L.I. solutions of Ax = 0 is “n-r”

Eigen values & Eigen Vectors

If A is n × n square matrix, then the equation

Ax = λ x

is called Eigen value problem.

Where λ is called as Eigen value of A.

x is called as Eigen vector of A.

Characteristic polynomial

11 12 1n

21 22 2n

mnm1 m2

a a a

a a aA I

a a a

Characteristic equation A I 0

The roots of characteristic equation are called as characteristic roots or the Eigen values.

To find the Eigen vector, we need to solve

A I x 0

This is a system of homogenous linear equation.

We substitute each value of λ one by one & calculate Eigen vector corresponding to

each Eigen value.

Important Facts

a. If x is an eigenvector of A corresponding to λ, the KX is also an Eigenvector where K

is a constant.

b. If a n × n matrix has ‘n’ distinct Eigen values, we have ‘n’ linearly independent Eigen

vectors.

c. Eigen Value of Hermitian/Symmetric matrix are real.

d. Eigen value of Skew - Hermitian / Skew – Symmetric matrix are purely imaginary or

zero.

e. Eigen Value of unitary or orthogonal matrix are such that | λ | = 1.

f. If n1 2, ......., are Eigen value of A, n1 2k ,k .......,k are Eigen values of kA.

g. Eigen Value of 1A are reciprocal of Eigen value of A.

13

h. If n1 2, ...., are Eigen values of A, n1 2

A A A, , ........., are Eigen values of Adj(A).

i. Sum of Eigen values = Trace (A)

j. Product of Eigen values = |A|

k. In triangular or diagonal matrix, Eigen values are diagonal elements.

Cayley - Hamiltonian Theorem

Every matrix satisfies its own Characteristic equation.

e.g., If characteristic equation is

n n 1

n1 2C C ...... C 0

Then

n n 1 n1 2C A C A ...... C I O

Where I is identity matrix

O is null matrix

14

CALCULUS

Important Series Expansion

a. n

n n

r 0

rr1 x C x

b. 1 21 x 1 x x ............

c. 2 3

2 3x x xa 1 x log a xloga xloga ................2! 3!

d. 3 5x xsinx x .................

3! 5!

e. 2 4x xcosx 1 + ......................

2! 4!

f. tan x = 3 52xx + x + .........

3! 15

g. log (1 + x) = 2 3x xx + + ............, x < 1

2 3

Important Limits

a. lt sinx

1x 0 x

b. lt tanx

1x 0 x

c. 1 nx

lt 1 nx e

x 0

d. lt

cos x 1x 0

e. 1

xlt

1 x ex 0

f.

xlt 1 1 exx

15

L – Hospitals Rule

If f (x) and g(x) are to function such that

lt

f x 0x a

and lt

g x 0x a

Then

lt ltf x f' x

x a x ag x g' x

If f’(x) and g’(x) are also zero as x a , then we can take successive derivatives till this

condition is violated.

For continuity, lim

f x =f ax a

For differentiability, 00f x h f xlim

h 0 h

exists and is equal to 0f ' x

If a function is differentiable at some point then it is continuous at that point but converse

may not be true.

Mean Value Theorems

Rolle’s Theorem

If there is a function f(x) such that f(x) is continuous in closed interval a ≤ x ≤ b and f’(x)

is existing at every point in open interval a < x < b and f(a) = f(b).

Then, there exists a point ‘c’ such that f’(c) = 0 and a < c < b.

Lagrange’s Mean value Theorem

If there is a function f(x) such that, f(x) is continuous in closed interval a ≤ x ≤ b; and f(x) is

differentiable in open interval (a, b) i.e., a < x < b,

Then there exists a point ‘c’, such that

f b f af ' c

b a

16

Differentiation

Properties: (f + g)’ = f’ + g’ ; (f – g)’ = f’ – g’ ; (f g)’ = f’ g + f g’

Important derivatives

a. nx → n

n 1x

b. 1nx

x

c. a a 1log x (log e) x

d. x xe e

e. x x

ea a log a f. sin x → cos x

g. cos x → -sin x

h. tan x → 2sec x

i. sec x → sec x tan x

j. cosec x → - cosec x cot x

k. cot x → - cosec2 x

l. sin h x → cos h x

m. cos h x → sin h x

n. 12

1sin x

1 - x

o. 2

1 -1cos x 1 x

p.

21 1tan x

1 x

q. 2

1 -1cosec x x x 1

r.

21 1sec x

x x 1

s.

1

2

-1cot x

1 x

17

Increasing & Decreasing Functions

f ' x 0 V x a, b , then f is increasing in [a, b]

f ' x 0 V x a, b , then f is strictly increasing in [a, b]

f ' x 0 V x a, b , then f is decreasing in [a, b]

f ' x 0 V x a, b , then f is strictly decreasing in [a, b]

Maxima & Minima

Local maxima or minima

There is a maximum of f(x) at x = a if f’(a) = 0 and f”(a) is negative.

There is a minimum of f (x) at x = a, if f’(a) = 0 and f” (a) is positive.

To calculate maximum or minima, we find the point ‘a’ such that f’(a) = 0 and then decide

if it is maximum or minima by judging the sign of f”(a).

Global maxima & minima

We first find local maxima & minima & then calculate the value of ‘f’ at boundary points of

interval given eg. [a, b], we find f(a) & f(b) & compare it with the values of local maxima &

minima. The absolute maxima & minima can be decided then.

Taylor & Maclaurin series

Taylor series

f(a + h) = f(a) + h f’(a) +

2h

2 f”(a) + ………………..

Maclaurin

f(x) = f(0) + x f’(0) +

2x

2 f“(0)+……………..

Partial Derivative

If a derivative of a function of several independent variables be found with respect to any

one of them, keeping the others as constant, it is said to be a partial derivative.

18

Homogenous Function

n 2 2n n 1 nn0 1 2a x a x y a x y ............. a y is a homogenous function

of x & y, of degree ‘n’

= 2 n

n0 1 2n y y yx a a a .................... a

x x x

Euler’s Theorem

If u is a homogenous function of x & y of degree n, then

u u

x y nux y

Maxima & minima of multi-variable function

2

2

x a

y b

flet r

x

; 2

x a

y b

fs

x y

; 2

2

x a

y b

ft

y

Maxima

rt > 2s ; r < 0

Minima

rt > 2s ; r > 0

Saddle point

rt < 2s

Integration

Indefinite integrals are just opposite of derivatives and hence important derivatives must

always be remembered.

Properties of definite integral

a. b b

a a

f x dx f t dt

b. b a

a b

f x dx f x dx

c. b c b

a a c

f x dx f x dx f x dx

19

d. b b

a a

f x dx f a b x dx

e.

t

t

df x dx f t ' t f t ' t

dt

VECTOR CALCULUS Vectors

Addition of vector

a b of two vector a = 1 2 3a ,a ,a and b = 1 2 3b ,b ,b

1 1 2 2 3 3 a + b = a b ,a b ,a b

Scalar Multiplication

1 2 3ca = ca , ca , ca

Unit vector

a

â a

Dot Product

a . b = a b cos γ, where ‘γ’ is angle between a & b .

1 2 2 3 31a . b = a b a b a b

Properties

a. | a . b | ≤ |a| |b| (Schwarz inequality)

b. |a + b| ≤ |a| + |b| (Triangle inequality)

c. |a + b|2 + |a – b|2 = 2 2 2 a b (Parallelogram Equality) Vector cross product

v a b a b sin γ

= 1 2

1 2 3

3

ˆ ˆ ˆi j k

a a a

b b b

where 1 2 3a a ,a ,a ; 1 2 3b b ,b ,b

Properties:

a. a b b a

b. a b c a b c

20

Scalar Triple Product

(a, b, c) = a . b c

Vector Triple product

a b c a . c b a . b c

Gradient of scalar field

Gradient of scalar function f (x, y, z)

grad f f fˆ ˆ ˆf i j kx y z

Directional derivative

Derivative of a scalar function in the direction of b

b

bD f

b . grad f

Divergence of vector field

31 2vv v

.vx y z

; where 1 2 3v v , v , v

Curl of vector field

Curl 1 2 3

2 31

i j k

v v v v , v , vx y z

v v v

Some identities

a. Div grad f = 2 2 2

2

2 2 2

f f f

x y z

b. Curl grad f = f 0

c. Div curl f = . f 0

d. Curl curl f = grad div f – 2 f

21

Line Integral

b

C a

drF r .dr F r t . dt

dt

Hence curve C is parameterized in terms of t ; i.e. when ‘t’ goes from a to b, curve C is

traced.

b

2 3C a

1F r .dr F x' F y ' F z ' dt

2 31F F ,F ,F

Green’s Theorem

2 1 1 2CR

F Fdx dy F dx F dy

x y

This theorem is applied in a plane & not in space.

Gauss Divergence Theorem

ST

ˆdiv F. dv F . n d A

Where n̂ is outer unit normal vector of s .

T is volume enclosed by s.

Stoke’s Theorem

S C

ˆcurl F . n d A F. r ' s ds

Where n̂ is unit normal vector of S

C is the curve which enclosed a plane surface S.

22

DIFFERENTIAL EQUATIONS

The order of a deferential equation is the order of highest derivative appearing in it.

The degree of a differential equation is the degree of the highest derivative occurring in it,

after the differential equation is expressed in a form free from radicals & fractions.

For equations of first order & first degree

Variable Separation method

Collect all function of x & dx on one side.

Collect all function of y & dy on other side.

like f(x) dx = g(y) dy

solution: f x dx g y dy c

Exact differential equation

An equation of the form

M(x, y) dx + N (x, y) dy = 0

For equation to be exact.

M N

y x

; then only this method can be applied.

The solution is

a = M dx (termsofNnotcontainingx)dy

Integrating factors

An equation of the form

P(x, y) dx + Q (x, y) dy = 0

This can be reduced to exact form by multiplying both sides by IF.

If

1 P Q

Q y x is a function of x, then

R(x) =

1 P Q

Q y x

Integrating Factor

IF = exp R x dx

23

Otherwise, if Q P1

P x y

is a function of y

S(y) = Q P1

P x y

Integrating factor, IF = exp S y dy

Linear Differential Equations

An equation is linear if it can be written as:

y ' P x y r x

If r(x) = 0 ; equation is homogenous

else r(x) ≠ 0 ; equation is non-homogeneous

y(x) = p x dx

ce is the solution for homogenous form

for non-homogenous form, h = P x dx

hhy x e e rdx c

Bernoulli’s equation

The equation ndy

Py Qydx

Where P & Q are function of x

Divide both sides of the equation by ny & put

1 ny z

dz

P 1 n z Q 1 ndx

This is a linear equation & can be solved easily.

Clairaut’s equation

An equation of the form y = Px + f (P), is known as Clairaut’s equation where P = dy dx The solution of this equation is

y = cx + f (c) where c = constant

24

Linear Differential Equation of Higher Order

Constant coefficient differential equation

n

n n 1

n 1

n1

d y d yk .............. k y X

dx dx

Where X is a function of x only

a. If n1 2y , y , .........., y are n independent solution, then

n n1 1 2 2c y c y .......... c y x is complete solution

where 1 2 nc ,c ,..........,c are arbitrary constants.

b. The procedure of finding solution of nth order differential equation involves

computing complementary function (C. F) and particular Integral (P. I).

c. Complementary function is solution of

n

n n 1

n 1

n1

d y d yk ............. k y 0

dx dx

d. Particular integral is particular solution of

n

n n 1

n 1

n1

d y d yk ............ k y x

dx dx

e. y = CF + PI is complete solution

Finding complementary function

Method of differential operator

Replace d

dx by D →

dyDy

dx

Similarly

n

n

d

dx by nD →

nn

n

d yD y

dx

n

n n 1

n 1

n1

d y d yk ............ k y 0

dx dx

becomes

n n 1 n1D k D ........... k y 0 Let 1 2 nm ,m ,............,m be roots of

n1

n 1nD k D ................ K 0

………….(i)

25

Case I: All roots are real & distinct

n1 2D m D m ............ D m 0 is equivalent to (i)

y = 1 2 nm xm x m x

n1 2c e c e ........... c e

is solution of differential equation

Case II: If two roots are real & equal

i.e., 1 2m m m

y = 2 3 nm x m xmx

n1 3c c x e c e .......... c e

Case III: If two roots are complex conjugate

1m j ; 2m j

y = 1 2nm x

nxe c 'cos x c 'sin x .......... c e

Finding particular integral

Suppose differential equation is

n n 1

nn 1 n 1

d y d yk .......... k y X

dx dx

Particular Integral

PI =

n1 2

n1 2

W x W x W xy dx y dx .......... y dx

W x W x W x

Where n1 2y , y ,............y are solutions of Homogenous from of differential equations.

n1 2

n1 2

n n nn1 2

y y y

y ' y ' y 'W x

y y y

n1 2 i 1

n1 2 i 1

i

n n n nn1 2 i 1

y y y 0 y

y ' y ' y '0 y 'W x

0

y y y 1 y

iW x is obtained from W(x) by replacing ith column by all zeroes & last 1.

26

Euler-Cauchy Equation

An equation of the form

n n 1

n n 1nn 1 n 1

d y d yx k x .......... k y 0

dx dx

is called as Euler-Cauchy theorem

Substitute y = xm

The equation becomes

m

n1m m 1 ........ m n k m(m 1)......... m n 1 ............. k x 0

The roots of equation are

Case I: All roots are real & distinct

1 2 nm m m

n1 2y c x c x ........... c x

Case II: Two roots are real & equal

1 2m m m

3m mm

n1 2 3ny c c nx x c x ........ c x

Case III: Two roots are complex conjugate of each other

1m j ; 2m j

y = 3m nm

n3x Acos nx Bsin nx c x ........... c x

27

COMPLEX FUNCTIONS

Exponential function of complex variable

x iyzf z e e

iyx xf z e e e cosy i siny = u + iv

Logarithmic function of complex variable

If we z ; then w is logarithmic function of z

log z = w + 2inπ

This logarithm of complex number has infinite numbers of values.

The general value of logarithm is denoted by Log z & the principal value is log z & is

found from general value by taking n = 0.

Analytic function

A function f(z) which is single valued and possesses a unique derivative with respect to z at

all points of region R is called as an analytic function.

If u & v are real, single valued functions of x & y s. t. u u v v

, , ,x y x y

are continuous

throughout a region R, then Cauchy – Riemann equations u v v u

; x y x y

are necessary & sufficient condition for f(z) = u + iv to be analytic in R.

Line integral of a complex function

b

C a

f z dz f z t z ' t dt

where C is a smooth curve represented by z = z(t), where a ≤ t ≤ b.

Cauchy’s Theorem

If f(z) is an analytic function and f’(z) is continuous at each point within and on a

closed curve C. then

C

f z dz 0

28

Cauchy’s Integral formula

If f(z) is analytic within & on a closed curve C, & a is any point within C.

C

f z1f a dz

2 i z a

C

n

n 1

f zn!f a dz

2 i z a

Singularities of an Analytic Function

Isolated singularity

n

nn

f z a z a

;

n n 1

f t1a dt

2 i t a

z = 0

z is an isolated singularity if there is no singularity of f(z) in the neighborhood of z

= 0

z .

Removable singularity

If all the negative power of (z – a) are zero in the expansion of f(z),

f(z) = n

nn 0

a z a

The singularity at z = a can be removed by defined f(z) at z = a such that f(z) is

analytic at z = a.

Poles

If all negative powers of (z – a) after nth are missing, then z = a is a pole of order ‘n’.

Essential singularity

If the number of negative power of (z – a) is infinite, the z = a is essential

singularity & cannot be removed.

RESIDUES

If z = a is an isolated singularity of f(z)

22 1 2

0 1 2 1f z a a z a a z a ............. a z a a z a ...........

Then residue of f(z) at z = a is 1a

29

Residue Theorem

c

f z dz 2 i (sum of residues at the singular points within c )

If f(z) has a pole of order ‘n’ at z=a

n 1

n

n 1

z a

1 dRes f a z a f z

n 1 ! dz

Evaluation Real Integrals

I=

2

0

F cos ,sin d

i ie ecos

2

;

i ie esin

2i

Assume z= ie

1z+

zcos

2;

1 1sin z

2i z

I= n

k

k=1c

dzf z 2 i Res f z

iz

Residue should only be calculated at poles in upper half plane.

Residue is calculated for the function: f z

iz

f x dx 2 i Res f z

Where residue is calculated at poles in upper half plane & poles of f(z) are found

by substituting z in place of x in f(x).

30

PROBABILITY AND STATISTICS

Types of events

Complementary events

cE s E

The complement of an event E is set of all outcomes not in E.

Mutually Exclusive Events

Two events E & F are mutually exclusive iff P(E ∩ F) = 0.

Collectively exhaustive events

Two events E & F are collectively exhaustive iff (E U F) = S

Where S is sample space.

Independent events

If E & F are two independent events

P(E ∩ F) = P (E) * P(F)

De Morgan’s Law

C

Ci

nn

ii 1 i 1

E = EU

C

Ci

n n

ii 1 i 1

E = E

Axioms of Probability

n1 2E ,E ,...........,E are possible events & S is the sample space.

a. 0 ≤ P (E) ≤ 1

b. P(S) = 1

c. n n

i ii=1i 1

P E = P E

for mutually exclusive events

31

Some important rules of probability

P(A U B) = P(A) + P(B) – P(A B)

P(A B) = P(A)* P B | A = P(B) * P A | B

P A | B is conditional probability of A given B.

If A & B are independent events

P(A B) = P(A) * P(B)

P(A | B) = P(A)

P(B | A) = P(B)

Total Probability Theorem

P(A B) = P (A E) + P (B E)

= P(A) * P(E |A) + P(B) * P(E |B)

Baye’s Theorem

P(A |E) = P(A E) + P (B E)

= P(A)* P(E | A) + P(B) * P(E | B)

Statistics

Arithmetic Mean of Raw Data

x

xn

x = arithmetic mean; x = value of observation ; n = number of observations

Arithmetic Mean of grouped data

fxx

f ; f = frequency of each observation

Median of Raw data

Arrange all the observations in ascending order

n1 2x x ............ x

If n is odd, median = n 1

2

th value

If n is even, Median =

th thn n value + 1 value

2 2

2

32

Mode of Raw data

Most frequently occurring observation in the data.

Standard Deviation of Raw Data

i i

22

2

n x x

n

n = number of observations

variance = 2

Standard deviation of grouped data

2i i i i

22N f x f x

N

fi = frequency of each observation

N = number of observations.

variance = 2

Coefficient of variation = CV =

Properties of discrete distributions

a. P x 1

b. E X x P x

22c. V x E x E x

Properties of continuous distributions

f x dx 1

x

F x f x dx = cumulative distribution

E x xf x dx = expected value of x

22V x E x E x = variance of x

33

Properties Expectation & Variance

E(ax + b) = a E(x) + b

V(ax + b) = a2 V(x)

1 2 1 2E ax bx aE x bE x

2 21 2 1 2V ax bx a V x b V x

cov (x, y) = E (x y) – E (x) E (y)

Binomial Distribution

no of trials = n

Probability of success = P

Probability of failure = (1 – P)

n xn x

xP X x C P 1 P

Mean = E(X) = nP

Variance = V[x] = nP(1 – P)

Poisson Distribution

A random variable x, having possible values 0,1, 2, 3,……., is poisson variable if

xe

P X xx!

Mean = E(x) = λ

Variance = V(x) = λ

Continuous Distributions

Uniform Distribution

1 if a x b

f x b a

0 otherwise

Mean = E(x) = b a

2

Variance = V(x) =

2b a

12

34

Exponential Distribution

xe if x 0

f x0 if x 0

Mean = E(x) = 1

Variance = V(x) = 21

Normal Distribution

2

22

x1f x e p , x

22

Means = E(x) = μ

Variance = v(x) = 2

Coefficient of correlation

cov x, y

var x var y

x & y are linearly related, if ρ = ± 1

x & y are un-correlated if ρ = 0

Regression lines

x x = xyb y y

y y = yxb x x

Where x & y are mean values of x & y respectively

xy

b =

cov x, y

var y ;

yxb =

cov x, y

var x

xy yxb b

35

NUMERICAL METHODS

Numerical solution of algebraic equations

Descartes Rule of sign:

An equation f(x) = 0 cannot have more positive roots then the number of sign

changes in f(x) & cannot have more negative roots then the number of sign

changes in f(-x).

Bisection Method

If a function f(x) is continuous between a & b and f(a) & f(b) are of opposite sign,

then there exists at least one roots of f(x) between a & b.

Since root lies between a & b, we assume root

0

a bx

2

If 0f x 0 ; 0x is the root

Else, if 0f x has same sign as f a , then roots lies between 0x & b and

we assume

01

x bx

2

, and follow same procedure otherwise if 0f x has same sign

as f b , then

root lies between a & 0x & we assume 0

1

a xx

2

& follow same procedure.

We keep on doing it, till nf x , i.e., nf x is close to zero.

No. of step required to achieve an accuracy

e

e

b alog

nlog 2

Regula-Falsi Method

This method is similar to bisection method, as we assume two value 10x & x

such that

10f x f x 0 .

36

1 0

0

0 12

1

f x .x f x .xx

f x f x

If f(x2)=0 then x2 is the root , stop the process.

If f(x2)>0 then

2 0

0

0 23

2

f x .x f x .xx

f x f x

If f(x2)

37

Order of convergence

Bisection = Linear

Regula Falsi = Linear

Secant = superlinear

Newton Raphson = quadratic

Numerical Integration

Trapezoidal Rule

b

a

f x dx , can be calculated as

Divide interval (a, b) into n sub-intervals such that width of each interval

b a

hn

we have (n + 1) points at edges of each intervals

1 2 n0x ,x ,x ,.........., x

n n0 0 1 1y f x ; y f x ,..................., y f x

1 2b

n0 n 1a

hf x dx y 2 y y .......... y y

2

Simpson’s 13

rd Rule

Here the number of intervals should be even

b a

hn

5 41 3 2b

n0 n 1 n 2a

hf x dx y 4 y y y .......... y 2 y y ................ y y

3

38

Simpson’s 38

th Rule

Here the number of intervals should be even

b

n4 5........0 1 2 n 1 3 6 9 n 3a

3hf x dx y 3(y y y y y ) 2(y y y ..............y ) y

8

Truncation error

Trapezoidal Rule:

2bound

(b a)T h max f "

12 and order of error =2

Simpson’s 13

Rule:

4

iv

bound

(b a)T h max f

180 and order of error =4

Simpson’s 38

th Rule:

4

iv

bound

3(b a)T h max f

n80 and order of error =5

where n0x x

Note : If truncation error occurs at nth order derivative then it gives exact result while

integrating the polynomial up to degree (n-1).

Numerical solution of Differential equation

Euler’s Method

dy

f x, ydx

To solve differential equation by numerical method, we define a step size h

We can calculate value of y at 0 0 0x h, x 2h,.........., x nh & not any

intermediate points.

i 1 i i iy y hf x , y

iiy y x ; i 1i 1y y x ; ii 1X X h

Modified Euler’s Method (Heun’s method)

01 0 0 0 0h

y y f x , y f x h, y h2

Runge – Kutta Method

1 0y y k

39

41 2 31k k 2k 2k k6

0 01k hf x , y

10 02kh

k hf x , y2 2

20 03kh

k hf x , y2 2

0 0 34k hf x h, y k

Similar method for other iterations

1

Contents

Manual for Kuestion .......................................................................... 2

Type 1: Linear Algebra ........................................................................ 3

Type 2: Differential Equations ............................................................. 5

Type 3: Probability and Statistics ........................................................ 7

Type 4: Numerical Methods ................................................................ 9

Type 5: Complex Functions ............................................................... 11

Type 6: Laplace Transform ................................................................ 14

Type 7: Calculus ................................................................................ 16

Type 8: Vector Calculus ..................................................................... 18

Answer Key ....................................................................................... 20

© 2014 Kreatryx. All Rights Reserved.

3

Type 1: Linear Algebra

For Concepts, please refer to Engineering Mathematics K-Notes, Linear Algebra

Unsolved Problems:

Q.1 If for a matrix, rank equals both the number of rows and number of columns, then the

matrix is called

(A) Non-singular (B) singular (C) transpose (D) minor

Q.2 The equation 2

2 1 1

1 1 1 0

y x x

represents a parabola passing through the points.

(A) (0, 1), (0, 2), (0, -1) (B) (0, 0), (-1, 1), (1, 2)

(C) (1, 1), (0, 0), (2, 2) (D) (1, 2), (2, 1), (0, 0)

Q.3 If matrix 2

a 1X

a a 1 1 a

and 2X X I 0 then the inverse of X is

(A) 2

1 a 1

a a

(B) 2

1 a 1

a a 1 a

(C) 2

a 1

a a 1 a 1

(D) 2a a 1 a

1 1 a

Q.4 Consider the matrices 4 3 4 3 2 3

X , Y andP . The order of T1T TP X Y P

will be

(A) 2x2 (B) 3x3 (C) 4x3 (D) 3x4

Q.5 The rank of the matrix

1 1 1

1 1 0

1 1 1

is

(A) 0 (B) 1 (C) 2 (D) 3

Q.6 The matrix

2 2 3

M 2 1 6

1 2 0

has given values -3, -3, 5. An eigen vector corresponding

to the eigen value 5 is T

1 2 1 . One of the eigen vector of the matrix 3M is

(A) T

1 8 1 (B) T

1 2 1

(C) T

31 2 1 (D) T

1 1 1

4

Q.7 The system of equations x + y + z = 6, x + 4y + 6z = 20, x + 4y + z has no solutions

for values of and given by

(A) 6, 20 (B) 6, 20

(C) 6, 20 (D) 6, 20

Q.8 Let P be 2x2 real orthogonal matrix and x is a real vector T

1 2x x with length

1 2

2 21 2

x x x . Then which one of the following statement is correct?

(A) px x where at least one vector satisfies px x

(B) px x for all vectors x

(C) px x where at least one vector satisfies px x

(D) No relationship can be established between x and px

Q.9 The characteristics equation of a 3 x 3 matrix P is defined as

3 2I P 2 1 0 . If I denotes identity matrix then the inverse of P will be

(A) 2P P 2I (B) 2P P I

(C) 2P P I (D) 2P P 2I

Q.10 A set of linear equations is represented by the matrix equations Ax = b. The necessary

condition for the existence of a solution for the system is

(A) A must be invertible

(B) b must be linearly dependent on the columns of A

(C) b must be linearly independent on the columns of A

(D) None.

Q.11 Consider a non-homogeneous system of linear equations represents mathematically an

over determined system. Such a system will be

(A) Consistent having a unique solution.

(B) Consistent having many solution.

(C) Inconsistent having a unique solution.

(D) Inconsistent having no solution.

Q.12 The trace and determinant of a 2x2 matrix are shown to be -2 and -35 respectively. Its

eigen values are

(A) -30, -5 (B) -37, -1 (C) -7, 5 (D) 17.5, -2

5

Type 2: Differential Equations For Concepts, please refer to Engineering Mathematics K-Notes, Differential Equations.

Unsolved Problems:

Q.1 For the differential equation dy

f x, y g x,y 0dx

to be exact is

(A) gf

y x

(B)

gf

x y

(C) f = g (D)2

2

2

2

gf

x y

Q.2 The differential equation dy

py Qdx

, is a linear equation of first order only if,

(A) P is a constant but Q is a function of y

(B) P and Q are functions of y (or) constants

(C) P is function of y but Q is a constant

(D) P and Q are function of x (or) constants

Q.3 Biotransformation on of an organic compound having concentration (x) can be modelled

using an ordinary differential equation 2dx kx 0

dt , where k is reaction rate constant. If x =

a at t = 0 then solution of the equation is

(A) kxx a e (B)

1 1 k tax

(C) ktx a 1 e (D) x = a + k t Q.4 Transformation to linear form by substituting v=

1 ny of the equation

ndy

p t y q t ydt

, n > 0 will be

(A) dv

1 n pv 1 n qdt

(B) dv

1 n pv 1 n qdt

(C) dv

1 n pv 1 n qdt

(D) dv

1 n pv 1 n qdt

Q.5 For

22x

2

d y dy4 3y 3e

dxdx , the particular integral is

(A) 2x1 e

15 (B)

2x1 e5

(C) 2x3e (D)

3x1 2

xC e C e

6

Q.6 Solution of the differential equation dy

3y 2x 0dx

represents a family of

(A) ellipses (B) circles (C) parabolas (D) hyperbolas

Q.7 Which one of the following differential equation has a solution given by the function

y 5sin 3x3

(A) dy 5

cos 3x 0dx 3

(B) dy 5

cos3x 0dx 3

(C)

2

2

d y9y 0

d x (D)

2

2

d y9y 0

dx

Q.8 The order and degree of a differential equation

3

3

32d y dy4 y 0

dxdx

are

respectively

(A) 3 and 2 (B) 2 and 3 (C) 3 and 3 (D) 3 and 1

Q.9 The maximum value of the solution y (t) of the differential equation y(t)+ y..

(t) = 0 with

initial conditions y 0 1.

and y(0) = 1, for t ≥ 0 is

(A) 1 (B) 2 (C) (D) 2

Q.10 It is given that y 2y y 0 , y(0) = 0 & y(1) = 0. What is y(0.5) ?

(A) 0 (B) 0.37 (C) 0.82 (D) 1.13

Q.11 A body originally at 60° cools down to 40 in 15 minutes when kept in air at a temperature

of 25°c. What will be the temperature of the body at the and of 30 minutes?

(A) 35.2° C (B) 31.5°C (C) 28.7°C (D) 15°C

Q.12 The solution

2

2

d y dy2 17y 0

dxdx ;

x4

dyy 0 1, 0

dx

in the range 0 x

4

is given by

(A) x1

e cos4x sin4x4

(B) x

1e cos4x sin4x

4

(C) 4x1

e cos4x sin x4

(D) 4x

1e cos4x sin4x

4

7

Type 3: Probability and Statistics

For Concepts, please refer to Engineering Mathematics K-Notes, Probability and Statistics.

Unsolved Problems:

Q.1 The probability that it will rain today is 0.5. the probability that it will rain tomorrow is 0.6.

The probability that it will rain either today or tomorrow is 0.7. What is the probability that it

will rain today and tomorrow?

(A) 0.3 (B) 0.25 (C) 0.35 (D) 0.4

Q.2 Let P(E) denote the probability of an event E. Given P(A) = 1. P(B) = 1

2 the values of

P(A/B) and P(B/A) respectively are

(A) 1 1

,4 2

(B) 1 1

,2 4

(C) 1

,12

(D) 1

1,2

Q.3 If P and Q are two random events, then which of the following is true?

(A) Independence of P and Q implies that probability P P Q 0

(B) Probability P Q Probability (P) + probability (Q)

(C) If P and Q are mutually exclusive then they must be independent

(D) Probability P Q Probability (P)

Q.4 The random variable X takes on the values 1, 2 (or) 3 with probabilities 2 5P 1 3P

,5 5

and

1.5 2P

5

respectively the values of P and E(X) are respectively

(A) 0.05, 1.87 (B) 1.90, 5.87 (C) 0.05, 1.10 (D) 0.25, 1.40

Q.5 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

Q.6 2 x 3 x

XP X Me Ne

is the probability density function for the real random variable

X, over the entire x – axis, M and N are both positive real numbers. The equation relating M

and N is

(A) 2

M N 13

(B) 1

2M N 13

(C) M +N = 1 (D) M+N = 3

8

Q.7 There value of x and y are to be fitted in a straight line in the form y= a + bx by the

method of least squares. Given x 6 , y 21 , 2x 14 , xy 46 , the values of a

and b are respectively

(A) 2, 3 (B) 1, 2 (C) 2, 1 (D) 3, 2

Q.8 A fair coin is tossed 10 times. What is the probability that only the first two tosses will yield

heads?

(A)

21

2

(B)

2

2

110c

2

(C)

101

2

(D)

10

2

110c

2

Q.9 A discrete random variable X takes value from 1 to 5 with probabilities as shown in the

table. A student calculates the mean of X as 3.5 and her teacher calculates the variance to X as

1.5. Which of the following statements is true?

K 1 2 3 4 5

P X K 0.1 0.2 0.4 0.2 0.1

(A) Both the student and the teacher are right

(B) Both the student and the teacher are wrong

(C) The student is wrong but the teacher is right

(D) The student is right but the teacher is wrong

Q.10 It is estimated that the average number of events during a year is three. What is the

probability of occurrence of not more than two events over a two-year duration? Assume that

the number of events follow a poisson distribution.

(A) 0.052 (B) 0.062 (C) 0.072 (D) 0.082

Q.11 The annual precipitation data of a city is normally distributed with mean and standard

deviation as 1000 mm and 200 mm. respectively. The probability that the annual precipitation

will be more than 1200 mm is

(A) < 50 % (B) 50% (C) 75% (D) 100%

Q.12 An unbiased coin is tossed five times. The outcome of each loss is either a head or a tail.

Probability of getting at least one head is __________

(A) 1

32 (B)

13

32 (C)

16

32 (D)

31

32

9

Type 4: Numerical Methods For Concepts, please refer to Engineering Mathematics K-Notes, Numerical Methods.

Unsolved Problems:

Q.1The formula used to compute an approximation for the second derivative of function f at

a point 0x is

(A) 0 0f x h f x h

2

(B)

0 0f x h f x h

2h

(C) 0 0 0

2

f x h 2f x f x h

h

(D)

0 0 02

f x h 2f x f x h

h

Q.2 The Newton-Raphson method is used to find the root of the equation 2x 2 0 . If the

iterations are started from -1, then the iteration will

(A) converge to –1 (B) converge to 2

(C) converge to 2 (D) not converge.

Q.3 In the interval 0, the equations x = cos x has

(A) No solution (B) Exactly one solution

(C) Exactly 2 solutions (D) An infinite number of solutions

Q.4 The polynomial 5p x x x 2 has

(A) all real roots. (B) 3 real and 2 complex roots.

(C) 1 real and 4 complex roots. (D) all complex roots.

Q.5 Match the following and choose the correct combination

Group – I

E. Newton-Raphson method

F. Runge-Kutta method

G. Simpson’s Rule

H. Gauss elimination

10

Group – II

1) Solving non-linear equations.

2) Solving linear simulations equations.

3) Solving ordinary differential equations.

4) Numerical integration method.

5) Interpolation.

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.

Q.6 The real root of the equation xxe 2 is evaluated using Newton-Raphson’s method. If the

first approximation of the value of x is 0.8679, the 2nd approximation of the value of x correct

to three decimal places is

(A) 0.865 (B) 0.853 (C) 0.849 (D) 0.838

Q.7 The estimate of 1.5

0.5

dx

x obtained using Simpson’s rule with three-point function evaluation

exceeds the exact value by

(A) 0.235 (B) 0.068 (C) 0.024 (D) 0.012

Q.8 During the numerical solution of a first order differential equation using the Euler (also

known as Euler Cauchy) method with step size h, the local truncation error is of the order of

(A)2h (B)

3h (C)4h (D)

5h

Q.9 For solving algebraic and transcendental equation which one of the following used?

(A) Coulomb’s theorem

(B) Newton – Raphson method

(C) Euler’s method

(D) Stoke’s method

Q.10 The Newton-Raphson iteration can be used to compute

(A) Square of R (B) reciprocal of R

(C) Square root of R (D) Logarithm of R

11

Q.11 Equation xe 1 0 is required to be solved using Newton’s method with an initial guess

0x 1 . Then after one step of Newton’s method estimate 1x of the solution will be given by

(A) 0.71828 (B) 0.36784 (C) 0.20587 (D) 0.0000

Q.12 A numerical solution of the equation f x x x 3 0 can be obtained using

Newton-Raphson method. If the starting value is x=2 for the iteration then the value of x that

is to be used in the next step

(A) 0.306 (B) 0.739 (C) 1.8124 (D) 2.306

Type 5: Complex Functions

For Concepts, please refer to Engineering Mathematics K-Notes, Complex Functions.

Unsolved Problems:

Q.1 Consider the circle z 5 5i 2 in the complex number plane (x, y) with z = x + iy. The

minimum distance from the origin to the circle is

(A) 5 2 2 (B) 54 (C) 34 (D) 5 2

Q.2 For the function of a complex variable W lnz (Where W = u + jv and z = x + jy) the u

= constant lines get mapped in the z – plane as

(A) set of redial straight lines

(B) set of concentric circles

(C) set of confocal hyperbolas 2

(D) set of confocal ellipses.

Q.3 Potential function is given as 2 2x y . What will be the stream function with

the condition = 0 at x = 0, y = 0?

(A) 2xy (B) 2 2x y (C) 2 2x y (D) 2 22x y

Q.4 If (x, y) and (x, y) are function with continuous 2nd derivatives then

x,y i x,y can be expressed as an analytic function of x + iy i 1 when

(A) ,x x y y

(B) ,

y x x y

(C) 2

2 2 2

2

2

2 21

xx y y

(D) 0

x y x y

12

Q.5 If the semi-circular contour D of radius 2 is as shown in the figure. Then the value of the

integral D

2

1 ds

s 1 is

(A) i π

(B) –i π

(C) – π

(D) π

Q.6 Given

2

zX z

z a

with | z | > a, the residue of n 1X z z at z = a for n ≥ 0 will be

(A) n 1a (B) na (C) n na (D) n n 1a

Q.7 An analytic function of a complex variable z = x + i y is expressed as f(z) = u(x, y)+I v (x,

y) where i = 1 , If u = xy then the expression for v should be

(A)

2x y

k2

(B)

2x yk

2

(C)

2 2y xk

2

(D)

2

x yk

2

Q.8 The value of

C

2

4

z dz

z 1 , using Cauchy’s integral around the circle |z + 1| = 1 where

z x i y is

(A) 2 i (B) i2

(C) 3 i2

(D) 2i

Q.9 Consider likely applicability of cauchy’s Integral theorem to evaluate the following

integral counter clock wise around the unit circle C

C

I secz dx, z being a complex

variable. The value of I will be

(A) I = 0 ; Singularities set =

(B) I = 0 ; Singularities set = 2n 1

/n 0,1,2,2

(C) I2

; Singularities set = n ; n 0,1,2,

(D) None of the above.

13

Q.10 Given 1 2

f zz 1 z 3

. If C is a counter-clockwise path in the z-plane such that

z 1 1 , the value of C

1f z dz

2 j is

(A) –2 (B) –1 (C) 1 (D) 2

Q.11 For an analytic function f(x + I y) =u (x, y) + i v (x, y), u is given by 22u 3x 3y . The

expression for v, considering K is to be constant is

(A) 223y 3x k (B) 6x – 6y + k

(C) 6y – 6x = k (D) 6xy + k

Q.12 The contour C in the adjoining figure is described by 2 2x y 16 . Then the value of

2

C

z 8dz

0.5 z 1.5 j

(A) 2 j (B) 2 j (C) 4 j (D) 4 j

14

Type 6: Laplace Transform

For Concepts, please refer to Signals and Systems K-Notes, Laplace Transform.

Unsolved Problems:

Q.1 The Laplace transform of f(t) is F(s). Given 2 2

F ss

, the final value of f(t) is ____.

(A) two (B) zero (C) one (D) none

Q.2 The inverse Laplace transform of the function

s 5

s 1 s 3

is ____________

(A) t 3t2e e (B) t 3t2e e

(C) t 3te 2e (D) t 3te 2e

Q.3 The Laplace transform of 2t 2t u t 1 is ________________.

(A) s s3 2

2 2e e

s s

(B) 2s s3 2

2 2e e

s s

(C) s s3

2 2e e

ss

(D) None

Q.4 If

t2

2

0

s 2 s 1L f t , L g t , h t f T g t T dT

s 3 s 2s 1

then L{h(t)} is _________.

(A) 2s 1

s 3

(B)

1

s 3

(C)

2

2

s 1 s 2

s 3 s 2 s 1 (D) None.

Q.5 The Laplace transform of the following function is

sint for 0 tf t

0 for t

(A) 21

for all s 0(1 s )

(B) 21

for all s(1 s )

(C)

s

2

(1 e )for all s 0

(1 s ) (D)

s

2

efor all s 0

(1 s )

Q.6 The Laplace transform of f(s) is 21

s s 1. The function

(A)tt 1 e (B)

tt 1 e (C)t1 e (D)

t2t e

15

Q.7 Given f(t) and g(t) as shown below

g(t) can be expressed as

(A) g t f 2t 3 (B) t

g t f 32

(C) 3

g t f 2t2

(D)

t 3g t f

2 2

Q.8 If n n

x n 1 3 1 2 u n , then the region of convergence (ROC) of its Z-transform in

the Z-plane will be

(A) 1

z 33 (B)

1 1z

3 2

(C) 1

z 32 (D)

1z

3

Q.9 The function f(t) satisfies the differential equation 2

2

d ff 0

dt and the auxiliary

conditions, df

f 0 0, 0 4dt

. The Laplace transform of f(t) is given by

(A) 2

s 1 (B)

4

s 1 (C)

2

4

s 1 (D)

2

2

s 1

Q.10 In what range should Re(s) remain so that the Laplace transform of the function

a 2 t 5e exist?

(A) Re (s) > a + 2 (B) Re (s) > a + 7

(C) Re (s) < 2 (D) Re (s) > a + 5

Q.11 Laplace transform of f(x) = cos h(ax) is

(A) 2 2

a

s a (B)

2 2

s

s a

(C) 2 2

a

s a (D)

2 2

s

s a

16

Q.12 If Z=x+iy where x,y are real then the value of i ze

(A)1 (B) 2 2x y

e (C) ye (D) ye

Type 7: Calculus

For Concepts, please refer to Engineering Mathematics K-Notes, Calculus.

Unsolved Problems:

Q.1 If an every point of a certain curve, the slope of the tangent equal 2x

y, the curve is

(A) A straight line (B) A parabola

(C) A circle (D) An Ellipse

Q.2 If f(0) = 2 and f’(x) = 21

5 x, then the lower and upper bounds of f(1) estimated by the

mean value theorem are ________

(A) 1.9, 2.2 (B) 2.2, 2.25

(C) 2.25, 2.5 (D) None of the above

Q.3 Number of inflection points for the curve 4y x 2x is _______________

(A) 3 (B) 1 (C) n (D) 2

n 1

Q.4 Limit of the function,

n 2

nLim

n n is ______________

(A) 1 2 (B) 0 (C) (D) 1

Q.5 Consider the following integral

4

a1

a

Lim x dx ___________

(A) diverges (B) converges to 1/3

(C) converges to 31

a (D) converges to 0

Q.6 The function 3 2f x 2x 3x 36x 2 has its maxima at

(A) x = – 2 only (B) x = 0 only

(C) x = 3 only (D) both x = –2 and x = 3

17

Q.7 Changing the order order of integration in the double integral 8 2

x4

0

I f x, y dy dx

leads to q8

Pr

I f x, y dy dx. What is q ?

(A) 4y (B) 216y (C) x (D)8

Q.8 By a change of variables x, (u, v) = uv, vy u,v u in a double integral, the integral

f(x, y) changes to uf uv, v . Then u,v is _______.

(A) 2v

u (B) 2 uv (C)

2V (D) 1

Q.9 Consider the function f(x) = |x|3, where x is real. Then the function f(x) at x = 0 is

(A) Continuous but not differentiable

(B) Once differentiable but not twice.

(C) Twice differentiable but not thrice.

(D) thrice differentiable

Q.10 The series 2m

mm 0

1x 1

4

converges for

(A) – 2 < x < 2 (B) –1 < x < 3

(C) –3 < x < 1 (D) x < 3

Q.11 A political party orders an arch for the entrance to the ground in which the annual

convention is being held. The profile of the arch follows the equation y = 2x – 0.1x2 where y is

the height of the arch in meters. The maximum possible height of the arch is

(A) 8 meters (B) 10 meters

(C) 12 meters (D) 14 meters

Q.12 Given i 1 , wheat will be the evaluation of the definite integral 0

2 cosx i sinxdx

cosx i sinx

(A) 0 (B) 2 (C) –i (D) i

18

Type 8: Vector Calculus

For Concepts, please refer to Engineering Mathematics K-Notes, Vector Calculus.

Unsolved Problems:

Q.1 The directional derivative of 2 2 2f x, y 2x 3y z at point P(2, 1, 3) in the direction of

the vector a 1 2k is

(A) 4 5 (B) 4 5 (C) 5 4 (D) 5 4

Q.2 For the function 2 3ax y y to represent the velocity potential of an ideal fluid, 2

should be equal to zero. In that carve the value of ‘a’ has to be

(A) -1 (B) 1 (C) –3 (D) 3

Q.3 Value of the integral 2

cxydy y dx , where c is the square cut from the first quadrant by

the line x = 1 and y = 1 will be

(A) 1/2 (B) 1 (C) 3/2 (D) 5/3

Q.4 For the scalar field 22 yx

u2 3

, the magnitude of the gradient at the point (1,3) is

(A) 13

9 (B)

9

2 (C) 5 (D)

9

2

Q.5 If r is the position vector of any point on a closed surface S that enclosed the volume V

then S

r .ds is equal to

(A) 1

V2

(B) V (C) 2V (D) 3V

Q.6 The line integral 1

2

P

P

ydx xdy from 1 1 1 2 2 2P x , y to P x , y along the semi-circle 1 2P P

shown in the figure is

(A) 2 2 2 1x y x y

(B) 2 2 2 22 1 2 1y y x x (C) 2 1 2 1x x y y

(D) 2 2

2 1 2 1y y x x

19

Q.7 For the parallelogram OPQR shown in the sketch, ˆ ˆOP a i b j and ˆ ˆQR c i d j . The

area of the parallelogram is

(A) ad – bc

(B) ac + bd

(C) ad + bc

(D) ab – cd

Q.8 2 2x yF x, y x xy a y xy a . Its line integral over the straight line from

x, y 0,2 to x, y 2,0 evaluates to

(A) -8 (B) 4 (C) 8 (D) 0

Q.9 2x yA xy a x a then A.d lover the path shown in the figure is

(A) 0

(B) 2

3

(C) 1

(D) 2 3

Q.10 The two vectors [1,1,1] and 21,a,a where 1 3

a j2 2

are

(A) Orthonormal

(B) Orthogonal

(C) Parallel

(D) Collinear

Q.11 Divergence of the 3 – dimensional radial vector r is

(A) 3 (B) 1

r

(C) ˆ ˆi j k (D) ˆ ˆ3 i j k

Q.12 The angle (in degrees) between two planar vectors 3 1

a i j2 2

and 3 1

b i j2 2

(A)30 (B)60 (C)90 (D)120

20

Answer Key

1 2 3 4 5 6 7 8 9 10 11 12

Type 1 A B B A C B B B D B D C

Type 2 B D B A B A C A D A B A

Type 3 D D D A C A D C B B A D

Type 4 D C B C C B D A B C A C

Type 5 A B A B A D C B A C D D

Type 6 D A C B C A D C C A B D

Type 7 D B B D B A A A C B B D

Type 8 B D C C D A D D C B A D

Kreatryx

Subject Test

Engineering

Maths

www.kreatryx.com

1

KST- General Instructions during Examination

1. Total Duration of KST is 60 minutes.

2. The question paper consists of 2 parts. Questions 1-10 carry one mark each and Question

11-20 carry 2 marks each.

3. The question paper may consist of questions of Multiple Choice Type (MCQ) and

Numerical Answer Type.

4. Multiple choice type questions will have four choices against A, B, C, D, out of which

only ONE is the correct answer.

5. All questions that are not attempted will result in zero marks. However, wrong

answers for multiple choice type questions (MCQ) will result in NEGATIVE marks.

For all MCQ questions a wrong answer will result in deduction of 𝟏/𝟑 marks for a 1-mark

question and 𝟐/𝟑 marks for a 2-mark question.

6. There is NO NEGATIVE MARKING for questions of NUMERICAL ANSWER TYPE.

7. Non-programmable type Calculator is allowed. Charts, graph sheets, and

mathematical tables are NOT allowed during the Examination.

2

Q.1 The solution of the differential equation 2

222

d yk y y

dx under the boundary condition

(i) 1

y y at x = 0 and (ii) 2

y y at x where k, 1

y and 2

y are constant is

(A) 2xk

1 2 2y y y e y

(B) x

k2 1 1

y y y e y

(C) 1 2 1x

y y y sin h yk

(D) 1 2 2

xky y y e y

Q.2 In a class of 200 students 125 students have taken programming languages course, 85

students have taken data structures course, 65 students have taken computer organization

course, 50 students have taken both programming languages and data structures, 35 students

have taken both programming languages and computer organization, 30 students have taken

both data structures and computer organization, 15 students have taken all the three courses.

How many students have not taken any of the three courses?

(A) 15 (B) 20

(C) 25 (D) 35

Q.3 For the spherical surface 2 2 2x y z 1 , the unit outward normal vector at the point

1 1, ,0

2 2

is given by

(A) 1 1ˆ ˆi j2 2

(B) 1 1ˆ ˆi j2 2

(C) k̂ (D) 1 1 1ˆ ˆ ˆi j k3 3 3

Q.4 Consider the shaded triangular region P shown in the figure. What is

P

xy dx dy ?

(A) 1

6

(B) 2

9

(C) 7

16

(D) 1

3

Q.5 The value of C

Sin Z dz

z, where the contour of the integration is a simple closed curve

around the origin is

(A) 0 (B) 2 j

(C) (D) 1

2 j

Q.6 In matrix algebra AS = AT (A, S, T, are matrices of appropriate order) implies S = T only if

(A) A is symmetric (B) A is singular

(C) A is non-singular (D) A is skew-symmetric

Q.7 Choose the CORRECT set of functions, which are linearly dependent.

(A) 2 2sinx, sin x and cos x (B) cosx, sinx and tanx

(C) 2 2 2cos x, sin x and cos x (D) 2cos x, sinx and cos x

Q.8 The partial differential equation 2

2u u uu

t x t

is a

(A) Linear equation of order 2

(B) Non-linear equation of order 1

(C) Liner equation of order 1

(D) Non-linear equation of order 2

Q.9 With initial condition x(1) = 0.5, the solution of the differential equation dx

t x tdt

is

(A) 1

x t2

(B) 2 1x t

2

(C)

2txt

2 (D)

tx

2

Q.10 Suppose p is the number of cars per minute passing through a certain road junction

between 5PM and 6PM, and p has a poisson distribution with mean 3. What is the probability

of observing fewer than 3 cars during any given minute in this interval?

(A) 38 / 2e (B) 39 / 2e (C) 317 / 2e (D) 326 / 2e

Q.11 The solution for 4 3

0

/6

cos 3 sin 60d

(A) 0 (B) 1

15 (C) 1 (D)

8

3

4

Q.12 Given that one root of the equation 3 2x 10x 31x 30 0 is 5 then other roots are

(A)2 and 3 (B)2 and 4 (C)3 and 4 (D)-2 and -4

Q.13 The value of the integral of the function g(x, y) = 434x 10y along the straight line

segment from the point (0, 0) to the point (1, 2) in the xy-plane is

(A) 33 (B) 35 (C) 40 (D) 56

Q.14 A box contains 4 white balls and 3 red balls. In succession, two balls are randomly

selected and removed from the box. Given that first removed ball is white, the probability that

the 2nd removed ball is red is

(A) 1/3 (B) 3/7 (C) 1/2 (D) 4/7

Q.15 All the four entries of 2 x 2 matrix 11 12

21 22

P PP

P P

are non-zero and one of the eigen values

is zero. Which of the following statement is true?

(A) 11 22 12 21P P P P 1 (B) 11 22 12 21P P P P 1

(C) 11 22 21 12P P P P 0 (D) 11 22 12 21P P P P 0

Q.16 For the function f(x, y) = 2 2X y defined on

2R , the point (0, 0) is

(A) a local minimum

(B) Neither a local minimum (nor) a local maximum.

(C) a local maximum

(D) Both a local minimum and a local maximum.

Q.17 Consider the differential equation y 2y y 0 with boundary conditions y(0) = 1 &

y(1)=0. The value of y(2) is

(a) – 1 (B) 1e (C) 2e (D) 2e

Q.18

x 2y z 4

2x y 2z 5

x y z 1

The system of algebraic equations given above has

(A) a unique solution of x=1, y=1 and z=1

(B) only the two solutions of ( x=1, y=1, z=1 ) and (x=2, y=1, z=0)

(C) infinite number of solution

(D) no feasible solution

5

Q.19 The value of the following definite integral in

2

2 sin2xdx

1 cos x= ________________

(A) – 2 log 2 (B) 2 (C) 0 (D) None

Q.20 Which one of the following does NOT equal

2

2

2

1 x x

1 y y ?

1 z z

(A)

1 x x 1 x 1

1 y y 1 y 1

1 z z 1 z 1

(B)

2

2

2

1 x 1 x 1

1 y 1 y 1

1 z 1 z 1

(C)

2 2

2 2

2

0 x 1 x y

0 y 1 y z

1 z z

(D)

2 2

2 2

2

2 x 1 x y

2 y z y z

1 z z

6

Electrical and Electronic Measurements KST

Answer Key

1 C

2 B

3 C

4 C

5 A

6 D

7 A

8 B

9 C

10 3.34(3.33-3.35)

11 3.22(3.21-3.23)

12 D

13 A

14 D

15 B

16 D

17 C

18 1.31(1.30-1.32)

19 14.49(14.48-14.50)

20 1.06(1.05-1.07)

Engineering Maths.pdfEng maths_K-Notes.pdfBack Page.pdfEngineering Maths Kuestion.pdfEngg Maths Kuestion Front.pdfEng maths _Kuestions.pdfBack Page.pdf

Engineering Maths KST.pdfFront - KST (Network Theory).pdfEng maths_KST.pdfBack Page.pdf