OPG S. LIST OF FORMULAE [ For Class XII ] OP GUPTA. Electronics & Communications Engineering. Indira Award Winner

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1 OPG S MAHEMAICS LIS OF FORMULAE [ For Class XII ] Covering all the topics of NCER Mathematics et Book Part I For the session 0-4 By OP GUPA Electronics & Communications Engineering Inira Awar Winner Visit at M ,

2 Relations, Functions & Binary Operations Important erms, Definitions & Formulae 0 YPES OF INERVALS a) Open interval: If a an b be two real numbers such that a b then, the set of all the real numbers lying strictly between a an b is calle an open interval It is enote by ] a, b [ or a, b ie, R : a b b) Close interval: If a an b be two real numbers such that a b then, the set of all the real numbers lying between a an b such that it inclues both a an b as well is known as a close interval It is enote by a, b ie, R : a b c) Open Close interval: If a an b be two real numbers such that a b then, the set of all the real numbers lying between a an b such that it eclues a an inclues only b is known as an open close interval It is enote by a, b or a, b ie, R : a b ) Close Open interval: If a an b be two real numbers such that a b then, the set of all the real numbers lying between a an b such that it inclues only a an eclues b is known as a close open, or, R : a b interval It is enote bya b a b ie, A RELAIONS 0 Defining the Relation: A relation R, from a non-empty set A to another non-empty set B is mathematically efine as an arbitrary subset of A B Equivalently, any subset of A B is a relation from A to B hus, R is a relation from A to B R A B R a, b : a A, b B Illustrations: a) Let A,,4, B 4,6 Let R (,4),(,6),(,4),(,6),(4,6) Here R A B an therefore R is a relation from A to B b) Let A,,, B,,5,7 Let R (,),(,5),(5,7) Here R A B an therefore R is not a relation from A to B Since (5,7) R but (5,7) A B Let a R b means c) Let A,,, B,4,9,0 a b then, R (,),(,),(,4) Note the followings: A relation from A to B is also calle a relation from A into B ( a, b) R is also written as arb (rea as a is R relate to b) Let A an B be two non-empty finite sets having p an q elements respectively hen n A B n A n B pq hen total number of subsets of A B pq Since each subset of A B is a relation from A to B, therefore total number of relations from A to B is pq given as 0 DOMAIN & RANGE OF A RELAION a) Domain of a relation: Let R be a relation from A to B he omain of relation R is the set of all those elements a A such that ( a, b ) R for some b B Domain of R is precisely written as Dom( R ) symbolically hus, Dom(R) a A : a, b R for some b B hat is, the omain of R is the set of first component of all the orere pairs which belong to R b) Range of a relation: Let R be a relation from A to B he range of relation R is the set of all those elements b B such that ( a, b ) R for some a A hus, Range of R b B: a, b R for some a A hat is, the range of R is the set of secon components of all the orere pairs which belong to R Page - []

3 List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering) c) Coomain of a relation: Let R be a relation from A to B hen B is calle the coomain of the relation R So we can observe that coomain of a relation R from A into B is the set B as a whole Illustrations: A,,,7, B,6 Let arb means a) Let a b hen (,),(,6),(,),(,6),(,6) Here Dom( R),,, Range of R,6, Coomain of R B,6 b) Let A,,, B, 4,6,8 Let R (,),(,4),(,6), R (,4),(,6),(,8),(,6) R hen both R an R are relations from A to B because R A B, R A B Dom( R ),,, Range of R,4,6 ; Dom( R ),,, Range of R 4,6,8 Here 04 YPES OF RELAIONS FROM ONE SE O ANOHER SE a) Empty relation: A relation R from A to B is calle an empty relation or a voi relation from A to B if R φ For eample, let A,4,6, B 7, Let R ( a, b) : a A, b B an a b is even Here R is an empty relation b) Universal relation: A relation R from A to B is sai to be the universal relation if R A B For eample, let A,, B, Let R (,),(,),(,),(,) Here R = A B, so relation R is a universal relation 05 RELAION ON A SE & IS VARIOUS YPES A relation R from a non-empty set A into itself is calle a relation on A In other wors if A is a nonempty set, then a subset of A A A is calle a relation on A Illustrations:,, R (,),(,),(,) Here R is relation on set A Let A an NOE If A be a finite set having n elements then, number of relations on set A is nn ie, n a) Empty relation: A relation R on a set A is sai to be empty relation or a voi relation if R φ In other wors, a relation R in a set A is empty relation, if no element of A is relate to any element of A, ie, R φ A A A R a b a A b A an a b is o Here R contains no element, therefore it is an empty relation on set A b) Universal relation: A relation R on a set A is sai to be the universal relation on A if R A A ie, R A In other wors, a relation R in a set A is universal relation, if each element of A is relate to every element of A, ie, R A A For eample, let A, Let R (,),(,),(,),(,) Here R A A, so relation R is universal relation on A NOE he voi relation ie, φ an universal relation ie, A A on A are respectively the smallest an largest relations efine on the set A Also these are sometimes calle trivial relations An, any other relation is calle a non-trivial relation For eample, let,, (, ) :, he relations R φ an R A A are two etreme relations c) Ientity relation: A relation R on a set A is sai to be the ientity relation on A if R ( a, b) : a A, b A an a b hus ientity relation R (, ) : A a a a he ientity relation on set A is also enote by I A Page - []

4 MAHEMAICS List Of Formulae for Class XII By OP Gupta ( ) For eample, let A,,, 4 hen I A (,),(,),(,),(4,4) by (,),(,),(,),(4,4) But the relation given R is not an ientity relation because element is relate to elements an NOE In an ientity relation on A every element of A shoul be relate to itself only ) Refleive relation: A relation R on a set A is sai to be refleive if a R a a A ie, ( a, a) R a A For eample, let,, R (,),(,),(,), (,),(,),(,),(,),(,),(,) R (,),(,),(,),(,) A, an R, R, R be the relations given as R an Here R an R are refleive relations on A but R is not refleive as A but (,) R NOE he ientity relation is always a refleive relation but the opposite may or may not be true As shown in the eample above, R is both ientity as well as refleive relation on A but R is only refleive relation on A e) Symmetric relation: A relation R on a set A is symmetric if a, b R b, a R a, b A ie, a R b b R a (ie, whenever arb then, bra ) For eample, let A,,, R (,),(,), R (,),(,),(,),(,), R i e R an (,),(,),(,),(,) (,),(,),(,) R 4 (,),(,),(,) Here R, R an R are symmetric relations on A But R 4 is not symmetric because (,) R 4 but (,) R 4 f) ransitive relation: A relation R on a set A is transitive if, R b, c R a, c R ie, a R b an b R c a R c For eample, let A,,, R an (,),(,),(,),(,) a b an R (,),(,),(,) Here R is transitive relation whereas R is not transitive because (,) R an (,) R but (,) R g) Equivalence relation: Let A be a non-empty set, then a relation R on A is sai to be an equivalence relation if (i) R is refleive ie ( a, a) R a A ie, ara (ii) R is symmetric ie, R, R, A (iii) R is transitive ie a, b R an brc arc a b b a a b ie, arb bra b, c R a, c R a, b, c A ie, arb an For eample, let A,,, R (,),(,),(,),(,),(,) symmetric an transitive So R is an equivalence relation on A Here R is refleive, Equivalence classes: Let A be an equivalence relation in a set A an let a A hen, the set of all those elements of A which are relate to a, is calle equivalence class etermine by a an it is enote by a hus, a b A : a, b A NOE (i) wo equivalence classes are either isjoint or ientical (ii) An equivalence relation R on a set A partitions the set into mutually isjoint equivalence classes 06 ABULAR REPRESENAION OF A RELAION In this form of representation of a relation R from set A to set B, elements of A an B are written in the first column an first row respectively If a, b R then we write in the row containing a an column containing b an if a, b R then we write 0 in the same manner Page - []

5 List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering) For eample, let A,,, B,5 an (,),(,5),(,) R then, R INVERSE RELAION Let R A B be a relation from A to B hen, the inverse relation of R, to be enote by R, is a relation from B to A efine by R ( b, a) : ( a, b ) R hus ( a, b) R ( b, a) R a A, b B Clearly, Dom R Range of R, Range of R Dom R Also, R R For eample, let A,,4, B,0 an let (,),(4,0),(,) R (,),(0,4),(,) R be a relation from A to B then, Summing up all the iscussion given above, here is a recap of all these for quick grasp: 0 a) A relation R from A to B is an empty relation or voi relation iff R φ b) A relation R on a set A is an empty relation or voi relation iff R φ 0 a) A relation R from A to B is a universal relation iff R=A B b) A relation R on a set A is a universal relation iff R=A A 0 A relation R on a set A is refleive iff ar a, a A 04 A relation R on set A is symmetric iff whenever arb, then bra for all a, b A 05 A relation R on a set A is transitive iff whenever arb an brc, then arc 06 A relation R on A is ientity relation iff R={( a, a), a A} ie, R contains only elements of the type ( a, a) a A an it contains no other element 07 A relation R on a non-empty set A is an equivalence relation iff the following conitions are satisfie: i) R is refleive ie, for every a A, ( a, a ) R ie, ara ii) R is symmetric ie, for a, b A, arb bra ie, a, b R b, a R iii) R is transitive ie, for all a, b, c A we have, arb an R R a, b R an b, c R a, c R b c a c ie, B FUNCIONS 0 CONSAN & YPES OF VARIABLES a) Constant: A constant is a symbol which retains the same value throughout a set of operations So, a symbol which enotes a particular number is a constant Constants are usually enote by the symbols a, b, c, k, l, m, etc b) Variable: It is a symbol which takes a number of values ie, it can take any arbitrary values over the interval on which it has been efine For eample if is a variable over R (set of real numbers) then we Page - [4]

6 MAHEMAICS List Of Formulae for Class XII By OP Gupta ( ) mean that can enote any arbitrary real number Variables are usually enote by the symbols, y, z, u, v, etc c) Inepenent variable: hat variable which can take an arbitrary value from a given set is terme as an inepenent variable ) Depenent variable: hat variable whose value epens on the inepenent variable is calle a epenent variable 0 Defining A Function: Consier A an B be two non- empty sets then, a rule f which associates each element of A with a unique element of B is calle a function or the mapping from A to B or f maps A to B If f is a mapping from A to B then, we write f : A B which is rea as f is a mapping from A to B or f is a function from A to B If f associates a A to b B, then we say that b is the image of the element a uner the function f or b is the f - image of a or the value of f at a an enote it by f ( a ) an we write b f ( a ) he element a is calle the pre-image or inverse-image of b hus for a function from A to B, (i) A an B shoul be non-empty (ii) Each element of A shoul have image in B (iii) No element of A shoul have more than one image in B (iv) If A an B have respectively m an n number of elements then the number of functions m efine from A to B is n 0 Domain, Co-omain & Range of a function he set A is calle the omain of the function f an the set B is calle the co- omain he set of the images of all the elements of A uner the function f is calle the range of the function f an is enote as f (A) hus range of the function f is f (A) { f ( ) : A} Clearly f (A) B Note the followings: i) It is necessary that every f -image is in B; but there may be some elements in B which are not the f- images of any element of A ie, whose pre-image uner f is not in A ii) wo or more elements of A may have same image in B iii) f : y means that uner the function f from A to B, an element of A has image y in B iv) Usually we enote the function f by writing y f ( ) an rea it as y is a function of POINS O REMEMBER FOR FINDING HE DOMAIN & RANGE Domain: If a function is epresse in the form y f ( ), then omain of f means set of all those real values of for which y is real (ie, y is well - efine) Remember the following points: i) Negative number shoul not occur uner the square root (even root) ie, epression uner the square root sign must be always 0 ii) Denominator shoul never be zero iii) For logba to be efine a 0, b 0 an b Also note that logb is equal to zero ie 0 Range: If a function is epresse in the form y f ( ), then range of f means set of all possible real values of y corresponing to every value of in its omain Remember the following points: i) Firstly fin the omain of the given function ii) If the omain oes not contain an interval, then fin the values of y putting these values of from the omain he set of all these values of y obtaine will be the range iii) If omain is the set of all real numbers R or set of all real numbers ecept a few points, then epress in terms of y an from this fin the real values of y for which is real an belongs to the omain Page - [5]

7 List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering) 04 Function as a special type of relation: A relation f from a set A to another set B is sai be a function (or mapping) from A to B if with every element (say ) of A, the relation f relates a unique element (say y) of B his y is calle f -image of Also is calle pre-image of y uner f 05 Difference between relation an function: A relation from a set A to another set B is any subset of A B ; while a function f from A to B is a subset of A B satisfying following conitions: i) For every A, there eists y B such that (, y) f ii) If (, y) f an (, z) f then, y z Sl No Function Relation 0 Each element of A must be relate to some element of B 0 An element of A shoul not be relate to more than one element of B here may be some element of A which are not relate to any element of B An element of A may be relate to more than one elements of B 06 Real value function of a real variable: If the omain an range of a function f are subsets of R (the set of real numbers), then f is sai to be a real value function of a real variable or a real function 07 Some important real functions an their omain & range FUNCION REPRESENAION DOMAIN RANGE a) Ientity function I( ) R R R b) Moulus function or Absolute value function c) Greatest integer function or Integral function or Step function ) Smallest integer function if 0 f ( ) if 0 ( ) R 0, f or f ( ) R R Z f ( ) R R Z, 0 e) Signum function if 0 f ( ) ie, f ( ) 0, 0 0 if 0, 0 f) Eponential function ( ), 0 R {, 0,} f a a a R 0, g) Logarithmic function f ( ) loga a, a 0 an 0 0, R 08 YPES OF FUNCIONS a) One-one function (Injective function or Injection): A function f : A B is one- one function or injective function if istinct elements of A have istinct images in B hus, f : A B is one-one f ( a) f ( b) a b a, b A a b f ( a) f ( b) a, b A If A an B are two sets having m an n elements respectively such that m n, then total number of n one-one functions from set A to set B is C m! ie, n P m ALGORIHM O CHECK HE INJECIVIY OF A FUNCION SEP- ake any two arbitrary elements a, b in the omain of f SEP- Put f ( a) f ( b ) SEP- Solve f ( a) f ( b ) If it gives a b only, then f is a one-one function m Page - [6]

8 MAHEMAICS List Of Formulae for Class XII By OP Gupta ( ) b) Onto function (Surjective function or Surjection): A function f : A B is onto function or a surjective function if every element of B is the f-image of some element of A hat implies f (A) B or range of f is the co-omain of f hus, f : A B is onto f (A) B i e, range of f co-omain of f ALGORIHM O CHECK HE SURJECIVIY OF A FUNCION SEP- ake an element b B SEP- Put f ( ) b SEP- Solve the equation f ( ) b for an obtain in terms of b Let g( b ) SEP4- If for all values of b B, the values of obtaine from g( b ) are in A, then f is onto If there are some b B for which values of, given by g( b ), is not in A hen f is not onto c) One-one onto function (Bijective function or Bijection): A function f : A B is sai to be oneone onto or bijective if it is both one-one an onto ie, if the istinct elements of A have istinct images in B an each element of B is the image of some element of A Also note that a bijective function is also calle a one-to-one function or one-to-one corresponence If f : A B is a function such that, i) f is one-one n(a) n (B) ii) f is onto n( B) n (A) iii) f is one-one onto n(a) n (B) For an orinary finite set A, a one-one function f : A A is necessarily onto an an onto function f : A A is necessarily one-one for every finite set A ) Ientity Function: he function I A : A A ; I ( ) A A is calle an ientity function on A NOE Domain I A an Range I A A A e) Equal Functions: wo function f an g having the same omain D are sai to be equal if f ( ) g( ) for all D 09 COMPOSIION OF FUNCIONS Let f : A B an g : B C be any two functions hen f maps an element A to an element f ( ) y B; an this y is mappe by g to an element C z g( y) g f ( ) herefore, we have a rule which associates with each A, a unique element z g f ( ) of C his rule is therefore a mapping from A to C We enote this mapping by gof (rea as g composition f ) an call it the composite mapping of f an g z hus, Definition: If f : A B an g : B C be any two mappings (functions), then the composite mapping gof of f an g is efine by gof : A C such that gof ( ) g f ( ) for all A he composite of two functions is also calle the resultant of two functions or the function of a function Note that for the composite function gof to eist, it is essential that range of f must be a subset of the omain of g Observe that the orer of events occur from the right to left ie, gof reas composite of f an g an it means that we have to first apply f an then, follow it up with g Dom( gof ) Dom( f ) an Co- om( gof ) Co- om( g ) Remember that gof is well efine only if Co- om( f ) Dom( g ) If gof is efine then, it is not necessary that fog is also efine Page - [7]

9 List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering) If gof is one-one then f is also one-one function Similarly, if gof is onto then g is onto function 0 INVERSE OF A FUNCION Let f : A B be a bijection hen a function g : B A which associates each element y B to a unique element A such that f ( ) y is calle the inverse of f ie, f ( ) y g( y) he inverse of f is generally enote by f hus, if f : A B is a bijection, then a function f : B A is such that f ( ) y f ( y) ALGORIHM O FIND HE INVERSE OF A FUNCION SEP- Put f ( ) y where y B an A SEP- Solve f y to obtain in terms of y SEP- Replace by f ( y ) in the relation obtaine in SEP SEP4- In orer to get the require inverse of f ie f ( ), replace y by in the epression obtaine in SEP ie in the epression f ( y ) C BINARY OPERAIONS 0 Definition & Basic Unerstaning An operation is a process which prouces a new element from two given elements; eg, aition, subtraction, multiplication an ivision of numbers If the new element belongs to the same set to which the two given elements belong, the operation is calle a binary operation DEFINIION: A binary operation (or binary composition) on a non-empty set A is a mapping which associates with each orere pair ( a, b ) A A a unique element of A his unique element is enote by ab hus, a binary operation on A is a mapping : A A A efine by a, b a b Clearly a A, ba a b A mm NOE i) If set A has m elements, then number of binary operations on A is m ie, m m( m) ii) If set A has m elements, then number of Commutative binary operations on A is m 0 erms relate to binary operations A binary operation on a set A is a function from A A to A An operation on A is commutative if a b b a a, b A An operation on A is associative if a b c a b c a, b, c A An operation on A is istributive if a b c a b a c a, b, c A Also an operation on A is istributive if b c a b a c a a, b, c A An ientity element of a binary operation on A is an element e A such that e a a e a a A Ientity element for a binary operation, if it eists is unique 0 Inverse of an element Let be a binary operation on a non-empty set A an e be the ientity element for the binary operation An element b A is calle the inverse element or simply inverse of a for binary operation if a b b a e hus, an element a A is invertible if an only if its inverse eists m Page - [8]

10 MAHEMAICS List Of Formulae for Class XII By OP Gupta ( ) Inverse rigonometric Functions Important erms, Definitions & Formulae 0 rigonometric Formulae: Relation between trigonometric ratios sin a) tan b) tan c) tan cot cos cot cos ) cot e) cosec f) sec sin sin cos rigonometric ientities a) sin cos b) tan sec c) cot cosec Aition / subtraction formulae & some relate results sin A B sin Acos B cos Asin B a) cos A B cos Acos B sin Asin B b) cos cos cos sin cos sin A B A B A B B A c) sin sin sin sin cos cos A B A B A B B A ) e) tan A B f) cot A B tan A tan B tan Atan B cot B cot A cot B cot A ransformation of sums / ifferences into proucts & vice-versa C D C D a) sinc sin D sin cos C D C D b) sinc sin D cos sin C D C D c) cosc cos D cos cos C D C D ) cosc cos D sin sin sin Acos B sin A B sin A B cos Asin B sin A B sin A B e) f) g) cos AcosB cos A B cos A B h) sin Asin B cos A B cos A B Multiple angle formulae involving A an A a) sin A sin Acos A A A b) sin A sin cos c) cosa cos A sin A A A ) cos A cos sin e) cos A cos A f) cos A cosa g) cos A sin A h) sin A cos A tan A i) sin A tan A tan A j) cos A tan A tan A k) tana tan A l) sin A sin A 4sin A m) cos A 4cos A cos A tan A tan A n) tana tan A Relations in Different Measures of Angle Angle in Raian Measure = Angle in Degree Measure Angle in Degree Measure = Angle in Raian Measure ( in raian measure) l r Also followings are of importance as well: o o Right angle 90 = 60, = 60 Page - [9]

11 List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering) 80 o = = raians approimately General Solutions n a) sin sin y n ( ) y, where n Z b) cos cos y n y, where n Z c) tan tan y n y, where n Z Relation in Degree & Raian Measures o raian = or 0665 secons Angles in Degree Angles in Raian 0 c 6 c c 4 c c c c c In actual practice, we omit the eponent c an instea of writing c we simply write an similarly for others rigonometric Ratio of Stanar Angles Degree /Raian Ratios 0 sin 0 6 cos tan 0 4 cosec sec cot 0 0 rigonometric Ratios of Allie Angles Angles - Ratios OR sin cos cos sin sin cos cos sin sin cos sin sin cos cos sin sin cos cos tan cot cot tan tan cot cot cot tan tan cot cot tan tan tan tan cot cot sec cosec cosec sec sec cosec cosec sec sec cosec sec sec cosec cosec sec sec cosec cosec Page - [0]

12 MAHEMAICS List Of Formulae for Class XII By OP Gupta ( ) 0 a) sin cosec,, b) cosec sin,,, c) cos sec,, ) sec cos,,, e) tan cot, 0 π cot, 0 f) cot tan, 0 π tan, 0 0 a) sin sin,, b) cos π cos,, c) tan tan, R ) cosec cosec, e) sec π sec, f) cot π cot, R 04 a) sin sin, b) π π tan tan, π π cos cos, 0 π c) ) e) sec sec, 0 π, f) π 05 a) sin cos,, π π b) tan cot, R π c) cosec sec, i e, or cosec cosec, π π, 0 cot cot, 0 π 06 a) sin sin y sin y y b) cos cos y cos y y y tan, y y y c) tan tan y π tan, 0, y 0, y y y π tan, 0, y 0, y y y tan, y y y ) tan tan y π tan, 0, y 0, y y y π tan, 0, y 0, y y y z yz e) tan tan y tan z tan y yz z 07 a) tan sin, Page - []

13 List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering) b) tan cos, 0 c) tan tan, 08 Principal Value: Numerically smallest angle is known as the principal value Fining the principal value: For fining the principal value, following algorithm can be followe SEP Firstly, raw a trigonometric circle an mark the quarant in which the angle may lie SEP Select anticlockwise irection for st an n quarants an clockwise irection for r an 4 th quarants SEP Fin the angles in the first rotation SEP4 Select the numerically least (magnitue wise) angle among these two values he angle thus foun will be the principal value SEP5 In case, two angles one with positive sign an the other with the negative sign qualify for the numerically least angle then, it is the convention to select the angle with positive sign as principal value he principal value is never numerically greater than 09 able emonstrating omains an ranges of Inverse rigonometric functions: Inverse rigonometric Functions ie, f ( ) Domain/ Values of Range/ Values of f ( ) sin [, ] π π, cos [, ] [0, π] cosec R (, ) π π, {0} sec R (, ) π [0, π] tan R π π, cot R (0, π) Discussion about the range of inverse circular functions other than their respective principal value branch We know that the omain of sine function is the set of real numbers an π π range is the close interval [, ] If we restrict its omain to,, π π,, π π, etc then, it becomes bijective with the range [, ] So, we can efine the inverse of sine function in each of these intervals Hence, all the intervals of sin function, ecept principal value branch π π (here ecept of, for sin function) are known as the range of sin other than its principal value branch he same iscussion can be etene for other inverse circular functions Page - []

14 MAHEMAICS List Of Formulae for Class XII By OP Gupta ( ) 0 o simplify inverse trigonometrical epressions, following substitutions can be consiere: Epression Substitution a or a a tanθ or a cot θ a or a a sin θ or a cosθ a or a a secθ or a cosecθ a a or a a a a or a a a or a a or a a cos θ a cos θ a sin θ or a cos θ a tan θ or a cot θ Note the followings an keep them in min: he symbol sin is use to enote the smallest angle whether positive or negative, the sine of this angle will give us Similarly cos, tan, cosec, sec, an cot are efine You shoul note that sin can be written as arcsin Similarly other Inverse rigonometric Functions can also be written as arccos, arctan, arcsec etc Also note that sin (an similarly other Inverse rigonometric Functions) is entirely ifferent from ( sin ) In fact, sin is the measure of an angle in Raians whose sine is whereas ( sin ) is (which is obvious as per the laws of eponents) sin Keep in min that these inverse trigonometric relations are true only in their omains ie, they are vali only for some values of for which inverse trigonometric functions are well efine! Page - []

15 List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering) Algebra Of Matrices & Determinants Important erms, Definitions & Formulae 0 Matri - a basic introuction: A matri having m rows an n columns is calle a matri of orer m n (rea as m by n matri) An a matri A of orer Also in general, a ij means an element lying in the m n is epicte as A th i row an No of elements in the matri A a ij is given as ( m)( n ) m n 0 ypes of Matrices: a) Column matri: A matri having only one column is calle a column matri or column vector eg 0, 8 5 General notation: A a ij m th j column a ; i, j N ij m n b) Row matri: A matri having only one row is calle a row matri or row vector eg 4 4, 5 0 General notation: A a ij n c) Square matri: It is a matri in which the number of rows is equal to the number of columns ie, an m n matri is sai to constitute a square matri if m n an is known as a square matri of orer n 5 eg 7 4 is a square matri of orer 0 General notation: A a ij nn ) Diagonal matri: A square matri A a ij is sai to be a iagonal matri if a 0 mm ij, when i j ie, all its non- iagonal elements are zero 0 0 eg is a iagonal matri of orer Also there is one more notation specifically use for the iagonal matrices For instance, consier the matri epicte above, it can be also written as iag 5 4 Note that the elements a,a,a,,a mm of a square matri A a ij of orer m are sai to m n constitute the principal iagonal or simply the iagonal of the square matri A An these elements are known as iagonal elements of matri A e) Scalar matri: A iagonal matri equal ie, a eg ij A a ij is sai to be a scalar matri if its iagonal elements are mm 0, when i j k, when i j for some constant k is a scalar matri of orer f) Unit or Ientity matri: A square matri a ij, if i j 0, if i j A a ij is sai to be an ientity matri if mn Page - [4]

16 MAHEMAICS List Of Formulae for Class XII By OP Gupta ( ) A unit matri can also be efine as the scalar matri each of whose iagonal elements is unity We enote the ientity matri of orer m by I m or I eg 0 0 I 0 0, I 0 g) Zero matri or Null matri: A matri is sai to be a zero matri or null matri if each of its elements is zero eg , , 0 0 h) Horizontal matri: A m n matri is sai to be a horizontal matri if m < n 0 eg j) riangular matri: Lower triangular matri: A square matri is calle a lower triangular matri if a 0 when i j eg , 0 0 0, ij i) Vertical matri: A m n matri is sai to be a vector matri if m > n 5 eg 0 7 Upper triangular matri: A square matri is calle an upper triangular matri if a 0 when i j eg , Properties of matri aition: Commutative property: A B B A Associative property: A B C A B C Cancellation laws: i) Left cancellation- A B A C B C ii) Right cancellation- B A C A B C 04 Properties of matri multiplication: Note that the prouct AB is efine only when the number of columns in matri A is equal to the number of rows in matri B If A an B are m n an n p matrices respectively then matri AB will be an m p matri matri ie, orer of matri AB will be m p In the prouct AB, A is calle the pre-factor an B is calle the post-factor If the prouct AB is possible then it is not necessary that the prouct BA is also possible If A is a m n matri an both AB an BA are efine then B will be a n m matri If A is a n n matri an I n be the unit matri of orer n then, A In In A A Matri multiplication is associative ie, ABC ABC Matri multiplication is istributive over the aition ie, AB C AB AC Iempotent matri: A square matri A is sai to be an iempotent matri if 0 0 For eample, 0 0 0, A ij A 05 ranspose of a Matri: If A a ij be an m n matri, then the matri obtaine by interchanging m n the rows an columns of matri A is sai to be a transpose of matri A he transpose of A is enote by c A or A or A ie, if A a ij then, A mn a ji nm 0 For eample, Page - [5]

17 List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering) Properties of ranspose of matrices: A B A B A B A B A A ka ka where, k is any constant AB B A ABC C B A 06 Symmetric matri: A square matri A a ij is sai to be a symmetric matri if A A hat is, if A a ij then, A a ji a ij A A For eample: a h g i h b f, i g f c i 4 07 Skew-symmetric matri: A square matri A a ij is sai to be a skew-symmetric matri if A A ie, if A a ij then, A a ji aij A A 0 For eample: 0 5 0, Facts you shoul know: Note that a ji aij aii aii a ii 0 {Replacing j by i } hat is, all the iagonal elements in a skew-symmetric matri are zero he matrices AA an A A are symmetric matrices For any square matri A, A A is a symmetric matri an A A is a skew- symmetric matri always Also any square matri can be epresse as the sum of a symmetric an a skew-symmetric matri ie, A P Q, where P A A is a symmetric matri an Q A A is a skew- symmetric matri 08 Orthogonal matri: A matri A is sai to be orthogonal if A A I where A is the transpose of A 09 Invertible Matri: If A is a square matri of orer m an if there eists another square matri B of the same orer m, such that AB BA I, then B is calle the inverse matri of A an it is enote by A A matri having an inverse is sai to be invertible It is to note that if B is inverse of A, then A is also the inverse of B In other wors, if it is known that AB BA I then, A B B A 0 Determinants, Minors & Cofactors: a) Determinant: A unique number (real or comple) can be associate to every square matri A a ij of orer m his number is calle the eterminant of the square matri A, where, ij th a i j element of A a b a b For instance, if A then, eterminant of matri A is written as A et A an its c c value is given by a bc b) Minors: Minors of an element a ij of a eterminant (or a eterminant corresponing to matri A) is the eterminant obtaine by eleting its i th row an j th column in which a ij lies Minor of a ij is enote by M Hence we can get 9 minors corresponing to the 9 elements of a thir orer i e, eterminant ij Page - [6]

18 MAHEMAICS List Of Formulae for Class XII By OP Gupta ( ) c) Cofactors: Cofactor of an element a ij, enote by ij, minor of a ij Sometimes C ij is use in place of i j ij A is efine by, A ij to enote the cofactor of element A M, where M ij is Ajoint of a square matri: Let A a ij be a square matri Also assume B A ij where A ij is the cofactor of the elements a ij in matri A hen the transpose B of matri B is calle the ajoint of matri A an it is enote by aja o fin ajoint of a matri: Follow a b b A aja c c a For eample, consier a square matri of orer as A 4 then, in orer to fin the ajoint of matri 0 5 A, we fin a matri B (forme by the cofactors of elements of matri A as mentione above in the efinition) ie, B 0 4 Hence, aja B 6 4 Singular matri & Non-singular matri: a) Singular matri: A square matri A is sai to be singular if A 0 ie, its eterminant is zero eg 4 5, 4 4 a ij b) Non-singular matri: A square matri A is sai to be non-singular if A 0 eg 0 0, 0 A square matri A is invertible if an only if A is non-singular 4 Elementary Operations or ransformations of a Matri: he following three operations applie on the row (or column) of a matri are calle elementary row (or column) transformationsth a) Interchange of any two rows (or columns): When i row (or column) of a matri is interchange th with the j row (or column), it is enote as Ri R j (or Ci C j ) b) Multiplying all elements of a row (or column) of a matri by a non-zero scalar: When the row (or column) of a matri is multiplie by a scalar k, it is enote as Ri kri (or Ci kci ) c) Aing to the elements of a row (or column), the corresponing elements of any other row (or th column) multiplie by any scalar k: When k times the elements of j row (or column) is ae to th the corresponing elements of the i row (or column), it is enote as Ri Ri krj (or C C kc ) i i j NOE: In case, after applying one or more elementary row (or column) operations on A = IA (or A = AI), if we obtain all zeros in one or more rows of the matri A on LHS, then A oes not eist 4 Inverse or reciprocal of a square matri: If A is a square matri of orer n, then a matri B (if such a matri eists) is calle the inverse of A if AB BA I n Also note that the inverse of a square matri A is enote by A an we write, A B Inverse of a square matri A eists if an only if A is non-singular matri ie, A 0 If B is inverse of A, then A is also the inverse of B 5 Algorithm to fin Inverse of a matri by Elementary Operations or ransformations: By Row ransformations: SEP- Write the given square matri as A I n A ij th i Page - [7]

19 List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering) SEP- Perform a sequence of elementary row operations successively on A on the LHS an prefactor factor I n on the RHS till we obtain the result I n BA (or I n AB ) SEP- Write A B By Column ransformations: SEP- Write the given square matri as A A I n SEP- Perform a sequence of elementary column operations successively on A on the LHS an post post-factor I n on the RHS till we obtain the result I n AB SEP- Write A B 6 Algorithm to fin A by Determinant metho: SEP- Fin A SEP- If A 0 then, write A is a singular matri an hence not invertible Else write A is a non-singular matri an hence invertible SEP- Calculate the cofactors of elements of matri A SEP4- Write the matri of cofactors of elements of A an then obtain its transpose to get aja SEP5- Fin the inverse of A by using the relation A aja A 7 Properties associate with the Inverse of Matri & the Determinants: a) AB I BA b) A A I or A c) AB B A ) ABC I A C B A e) A A f) A A g) AajA aja A A I h) aj AB ajbaj A i) aj A ( aj A) j) aja aja k) n aj A A if A 0, where n is orer of A A aj l) AB A B m) A A n) A A provie matri A is invertible o) A A k A k n A where n is orer of square matri A an k is any scalar If A is a non-singular matri of orer n, then If A is a non-singular matri of orer n, then n, where n is orer of A n aj A A [ point ( k) given above] n aj aj A A A 8 Properties of Determinants: a) If any two rows or columns of a eterminant are proportional or ientical, then its value is zero a b c eg a b c 0 [As R an R are the same a b c b) he value of a eterminant remains unchange if its rows an columns are interchange a b c a a a eg a b c b b b a b c c c c Here rows an columns have been interchange, but there is no effect on the value of eterminant c) If each element of a row or a column of a eterminant is multiplie by a constant k, then the value of new eterminant is k times the value of the original eterminant Page - [8]

20 eg MAHEMAICS List Of Formulae for Class XII By OP Gupta ( ) a b c a b c, a b c k a b c k a b c a b c a b c ka kb kc a b c ) If any two rows or columns are interchange, then the eterminant retains its absolute value, but its sign is change a b c a b c eg a b c, a b c [Here R R a b c a b c e) If every element of some column or row is the sum of two terms, then the eterminant is equal to the sum of two eterminants; one containing only the first term in place of each sum, the other only the secon term he remaining elements of both eterminants are the same as given in the original eterminant a b c a b c b c eg a b c a b c b c a b c a b c b c 9 Area of triangle: Area of a triangle whose vertices are, y,, y an, y y sq units y y is given by, As the area is a positive quantity, we take absolute value of the eterminant given above y are collinear then 0 he equation of a line passing through the points, y an, y can be obtaine by the epression given here: y y 0 y If the points, y,, y an, 0 Solutions of System of Linear equations: a) Consistent an Inconsistent system: A system of equations is consistent if it has one or more solutions otherwise it is sai to be an inconsistent system In other wors an inconsistent system of equations has no solution b) Homogeneous an Non-homogeneous system: A system of equations AX B is sai to be a homogeneous system if B 0 Otherwise it is calle a non-homogeneous system of equations Solving of system of equations by Matri metho Inverse Metho : therefore Consier the following system of equations, a b y c z SEP- Assume,, a b y c z a b y c z a b c A a b c a b c, B an X SEP- Fin A Now there may be following situations: a) y z A 0 A eists It implies that the given system of equations is consistent an therefore, the system has unique solution In that case, write AX B Page - [9]

21 List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering) X A B where A aja A hen by using the efinition of equality of matrices, we can get the values of, y an z b) A 0 A oes not eist It implies that the given system of equations may be consistent or inconsistent In orer to check procee as follow: Fin aja B Now we may have either aja B 0 or aja B 0 If aja B 0, then the given system may be consistent or inconsistent o check, put z k in the given equations an procee in the same manner in the new two variables system system of equations assuming c k, i as constant An if aja B 0 i i, then the given system is inconsistent with no solutions Page - [0]

22 MAHEMAICS List Of Formulae for Class XII By OP Gupta ( ) Continuity & Differential Calculus Important erms, Definitions & Formulae 0 Formulae for Limits & Differential Calculus: a) lim cos 0 sin b) lim 0 tan c) lim 0 sin ) lim 0 tan e) lim 0 a) n n n LIMIS FOR SOME SANDARD FORMS a f) lim log e a, a 0 0 e g) lim 0 loge h) lim 0 n n a n i) lim na a a / lim k e k, where k is any constant j) 0 DERIVAIVE OF SOME SANDARD FUNCIONS b) ( k ) 0, where k is any constant a a e a a c) log, 0 e e ) e) log a log log a log e f) g) sin h) cos tan cos e sin sec i) a e sec sec tan j) cot cosec k) cosec cosec cot l) m) sin,, n) cos,, o) tan, R cot, R p) q) sec, where,, r) cosec Do you know for trigonometric functions, angle is in Raians?,where,, Following erivatives shoul also be memorize by you for quick use: log 0 Important terms an facts about Limits an Continuity of a function: For a function lim lim f lim f f, f eists iff m m m Page - []

23 List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering) A function lim m f at m A function f is continuous at a point m iff lim f lim f f m is Left Han Limit of Also f at m f m is the value of function f is continuous at m m m an lim f m f at m (say) if, lim, where is Right Han Limit of f m f ie, a function is continuous at m a point in its omain if the limit value of the function at that point equals the value of the function at the same point For a continuous function Ineterminate forms or meaningless forms: 0,, 0 0 0,,, 0, 0 he first two forms ie 0, 0 f at m, lim f can be irectly obtaine by evaluating m are the forms which are suitable for L Hospital Rule 0 Important terms an facts about Derivatives an Differentiability of a function: Left Han Derivative of f at m, f f m Lf m lim m m Right Han Derivative of Rf m lim m an, f at m, f f m m For a function to be ifferentiable at a point, the LHD an RHD at that point shoul be equal y y Derivative of y wrt : lim 0 Also for very- very small value h, f f h f, as h 0 h 04 Relation between Continuity an Differentiability: a) If a function is ifferentiable at a point, it is continuous at that point as well b) If a function is not ifferentiable at a point, it may or may not be continuous at that point c) If a function is continuous at a point, it may or may not be ifferentiable at that point ) If a function is iscontinuous at a point, it is not ifferentiable at that point 05 Rules of erivatives: Prouct or Leibnitz Rule of erivatives: u v u v v u u v u u v Quotient Rule of erivatives: v v f f m Page - []

24 MAHEMAICS List Of Formulae for Class XII By OP Gupta ( ) Applications Of Derivatives Important erms, Definitions & Formulae Rolle s heorem 0 he statement of Rolle s heorem: If a function f ( ) is, a) continuous in the close interval [ a, b ], b) ifferentiable in the open interval ( a, b ), an c) f ( a) f ( b) hen, there will be at least one point c in ( a, b ) such that f ( c) 0 Note that the converse of Rolle s heorem is not true ie, if a function f ( ) is such that f ( c) 0 for at least one c in the open interval ( a, b ) then it is not necessary that a) f ( ) is continuous in [ a, b ], b) f ( ) is ifferentiable in ( a, b ), c) f ( a) f ( b) Mean Value heorem 0 he statement of Mean Value heorem: If a function f ( ) is, a) continuous in the close interval [ a, b ], b) ifferentiable in the open interval ( a, b ), hen, there will be at least one point c, such that a c b ie f ( b) f ( a) c ( a, b) for which, f ( c) b a Note that the Mean Value heorem is an etension of Rolle s heorem On Rolle s heorem & Mean Value heorem, we are generally aske three types of problems: (i) o check the applicability of Rolle s heorem to a given function on a given interval, (ii) o verify Rolle s theorem for a given function on a given interval, an (iii) Problems base on the geometrical interpretation of Rolle s theorem In the first two cases, we first check whether the function satisfies conitions of Rolle s theorem or not he following results may be very helpful in oing so: (a) A polynomial function is everywhere continuous an ifferentiable (b) he eponential function, sine an cosine functions are everywhere continuous an ifferentiable (c) Logarithmic function is continuous an ifferentiable in its omain () tan is not continuous at /, /, 5 /, (e) is not ifferentiable at 0 (f) he sum, ifference, prouct an quotient of continuous (ifferentiable) function is continuous (ifferentiable) Approimate Values 0 Approimate change in the value of function y f : Given function is y f y y From the efinition of erivatives, Lt 0 by efinition of limit as 0, y y if is very near to zero, then we have y y (approimately) Page - []

25 List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering) herefore, y y, where y respresents the approimate change in y In case is relatively small when compare with, y is a goo approimation of y an we enote it by y y 0 Approimate value: By the efinition of erivatives (first principle), f h f f Lt h0 h by the efinition of limit when h 0, we have f h f h f if h is very near to zero, then we have f h f f (approimately) h f h f h f approimately as h 0 or Rate Of Change 0 Interpretation of y as a rate measurer: If two variables an y are varying with respect to another variable say t, ie, if f ( t ) an y g( t ), then y y / t by the Chain Rule, we have / t, t 0 hus, the rate of change of y with respect to can be calculate using the rate of change of y an that of both with respect to t y Also, if y is a function of an they are relate as y f then, f ie, represents the at rate of change of y with respect to at the instant when angents & Normals 0 Slope or graient of a line: If a line makes an angle with the positive irection of X-ais in anticlockwise irection, then tan is calle slope or graient of the line [Note that is taken as positive or negative accoring as it is measure in anticlockwise (ie, from positive irection of X-ais to the positive irection of Y-ais) or clockwise irection respectively] 0 Facts about the slope of a line: a) If a line is parallel to - ais (or perpenicular to y - ais) then, its slope is 0 (zero) b) If a line is parallel to y- ais (or perpenicular to - ais) then, its slope is ie not efine 0 c) If two lines are perpenicular then, prouct of their slopes equals ie, m m Whereas for two parallel lines, their slopes are equal ie, m m (Here in both the cases, m an m represent the slopes of respective lines) 0 Equation of angent at, y : y y m where, 04 Equation of Normal at, y : y y mn where, N m is the slope of tangent such that m is the slope of normal such that m m N y a t, y y a t, y Page - [4]

26 MAHEMAICS List Of Formulae for Class XII By OP Gupta ( ) Note that m m N, which is obvious because tangent an normal are perpenicular to each other In other wors, the tangent an normal lines are incline at right angle on each other 05 Acute angle between the two curves whose slopes m an m are known: tan m m m m m m tan m m It is absolutely sufficient to fin one angle (generally the acute angle) between the two curves Other angle between the two curve is given by Note that if the curves cut orthogonally (ie, they cut each other at right angles) then, it means m m where m an m represent slopes of the tangents of curves at the intersection point 06 Fining the slope of a line a by c 0 : SEP- Epress the given line in the stanar slope-intercept form y m c ie, SEP- By comparing to the stanar form y m c, we can conclue a b a by c 0 a c y b b as the slope of given line Maima & Minima 0 Unerstaning maima an minima: Consier y f be a well efine function on an interval I hen a) f is sai to have a maimum value in I, if there eists a point c in I such that f(c) > f(), for all I he value corresponing to f(c) is calle the maimum value of f in I an the point c calle a point of maimum value of f in I b) f is sai to have a minimum value in I, if there eists a point c in I such that f(c) < f(), for all I he value corresponing to f(c) is calle the minimum value of f in I an the point c calle a point of minimum value of f in I c) f is sai to have an etreme value in I, if there eists a point c in I such that f(c) is either a maimum value or a minimum value of f in I he value f(c) in this case, is calle an etreme value of f in I an the point c calle an etreme point 0 Meaning of local maima an local minima: Let f be a real value function an also take a point c from its omain hen a) c is calle a point of local maima if there eists a number h>0 such that f(c) > f(), for all in c h, c h he value f(c) is calle the local maimum value of f b) c is calle a point of local minima if there eists a number h>0 such that f(c) < f(), for all in c h, c h he value f(c) is calle the local minimum value of f Critical points: It is a point c (say) in the omain of a function f() at which either 0 f c or f is not ifferentiable f vanishes ie, 0 First Derivative est: Consier y f be a well efine function on an open interval I Now procee as have been mentione in the following algorithm: SEP- Fin y y SEP- Fin the critical point(s) by putting 0 Suppose c I (where I is the interval) be any critical point an f be continuous at this point c hen we may have following situations: Page - [5]

27 List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering) y changes sign from positive to negative as increases through c, then the function attains a local maimum at c y changes sign from negative to positive as increases through c, then the function attains a local minimum at c y oes not change sign as increases through c, then c is neither a point of local maimum nor a point of local minimum Rather in this case, the point c is calle the point of inflection 04 Secon Derivative est: Consier y f be a well efine function on an open interval I an twice ifferentiable at a point c in the interval hen we observe that: c is a point of local maima if f c 0 an f c 0 he value f c is calle local maimum value of f c is a point of local minima if 0 he value his test fails if f c 0 an 0 para 0 f c f c an 0 f c is calle local minimum value of f f c In such a case, we use first erivative test as iscusse in the 05 Absolute maima an absolute minima: If f is a continuous function on a close interval I then, f has the absolute maimum value an f attains it at least once in I Also f has the absolute minimum value an the function attains it at least once in I ALGORIHM SEP- Fin all the critical points of f in the given interval, ie, fin all points where either 0 f or f is not ifferentiable SEP- ake the en points of the given interval SEP- At all these points (ie, the points foun in SEP an SEP), calculate the values of f SEP4- Ientify the maimum an minimum values of f out of the values calculate in SEP his maimum value will be the absolute maimum value of f an the minimum value will be the absolute minimum value of the function f Absolute maimum value is also calle as global maimum value or greatest value Similarly absolute minimum value is calle as global minimum value or the least value Increasing & Decreasing 0 A function f is sai to be an increasing function in [ a, b ] if as increases, if, [ a, b ] an f f If f 0 lies in, a an b a b then, f is an increasing function in [ a, b ] provie 0 A function f is sai to be a ecreasing function in [ a, b ] if as increases,, [ a, b ] an f f If f 0 lies in, a an b A function a b then, f is a constant function in [, ] f is a ecreasing function in [ a, b ] provie a b if f 0 for each, f a b also increases ie, f f f is continuous at ecreases ie if is continuous at By monotonic function f in interval I, we mean that f is either only increasing in I or only ecreasing in I 0 Fining the intervals of increasing an/ or ecreasing of a function: Page - [6]