Synopsis Grade 11 Math

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1 Syopsis Grade Math Chapter : Sets A set is a well-defied collectio of objects. Example: The collectio of all ratioal umbers less tha 0 is a set whereas the collectio of all the brilliat studets i a class is ot a set. Sets are usually deoted by capital letters A, B, S, etc. The elemets of a set are usually deoted by small letters a, b, t, u, etc. If x is a elemet of a set S, the we say that x belogs to S. Mathematically, we write it as x S. If y is ot a elemet of a set S, the we say that y does ot belog to S. Mathematically, we write it as y S. Methods for represetig a set Roster or tabular form: I this form, all the elemets of a set are listed, separated by commas ad eclosed withi braces { }. I this form, the order i which the elemets are listed is immaterial, ad the elemets are ot repeated. Example: The set of letters formig the word TEST is {T, E, S}. Set-builder form: I this form all the elemets of a set possess a sigle commo property which is ot possessed by ay elemet outside the set. Example: The set {, } ca be writte i the set-builder form as {x : x is a iteger ad x 4 = 0}. Types of sets A set which does ot cotai ay elemet is called a empty set or a ull set or a void set. It is deoted by the symbol or { }. Example: The set {x : x N, x is a eve umber ad 8 < x < 0} is a empty set. A set which is empty or cosists of a defiite umber of elemets is called fiite; otherwise, the set is called ifiite. Example: The set {x : x N ad x is a square umber} is a ifiite set. The set {x : x N ad x x 3 = 0} is a fiite set as it is equal to {, 3}. All ifiite sets caot be described i the roster form. For example, the set of ratioal umbers caot be described i this form. This is because the elemets of this set do ot follow ay particular patter. Two sets A ad B are said to be equal if they have exactly the same elemets ad we write A = B; otherwise, the sets are said to be uequal ad we write A B. A x : xn ad ( x )( x 4) 0 ad B = { 4, } are equal sets. Example: The sets A set does ot chage if oe or more elemets of the set are repeated.

2 Example: The sets A = {E, L, E, M, E, N, T, S} ad B = {E, L, M, N, T, S} are equal sice each elemet of A is i B, ad vice-versa. Subsets A set A is said to be a subset of a set B if every elemet of A is also a elemet of B ad we write A B. I other words, A B if a A a B. A B ad B A A = B A empty set is a subset of every set. Every set is a subset of itself. If A B ad A B, the A is called a proper subset of B, ad B is called a superset of A. If a set has oly oe elemet, the it is called a sigleto set. Example: A = {7} is a sigleto set. Itervals as subsets of R Let a, b R ad a < b. The, {y : a < y < b} is called a ope iterval ad is deoted by (a, b). I the ope iterval (a, b), all the poits betwee a ad b belog to the ope iterval (a, b), but a, b themselves do ot belog to this iterval. { y : a y b} is called a closed iterval ad is deoted by [a, b]. I this iterval, all the poits betwee a ad b as well as the poits a ad b are icluded. [ a, b) { y : a y b} is a ope iterval from a to b, icludig a, but excludig b. ( a, b] { y : a y b} is a ope iterval from a to b, icludig b, but excludig a. Power set The collectio of all subsets of a set A is called the power set of A. It is deoted by P(A). If A is a set with (A) = m, the [P(A)] = m. Ve diagrams Most of the relatioships betwee sets ca be represeted by meas of diagrams kow as Ve diagrams. A uiversal set is the super set of all sets uder cosideratio ad is deoted by U. Uio of sets The uio of two sets A ad B is the set which cotais all those elemets which are oly i A, oly i B ad i both A ad B, ad this set is deoted by A B. A B { x : x A or x B} Properties of uio of sets (i) A B B A (Commutative Law) (ii) ( A B) C A( B C) (Associative Law) (iii) A A ( is the idetity of ) (iv) AA A (Idempotet Law)

3 (v) U A U (Law of U) Itersectio of sets The itersectio of two sets A ad B is the set of all those elemets which belog to both A ad B. It is deoted by A B. A B={ x : x A ad x B} Properties of itersectio of sets (i) A B B A (Commutative Law) (ii) ( A B) C A( B C) (Associative Law) (iii) A (Law of ) (iv) U A A (Law of U) (v) AA A ( Idempotet Law) (vi) A( B C) ( A B) ( A C) (Distributive law of o ) (vii) A (B C) = (A B) (A C) Differece of sets The differece betwee the sets A ad B (i.e., A B, i this order) is the set of the elemets which belog to A, but ot to B. A B = {x: x A ad x B} Properties of operatio of differece of sets (i) A B =A (A B) (ii) For A B, A B B A (iii)for A = B, A B = B A = (iv) For B =, A B = A ad B A = (v) A U = (vi) For A B, A B = ; for this reaso, A U = The sets A B, A B ad B A are mutually disjoit sets, i.e., the itersectio of ay of these two sets is a ull set. Complemet of a set If U is a uiversal set ad A is a subset of U, the the complemets of A are deoted by the set A. This is the set of all elemet of U which are ot the elemets of A. A { x : xu ad x A} U A A is also the subset of U Properties of complemet of a set (i) ( A) A (ii) AA U

4 (iii) AA (iv) U ad U' De Morga s laws For ay sets A ad B, (i) ( A B) A B (ii) ( A B) A B If A ad B are fiite sets, such that AB, the ( A B) ( A) ( B) If A ad B are fiite sets, such that AB, the, ( A B) ( A) ( B) ( A B) If A, B ad C are fiite sets, the ( A B C) ( A) ( B) ( C) ( A B) ( B C) ( AC) ( AB C) Chapter : Relatios ad Fuctios Cartesia product of sets Two o-empty sets P ad Q are give. The Cartesia product P Q is the set of all ordered pairs of elemets from P ad Q, i.e., P Q ( p, q) : pp ad q Q If either P or Q is a ull set, the P Q will also be a ull set, i.e., PQ. I geeral, if A is ay set, the A. Two ordered pairs are equal if ad oly if the correspodig first elemets are equal ad the secod elemets are also equal. I other words, if (a, b) = (x, y), the a = x ad b = y. For ay two sets A ad B A B B A If (A) = p, (B) = q, the (A B) = pq If A ad B are o-empty sets ad either A or B is a ifiite set, the so is the case with A B A A A {( a, b, c): a, b, c A}. Here, (a, b, c) is called a ordered triplet. Relatio A relatio R from a set A to a set B is a subset of the Cartesia product A B, obtaied by describig a relatioship betwee the first elemet x ad the secod elemet y of the ordered pairs (x, y) i A B. The image of a elemet x uder a relatio R is y, where ( x, y) R.

5 The set of all the first elemets of the ordered pairs i a relatio R from a set A to a set B is called the domai of the relatio R. The set of all the secod elemets i a relatio R from a set A to a set B is called the rage of the relatio R. The whole set B is called the co-domai of the relatio R. Rage Co-domai The total umber of relatios that ca be defied from a set A to a set B is the umber of possible subsets of A B. If (A) = p ad (B) = q, the (A B) = pq ad the total umber of relatios is pq. Fuctios A relatio f from a set A to a set B is said to be a fuctio if every elemet of set A has oe ad oly oe image i set B. I other words, a fuctio f is a relatio from a o-empty set A to aother o-empty set B, such that the domai of f is A ad o two distict ordered pairs i f have the same first elemet. The fuctio f from A to B is deoted by f : A B. Here, A is the domai ad B is the co-domai of f. If f is a fuctio from A to B ad ( a, b) f, the f (a) = b, where b is called the image of a uder f, ad a is called the pre-image of b uder f. A fuctio havig either R (real umbers) or oe of its subsets as its rage is called a real-valued fuctio. Further, if its domai is also either R or a subset of R, it is called a real fuctio. Types of fuctios Idetity fuctio: The fuctio f: R R defied by y f ( x) x, for each x R, is called the idetity fuctio. Here, R is the domai ad rage of f. Costat fuctio: The fuctio f: R R defied by y f ( x) c, for each x R, where c is a costat, is a costat fuctio. Here, the domai of f is R ad its rage is {c}. Polyomial fuctio: A fuctio f: R R is said to be a polyomial fuctio if for each x R, y f ( x) a0 ax... ax, where is a o-egative iteger ad a,... 0, a a R. f( x) Ratioal fuctio: The fuctios of the type, where f(x) ad g(x) are polyomial gx ( ) fuctios of x defied i a domai ad where gx ( ) 0, are called ratioal fuctios. Modulus fuctio: The fuctio f: R R defied by f ( x) called the modulus fuctio. x, x 0 I other words, f( x) xx, 0 Sigum fuctio: The fuctio f: R R defied by x, for each x R, is

6 , if x 0 f ( x) 0, if x 0, if x 0 is called the sigum fuctio. Its domai is R ad its rage is the set {, 0, }. Greatest Iteger fuctio: The fuctio f: R R defied by f(x) = [x], x R, assumig the value of the greatest iteger less tha or equal to x, is called the greatest iteger fuctio. Liear fuctio: The fuctio f defied by f ( x) mx c, for x R, where m ad c are costats, is called the liear fuctio. Here, R is the domai ad rage of f. Algebra of fuctios For fuctios f: X R ad g: X R, we defie (f + g): X R by ( f g)( x) f ( x) g( x), x X (f g): X R by ( f g)( x) f ( x) g( x), x X (fg): X R by ( fg)( x) f ( x). g( x), x X (f): X R by ( f )( x) f ( x), x X ad is a real umber f g : X R by f f ( x) ( x), x X ad g(x) 0. g g( x) Chapter 3: Trigoometric Fuctios Cosider a circle of radius r havig a arc of legth l that subteds a agle of radias. The, l = r. Relatio betwee radia ad degree measure Radia measure = π Degree measure 80 A degree is divided ito 60 miutes ad a miute is divided ito 60 secods. Oe sixtieth of a degree is called a miute, writte as, ad oe sixtieth of a miute is called a secod, writte as. Thus, 0 60ad 60 Domai ad rage of trigoometric fuctios Trigoometric fuctio Domai Rage si x R [, ] cos x R [, ] ta x ( )π R x : x, Z R

7 cot x R {x : x =, Z} R sec x ( )π R x : x, Z R [, ] cosec x R {x : x =, Z} R [, ] Sigs of trigoometric fuctios i differet quadrats Trigoometric fuctio si x cos x ta x cot x sec x cosec x Quadrat I + ve (Icreases from 0 to ) + ve (Decreases from to 0) + ve (Icreases from 0 to ) + ve (Decreases from to 0) + ve (Icreases from to ) + ve (Decreases from to ) Quadrat II + ve (Decreases from to 0) ve (Decreases from 0 to ) ve (Icreases from to 0) ve (Decreases from 0 to ) ve (Icreases from to ) + ve (Icreases from to ) Quadrat III ve (Decreases from 0 to ) ve (Icreases from to 0) + ve (Icreases from 0 to ) + ve (Decreases from to 0) ve (Decreases from to ) ve (Icreases from to ) Quadrat IV ve (Icreases from to 0) + ve (Icreases from 0 to ) ve (Icreases from to 0) ve (Decreases from 0 to ) + ve (Decreases from to ) ve (Decreases from to ) Trigoometric idetities cosec x si x sec x cos x si x ta x cos x cos x cot x ta x si x cos x + si x = + ta x = sec x cot x cos ec x

8 Trigoometric ratios of allied agles si ( x) = si x cos ( x) = cos x π cos xsi x π si xcos x π cos x si x π si xcos x cos(π x) cos x si(π x) si x cos(π x) cos x si(π x) si x cos(π x) cos x si(π x) si x cos(π x) cos x, Z si(π x) si x, Z Sum ad differece of two agles si (x + y) = si x cos y + cos x si y cos (x + y) = cos x cos y si x si y si (x y) = si x cos y cos x si y cos (x y) = cos x cos y + si x si y If oe of the agles x, y ad ( x y) is a odd multiple of π, the ta x ta y ta( x y), ad ta xta y ta x ta y ta( xy) ta xta y If oe of the agles x, y ad ( x y) is a multiple of, the cot xcot y cot( x y), ad cot y cot x cot xcot y cot( xy) cot y cot x Trigoometric ratios of multiple agles ta x cos x cos x si x cos x si x ta x

9 ta x ta x ta x ta x ta x si 3x = 3 si x 4 si 3 x cos 3x = 4 cos 3 x 3 cos x 3 3ta x ta x ta 3x 3ta x si x si x cos x Some more trigoometric idetities x y x y cos xcos y cos cos x y x y cos x cos y si si x y x y si xsi y si cos x y x y si xsi y cos si cos x cosy = cos (x + y) + cos (x y) si x si y = cos (x + y) cos (x y) si x cos y = si (x + y) + si (x y) cos x si y = si (x + y) si (x y) Geeral solutios of trigoometric equatios si x 0 x π, where Z π cos x 0 x ( ), where Z si x si y x π ( ) y, where Z cos x cos y x π y, where Z ta x ta y x π y, where Z Chapter 4: Priciple of Mathematical Iductio There are some mathematical statemets or results that are formulated i terms of, where is a positive iteger. To prove such statemets, the well-suited priciple that is used, based o the specific techique, is kow as the priciple of mathematical iductio. To prove a give statemet i terms of, we assume the statemet to be P().

10 . Thereafter, we examie the correctess of the statemet for = i.e., for P() to be true.. The, assumig that the statemet is true for = k, where k is a positive iteger, we prove that the statemet is true for = k + i.e., the truth of P(k) implies the truth of P(k + ). The, we say that P() is true for all atural umbers. Example: For all N, prove that Solutio: Let the give statemet be P(), i.e., P( ) : For =, P( ) : , which is true. 3 3 Now, assume that P(x) is true for some positive iteger k. This meas k k () We shall ow prove that P(k + ) is also true. Now, we have k k k k k k k k k k k

11 Thus, P(k + ) is true wheever P(k) is true. Hece, from the priciple of mathematical iductio, the statemet P() is true for all atural umbers. Chapter 5: Complex Numbers ad Quadratic Equatios A umber of the form a + ib, where a ad b are real umbers ad i, is defied as a complex umber. For the complex umbers z = a + ib, a is called the real part (deoted by Re z) ad b is called the imagiary part (deoted by Im z) of the complex umber z. Two complex umbers z = a + ib ad z = c + id are equal if a = c ad b = d. Additio of complex umbers Let z = a + ib ad z = c + id be ay two complex umbers. The sum is defied as z + z = (a + c) + i (b + d). Properties of additio of complex umbers (i) Closure law: Sum of two complex umbers is also a complex umber. (ii) Commutative law: For two complex umbers z ad z, z + z = z + z (iii)associative law: For ay three complex umbers z, z ad z 3, (z + z ) + z 3 = z + (z + z 3 ) (iv) Existece of additive idetity: There exists a complex umber 0 + i0 (deoted by 0), called the additive idetity or zero complex umber, such that for every complex umber z, z + 0 = z (v) Existece of additive iverse: For every complex umber z = a+ ib, there exists a complex umber a + i( b) [deoted by z], called the additive iverse or egative of z, such that z + ( z) = 0 Give ay two complex umbers z, ad z, the differece z z is defied as z z = z + ( z ) Multiplicatio of complex umbers For two complex umbers z ad z, such that z = a + ib ad z = c + id, the multiplicatio is defied as z z = (ac bd) + i(ad + bc). Properties of multiplicatio of complex umbers (i) Closure law: The product of two complex umbers is also a complex umber. (ii) Commutative law: For ay two complex umbers z ad z, z z = z z. (iii)associative law: For ay three complex umbers z, z ad z 3, (z z ) z 3 = z (z z 3 ) (iv) Existece of multiplicative idetity: There exist a complex umber + i 0 (deoted as ), called the multiplicative idetity, such that for every complex umbers z, z. = z

12 (v) Existece of multiplicative iverse: For every o-zero complex umber z = a + ib ( a0, b 0), we have the complex umber a b i (deoted by a b a b z or z ), called the multiplicative iverse of z, such that z. z (vi) Distributive law: For ay three complex umbers z, z ad z 3, z (z + z 3 ) = z z + z z 3 (z + z ) z 3 = z z 3 + z z 3 z Give ay two complex umber z ad z, where z 0, the quotiet z z z z z is defied as Powers of i For ay iteger k, 4 k 4 k,, 4 k, 4 k i i i i i 3 i If a ad b are egative real umbers, the a b ab. Idetities for complex umbers For two complex umbers z ad z ( z z ) z z z z ( z z ) z z z z z z ( z z )( z z ) ( z z ) z 3z z 3z z z ( z z ) z 3z z 3z z z Modulus ad cojugate of complex umbers The modulus of a complex umber z = a + ib, is deoted by z, ad is defied as the oegative real umber a b, i.e., z a b. The cojugate of a complex umber z = a + ib, is deoted by z, ad is defied as the complex umber a ib, i.e., z a ib. Properties of modulus ad cojugate of complex umbers For ay three complex umbers z, z, z, z z or z. z z z

13 zz z z z z z z z, provided z 0 z z z z z z z z z z z, provided z 0 The polar form of the complex umber z = x + iy, is r cos si, where r x y x y (modulus of z) ad cos,si ( is kow as the argumet of z). r r The value of is such that, which is called the priciple argumet of z. Example: Represet the complex umber z ii polar form. Solutio: z i Let r cos ad r si By squarig ad addig them, we have r (cos si ) r 4 r 4 Thus, cos π si si π 4 π 7π π 4 4 7π 7π Thus, the required polar form is cos si 4 4. A polyomial equatio of degree has roots. The solutios of the quadratic equatio ax + bx + c = 0, where a, b, c R, a b ac 0 ad 4 0, are give by b 4ac b i x. a

14 Chapter 6: Liear Iequalities Two real umbers or two algebraic expressios related by the symbol <, >, or form a iequality. The solutio of a iequality i oe variable is a value of the variable which makes it a true statemet. Solutio of liear iequalities i oe variable Solvig a iequality algebraically (i) Equal umbers may be added to or subtracted from both sides of a iequality without affectig the sig of the iequality. (ii) Both sides of a iequality ca be multiplied with or divided by the same positive umber. But whe both sides are multiplied with or divided by a egative umber, the sig of iequality is reversed. Graphical represetatio of Solutio (i) To represet x < a (or x > a) o a umber lie, ecircle the umber a, ad darke the lie to the left (or the right) of a. (ii) To represet x a (or x a) o a umber lie, ecircle the umber a, ad darke the lie to the left (or the right) of a. Example: Solve 5(x 3) x + 9 Solutio: 5(x 3) x + 9 5x 5 x + 9 5x 5 x x + 9 x 3x 5 9 3x x 4 x 8 Thus, the solutio of the give iequality ca be represeted o the umber lie as show below: Graphical solutio of liear iequalities i two variables The solutio regio of a system of iequalities is the regio which satisfies all the give iequalities i the system simultaeously. I order to idetify the half plae represeted by a iequality, it is sufficiet to take ay poit (a, b) (ot o the lie) ad check whether it satisfies the iequality or ot. If it satisfies, the the iequality represets the half plae cotaiig the poit ad we shade this regio. If ot, the the iequality represets the half plae which does ot cotai the poit. For coveiece, the poit (0, 0) is preferred.

15 I a iequality of the type ax + by c or ax + by c, the poits o the lie ax + by = c are to be icluded i the solutio regio. So, we darke the lie i the solutio regio. I a iequality of the type ax + by > c or ax + by < c, the poits o the lie ax + by = c are ot to be icluded i the solutio regio. So, we draw a broke or dotted lie i the solutio regio. Example: Solve the followig system of liear iequalities graphically: x y 4, x < Solutio: The give liear iequalities are x y 4 (i) x < (ii) The graphs of the lies x y = 4 ad x = are draw i the figure below. Iequality (i) represets the regio o the left of the lie x y = 4 (icludig the lie x y = 4). Iequality (ii) represets the regio o the left of the lie x = (excludig the lie x = ). Hece, the solutio of the give system of liear iequalities is represeted by the commo shaded regio, icludig the poits o the lie x y = 4. Chapter 7: Permutatios ad Combiatios If a evet occurs i m differet ways, followig which aother evet occurs i differet ways, the the total umber of occurrece of the evets i the give order is m. This is called the fudametal priciple of coutig. A permutatio is a arragemet i a defiite order of a umber of objects take some or all at a time.

16 Factorial otatio The otatio! represets the product of the first atural umbers, i.e.,! = 3 0! = The umber of permutatios of differet thigs take r at a time, Whe repetitio is ot allowed, is deoted by P ad is give by! r Pr, ( r)! where 0 r. Whe repetitio is allowed, is r. Permutatios whe all the objects are ot distict The umber of permutatios of objects, whe p objects are of the same kid ad the! rest are all differet, is p!. The umber of permutatios of o objects, where p objects are of oe kid, p are of the secod kid,., p k are of the k th kid ad the rest, if ay, are of differet kids,! is p! p!... p!. k A combiatio is differet selectios of a umber of objects take a few or all at a time, irrespective of their arragemets. The umber of combiatios of differet thigs take r at a time is deoted by C, which r is give by! C r,0 r r!( r)! Properties of Cr C C r r Ca Cb a b or a b C r C r Cr Chapter 8: Biomial Theorem The coefficiets of the expasios of a biomial are arraged i a array. This array is called Pascal s triagle. It ca be writte as:

17 Idex Coefficiet(s) 0 C C 0 ( ) C 0 ( ) ( ) C C C 0 ( ) ( ) ( ) C C C C ( ) ( 3) ( 3) ( ) C C C C C ( ) ( 4) ( 6) ( 4) ( ) C C C C C C ( ) ( 5) ( 0) ( 0) ( 5) ( ) Biomial theorem for ay positive iteger The expasio of a biomial for ay positive itegral is as follows: ( a b) C a C a b C a b... C a b C b 0 Some poits about Biomial Theorem k C k a k b stads for k0 0 r r C0 a b C a b... C r a b... C a b (i) The otatio Hece, the biomial theorem ca also be stated as follows: ( a b) C a b k k k k0 (ii) The coefficiets of Ck occurrig i the biomial theorem are kow as biomial coefficiets. (iii)i the expasio of (a + b), there are ( + ) terms i.e., oe more tha the idex. (iv) I the successive terms of the expasio of (a + b), the idex of a goes o decreasig by uity, startig with i the first term ad edig with 0 i the last term. Also, the idex of b icreases by uity, startig with 0 i the first term ad edig with i the last term. (v) I every term of the expasio, the sum of the idices of a ad b is. I the biomial expasio of (a + b), takig a = x ad b = y, we have 3 3 ( x y) x ( y) C x C x ( y) C x ( y) C x ( y)... C ( y) Thus, 0 3 ( x y) C x C x y C x y C x y... ( ) C y The (r + ) th term (deoted by T r + ) is kow as the geeral term of the expasio (a + b) ad it is give by T a r b r r Cr

18 Middle term i the expasio of (a + b) If is eve, the the umber of terms i the expasio will be +. Sice is eve, + is odd. Therefore, the middle term is term. If is odd, the + is eve. So, there will be two middle terms i the expasio. They are th term ad th term. th Chapter 9: Sequeces ad Series A sequece is a arragemet of umbers i defiite order accordig to some rule. A sequece may be defied as a fuctio whose domai is the set of atural umbers or some subset of the type {,, 3 k}. A sequece cotaiig fiite umber of terms is called a fiite sequece, whereas a sequece cotaiig ifiite umber of terms is called a ifiite sequece. The geeral form of a sequece is a, a, a 3, a, a, a, a etc. are called the terms of the sequece. The th term of the sequece, a, is called the geeral term of the sequece. A arragemet of umbers such as,, 4, 6, 0 has o visible patter. However, the sequece is geerated by the recurrece relatio give by a, a, a 4 3 a a a, 3 This sequece is called the Fiboacci sequece. Let a, a,... a, be a give sequece. Accordigly, the sum of this sequece is give by the expressio a a... a... This is called the series associated with the give sequece. The series is fiite or ifiite accordig as the give sequece. A series is usually represeted i a compact form usig sigma otatio (). This meas the series a a... a aca be writte as Arithmetic progressio The sequece a, a,... a is called a arithmetic sequece or a arithmetic progressio (A.P.) if a a d, N. k a k

19 Here, a is called the first term ad d is called the commo differece of the A.P. I stadard form, the A.P. is writte as a, a + d, a + d The th or the geeral term of a A.P. is give by a = a + ( ) d. If a is the first term, d is the commo differece, l is the last term ad is the umber of terms, the l = a + ( ) d. The sum of terms of a A.P. (deoted by S ) is give by S [ a ( ) d ] Also, S ( a l ) Properties of a A.P. If a costat is added or subtracted or multiplied to each term of a A.P., the the resultig sequece is also a A.P. If each term of a A.P. is multiplied or divided by a o-zero costat, the the resultig sequece is also a A.P. Arithmetic mea For ay two umbers a ad b, we ca isert a umber A betwee them such that a, A, b is a A.P. Here, A is called the arithmetic mea (A.M.) of umbers a ad b ad a b A. Let A, A A be umbers betwee a ad b such that a, A, A,, A, b is a A.P. b a Accordigly, commo differece (d) is give by. The umbers A, A A are give as follows: b a A a d a ( b a) A a d a... ( b a) A a d a Geometric progressio

20 A sequece is said to be a geometric progressio (G.P.) if the ratio of ay term to its precedig term is the same throughout. This costat factor is called the commo ratio ad it is deoted by r. I stadard form, the G.P. is writte as a, ar, ar where, a is the first term ad r is the commo ratio. The th term (or geeral term) of a G.P. is give by a = ar The sum of terms (S ) of a G.P. is give by a( r ) a( r ) or, if r S r r a, if r Geometric mea For ay two positive umbers a ad b, we ca isert a umber G betwee them such that a, G, b is a G.P. G is called a geometric mea (G.M.) ad is give by G = ab Let G, G,,G be umbers betwee positive umbers a ad b such that a, G, G,, G, b is a G.P. Commo ratio of G.P., Therefore, G = ar, G = ar,, G = ar b r a Let A ad G be the respective A.M. ad G.M. of two give positive real umbers a ad b. a b A = ad G ab A G Sum of -terms of some special series Sum of first atural umbers ( ) 3... Sum of squares of the first atural umbers ( )( ) Sum of cubes of the first atural umbers ( ) 3...

21 Chapter 0: Straight Lies Slope of a lie If is the icliatio of a lie l (the agle betwee positive x-axis ad lie l), the ta is called the slope or gradiet of lie l. The slope of a lie is deoted by m. Thus, m = ta, 90 The slope of a lie whose icliatio is 90º is ot defied. Hece, the slope of the vertical lie (y-axis) is udefied. The slope of the horizotal lie (x-axis) is zero. The slope (m) of a o-vertical lie passig through the poits (x, y ) ad (x, y ) is give y y by m, x x. x x Suppose l ad l are o-vertical lies havig slopes m ad m respectively l is parallel to l if ad oly if m = m, i.e., their slopes are equal. l is perpedicular to l if ad oly if m m =, i.e., the product of their slopes is. A acute agle,, betwee l ad l is give by m m ta, mm 0. mm Three poits A, B ad C are colliear if ad oly if Slope of AB = Slope of BC Horizotal ad vertical lies The equatio of a horizotal lie at distace a from the x-axis is either y = a (above x- axis) or y = a (below y-axis). The equatio of a vertical lie at distace b from the y-axis is either x = b (right of y-axis) or x = b (left of y-axis). Poit-slope form of a lie

22 The poit (x, y) lies o the lie with slope m through the fixed poit (x 0, y 0 ) if ad oly if its coordiates satisfy the equatio. This meas y y 0 = m (x x 0 ). Two-poit form of a lie The equatio of the lie passig through the poits (x, y ) ad (x, y ) is give by y y y y ( x x). x x Slope-itercept form of a lie The equatio of the lie, with slope m, which makes y-itercept c is give by y = mx + c. The equatio of the lie, with slope m, which makes x-itercept d is give by y = m(x d). Itercept form of a lie The equatio of the lie makig itercepts a ad b o x-axis ad y-axis respectively is x y a b Normal form of a lie The equatio of the lie at ormal distace p from the origi ad agle which the ormal makes with the positive directio of the x-axis is give by x cos y si p Ay equatio of the form Ax + By + C = 0, where A ad B are ot zero simultaeously is called the geeral liear equatio or geeral equatio of lie. The perpedicular distace (d) of a lie Ax + By + C = 0 from a poit (x, y ) is AxBy C d. A B The distace (d) betwee two parallel lies i.e., Ax + By + C = 0 ad Ax + By + C = 0 is give by C C d A B Chapter : Coic Sectios Coic sectios or coics are the curves that are obtaied by itersectig a plae with a double-apped right circular coe. Circles, ellipses, parabolas ad hyperbolas are examples of coic sectios.

23 A double-apped coe ca be obtaied by rotatig a lie (let us say m) about a fixed vertical lie (let us say l). Here, the fixed lie l is called the axis of the coe ad m is called the geerator of the coe. The itersectio (V) of l ad m is called the vertex of the coe. Differet coics formed by itersectig a plae ad a double-apped coe If is the agle betwee the axis ad the geerator ad is the agle betwee the plae ad the axis, the, for differet coditios of ad, we get differet coics, which are described with the help of a table as show below. Coditio Coic Formed Figure = 90 (Oly oe appe of the coe is etirely cut by the plae) A circle < < 90 (Oly oe appe of the coe is etirely cut by the plae) A ellipse

24 = (Oly oe appe of the coe is etirely cut by the plae) A parabola 0 < < (Both the appes of the coe are etirely cut by the plae) A hyperbola Degeerated coics The coics obtaied by cuttig a plae with a double-apped coe at its vertex are kow as degeeratig coic sectios. If is the agle betwee the axis ad the geerator ad is the agle betwee the plae ad the axis, the, for differet coditios of ad, we get differet coics, which are described with the help of a table as show below. Coditio Coic Formed Figure = 90 A poit

25 = A straight lie 0 < A hyperbola Circle A circle is the set of all poits i a plae that are equidistat from a fixed poit i the plae. The fixed poit is called the cetre of the circle ad the fixed distace from the cetre is called the radius of the circle. Equatio of a circle with cetre at (h, k) ad radius r is (x h) + (y k) = r with cetre is at origi ad radius r is x + y = r Parabola A parabola is the set of all poits i a plae that are equidistat from a fixed lie ad a fixed poit (ot o the lie i the plae). The fixed lie is called the directrix. The fixed poit F is called the focus. The lie through the focus ad perpedicular to the directrix is called the axis of the parabola. The poit of itersectio of parabola with the axis is called the vertex of the parabola. The lie segmet that is perpedicular to the axis of the parabola through the focus ad whose ed poits lie o the parabola is called the latus rectum of the parabola. Stadard equatios of parabola

26 Ope Towards Right Stadard Equatio y = 4ax, a > 0 Coordiates of Focus (a, 0) Coordiates of Vertex (0, 0) Equatio of Directrix Legth of Latus Rectum Axis of Parabola x = a 4a Positive x-axis Ope Towards Left Stadard Equatio y = 4ax, a > 0 Coordiates of Focus (a, 0) Coordiates of Vertex (0, 0) Equatio of Directrix Legth of Latus Rectum Axis of Parabola x = a 4a Negative x-axis Ope Towards Upward Stadard Equatio x = 4ay, a > 0 Coordiates of Focus (0, a) Coordiates of Vertex (0, 0) Equatio of Directrix Legth of Latus Rectum Axis of Parabola y = a 4a Positive y-axis

27 Ope Towards Dowward Stadard Equatio x = 4ay, a > 0 Coordiates of Focus (0, a) Coordiates of Vertex (0, 0) Equatio of Directrix Legth of Latus Rectum Axis of Parabola y = a 4a Negative y-axis If the fixed poit lies o the fixed lie, the the set of poits i the plae that are equidistat from the fixed poit ad the fixed lie is a straight lie through the fixed poit ad perpedicular to the fixed lie. We call this straight lie the degeerate case of parabola. Ellipse A ellipse is the set of all poits i a plae, the sum of whose distaces from two fixed poits i the plae is a costat. The two fixed poits are called the foci. The costat, which is the sum of the distaces of a poit o the ellipse from the two fixed poits, is always greater tha the distace betwee the two fixed poits. The mid-poit of the lie segmet joiig the foci is called the cetre of the ellipse. The lie segmet through the foci of the ellipse is called the major axis ad the lie segmet through the cetre ad perpedicular to the major axis is called the mior axis. The ed poits of the major axis are called the vertices of the ellipse. The eccetricity of the ellipse is the ratio of the distaces from the cetre of the ellipse to oe of the foci ad to oe of the vertices of the ellipse. A ellipse is symmetric with respect to both the coordiate axes. The lie segmet that is perpedicular to the major axis of the ellipse through the focus ad whose ed poits lie o the ellipse is called the latus rectum of the ellipse.

28 Stadard equatios of ellipse Stadard Equatio x y, a b a b Cetre (0, 0) Vertex (a, 0) Ed poits of mior axis (0, b) Foci (c, 0) Legth of major axis a alog x-axis Legth of mior Axis b alog y-axis Legth betwee foci c alog x-axis Relatio betwee a, b ad c a b c Legth of latus rectum b a Eccetricity (e ) c a Stadard Equatio x y, a b b a Cetre (0, 0) Vertex (0, a) Ed poits of mior axis (b, 0) Foci (0, c) Legth of major axis a alog y-axis Legth of mior Axis b alog x-axis Legth betwee foci c alog y-axis Relatio betwee a, b ad c a b c Legth of latus rectum b a Eccetricity (e ) c a Special cases of a ellipse Whe c = 0, i.e., both the foci merge together, the the ellipse becomes a circle. Whe c = a, the b = 0. I such a case, the ellipse reduces to a lie segmet joiig the foci. This meas the legth of the lie segmet is c. Hyperbola A hyperbola is the set of all poits i a plae, the differece of whose distaces from two fixed poits i the plae is a costat. The two fixed poits are called the foci.

29 The costat, which is the differece of the distaces of a poit o the hyperbola from the two fixed poits, is always less tha the distace betwee the two fixed poits. The mid-poit of the lie segmet joiig the foci is called the cetre of the hyperbola. The lie through the foci is called the trasverse axis ad the lie through the cetre ad perpedicular to the trasverse axis is called the cojugate axis. The poits at which the hyperbola itersects the trasverse axis are called the vertices of the hyperbola. The eccetricity of the hyperbola is the ratio of the distaces from the cetre of the ellipse to oe of the foci ad to oe of the vertices of the ellipse. A hyperbola is symmetric with respect to both the coordiate axes. The lie segmet that is perpedicular to the trasverse axis through the focus ad whose ed poits lie o the hyperbola is called the latus rectum of the hyperbola. Stadard equatios of hyperbola x y Stadard Equatio a b Cetre (0, 0) Vertices (a, 0) Foci (c, 0) Cojugate axis y-axis Trasverse axis x-axis Legth of cojugate axis b Legth of trasverse axis a Legth betwee foci c Relatio betwee a, b ad c c a b Legth of latus rectum b a Eccetricity ( e ) c a y x Stadard Equatio a b Cetre (0, 0) Vertices (0, a) Foci (0, c) Cojugate axis x-axis Trasverse axis y-axis

30 Legth of cojugate axis b Legth of trasverse axis a Legth betwee foci c Relatio betwee a, b ad c c a b Legth of latus rectum b a Eccetricity ( e ) c a A hyperbola havig equal legths of both the axes, i.e., trasverse ad cojugate (a = b) is called a equilateral hyperbola. Chapter : Itroductio to Three Dimesioal Geometry Coordiate axes ad coordiate plaes I three-dimesios, the coordiate axes of a rectagular Cartesia coordiate system are three mutually perpedicular lies. The axes are called x, y, ad z-axes. The three plaes determied by the pair of axes are the coordiate plaes, called XY, YZ ad ZX-plaes. The three coordiate plaes divide the space ito eight parts kow as octats. Coordiates of a poit I three-dimesioal geometry, the coordiates of a poit P are always writte i the form of triplets i.e., (x, y, z). Here, x, y, ad z are the distaces from the YZ, ZX ad XYplaes. Also, the coordiates of the origi are (0, 0, 0)

31 The sig of the coordiates of a poit determie the octat i which the poit lies. The followig table shows the sigs of the coordiates i the eight octats. Octats Coordiates I II III IV V VI VII VIII x y z Coordiates of poits lyig o differet axes (i) Ay poit o the x-axis is of the form (x, 0, 0) (ii) Ay poit o the y-axis is of the form (0, y, 0) (iii)ay poit o the z-axis is of the form (0, 0, z) Coordiates of poits lyig i differet plaes (i) Coordiates of a poit i the YZ-plae are of the form (0, y, z) (ii) Coordiates of a poit i the XY-plae are of the form (x, y, 0) (iii)coordiates of a poit i the ZX-plae are of the form (x, 0, z) Distace betwee two poits P ( x, y, z) ad Q ( x, y, z) is give by PQ ( x x ) ( y y ) ( z z ) This is kow as the distace formula. Sectio formula The coordiates of poit R, which divides the lie segmet joiig two poits P ( x, y, z ) ad Q ( x, y, z ) iterally i the ratio m: are mx x my y mz z,, m m m exterally i the ratio m: are mx x my y mz z,, m m m The coordiates of the mid-poit of the lie joiig the poits ( x, y, z) ad ( x, y, z) are x x y y z z,,. The coordiates of the cetroid of a triagle whose vertices are ( x, y, z), ( x, y, y) ad ( x3, y3, z3) are x x x, y y y, z z z 3 3 3

32 Chapter 3: Limits ad Derivatives Limit of a fuctio lim f( x) is the expected value of f at x = a, give the values of f ear x to the left of a. xa This value is called the left had limit of f(x) at a. lim f( x) is the expected value of f at x = a, give the values of f ear x to the right of a. xa This value is called the right had limit of f(x) at a. If the right ad left had limits coicide, we call that commo value the limit of f(x) at x = a, ad deote it by lim f( x). xa For a fuctio f ad a real umber a, lim f( x) ad f(a) may ot be the same. Algebra of limits Let f ad g be two fuctios such that both lim f( x) ad lim gx ( ) exist. The, xa lim [ f ( x) g( x)] lim f ( x) lim g( x) xa xa xa lim [ f ( x) g( x)] lim f ( x) lim g( x) xa xa xa xa lim [( f )( x)] lim f ( x), where is a costat xa xa f( x) lim f( x) xa lim, where lim gx ( ) 0 xa g ( x ) lim g ( x ) xa xa Some stadard limits For a polyomial fuctio f(x), lim f ( x) f ( a). xa gx ( ) If f(x) is a ratioal fuctio of the form f( x), where g(x) ad h(x) are polyomial hx ( ) ga ( ) fuctios, such that h(x) 0, the lim f( x). xa ha ( ) x a lim a xa x a si x lim x0 x cos x lim 0 x0 x xa, where is a positive iteger or ay ratioal umber ad a is positive Let f ad g be two real-valued fuctios with the same domai, such that f(x) g(x) for all x i the domai of defiitio. For some a, if both lim f( x) ad lim gx ( ) exist, the lim f( x) lim gx ( ). xa xa xa xa

33 Sadwich theorem Let f, g, ad h be real fuctios such that f(x) g(x) h(x) for all x i the commo domai of defiitio. For some real umber a, if lim f ( x) l lim h( x), the lim g( x) l. xa xa Derivatives Suppose f is a real-valued fuctio ad a is a poit i its domai of defiitio. The derivative of f at a [deoted by f (a)] is defied as f ( a h) f ( a) f '( a) lim, provided the limit exists. h0 h Derivative of f(x) at a is deoted by f (a). d Suppose f is a real-valued fuctio. The derivative of f {deoted by f ( x) or [ f ( x )] } dx is defied as d f ( x h) f ( x) [ f ( x)] f ( x) lim, provided the limit exists. dx h0 h This defiitio of derivative is called the first priciple of derivative. Algebra of derivatives of fuctios For the fuctios u ad v (provided u ad v are defied i a commo domai), ( u v)' u' v' ( uv)' u' v uv ' u u ' v uv ' v v Some stadard derivatives d ( x ) x, for ay positive iteger dx d ( a x a x... a x a ) a x ( ) a x... a dx d (si x) cos x dx d (cos x) si x dx d (ta x) sec x dx xa 0

34 Chapter 4: Mathematical Reasoig A setece is called a mathematically acceptable statemet if it is either true or false, but ot both. Statemets are deoted by small letters p, q, r, s, etc. Example: Cosider the followig seteces: The value of iota is. It is a statemet. Sice we kow that i =, the give setece is false. The umber 0 is the smallest whole umber. It is a statemet as this setece is always true. Do you have iteret at your home? It is ot a statemet as it is a questio. Remember Seteces ivolvig variable time such as today, tomorrow ad yesterday are ot statemets. Seteces ivolvig a exclamatio, a questio or a order are ot statemets. Seteces ivolvig proous such as he or she, uless a particular perso is referred to, are ot statemets. Seteces ivolvig proous for variable places such as here ad there are ot statemets. The deial of a statemet is called the egatio of the statemet. If p is a statemet, the the egatio of p is also a statemet, ad is deoted by p, ad is read as ot p. While writig the egatio of a statemet, phrases such as It is ot the case or It is false that are used. Example: Write the egatio of the followig statemet: p: The square root of every positive umber is positive. Solutio: The egatio of the give statemet ca be writte as: The square root of every positive umber is ot positive. Or It is false that the square root of every positive umber is positive. Or It is ot the case that the square root of every positive umber is positive. Or There exists a positive umber whose square root is ot positive. A compoud statemet is oe that is made up of two or more statemets. Each smaller statemet is called the compoet of the compoud statemet. These compoets are joied by words such as Ad ad Or. These are called coectors or coectig words. Example: The statemet 7 is a multiple of 9 ad it is eve is a compoud statemet.

35 Its compoet statemets are: p: 7 is a multiple of 9. q: 7 is eve. Here, the coectig word is Ad. A statemet with Ad is ot always a compoud statemet. Example: Water ca be prepared by the mixture of hydroge ad oxyge i a certai ratio. This statemet is ot a compoud statemet. Rules regardig the coector Ad The compoud statemet with the coector Ad is true if all its compoet statemets are true. The compoud statemet with the coector Ad is false if ay/both of its compoet statemets is/are false. Rules regardig the coector Or A compoud statemet with the coector Or is true whe oe compoet statemet is true, or both the compoet statemets are true. A compoud statemet with the coector Or is false whe both the compoet statemets are false. Types of Or Exclusive Or : A compoud statemet with the coector Or i which either of the compoet statemets may be true, but ot both Example: A studet ca take home sciece or paitig as his/her additioal subject i class XI. Iclusive Or : A compoud statemet with the coector Or i which either of the compoet statemets or both may be true. Example: I a equilateral triagle, all the three sides are of equal legth or all the three agles are of equal measure. Some statemets may cotai special phrases such as There exists, For all, For every. These are called quatifiers. Implicatio statemet if-the : The setece if p, the q says that i the evet if p is true, the q must be true. It does ot say aythig for q whe p is false. A setece if p, the q ca be writte i the followig ways: (i) p implies q (deoted by p q) (ii) p is a sufficiet coditio for q (iii)p oly if q (iv) q is a ecessary coditio for p (v) q implies p Cotrapositive ad coverse of a statemet The cotrapositive of the statemet p q is the statemet q p.

36 The coverse of a statemet p q is the statemet q p. Example: Write the coverse ad the cotrapositive of the followig statemet: If a object is made up of oly lie segmets, the it is a polygo. Solutio: The coverse of the give statemet ca be writte as follows: If a object is a polygo, the it is made up of oly lie segmets. The cotrapositive of the give statemet ca be writte as follows: If a object is ot a polygo, the it is ot made up of oly lie segmets. The equivalet forms of the statemet p if ad oly if q (deoted by p q) are as follows: q if ad oly if p p if ad oly if q p is a ecessary ad sufficiet coditio for q ad vice-versa Validatig statemets I order to show that the statemet p ad q is true, the followig steps are followed:. Show that statemet p is true.. Show that statemet q is true. I order to show that the statemet p or q is true, the followig cases are to be cosidered.. Assumig that p is false, show that q must be true.. Assumig that q is false, show that p must be true. I order to prove the statemet if p, the q, we eed to show that ay oe of the followig cases is true.. Assumig that p is true, prove that q must be true. (Direct method). Assumig that q is false, prove that p must be false. (Cotrapositive method) I order to prove the statemet p if ad oly if q, we eed to follow the followig steps:. If p is true, the q is true.. If q is true, the p is true. Method of cotradictio To check whether a statemet p is true, we assume that p is ot true, i.e., p is true, ad the we arrive at some result which cotradicts our assumptio. Therefore, we coclude that p is true. This method of provig a give statemet to be true is called the cotradictio method. Usig a couter example I order to disprove a statemet, we give a example of the situatio where the statemet is ot valid. Such a example is called a couter example. I mathematics, couter examples are used for disprovig a statemet. However, geeratig examples i favour of a statemet does ot provide validity to the statemet. ( ) Example: The statemet ( ), N is true for =,, but it is ot true for = 3, 4. Here, = 3 ad 4 are the couter examples of the give statemet. Sice the statemet is ot true for each N, the give statemet is false.

37 Chapter 5: Statistics The measures of cetral tedecy, mea, media ad mode give us a rough idea where data poits are cetered. The dispersio or scatter i a data is measured o the basis of observatios ad the types of measure of cetral tedecy used. The measures of dispersio are as follows: Rage Quartile deviatio Mea deviatio Stadard deviatio Rage of a series = Maximum value Miimum value Mea deviatio about mea[ M.D. ( x ) ] For ugrouped data: M.D.( x) x x, where x is the mea give by x xi i i i i For grouped data: M.D. ( x) fi xi x, where x is the mea give by N i x fixi ad N fi N i Mea deviatio about media [M.D.(M)] For ugrouped data: M.D.( M ) x M, where M is the media For grouped data: i i i M.D.( M ) fi xi M, where M is the media ad N N f i i The mea of the squares of the deviatios from mea is called the variace ad it is deoted by. Variace of data For ugrouped data: (I direct method) or ( xi x) i shortcut method), where x is the mea. (I xi ( x) i

38 For discrete frequecy distributio: N fixi i i N f x i i ad N f. i (I direct method) or fi( xi x) N i (I shortcut method), where x is the mea i For cotiuous frequecy distributio: N fixi i i N f x i i h N or fi( xi x) N i (I direct method) or f i yi i i N f y i i (I shortcut method), where x i = class marks of the class itervals, x = mea, itervals, xi y i h A, where A is the assumed mea. N f, h = width of the class i i Addig (or subtractig) a positive umber to (or from) each observatio does ot affect the variace. If each observatio is multiplied with a costat k, the the variace of the resultig observatios becomes k times the origial variace. Stadard deviatio is the square root of variace ad it is deoted by. This meas: Stadard Deviatio Variace Coefficiet of variatio The measure of variability, which is idepedet of uits, is called the coefficiet of variatio. The coefficiet of variatio (C.V.) is defied as C.V. 00, x 0 x Where, ad x are stadard deviatio ad mea of the data respectively. For comparig the variability or dispersio of two series, we first calculate the C.Vs of each series. The series havig higher C.V. is said to be more variable tha the other ad the series havig lower C.V. is said to be more cosistet tha the other. For two series with equal mea values, the series with greater stadard deviatio (or variace) is more variable or dispersed tha the other. Also, the series with lower value of stadard deviatio (or variace) is said to be more cosistet or less scattered tha the other.

39 Chapter 6: Probability A experimet is called a radom experimet if it satisfies the followig two coditios: It has more tha oe possible outcome It is ot possible to predict the outcome i advace A possible result of a radom experimet is called its outcome. The set of all possible outcomes of a radom experimet is called the sample space associated with the experimet. It is deoted by S. Each elemet of the sample space i.e., each outcome of the radom experimet is called the sample poit. Ay subset (E) of a sample space is called a evet. The evet (E) of a sample space (S) is said to have occurred if the outcome () of the experimet is such that E. If the outcome () is such that E, we say that the evet (E) has ot occurred. Types of evets Impossible evet: A empty set () is called a impossible evet. Sure evet: The whole sample space (S) is called a sure evet. Simple evet: If a evet (E) has oly oe sample poit of a sample space, the it is called a simple (or elemetary) evet. I a sample space cotaiig distict elemets, there are exactly simple evets. Compoud evet: If a evet has more tha oe sample poit, it is called a compoud evet. Example: Cosider a experimet of selectig umbers from first 50 atural umbers, the the sample space (S) is give as S = {,, 3, 4, 49, 50} I this case, (i) The evet Gettig a umber that is a multiple of both 7 ad 9 is a impossible evet as there is o outcome related to it. (ii) The evet Gettig a perfect square that is more tha 40 is a simple evet sice E = {49}. (iii)the evet Gettig a odd multiple of 3 is a compoud evet sice E = {3, 39}. Algebra of evets Complemetary evet: For every evet A, there correspods aother evet A called the complemetary evet to A. It is also called the evet ot A. A = { : S ad A} = S A. Evet A or B : Whe sets A ad B are two evets associated with a sample space, the the set A B is the evet either A or B or both. That is, evet A or B = A B = { : A or B}

40 Evet A ad B : Whe sets A ad B are two evets associated with a sample space, the the set A B is the evet A ad B. That is, evet A ad B = A B = { : A ad B} Evet A but ot B : Whe sets A ad B are two evets associated with a sample space, the the set A B is the evet A but ot B. That is, evet A but ot B = A B = A B = { : A ad B} Example: Cosider the experimet of tossig cois. Let A be the evet gettig at least oe head ad B be the evet gettig exactly two heads. Fid the sets represetig the evets (i) complemet of A or B (ii) A ad B (iii) A but ot B Solutio: Here, S = {HH, HT, TH, TT} A = {HH, HT, TH}, B = {HH} (i) A or B = A B = {HH, HT, TH} Hece, complemet of A or B = (A or B) = (A B) = {TT} (ii) A ad B = A B = {HH} (iii) A but ot B = A B = {HT, TH} Two evets, A ad B, are called mutually exclusive evets if the occurrece of ay oe of them excludes the occurrece of the other evet i.e., if they caot occur simultaeously. I this case, sets A ad B are disjoit i.e., A B = If E, E, E are evets of a sample space S, ad if E E. E = E S, the i i E, E E are called mutually exhaustive evets. I other words, at least oe of E, E E ecessarily occurs wheever the experimet is performed. The evets E, E, E, i.e., evets of a sample space (S) are called mutually exclusive ad exhaustive evets if E i E j = for i j i.e., evets E i ad E j are pairwise disjoit, ad E S i i The umber P ( i ) i.e., the probability of the outcome i, is such that 0 P( i ) P( i ) = for all i S For ay evet A, P(A) = P( i ) for all i A For a fiite sample space, S, with equally likely outcomes, the probability of a evet A is deoted as P (A) ad it is give by