LIST OF MEMBERS WHO PREPARED QUESTION BANK FOR MATHEMATICS FOR CLASS XII TEAM MEMBERS. Sl. No. Name Designation

Transcription

1 LIST OF MEMBERS WHO PREPARED QUESTION BANK FOR MATHEMATICS FOR CLASS XII TEAM MEMBERS Sl. No. Name Designation. Sh. S.B. Tripathi R.S.B.V., Jheel Khuranja (Group Leader) Delhi. (M ). Sh. Sanjeev Kumar R.S.V.V., Raj Niwas Marg, (Lecturer Maths) Delhi. (M ). Dr. R.P. Singh R.P.V.V., Gandhi Nagar (Lecturer Maths) Delhi (M ) 4. Sh. Joginder Arora R.P.V.V., Hari Nagar (Lecturer Maths) Delhi. (M ) 5. Sh. Manoj Kumar R.P.V.V., Kishan Kunj (Lecturer Maths) Delhi. (M ) 6. Miss Saroj G.G.S.S.S., No., Roop Nagar (Lecturer Maths) Delhi-0007 (M ) REVIEWED BY. Dr. Reva Dass G.G.S.S. School. Vivek Vihar Retd. Principal Delhi - 95 (M ). Mr. Sanjeev Kumar R.P.V.V. Raj Niwas Marg, (M ) Delhi. Mr. Jaginder Arora R.P.V.V. Hari Nagar (M ) Delhi

2 XII Maths

3 CLASS XII MATHEMATICS Units Weightage (Marks) (i) Relations and Functions 0 (ii) Algebra (iii) Calculus 44 (iv) Vector and Three Dimensional Geometry 7 (v) Linear Programming 06 (vi) Probability 0 Total : 00 Unit I : RELATIONS AND FUNCTIONS. Relations and Functions (0 Periods) Types of Relations : Refleive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function. Binary operations.. Inverse Trigonometric Functions ( Periods) Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions. Unit II : ALGEBRA. Matrices (8 Periods) Concept, notation, order, equality, types of matrices, zero matri, transpose of a matri, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication. Non-commutativity of multiplication XII Maths

4 of matrices and eistence of non-zero matrices whose product is the zero matri (restrict to square matrices of order ). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it eists; (Here all matrices will have real entries).. Determinants (0 Periods) Determinant of a square matri (up to matrices), properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle. adjoint and inverse of a square matri. Consistency, inconsistency and number of solutions of system of linear equations by eamples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matri. Unit III : CALCULUS. Continuity and Differentiability (8 Periods) Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function. Concept of eponential and logarithmic functions and their derivatives. Logarithmic differentiation. Derivative of functions epressed in parametric forms. Second order derivatives. Rolle s and Lagrange s mean Value Theorems (without proof) and their geometric interpretations.. Applications of Derivatives (0 Periods) Applications of Derivatives : Rate of change, increasing/decreasing functions, tangents and normals, approimation, maima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Sample problems (that illustrate basic principles and understanding of the subject as well as real-life situations).. Integrals (0 Periods) Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, only simple integrals of the type to be evaluated.,,,, a a b c a a a b c p q p q,, a and a a b c a b c XII Maths 4

5 Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals. 4. Applications of the Integrals (0 Periods) Application in finding the area under simple curves, especially lines, area of circles/parabolas/ellipses (in standard form only), area between the two above said curves (the region should be clearly identifiable). 5. Differential Equations (0 Periods) Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given. Solution of differential equations by method of separation of variables, homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type : dy p y q, where p and q are function of. Unit IV : VECTORS AND THREE-DIMENSIONAL GEOMETRY. Vectors ( Periods) Vectors and scalars, magnitude and direction of a vector. Direction cosines/ ratios of vectors. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of vectors.. Three-Dimensional Geometry ( Periods) Direction cosines/ratios of a line joining two points. Cartesian and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane. 5 XII Maths

6 Unit V : LINEAR PROGRAMMING ( Periods). Linear Programming : Introduction, definition of related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints). Unit VI : PROBABILITY. Probability (8 Periods) Multiplication theorem on probability. Conditional probability, independent events, total probability, Baye s theorem, Random variable and its probability distribution, mean and variance of haphazard variable. Repeated independent (Bernoulli) trials and Binomial distribution. XII Maths 6

7 CONTENTS S.No. Chapter Page. Relations and Functions 9. Inverse Trigonometric Functions 8 & 4. Matrices and Determinants 4 5. Continuity and Differentiation Applications of Derivatives Integrals 6 8. Applications of Integrals Differential Equation Vectors 0. Three-Dimensional Geometry 0. Linear Programming. Probability 6 Model Papers 5 7 XII Maths

8 XII Maths 8

9 CHAPTER RELATIONS AND FUNCTIONS IMPORTANT POINTS TO REMEMBER Relation R from a set A to a set B is subset of A B. A B = {(a, b) : a A, b B}. If n(a) = r, n (B) = s from set A to set B then n (A B) = rs. and no. of relations = rs is also a relation defined on set A, called the void (empty) relation. R = A A is called universal relation. Refleive Relation : Relation R defined on set A is said to be refleive iff (a, a) R a A Symmetric Relation : Relation R defined on set A is said to be symmetric iff (a, b) R (b, a) R a, b, A Transitive Relation : Relation R defined on set A is said to be transitive if (a, b) R, (b, c) R (a, c) R a, b, c R Equivalence Relation : A relation defined on set A is said to be equivalence relation iff it is refleive, symmetric and transitive. One-One Function : f : A B is said to be one-one if distinct elements in A has distinct images in B. i.e., A s.t. f( ) f( )., A s.t. f( ) = f( ) = OR One-one function is also called injective function. 9 XII Maths

10 Onto function (surjective) : A function f : A B is said to be onto iff R f = B i.e. b B, there eist a A s.t. f(a) = b A function which is not one-one is called many-one function. A function which is not onto is called into. Bijective Function : A function which is both injective and surjective is called bijective. Composition of Two Function : If f : A B, g : B C are two functions, then composition of f and g denoted by gof is a function from A to C given by, (gof) () = g (f ()) A Clearly gof is defined if Range of f defined. domain of g. Similarly fog can be Invertible Function : A function f : X Y is invertible iff it is bijective. If f : X Y is bijective function, then function g : Y X is said to be inverse of f iff fog = I y and gof = I when I, I y are identity functions. g is inverse of f and is denoted by f. Binary Operation : A binary operation * defined on set A is a function from A A A. * (a, b) is denoted by a * b. Binary operation * defined on set A is said to be commutative iff a * b = b * a a, b A. Binary operation * defined on set A is called associative iff a * (b * c) = (a * b) * c a, b, c A If * is Binary operation on A, then an element e A is said to be the identity element iff a * e = e * a a A Identity element is unique. If * is Binary operation on set A, then an element b is said to be inverse of a A iff a * b = b * a = e Inverse of an element, if it eists, is unique. XII Maths 0

11 VERY SHORT ANSWER TYPE QUESTIONS. If A is the set of students of a school then write, which of following relations are. (Universal, Empty or neither of the two). R = {(a, b) : a, b are ages of students and a b 0} R = {(a, b) : a, b are weights of students, and a b < 0} R = {(a, b) : a, b are students studying in same class} R 4 = {(a, b) : a, b are age of students and a > b}. Is the relation R in the set A = {,,, 4, 5} defined as R = {(a, b) : b = a + } refleive?. If R, is a relation in set N given by R = {(a, b) : a = b, b > 5}, then does elements (5, 7) R? 4. If f : {, } {,, 5} and g : {,, 5} {,,, 4} be given by f = {(, ), (, 5)}, g = {(, ), (, ), (5, )} Write down gof. 5. Let g, f : R R be defined by g, f. Write fog. 6. If f : R R defined by f 5 be an invertible function, write f (). 7. If f, Write fo f. 8. Let * is a Binary operation defined on R, then if (i) a * b = a + b + ab, write * XII Maths

12 (ii) a b a * b, Write * * 4. (iii) a * b = 4a 9b, Write ( * ) *. 9. If n(a) = n(b) =, Then how many bijective functions from A to B can be formed? 0. If f () = +, g() =, Then (gof) () =?. Is f : N N given by f() = is one-one? Give reason.. If f : R A, given by f() = + is onto function, find set A.. If f : A B is bijective function such that n (A) = 0, then n (B) =? 4. If n(a) = 5, then write the number of one-one functions from A to A. 5. R = {(a, b) : a, b N, a b and a divides b}. Is R refleive? Give reason? 6. Is f : R R, given by f() = is one-one? Give reason? 7. f : R B given by f() = sin is onto function, then write set B. 8. Is f : R R, f () = is bijective function? 9. If * is a binary operation on set Q of rational numbers given by a * b then write the identity element in Q. ab 5 0. If * is Binary operation on N defined by a * b = a + ab a, b N. Write the identity element in N if it eists. SHORT ANSWER TYPE QUESTIONS (4 Marks). Check the following functions for one-one and onto. (a) f : R R, f ( ) 7 (b) f : R R, f() = + XII Maths

13 (c) f : R R, f() = + (d) f : R {} R, f (e) f : R R, f () = sin (f) f : R [, ], f() = sin (g) f : R R, f () = +. Show f : R R given by f is bijective. Also find f See f : R R be a function given by 4 that f is invertible with f. 4 f 4. 4 Show 4. Let R be the relation on set A = { : Z, 0 0} given by R = {(a, b) : (a b) is multiple of 4}, is an equivalence relation. Also, write all elements related to Show that function f : A B defined as f A R, B R is invertible and hence find f Let * be a binary operation on Q. Such that a * b = a + b ab. where (i) (ii) Prove that * is commutative and associative. Find identify element of * in Q (if it eists). 7. If * is a binary operation defined on R {0} defined by check * for commutativity and associativity. a a * b, b then 8. If A = N N and binary operation * is defined on A as (a, b) * (c, d) = (ac, bd). (i) (ii) Check * for commutativity and associativity. Find the identity element for * in A (If it eists). XII Maths

14 9. Show that the relation R defined by (a, b) R(c, d) a + d = b + c on the set N N is an equivalence relation. 0. Let * be a binary operation on set Q defined by ab a * b, show that 4 (i) 4 is the identity element of * on Q. (ii) Every non zero element of Q is invertible with 6 a, a Q 0. a. Show that f : R + R + defined by f set of all non-zero positive real numbers. is bijective where R + is the. Consider f : R + [ 5, ) given by f() = show that f is invertible with 6 f.. If * is a binary operation on R defined by a * b = a + b + ab. Prove that * is commutative and associative. Find the identify element. Also show that every element of R is invertible ecept. 4. If f, g : R R defined by f() = and g() = + find (fog) () and (gof) (). Are they equal? 5. Prove that composition of two one-one functions is also one-one? 6. f : R R, g : R R given by f() = [], g() = then find fog and gof. H.O.T.S. 7. f : [, ) [, ) is given by, f then find f () f 4 7, find f 9. If f log show that f f. XII Maths 4

15 ANSWERS VERY SHORT ANSWER TYPE QUESTIONS. R : is universal relation. R : is empty relation. R : is neither universal nor empty. R 4 : is neither universal nor empty.. No, R is not refleive.. (5, 7) R 4. gof = {(, ), (, )} 5. (fog)() = R 6. f 5 7. fof, 8. (i) * = (ii) 69 7 (iii) Yes, f is one-one, N.. A = [, ) because R f = [, ). n(b) = XII Maths

16 5. No, R is not refleive a, a R a N 6. f is not one-one functions f() = f ( ) = i.e. distinct element has same images. 7. B = [, ] 8. Yes 9. e = 5 0. Identity element does not eist. SHORT ANSWER TYPE QUESTIONS (4 MARKS). (a) Bijective (b) (c) (d) (e) (f) (g) Neither one-one nor onto. Neither one-one nor onto. One-one, but not onto. Neither one-one nor onto. Neither one-one nor onto. Neither one-one nor onto. 5. f 4. Elements related to 4 are 0, 4, f is the identity element. 7. Neither commutative nor associative. 8. (i) Commutative and associative. (ii) (, ) is identity in N N XII Maths 6

17 . 0 is the identity element. 4. (fog) () = + (gof) () = + Clearly, they are unequal. 5. Hint : Let, R s.t. (fog) ( ) = (fog) ( ) f(g ()) = f (g( )) g( ) = g ( ) = ( f is one-one) g is one-one). Hence (fog) () is one-one function. Similarly, (gof) () is also one-one function. 6. fog 0 gof 7. f 4 8. f XII Maths

18 CHAPTER INVERSE TRIGONOMETRIC FUNCTIONS IMPORTANT POINTS sin, cos,... etc., are angles. If sin and, then = sin etc. Function Domain Range (Principal Value Branch) sin [, ], cos [, ] [0, ] tan R, cot R (0, ) sec R (, ) 0, cosec R (, ), 0 sin (sin ) =, cos (cos ) = [0, ] etc. sin (sin ) = [, ] cos (cos ) = [, ] etc. XII Maths 8

19 sin cosec, tan = cot (/) > 0 sec = cos (/), sin ( ) = sin [, ] tan ( ) = tan R cosec ( ) = cosec cos ( ) = cos [, ] cot ( ) = cot R sec ( ) = sec sin cos,, tan cot R sec cosec tan tan y tan ; y. y y tan tan y tan ; y. tan tan, y y VERY SHORT ANSWER TYPE QUESTIONS ( MARK). Write the principal value of (i) sin (ii) sin. 9 XII Maths

20 (iii) (v) cos (iv) tan (vi) cos. tan. (vii) cosec ( ). (viii) cosec () (i) cot () cot. (i) sec ( ). (ii) sec (). (iii) sin cos tan. What is value of the following functions (using principal value). (i) tan sec. (ii) sin cos. (iii) tan () cot ( ). (iv) cos sin. (v) tan cot. (vi) cosec sec. (vii) tan () + cot () + sin (). (viii) cot sin. (i) 4 sin sin. 5 () 7 cos cos. 5 (i) 5 tan tan. 6 (ii) cosec cosec. 4 XII Maths 0

21 SHORT ANSWER TYPE QUESTIONS (4 MARKS). Show that 4. Prove cos cos tan. cos cos 4 [0, ] cos cos tan cot 0,. sin cos 4 5. Prove 6. Prove a tan sin cos. a a a cot tan cos tan tan sin tan Prove tan cos Solve cot cot Prove that m m n tan tan, m, n 0 n m n 4 0. Prove that y y tan sin cos y y. Solve for, cos tan. Prove that tan tan tan tan XII Maths

22 . Solve for, tan cos sin tan ; 0 4. Prove that tan tan tan Evaluate tan cos H.O.T.S. 6. Prove that 7. Prove that tan a cos b sin a tan b cos a sin b cot tan tan cos cos, 0 8. Prove that c > 0 a b b c c a tan tan tan 0 ab bc ca where a, b, 9. Solve for, tan (cos ) = tan ( cosec ) 0. Epress sin in simplest form.. If tan a + tan b + tan c = then prove that a + b + c = abc. If sin > cos, then belongs to which interval? ANSWERS. (i) (ii) (iii) 5 6 (iv) 6 (v) 6 (vi) 6 (vii) 6 (viii) 6 XII Maths

23 (i) () (i) (ii) (iii). 6. (i) 0 (ii) (iii) (iv) (v) (vi) (vii) (viii) (i) 5 () 5 (i) 6 (ii) tan sin sin. 4., XII Maths

24 CHAPTER & 4 MATRICES AND DETERMINANTS POINTS TO REMEMBER Matri : A matri is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements of the matri. Order of Matri : A matri having m rows and n columns is called the matri of order mn. Zero Matri : A matri having all the elements zero is called zero matri or null matri. Diagonal Matri : A square matri is called a diagonal matri if all its non diagonal elements are zero. Scalar Matri : A diagonal matri in which all diagonal elements are equal is called a scalar matri. Identity Matri : A scalar matri in which each diagonal element is, is called an identity matri or a unit matri. It is denoted by I. I = [e ij ] n n where, e ij if 0 if i i j j Transpose of a Matri : If A = [a i j ] m n be an m n matri then the matri obtained by interchanging the rows and columns of A is called the transpose of the m atri.transpose of A is denoted by A or A T. Properties of the transpose of a matri. (i) (A ) = A (ii) (A + B) = A + B (iii) (ka) = ka, k is a scalar (iv) (AB) = B A XII Maths 4

25 Symmetri Matri : A square matri A = [a ij ] is symmetri if a ij = a ji i, j. Also a square matri A is symmetri if A = A. Skew Symmetri Matri : A square matri A = [a ij ] is skew-symmetri, if a ij = a ji i, j. Also a square matri A is skew - symmetri, if A = A. Determinant : To every square matri A = [a ij ] of order n n, we can associate a number (real or comple) called determinant of A. It is denoted by det A or A. Properties (i) (ii) AB = A B ka n n = k n A n n where k is a scalar. Area of triangles with vertices (, y ), (, y ) and (, y ) is given by y y y The points (, y ), (, y ), (, y ) are collinear 0 Adjoint of a square matri A is the transpose of the matri whose elements have been replaced by their cofactors and is denoted as adj A. y y y Let A = [a ij ] n n adj A = [A ji ] n n Properties (i) A(adj A) = (adj A) A = A I (ii) If A is a square matri of order n then adj A = A n (iii) adj (AB) = (adj B) (adj A). Singular Matri : A square matri is called singular if A = 0, otherwise it will be called a non-singular matri. 5 XII Maths

26 Inverse of a Matri : A square matri whose inverse eists, is called invertible matri. Inverse of only a non-singular matri eists. Inverse of a matri A is denoted by A and is given by A. adj. A A Properties (i) (ii) AA = A A = I (A ) = A (iii) (AB) = B A (iv) (A T ) = (A ) T Solution of system of equations using matri : If AX = B is a matri equation then its solution is X = A B. (i) If A 0, system is consistent and has a unique solution. (ii) If A = 0 and (adj A) B 0 then system is inconsistent and has no solution. (iii) If A = 0 and (adj A) B = 0 then system is consistent and has infinite solution. VERY SHORT ANSWER TYPE QUESTIONS ( Mark). If 4 5 4, y 4 y 9 find and y.. If i 0 0 i A and B, find AB. 0 i i 0. Find the value of a + a in the matri A = [a ij ] where a ij i j if i j i j if i j. XII Maths 6

27 4. If B be a 4 5 type matri, then what is the number of elements in the third column. 5. If 5 6 A and B find A B If 0 A and B find A B If A = [ 0 4] and B 5 find AB If A 4 is symmetri matri, then find. 9. For what value of the matri is skew symmetri matri. 0. If A P Q where P is symmetric and Q is skew-symmetric 0 matri, then find the matri Q.. Find the value of a ib c id c id a ib. If 5 0, find For what value of k, the matri k 4 has no inverse. 4. If sin 0 cos 0 A, what is A. sin 60 cos 60 7 XII Maths

28 5. Find the cofactor of a in 6. Find the minor of a in Find the value of P, such that the matri 4 P is singular. 8. Find the value of such that the points (0, ), (, ) and (, ) are collinear. 9. Area of a triangle with vertices (k, 0), (, ) and (0, ) is 5 unit. Find the value (s) of k. 0. If A is a square matri of order and A =, find the value of A.. If A = B where A and B are square matrices of order and B = 5, what is A?. What is the condition that a system of equation AX = B has no solution.. Find the area of the triangle with vertices (0, 0), (6, 0) and (4, ). 4. If 4 6, find. y y z z 5. If A z y, write the value of det A. a a 6. If A such that A = 5, find a a a C + a C where C ij is cofactors of a ij in A = [a ij ]. 7. If A is a non-singular matri of order and A = find adj A. 8. If A 5 find 6 8 adj A XII Maths 8

29 9. Given a square matri A of order such that A = find the value of A adj A. 0. If A is a square matri of order such that adj A = 8 find A.. Let A be a non-singular square matri of order find adj A if A = 0.. If A find A. 4. If A and B 4 find AB. 0 SHORT ANSWER TYPE QUESTIONS (4 MARKS) 4. Find, y, z and w if y z 5. y w 0 5. Construct a matri A = [a ij ] whose elements are given by a ij = i j if i j i j if i j 6. Find A and B if A + B = 0 and A B If A and B 4, verify that (AB) = B A. 8. Epress the matri 4 5 P Q where P is a symmetric and Q is a skew-symmetric matri. 9 XII Maths

30 cos sin 9. If A =, then prove that sin cos where n is a natural number. n cosn sinn A sinn cosn 40. Let 5 5 A, B, C, find a matri D such that CD AB = O. 4. Find the value of such that Prove that the product of the matrices cos cos sin cos sin sin cos cos sin and cos sin sin is the null matri, when and differ by an odd multiple of. 4. If 44. If 5 A show that A A I = 0. Hence find A. 7 A find f(a) where f() = If 4 A, find and y such that A A + yi = Find the matri X so that X If A and B then show that (AB) = B A Test the consistency of the following system of equations by matri method : y = 5; 6 y = XII Maths 0

31 49. Using elementary row transformations, find the inverse of the matri 6 A, if possible. 50. By using elementary column transformation, find the inverse of A If A cos sin sin cos and A + A = I, then find the general value of. Using properties of determinants, prove the following : Q 5 to Q a b c a a b b c a b a b c c c c a b 5. a 4 b 0 if a, b, c are in A. P. 4 5 c sin cos sin sin cos sin 0 sin cos sin b c a a b c a b 4 a b c. c c a b b c c a a b a b c 56. q r r p p q p q r. y z z y y z 57. a bc ac c a ab b ac 4 a b c. ab b bc c XII Maths

32 58. a b c a b c a b c a b c 59. Show that :. y z y z y z z y yz z y yz z y. 60. (i) If the points (a, b) (a, b ) and (a a, b b ) are collinear. Show that ab = a b. (ii) If 5 4 A and B verity that AB A B Given 0 0 A and B 0. Find the product AB and 0 also find (AB). 6. Solve the following equation for. a a a a a a a a a Verify that (AB) = B A for the matrices A 4 5 and B Use matri method to solve the following system of equations : 5 7y =, 7 5y =. LONG ANSWER TYPE QUESTIONS (6 MARKS) 65. Obtain the inverse of the following matri using elementary row operations A 0. XII Maths

33 66. Use product y + z =, y z =, y + 4z =. to solve the system of equations 67. Solve the following system of equations by matri method, where 0, y 0, z 0 0, 0,. y z y z y z 68. Find A, where A 4, hence solve the system of linear equations : + y z = 4 + y + z = y 4z = 69. The sum of three numbers is. If we subtract the second number from twice the first number, we get. By adding double the second number and the third number we get 0. Represent it algebraically and find the numbers using matri method. 70. Compute the inverse of the matri. A and verify that A A = I If the matri 0 A 0 and B 0, then 4 0 compute (AB). XII Maths

34 7. Using matri method, solve the following system of linear equations : y = 4, y + z = 5, z + = Find 0 A if A 0. Also show that 0 A A I. 74. Find the inverse of the matri column transformations. A 0 by using elementary Let A and f() = Show that f (A) = 0. Use this result to find A If cos sin 0 A sin cos 0, verify that A. (adj A) = (adj A). A = A I For the matri find A. A, verify that A 6A + 9A 4I = 0, hence 78. Find the matri X for which. X By using properties of determinants prove the following : a b ab b ab a b a a b. b a a b XII Maths 4

35 y z y z y z yz yz y z. z yz y a a b a b c a a b 4a b c a. a 6a b 0a 6b c 8. If, y, z are different and y y y 0. z z z H.O.T.S. Show that yz =. 8. If a matri A has elements, what is its possible order? 84. Given a square matri A of order, such that A = 5, find the value of A. adj A. 85. If A = and A = [a ij ] and C ij the cofactors of a ij then what is the value of a C + a C + a C. 86. What is the number of all possible matrices of order with each entry 0, or. SHORT ANSWER TYPE QUESTIONS (4 MARKS) 87. If 0 tan A and I is the identity matri of order, show that tan 0 I A I A cos sin sin cos 5 XII Maths

36 88. If cos sin 0 F sin cos 0, show that F ( ) F ( ) = F( + ) Let 0 A, show that (ai + ba) n = a n I + na n ba, where I is the 0 0 identity matri of order and n N. 90. If, y, z are the 0 th, th and 5 th terms of a G.P. find the value of log 0 logy. logz 5 9. Using properties of determinants, show that b c a bc c a b ca a b b c c a a b c a b c a b c ab. LONG ANSWER TYPE QUESTIONS (6 MARKS) 9. Using properties of determinants prove that bc b bc c bc a ac ac c ac ab bc ca a ab b ab ab 9. If A 4, find A and hence solve the system of equations 7 + 4y + 7z = 4, y + z = 4, + y z = 0. ANSWERS. =, y = XII Maths 6

37 AB = [6]. 8. = 5 9. = a + b + c + d.. =. k 4. A = P = k A = 0, (adj A). B 0. 9 sq. units 4. = ± A = AB = 4. =, y =, z =, w = 4 7 XII Maths

38 A , B D. 4. = or A. 44. f(a) = = 9, y = Inconsistent 49. Inverse does not eist. 50. A. 5. n, n z 5 6. AB, AB , a 64., y A = 0, y = 5, z = 5 XII Maths 8

39 67., y, z A =, y =, z = 70. A AB =, y =, z = A. 74. A A A , =, y =, z =. 9 XII Maths

40 CHAPTER 5 CONTINUITY AND DIFFERENTIATION POINTS TO REMEMBER A function f() is said to be continuous at = c iff lim c f f c i.e., lim f lim f f c c c f() is continuous in ] a, b [ iff it is continuous at c c a, b. f() is continuous in [a, b] iff (i) f() is continuous in (a, b) (ii) lim f f a, a (iii) lim f b f b Trigonometric function are continous in their resp. domains. Every polynomial function is continuous on R. If f () and g () are two continuous functions and c R then at = a (i) f () ± g () are also continuous functions at = a. (ii) g (). f (), f () + c, cf (), f () are also continuous at = a. (iii) f g is continuous at = a provided g(a) 0. f () is derivable at = c iff L.H.D. (c) = R.H.D. (c) XII Maths 40

41 f f c f f c i.e. lim lim c c c c and value of above limit is denoted by f (c) and is called the derivative of f() at = c. d dv du u v u v du dv v u d u v v If y = f(u), = g(u) then dy f u g u. dy dy du If y = f(u) and u = g(t) then f u. g t Chain Rule dt du dt f () = [] is discontinuous at all integral points and continuous for all R Z. Rolle s theorem : If f () is continuous in [ a, b ], derivable in (a, b) and f (a) = f (b) then there eists atleast one real number c (a, b) such that f (c) = 0. Mean Value Theorem : If f () is continuous in [a, b] and derivable in (a, b) then there eists atleast one real number c (a, b) such that f c f b f a b a f () = log e, ( > 0) is continuous function.. VERY SHORT ANSWER TYPE QUESTIONS ( MARK). For what value of, f() = 7 is not derivable.. Write the set of points of continuity of g() = XII Maths

42 . What is derivative of at =. 4. What are the points of discontinuity of f Write the number of points of discontinuity of f() = [] in [, 7]. 6. The function, f if 4 if if is a continuous function for all R, find. 7. For what value of K, f tan, sin 0 K, 0 is continuous R. 8. Write derivative of sin w.r.t. cos. 9. If f() = g() and g() = 6, g () = find value of f (). 0. Write the derivatives of the following functions : (i) log ( + 5) (ii) e 6 log (iii) e e, log (iv) sec cosec,. (v) 7 sin (vi) log 5, > 0. SHORT ANSWER TYPE QUESTIONS (4 MARKS). Discuss the continuity of following functions at the indicated points. (i), 0 f at 0., 0 XII Maths 4

43 (ii) (iii) sin, 0 g at 0. 0 cos 0 f at (iv) f() = + at =. (v), f at. 0. For what value of k, f k 5, 0 is continuous 0,.. For what values of a and b a if f a b if b if is continuous at =. 4. Prove that f() = + is continuous at =, but not derivable at =. 5. For what value of p, f p sin is derivable at = 0. dy 6. If y tan tan, 0 find. 4 XII Maths

44 7. If y dy sin tan then? dy 8. If y = 5 +y y then prove that If y y a then show that dy y. 0. If y a y then show that dy y. dy y. If ( + y) m + n = m. y n then prove that.. Find the derivative of tan w.r.t. sin.. Find the derivative of log e (sin ) w.r.t. log a (cos ). dy 4. If y + y + = m n, then find the value of. 5. If = a cos, y = a sin then find d y. 6. If = ae t (sint cos t) dy y = ae t (sint + cost) then show that at is If 8. If dy y sin then find. loge dy y loge then find. XII Maths 44

45 9. Differentiate 0. Differentiate w.r.t.. e e w.r.t.. H.O.T.S.. If y sin sin tan where sin sin dy find.. If sin log e y a then show that ( ) y y a y = 0.. If y 5 dy sin,? 4. If sin y = sin (a + y) then show that dy sin a sina y. 5. If y = sin, find d y in terms of y. 6. If a y, b then show that 4 b. d y a y 7. If a + hy + by = then prove that d y h ab. h by 8. If y = a then prove that d y a 5 y. ANSWERS VERY SHORT ANSWER TYPE QUESTIONS ( MARK). = 7/. R. 4. = 6, 7 45 XII Maths

46 5. Points of discontinuity of f() are 4, 5, 6, 7 i.e. four points. Note : At =, f() = [] is continuous. because lim f f k cot (i) log 5 e log (ii) e.log. e (iii) 6 ( ) 5 (iv) 0 (v) 7 7. loge 5 (vi). log e SHORT ANSWER TYPE QUESTIONS (4 MARKS). (i) Discontinuous (ii) Discontinuous (iii) Continuous (iv) continuous (v) Discontinuous. k =. a = 0, b =. 5. p > cot log e a 4. y dy log y y log y y. log y 5. d y cosec sec 4. a 7. dy. XII Maths 46

47 log log log log log. log dy. log log. log e e e. e log. H.O.T.S.. dy. Hint. : sin cos for, sec y tany. 47 XII Maths

48 CHAPTER 6 APPLICATIONS OF DERIVATIVES POINTS TO REMEMBER Rate of Change : Let y = f() be a function then the rate of change of y dy with respect to is given by f another quantity. where a quantity y varies with dy 0 or f If = f(t) and y = g (t) By chain rule 0 represents the rate of change of y w.r.t. at = 0. dy dy if 0. dt dt dt (i) A function f() is said to be non-decreasing on an interval (a, b) if in (a, b) f( ) f ( ), (a, b). Alternatively if f 0 0 a, b, then f() is increasing function in (a, b). (ii) A function f() is said to be non-increasing on an interval (a, b). If in (a, b) f( ) f( ), (a, b). Alternatively if f () 0 (a, b), then f() is decreasing function in (a, b). The equation of tangent at the point ( 0, y 0 ) to a curve y = f() is given by dy y y 0 0 0, y0. XII Maths 48

49 where dy, y 0 0 slope of the tangent of the point, y. 0 0 Slope of the normal to the curve at the point ( 0, y 0 ) is given by dy 0 Equation of the normal to the curve y = f() at a point ( 0, y 0 ) is given by y y dy 0 0 0, y 0. If If dy, y of the normal is = 0., y 0 0 Then the tangent is parallel to -ais at ( 0, y 0 ). Equation dy does not eit, then the normal is parallel to -ais and the equation of the normal is y = y 0. Let y = f() = the small increment in and y be the increment in y corresponding to the increment in Then approimate change in y is given by dy dy or dy = f () The approimate change in the value of f is given by f f f Let f be a function. Let point c be in the domain of the function f at which either f () = 0 or f is not derivable is called a critical point of f. First Derivative Test : Let f be a function defined on an open interval I. Let f be continuous at a critical point c I. Then 49 XII Maths

50 (i) f () changes sign from positive to negative as we pass through c, then c is called the point of the local maima. (ii) (iii) If f () changes sign from negative to positive as we pass through c, then c is a point of local maima. If f () does not change sign as we pass through c, then c is neither a point of local maima nor a point of local minima. Such a point is called a point of infleion. Second Derivative Test : Let f be a functions defined on an interval I and let c I. (i) = c is a point of local maima if f (c) = 0 and f (c) < 0. Then f (c) is the local maimum value of f. (ii) = c is a point of local minima if f (c) = 0 and f"(c) > 0. Then f(c) is the local minimum value of f. (iii) The test fails if f (c) = 0 and f (c) = 0. VERY SHORT ANSWER TYPE QUESTIONS. The side of a square is increasing at a rate of 0. cm/sec. Find the rate of increase of perimeter of the square.. The radius of the circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference?. If the radius of a soap bubble is increasing at the rate of cm sec. what rate its volume is increasing when the radius is cm. At 4. A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 0 cm, how fast is the enclosed area increasing? 5. The total revenue in rupees received from the sale of units of a product is given by R() = Find the marginal revenue when = 7. XII Maths 50

51 6. Find the maimum and minimum values of function f() = sin Find the maimum and minimum values (if any) of the function f() = + 7 R. 8. Find the value of a for which the function f() = a + 6, > 0 is strictly increasing. 9. Write the interval for which the function f() = cos, 0 is decreasing. 0. What is the interval on which the function increasing? f log, 0, is. For which values of, the functions 4 4 y is increasing?. Write the interval for which the function f is strictly decreasing.. Find the sub-interval of the interval (0, /) in which the function f() = sin is increasing. 4. Without using derivatives, find the maimum and minimum value of y = sin If f () = a + cos is strictly increasing on R, find a. 6. Write the interval in which the function f() = is increasing. 7. What is the slope of the tangent to the curve f = 5 + at the point whose co-ordinate is? 8. At what point on the curve y = does the tangent make an angle of 45 with positive direction of the -ais? 9. Find the point on the curve y = + 9 at which the tangent is parallel to -ais. 0. What is the slope of the normal to the curve y = 5 4 sin at = 0.. Find the point on the curve y = + 4 at which the tangent is perpendicular to the line with slope. 6 5 XII Maths

52 . Find the point on the curve y = where the slope of the tangent is equal to the y co-ordinate.. If the curves y = e and y = ae intersect orthogonally (cut at right angles), what is the value of a? 4. Find the slope of the normal to the curve y = 8 at 5. Find the rate of change of the total surface area of a cylinder of radius r and height h with respect to radius when height is equal to the radius of the base of cylinder. 6. Find the rate of change of the area of a circle with respect to its radius. How fast is the area changing w.r.t. its radius when its radius is cm? 7. For the curve y = ( + ) find the rate of change of slope at =. 8. Find the slope of the normal to the curve. 4 = a sin ; y = b cos at 9. If a manufacturer s total cost function is C() = , where is the out put, find the marginal cost for producing 0 units. 0. Find a for which f () = ( + sin ) + a is increasing (strictly). ANSWERS. 0.8 cm/sec cm/sec.. cm /sec cm /sec. 5. Rs Minimum value = 4, maimum value = Maimum value = 7, minimum value does not eist. 8. a (0, ] 0. (0, e].. (, 0) U (0, ). 0,. 6 XII Maths 5

53 4. Maimum value = 4, minimum valve = a >. 6. R , (, ). (, 7). (0, 0), (, 4) r 6. cm /cm 7. 7 a 8.. b 9. Rs a > 0. VERY SHORT ANSWER TYPE QUESTIONS. A particle cover along the curve 6y = +. Find the points on the curve at which the y co-ordinate is changing 8 times as fast as the coordinate.. A ladder 5 metres long is leaning against a wall. The bottom of the ladder is pulled along the ground away from the wall as the rate of cm/sec. How fast is its height on the wall decreasing when the foot of the ladder is 4 metres away from the wall?. A balloon which always remain spherical is being inflated by pumping in 900 cubic cm of a gas per second. Find the rate at which the radius of the balloon increases when the radius is 5 cm. 4. A man meters high walks at a uniform speed of 5 km/hr away from a lamp post 6 metres high. Find the rate at which the length of his shadow increases. 5. Water is running out of a conical funnel at the rate of 5 cm /sec. If the 5 XII Maths

54 radius of the base of the funnel is 0 cm and attitude is 0 cm, find the rate at which the water level is dropping when it is 5 cm from the top. 6. The length of a rectangle is decreasing at the rate of 5 cm/sec and the width y is increasing as the rate of 4 cm/sec when = 8 cm and y = 6 cm. Find the rate of change of (a) Perimeter (b) Area of the rectangle. 7. Sand is pouring from a pipe at the rate of c.c/sec. 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 height is 4 cm? 8. The area of an epanding rectangle is increasing at the rate of 48 cm /sec. The length of the rectangle is always equal to the square of the breadth. At what rate is the length increasing at the instant when the breadth is 4.5 cm? 9. Find a point on the curve y = ( ) where the tangent is parallel to the line joining the points (4, ) and (, 0). 0. Find the equation of all lines having slope zero which are tangents to the curve y.. Prove that the curves = y and y = k cut at right angles if 8k =.. Find the equation of the normal at the point (am, am ) for the curve ay =.. Show that the curves 4 = y and 4y = k cut as right angles if k = Find the equation of the tangent to the curve y which is parallel to the line 4 y + 5 = Find the equation of the tangent to the curve y a at the point a a, Find the points on the curve 4y = where slope of the tangent is 6. XII Maths 54

55 y 7. Show that touches the curve y = be /a at the point where the a b curve crosses the y-ais. 8. Find the equation of the tangent to the curve given by = a sin t, y = b cos t at a point where t. 9. Find the intervals in which the function f() = log ( + ), is increasing or decreasing. 0. Find the intervals in which the function f() = is (a) Increasing (b) Decreasing.. Prove that the function f() = + is neither increasing nor decreasing in [0, ].. Find the intervals on which the function f is decreasing.. Prove that f 9,, is strictly increasing. Hence find the minimum value of f (). 4. Find the intervals on which the function f or decreasing. log, 0, is increasing 5. Find the intervals in which the function f() = sin 4 + cos 4, 0 increasing or decreasing. is 6. Find the least value of a such that the function f() = + a + is strictly increasing on (, ). 7. Find the interval in which the function decreasing. 5 f 5, 0 is strictly 55 XII Maths

56 8. S how that the function f() = tan (sin + cos ), is strictly increasing on the interval 0, Show that the function f cos is strictly increasing on 4 7, Show that the function f sin is strictly decreasing on 0,. Using differentials, find the approimate value of (Q. No. to 7) Find the approimate value of f (5.00) where f() = Find the approimate value of f (.0) where f () = Find the approimate value of f (.998) where f() = ANSWERS. 4, and 4,.. 8 cm sec.. cm sec km/hr cm sec (a) cm/min, (b) cm /min XII Maths 56

57 7. cm sec cm/sec. 9. 7, y.. + my = am ( + m ) y = 5. + y = a ,,, y = 0 9. Increasing in (0, ), decreasing in (, 0). 0. Increasing in (, ) (6, ), Decreasing in (, 6).. (, ) and (, ) Increasing in (0, e] and Decreasing in [e, ). 5. Increasing in, 4 Decreasing in 0, a =. 7. Strictly decreasing in (, ) LONG ANSWER TYPE QUESTION (6 MARKS). Show that of all rectangles inscribed in a given fied circle, the square has the maimum area.. Find two positive numbers and y such that their sum is 5 and the product y 5 is maimum. 57 XII Maths

58 . Show that of all the rectangles of given area, the square has the smallest perimeter. 4. Show that the right circular cone of least curved surface area and given volume has an altitude equal to times the radium of the base. 5. Show that the semi vertical angle of right circular cone of given surface area and maimum volume is sin. 6. A point on the hypotenuse of a triangle is at a distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is a b. 7. Prove that the volume of the largest cone that can be inscribed in a sphere 8 of radius R is of the volume of the sphere Find the interval in which the function f given by f() = sin + cos, 0 is strictly increasing or strictly decreasing. 9. Find the intervals in which the function f() = ( + ) ( ) is strictly increasing or strictly decreasing. 0. Find the local maimum and local minimum of f() = sin,.. Find the intervals in which the function f() = is strictly increasing or decreasing. Also find the points on which the tangents are parallel to -ais.. A solid is formed by a cylinder of radius r and height h together with two hemisphere of radius r attached at each end. It the volume of the solid is constant but radius r is increasing at the rate of metre min. must h (height) be changing when r and h are 0 metres. How fast. Find the equation of the normal to the curve = a (cos + sin ) ; y = a (sin cos ) at the point and show that its distance from the origin is a. XII Maths 58

59 4. For the curve y = 4 5, find all the points at which the tangent passes through the origin. 5. Find the equation of the normal to the curve = 4y which passes through the point (, ). 6. Find the equation of the tangents at the points where the curve y = 8 cuts the -ais and show that they make supplementary angles with the -ais. 7. Find the equations of the tangent and normal to the hyperbola a at the point ( 0, y 0 ). 8. A window is in the form of a rectangle surmounted by an equilateral triangle. Given that the perimeter is 6 metres. Find the width of the window in order that the maimum amount of light may be admitted. 9. A jet of an enemy is flying along the curve y = +. A soldier is placed at the point (, ). What is the nearest distance between the soldier and the jet? 0. Find a point on the parabola y = 4 which is nearest to the point (, 8).. A square piece of tin of side 8 cm is to be made into a bo without top by cutting a square from each cover and folding up the flaps to form the bo. What should be the side of the square to be cut off so that the valume of the bo is the maimum.. A window in the form of a rectangle is surmounted by a semi circular opening. The total perimeter of the window is 0 metres. Find the dimensions of the rectangular part of the window to admit maimum light through the whole opening.. An open bo with square base is to be made out of a given iron sheet of area 7 sq. meter, show that the maimum value of the bo is.5 cubic metres. 4. A wire of length 8 cm is to be cut into two pieces. One of the two pieces is to be made into a square and other in to a circle. What should be the length of two pieces so that the combined area of the square and the circle is minimum? 5. Show that the height of the cylinder of maimum volume which can be inscribed in a sphere of radius R is R. Also find the maimum volume. 59 XII Maths y b

60 6. Show that the altitude of the right circular cone of maimum volume that can be inscribed is a sphere of radius r is 4 r. 7. Prove that the surface area of solid cuboid of a square base and given volume is minimum, when it is a cube. 8. Show that the volume of the greatest cylinder which can be inscribed in a right circular cone of height h and semi-vertical angle is 4 7 h tan. 9. Show that the right triangle of maimum area that can be inscribed in a circle is an isosceles triangle. 0. A given quantity of metal is to be cast half cylinder with a rectangular bo and semicircular ends. Show that the total surface area is minimum when the ratio of the length of cylinder to the diameter of its semicircular ends is : ( + ).. 5, 0 ANSWERS 8. Strictly increasing in Strictly decreasing is 5 0,, 4 4 5, Strictly increasing in (, ) (, ) Strictly decreasing in (, ) (, ). 0. Local maima at Local ma. value 6 6 Local minima at 6 XII Maths 60

61 Local minimum value 6. Strictly increasing in (, ] [, ) Strictly decreasing in (, ). Points are (, 9) and (, 8).. metres min.. + y tan a sec = (0, 0), (, ) and (, ) y = 6. 5 y 0 = 0 and 5 + y + 0 = 0 7. a yy 0 0, b y y b 0 a y (4, 4). cm. 60 0, cm, cm XII Maths

62 CHAPTER 7 INTEGRALS POINTS TO REMEMBER Integration is the reverse process of Differentiation. Let d F f then we write f F c. These integrals are called indefinite integrals and c is called constant of integrations. From geometrical point of view an indefinite integral is collection of family of curves each of which is obtained by translating one of the curves parallel to itself upwards or downwards along with y-ais. STANDARD FORMULAE. n n c n n log c n. n a b n a b c n n a log a b c n a. sin cos c. 4. cos sin c. 5. tan. log cos c log sec c. XII Maths 6

63 6. cot log sin c. 7. sec. tan c. 8. cosec. cot c. 9. sec. tan. sec c. 0. cosec cot cosec c.. sec log sec tan c.. cosec log cosec cot c.. e e c. 4. a a loga c 5. sin c,. 6. tan c. 7. sec c,. 8. a a log c. a a a a log c. a a a tan c. a a 6 XII Maths

64 . a sin c. a. a log a c.. a log a c. 4. a a a sin c. a 5. a a a log a c. 6. a a a log a c. RULES OF INTEGRATION. k. f k f.. k f g k f k g. INTEGRATION BY SUBSTITUTION.. f log f c. f n n f f f c. n XII Maths 64

65 . n f f n f n c. INTEGRATION BY PARTS f. g f. g f. g. b a DEFINITE INTEGRALS f F b F a, where F f. DEFINITE INTEGRAL AS A LIMIT OF SUMS. b a f lim h h 0 f a f a h f a h... f a n h where b n b a h. or f lim h f a rh h h 0 a r PROPERTIES OF DEFINITE INTEGRAL b a. f f.. f f t dt. b b a b a a b c b. f f f. a a c b b 4. (i) f f a b. (ii) a a f f a. a a XII Maths

66 5. a a f 0; if f is odd function. 6. a a a f f, if f() is even function. 0 a f a f, if f a f 0, if f a f VERY SHORT ANSWER TYPE QUESTIONS ( MARK) Evaluate the following integrals. sin cos.. e.. sin cos. 6. log log log sin log. 4 cos 8. a log loga e e. 9. cos sin. cos 0. 7 sin. XII Maths 66

67 c d. c.. f.. sin. cos loge e e. 6.. a 7. e e 0... cos.. cos.. sec.log sec tan. 4. cos sin. 5. cot.logsin log. 8. sin. cos 9. cos. sin 0. e e e e.. log.. a. a. cos XII Maths

68 4. 0 where [ ] is greatest integer function where [ ] is greatest integer function. b f 6.. a f f a b a 9. If 0, then what is value of a. 4 b 40. f f. a a b SHORT ANSWER TYPE QUESTIONS (4 MARKS) 4. (i) cosec tan 4. (ii). (iii) sin a sin b. (iv) cos cos a. a (v) cos cos cos. (vi) 5 cos. (vii) 4 sin cos. (viii) 4 cot cosec. (i) sin cos a sin b cos. () cos cos a. (i) 6 6 sin cos. sin cos (ii) sin cos. sin XII Maths 68

69 4. Evaluate : (i) 4. *(ii) 6 log 7 log. (iii). (iv) 9 8. (v) a b. (vi) sin sin. (vii) 5. (viii). 6 (i) 4. (). (ii). (iii) sec. 4. Evaluate : (i) 7. (ii) sin cos cos. (iii) sin cos d cos cos. 69 XII Maths

70 (iv). (v). (vi) 4. (vii) 4. (viii). sin cos (i) sin. sin 4 () 4. (i) tan. (ii) Evaluate : (i) 5 sin. (ii) sec. a (iii) e cos b c. (iv) 6 sin. 9 (v) cos. (vi) tan. (vii) sin e cos. (viii) e. (i) e. () e. (i) e sin cos. (ii) log log. log XII Maths 70

71 45. Evaluate the following definite integrals : (i) 4 0 sin cos. 9 6 sin (ii) 0 cos log sin. (iii) 0. (iv) 0 sin. (v) 0 sin sin 4 4 cos. (vi) 5. 4 (vii) 0 sin cos. 46. Evaluate : (i). (ii) 0 sin. (iii) 4 log tan. (iv) log sin. 0 0 (v) 0 sin cos. (vi) when f where f when when. 7 XII Maths

72 (vii) 0 sin cos sin 4 4 cos. (viii) 0 a cos b sin. 47. Evaluate the following integrals (i). (ii) 0 sin. (iii) sin log. sin (iv) 0 e e cos cos e cos. LONG ANSWER TYPE QUESTIONS (6 MARKS) 48. Evaluate the following integrals : (i) (ii) 4 (iii) (iv) (v) 0 tan cot. (vi) 4. (vii) 0 tan. XII Maths 7

73 49. Evaluate the following integrals as limit of sums : (i) 4. (ii). 0 (iii) 4. (iv) 4 e. 0 (v) 5. H.O.T.S. VERY SHORT ANSWER TYPE QUESTIONS ( MARK EACH) where [ ] is greatest integer function. 5. log log sin e. 5. e log log. 5. sin. sin 54. sin sin b a sin. 56. f f a b. a b 57. sec tan. 58. sin. cos 59. tan. tan a b 60.. c 7 XII Maths

74 SHORT ANSWER TYPE QUESTIONS (4 MARKS) 6. Evaluate (i) sin sin cos, 0, cos (ii) (iii) log log 4 (iv) sin cos (v) sin a (vi) sin cos sin 6 (vii) sin cos (viii), where [] is greatest integer function (i) sin. XII Maths 74

75 LONG ANSWER TYPE QUESTIONS (6 MARKS) 6. Evaluate (i) 0 cot (ii) sin cos sin cos (iii) 0 log (iv) 0 log sin log sin. 6. sin sin. sin cos cos 4sin d. 65. sec. 66. e cos. ANSWERS. c.. e. tan + c log c. log log log (log ) + c a a a loga c 9. tan + c XII Maths

76 . c c c logc c. f() + c. tan cot + c 4. c e 5. log + c 6. log a e a c 7. e log e c 8.. c 9. log c. 0. e c. cos + c. log cos cos c.. log sec tan c 4. log cos sin sin c 5. logsin c log c. 7. log log c. 8. log + cos c 9. log sec / + c. 0. log e e c. e. log c. log a a c. a XII Maths 76