SUPPORT STUDY MATERIAL

Transcription

1 SUPPORT STUDY MATERIAL XII Maths Support Material, Key Points, HOTS and VBQ

2 SUBJECT: MATHEMATICS CLASS :XII

3 INDEX Sl. No Topics Page No.. Detail of the concepts 4. Relations & Functions 9. Inverse Trigonometric Functions 5 4. Matrices & Determinants 9 5. Continuity& Differentiability 7 6. Application of derivative 7. Indefinite Integrals 9 8. Applications of Integration Differential Equations 5. Vector Algebra 54. Three Dimensional Geometry 6. Linear Programming 69. Probability 7 4. Answers Bibliography 99

4 Topic wise Analysis of Eamples and Questions NCERT TEXT BOOK Chapters Concepts Number of Questions for revision Total Questions From Solved Eamples Questions From Eercise Relations & Functions Inverse Trigonometric Functions Matrices & Determinants Continuity& Differentiability Application of Derivative Indefinite Integrals Applications of Integration Differential Equations Vector Algebra Three Dimensional Geometry 7 9 Linear Programming 9 Probability TOTAL 6 47

5 Detail of the concepts to be mastered by every child of class XII with eercises and eamples of NCERT Tet Book. SYMBOLS USED * : Important Questions, ** :Very Important Questions, *** : Very-Very Important Questions S.No Topic Concepts Degree of importance Refrences NCERT Tet Book XII Ed. 7 Relations & Functions Inverse Trigonometric Functions Matrices & Determinants 4 Continuity& Differentiability (i).domain, Co-domain & * (Previous Knowledge) Range of a relation (ii).types of relations *** E. Q.No- 5,9, (iii).one-one, onto & inverse ofa *** E. Q.No- 7,9 function (iv).composition of function * E. QNo- 7,9, (v).binary Operations *** Eample 45 E.4 QNo- 5, (i).principal value branch Table ** E. QNo-, 4 (ii). Properties of Inverse *** E. QNo- 7,, 5 Trigonometric Functions Misc E Q.No.9,,, (i) Order, Addition, *** E. Q.No 4,6 Multiplication and transpose of E. Q.No 7,9,,7,8 matrices E. Q.No (ii) Cofactors &Adjoint of a matri (iii)inverse of a matri & applications (iv)to find difference between A, adj A, ka, A.adjA 4 ** E 4.4 Q.No 5 E 4.5 Q.No,,7,8 *** E 4.6 Q.No 5,6 Eample 9,,, MiscE 4,Q.No4,5,8,,5 * E 4. Q.No,4,7,8 (v) Properties of Determinants ** E 4. Q.No,, Eample 6,8 (i).limit of a function * (ii).continuity *** E 5. Q.No-, 6, (iii).differentiation * E 5. Q.No- 6 E 5. Q.No- 4,7, (iv).logrithmic Differentiation *** E 5.5 Q.No- 6,9,,5 (v) Parametric Differentiation *** E 5.6 Q.No- 7,8,, (vi). Second order derivatives *** E 5.7 Q.No- 4,6,7

6 5 Application of Derivative. 6 Indefinite Integrals (vii). M. V.Th ** E 5.8 Q.No-,4 (i).rate of change * Eample 5E 6. Q.No- 9, (ii).increasing & decreasing *** E 6.,Q.No- 6 Eample, functions (iii).tangents & normal ** E 6.,Q.No- 5,8,,5, (iv).approimations * E 6.4,Q.No-, (v) Maima & Minima *** E 6.5, Q.No- 8,,,5 Eample 5,6,7 (i) Integration by substitution * Ep 5&6 Page, (ii) Application of trigonometric ** E 7 Page 6, Eercise function in integrals 7.4Q&Q4 (iii) Integration of some particular function,, a a, a a b,, a b c (p q), a b c (p q) c *** Edition Ep 8, 9, Page,Eercise 7.4 Q,4,8,9,& a b c (iv) Integration using Partial *** EditionEp & Page 8 Fraction Ep 9,Ep 4 & 5 Page (v) Integration by Parts ** Ep 8,9& Page 5 Definite Integrals (vi)some Special Integrals *** Ep &4 Page 9 a, a (vii) Miscellaneous Questions *** Solved E.4 (i) Definite integrals as a limit ** Ep 5 &6 Page, 4 of sum Q, Q5 & Q6 Eercise 7.8 () Properties of definite *** Ep Page 4*,Ep Integrals *,4&5 page 44 Ep 6*Ep 46 Ep 44 page5 Eercise 7. Q7 & (i) Integration of modulus ** Ep Page 4,Ep 4 Page 5

7 7 Applications of Integration 8. Differential Equations 9. Vector Algebra function 5 Q5& Q6 Eercise 7. (i)area under Simple Curves * E.8. Q.,,5 (ii) Area of the region enclosed *** E. 8. Q, Misc.E. Q 7 between Parabola and line (iii) Area of the region enclosed *** Eample 8, page 69Misc.E. between Ellipse and line 8 (iv) Area of the region enclosed *** E. 8. Q 6 between Circle and line (v) Area of the region enclosed *** E 8. Q, Misc.E.Q 5 between Circle and parabola (vi) Area of the region enclosed between Two Circles *** Eample, page7e 8. Q (vii) Area of the region *** Eample 6, page6 enclosed between Two parabolas (viii) Area of triangle when *** Eample 9, page7e 8. Q4 vertices are given (i) Area of triangle when sides *** E 8. Q5,Misc.E. Q 4 are given () Miscellaneous Questions *** Eample, page74misc.e.q 4, (i) Order and degree of a *** Q.,5,6 pg 8 differential equation.general and particular ** E., pg84 solutions of a differential equation.formation of differential * Q. 7,8, pg 9 equation whose general solution is given 4.Solution of differential * Q.4,6, pg 96 equation by the method of separation of variables 5.Homogeneous differential ** Q.,6, pg 46 equation of first order and first degree Solution of differential equation *** Q.4,5,,4 pg 4,44 of the type dy/ +py=q where p and q are functions of And solution of differential equation of the type /dy+p=q where p and q are functions of y (i)vector and scalars * Q pg 48 (ii)direction ratio and direction * Q, pg 44 6

8 Three Dimensional Geometry Linear Programmin g cosines (iii)unit vector * * E 6,8 Pg 46 (iv)position vector of a point and collinear vectors * * Q 5 Pg 44 Q Pg44 Q 6 Pg448 (v)dot product of two vectors ** Q6, Pg445 (vi)projection of a vector * * * E 6 Pg 445 (vii)cross product of two * * Q Pg458 vectors (viii)area of a triangle * Q 9 Pg 454 (i)area of a parallelogram * Q Pg 455 (i)direction Ratios and * E No Pg -466 Direction Cosines E No 5 Pg 467 E No 4 Pg - 48 (ii)cartesian and Vector equation of a line in space & conversion of one into another form ** E No 8 Pg -47 Q N. 6, 7, - Pg 477 QN 9 Pg 478 (iii) Co-planer and skew lines * E No 9 Pg -496 (iv)shortest distance between two lines *** E No Pg -476 Q N. 6, 7 - Pg 478 (v)cartesian and Vector equation of a plane in space & conversion of one into another form ** E No 7 Pg -48 E No 8 Pg 484 E No 9 Pg 485 E No 7 Pg 495 (vi)angle Between (i) Two lines (ii) Two planes (iii) Line & plane (vii)distance of a point from a plane (viii)distance measures parallel to plane and parallel to line (i)equation of a plane through the intersection of two planes ()Foot of perpendicular and image with respect to a line and plane (i) LPP and its Mathematical Formulation (ii) Graphical method of solving LPP (bounded and unbounded solutions) * * ** Q N. 9, - Pg 499 E No 9 Pg -47 Q N. - Pg 478 E No 6 Pg 494 Q N. - Pg 494 E No 5 Pg - 49 ** Q No 8 Pg -499 Q No 4 Pg 494 ** *** Q No Pg -49 ** E. N 6 Pg 48 ** Articles. and.. ** Article.. Solved Eamples to 5 Q. Nos 5 to 8 E.. 7

9 (iii) Types of problems (a) Diet Problem *** Q. Nos, and 9 E.. Solved Eample 9 Q. Nos and Misc. E. (b) Manufacturing Problem *** Solved Eample 8 Q. Nos,4,5,6,7 of E.. Solved Eample Q. Nos 4 & Misc. E. (c) Allocation Problem ** Solved Eample 7 Q. No E.., Q. No 5 & 8 Misc. E. (d) Transportation Problem * Solved Eample Q. Nos 6 & 7 Misc. E. (e) Miscellaneous Problems ** Q. No 8 E.. Probability (i) Conditional Probability *** Article. and.. Solved Eamples to 6 Q. Nos and 5 to 5 E.. (ii)multiplication theorem on probability ** Article. SolvedEamples 8 & 9 Q. Nos,, 4 & 6 E.. (iii) Independent Events *** Article.4 Solved Eamples to 4 Q. Nos, 6, 7, 8 and E.. (iv) Baye s theorem, partition of sample space and Theorem of total probability (v) Random variables & probability distribution Mean & variance of random variables (vi) Bernoulli,s trials and Binomial Distribution *** Articles.5,.5.,.5. Solved Eamples 5 to, & 7,Q. Nos to E.. Q. Nos & 6 Misc. E. *** Articles.6,.6.,.6. &.6. Solved Eamples 4 to 9 Q. Nos & 4 to 5 E..4 *** Articles.7,.7. &.7. Solved Eamples & Q. Nos to E..5 8

10 TOPIC RELATIONS & FUNCTIONS SCHEMATIC DIAGRAM Topic Concepts Degree of importance References NCERT Tet Book XII Ed. 7 Relations & (i).domain, Co domain & * (Previous Knowledge) Functions Range of a relation (ii).types of relations *** E. Q.No- 5,9, (iii).one-one, onto & inverse *** E. Q.No- 7,9 of a function (iv).composition of function * E. QNo- 7,9, (v).binary Operations *** Eample 45 E.4 QNo- 5, SOME IMPORTANT RESULTS/CONCEPTS ** A relation R in a set A is called (i) refleive, if (a, a) R, for every a A, (ii) symmetric, if (a, a ) R implies that (a, a ) R, for all a, a A. (iii)transitive, if (a, a ) R and (a, a ) R implies that (a, a ) R, for all a, a, a A. ** Equivalence Relation : R is equivalence if it is refleive, symmetric and transitive. ** Function :A relation f : A B is said to be a function if every element of A is correlatedto unique element in B. * A is domain * B is codomain * For any element A, function f correlates it to an element in B, which is denoted by f()and is called image of under f. Again if y= f(), then is called as pre-image of y. * Range = {f() A }. Range Codomain * The largest possible domain of a function is called domain of definition. **Composite function : Let two functions be defined as f : A B and g : B C. Then we can define a function : A C by setting () = g{f()} where A, f () B, g{f()} C. This function : A C is called the composite function of f and g in that order and we write. = gof. A B C () f g f() g{f()} 9

11 ** Different type of functions : Let f : A B be a function. * f is one to one (injective) mapping, if any two different elements in A is always correlated to different elements in B, i.e. f( ) f( )or, f( ) = f( ) = * f is many one mapping, if at least two elements in A such that their images are same. * f is onto mapping (subjective), if each element in B is having at least one preimage. *f is into mapping if range codomain. * f is bijective mapping if it is both one to one and onto. ** Binary operation : A binary operation * on a set A is a function * : A A A. We denote *(a, b) by a *b. * A binary operation * on A is a rule that associates with every ordered pair (a, b) of A A a unique element a *b. * An operation * on a is said to be commutative iff a * b = b * a a, b A. * An operation * on a is said to be associative iff (a * b) * c = a * (b * c) a, b, c A. * Given a binary operation * : A A A, an element e A, if it eists, is called identity for the operation *, if a *e = a = e *a, a A. * Given a binary operation * : A A A with the identity element e in A, an element a A is said to be invertible with respect to the operation*, if there eists an element b in A such that a b = e = b a and b is called the inverse of a and is denoted by a. ASSIGNMENTS (i) Domain, Co domain & Range of a relation LEVEL I. If A = {,,,4,5}, write the relation a R b such that a + b = 8, a,b A. Write the domain, range & co-domain.. Define a relation R on the set N of natural numbers by R={(, y) : y = +7, is a natural number lesst han 4 ;, y N}. Write down the domain and the range.. Types of relations LEVEL II. Let R be the relation in the set N given by R = {(a, b) a = b, b > 6} Whether the relation is refleive or not?justify your answer.. Show that the relation R in the set N given by R = {(a, b) a is divisible by b, a, b N} is refleive and transitive but not symmetric.. Let R be the relation in the set N given by R = {(a,b) a > b} Show that the relation is neither refleive nor symmetric but transitive. 4. Let R be the relation on R defined as (a, b) R iff + ab> a,br. (a) Show that R is symmetric. (b) Show that R is refleive. (c) Show that R is not transitive. 5. Check whether the relation R is refleive, symmetric and transitive. R = { (, y) y = } on A ={,,., 4}.

12 LEVEL III. Show that the relation R on A,A = { Z, }, R = {(a,b): a - b is multiple of.} is an equivalence relation..let N be the set of all natural numbers & R be the relation on N N defined by { (a, b) R (c, d) iff a + d = b + c}. Show that R is an equivalence relation.. Show that the relation R in the set A of all polygons as: R ={(P,P ), P & P have the same number of sides} is an equivalence relation. What is the set of all elements in A related to the right triangle T with sides,4 & 5? 4. Show that the relation R on A,A = { Z, }, R = {(a,b): a - b is multiple of.} is an equivalence relation. 5. Let N be the set of all natural numbers & R be the relation on N N defined by { (a, b) R (c,d) iff a + d = b + c}. Show that R is an equivalence relation. [CBSE ] 6. Let A = Set of all triangles in a plane and R is defined by R={(T,T ) : T,T A & T ~T } Show that the R is equivalence relation. Consider the right angled s, T with size,4,5; T with size 5,,; T with side 6,8,; Which of the pairs are related? (iii)one-one, onto & inverse of a function LEVEL I. If f() =, then find f(/). Show that the function f: RR defined by f()= is neither one-one nor onto. Show that the function f: NN given by f()= is one-one but not onto., if 4 Show that the signum function f: RR given by: f (), if, if is neither one-one nor onto. 5 Let A = {-,,} and B = {,}. State whether the function f : A B defined by f() = isbijective. 6. Let f() =, -,then find f - () LEVEL II. Let A = {,,}, B = {4,5,6,7} and let f = {(,4),(,5), (,6)} be a function from A to B. State whether f is one-one or not. [CBSE] 7. If f : RR defined as f() = is an invertible function. Find f - (). 4. Write the number of all one-one functions on the set A={a, b, c} to itself. 4. Show that function f :RR defined by f()= 7 for all R is bijective If f: RR is defined by f()=. Find f -.

13 LEVEL III. Show that the function f: RR defined by f() =. R is one- one & onto function. Also find the f -.. Consider a function f :R + [-5, ) defined f() = Show that f is invertible & f - y 6 (y) =, where R + = (, ).. Consider a function f: RR given by f() = 4 +. Show that f is invertible & f - : RR with f - (y)=. 4. Show that f: RR defined by f()= +4 is one-one, onto. Show that f - ()=( 4) /. 5. Let A R {} and B R {}. Consider the function f : A Bdefined by f (). Show that f is one one onto and hence find 6. Show that f : N N defined by (iv) Composition of functions f. [CBSE], if is odd f () is both one one onto., if is even [CBSE] LEVEL I. If f() = e and g() = log, >, find (a) (f + g)() (b) (f.g)() (c) f o g ( ) (d) g o f ( ).. If f() =, then show that (a) f = f() (b) f = LEVEL II f (). Let f, g : RR be defined by f()= & g() = [] where [] denotes the greatest integer function. Find f o g ( 5/ ) & g o f (- ).. Let f() =. Then find f(f()) 4. If y = f() =, then find (fof)() i.e. f(y) 5 4. Let f : R R be defined as f() = +7.Find the function g : R Rsuch that g f ()= f g() = I R [CBSE] 5. If f : R R be defined as f() =, then find f f(). [CBSE] 6. Let f :RR& g : RR be defined as f() =, g() =. Find fog().

14 (v)binary Operations LEVEL I. Let * be the binary operation on N given by a*b = LCM of a &b. Find *5.. Let *be the binary on N given by a*b =HCF of {a,b}, a,bn. Find *6. ab. Let * be a binary operation on the set Q of rational numbers defined as a * b =. 5 Write the identity of *, if any. 4. If a binary operation * on the set of integer Z, is defined by a * b = a + b Then find the value of * 4. LEVEL. Let A= N N & * be the binary operation on A defined by (a,b) * (c,d) = (a+c, b+d ) Show that * is (a) Commutative (b) Associative (c) Find identity for * on A, if any.. Let A = Q Q. Let * be a binary operation on A defined by (a,b)*(c,d)= (ac, ad+b). Find: (i) the identity element of A (ii) the invertible element of A.. Eamine which of the following is a binary operation a b a b (i) a * b = ; a, b N (ii) a*b = a, b Q For binary operation check commutative & associative law. LEVEL.Let A= N N & * be a binary operation on A defined by (a, b) (c, d) = (ac, bd) (a, b),(c, d) N N (i) Find (,) * (4,) (ii) Find [(,)*(4,)]*(,5) and (,)*[(4,)* (,5)] & show they are equal (iii) Show that * is commutative & associative on A. a b, if a b 6. Define a binary operation * on the set {,,,,4,5} as a * b = a b 6, a b 6 Show that zero in the identity for this operation & each element of the set is invertible with 6 a being the inverse of a. [CBSE]. Consider the binary operations :R R Rand o : R R R defined as a b = a b and a o b = a, a, b R. Show that is commutative but not associative, o is associative but not commutative. [CBSE] Questions for self evaluation. Show that the relation R in the set A = {,,, 4, 5} given by R = {(a, b) : a b is even}, is an equivalence relation. Show that all the elements of {,, 5} are related to each other and all the elements of {, 4} are related to each other. But no element of {,, 5} is related to any element of {, 4}.. Show that each of the relation R in the set A = { Z : }, given by R = {(a, b) : a b is a multiple of 4} is an equivalence relation. Find the set of all elements related to.

15 . Show that the relation R defined in the set A of all triangles as R = {(T, T ) : T is similar to T }, is equivalence relation. Consider three right angle triangles T with sides, 4, 5, T with sides 5,, and T with sides 6, 8,. Which triangles among T, T and T are related? 4. If R and R are equivalence relations in a set A, show that R R is also an equivalence relation. 5. Let A = R {} and B = R {}. Consider the function f : A B defined by f () =. Is f one-one and onto? Justify your answer. 6. Consider f :R+ [ 5, ) given by f () = Show that f is invertible and findf. 7. On R {} a binary operation * is defined as a * b = a + b ab. Prove that * is commutative and associative. Find the identity element for *.Also prove that every element of R {} is invertible. 8. If A = Q Q and * be a binary operation defined by (a, b) * (c, d) = (ac, b + ad), for (a, b), (c, d) A.Then with respect to * on A (i) eamine whether * is commutative & associative (i) find the identity element in A, (ii) find the invertible elements of A. 4

16 TOPIC INVERSE TRIGONOMETRIC FUNCTIONS SCHEMATIC DIAGRAM Topic Concepts Degree of importance References NCERT Tet Book XI Ed. 7 Inverse (i).principal value branch ** E. QNo-, 4 Trigonometric Functions Table (ii). Properties of Inverse Trigonometric Functions *** E. Q No- 7,, 5 Misc E Q.No. 9,,, * Domain & Range of i. ii. iii. iv. v. vi. SOME IMPORTANT RESULTS/CONCEPTS the Inverse Trigonometric Function : Functions sin cos sec tan cot cosec : : : : : : Domain Range, /, /,, R, /, / R,, / R /, / R, Principal value Branch * Properties of Inverse Trigonomet ric i sin sin & sin sin ii.cos cos & cos cos iii.tan tan & tan tan iv.cot cot & cot cot v. sec sec & sec sec vi.cos ec cos ec & cos ec cosec iii. tan i i. i ii. tan iii. cos ec iii. sin sin cos ec cot & cot sin iv cos tan v sec cos ec vi cot sin cos cos ec sec & sin cos ec tan Function ii. cos ii. tan sec cot & cos sec cot sec cos 5

17 5. tan 6. tan tan tan tan tan tan tan y tan y tan y tan cos y y y y y y if if if sin y y y ASSIGNMENTS (i). Principal value branch Table LEVEL I Write the principal value of the following :.cos.tan Write the principal value of the following : LEVEL II.sin 4.cos π π 4π. cos cos sin sin [CBSE ]. sin sin 5 7. cos cos 6 (ii). Properties of Inverse Trigonometric Functions. Evaluate cot[tan a cot a].prove sin sin 4 π. Find ifsec cos ec LEVEL I LEVEL II. Write the following in simplest form : tan, 6

18 8 77. Prove that sin sin tan Prove that tan tan tan tan Prove that tan tan tan Prove thatsin sin cos π 4. [CBSE ] [CBSE ] LEVEL III sin sin. Prove that cot,, sin sin 4. Prove that tan cos 4. Solve tan tan π / 4 4. Solve tan tan tan π 5.Solve tan tan 4 8 [CBSE ] cos 6. Prove that tan,, [CBSE ] sin 4. Prove that sin 5 cos Questions for self evaluation tan 5. Prove that tan cos,, Prove that sin cos tan Prove that tan tan tan 5. Prove that y tan tan y y 4 6. Write in the simplest form cos tan 7

19 7. Solve tan tan 4 8. Solve tan tan / 4 8

20 TOPIC MATRICES & DETERMINANTS SCHEMATIC DIAGRAM Topic Concepts Degree of importance Matrices & Determinants (i) Order, Addition, Multiplication and transpose of matrices (ii) Cofactors &Adjoint of a matri (iii)inverse of a matri & applications (iv)to find difference between A, adj A, ka, A.adjA (v) Properties of Determinants References NCERT Tet Book XI Ed. 7 *** E. Q.No 4,6 E. Q.No 7,9,,7,8 E. Q.No ** E 4.4 Q.No 5 E 4.5 Q.No,,7,8 *** E 4.6 Q.No 5,6 Eample 9,,, MiscE 4 Q.No 4,5,8,,5 * E 4. Q.No,4,7,8 ** E 4. Q.No,, Eample 6,8 SOME IMPORTANT RESULTS/CONCEPTS A matri is a rectangular array of m n numbers arranged in m rows and n columns. a a a n a a a n A OR A = [a ] ij m n, where i =,,., m ; j =,,.,n. am am a mnmn * Row Matri : A matri which has one row is called row matri. A [a ] ij n * Column Matri : A matri which has one column is called column matri. A [a ] ij m. * Square Matri: A matri in which number of rows are equal to number of columns, is called a square matri A [a ] ij mm * Diagonal Matri : A square matri is called a Diagonal Matri if all the elements, ecept the diagonal elements are zero. A [a ] ij n n, where a ij =, i j. aij, i = j. * Scalar Matri: A square matri is called scalar matri it all the elements, ecept diagonal elements are zero and diagonal elements are same non-zero quantity. A [a ij ] n n, where a ij =, i j. aij, i = j. * Identity or Unit Matri : A square matri in which all the non diagonal elements are zero and diagonal elements are unity is called identity or unit matri. 9

21 * Null Matrices : A matrices in which all element are zero. * Equal Matrices : Two matrices are said to be equal if they have same order and all their corresponding elements are equal. * Transpose of matri : If A is the given matri, then the matri obtained by interchanging the rows and columns is called the transpose of a matri.\ * Properties of Transpose : If A & B are matrices such that their sum & product are defined, then T T (i). A A T T (ii). T T T T A B A B (iii). KA K.A where K is a scalar. T T T T T T (iv). AB B A (v). ABC C B A. * Symmetric Matri : A square matri is said to be symmetric if A = A T i.e. If A [a ] ij mm, then a ij a ji for all i, j. Also elements of the symmetric matri are symmetric about the main diagonal * Skew symmetric Matri : A square matri is said to be skew symmetric if A T = -A. If A [a ] ij mm, then aij a ji for all i, j. *Singular matri:a square matri A of order n is said to be singular, if A =. * Non -Singular matri : A square matri A of order n is said to be non-singular, if A. *Product of matrices: (i) If A & B are two matrices, then product AB is defined, if Number of column of A = number of rows of B. (ii) (iii) i.e. A [a ] ij mn, B [b j k] n p then AB = AB [C ] ik mp. Product of matrices is not commutative. i.e. AB BA. Product of matrices is associative. i.e A(BC) = (AB)C (iv) Product of matrices is distributive over addition. *Adjoint of matri : If A [a ] be a n-square matri then transpose of a matri [A ij ], where ij A ij is the cofactor of A ij element of matri A, is called the adjoint of A. T Adjoint of A = Adj. A = [A ]. ij A(Adj.A) = (Adj. A)A = A I. *Inverse of a matri :Inverse of a square matri A eists, if A is non-singular or square matri A is said to be invertible and A - = A Adj.A *System of Linear Equations : a + b y + c z = d. a + b y + c z = d. a + b y + c z = d.

22 a b c d a b c y d A X = B X = A - B ; { A }. a b c z d *Criteria of Consistency. (i) If A, then the system of equations is said to be consistent & has a unique solution. (ii) If A = and (adj. A)B =, then the system of equations is consistent and has infinitely many solutions. (iii) If A = and (adj. A)B, then the system of equations is inconsistent and has no solution. * Determinant : To every square matri we can assign a number called determinant If A = [a ], det. A = A = a. a a If A =, A = a a a a. a a * Properties : (i) The determinant of the square matri A is unchanged when its rows and columns are interchanged. (ii) The determinant of a square matri obtained by interchanging two rows(or two columns) is negative of given determinant. (iii) If two rows or two columns of a determinant are identical, value of the determinant is zero. (iv) If all the elements of a row or column of a square matri A are multiplied by a non-zero number k, then determinant of the new matri is k times the determinant of A. If elements of any one column(or row) are epressed as sum of two elements each, then determinant can be written as sum of two determinants. Any two or more rows(or column) can be added or subtracted proportionally. If A & B are square matrices of same order, then AB = A B ASSIGNMENTS (i). Order, Addition, Multiplication and transpose of matrices: LEVEL I. If a matri has 5 elements, what are the possible orders it can have? [CBSE ]. Construct a matri whose elements are given by a ij = i j. If A =, B =, then find A B. 4. If A = and B = [ ], write the order of AB and BA. LEVEL II. For the following matrices A and B, verify (AB) T = B T A T, where A = [ ], B =, -. Give eample of matrices A & B such that AB = O, but BA O, where O is a zero matri and

23 A, B are both non zero matrices.. If B is skew symmetric matri, write whether the matri (ABA T ) is Symmetric or skew symmetric. 4. If A = and I =, find a and b so that A + ai = ba LEVEL III. If A = [ ], then find the value of A A + I. Epress the matri A as the sum of a symmetric and a skew symmetric matri, where: A = [ ]. If A =, prove that A n = [ ( ) ], n N (ii) Cofactors &Adjoint of a matri LEVEL I. Find the co-factor of a in A =. Find the adjoint of the matri A = LEVEL II Verify A(adjA) = (adja) A = I if. A =. A = [ ] (iii)inverse of a Matri & Applications LEVEL I. If A =, write A - in terms of A CBSE. If A is square matri satisfying A = I, then what is the inverse of A?. For what value of k, the matri A = is not invertible? LEVEL II. If A =, show that A 5A 4I =. Hence find A -. If A, B, C are three non zero square matrices of same order, find the condition on A such that AB = AC B = C.

24 . Find the number of all possible matrices A of order with each entry or and for which A [ ] = [ ] has eactly two distinct solutions. LEVEL III If A = [ ], find A - and hence solve the following system of equations: y + 5z =, + y 4z = - 5, + y z = -. Using matrices, solve the following system of equations: a. + y - z = y + z = - y 4z = [CBSE ] b. 4 + y + z = 6 + y + z = y + z = 7 [CBSE ]. Find the product AB, where A = [ ], B = [ ] and use it to solve the equations y =, + y + 4z = 7, y + z = 7 4. Using matrices, solve the following system of equations: - + = = + + = 5. Using elementary transformations, find the inverse of the matri [ ] (iv)to Find The Difference Between LEVEL I. Evaluate [CBSE ]. What is the value of, where I is identity matri of order?. If A is non singular matri of order and =, then find 4. For what valve of a, is a singular matri? LEVEL II. If A is a square matri of order such that = 64, find. If A is a non singular matri of order and = 7, then find

25 LEVEL III. If A = and = 5, then find a.. A square matri A, of order, has = 5, find (v).properties of Determinants LEVEL I. Find positive valve of if =. Evaluate LEVEL II. Using properties of determinants, prove the following : b c b c a a c a c a b a b 4abc. ab b b a ab b a a b a b a b [CBSE ]. = ( + pyz)( - y)(y - z) (z - ) 4. a b c (a b)(b c)(c a)(a b c) [CBSE ] a b c LEVEL III. Using properties of determinants, solve the following for : a. = [CBSE ] b. = [CBSE ] c. = [CBSE ]. If a, b, c, are positive and unequal, show that the following determinant is negative: = 4

26 5. c b a c cb ca bc b ab ac ab a 4. abc c b a b a a c c b a c c b b a c b a [CBSE ] 5. b a ab b a a c ca a c c b bc c b 6. ca) bc (ab ab ab b ab a ac c ac ac a bc c bc b bc 7. (b+c) ab ca ab (a+c) bc ac bc (a+b) = abc( a + b + c) 8. If p, q, r are not in G.P and r p p that show, r q q p q r q r p q p q. 9. If a, b, c are real numbers, and a c c b b a c b b a a c b a a c c b Show that either a + b +c = or a = b = c. QUESTIONS FOR SELF EVALUTION. Using properties of determinants, prove that : y q p b a z p r a c z y r q c b z r c y q b p a

27 6. Using properties of determinants, prove that : b a b a a b a b a ab b ab b a. Using properties of determinants, prove that : c b a c cb ca bc b ab ac ab a 4..Epress A = as the sum of a symmetric and a skew-symmetric matri. 5. Let A = 4, prove by mathematical induction that : n n 4n n A n. 6. If A = 5 7, find and y such that A + I = ya. Hence find A. 7. Let A= tan tan I and. Prove that cos sin sin cos A) (I A I. 8. Solve the following system of equations : + y + z = 7, + z =, y =. 9. Find the product AB, where A = B and and use it to solve the equations y + z = 4, y z = 9, + y + z =.. Find the matri P satisfying the matri equation 5 P.

28 TOPIC 4 CONTINUITY AND DIFFRENTIABILITY SCHEMATIC DIAGRAM Topic Concepts Degree of importance Refrences NCERT Tet Book XII Ed. 7 Continuity& Differentiability.Limit of a function.continuity *** E 5. Q.No-, 6,.Differentiation * E 5. Q.No- 6 E 5. Q.No- 4,7, 4.Logrithmic Differentiation *** E 5.5 QNo- 6,9,,5 5 Parametric Differentiation *** E 5.6 QNo- 7,8,, 6. Second order derivatives *** E 5.7 QNo- 4,6,7 7. Mean Value Theorem ** E 5.8 QNo-,4 SOME IMPORTANT RESULTS/CONCEPTS * A function f is said to be continuous at = a if Left hand limit = Right hand limit = value of the function at = a i.e. lim f () lim f () f (a) a a i.e. lim f (a h) lim f (a h) f (a). h h * A function is said to be differentiable at = a if Lf (a) Rf (a) i.e f (a h) f (a) f (a h) f (a) lim lim h h h h (i) d ( n ) = n n -. (ii) d () = (iii) d (c) =, c R (iv) d (a ) = a log a, a >, a. (v) d (e ) = e. (vi) d (loga ) =, log a (vii) d (log ) =, > a >, a, 7 (iii) d (cot ) = cosec, R. (iv) d (sec ) = sec tan, R. (v) d (cosec ) = cosec cot, R. (vi) d (sin - ) = -. d (vii) (cos - ) = - -. d (viii) (tan - ) =, R d (i) (cot - ) =, R. d () (sec - ) =,. d (i) (cosec - ) =. d (ii) ( ) =, d du (iii) (ku) = k d (iv) u v du dv

29 (viii) d (loga ) = log a (i) d (log ) =, () d (sin ) = cos, R., a >, a, d dv du (v) (u.v) = u v du dv v u d u (vi) v v (i) d (cos ) = sin, R. (ii) d (tan ) = sec, R..Continuity LEVEL-I.Eamine the continuity of the function f()= + 5 at =-.. Eamine the continuity of the function f()=, R.. Show that f()=4 is a continuous for all R. LEVEL-II. Give an eample of a function which is continuous at =,but not differentiable at =. k, if. For what value of k,the function is continuous at =., if.find the relationship between a and b so that the function f defined by: a + f()= b + if is continuous at =. if > sin,when 4. If f()=. Find whether f() is continuous at =.,when [CBSE ] LEVEL-III cos 4,.For what value of k, the function f()= 8 is continuous at =? k, sin. If function f()=, for is continuous at =, then Find f(). sin 8

30 sin, if cos.let f() = If f() be a continuous function at =, find a and b. a if b( sin ) if ( ) 4.For what value of k,is the function f() =.Differentiation sin cos k LEVEL-I, when, when. Discuss the differentiability of the function f()=(-) / at =..Differentiate y=tan -. If y= ( )( 4) 4 5 dy. Find, y = cos(log ). dy. Find of y= tan. dy, Find..If y=e a sin b, then prove that 4.Find d y, if y= at t, = d y at t. LEVEL-II a dy +(a +b )y=. LEVEL-III continuous at =? dy.find, if y = tan dy.find y = sin sin cot, <<. sin sin a bcos. If y sin dy b a b a cos, show that =. b a cos 9

31 4.Prove that d 4 4.Logrithmic Differentiation.Differentiate y=log 7 (log ).. Differentiate, sin(log ),with respect to..differentiate y= ( ).If log tan 4 LEVEL-I LEVEL-II. If y. =log[ -],show that ( dy +) +y+=. dy. Find, y = cos(log ). dy. Find if (cos) y = (cosy) [CBSE ] LEVEL-III p.y q. y = log ( y) cos pq, prove that dy, find dy. If Show that = * ( )+ [CBSE ] y. 4. Find dy cot when y [CBSE ] 5 Parametric Differentiation LEVEL-II.If y = tan, prove that..if = a cos log tan and y a sin find at. 4. If = tan. /, show that ( ) ( ) [CBSE ] 6. Second order derivatives LEVEL-II

32 d y dy. If y = a cos (log ) + b sin(log ), prove that y..if y=(sin - ), prove that (- ) d y - dy =.If ( ) + ( ) = c for some c>.prove that 7. Mean Value Theorem dy d y LEVEL-II / is a constant, independent.it is given that for the function f()= -6 +p+q on[,], Rolle s theorem holds with c=+. Find the values p and q.. Verify Rolle s theorem for the function f() = sin, in [, ].Find c, if verified.veifylagrange s mean Value Theorem f() = in the interval [,4] Questions for self evaluation.for what value of k is the following function continuous at =? f () ; k ; ; a b, if.if f() = if, continuous at =, find the values of a and b.[cbse Comptt.] 5a - b, if. Discuss the continuity of f() = at = & =. 4. If f(), defined by the following is continuous at =, find the values of a, b, c sin(a ) sin f () c b / b,,, 5.If = a cos log tan and y a sin find dy at If y = log cos dy, find.

33 7. If y + y dy = tan + y, find. 9.If 8. If y = y log = a( y), prove that dy, find. dy y =. dy. Find if (cos) y = (cosy) d y dy.if y = a cos (log ) + b sin(log ), prove that y..if p.y q ( y) pq, prove that dy y.

34 TOPIC 5 APPLICATIONS OF DERIVATIVES SCHEMATIC DIAGRAM Topic Concepts Degree of importance Application of Derivative. Refrences NCERT Tet Book XII Ed. 7.Rate of change * Eample 5 E 6. Q.No- 9,.Increasing & decreasing *** E 6. Q.No- 6 Eample, functions.tangents & normals ** E 6. Q.No- 5,8,,5, 4.Approimations * E 6.4 QNo-, 5 Maima & Minima *** E 6.5Q.No- 8,,,5 Eample 5,6,7, SOME IMPORTANT RESULTS/CONCEPTS dy ** Whenever one quantity y varies with another quantity, satisfying some rule y = f (), then (or f ()) dy represents the rate of change of y with respect to and (or f ( )) represents the rate of change o of y with respect to at =. ** Let I be an open interval contained in the domain of a real valued function f. Then f is said to be (i) increasing on I if < in I f ( ) f ( ) for all, I. (ii) strictly increasing on I if < in I f ( ) < f ( ) for all, I. (iii) decreasing on I if < in I f ( ) f ( ) for all, I. (iv) strictly decreasing on I if < in I f ( ) > f ( ) for all, I. ** (i) f is strictly increasing in (a, b) if f () > for each (a, b) (ii) f is strictly decreasing in (a, b) if f () < for each (a, b) (iii) A function will be increasing (decreasing) in R if it is so in every interval of R. ** Slope of the tangent to the curve y = f () at the point (, y ) is given by dy ( ) (,y). ** If slope of the tangent line is zero, then tan θ = and so θ = which means the tangent line is parallel to the f ** The equation of the tangent at (, y ) to the curve y = f () is given by y y = ( ) ( ). ** Slope of the normal to the curve y = f () at (, y ) is. f ( ) ** The equation of the normal at (, y ) to the curve y = f () is given by y y = f ( ). f ( )

35 -ais. In this case, the equation of the tangent at the point (, y) is given by y = y. ** If θ, then tan θ, which means the tangent line is perpendicular to the -ais, i.e., parallel to the y-ais. In this case, the equation of the tangent at (, y ) is given by =. dy ** Increment y in the function y = f() corresponding to increment in is given by y =. y ** Relative error in y =. y y ** Percentage error in y =. y ** Let f be a function defined on an interval I. Then (a) f is said to have a maimum value in I, if there eists a point c in I such that f (c) f (), for all I. The number f (c) is called the maimum value of f in I and the point c is called a point of maimum value of f in I. (b) f is said to have a minimum value in I, if there eists a point c in I such that f (c) f (), for all I. The number f (c), in this case, is called the minimum value of f in I and the point c, in this case, is called a point of minimum value of f in I. (c) f is said 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. The number f (c), in this case, is called an etreme value of f in I and the point c is called an etreme point. * * Absolute maima and minima Let f be a function defined on the interval I and ci. Then f(c) for all I. (b) f(c) is absolute maimum if f() f(c) for all I. (c) c I is called the critical point off if f (c) = (d) Absolute maimum or minimum value of a continuous function f on [a, b] occurs at a or b or at critical points off (i.e. at the points where f is zero) If c,c,, c n are the critical points lying in [a, b], then absolute maimum value of f = ma{f(a), f(c ), f(c ),, f(c n ), f(b)} and absolute minimum value of f = min{f(a), f(c ), f(c ),, f(c n ), f(b)}. ** Local maima and minima (a)a function f is said to have a local maima or simply a maimum vajue at a if f(a ± h) f(a) for sufficiently small h (b)a function f is said to have a local minima or simply a minimum value at = a if f(a ± h) f(a). ** First derivative test : A function f has a maimum at a point = a if (i) f (a) =, and (ii) f () changes sign from + ve to ve in the neighbourhood of a (points taken from left to right). However, f has a minimum at = a, if (i) f (a) =, and (ii) f () changes sign from ve to +ve in the neighbourhood of a. If f (a) = and f () does not change sign, then f() has neither maimum nor minimum and the point a is called point of inflation. The points where f () = are called stationary or critical points. The stationary points at which the function attains either maimum or minimum values are called etreme points. ** Second derivative test 4

36 (i) a function has a maima at a if f () and f (a) < (ii) a function has a minima at = a if f () = and f (a) >..Rate of change ASSIGNMENTS LEVEL -I. A balloon, which always remains spherical, has a variable diameter. Find the rate of change of its volume with respect to..the side of a square sheet is increasing at the rate of 4 cm per minute. At what rate is the area increasing when the side is 8 cm long?. The radius of a circle is increasing at the rate of.7 cm/sec. what is the rate of increase of its circumference? LEVEL II. Find the point on the curve y = 8 for which the abscissa and ordinate change at the same rate?. A man metre high walks at a uniform speed of 6km /h away from a lamp post 6 metre high. Find the rate at which the length of his shadow increases. Also find the rate at which the tip of the shadow is moving away from the lamp post.. The length of a rectangle is increasing at the rate of.5 cm/sec and its breadth is decreasing at the rate of cm/sec. find the rate of change of the area of the rectangle when length is cm and breadth is 8 cm LEVEL III. A particle moves along the curve 6 y = +., Find the points on the curve at which y- coordinate is changing 8 times as fast as the -coordinate.. Water is leaking from a conical funnel at the rate of 5 cm /sec. If the radius of the base of the funnel is cm and altitude is cm, Find the rate at which water level is dropping when it is 5 cm from top.. From a cylinder drum containing petrol and kept vertical, the petrol is leaking at the rate of ml/sec. If the radius of the drum is cm and height 5cm, find the rate at which the level of the petrol is changing when petrol level is cm.increasing & decreasing functions LEVEL I. Show that f() = is an increasing function for all R.. Show that the function + is neither increasing nor decreasing on (,). Find the intervals in which the function f() = sin cos, < < isincreasing or 5

37 decreasing. LEVEL II. Indicate the interval in which the function f() = cos, is decreasing. sin.show that the function f() = is strictly decreasing on (, /) log. Find the intervals in which the function f() = increasing or decreasing. LEVEL III. Find the interval of monotonocity of the function f() = log, 4sin θ. Prove that the function y = θ is an increasing function of in [, /] cos θ [CBSE ].Tangents &Normals LEVEL-I.Find the equations of the normals to the curve y = 8 which are parallel to the line + y = 4.. Find the point on the curve y = where the slope of the tangent is equal to the -coordinate of the point.. At what points on the circle + y 4y + =, the tangent is parallel to ais? LEVEL-II. Find the equation of the normal to the curve ay = at the point ( am, am ). For the curve y = + + 8, find all the points at which the tangent passes through the origin.. Find the equation of the normals to the curve y = which are parallel to the line + 4y + 4= 4. Show that the equation of tangent at (, y ) to the parabola yy =a( + ). [CBSE Comptt.] LEVEL- III.Find the equation of the tangent line to the curve y = 5 which is parallel to the line 4 y + =. Show that the curve +y = and +y y = cut orthogonally at the point (,) 6

38 y. Find the condition for the curves and y = c to intersect orthogonally. a b 4.Approimations LEVEL-I Q. Evaluate 5. Q. Use differentials to approimate the cube root of 66 Q. Evaluate. 8 Q.4 Evaluate [CBSE ] LEVEL-II. If the radius of a sphere is measured as 9 cm with an error of. cm, then find the approimate error in calculating its surface area 5 Maima & Minima LEVEL I. Find the maimum and minimum value of the function f() = sin. Show that the function f() = has neither a maimum value nor a minimum value. Find two positive numbers whose sum is 4 and whose product is maimum LEVEL II. Prove that the area of a right-angled triangle of given hypotenuse is maimum when the triangle is isosceles..a piece of wire 8(units) long is cut into two pieces. One piece is bent into the shape of a circle and other into the shape of a square. How should the wire be cut so that the combined area of the two figures is as small as possible.. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is m. Find the dimensions of the window to admit maimum light through the whole opening. LEVEL III.Find the area of the greatest isosceles triangle that can be inscribed in a given ellipse having its verte coincident with one etremity of major ais..an open bo with a square base is to be made out of a given quantity of card board of area c square units. Show that the maimum volume of the bo is 6 7 c cubic units.[cbse Comptt.]

39 .A window is in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is m, find the dimensions of the rectangle that will produce the largest area of the window. [CBSE ] Questions for self evaluation.sand is pouring from a pipe at the rate of cm /s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sith of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?. The two equal sides of an isosceles triangle with fied base b are decreasing at the rate of cm per second. How fast is the area decreasing when the two equal sides are equal to the base?. Find the intervals in which the following function is strictly increasing or decreasing: f() = Find the intervals in which the following function is strictly increasing or decreasing: f() = sin + cos, 5. For the curve y = 4 5, find all the points at which the tangent passes through the origin. 6. Find the equation of the tangent line to the curve y = +7 which is (a) parallel to the line y + 9 = (b) perpendicular to the line 5y 5 =. 7. Prove that the curves = y and y = k cut at right angles if 8k =. 8. Using differentials, find the approimate value of each of the following up to places of decimal : (i) 6 (ii) Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is of the 7 volume of the sphere.. An open topped bo is to be constructed by removing equal squares from each corner of a metre by 8 metre rectangular sheet of aluminium and folding up the sides. Find the volume of the largest such bo. 8

40 TOPIC 6 INDEFINITE & DEFINITE INTEGRALS SCHEMATIC DIAGRAM Topics Concept Degree of Importance References Tet book of NCERT, Vol. II 7 Edition Indefinite (i) Integration by substitution * Ep 5&6 Page, Integrals (ii) ) Application of trigonometric ** E 7 Page 6, Eercise 7.4Q&Q4 function in integrals (iii) Integration of some particular function *** Ep 8, 9, Page, Eercise 7.4 Q,4,8,9,& a, a,,, a a b c, a b c (p q), a b c (p q) Definite Integrals a b c (iv) Integration using Partial *** Ep & Page 8 Fraction Ep 9,Ep 4 & 5 Page (v) Integration by Parts ** Ep 8,9& Page 5 (vi)some Special Integrals *** Ep &4 Page 9 a, a (vii) Miscellaneous Questions *** Solved E.4 (i) Definite Integrals based upon types of indefinite integrals * Eercise 7 Page 6, Q,,4,5,9,,6 Eercise 7.9 (ii) Definite integrals as a limit of sum ** Ep 5 &6 Page, 4 Q, Q5 & Q6 Eercise 7.8 (iii) Properties of definite Integrals *** Ep Page 4*,Ep *,4&5 page 44 Ep 6***Ep 46 Ep 44 page5 Eercise 7. Q7 & (iv) Integration of modulus function ** Ep Page 4,Ep 4 Page 5 Q5& Q6 Eercise 7. 9

41 SOME IMPORTANT RESULTS/CONCEPTS n n * c n. c * n n * * * c e e * * c c c a a c log a * sin cos c * sin cos c * cos sin c * sec tan c * cos ec cot c * sec.tan sec c * cos ec.cot cos ec c * tan log cos c log sec * cot = log sin + C * sec log sec tan C =log tan C 4 * cosec log cosec - cot = log cosec + cot + C = log tan + C a * log C, if > a a a a a * log C, if > a a a a a * log C, if > a a a a C c * sin c = - cos - C` a a a * log a C a * log a C a * a a a log a * a a a log a a * a a sin C a * f () f()...f n () 4 =f() f ()... * f () f () C * u.v u. v. v. b * f () a du. = F(b) F(a), where F() = f() f n () * General Properties of Definite Integrals. b b * f () = f (t) a b b * f () = - f () a b * f () a b * a a * a a c = a b f() = f(a a a a f() + f() + b - ) f() = f(a - ) a a * f() = f(), b c if f() is an even function of if f() is an odd function of C C.

42 * tan C, cot C` a a a a a a a * f() = f(), if f(a - ) f().. if f(a - ) -f() (i) Integration by substitution sec. (log ) e.. tan. sin.cos. Assignments mtan LEVEL I LEVEL II 6 LEVEL III tan. sec cos. e (ii) Application of trigonometric function in integrals LEVEL I sin e.. sin.cos. sin.. cos.. cos.cos.cos. LEVEL II. sec 4 sin 4.tan.. sin LEVEL III. cos 5.. sin.cos. (iii) Integration using standard results LEVEL I LEVEL II cos. sin 4sin

43 LEVEL III (iv) Integration using Partial Fraction LEVEL I. ( )( ) 8. ( )( ) 8. ( )( 4) (v) Integration by Parts..sec.. [CBSE ] ( )( )( ) ( ) ( ) LEVEL II. ( ) LEVEL III. sin sin.sin...sin 4. cos. LEVEL I. ( ) ( ).. log.. e (tan logsec) LEVEL II 5. sec...sin.. log LEVEL III e ( ) cos. ( ) sin 4. e cos. (vi) Some Special Integrals e.cos. LEVEL I. 4. LEVEL II LEVEL III log. ( log ) 4

44 . ( ). (vii) Miscellaneous Questions. cos 4. sin 8cos sin cos. cos sin tan. ( 5) LEVEL II.. sin 4sin 5cos sin sec sin cos 5sec 4 tan LEVEL III 5. 4 Definite Integrals tan. (i) Definite Integrals based upon types of indefinite integrals LEVEL I / 5. sin.cos. LEVEL II. e. (ii) Definite integrals as a limit of sum LEVEL I. Evaluate ( ) as the limit of a sum. 4. Evaluate ( ) definite integral as the limit of a sum. LEVEL II 4

45 . Evaluate ( ) as the limit of a sum.. Evaluate ( ) as the limit of a sum.. Evaluate ( ) as the limit of a sum. 4. Evaluate e as the limit of a sum. LEVEL III (iii) Properties of definite Integrals LEVEL I / tan. tan π/. sin cos 4. 4 LEVEL II sin. cos / 4 sin. 4 4 sin cos tan. sec.cos ec / 4. [CBSE ] tan / 6 LEVEL III / / 4 sin. [CBSE ]. cos log sin. log tan [CBSE ] (iv) Integration of modulus function LEVEL III 5 /. ( 4).. sin cos / Questions for self evaluation ( ). Evaluate. Evaluate 8 44 ( ). 5

46 . Evaluate cos Evaluate sin cos sin cos 5. Evaluate sin 4cos / 5 7. Evaluate sin.cos. / 9. Evaluate log sin.sin 6. Evaluate / 8. Evaluate sin 4. Evaluate 45

47 TOPIC 7 APPLICATIONS OF INTEGRATION SCHEMATIC DIAGRAM Topic Concepts Degree of Importance Reference NCERT Tet BookEdition 7 Applications of (i)area under Simple Curves * E.8. Q.,,5 Integration (ii) Area of the region enclosed *** E. 8. Q, Misc.E.Q 7 between Parabola and line (iii) Area of the region enclosed between Ellipse and line *** Eample 8, page 69 Misc.E. 8 (iv) Area of the region enclosed *** E. 8. Q 6 betweencircle and line (v) Area of the region enclosed *** E 8. Q, Misc.E.Q 5 between Circle and parabola (vi) Area of the region enclosed between Two Circles *** Eample, page7 E 8. Q (vii) Area of the region enclosed *** Eample 6, page68 between Two parabolas (viii) Area of triangle when vertices are given *** Eample 9, page7 E 8. Q4 (i) Area of triangle when sides *** E 8. Q5,Misc.E. Q 4 are given () Miscellaneous Questions *** Eample, page74 Misc.E.Q 4, SOME IMPORTANT RESULTS/CONCEPTS b ** Area of the region PQRSP = a b da = a y b = a f (). ** The area A of the region bounded by the curve = g (y), y-ais and d the lines y = c, y = d is given by A= c dy d = c g (y) dy 46

48 ASSIGNMENTS (i) Area under Simple Curves LEVEL I y. Sketch the region of the ellipse and find its area, using integration, Sketch the region {(, y) : 4 + 9y = 6} and find its area, using integration. (ii) Area of the region enclosed between Parabola and line LEVEL II. Find the area of the region included between the parabola y = and the line + y =.. Find the area of the region bounded by = 4y, y =, y = 4 and the y-ais in the first quadrant. LEVEL III. Find the area of the region :(, y): y, y, (iii) Area of the region enclosed between Ellipse and line LEVEL II y y. Find the area of smaller region bounded by the ellipse and the straight line (iv) Area of the region enclosed between Circle and line LEVEL II. Find the area of the region in the first quadrant enclosed by the -ais, the line y = and the circle + y =. LEVEL III. Find the area of the region :(, y): y y (v) Area of the region enclosed between Circle and parabola LEVEL III. Draw the rough sketch of the region {(, y): 6y, + y 6} an find the area enclosed by the region using the method of integration.. Find the area lying above the -ais and included between the circle + y = 8 and the parabola y = 4. (vi) Area of the region enclosed between Two Circles LEVEL III. Find the area bounded by the curves + y = 4 and ( + ) + y = 4 using integration. (vii) Area of the region enclosed between Two parabolas LEVEL II. Draw the rough sketch and find the area of the region bounded by two parabolas 47

49 4y = 9 and = 6y by using method of integration. (viii) Area of triangle when vertices are given LEVEL III. Using integration compute the area of the region bounded by the triangle whose vertices are (, ), (, 4), and (5, ).. Using integration compute the area of the region bounded by the triangle whose vertices are (, ), (, 5), and (, ). (i) Area of triangle when sides are given LEVEL III. Using integration find the area of the region bounded by the triangle whose sides are y = +, y = +, = 4.. Using integration compute the area of the region bounded by the lines + y =, y =, and + y = 7. () Miscellaneous Questions LEVEL III. Find the area of the region bounded by the curves y = and y = +.. Find the area bounded by the curve y = and y =.. Draw a rough sketch of the curve y = sin and y = cos as varies from = to = and find the area of the region enclosed by them and -ais 4. Sketch the graph of y =.Evaluate.What does this value represent on the graph. 5. Find the area bounded by the curves y = 6 and y =. 6. Sketch the graph of y = and evaluate the area under the curve y = above -ais and between = 6 to =. [CBSE ] Questions for self evaluation. Find the area bounded by the curve = 4y and the line = 4y.. Find the area bounded by the parabola y = and y =.. Find the area of the region :(,y): y, y, y y 4. Find the area of the smaller region bounded by the ellipse and the line Find the area of the region :(,y): y, y 6. Find the area lying above the -ais and included between the circle + y = 8 and the parabola y = Find the area bounded by the curves + y = 4 and ( + ) + y = 4 using integration. 48