GATE-O-PEDIA Computer Science & Information Technology PDF

OPEDIA GATE COMPUTER SCIENCE & INFORMATION TECHNOLOGY Published By: Physics Wallah ISBN: 978-93-94342-51-4 Mobile App: Physics Wallah (Available on Play Store) Website: www.pw.live Email: [email protected] Rights All rights will be reserved by Publisher. No part of this book may be used or reproduced in any manner whatsoever without the written permission from author or publisher. In the interest of student's community: Circulation of soft copy of Book(s) in PDF or other equivalent format(s) through any social media channels, emails, etc. or any other channels through mobiles, laptops or desktop is a criminal offence. Anybody circulating, downloading, storing, soft copy of the book on his device(s) is in breach of Copyright Act. Further Photocopying of this book or any of its material is also illegal. Do not download or forward in case you come across any such soft copy material. Disclaimer A team of PW experts and faculties with an understanding of the subject has worked hard for the books. While the author and publisher have used their best efforts in preparing these books. The content has been checked for accuracy. As the book is intended for educational purposes, the author shall not be responsible for any errors contained in the book. The publication is designed to provide accurate and authoritative information with regard to the subject matter covered. This book and the individual contribution contained in it are protected under copyright by the publisher. (This Module shall only be Used for Educational Purpose.) Design Against Static Load INDEX 1. Engineering Mathematics (EM)........................................................................................... 1.1 – 1.40 2. Discrete Mathematics.......................................................................................................... 2.1 – 2.40 3. Digital Logic......................................................................................................................... 3.1 – 3.75 4. Computer Organization & Architecture.............................................................................. 4.1 – 4.56 5. Programming & Data Structures.......................................................................................... 5.1 – 5.41 6. Algorithms............................................................................................................................ 6.1 – 6.34 7. Theory of Computation....................................................................................................... 7.1 – 7.39 8. Compiler Design.................................................................................................................. 8.1 – 8.23 9. Operating System................................................................................................................. 9.1 – 9.44 10. Database Management Systems......................................................................................... 10.1 – 10.46 11. Computer Networks............................................................................................................. 11.1 – 11.37 12. General Aptitude................................................................................................................. 12.1 – 12.25 Engineering Mathematics Design Against Static Load INDEX 1. Basic Calculus...................................................................................................................... 1.1 – 1.16 2. Linear Algebra...................................................................................................................... 1.17 – 1.28 3. Probability and Statistics..................................................................................................... 1.29 – 1.40 GATE-O-PEDIA COMPUTER SCIENCE & INFORMATION TECHNOLOGY Design Against Static Load 1 BASIC CALCULUS 1.1. Introduction 1.1.1 Limits, Continuity and Differentiability (a) As x tends to a (x → a)  x is moving towards a A value l is said to be limit of a function f(x) at x → a if f(x) → l as x → a. It is mathematically defined as Y𝑖 Z S( e) = Y = Y𝑖 Z− S( e) = Y𝑖 Z+ S( e) 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎 That is, Limit exist at any point, if LHL = RHL A function f(x) is said to be continuous at x = a, if Y𝑖 Z S( e) = Y = S( N) = S( e)|𝑥=𝑎 𝑥→𝑎 That is, for a function to be continuous at any point, RHL = LHL = Value of function at point x = a. Note: For Y𝑖 Z S( e) to exist, the function need not be continuous at x = a. 𝑥→𝑎 But for f(x) to be continuous at x = a, Y𝑖 Z S( e) should exist. 𝑥→𝑎 Continuity from Left : lim f ( x ) = f ( a ) x→a− Continuity from Right : If lim f ( x) = f (a) x→a + Continuity in an Open Interval A function ‘f ’ is said to be continuous in open interval (a, b), if it is continuous at each point of open interval. Continuity in a Closed Interval Let ‘f ’ be a function defined on the closed interval (a, b) then ‘f ’ is said to be continuous on the closed interval [a, b], if it is : 1. Continuous from the right at a and 2. Continuous from the left at b and 3. Continuous on the open interval (a, b). GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.1 Engineering Mathematics Fig. 1.1 (b) Concept of differentiability S(𝑥)− S(𝑎) A continuous function f(x) is said to be differentiable at x = a, if Y𝑖 Z exists, that is, RHL and LHL exist at a point 𝑥→𝑎 𝑥−𝑎 under consideration in S′( e). S(𝑥)− S(𝑎) S′( e)|𝑥=𝑎 = S′( N) = Y𝑖 Z 𝑥→𝑎 𝑥−𝑎 S′( N) = a N [ 𝜃, where 𝜃 is the angle made by the tangent to the curve at x=a with x – axis. (c) Some Standard Derivatives Q (i) ( e [ ) = [. e [−1 Q𝑥 Q (ii) ( `𝑖 [ e) = P \ ` e Q𝑥 Q (iii) ( P \ ` e) = − `𝑖 [ e Q𝑥 Q (iv) ( a N [ e) = ` R P 2 e Q𝑥 Q (v) ( P \ a e) = − P \ ` R P 2 e Q𝑥 Q (vi) ( ` R P e) = ` R P e. a N [ e Q𝑥 Q (vii) ( P \ ` R P e) = – P \ ` R P e P \ a e Q𝑥 Q 1 (viii) ( `𝑖 [−1 e) = ; −1 < e < 1 Q𝑥 √1−𝑥 2 Q −1 (ix) ( P \ ` −1 e) = , −1 < e < 1 Q𝑥 √1−𝑥 2 Q 1 (x) ( a N [−1 e) = Q𝑥 1+𝑥 2 GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.2 Engineering Mathematics Q −1 (xi) ( P \ a −1 e) = Q𝑥 1+𝑥 2 Q 1 (xii) ( ` R P −1 e) = Q𝑥 |𝑥|√𝑥 2 −1 Q −1 (xiii) ( P \ ` R P −1 e) = ; |x| > 1 Q𝑥 |𝑥|√𝑥 2 −1 Q 1 (xiv) ( Y \ T𝑎 e) = Q𝑥 𝑥 Y \ T𝑒 𝑎 Q 1 (xv) ( Y \ T R e) = Q𝑥 𝑥 Q (xvi) ( N 𝑥 ) = N 𝑥. Y \ T R N Q𝑥 Q (xvii) ( R 𝑥 ) = R 𝑥 Q𝑥 Q |𝑥| (xviii) (| e|) = , ( e ≠ 0) Q𝑥 𝑥 Q (xix) ( e 𝑥 ) = e 𝑥 (1 + Y \ T R e) Q𝑥 Q (xx) ( `𝑖 [ℎ e) = P \ ` ℎ e Q𝑥 (d) Product rule of differentiation Q (i) ( S( e). T( e)) = S( e). T′( e) + S′( e). T( e) Q𝑥 (ii) Q( b c d) = b c d′ + b c′ d + b′ c d (e) Quotient rule of differentiation Q S(𝑥) T(𝑥). S′(𝑥)− S(𝑥). T′(𝑥) ( )= 2 , ( T( e) ≠ 0) Q𝑥 T(𝑥) ( T(𝑥)) (f) Logarithmic differentiation: Taking log might help in differentiation of a function. For example if y = vu then we can take log both side and dy differentiable to get dx (g) Differentiation in parametric from : dy dy / dt If we write x and y in term of find variable ‘t’ that is x = f(t), y = (t), then = dx dx / dt (h) Greatest Integer function / step function / integer part function S( e) = [ e ] = [, ∀ [ ≤ e < [ + 1where, [ ∈ 𝑍 Y𝑖 Z [ e] = ∄ if a is an integer ( ∄ = do not exist) 𝑥→𝑎 L.H.L. = Y𝑖 Z− [ e] = N − 1 𝑥→𝑎 R.H.L. = Y𝑖 Z+ [ e] = N 𝑥→𝑎 GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.3 Engineering Mathematics Fig.1.2. Greatest Integer (i) Properties of Limits (i) Y𝑖 Z ( S( e) ± T( e)) = Y𝑖 Z S( e) ± Y𝑖 Z T( e) 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎 (ii) Y𝑖 Z ( S( e). T( e)) = Y𝑖 Z S( e). Y𝑖 Z T( e) 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎 S(𝑥) Y𝑖 Z S(𝑥) (iii) Y𝑖 Z = 𝑥→𝑎 , ( Y𝑖 Z T( e) ≠ 0) 𝑥→𝑎 T(𝑥) Y𝑖 Z T(𝑥) 𝑥→𝑎 𝑥→𝑎 (iv) If Y𝑖 Z S( e) exists and Y𝑖 Z T( e) = ∄, then Y𝑖 Z S( e). T( e) may exist 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎 1 1 Example: Let S( e) = `𝑖 [ e, T( e) = , Y𝑖 Z S( e) = 0, Y𝑖 Z = ∄ 𝑥 𝑥→0 𝑥→0 𝑥 1 But Y𝑖 Z `𝑖 [ e. = 1 𝑥→0 𝑥 (v) Indeterminate form III (00, 1, 0) ( x) If y = lim  f ( x) x→a Then, log y = lim ( x)log  f ( x) x→a Thus 00, 1, 0 will convert into ×0 from which can be solved easily. S(𝑥) 0 ∞ S(𝑥) S′(𝑥) 0 (vi) If Y𝑖 Z = (or) , then Y𝑖 Z = Y𝑖 Z ≠( ) 𝑥→𝑎 T(𝑥) 0 ∞ 𝑥→𝑎 T(𝑥) 𝑥→𝑎 T′(𝑥) 0 S′(𝑥) 0 ∞ S′(𝑥) S′′(𝑥) If Y𝑖 Z = (or) , then Y𝑖 Z = Y𝑖 Z and so on 𝑥→𝑎 T′(𝑥) 0 ∞ 𝑥→𝑎 T′(𝑥) 𝑥→𝑎 T′′(𝑥) S(𝑥) 0 (vii) If Y𝑖 Z ( S( e). T( e)) = 0 × ∞ ⇒ Y𝑖 Z 1 = (Apply L- Hospital Rule again) 𝑥→𝑎 𝑥→𝑎 ( ) 0 𝑔(𝑥) GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.4 Engineering Mathematics (j) Some Standard Limits `𝑖 [ 𝑥 (i) Y𝑖 Z =1 𝑥→0 𝑥 a𝑎 [ 𝑥 (ii) Y𝑖 Z =1 𝑥→0 𝑥 1− P \ ` 𝑎𝑥 𝑎2 (iii) Y𝑖 Z = 𝑥→0 𝑥2 2 `𝑖 [ 𝑥 (iv) Y𝑖 Z =0 𝑥→∞ 𝑥 P \ ` 𝑥 (v) Y𝑖 Z =0 𝑥→∞ 𝑥 (vi) Y𝑖 Z(1 + N e) O/𝑥 = R 𝑎 O 𝑥→0 𝑎 O𝑥 (vii) Y𝑖 Z (1 + ) = R 𝑎 O 𝑥→∞ 𝑥 𝑎𝑥 + O 𝑥 1/𝑥 (viii) Y𝑖 Z ( ) = √ N O 𝑥→0 2 1𝑥 +2𝑥 +3𝑥 +....+ [𝑥 1/𝑥 [ (ix) Y𝑖 Z ( ) = √ [! 𝑥→0 [ 𝑎𝑥 −1 R 𝑥 −1 (x) Y𝑖 Z = Y \ T R N ; Y𝑖 Z =1 𝑥→0 𝑥 𝑥→0 𝑥 1 (xi) Y𝑖 Z e. `𝑖 [ ( ) = 0 𝑥→0 𝑥 1.2 Mean Value Theorems 1.2.1 Lagrange’s Mean Value Theorem (LMVT) If S( e) is continuous in [a, b] and it is differentiable in (a, b) then ∃ at least one point ‘c’ such that c  (a, b) and S( O) − S( N) S′( P) = O − N Here S′( P) slope of tangent to f(x) at x = c. Tangent at x = c is parallel to the line connecting the points A and B Fig.1.3. LMVT GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.5 Engineering Mathematics 1.2.2 Rolle’s Mean Value Theorem If S( e) is continuous in [a, b] and differentiable in (a, b) and f(a) = f(b) then  at least one-point c  (a, b) such that S′( P) = 0. Fig. 1.4. Rolle’s mean value 1.2.3 Cauchy’s Mean Value Theorem If f(x) and g(x) are continuous in [a, b] and differentiable in (a, b) then  at least one value of ‘c’ such that c  (a, b) and T′( 6) T( O)− T(𝑎) = S′( 6) S( O)− S(𝑎) 1.3 Increasing and Decreasing Functions 1.3.1 Increasing Functions A function f(x) is said to be increasing, if S( e1 ) < S( e2 ) ∀ e1 < e2 Or A function f(x) is said to be increasing, if f(x) increases as x increases. For a function S( e) to be increasing at the point x=a, S ′ ( N) > 0. Example: ex, log ex → Monotonically increasing functions sin x in (0, /2) → non-monotonic functions 1.3.2 Decreasing Functions A function f(x) is said to be a decreasing function, if S( e1 ) > S( e2 )∀ e1 < e2 A function S( e) is said to be decreasing function, if S( e) decreases as x increases. 𝜋 Example: R −𝑥 →Monotonically decreasing function, sin e in ( , 𝜋) 2 1.4. Concept of Maxima and Minima Let f(x) be a differentiable function, then to find the maximum (or) minimum of f(x). (1) Find f (x) and equate to zero. GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.6 Engineering Mathematics (2) Solve the resulting equation for x. Let its roots be a1, a2,... then f(x) is stationary at x = a1, a2,...... Thus x = a1, a2,...... are the only points at which f(x) can be maximum or a minimum. (3) Find f (x) and substitute in it by terms x = a1, a2,...... wherever f (x) is negative, we have a maximum and wherever f (x) is positive, we have a minimum. (4) If f (a1) = 0, find f (x) put x = a1 in it. If f (a1)  0, there is neither a maximum nor a minimum at x = a1. If f (a1) = 0, find f iv(x) and put x = a1 in it. If f iv(a1) is negative, we have maximum at x = a1, if it is positive there is a minimum at x = a1. If f iv(a1) is zero, we must find f v(x), and so on. Repeat the above process for each root of the equation f (x) = 0. Example: x = 0 is a critical point of f(x) = x3 Fig. 1.5. Graph of 𝒙𝟑 S( e) = e 3  S′( e) = 3 e 2 = 0  x=0 S′′( e) = 6 e ⇒ S′′(0) = 6(0) = 0 Global maxima and minima : We first find local maxima and minima and then calculate the value of ‘f’ at boundary points of interval given e.g. (a, b) we find f(a) and f(b) and compare it with the values of local maxima and minima. The absolute maxima and minima can be decided then. 1.5. Taylor Series If f(x) is continuously differentiable ( S′( e), S"( e), S′′′( e),..... exists) then the Taylor series expansion of f(x) about the point x = a is given by S′(𝑎) S′′(𝑎) S′′′(𝑎) S( e) = S( N) + ( e − N) + ( e − N)2 + ( e − N)3. +... ∞ 1! 2! 3! S′(0) S′′(0) S′′′(0) If a = 0, then S( e) = S(0) + e + e 2 + e 3 +..... ∞ (Remember that Mc-Lauren Series is same as Taylor 1! 2! 3! Series if a = 0) S [ (𝑎) The coefficient of (x – a)n in the Taylor series expansion of f(x) is. [! S′(𝑥) S′′(𝑥) S′′′(𝑥) The general expansion of Taylor series is given by S( e + ℎ) = S( e) + ℎ. + ℎ2. + ℎ3. +...... ∞ 1! 2! 3! GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.7 Engineering Mathematics Finding the expansion of ex about x = 0 S( e) = R 𝑥 ⇒ S(0) = R 0 = 1 S′( e) = R 𝑥 ⇒ S′(0) = R 0 = 1; S"(0) = S′′′(0) = S′′′′(0) =..... = 1 1 1 1 S( e) = R 𝑥 = 1 + ( e − 0) + ( e − 0)2. + ( e − 0)3. +...... 1! 2! 3! 𝑥 𝑥2 𝑥3  R 𝑥 = 1 + + + +..... 1! 2! 3! 1.6 Integral Calculus 𝑥= O If F(x) is anti-derivative of f(x). That is, continuous and differentiable in (a, b), then we write ∫𝑥=𝑎 S( e) Q e = 9( O) − 9( N). Here f(x) is integrand O If S( e) > 0 ∀ N ≤ e ≤ O, aℎ R [ ∫𝑎 S( e) Q e represents the shaded area in the given figure. y = f(x) x=a x=b Fig.1. 6. Integration of continuous function 1.6.1 Mean Value Theorem of Integration If f(x) is continuous in [a, b] and differentiable in (a, b) then ‘’ atleast one-point c (a, b) such that O ∫𝑎 S(𝑥) Q𝑥 S( P) = ( O−𝑎) Fig. 1.7. Mean value of integration GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.8 Engineering Mathematics 1.7. Newton-Leibnitz Rule If f(x) is continuously differentiable and (x), (x) are two functions for which the 1st derivative exists, then 𝜓(𝑥) Q (∫ S( e) Q e ) = S(𝜓( e)). 𝜓′( e) − S(𝜙( e)). 𝜙′( e) Q e 𝜙(𝑥) Q 𝑥2 Example: (∫𝑥 `𝑖 [ e Q e) = `𝑖 [( e 2 ). 2 e − `𝑖 [ e. 1 = 2 e `𝑖 [( e 2 ) − `𝑖 [ e Q𝑥 1.8. Some Standard Integrals 𝑥 [+1 1. ∫ e [ Q e = [+1 + 6, ( [ ≠ −1) 1 2. ∫ 𝑥 Q e = Y \ T R | e| + 6 3. ∫ `𝑖 [ e Q e = − P \ ` e + 6 4. ∫ P \ ` e Q e = `𝑖 [ e + 6 S′(𝑥) 5. ∫ S(𝑥) Q e = Y \ T R | S( e)| + 6 `𝑖 [ 𝑥 6. ∫ a N [ e Q e = − ∫ − P \ ` 𝑥 Q e = − Y \ T R | P \ ` e| + 6  ∫ a N [ e Q e = Y \ T R | ` R P e | + 6 P \ ` 𝑥 7. ∫ P \ a e Q e = ∫ `𝑖 [ 𝑥 Q e = Y \ T R | `𝑖 [ e| + 6 = − Y \ T R | P \ ` R P e| + 6 ` R P 𝑥( ` R P 𝑥+ a𝑎 [ 𝑥) 8. ∫ ` R P e Q e = ∫ ( ` R P 𝑥+ a𝑎 [ 𝑥) Q e = Y \ T R | ` R P e + a N [ e| + 6 9. ∫ P \ ` R P e Q e = Y \ T R | P \ ` R P e − P \ a e| + 6 𝑎𝑥 10. ∫ N 𝑥 Q e = + 6 Y \ T 𝑒𝑎 1 11. ∫ Q e = Y \ T𝑎 e + 6 𝑥. Y \ T 𝑒𝑎 12. ∫ e 𝑥 (1 + Y \ T R e) Q e = e 𝑥 + 6 1 2 13. ∫ S( e). S′( e) Q e = ( S( e)) + 6 2 S′(𝑥) 14. ∫ Q e = 2. √ S( e) + 6 √ S(𝑥) 15. If f(x), g(x) are two functions. that are differentiable, then ∫ S( e) T( e) Q e = S( e). ∫ T( e) Q e − ∫[ S′( e) T( e)] Q e + 6 GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.9 Engineering Mathematics Before integrating the product, the functions f(x) and g(x) are to be arranged according to the ILATE Principle. Here, ILATE stands for INVERSE LOGARITHMIC ALGEBRAIC TRIGONOMETRIC EXPONENTIAL. 1.9 Properties of Definite Integrals O 𝑎 1. If f(x) is differentiable in interval (a, b), then ∫𝑎 S( e) Q e = − ∫ O S( e) Q e 2. If  a point c  (a, b) such that f(x) is not differentiable, then O P O ∫𝑎 S( e) Q e = ∫𝑎 S( e) Q e + ∫ P S( e) Q e 3. If f(x) is continuously differentiable function, 𝑎 𝑎 ∫−𝑎 S( e) Q e = 2 × ∫0 S( e) Q e; if S(− e) = S( e), (“f(x) is even function”) = 0; if S(− e) = − S( e), (" S( e) is odd function") 2𝑎 𝑎 4. ∫0 S( e) Q e = 2 × ∫0 S( e) Q e, if S(2 N − e) = S( e) O O 5. ∫𝑎 S( e) Q e = ∫𝑎 S( N + O − e) Q e O S(𝑥) O−𝑎 6. ∫𝑎 =( ) S(𝑥)+ S(𝑎+ O−𝑥) 2 Example: 𝜋/2 `𝑖 [ 𝑥 𝜋 (i) ∫0 = `𝑖 [ 𝑥+ P \ ` 𝑥 4 𝜋/2 1 𝜋/2 1 𝜋/2 √ P \ ` 𝑥 𝜋 (ii) ∫0 Q e = ∫0 √ `𝑖 [ 𝑥 Q e = ∫0 Q e = 1+√ a𝑎 [ 𝑥 1+( ) √ P \ ` 𝑥+√ `𝑖 [ 𝑥 4 √ P \ ` 𝑥 3 √𝑥 3−2 1 (iii) ∫2 =( )= √𝑥+√5−𝑥 2 2 𝜋/2 √ a𝑎 [ 𝑥 𝜋 (iv) ∫0 Q e = √ a𝑎 [ 𝑥+√ P \ a 𝑥 4 𝜋/2 𝜋/2 ( Z−1)×( Z−3)×( Z−5) 1 2 7. ∫0 `𝑖 [ Z e Q e = ∫0 P \ ` Z e Q e = ×... ( ) (or) × 𝐾 Z×( Z−2)×( Z−4) 2 3 Where K = /2 if m is even = 1 if m is odd. 𝜋 Q𝑥 𝜋 8. ∫0 = 𝑎2 P \ ` 2 𝑥+ O 2 `𝑖 [2 𝑥 𝑎 O 𝜋/2 Q𝑥 𝜋 9. ∫0 = 𝑎2 P \ ` 2 𝑥+ O2 `𝑖 [2 𝑥 2𝑎 O 1.10 Length of a Curve b   dy 2  (a) The length of the arc of the curve y = f(x) between the points where x = a and x = b is s =  1 +    dx a   dx   GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.10 Engineering Mathematics Fig.1.8. Length of the curve   dx 2  b (b) The length of the arc of the curve x = f ( y ) between the points where y = a and y = b, is s =  1 +    dy a      dy (c) The length of the arc of the curve x = f (t ), y = f (t ) between the points where t = a and t = b, is 2 2 b  dx   dy  s=  dt  +  dt  dt a       2  dr 2  (d) The length of the arc of the curve r = f (), between the points where  =  and  = , is s =  r +   d     d    1.11 Surface Area of Solid generated by revolving a curve about a fixed axis Elemental Surface Area Q𝐴 = 2𝜋 f × Q ` = 2𝜋 f Q ` 𝑥= O Q f 2  Total surface area = A = ∫𝑥=𝑎 2𝜋 f √1 + ( ) Q e Q𝑥 Fig.1.9. Surface area GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.11 Engineering Mathematics 1.12 Volume of the solid A. The volume of the solid obtained by revolving the curve y = f (x) between the lines x = a and x = b is given by 𝑥= O  𝑉 ≈ ∫𝑥=𝑎 𝜋 f 2 Q e Fig. 1.10. Volume of the solid B. Revolution about the y-axis. Interchanging x and y in the above formula, we see that the volume of the solid generated by the revolution, about y-axis, of the area, bounded by the curve x = f ( y), the y-axis and the abscissa y = a, y = b is b a x dy. 2 1.13 Gamma Function ∞ ∞ The integral ∫0 R −𝑥. e [−1 Q e, ( [ > 0) is called Gamma function of n. It is denoted by Γ [ = ∫0 R −𝑥 e [−1 Q e.  m +1  n +1   2   2  /2 Note :  sin m x cosn xdx =  m + n + 2 0 2    2 Where (x) is called the gamma function. 1.13.1 Properties of Gamma Function (i) Γ [ = ( [ − 1)! (ii) Γ( [ + 1) = ( [)! 1 (iii) Γ( [ + 1) = [Γ [ (iv) Γ ( ) = √𝜋 2 1.14 Beta Function 1 The function  (m, n) = ∫0 e Z−1. (1 − e) [−1 Q e (m, n > 0) is called  function of m and n. GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.12 Engineering Mathematics 1.14.1 Properties of  function Γ Z.Γ [ (i) 𝛽( Z, [) = Γ(m+ [) (ii) 𝛽( Z, [) = 𝛽( [, Z) ∞ 𝑥 𝑚−1 (iii) 𝛽( Z, [) = ∫0 Q e (1+𝑥)𝑚+ [ ∞ 𝑥 [−1 𝛽( [, Z) = ∫0 Q e (1+𝑥)𝑚+ [ 1 ]+1 ^+1 (iv) `𝑖 [ ] 𝜃. P \ ` ^ 𝜃 Q e = 𝛽 ( , ) , ( ], ^ > −1) 2 2 2 1.15 Area between the curves If the function f(x) > g(x) for all values of x between x=a and x=b then O O O 𝐴 = ∫𝑎 S( e) Q e − ∫𝑎 T( e) Q e ⇒ 𝐴 = ∫𝑎 ( S( e) − T( e)) Q e Fig. 1.11. Area under curve 1  2 2  Note : Area bounded by curve r = f () between  =  and  is r d 1.16 Multi Variable Calculus (a) Continuity of a function A function f(x, y) is said to be continuous at (a, b), if Y𝑖 Z S( e, f) = S( N, O) 𝑥→𝑎 f→ O (b) Differentiation of a two-variable function If f(x, y) is a continuous function, then the derivative of f(x, y) with respect to x treating y as constant is given by 𝜕 S S(𝑥+ℎ, f)− S(𝑥, f) p= = Y𝑖 Z 𝜕𝑥 ℎ→0 ℎ The derivative of f(x, y) with respect to y treating x as constant is given by 𝜕 S S( e, f + X) − S( e, f) ^ = = Y𝑖 Z 𝜕 f X→0 X (c) Homogenous Function A function f (x, y) is said to be homogenous function of degree ‘n’ if S( X e, X f) = X [. S( e, f). GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.13 Engineering Mathematics Example: S( e, f) = e 3 − 3 e 2 f + 3 e f 2 + f 3  S( X e, X f) = ( X e)3 − 3( X e)2 ( X f) + 3( X e). ( X f)2 + ( X f)3 = X 3 ( e 3 − 3 e 2 f + 3 e f 2 + f 3 ) = X 3. S( e, f)  S( e, f) is a homogenous function of degree ‘3’. (d) Euler’s Theorem If f (x, y) is a homogeneous function of degree ‘n’ then 𝜕 S 𝜕 S (i) e. + f. = [ S 𝜕𝑥 𝜕 f 𝜕2 S 𝜕2 S 𝜕2 S (ii) e 2. + 2 e f + f 2 = [( [ − 1) S 𝜕𝑥 2 𝜕𝑥𝜕 f 𝜕 f 2 If f(x, y) = g(x, y) + h(x, y) + (x, y) where g (x, y), h (x, y) and (x, y) are homogenous functions of degrees m, n and p respectively, then 𝜕 S 𝜕 S e. + f. = Z. T( e, f) + [. ℎ( e, f) + ]. 𝜙( e, f) 𝜕𝑥 𝜕 f 𝜕2 S 𝜕2 S 𝜕2 S e 2. + 2 e f. + f 2. = Z( Z − 1). T( e, f) + [( [ − 1). ℎ( e, f) + ]( ] − 1). 𝜙( e, f) 𝜕𝑥 2 𝜕𝑥𝜕 f 𝜕 f 2 (e) Total derivative: du u dx u dy (i) If u = f(x, y) and if x = (t), y = v(t) then =. +. dt x dt y dt du u u dy (ii) If u be a function of x and y, where y is a function of x, then = +. dx x y dx (iii) If u = f(x, y) and x = f1 (t1, t2 ) and y = f 2 (t1 , t2 ), then u u x u y u u x u y =. +. and =. +. t1 x t1 y t1 t2 x t2 y t2 dy f / x (iv) If x and y are connected by an equation of the form f(x, y) = 0, then =− dx f / y (f) Concept of Maxima and Minima in Two Variables If f(x, y) is a two-variable differentiable function, then to find the maxima (or) minima. 𝜕 S 𝜕 S Step-1: Calculate ] = and ^ = and equate p = 0, q = 0 𝜕𝑥 𝜕 f Let (x0, y0) be a stationary point. 𝜕2 S 𝜕2 S 𝜕2 S Step-2: Calculate r, s, t where _ = | ; ` = | ; a = | 𝜕𝑥 2 (𝑥0 , f0 ) 𝜕𝑥.𝜕 f (𝑥 , f ) 𝜕 f 2 (𝑥 , f ) 0 0 0 0 Case (i): If _ a − ` 2 > 0 and r > 0, then the function f (x, y) has minimum at (x0, y0) and the minimum value is f(x0, y0). Case (ii): If _ a − ` 2 > 0 and r < 0, then the function f (x, y) has maximum at (x0, y0) and the maximum value is GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.14 Engineering Mathematics f(x0, y0). Case (iii): If _ a − ` 2 < 0 ; then we cannot comment on the existence of maxima and minima. Such stationary points where _ a − ` 2 = 0 are called saddle points. (g) Concept of Constraint Maxima and Minima (Method of Lagrange’s unidentified multipliers). If f(x, y, z) is a continuous and differentiable function, such that the variables x, y and z are related/constrained by the equation (x, y, z) = C then to calculate the extreme value of f(x, y, z) using Lagrange’s Method of unidentified multipliers. Step-1: Form the function F(x, y, z) = f(x, y, z) + {(x, y, z) – C}, where 𝜆 is a multiplier. 𝜕𝐹 𝜕𝐹 𝜕𝐹 Step-2: Calculate , and and equate them to zero 𝜕𝑥 𝜕 f 𝜕 g Step-3: Equate the values of  from the above 3 equations and obtain the relation between the variables x, y and z. Step-4: Substitute the relation between x, y and z in (x, y, z) = C and get the values of x, y, z. Let they be (x0, y0, z0). Step-5: Calculate f(x0, y0, z0) The value f(x0, y0, z0) is the extreme value of f(x, y, z). (h) Multiple Integrals If f(x, y) is continuous and differentiable at every point within a region ‘R’ bounded by some curves is given by 𝐼 = ∬𝑅 S( e, f) Q e Q f If there is a double integral, 𝑥= O f=𝜓(𝑥) 𝐼 = ∫𝑥=𝑎 ∫ f=𝜙(𝑥) S( e, f) Q f Q e [Let (x) > (x)] Then I = area of the region bounded by the curves, y = (x); y = (x), x = a and x = b if f(x, y) = 1 Value of x – co-ordinate of centroid of the region bounded by y = (x); y = (x); x = a, x = b if f(x, y) = x (i) Change of Orders of Integration 𝑥= O f=𝜓(𝑥) f= Q 𝑥=ℎ( f) 𝐼 = ∫𝑥=𝑎 ∫ f=𝜙(𝑥) S( e, f) Q f Q e → 𝐼 = ∫ f= P ∫𝑥= T( f) S( e, f) Q e Q f In change of order of Integration, the order of the Integrating variables changes and the limits as well. Note : When limits are constants, the order of integration does not matter, y =d x=b x=b y =d   f ( x, y) dxdy =   f ( x, y)dydx y =c x=a x=a y =c 1.17 Jacobian of the Transformation (i) The Jacobian of the transformation, x = f1 (u, v ) , y = f2 (u, v) is defined as, ( x, y) xu xv J= = (u, v) yu yv GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.15 Engineering Mathematics (ii) The Jacobian of the transformation, x = f1 (u, v, w), y = f 2 (u, v, w), z = f3 (u, v, w) is defined as xu xv xw ( x, y, z) J= = yu yv yw (u, v, w) zu zv zw 1.18 Change of Variables Formula (i)  f ( x, y)dx dy =  f ( f1(u, v), f2 (u, v) | J | du dv R R (ii)  f ( x, y, z)dxdydz =  f ( f1(u, v, w), f2 (u, v, w), f3 (u, v, w)) | J | du dv dw R R 1.19 Change of Variables (i) Cartesian to polar co-ordinates : x = r cos  y = r sin  J=r dx dy = rdrd  (ii) Cartesian to cylindrical polar co-ordinate : x = r cos  y = r sin  z=z J=r dxdydz = rdr d  dz (iii) Cartesian to spherical polar co-ordinates : x =  sin  cos  y =  sin  sin  z =  cos  J = 2 sin  dx dy dz = 2 sin  d d  d  ❑❑❑ GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.16 Design Against Static Load 2  Matrix LINEAR ALGEBRA An array of elements in horizontal lines (Rows) and Vertical Lines (Columns) is called a Matrix. 𝑖 [ Q 𝑖 N Example: 𝐴= [ ] W N ] N [ 2.1.1 Size of Matrix If a matrix has 'm' rows and 'n' columns, then we say that the size of the matrix is m × n (read as m by n) N11 N12 N13......... N1 [ N21 N22 N23......... N2 [ 𝐴=.... ; 𝐴 = [ N𝑖 W ] such that 1 ≤ 𝑖 ≤ Z, 1 ≤ W ≤ [ and N𝑖 W = S(𝑖, W) Z× [.... [ N Z1 N Z2 N Z3......... N Z [ ] 2.1.2 Addition of Matrices (i) Two matrices 𝐴 = [ N𝑖 W ] & 5 = [ O𝑖 W ] can be added only if m = p & n = q. Z× [ ]× ^ (ii) Matrix Addition is commutative (A + B = B + A) (iii) Matrix Addition is Associative. A + (B + C) = (A + B) + C (iv) Existence of additive identity : If O be m × n matrix each of whose elements are zero. Then, A + O = A = O + A for every m × n matrix A. (v) Existence of additive inverse : Let A = aij  then the negative of matrix A is defined as matrix −aij  and is mn mn denoted by –A.  Matrix –A is additive inverse of A. Because (–A) + A = O = A + (–A). Here O is null matrix of order m × n. (vi) Cancellation laws holds good in case of addition of matrices, which is X = –A.  A+X=B+XA=B  X+A=X+BA=B (vii) The equation A + X = 0 has a unique solution in the set of all m × n matrices. 2.1.3 Multiplication of Matrices The multiplication of two matrices 𝐴 = [ N𝑖 W ] and 5 = [ O𝑖 W ] (⇒ 𝐴 5 Z× ^ ) is feasible only if n = P. Z× [ ]× ^ 𝐴 Z× [ ⋅ 5 ]× ^ = 6 GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.17 Engineering Mathematics N11 N12 N13 O11 O12 N A = [ 21 N22 N23 ] O 5 = [ 21 O22 ] 𝐴3×3 × 53×2 N31 N32 N33 3×3 O31 O32 3×2 N11. O11 + N12 ⋅ O21 + N13. O31 N11 O12 + N12 O22 + N13 O32 ⇒ [ N21. O11 + N22. O21 + N23. O31 N21 O12 + N22 O22 + N23 O32 ] N31. O11 + N32 ⋅ O21 + N33 ⋅ O31 N31 O12 + N32 O22 + N33 O32 3×2 2.1.4 Properties of Multiplication of Matrices (i) Matrix Multiplication Need not be commutative. (ii) Matrix Multiplication is Associative (A(BC)) = ((AB)C) (iii) Matrix Multiplication is distributive A(B + C) = (AB + AC) (iv) The product of two Matrices 𝐴 Z× [ , 5 [× ^ (i.e. 𝐴 5 Z× ^ ) can be a zero matrix even if 𝐴 ≠ 𝑂& 5 ≠ 𝑂. 3 0 0 0 0 0 Example: 𝐴 = [ ] ; 5 = [ ] ⇒ 𝐴 5 = [ ] 0 0 0 4 0 0 For the multiplication of two matrices 𝐴 Z× [ & 5 [× ^ (i) The No. of Multiplications required = m n q (ii) The number of Additions required = m (n –1) q 2.2 Types of Matrices (1) Upper triangular Matrix: A matrix 𝐴 = [ N𝑖 W ]; 1 ≤ 𝑖, W ≤ [ is said to be an upper triangular matrix if N𝑖 W = 0 ∀ 𝑖 > W (2) Lower Triangular Matrix: A matrix 𝐴 = [ N𝑖 W ] ; 1 ≤ 𝑖, W ≤ [ is said to be a lower Triangular Matrix [× [ if N𝑖 W = 0 ∀ 𝑖 < W (3) Diagonal Matrix: A matrix 𝐴 = [ N𝑖 W ], ∀ 1 ≤ 𝑖, W ≤ [ is said to be a diagonal matrix if N𝑖 W = 𝑂 ∀ 𝑖 ≠ W Q1 0 0 Example: 𝐴 = [ 0 Q2 0 ]. The diagonal Matrix is also denoted as 𝐴 = Q𝑖 N T [ Q1 , Q2 , Q3 ] 0 0 Q3 𝐾; 𝑖 = W (4) Scalar Matrix: A Matrix 'A' = [ N𝑖 W ] ; 1 ≤ 𝑖, W ≤ [ is said to be a scalar Matrix if N𝑖 W = { 0; 1 ≠ W If K = 1, then A → Identity Matrix, If K = 0, then A → Null Matrix. (5) Idempotent Matrix: A Matrix '𝐴 [× [ ' is said to be an idempotent matrix if 𝐴2 = 𝐴. 4 −1 Example: 𝐴 = [ ] 12 −3 4 −1 4 −1 4 −1 ⇒𝐴⋅𝐴=[ ][ ]=[ ]=𝐴 12 −3 12 −3 12 −3 GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.18 Engineering Mathematics (6) Nilpotent Matrix: A matrix A is said to be nilpotent of class x or index if Ax = 0 and Ax– 1  0 i.e. x is the smallest index which makes Ax = 0. 1 1 3 Example: The matrix A =  5 2 6  is nilpotent class 3, since A  0 and A2  0, but A3 = 0.   −2 −1 −3 (7) Orthogonal Matrix: A matrix A is said to be orthogonal if A. AT = I cos 𝜃 − sin 𝜃 Example: [ ] sin 𝜃 cos 𝜃 (8) Involutory Matrix: A matrix A is said to be involutory if A2 = I 2 3 Example: [ ] −1 −2 2.3 Transpose of a Matrix For a given matrix = [ N𝑖 W ]; 1 ≤ 𝑖 ≤ Z, 1 ≤ W ≤ [, we can say that 'B' where 5 = [ O𝑖 W ], 𝑖 ≤ 𝑖 ≤ [ 𝑖 ≤ W ≤ Z is the transpose of the Matrix 'A' if N𝑖 W = O W𝑖 2.3.1 Properties of Transpose of a Matrix (i) (𝐴𝑇 )𝑇 = 𝐴 (ii) (𝐴 5)𝑇 = 5𝑇 ⋅ 𝐴𝑇 (iii) (𝐾𝐴)𝑇 = 𝐾𝐴𝑇 where 'K' is a scalar. 2.4 Conjugate of a matrix The matrix obtained by replacing each element of matrix by its complex conjugate. 2.4.1 Properties of conjugate matrix (a) (A) = A (b) ( A + B) = A + B (c) (KA) = K A (d) ( AB) = AB (e) A = A if A is real matrix A = − A if A is purely imaginary matrix 2.5 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 (K : Complex number) (d) ( AB) = B A 2.6 Trace of a Matrix Trace is simply sum of all diagonal elements of a matrix. GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.19 Engineering Mathematics 2.6.1 Properties of Trace of a matrix Let A and B be two square matrices of order  and  is scalar then 1. Tr (A) = Tr ( A) 2. Tr ( A + B) = Tr ( A) + Tr ( B) 3. Tr ( AB) = Tr ( BA) [If both AB and BA are defined] 2.7 Type of Real Matrix (a) Symmetric matrix : (A)T = A (b) Skew symmetric matrix : (AT) = –A (c) Orthogonal matrix : (AT = A–1, AAT = I) Note : (a) If A and B are symmetric, then (A + B) and (A – B) are also symmetric. (b) For any matrix AAT is always symmetric.  A + AT   A − AT  (c) For any matrix,   is symmetric and   is skew symmetric.  2    2  (d) For orthogonal matrices, |A| =  1 A + AT A − AT (e) We can write any matrix A as a sum of symmetric and skew symmetric matrix A = + 2 2 2.8 Type of complex matrix (a) Hermitian matrix : (A = A) (b) Skew-Hermitian matrix: A = –A (c) Unitary matrix : (A = A–1, AA = I) A + A A − A Note : (a) is Hermitian and is skew Hermitian matrix. 2 2 A + A A − A (b) We can write any matrix as a sum of Hermitian and skew Hermitian matrix A = + 2 2 2.9 Determinant The summation of the product of elements of a row(or) column of a matrix with their corresponding Co-factors. 𝐴 ⋅ N Q W(𝐴) = |𝐴| ⋅ I Determinant can be calculated only if matrix is a square matrix. Suppose, we need to calculate a 3 × 3 determinant, 3 3 3  =  a1 j cof (a1 j ) =  a2 j cof (a2 j ) =  a3 j cof (a3 j ) j =1 j =1 j =1 We can calculate determinant along any row or column of the matrix. GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.20 Engineering Mathematics 2.9.1 Properties of Determinants (i) If 'A' is a Square Matrix of size ' [ × [ ' and 'k' is a Scalar then |𝐾 ⋅ 𝐴 [× [ | = 𝐾 [ ⋅ |𝐴 [× [ | (ii) | N Q W(𝐴)| = |𝐴|( [−1) 2 (iii) | N Q W( N Q W(𝐴))| = (|𝐴|)( [−1) (iv) |𝐴 5| = |𝐴| ⋅ | 5| (v) |(𝐴 5)𝑇 | = | 5𝑇 | ⋅ |𝐴𝑇 | (vi) If two rows (or) two columns of a determinant are interchanged, then the determinant changes its sign. (vii) The determinant of an upper triangular Matrix/a lower triangular Matrix/a diagonal Matrix is the product of the principal diagonal elements of the Matrix. (viii) The determinant of Every Skew-Symmetric Matrix of odd order (𝐴 [× [ )(′ [′′ 𝑖 ` \ Q Q) is zero. (ix) The determinant of an orthogonal Matrix 𝐴 [× [ is ±1 (x) The determinant of an Idempotent Matrix is either 0 (or) 1. (xi) The determinant of an Involuntary Matrix is ±1 (xii) The determinant of a Nilpotent Matrix is always zero. (xiii) If the product of two Non-zero Matrices 𝐴 [× [ ≠ 0; 5 [× [ ≠ 0 is a zero Matrix ((𝐴 5) [× [ = 0), then both |𝐴| = 0 & | 5| = 0. (xiv) If two rows (or) two columns of a Matrix are either equal or Proportional, then the determinant of the Matrix is equal to zero. (xv) The number of terms in the general expansion of an 'n × n' determinant is [! (xvi) Value of the determinant is invariant under row and column interchange i.e., AT = A (xvii) If any row or column is completely zero, then |A| = 0. (xviii) If any single row or column of the matrix is multiplied by k then the determinant the of new matrix = K|A| (xix) In a determinant the sum of the product of the element of any row or column with its cofactor gives a determinant of the matrix. (xx) In determinant the sum of the product of the element of any row or column with a cofactor of another row or column will give zero. (xxi) |AB| = |A| × |B| Ri =Ri +KR j (xxii) Elementary operations don’t effect the determinant that is A ⎯⎯⎯⎯⎯→ B then |A| = |B| C =C +KC A ⎯⎯⎯⎯⎯ i i j →B then |A| = |B| GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.21 Engineering Mathematics 2.10 Minors, Cofactor and Adjoint of a Matrix Minor of an element is equal to the determinant of the remaining elements of the matrix, after excluding the row and column containing the particular element. The cofactor of an element can be calculated from the minor of the element. The cofactor of an element is equal to the product of the minor of the element, and -1 to the power of position values of row and column of the element. i+ j Cofactor of an Element = ( −1)  Minor of an Element Here i and j are the positional values of the row and column of the element. Example : a11 a12 a13 If  = a21 a22 a23 a31 a32 a33 a12 a13 Minor of element a21 : M 21 = a32 a33 i+ j Co-factor of an element, aij = ( −1) M ij To design co-factor matrix, we replace each element by its co-factor. Adjoint of a matrix = transpose of cofactor matrix Adj ( A) A−1 = | A| 2.11 Inverse of a matrix Inverse of a matrix only exists for square matrices. ( A−1 ) = AdjA( A) and A 0 Properties: (a) AA−1 = A−1 A = I (b) ( AB)−1 = B−1 A−1 (c) ( ABC)−1 = C−1B−1 A−1 (d) ( AT )−1 = ( A−1)T (e) The inverse of 2 × 2 matrix should be remembered, −1 a b  1  d −b c d  =   ( ad − bc ) −c a  (i) Interchange the diagonal elements and put negative sign on the rest. (ii) Divide by determinant. GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.22 Engineering Mathematics 2.12 Rank of a Matrix The rank of the matrix refers to the number of linearly independent rows or columns in the matrix. ρ(A) is used to denote the rank of matrix A. A matrix is said to be of rank zero when all of its elements become zero. The rank of the matrix is the dimension of the vector space obtained by its columns. The rank of a matrix cannot exceed more than the number of its rows or columns. The rank of the null matrix is zero. The nullity of a matrix is defined as the number of vectors present in the null space of a given matrix. In other words, it can be defined as the dimension of the null space of matrix A called the nullity of A. Rank + Nullity is the number of all columns in matrix A. A real Number 'r' is said to be the rank of a matrix '𝐴 Z× [ ' if (1) There is at least one square sub-matrix of A of order r whose determinant is not equal to zero. (2) If the matrix A contains any square sub-matrix of order (r + 1) and above, then the determinant of such a matrix should be zero. It is mathematically denoted by (𝐴) = _ 2.12.1 Properties of Rank of a Matrix (i) (𝐴 Z× [ ) ≤ ( Z, [) (ii) (𝐴 5) ≤ Z𝑖 [{ (𝐴), ( 5)} (iii) Rank of transpose of matrix is equal to rank of matrix (iv) Elementary operations do-not affect the rank the matrix (v) (𝐴 + 5) ≤ { (𝐴) + ( 5)} 2.12.2 Row Echelon Form A Matrix 𝐴 Z× [ is said to be in row-echelon form if (i) Number of zeroes before the 1st Non-zero element in any row is less than the number of such zeroes in its succeeding row. (ii) Zero rows (if any) should lie at the bottom of the Matrix. (𝐴 Z× [ ) = Number of non-zero rows in the Row-Echelon form of A.  System of Equations The given system of equations N11 e1 + N12 e12 + N13 e3 = O1 N21 e1 + N22 e2 + N23 e3 = O2 N31 e1 + N32 e2 + N33 e3 = O3 can be written in Matrix form as GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.23 Engineering Mathematics N11 N12 N13 e1 O1 N [ 21 N22 N23 ] [ e2 ] = [ O2 ] N31 N32 N33 e3 O3 N11 N12 N13 e1 O1 N [ 21 N22 N23 ] [ e2 ] = [ O2 ] N31 N32 N33 e3 O3 Ax = B Coefficient Variable Constants Matrix Matrix Matrix The system Ax = B is said to be a homogeneous system if B = 0. The system of Ax = B is said to be a non-homogeneous system if 5 ≠ 0. 2.13.1 Consistency of a non-homogeneous system of Equations N11 N12 N13 O1 For the above system of non – homogeneous equations, Ax = B; Augmented Matrix = [A/B] = [ 21 N N22 N23 O2 ] N31 N32 N33 O3 (i) If (𝐴) = (𝐴/ 5) = Number of unknowns, then the system Ax = B has a unique solution. (ii) If (𝐴) = (𝐴/ 5) < Number of unknowns, then the system has infinitely many solutions. (iii) If (𝐴) ≠ (𝐴/ 5), then the system has no solution. Number of linearly independent solutions for a system of 'n' equations given by Ax = B is [ − (𝐴) 2.13.2 Consistency of Homogeneous System of Equations N11 e + N12 f + N13 g = 0 N21 e + N22 f + N23 g = 0 N31 e + N32 f + N33 g = 0 N11 N12 N13 e 0 N11 N12 N13 0 N Ax = 0⇒ [ 21 N22 N23 ] [ f] = [𝐴/ 5] = [ N21 N22 N23 0] N31 N32 N33 g 0 N31 N32 N33 0 3×4 If (𝐴) = (𝐴/ 5) = [ (𝑖. R |𝐴| ≠ 0); the system has a unique solution. (Trivial solution; x = 0, y = 0, z = 0) If (𝐴) = (𝐴/ 5) < [(|𝐴| = 0); the system has infinitely many solutions (Non-trivial solution exists for the system). 2.14 Linear Combination of Vectors If e1 , e2 , e3 ,...... , e [ are 'n' rows vectors, then the combination X1 e1 + X2 e2 + X3 e3 +..... + X [ e [ is called a linear combination of e1 , e2 ,.... , e [ ( X1 , X2 , X3 ,..... X [ are scalars) (1) The linear combination X1 e1 + X2 e2 + X3 e3 +..... + X [ e [ is said to be linearly dependent if X1 e1 + X2 e2 + X3 e3 +..... + X [ e [ = 0 when X1 , X2 , X3 ,..... , X [ (NOT All zeroes). GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.24 Engineering Mathematics If e1 = [ N1 O1 P1 ]; e2 [ N2 O2 P2 ]; e3 = [ N3 O3 P3 ], then the vectors e1 , e2 , e3 are said to be linearly dependent if N1 O1 P1 | N2 O2 P2 | = 0. N3 O3 P3 (2) The combination X1 e1 + X2 e2 +...... + X [ e [ is said to be linearly independent if X1 e1 + X2 e2 +........ + X [ e [ = 0 when X1 = X2 = X3 =..... X [ = 0 2.14.1 Eigen Values and Eigen Vectors For any square Matrix 𝐴 [× [ , the equation |𝐴 − 𝜆𝐼| = 0 where '' is a scalar is called the characteristic equation. The roots of the characteristic equation of a Matrix are called Eigen Values. 2.14.2 Properties of Eigen Values (i) If 𝜆1 , 𝜆2 , 𝜆3 ,...... , 𝜆 [ are 'n' Eigen Values of 𝐴 [× [ , then (a) Sum of Eigen Values of 'A' = 𝜆1 + 𝜆2 + 𝜆3 +.... +𝜆 [ = ∑ [𝑖=1 𝜆𝑖 = a _ N P R(𝐴) = Sum of Principal diagonal elements (b) Product of all the Eigen Values of 'A' = 𝜆1 ⋅ 𝜆2 ⋅ 𝜆3 ⋅...... 𝜆 [ = ∏ [𝑖=1 𝜆𝑖 = |𝐴| (c) Eigen Values of 𝐴 Z are 𝜆1 Z , 𝜆 Z Z Z 2 , 𝜆3 ,...... 𝜆 [ |𝐴| |𝐴| |𝐴| |𝐴| (d) Eigen Values of adj(A) are , , ,......, 𝜆1 𝜆2 𝜆3 𝜆 [ (e) Eigen Values of A & AT are the same. (f) Eigen Values of X1 𝐴 + X2 𝐼 (Where X1 and X2 are scalar) are X1 𝜆1 + X2 , X1 𝜆2 + X2 , X1 𝜆3 + X2 , X1 𝜆4 + X2 ,........ X1 𝜆 [ + X2 (ii) '0' is always an Eigen Value of an odd-order Skew-Symmetric Matrix. (iii) Eigen Values of a Real Symmetric Matrix are always real. (iv) Eigen Values of the Skew-Symmetric Matrix are either zero (or) purely Imaginary. (v) The Eigen values of an Orthogonal Matrix are of unit modulus. (vi) If the sum of all the elements in a row (or Column) is constant (= k) for all the rows (or columns) in the matrix respectively, then 'k' is an Eigen Value of the Matrix. N1 O1 P1 Example: If 𝐴 = [ N2 O2 P2 ] and if N1 + O1 + P1 = N2 + O2 + P2 = N3 + O3 + P3 = X, N3 O3 P3 then 'k' is an Eigen Value of 'A'. (vii) The Eigen Values of an upper triangular Matrix, a lower triangular Matrix, a diagonal Matrix are the Principal diagonal elements of the Matrix. GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.25 Engineering Mathematics  Eigen Vector A non-zero column vector 𝑋 [×1 is said to be an Eigen Vector of 𝐴 [× [ corresponding to the Eigen Value '', if 𝐴𝑋 = 𝜆𝑋(𝑋 ≠ 0). 2.15.1 Properties of Eigen Vectors (i) Eigen Vectors of A & AT are not the same. (ii) Eigen Vectors of A & AM are same. (iii) The Eigen Vectors of a Real Symmetric Matrix are always orthogonal. (iv) The number of linearly independent Eigen Vectors of '𝐴 [× [ ' is equal to the number of distinct Eigen Values of '𝐴 [× [ '. 2.15.2 Cayley Hamilton Theorem Every Matrix satisfies its characteristic equation. −1 This means that, if c0 + c1 +... + cn−1 + cn = 0 is the characteristic equation of a square matrix A of order n, then n c0 An + c1 An −1 +... + cn −1 A + cn I = 0 …(i) Note: When 𝜆 is replaced by A in the characteristic equation, the constant term cn should be replaced by cnI to get the result of the Cayley-Hamilton theorem, where I is the unit matrix of order n. Also, 0 in the R.H.S. of (i) is a null matrix of order n. 2.16. Subspace (Basis of Dimensions) 2.16.1 Vector An ordered n-tuple of numbers is called an n-vector. 2.16.2 Linearly Independent and Dependent Vector Let X1 and X2 be the non-zero vectors: {x1, x2, …., xk} are linearly independent if r1x1 + r2x2 + … + rk xk = 0 only for r1 = r2 = … + rk = 0. The vectors x1, r2, …., xk = are linearly dependent if they are not linearly independent; that is, if there exist scalars r 1, r2, … , rk which are not all zero such that r1x1 + r2x2 + …. + rk xk = 0 Note: Let X1, X2…….Xn be ‘n’ vector of matrix A. If rank (A) = number of vectors then vector X1, X2…..Xn are linearly independent. If rank (A) ≠ number of vectors then vector X1, X2 ….. Xn are linearly dependent. GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.26 Engineering Mathematics 2.16.3 Vector Space Rn If n positive integer, then an ordered n-tuple is a sequence of n real numbers (α1, α2,…. αn). the set of all ordered n-tuples is called n-space and is denoted by ℝn. 2.16.4 Subspaces of an N-vector space Vn A non-empty set S, of vectors of Vn(F) ,is called a subspace of Vn(F), if 𝜉 1, 𝜉 2 are any two members of S, then ξ1 + ξ2 is also a member of S; and 𝜉 is a member of S, and k is a scalar then kξ is also a member of S. Briefly, we may say that a set S of vectors Vn(F) is a subspace of Vn(F) it closed w.r.t. the compositions of “addition” and "multiplication with scalars". Every subspace of Vn contains the zero vector; being the product of any vector with the scalar zero. 2.16.5 Construction of Subspaces A subspace Spanned by a Set of Vectors: A subspace that arises as a set of all linear combinations of any given set of vectors is said to be spanned by the given set of vectors. Basis of a subspace: A set of vectors is said to be a basis of a subspace, if The subspace is spanned by the set, and The set is linearly independent. Note: If we have N vectors and they are independent then they span N-dimension space. But if they are dependent then they span only a subspace of N-dimension space. 2.16.6 Orthogonality of Vectors Two vectors are orthogonal if each is non-zero and X1T X 2 = 0 If n vectors X1, X2 …. Xn each of n dimensions is orthogonal then they are surely linearly independent and form the basis for n-dimension space. The set of the vector is orthonormal if they are orthogonal and have unit magnitude. 2.17 Similar Matrices Two matrix A and B are similar if there exist a non singular matrix P such that B = P–1AP Similar matrix has same eigen valves If A is similar to B then B is also similar to A If A is similar to B and B is similar to C then A is similar to C. 2.18 Diagonalization of a matrix Finding a matrix D which is a diagonal matrix and which is similar to A is called diagonalization i.e., we wish to find a non- singular matrix M such that A = M–1DM where D is a diagonal matrix. GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.27 Engineering Mathematics 2.18.1 Condition for a Matrix to be Diagonalizable 1. A necessary and sufficient condition for a matrix An × n to be diagonalizable is that the matrix must have n linearly independent eigen vectors. 2. A sufficient (but not necessary) condition for a matrix An × n to be diagonalizable is that the matrix must have n linearly independent eigen values. This is because if a matrix has n linearly independent eigen values then it surely has n linearly independent eigen vectors (although the converse of this is not true). ❑❑❑ GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.28 Engineering Mathematics 3 3.1 Random Experiment PROBABILITY AND STATISTICS The experiment in which the outcome is uncertain is called a Random Experiment (RE). Example: Flipping a coin, rolling a pair of dice, Picking a ball from a bag. 3.1.1 Sample Space The set contains all the possible outcomes of a random experiment. It is denoted by 'S'. If RE is flipping a coin, S = {Head, Tail} If RE is rolling a dice, S = {1,2,3,4,5,6} 3.2 Event Any subset of sample space 'S' is called an Event. Example: If RE is flipping a coin, then the occurring of a Head is an Event. If RE is rolling a dice, then getting an odd number is an Event. 3.2.1 Probability of an Event If 'A' is any event with in the sample space 'S' of a Random experiment, then the probability of event 'A' is given by No. of outcomes favouring event 'A' to happen n ( A) P ( A) = = Total number of elements in 'S' n(S ) Probability of getting an Even Number when a dice is rolled. 3 P(Even Number) = = 0.5 6 S = {1,2,3,4,5,6}, A = {2,4,6} Note: Probability can also be expressed as odds if favour and odds against an event: Odds is favour of an event: Odds in favour of an event = Number of successes : Number of failures = m: (n – m). Odds against an event: Odds against an event = Number of failures : Number of successes = (n – m) : m. GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.29 Engineering Mathematics 3.2.2 Axioms Probability (i) If 'A' is any event with in the sample space 'S' of a RE, then 0 ≤ C(𝐴) ≤ 1 0  n(A)  n(S) n(S) n(S) n(S) 0 ≤ C(𝐴) ≤ 1 (ii) P(S) = 1 When a RE is conducted the experiment yields a possible outcome. 3.2.3 Types of Events (i) Mutually Exclusive Events: If A, B are two events within a sample space 'S', then A & B are said to be mutually exclusive if A∩B = . Example: If 'A' is the event of getting a prime number when a dice is rolled and 'B' is the event of getting a composite number when a dice is rolled then S = {1,2,3,4,5,6}, A = {2,3,5},B = {4,6}  A  B =   P(A  B) = 0 Fig. 5.1. Mutually exclusive event (ii) Mutually Exhaustive Events: If 'A', and 'B' are two events within a sample space 'S', then 'A' & 'B' are said to be mutually exhaustive if A  B = S Example: If 'A' is the event of getting an odd number when a dice is rolled and 'B' is the event of getting an Even Number, then =S S = {1,2,3,4,5,6} A = {1,3,5}, B ={2,4,6}  B=S Fig. 5.2. Mutually exhaustive event GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.30 Engineering Mathematics (iii) Independent Events: Two events 'A' & 'B' within the sample space 'S' (or) within two different sample spaces 'S1' & 'S2' are said to be independent if P(  ) = P(A)  P(B). Fig. 5.3. Independent Event (iv) Impossible Event (): The event with zero probability is called an Impossible Event P() = 0. 3.3 Addition Theorem of Probability If A, and B are two events with a sample space 'S' of a Random Experiment, then P(  ) = P(A) + P(B) – P(  ) n(A  B) = n(A) + n(B) − n(A  B) n(S) n(S) n(S) n(S) Fig. 5.4. Addition theorem  P(A  B) = P(A) + P(B) – P(A  B) When A, and B are mutually exclusive events, A  B = .  P(A  B) = 0 P(  ) = P(A) + P(B) If E1, E2, E3,…….En are mutually exclusive events (Ei  j = ), then P(E1  E2  E3  …….  En ) = ∑ [𝑖=1 C(𝐸𝑖 ) = P(E1) + P(E2) + P(E3) + …… p(En) 3.3.1 De Morgan’s Law ( A  B )C = AC  BC ( A  B )C = AC  BC GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.31 Engineering Mathematics 3.3.2 Union and Intersection properties For any two events A and B: (a) P( A  B) = P( A) + P( B) − P( A  B) (b) P( A  B ) = 1 − P( A  B) c c For any three events A, B and C: (a) P( A  B  C) = P( A) + P(B) + P(C ) − P( A  B) − P(B  C ) − P(C  A) + P( A  B  C ) (b) P( A  B  C ) = 1 − P( A  B  C) c c c 3.3.3 Conditional Probability: The probability of the happening of event 'A' when it is known that event 'B' has already occurred is given by P(A/B) C(𝐴 ∩ 5) [(𝐴 ∩ 5) C(𝐴/ 5) = = C( 5) [( 5) 3.3.4 Joint Probability: f(x, y) is the joint probability of two RV’S x, y. If the two RV are Independent then f(x, y) = f(x) ⋅ f(y) b d P(a  x  b, c  y  d ) =  f ( x, y)dydx a c  f ( x) =  f ( x, y) dy −  f ( y) =  f ( x, y) dx − 3.3.5 Multiplication Theorem of Probability: If A, and B are two events within a sample space 'S', then P(A/B)  P(B) = P(B/A)  P(A) 𝑃(𝐴∩ 5) P(A/B) = ⇒ C(𝐴 ∩ 5) = C(𝐴/ 5) ⋅ C( 5) → (1) 𝑃( 5) 𝑃( 5∩𝐴) P(B/A) = ⇒ C( 5 ∩ 𝐴) = C( 5/𝐴) ⋅ C(𝐴) → (2) 𝑃(𝐴) From (1) & (2) C(𝐴/ 5) ⋅ C( 5) = C( 5/𝐴) ⋅ C(𝐴) 3.3.6 Total Theorem of Probability: If E1, E2, E3,……En are 'n' mutually exclusive (𝐸𝑖 ∩ 𝐸 W = 𝜙; ∀𝑖 ≠ W) and collectively exhaustive event (E1  E2  E3  ……  En = S) and 'A' is any event with in the sample space 'S', then C(𝐴) = C(𝐸1 ) ⋅ C(𝐴/𝐸1 ) + C(𝐸2 ) ⋅ C(𝐴/𝐸2 )+...... + C(𝐸 [ ) ⋅ C(𝐴/𝐸 [ ) n P( A) =  P( Ei ) P( A / Ei ) i =1 GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.32 Engineering Mathematics 3.3.7 Baye's Theorem If E1,E2,E3,……En are mutually exclusive (𝐸𝑖 ∩ 𝐸 W = 𝜙∀𝑖 ≠ W) and collectively exhaustive event (𝐸1 ∪ 𝐸2 ∪ 𝐸3 ∪.......∪ 𝐸 [ = 𝑆) and 'A' is any event with in the sample space 'S', then C(𝐸𝑖 ) ⋅ C(𝐴/𝐸𝑖 ) C(𝐸𝑖 /𝐴) = [ ∑𝑖=1 C(𝐸𝑖 ) ⋅ C(𝐴/𝐸𝑖 ) Fig. 5.5. Baye’s theorem 3.3.8 Use of permutation and combination What is combination? A combination of ‘n’ objects taken ‘r’ at a time (r-combination of ‘n’ objects is an unordered selection of ‘r’ of the objects). Number of ways of combining of ‘r’ object out of ‘n’ objects without repetition  n! Cr = (n − r )!r ! What is permutation? A combination of ‘n’ objects taken ‘r’ at a time (r-combination of ‘n’ objects is an ordered selection of ‘r’ of the objects). Number of ways of selection of r object out of n objects without repetition  n! Pr = (n − r )! Result: (i) n Cr = nCn−r (ii) n C0 + nC1 + nC2 +.... + nCn = 2n (iii) n C0 + nC2 + nC4 +.... += 2n−1 (iv) n C1 + nC3 + nC5 +.... += 2n−1 n−1 (v) 0. C0 + 1. C1 + 2. C2 +.... + n. Cn = n.2 n n n n Permutations with Repetition GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.33 Engineering Mathematics The number of permutations of n objects, where p objects are of one kind, q objects are of another kind and the rest, if any, are n Pr of a different kind is. p !q ! Combination with Repetition Number of combinations of ‘n’ distinct things taking ‘r’ at a time when each thing may be repeated any number of times is n−1+r given by Cr. 3.4 STATISTICS Statistics → Collection and Analysis of Data 3.4.1 Analysis of Ungrouped Data If x1, x2, x3, …….,xn are 'n' observations, then (1) The range of the data = R = max{x1, x2, …….,xn} – min{x1, x2, x3, ….., xn} (2) Arithmetic mean : Mean of the data is equal to sum of observaions divided by the total number of observations. e1 + e2 +...... + e [ ∑ [𝑖=1 e𝑖 ē ( \ _) = = = ē = [ [ [( [+1) ( ) [+1 2 The mean of 1st 'n' natural numbers = = [ 2 [2 The mean of 1st 'n' odd numbers = = [ [ The mean of 1st 'n' even numbers = n +1 3.4.2 Median The middle most observation of the data ( e1 , e2 , e3 ,..... e [ ) When the data is arranged in either ascending or descending order. If e1 , e2 , e3 , e4 ,....... e [ are 'n' observations that are arranged in ascending/descending order then [+1 aℎ (i) Median of the Data = ( ) observation, if 'n' is odd. 2 [ aℎ [ aℎ (ii) Median of the Data = Mean of ( ) & ( + 1) observations, if 'n' is even. 2 2 3.4.3 Mode The observation with highest frequency is called mode. Any Data with two Modes is called → Bimodel Data 𝑥1 +𝑥2 +.......+𝑥 [ If e1 , e2, e3 ,...... , e [ are 'n' data points, ē = = [ Mean Deviation of the observation ( e𝑖 ) = Q𝑖 = e𝑖 − ē GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.34 Engineering Mathematics Fig. 5.6. Discrete data Sum of derivations of all the observations = 𝛴 Q𝑖 = ( e1 − ē ) + ( e2 − ē )+......0 + ( e [ − ē ) = 𝛴 Q𝑖 = ( e1 + e2 +..... + e [ ) − [ ē 𝛴 Q𝑖 = 0 The sum of mean deviations of all the observations is equal to zero. 3.4.4 Absolute Mean Deviation If x1, x2, x3,…….,xn are 'n' data points with Mean = ē , then the absolute mean deviation of e𝑖 about ē is given by | Q𝑖 | = | e − ē | The sum of absolute mean derivations of given data is not zero. (𝛴| Q𝑖 | ≠ 0) ⇒ (| e1 − ē | + | e2 − ē |+........ +| e [ − ē | ≠ 0) 3.4.5 Standard Deviation If x1, x2, x3,……,xn ('n' is very large), then the standard deviation of the data is given by 1 Population Standard Deviation 𝜎 = √ 𝛴( e𝑖 − ē )2 , n → size of population [ 1 Sample Standard derivation: 𝜎 = √ 𝛴( e𝑖 − ē )2 , n→ size of sample ( [−1) Generally (n > 29 → population) (n < 29 →sample) Note: Measures of skewness (The degree of asymmetry) A lack of symmetry is skewness. For symmetric distribution mean (M) = Median (Md) = Mode (Me) For negatively skewed distribution mean (M) < Median (Md) < Mode (Me) For positively skewed distribution Mean (M) > Median (Md) > Mode Me). GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.35 Engineering Mathematics 3.5 Random Variables The variable that connects the outcome of a Random Experiment to a real number. Example: 'x' is the value of the number that a dice shows when it is rolled. Discrete RV → The RV whose value is obtained by counting, defined by PMF Random Variable Continuous RV → The RV whose value is obtained by Measuring, defined by PDF If a data consists of 'f1' data points with value ′ e1 ′, ′ S2 ′ data points with value ′ e2 ′...... ′ S [ ′ data point with value ′ e [′ , then (i) Expectation of 'x' = 𝐸( e) = ∑ [𝑖=1 e𝑖 C( e = e𝑖 ) (ii) Variance of ‘x’ = 𝜎 2 = 𝐸( e 2 ) − (𝐸( e))2 and σ is the standard deviation. 3.5.1 Probability Mass Function (PMF) The PMF p(x) of a discrete random variable X taking values x1 , x2 ,.....xn is defined such that, (i) p( xi )  0 n (ii)  p( xi ) = 1 i =1 (iii) p( xi ) = p( X = xi ) 3.5.2 Probability Density Function (PDF) The pdf f(x) of a continuous random variable X is defined such that, (i) f ( x)  0  (ii)  f ( x)dx = 1 − b (iii) P(a  X  b) =  f ( x)dx a 3.5.3 Expected Value   xp( x); X is discrete rv  1. Expected value of a random variable X, E [X], is defined as, E  X     xf ( x)dx; X is continuous rv − 2. Expected value of X2 is,   x2 p( x); X is discrete rv   E[ X ] =  2   x f ( x)dx; X is continuous rv 2  − n Note: E[ X ] is called nth moment. GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.36 Engineering Mathematics 3.5.4 Mean of Random Variable ‘X’ Mean =  = E[ X ] 3.5.5 Variance of a Random Variable ‘X’ Var ( X ) = E[( X −) ] 2 Or, Var ( X ) = E[ X ] − 2 2 3.5.6 Properties of Expectation (i) E[c] = c, c is a constant. (ii) E[ax] = aE[X] (iii) E(aX + b) = aE(X) + b (iv) If X and Y are random variable E[X  Y] = E(x)  E(Y). (v) If X and Y are random variables E(X. Y) = E(X). E(Y / X). (vi) If X and Y independent random variables E(X. Y) = E(X). E (Y). 3.5.7 Properties of Variance (i) Var[C] = 0, C is constant. (ii) Var (aX) = a2V(X) where X is random variable and ‘a’ constant. Var(–X) = (–1)2 Var(X) = Var(X) Variance is always positive. (iii) Var(ax + b) = a2 Var(X) + 0 (iv) If X and Y are independent random variables. Var(X + Y) = Var(X) + Var(Y) Var(X – Y) = Var(X) + Var(Y) (v) Var(ax + by) = a2 v(x) + b2 v(y) + 2ab Cov (x, y) (vi) Cov (x, y) = E(x, y) – E(x) E(y) (vii) For independent random variables Cov(x, y) = 0 3.5.8 Continuous RV The value of the Random Variable is obtained by Measuring. 3.6 Probability Distribution Function (PDF) A continuous & differentiable function P(x) is said to be a probability distribution/density function of a continuous random O variable 'x' if C( N ≤ e ≤ O) = ∫𝑎 C( e) Q e GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.37 Engineering Mathematics 3.6.1 Mean (or) Expectation ∞ If P(x) is a probability distribution/density function of a continuous Random Variable 'x' then the Mean of 'x' = E(x) = ∫−∞ e ⋅ C( e) Q e 3.6.2 Median The value of 'x' for which the total probability is exactly divided into two equal halves is called Median. 3.6.3 Mode The valueof 'x' at which P(x) is maximum is called mode. 3.6.4 Variance = 𝜎 2 = 𝐸( e 2 ) − (𝐸( e))2 ( )   2   =  x  P ( x ) dx − 2 2 x  P ( x ) dx − − Fig. 5.7. Continuous random variables 3.7 Continous RV distributions (1) Gaussian/Normal Distributon: If 'x' is a continuous Random variable with mean '' and standard deviation '', then the probability distribution/density function of normally distributed variable 'x' is given by Fig. 5.8. Normal distribution GATE WALLAH COMPUTER SCIENCE & INFORMATION TECHNOLOGY HANDBOOK 1.38 Engineering Mathematics Mean = Median = Mode =  C( − 𝜎 ≤ e ≤ + 𝜎) = 0.6828 C( − 2𝜎 ≤ e ≤ + 2𝜎) = 0.9544 C( − 3𝜎 ≤ e ≤ + 3𝜎) = 0.9973 1 −(𝑥− )2 C( e) =. R 2𝜎2 𝜎 ⋅ √2𝜋 (2) Standard Normal Distribution: −𝑧2 𝑥− 1 Assuming g = ; = 0; 𝜎 = 1, C( g) = ⋅ R 2 𝜎 √2𝜋 C(−1 ≤ g ≤ 1) = 0.6828 C(−2 ≤ g ≤ 2) = 0.9544 C(−3 ≤ g ≤ 3) = 0.9973 Note: 1. The normal distribution curve is bell shaped curve 2. The points of infelection of the normal distribution curve are at e = + 𝜎 N [ Q e = − 𝜎. 3. The cumulative function graph is of ‘S’ Shape. 4. For a given normal distribution, Mean = median = Mode (3) Uniform Distribution: If 'x' is a uniformly distrbuted random variable such that N ≤ e ≤ O then the Pdf is given by

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