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 [XII Maths]
multiplication 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. d d d d d,,,, a a b c a a a b c 4 [XII Maths]
p q p q d, d, a d, a d, a b c a b c a b c d and p q a b c d 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 d 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. Scalar triple 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 5 [XII Maths]
(i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane. 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. 6 [XII Maths]
CONTENTS S.No. Chapter Page. Relations and Functions 9. Inverse Trigonometric Functions 7 & 4. Matrices and Determinants 5. Continuity and Differentiation 9 6. Applications of Derivatives 47 7. Integrals 6 8. Applications of Integrals 85 9. Differential Equations 90 0. Vectors 0. Three-Dimensional Geometry. Linear Programming. Probability 7 Model Papers 4 7 [XII Maths]
8 [XII Maths]
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 number 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 have distinct images in B. i.e., A such that f( ) f( ). OR, A such that f( ) = f( ) = One-one function is also called injective function. 9 [XII Maths]
Onto function (surjective) : A function f : A B is said to be onto iff R f = B i.e. b B, there eists a A such that 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 domain of g. Similarly fog can be defined. 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. 0 [XII Maths]
VERY SHORT ANSWER TYPE QUESTIONS ( MARK). 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}. 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 element (5, 7) R? 4. If f : {, } {,, 5} and g : {,, 5} {,,, 4} be given by f = {(, ), (, 5)}, g = {(, ), (, ), (5, )}, write 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 * be a Binary operation defined on R, then if (i) a * b = a + b + ab, write * [XII Maths]
(ii) a b a * b, write * * 4. 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() = 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() = one-one? Give reason 7. f : R B given by f() = sin is onto function, then write set B. 8. If f log, show that f f. ab 9. If * is a binary operation on set Q of rational numbers given by a * b 5 then write the identity element in Q. 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() = + (c) f : R {} R, f [XII Maths]
(d) f : R [, ], f() = sin. Consider the binary operation * on the set {,,, 4, 5} defined by a* b = H.C.F. of a and b. Write the operation table for the operation *. 4 4. Let : 4 f R R be a function given by f. 4 that f is invertible with 4 f. 4 Show 4. Let R be the relation on set A = { : Z, 0 0} given by R = {(a, b) : (a b) is divisible by 4}, is an equivalence relation. Also, write all elements related to 4. 4 5. Show that function f : A B defined as f 5 7 7 A R, B R is invertible and hence find f. 5 5 6. 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). 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 a * b ab, show that 4 (i) 4 is the identity element of * on Q. [XII Maths]
(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. Let f : N R be a function defined as f() = 4 + + 5. Show that f : N Range of f is invertible. Find the inverse of 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. f : [, ) [, ) is given by f, find f. 6. f : R R, g : R R given by f() = [], g() = then find and. fog gof. R : is universal relation. R : is empty relation. ANSWERS R : is neither universal nor empty.. No, R is not refleive.. (5, 7) R 4. gof = {(, ), (, )} 5. (fog)() = R 5 6. f 4 [XII Maths]
7. fof, 8. (i) * = (ii) 69 7 9. 6 0.. Yes, f is one-one, N.. A = [, ) because R f = [, ). n(b) = 0 4. 0. 5. No, R is not refleive a, a R a N 6. f is not one-one function f() = f ( ) = i.e. distinct elements have same images. 7. B = [, ] 9. e = 5 0. Identity element does not eists.. (a) Bijective (b) (c) (d) Neither one-one nor onto. One-one, but not onto. Neither one-one nor onto. 5 [XII Maths]
. * 4 5 4 4 5 5 4. Elements related to 4 are 0, 4, 8. 5. 7 4 f 5 6. 0 is the identity element. 7. Neither commutative nor associative. 8. (i) Commutative and associative. (ii) (, ) is identity in N N. f 6. 0 is the identity element. 4. (fog) () = + (gof) () = + Clearly, they are unequal. 4 5. f 6. fog 0 gof 6 [XII Maths]
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. 7 [XII Maths]
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 y tan tan y tan ; y. y y tan tan y tan ; y. tan tan, tan sin,, tan cos, 0. y 8 [XII Maths]
VERY SHORT ANSWER TYPE QUESTIONS ( MARK). Write the principal value of (i) sin (ii) cos. (iii) tan (iv) cosec ( ). (v) cot. (vi) sec ( ). (vii) sin cos tan. What is the value of the following functions (using principal value). (i) tan sec. (ii) sin cos. (iii) tan () cot ( ). (iv) (v) tan () + cot () + sin (). cosec sec. (vi) 4 sin sin. 5 (vii) 5 tan tan. 6 (viii) cosec cosec. 4 SHORT ANSWER TYPE QUESTIONS (4 MARKS). Show that cos cos tan. cos cos 4 [0, ] 9 [XII Maths]
4. Prove that tan cot 0,. sin cos 4 cos cos 5. Prove that 6. Prove that a tan sin cos. a a a 8 8 00 cot tan cos tan tan sin tan. 7 7 6 7. Prove that 8. Solve : 9. Prove that tan cos. 4 cot cot. 4 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 5 7 8 4. Solve for, tan cos sin tan ; 0 4. Prove that tan tan tan 5 4 4 0 [XII Maths]
5. Evaluate tan cos 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 cos + cos y + cos z =, prove that + y + z + yz = [Hint : Let cos = A, cos y = B, cos z = c then A + B + C = or A + B = c Take cos on both the sides]. ANSWERS. (i) (ii) 6 (iii) 6 (iv) 6 (v) (vi) (vii). 6. (i) 0 (ii) (iii) (iv) (v) (vi) 5 (vii) 6 (viii). 4 [XII Maths]
8.. tan. 5 5. 9.. 0 sin sin. 4.,. Hint: Let tan a = tan b = tan c = then given, take tangent on both sides, tan ( ) = tan [XII Maths]
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. Square Matri : An mn matri is said to be a square matri of order n if m = n. Column Matri : A matri having only one column is called a column matri i.e. A = [aij] m is a column matri of order m. Row Matri : A matri having only one row is called a row matri B bij is a row matri of order n. i.e. n 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 [XII Maths]
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 matri. 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 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 4 [XII Maths]
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) [Note : adj (AB) = (adj B) (adj A). Correctness of adj A can be checked by using A.(adj A) = (adj A). A = A I ] Singular Matri : A square matri is called singular if A = 0, otherwise it will be called a non-singular matri. 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) (ii) (iii) If A 0, system is consistent and has a unique solution. If A = 0 and (adj A) B 0 then system is inconsistent and has no solution. If A = 0 and (adj A) B = 0 then system is either consistent and has infinitely many solutions or system is inconsistent and has no solution. 5 [XII Maths]
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 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. 0 9 0 0 6. If A and B find A B. 7 5 6 7. If A = [ 0 4] and B 5 find AB. 6 4 8. If A is symmetric matri, then find. 0 9. For what value of the matri 0 4 is skew symmetri matri. 4 5 0. If A P Q 0 where P is symmetric and Q is skew-symmetric matri, then find the matri Q. 6 [XII Maths]
. Find the value of a ib c id c id a ib. If 5 0, find. 5 9. For what value of k, the matri k 4 has no inverse. 4. If A sin 0 cos 0, sin 60 cos 60 what is A. 5. Find the cofactor of a in 6. Find the minor of a in 5 6 0 4. 5 7 4 5 6. 5 7. 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 number of all possible matrices of order with each entry 0, or.. Find the area of the triangle with vertices (0, 0), (6, 0) and (4, ). 4. If 4 6, find. 7 [XII Maths]
y y z z 5. If A z y, write the value of det A. 6. Write the value of the following determinant 4 5 6 8 6 9 7. If A is a non-singular matri of order and A = find adj A. 5 find 6 8 8. If A adj A 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. A find A. 4. If. If A and B 4 0 find AB. 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 8 [XII Maths]
6. Find A and B if A + B = 0 and A B 0 6. and 4, 7. If A B verify that (AB) = B A. 8. Epress the matri P Q 4 5 a skew-symmetric matri. cos sin 9. If A =, sin cos then prove that where n is a natural number. 5 5 40. Let A, B, C, 4 7 4 8 CD AB = O. where P is a symmetric and Q n cosn sinn A sinn cosn find a matri D such that 4. Find the value of such that 4. Prove that the product of the matrices 5 0 5 cos cos sin cos cos sin and cos sin sin cos sin sin is the null matri, when and differ by an odd multiple of. 4. If A 5 7, show that A A I = 0. Hence find A. 9 [XII Maths]
44. Show that find A. A 4 satisfies the equation 6 + 7 = 0. Hence 45. If A 4, 5 find and y such that A A + yi = 0. 46. Find the matri X so that 7 8 9 X. 4 5 6 4 6 47. If A and B 4 then show that (AB) = B A. 48. Test the consistency of the following system of equations by matri method : y = 5; 6 y = 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. 5 cos sin 5. If A sin cos and A + A = I, then find the general value of. Using properties of determinants, prove the following : Q 5 to Q 59. a b c a a 5. b b c a b a b c c c c a b 5. 54. 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 0 [XII Maths]
55. 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 a b c 58. 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 verify that AB A B. 5 6. Given A also find (AB). 0 0 and B 0. 0 Find the product AB and 6. Solve the following equation for. a a a a a a a a a 0. [XII Maths]
0 tan 6. If A and I is the identity matri of order, show tan 0 that, I cos sin A I A sin cos 64. 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 operations 4 4 0. 7 66. If 0 4 A 4 and 4 4 B are two square matrices, find 0 5 AB and hence solve the system of linear equations : y =, + y + 4z = 7, y + z = 7. 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. F ind A, where equations : A, hence solve the system of linear 4 + y z = 4 + y + z = y 4z = [XII Maths]
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 5 6 5 5 5 and verify that A A = I. 7. If the matri A compute (AB). 0 0 and B 0, 4 0 then 7. Using matri method, solve the following system of linear equations : y = 4, y + z = 5, z + = 7. 7. Find A 0 if A 0. 0 Also show that A A I. 74. Show that b c ab ac ba c a bc 4a b c ca cb a b a b c c b a c b c a 75. Show that a b c a b c a b b a c 76. If cos sin 0 A sin cos 0, 0 0 verify that A. (adj A) = (adj A). A = A I. [XII Maths]
77. For the matri find A. A, verify that A 6A + 9A 4I = 0, hence 78. Find the matri X for which. X. 7 5 0 4 79. By using properties of determinants prove the following : a b ab b ab a b a a b. b a a b 80. 8. 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 show that yz =. 8. 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 4 [XII Maths]
84. Using the properties of determinants, show that : a b abc abc bc ca ab a b c c 85. Using properties of determinants prove that bc b bc c bc a ac ac c ac ab bc ca a ab b ab ab 86. If A 4, find A and hence solve the system of equations 7 + 4y + 7z = 4, y + z = 4, + y z = 0. ANSWERS. =, y = 7. 0 0.. 4. 4 5. 9 6 0 9. 6. 5. 7. AB = [6]. 8. = 5 9. = 5 0. 0. 0. a + b + c + d.. =. k 4. A =. 5. 46 6. 4 5 [XII Maths]
7. P = 8 8. 5. 9. 7 k,. 0. 54.. 40.. 79. 9 sq. units 4. = ± 5. 0 6. 0 7. 9 8. 8. 6 5 9. 78 0. A = ± 9. 00.. AB = 4. =, y =, z =, w = 0 5. 5 4 5. 5 6 7 6. 8 9 5 7 7 7 7 7 7 A, B 8 4 4 5 7 7 7 7 7 7 40. D 9 0. 77 44 4. = or 4 4. A 7. 5 44. 4 A. 7 6 [XII Maths]
45. = 9, y = 4 46.. 0 48. Inconsistent 49. Inverse does not eist. 50. A. 5 5. n, n z 6. AB, AB. 6 6 0, a 64., y. 4 4 65. A 6. 66. =, y =, z = 4 4 67., y, z 5 68. 6 7 A 4 5 8,,, 67 y z 5 9 69. =, y =, z = 70. A 0 5 0 0 6 7. AB 7. 7. =, y =, z =. 9 0 7 [XII Maths]
7. A. 77. A. 4 78. 6 X. 4 5 8. 0 86. =, y =, z =. 8 [XII Maths]
CHAPTER 5 CONTINUITY AND DIFFERENTIATION POINTS TO REMEMBER A function f() is said to be continuous at = c iff lim f f c 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 f b b,. Trigonometric functions are continuous in their respective 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 in its domain iff 9 [XII Maths]
f f c f f c lim lim, and is finite c c c c The value of above limit is denoted by f (c) and is called the derivative of f() at = c. d u v u dv v du d d d du dv v u d v v d u d d dy dy du If y = f(u) and u = g(t) then f u. g t Chain Rule If y = f(u), = g(u) then, dt du dt d d dy dy du f u. d du d g u d sin, cos d d d d d d tan, cot d d sec, cosec d d d e e, log d d 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. 40 [XII Maths]
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 b f a f c. 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() = + +.. What is derivative of at =. 4. What are the points of discontinuity of f. 7 6 5. Write the number of points of discontinuity of f() = [] in [, 7]. if 6. The function, f 4 if if R, find. is a continuous function for all tan, 0 7. For what value of K, f sin 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 derivative of the following functions : (i) log ( + 5) (ii) e log (iii) 6 loge e, 4 [XII Maths]
(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) f, 0, 0 at 0. (ii) g sin, 0 0 at 0. (iii) cos f 0 at 0. 0 0 (iv) f() = + at =., (v) f at. 0. For what value of k, f 0,. k 5, 0 is continuous. For what values of a and b a if f a b if b if is continuous at =. 4 [XII Maths]
4. Prove that f() = + is continuous at =, but not derivable at =. 5. For what value of p, p f sin 0 0 0 is derivable at = 0. 6. If y dy tan tan, 0, find. d 7. If y dy sin tan then? d dy 8. If 5 + 5 y = 5 +y y then prove that 5 0. d 9. If y y a then show that dy d y. 0. If y a y then show that dy d y. dy y. If ( + y) m + n = m. y n then prove that. d. 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. d 5. If = a cos, y = a sin then find d y d. 4 [XII Maths]
6. If = ae t (sint cos t) dy y = ae t (sint + cost) then show that at is. d 4 7. If dy y sin then find. d dy e then find. d log 8. If e y log 9. Differentiate w.r.t.. dy d y 0. Find, if cos cos y. If y sin sin tan where sin sin dy find. d Hint : sin cos for,.. If sin log e y a then show that ( ) y y a y = 0.. Differentiate log log, w. r. t. 4. If sin y = sin (a + y) then show that dy d sin a y sina. 5. If y = sin, find d y d in terms of y. 6. If a y d y then show that b d, 4 b. a y cos 7. If a d y dy y e,, show that a y 0 d d 8. If y = a then prove that d y a 5 d y. 44 [XII Maths]
9. Verify Rolle's theorem for the function, y = + in the interval [a, b] where a =, b =. 40. Verify Mean Value Theorem for the function, f() = in [, 4] ANSWERS. 7.. R. 4. = 6, 7 5. Points of discontinuity of f() are 4, 5, 6, 7 i.e. four points. 6. Note : At =, f() = [] is continuous. because lim f f. 7. 7. 8. cot 9. 5 k. 4 0. (i) log 5 e log (ii) e.log. e (iii) 6 ( ) 5 (iv) 0 (v) 7 7. (vi) loge 5. log. (i) Discontinuous (ii) Discontinuous (iii) Continuous (iv) continuous e (v) Discontinuous. k =. a = 0, b =. 5. p >. 6. 0 7.... cot log e a 45 [XII Maths]
4. y dy log y y log y y. d log y 5. 7. d y 4 cosec sec. d a dy d. log log log log log. log 8. 9. 0.. dy. log log. d log dy tan logcos y y d tany logcos dy. d. log log log log, 5. sec y tany. 46 [XII Maths]
CHAPTER 6 APPLICATIONS OF DERIVATIVES POINTS TO REMEMBER Rate of Change : Let y = f () be a function then the rate of change of dy y with respect to is given by f where a quantity y varies with d another quantity. dy d 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 d d if 0. d dt dt dt (i) A function f () is said to be increasing (non-decreasing) on an interval (a, b) if in (a, b) f ( ) f ( ), (a, b). Alternatively if f a b function in (a, b). 0,, then f () is increasing (ii) A function f() is said to be decreasing (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 y0 0. d, y 0 0 47 [XII Maths]
where (i) dy d, y 0 0 slope of the tangent at the point 0, y 0. dy If does not eist then tangent is parallel to y-ais at d 0, y 0 ( 0, y 0 ) and its equation is = 0. (ii) If tangent at = 0 is parallel to -ais then dy d 0 0 Slope of the normal to the curve at the point ( 0, y 0 ) is given by dy d 0. Equation of the normal to the curve y = f () at a point ( 0, y 0 ) is given by y y dy d, y 0 0 0 0. If dy d, y 0 0 0. then equation of the normal is = 0. If dy d, y 0 0 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 d or dy = f () The approimate change in the value of f is given by f f f 48 [XII Maths]
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 if, (i) (ii) (iii) f () changes sign from positive to negative as increases through c, then c is called the point of the local maima. f () changes sign from negative to positive as increases through c, then c is a point of local minima. f () does not change sign as increases 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 function defined on an interval I and let c I. Then (i) = c is a point of local maima if f (c) = 0 and f (c) < 0. f (c) is local maimum value of f. (ii) = c is a point of local minima if f (c) = 0 and f "(c) > 0. f (c) is local minimum value of f. (iii) The test fails if f (c) = 0 and f (c) = 0. VERY SHORT ANSWER TYPE QUESTIONS ( MARK). The side of a square is increasing at the 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? 49 [XII Maths]
5. T he totalrevenue in rupees received from the sale of units of a product is given by R() = + 6 + 5. Find the marginal revenue when = 7. 6. Find the maimum and minimum values of function f () = sin + 5. 7. 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. log, 0, is 0. What is the interval on which the function f increasing?. 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 +. 5. If f () = a + cos is strictly increasing on R, find a. 6. Write the interval in which the function f () = 9 + 7 + 64 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. 50 [XII Maths]
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. 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. 4 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 of the tangent at =. 8. Find the slope of the normal to the curve = a sin ; y = b cos at 9. If a manufacturer s total cost function is C() = 000 + 40 +, where is the out put, find the marginal cost for producing 0 units. 0. Find a for which f () = a ( + sin ) is strictly increasing on R. SHORT ANSWER TYPE QUESTIONS (4 MARKS). A particle moves 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 at 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? 5 [XII Maths]
. 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 metres high walks at a uniform speed of 6 metres per minute away from a lamp post 5 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 radius of the base of the funnel is 0 cm and altitude 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 cm/sec and the width y is increasing as the rate of cm/sec when = cm and y = 5 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). 40. Find the equation of all lines having slope zero which are tangents to the curve y. 4. Prove that the curves = y and y = k cut at right angles if 8k =. 4. Find the equation of the normal at the point (am, am ) for the curve ay =. 4. Show that the curves 4 = y and 4y = k cut as right angles if k = 5. 44. Find the equation of the tangent to the curve y which is parallel to the line 4 y + 5 = 0. 5 [XII Maths]
45. Find the equation of the tangent to the curve y a at the point a a,. 4 4 46. Find the points on the curve 4y = where slope of the tangent is 6. y 47. Show that touches the curve y = be /a at the point where the a b curve crosses the y-ais. 48. Find the equation of the tangent to the curve given by = cos, y = sin at a point where. 4 49. Find the intervals in which the function f () = log ( + ), is increasing or decreasing. 50. Find the intervals in which the function f () = + 6 + 7 is (a) Increasing (b) Decreasing. 5. Prove that the function f () = + is neither increasing nor decreasing in [0, ]. 5. Find the intervals on which the function f is decreasing. 5. Prove that 9,, f is strictly increasing. Hence find the minimum value of f (). 54. Find the intervals in which the function f () = sin 4 + cos 4, 0 is increasing or decreasing. 55. Find the least value of 'a' such that the function f () = + a + is strictly increasing on (, ). 5 [XII Maths]
5 56. Find the interval in which the function f 5, 0 is strictly decreasing. 57. Show that the function f () = tan (sin + cos ), is strictly increasing on the interval 0,. 4 58. Show that the function f cos 4 is strictly increasing on 7,. 8 8 59. sin Show that the function f is strictly decreasing on 0,. Using differentials, find the approimate value of (Q. No. 60 to 64). 60. 0.009. 6. 80 4. 6. 0.007. 6. 0.07. 64. 5.0. 65. Find the approimate value of f (5.00) where f() = 7 + 5. 66. Find the approimate value of f (.0) where f () = + 5 +. LONG ANSWER TYPE QUESTIONS (6 MARKS) 67. Show that of all rectangles inscribed in a given fied circle, the square has the maimum area. 68. Find two positive numbers and y such that their sum is 5 and the product y 5 is maimum. 69. Show that of all the rectangles of given area, the square has the smallest perimeter. 70. Show that the right circular cone of least curved surface area and given volume has an altitude equal to times the radius of the base. 54 [XII Maths]
7. Show that the semi vertical angle of right circular cone of given surface area and maimum volume is sin. 7. 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 8 sphere of radius R is of the volume of the sphere. 7 74. Find the interval in which the function f given by f () = sin + cos, 0 is strictly increasing or strictly decreasing. 75. Find the intervals in which the function f () = ( + ) ( ) is strictly increasing or strictly decreasing. 76. Find the local maimum and local minimum of f () = sin,. 77. Find the intervals in which the function f () = 5 + 6 + is strictly increasing or decreasing. Also find the points on which the tangents are parallel to -ais. 78. 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. fast must h (height) be changing when r and h are 0 metres. How 79. 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. 80. For the curve y = 4 5, find all the points at which the tangent passes through the origin. 8. Find the equation of the normal to the curve = 4y which passes through the point (, ). 55 [XII Maths]
8. 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. 8. Find the equations of the tangent and normal to the hyperbola a at the point ( 0, y 0 ). y 84. 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. 85. 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? 86. Find a point on the parabola y = 4 which is nearest to the point (, 8). 87. A square piece of tin of side 4 cm is to be made into a bo without top by cutting a square from each corner and folding up the flaps to form the bo. What should be the side of the square to be cut off so that the volume of the bo is the maimum. 88. 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. 89. 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. 90. A wire of length 6 m is to be cut into two pieces. One of the two pieces is to be made into a square and other into a circle. What should be the length of two pieces so that the combined area of the square and the circle is minimum? 9. 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. 9. 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. 56 [XII Maths] b
9. Prove that the surface area of solid cuboid of a square base and given volume is minimum, when it is a cube. 94. 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. 95. Show that the right triangle of maimum area that can be inscribed in a circle is an isosceles triangle. 96. 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 : ( + ). ANSWERS. 0.8 cm/sec.. 4.4 cm/sec.. cm /sec. 4. 80 cm /sec. 5. Rs. 08. 6. Minimum value = 4, maimum value = 6. 7. Maimum value = 7, minimum value does not eist. 8. a 0. 9. [0, ] 0. (0, e].. (, 0) U (0, ). 0,. 6 4. Maimum value = 4, minimum valve = 0. 5. a >. 6. R 7. 7 8. 0.,. 4 4 9. (, ). (, 7) 57 [XII Maths]
. (0, 0), (, 4).. 4.. 4 5. 6r 6. r cm /cm, 6 cm /cm 7. 7 8. a. b 9. Rs. 80. 0. a > 0.. 4, and 4,.. 8 cm sec.. cm sec. 4. 4 metres/minute 5. 4 cm sec. 45 6. (a) 0 cm/sec., (b) 4 cm /sec. 7. cm sec. 48 8. 7. cm/sec. 9. 7,. 4 40. y. 4. + my = am ( + m ) 44. 48 4y = 45. + y = a 46. 8 8 8 8,,,. 7 7. 4 48. y 49. Increasing in (0, ), decreasing in (, 0). 50. Increasing in (, ) (6, ), Decreasing in (, 6). 58 [XII Maths]
5. (, ) and (, ). 5. 5. 54. Increasing in, 4 Decreasing in 0, 4. 55. a =. 56. Strictly decreasing in (, ). 60. 0.08 6..9907 6. 0.0608 6. 0.95 64. 5.00 65. 4.995 66. 45.46 68. 5, 0 74. Strictly increasing in Strictly decreasing in 5 0,, 4 4 5,. 4 4 75. Strictly increasing in (, ) (, ) Strictly decreasing in (, ) (, ). 76. Local maima at Local ma. value 6 6 Local minima at 6 Local minimum value 6 77. Strictly increasing in (, ] [, ) Strictly decreasing in (, ). 59 [XII Maths]
Points are (, 9) and (, 8). 78. metres min. 79. + y tan a sec = 0. 80. (0, 0), (, ) and (, ). 8. + y = 8. 5 y 0 = 0 and 5 + y + 0 = 0 8. a yy 0 0, b y y 0 0 0 b 0 a y 0. 84. 6 6 85. 5 86. (4, 4) 87. 4cm 88. 60 0,. 4 4 90. 44 6 m, m. 4 4 9. 4R 60 [XII Maths]
CHAPTER 7 INTEGRALS POINTS TO REMEMBER Integration is the reverse process of Differentiation. d Let F f d then we write f d F c. These integrals are called indefinite integrals and c is called constant of integration. 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 y-ais. STANDARD FORMULAE. n d n c n n log c n n a b c n n. a b d n a log a b c n a. sin d cos c. 4. cos d sin c. 5. tan. d log cos c log sec c. 6 [XII Maths]
6. cot d log sin c. 7. sec. d tan c. 8. cosec. d cot c. 9. sec. tan. d sec c. 0. cosec cot d cosec c.. sec d log sec tan c.. cosec d log cosec cot c. a. e d e c. 4. a d c loga 5. d c sin,. 6. d tan c. 7. d sec c,. 8. a a d log c. a a 9. a d log c. a a a 0. a d tan c. a a 6 [XII Maths]
. a d sin c. a. a d log a c.. a d log a c. 4. a a d a sin c. a 5. a a d a log a c. 6. a a d a log a c. RULES OF INTEGRATION. k. f d k f d... k f g d k f d k g d INTEGRATION BY SUBSTITUTION. log. f d f c f n f. f f d c. n n 6 [XII Maths]
. f f n d n f n c. INTEGRATION BY PARTS b a b a f. g d f. g d f. g d d. DEFINITE INTEGRALS f d F b F a, where F f d. DEFINITE INTEGRAL AS A LIMIT OF SUMS. f a f a h f a h f d lim h h 0... f a n h b a b. or lim where h f d h f a rh h a h 0 n r PROPERTIES OF DEFINITE INTEGRAL b. f d f d. a a. f d f t dt. b b a b a b c b. f d f d f d. a a c b b. 4. (i) f d f a b d (ii) f d f a d. a a a a 0 0 64 [XII Maths]
a 5. f 0; if f is odd function. a a 6. f d f d, a a if f() is even function. 0 7. a 0 f d a f d, if f a f 0 0, if f a f VERY SHORT ANSWER TYPE QUESTIONS ( MARK) Evaluate the following integrals sin cos d... e d.. d. sin 4. 8 8 8 d. 8 5. 99 4 cos d. 6. log log log d. 7. 4 sin 4 cos log d. 8. a log loga 0 e e d. 9. cos sin. d 0. cos 7 sin d.. d d 0 4 d.. f d. 65 [XII Maths]
. d. 4. sin cos d. 5. log e e e d. 6.. d a 7. e d. 8. d. 9. d. e 0. d.. cos d.. cos d.. sec.logsec tan d. 4. cos sin d. 5. cot.logsin d. 6. d. 7. d. 8. log sin d. cos 9. cos d. 0. sin e e e e d.. log d.. a d. a. cos d. 0 4. 0 d where [ ] is greatest integer function. 66 [XII Maths]
5. d where [ ] is greatest integer function. 0 6. 8. b f d. 7. a f f a b d. d. a 9. If 0, then what is value of a. 4 b 40. f d f d. a a 4. b e log log d. 4. sin d. 4. sin sin d. sin 44. 4 sin d. 45. 4 b a a f d f a b d. b 46. sec tan d. 47. sin d. cos 48. tan a b d. 49.. d tan c SHORT ANSWER TYPE QUESTIONS (4 MARKS) 50. (i) cosec tan 4 d. (ii) d. (iii) sin a sin b d. (iv) cos cos a d. a 67 [XII Maths]
(v) cos cos cos d. (vi) 5 cos d. (vii) 4 sin cos d. (viii) 4 cot cosec d. (i) sin cos d. [Hint : put a sin + b cos = t ] a sin b cos () d. [Hint : Take sec as numerator] cos cos a (i) 6 6 sin cos. d sin cos (ii) sin cos d. sin 5. Evaluate : (i) (ii) d. [Hint : put = t] 4 d. 6 log 7 log [Hint : put log = t] (iii) d. (iv) 9 8 d. (v) a b d. (vi) sin sin d. Hint : sin sin sin sin sin (vii) 5 d. (viii) d. 6 68 [XII Maths]
(i) 4 d. () d. (i) d. (ii) sec d. 5. Evaluate : [Hint : Multiply and divide by sec ] (i) d. 7 (ii) sin d. cos cos (iii) sin cos d. cos cos (iv) d. (v) d. (vi) 4 d. [Hint : = t ] (vii) d 4. (viii) d sin cos. [Hint : Multiply Numerator and Denominator by sin and put cos = t] (i) sin d. sin 4 () d. 4 69 [XII Maths]
(i) tan d. 5. Evaluate : (i) 5 sin d. (ii) (ii) 9 d. 8 sec d. [Hint : Write sec = sec. sec and take sec as first function] a (iii) e cos b c d. (iv) 6 sin d. 9 (v) cos d. (vii) (i) sin e d. cos (vi) (viii) a d. () [Hint : put = tan ] e e tan d. d. d. (i) e sin d. cos log log d. [Hint : put log = t = et ] log (ii) (iii) 6 5 6 d. (iv) d. (v) 5 4 d. (vi) 4 8. d 70 [XII Maths]
54. Evaluate the following definite integrals : (i) 4 0 sin cos d. (ii) 9 6 sin 0 cos log sin d. (iii) 0 d. [Hint : put = t ] (iv) 0 sin d. (v) 0 sin sin 4 4 cos d. (vi) 5 d. 4 (vii) 0 55. Evaluate : sin d cos. Hint : Write sin sin as cos cos cos (i) (iii) d. tan e d. (ii) (iv) 0 0 d. sin log sin d. (v) 0 sin cos d. when (vi) f d where f when when. 7 [XII Maths]
Hint : f d f d f d f d (vii) 0 sin cos sin 4 4 cos d. (viii) 0 d. a cos b sin Hint : Use a a f d f a d 0 0 56. Evaluate the following integrals (i) d 6 tan (ii) 0 sin d. (iii) sin log d. sin (iv) 0 e e cos cos e cos d. (v) 0 tan sec cosec d. a a a (iv) d. a 6 d 57. 4 5 58. e log log sin d. 59. Evaluate sin cos d cos (i), 0, sin 7 [XII Maths]
(ii) d (iii) (iv) log log 4 sin cos d d (v) sin a d (vi) 6 sin cos d sin (vii) sin cos d (viii) d, where [] is greatest integer function (i) sin d. LONG ANSWER TYPE QUESTIONS (6 MARKS) 60. Evaluate the following integrals : (i) 5 5 4 d. (ii) d 4 d (iii) d (iv) 4 d 4 6 7 [XII Maths]
(v) 0 tan cot d. (vi) 4 d. (vii) 0 tan d. 6. Evaluate the following integrals as limit of sums : 4 (i) d. (ii) 0 d. 6. Evaluate (iii) 4 d. (iv) 5 (v) d. (i) 0 cot d d (ii) sin cos sin cos 4 0 e d. (iii) log d (iv) 0 0 log sin log sin d. 6. sin sin d. 64. sin cos. 5 cos 4sin d 74 [XII Maths]
65. tan d 66. e cos d. 0 ANSWERS.. c. e. tan + c. 4. 9 8 8log c. log8 9 6 5. 0 6. log log (log ) + c 7. 0 8. a a c a loga 9. tan + c 0. 0. 4 0 log 4 0 c. f () + c. tan cot + c 4. c e a 5. log + c 6. loge a c 7. e c loge 8.. c 9. log c. 0. e c. cos + c. log cos cos c. 75 [XII Maths]
. log sec tan c 4. log cos sin c sin 5. logsin c 6. 4 log c. 4 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. 0 4. 5. 6. b a 7. 8. 0 9. 40. 0 4. + log + c. 4. log sec tan c. 4. sin sin c 44. 45. 0 46. log + sin + c 47. sin + c 48. log cos + sin + c 49. a c log a c b c c log b c. 76 [XII Maths]