Maths 12 supporting material by cbse

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Maths 12 supporting material by cbse

  1. 1. LIST OF MEMBERS WHO PREPARED QUESTION BANK FOR MATHEMATICS FOR CLASS XII TEAM MEMBERS Sl. No. Name Designation 1. Sh. S.B. Tripathi (Group Leader) (M. 9810233862) R.S.B.V., Jheel Khuranja Delhi – 31. 2. Sh. Sanjeev Kumar (Lecturer Maths) (M. 9811458610) R.S.V.V., Raj Niwas Marg, Delhi. 3. Dr. R.P. Singh (Lecturer Maths) (M. 9818415348) R.P.V.V., Gandhi Nagar Delhi – 31 4. Sh. Joginder Arora (Lecturer Maths) (M. 9953015325) R.P.V.V., Hari Nagar Delhi. 5. Sh. Manoj Kumar (Lecturer Maths) (M. 9818419499) R.P.V.V., Kishan Kunj Delhi. 6. Miss Saroj (Lecturer Maths) (M. 9899240678) G.G.S.S.S., No. 1, Roop Nagar Delhi-110007
  2. 2. REVIEW OF SUPPORT MATERIAL : 2012 GROUP LEADER 1. Dr. Vandita Kalra (Vice Principal) GGSSS, Kirti Nagar, Delhi. TEAM MEMBERS 1. Sh. Sanjeev Kumar (PGT Maths) R.P.V.V., Raj Niwas Marg, Delhi. 2. Ms. Saroj (PGT Maths) G.G.S.S.S., No. 1, Roop Nagar Delhi. 3. Ms. Anita Saluja (PGT Maths) G.G.S.S.S., Kirti Nagar Delhi. XII – Maths 2
  3. 3. CLASS XII MATHEMATICS Units Weightage (Marks) (i) Relations and Functions 10 (ii) Algebra 13 (iii) Calculus 44 (iv) Vector and Three Dimensional Geometry 17 (v) Linear Programming 06 (vi) Probability 10 Total : 100 Unit I : RELATIONS AND FUNCTIONS 1. Relations and Functions (10 Periods) Types of Relations : Reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function. Binary operations. 2. Inverse Trigonometric Functions (12 Periods) Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions. Unit II : ALGEBRA 1. Matrices (18 Periods) Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication. Non-commutativity of 3 XII – Maths
  4. 4. multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries). 2. Determinants (20 Periods) Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle. adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix. Unit III : CALCULUS 1. Continuity and Differentiability (18 Periods) Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function. Concept of exponential and logarithmic functions and their derivatives. Logarithmic differentiation. Derivative of functions expressed in parametric forms. Second order derivatives. Rolle’s and Lagrange’s mean Value Theorems (without proof) and their geometric interpretations. 2. Applications of Derivatives (10 Periods) Applications of Derivatives : Rate of change, increasing/decreasing functions, tangents and normals, approximation, maxima 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). 3. Integrals (20 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. dx  x 2  a2 ,  XII – Maths dx 2 x a 2 , dx 2 a x 2 , 4 dx dx , 2 ax  bx  c ax  bx  c 2
  5. 5. px  q   ax 2  bx  c dx ,   px  q  2 ax  bx  c ax 2  bx  c dx and dx ,  a 2  x 2 dx ,  px  q   x 2  a 2 dx , ax 2  bx  c dx 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 (10 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 (10 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  x  y  q  x  , where p  x  and q  x  are function of x. dx Unit IV : VECTORS AND THREE-DIMENSIONAL GEOMETRY 1. Vectors (12 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. 2. Three-Dimensional Geometry (12 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
  6. 6. (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane. Unit V : LINEAR PROGRAMMING (12 Periods) 1. 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 1. Probability (18 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. 7. CONTENTS S.No. Chapter Page 1. Relations and Functions 9 2. Inverse Trigonometric Functions 3 & 4. Matrices and Determinants 17 23 5. Continuity and Differentiation 39 6. Applications of Derivatives 47 7. Integrals 61 8. Applications of Integrals 85 9. Differential Equations 90 10. Vectors 101 11. Three-Dimensional Geometry 111 12. Linear Programming 122 13. Probability 127 Model Papers 141 7 XII – Maths
  8. 8. XII – Maths 8
  9. 9. CHAPTER 1 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 = 2rs   is also a relation defined on set A, called the void (empty) relation.  R = A × A is called universal relation.  Reflexive Relation : Relation R defined on set A is said to be reflexive 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 reflexive, 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.  x1, x2  A s.t. x1  x2  f(x1)  f(x2). OR  x1, x2  A s.t. f(x1) = f(x2)  x1 = x2 One-one function is also called injective function. 9 XII – Maths
  10. 10.  Onto function (surjective) : A function f : A  B is said to be onto iff Rf = B i.e.  b  B, there exist 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) (x) = g (f (x))  x  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 = Iy and gof = Ix when Ix, Iy are identity functions. –1.  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 exists, is unique. XII – Maths 10
  11. 11. VERY SHORT ANSWER TYPE QUESTIONS (1 MARK) 1. If A is the set of students of a school then write, which of following relations are. (Universal, Empty or neither of the two). R1 = {(a, b) : a, b are ages of students and |a – b|  0} R2 = {(a, b) : a, b are weights of students, and |a – b| < 0} R3 = {(a, b) : a, b are students studying in same class} 2. Is the relation R in the set A = {1, 2, 3, 4, 5} defined as R = {(a, b) : b = a + 1} reflexive? 3. If R, is a relation in set N given by R = {(a, b) : a = b – 3, b > 5}, then does elements (5, 7)  R? 4. If f : {1, 3}  {1, 2, 5} and g : {1, 2, 5}  {1, 2, 3, 4} be given by f = {(1, 2), (3, 5)}, g = {(1, 3), (2, 3), (5, 1)} Write down gof. 5. Let g, f : R  R be defined by g x   6. If x 2 , f  x   3 x  2. Write fog. 3 f : R  R defined by f x   2x  1 5 be an invertible function, write f–1(x). x  x  1 Write , x 1 7. If f  x   8. Let * is a Binary operation defined on R, then if (i) fo f  x  . a * b = a + b + ab, write 3 * 2 11 XII – Maths
  12. 12. (ii) 9. a *b  a  b 2 3 , Write  2 * 3 * 4. If n(A) = n(B) = 3, Then how many bijective functions from A to B can be formed? 10. If f (x) = x + 1, g(x) = x – 1, Then (gof) (3) = ? 11. Is f : N  N given by f(x) = x2 is one-one? Give reason. 12. If f : R  A, given by f(x) = x2 – 2x + 2 is onto function, find set A. 13. If f : A  B is bijective function such that n (A) = 10, then n (B) = ? 14. If n(A) = 5, then write the number of one-one functions from A to A. 15. R = {(a, b) : a, b  N, a  b and a divides b}. Is R reflexive? Give reason? 16. Is f : R  R, given by f(x) = |x – 1| is one-one? Give reason? 17. f : R  B given by f(x) = sin x is onto function, then write set B. 18.  1 x   2x  , show that f   2f  x . If f  x   log   1 x    1 x2   19. If ‘*’ is a binary operation on set Q of rational numbers given by a * b  then write the identity element in Q. 20. If * is Binary operation on N defined by a * b = a + ab  a, b  N. Write the identity element in N if it exists. SHORT ANSWER TYPE QUESTIONS (4 Marks) 21. ab 5 Check the following functions for one-one and onto. 2x  3 7 (a) f : R  R , f (x )  (b) f : R  R, f(x) = |x + 1| (c) f : R – {2}  R, f  x   XII – Maths 3x  1 x 2 12
  13. 13. (d) f : R  [–1, 1], f(x) = sin2x 22. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a* b = H.C.F. of a and b. Write the operation table for the operation *. 23. Let f : R     4 4 R  3 3 be a function given by f  x   1 that f is invertible with f  x   4x . Show 3x  4 4x . 4  3x 24. Let R be the relation on set A = {x : x  Z, 0  x  10} given by R = {(a, b) : (a – b) is multiple of 4}, is an equivalence relation. Also, write all elements related to 4. 25. Show that function f : A  B defined as f  x   3x  4 where 5x  7 7 3 A  R    , B  R    is invertible and hence find f –1. 5  5 26. Let * be a binary operation on Q. Such that a * b = a + b – ab. (i) (ii) 27. Prove that * is commutative and associative. Find identify element of * in Q (if it exists). If * is a binary operation defined on R – {0} defined by a * b  check * for commutativity and associativity. 28. 2a b2 , then If A = N × N and binary operation * is defined on A as (a, b) * (c, d) = (ac, bd). (i) Check * for commutativity and associativity. (ii) Find the identity element for * in A (If it exists). 29. 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. 30. 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. 13 XII – Maths
  14. 14. (ii) Every non zero element of Q is invertible with a 1  16 , a a  Q   0 . 1 is bijective where R+ is the 2x set of all non-zero positive real numbers. 31. Show that f : R+  R+ defined by f  x   32. Consider f : R+  [–5, ) given by f(x) = 9x2 + 6x – 5 show that f is 1 invertible with f  x  6 1 . 3 33. 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 except –1. 34. If f, g : R  R defined by f(x) = x2 – x and g(x) = x + 1 find (fog) (x) and (gof) (x). Are they equal? 35. f : [1  )  [2,  ) is given by f  x   x  , 36. f : R  R, g : R  R given by f(x) = [x], g(x) = |x| then find fog     1 , find f 1  x . x 2   2   and gof    .   3 3 ANSWERS 1. R1 : is universal relation. R2 : is empty relation. R3 : is neither universal nor empty. 2. No, R is not reflexive. 3. (5, 7)  R 4. gof = {(1, 3), (3, 1)} 5. (fog)(x) = x  x  R XII – Maths 14
  15. 15. 5x  1 2 6. f 1  x   7. fof   x   8. x 1 ,x  2x  1 2 (i) 3 * 2 = 11 (ii) 1369 27 9. 6 10. 3 11. 2 2 Yes, f is one-one   x 1, x 2  N  x 1  x 2 . 12. A = [1, ) because Rf = [1, ) 13. n(B) = 10 14. 120. 15. No, R is not reflexive  a, a   R  a  N 16. f is not one-one functions  f(3) = f (–1) = 2 3  – 1 i.e. distinct element has same images. 17. B = [–1, 1] 19. e = 5 20. Identity element does not exist. 21. (a) Bijective (b) Neither one-one nor onto. (c) One-one, but not onto. (d) Neither one-one nor onto. 15 XII – Maths
  16. 16. 22. * 1 2 3 4 5 1 1 1 1 1 1 2 1 2 1 2 1 3 1 1 3 1 1 4 1 2 1 4 1 5 1 1 1 1 5 24. Elements related to 4 are 0, 4, 8. 25. f 1  x   26. 0 is the identity element. 27. Neither commutative nor associative. 28. 7x  4 5x  3 (i) Commutative and associative. (ii) (1, 1) is identity in N × N 33. 0 is the identity element. 34. (fog) (x) = x2 + x (gof) (x) = x2 – x + 1 Clearly, they are unequal. x  x2  4 2 35. f 1  x   36. fog     2   0 3 gof     2   1 3 XII – Maths 16
  17. 17. CHAPTER 2 INVERSE TRIGONOMETRIC FUNCTIONS IMPORTANT POINTS  sin–1 x, cos–1 x, ... etc., are angles.    If sin  x and    ,  then  = sin–1x etc.  2 2    Function Domain Range (Principal Value Branch) sin–1x     2 , 2    cos–1x [–1, 1] [0, ] tan–1x R     ,   2 2 cot–1x R (0, ) sec–1x R – (–1, 1) 0,    cosec –1x  [–1, 1] R – (–1, 1)       2 , 2   0     2 sin–1 (sin x) = x  x     ,    2 2   cos–1 (cos x) = x  x  [0, ] etc.  sin (sin–1x) = x  x  [–1, 1] cos (cos–1x) = x  x  [–1, 1] etc. 17 XII – Maths
  18. 18.   1 sin1x  cosec –1    x   1, 1 x tan–1x = cot–1 (1/x)  x > 0 sec–1x = cos–1 (1/x),  |x|  1  sin–1(–x) = – sin–1x  x  [–1, 1] tan–1(–x) = – tan–1x  x  R cosec–1(–x) = – cosec–1x  |x|  1  cos–1(–x) =  – cos–1x  ×  [–1, 1] cot–1(–x) =  – cot–1x  x  – R sec–1(–x) =  – sec–1x  |x|  1  sin1 x  cos 1 x  tan–1 x  cot –1 x   , x   1, 1 2   x R 2 sec –1 x  cosec –1x    x 1 2  x  y tan1 x  tan 1 y  tan 1  ;  1  xy   xy  1.  x  y tan1 x  tan1 y  tan 1  ;  1  xy   xy  1.   2x  2 tan–1 x  tan –1  , x 1  1 x 2    2x  2 tan1 x  sin1  , x 1 ,  1 x 2    1 x 2  2 tan1 x  cos 1  , x  0.  1 x 2   XII – Maths 18
  19. 19. VERY SHORT ANSWER TYPE QUESTIONS (1 MARK) 1. Write the principal value of –1 – –1 (i) sin (iii) tan (v) cot (vii) sin 3 2 –1  3 2. (ii) cos 1   –   3 (iv) cosec–1 (– 2). –1  1  .   3  (vi) sec–1 (– 2). 1  3 1  1 1    cos    tan  1 3     2  2 2. What is value of the following functions (using principal value). (i) tan –1 –1  2   1  .   – sec   3  3  –1 –1  3   1  .  –  – cos   2 2  (ii) sin (iv) cosec (iii) tan–1 (1) – cot–1 (–1). (v) sin  –1 5    tan .  6 2   sec –1  2 . tan–1 (1) + cot–1 (1) + sin–1 (1). (vi) –1 (viii) –1 4    sin .  5 cosec –1 (vii) tan 3    cosec .  4  SHORT ANSWER TYPE QUESTIONS (4 MARKS) 3. Show that tan –1  1  cos x    1  cos x – 19 1  cos x   x  .   1 – cos x  4 2 x  [0, ] XII – Maths
  20. 20. 4. Prove 1 tan 5. tan Prove 6.  1  1  cos x   cos x       cot   1  sin x   1  cos x  4 –1 x   0,  2 . 2 2   x a  x –1 x –1   .  sin  cos      a a  a2 – x 2  Prove cot –1  –1 8   –1  –1 8   –1  300    .  2 tan  cos    tan  2 tan  sin    tan       161   17 17   1 7. Prove tan 8. Solve cot –1 2  1 x     1  x2  2x  cot –1   1 1 2    cos x . 2  4 2 1 x  1 x 3x   2 . 4 9.  m  m n  Prove that tan1    tan 1   , m, n  0 n  m n 4  10. 1  1 y 2   x  y  2x  1  cos 1  Prove that tan  sin1     1 x 2  2   1  y 2   1  xy 2   11.  x 2  1 1  2x  2  Solve for x, cos1  2   tan 1   x  1 2  1 x 2   3 12. Prove that tan1 13. Solve for 14.  1  1  32  Prove that 2 tan–1    tan –1    tan –1    4  5  43  XII – Maths x, 1 1 1 1   tan1  tan1  tan1  3 5 7 8 4 tan cos 1 x   sin  tan1 2 ; x  0 20
  21. 21. 1  3  tan  cos –1   11   2  15. Evaluate 16.  a cos x  b sin x   a  tan –1    x Prove that tan–1   b cos x  a sin x   b 17. Prove that   1   cot  tan1 x  tan1     cos 11  2 x 2   cos 1 2 x 2  1  , x  0  x   18.  a b   b c   c a   tan –1   tan –1   0 where a, b, Prove that tan–1   1  ab    1  bc    1  ca   c > 0 19. Solve for x, 2 tan–1(cos x) = tan–1 (2 cosec x) 20. Express 21. If tan–1a + tan–1b + tan–1c =  then  sin–1 x 1  x  x 1  x 2  in simplest form. prove that a + b + c = abc 22. If sin–1x > cos–1x, then x belongs to which interval? ANSWERS (ii)  6 (iii) – 6  3 (vi) 2 3 (vii)  . 6 2. (i) 0 (ii)  3 (iii)  (v)  (vi)  5 (vii) (i) – (v) 1.  3 21 (iv) – 6  2 (iv)  2 – 6 (viii)  . 4 XII – Maths
  22. 22. 8. 1 11. 13. 5 3 19. x  22.  1  , 1   2   21. Hint: 15.  . 4 Let 20 tan–1 a =  tan–1 b =  tan–1 c =  then given,                take tangent on both sides, tan (  ) = tan    XII – Maths 22 tan   2 3 12 11  3 3  11 sin–1 x – sin–1 x.
  23. 23. CHAPTER 3 & 4 MATRICES AND DETERMINANTS POINTS TO REMEMBER  Matrix : A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements of the matrix.  Order of Matrix : A matrix having ‘m’ rows and ‘n’ columns is called the matrix of order mxn.  Square Matrix : An mxn matrix is said to be a square matrix of order n if m = n.  Column Matrix : A matrix having only one column is called a column matrix i.e. A = [aij]mx1 is a column matrix of order mx1.  Row Matrix : A matrix having only one row is called a row matrix i.e. B  bij 1xn is a row matrix of order 1xn.  Zero Matrix : A matrix having all the elements zero is called zero matrix or null matrix.  Diagonal Matrix : A square matrix is called a diagonal matrix if all its non diagonal elements are zero.  Scalar Matrix : A diagonal matrix in which all diagonal elements are equal is called a scalar matrix.  Identity Matrix : A scalar matrix in which each diagonal element is 1, is called an identity matrix or a unit matrix. It is denoted by I.  I = [eij]n × where,  1 if i  j eij   0 if i  j n 23 XII – Maths
  24. 24.  Transpose of a Matrix : If A = [ai j ]m × n be an m × n matrix then the matrix obtained by interchanging the rows and columns of A is called the transpose of the matrix. Transpose of A is denoted by A´ or AT. Properties of the transpose of a matrix. (i) (iii) (A´)´ = A (ii) (A + B)´ = A´ + B´ (kA)´ = kA´, k is a scalar (iv) (AB)´ = B´A´  Symmetrix Matrix : A square matrix A = [aij ] is symmetrix if aij = aji  i, j. Also a square matrix A is symmetrix if A´ = A.  Skew Symmetrix Matrix : A square matrix A = [aij] is skew-symmetrix, if aij = – aji  i, j. Also a square matrix A is skew - symmetrix, if A´ = – A.  Determinant : To every square matrix A = [aij] of order n × n, we can associate a number (real or complex) called determinant of A. It is denoted by det A or |A|. Properties (i) |AB| = |A| |B| (ii) |kA|n × n = kn |A|n × n where k is a scalar. Area of triangles with vertices (x1, y1), (x2, y2) and (x3, y3) is given by   x1 1 x2 2 x3 y1 1 y2 1 y3 1 x1 The points (x1, y1), (x2, y2), (x3, y3) are collinear  x 2 x3  y1 1 y2 1  0 y3 1 Adjoint of a Square Matrix A is the transpose of the matrix whose elements have been replaced by their cofactors and is denoted as adj A. Let n adj A = [Aji]n × XII – Maths A = [aij]n × n 24
  25. 25. Properties (i) A(adj A) = (adj A) A = |A| I (ii) If A is a square matrix of order n then |adj A| = |A|n–1 (iii) adj (AB) = (adj B) (adj A). [Note : Correctness of adj A can be checked by using A.(adj A) = (adj A) . A = |A| I ] Singular Matrix : A square matrix is called singular if |A| = 0, otherwise it will be called a non-singular matrix. Inverse of a Matrix : A square matrix whose inverse exists, is called invertible matrix. Inverse of only a non-singular matrix exists. Inverse of a matrix A is denoted by A–1 and is given by A 1  1 . adj . A A Properties (i) (ii) (A–1)–1 = A (iii) (AB)–1 = B–1A–1 (iv)  AA–1 = A–1A = I (AT)–1 = (A–1)T Solution of system of equations using matrix : If AX = B is a matrix equation then its solution is X = A–1B. (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 either consistent and has infinitely many solutions or system is inconsistent and has no solution. 25 XII – Maths
  26. 26. VERY SHORT ANSWER TYPE QUESTIONS (1 Mark) 1. 4  x  3 If   y  4 x  y  5 4 3 9  , find x and y.   2. i If A   0 3. Find the value of a23 + a32 in the matrix A = [aij]3 0 0  and B   i i   i , find AB . 0  × 3 if i  j  2i – j where aij   .  i  2 j  3 if i  j 4. If B be a 4 × 5 type matrix, then what is the number of elements in the third column. 5.  5 2 3 6  If A    and B  0 1 find 3 A  2B. 0 9   6.  2 3  1 0  If A    and B  2 6  find  A  B ´.  7 5    7.  2 If A = [1 0 4] and B  5 find AB .   6   8. x  2  4 If A    is symmetric matrix, then find x. 2x  3 x  1 9. 3  0 2  2 0 4 is skew symmetrix matrix. For what value of x the matrix    3 4 x  5   10. 2 3 If A     P  Q where P is symmetric and Q is skew-symmetric 1 0 matrix, then find the matrix Q. XII – Maths 26
  27. 27. 11. a  ib Find the value of c  id 12. If 13. k For what value of k, the matrix  3 14.  sin 30 If A     sin 60 15. 2x  5 5x  2 c  id a  ib 3  0, find x . 9 2 has no inverse. 4  cos 30  , what is |A|. cos 60   Find the cofactor of a12 in 2 6 1 3 0 5 5 4 7 . 1 3 2 4 5 6 . 3 5 2 16. Find the minor of a23 in 17. 1 2 Find the value of P, such that the matrix  is singular.  4 P   18. Find the value of x such that the points (0, 2), (1, x) and (3, 1) are collinear. 19. Area of a triangle with vertices (k, 0), (1, 1) and (0, 3) is 5 unit. Find the value (s) of k. 20. If A is a square matrix of order 3 and |A| = – 2, find the value of |–3A|. 21. If A = 2B where A and B are square matrices of order 3 × 3 and |B| = 5, what is |A|? 22. What is the number of all possible matrices of order 2 × 3 with each entry 0, 1 or 2. 23. Find the area of the triangle with vertices (0, 0), (6, 0) and (4, 3). 24. If 2x 4 6 3  , find x . 1 x 2 1 27 XII – Maths
  28. 28. 25. x  y  If A   z  1  y z x 1 z  x y  , write the value of det A.   1  26. a11 a12  If A    such that |A| = – 15, find a11 C21 + a12C22 where Cij is a21 a22  cofactors of aij in A = [aij]. 27. If A is a non-singular matrix of order 3 and |A| = – 3 find |adj A|. 28. 5 3  If A   6 8  find adj A    29. Given a square matrix A of order 3 × 3 such that |A| = 12 find the value of |A adj A|. 30. If A is a square matrix of order 3 such that |adj A| = 8 find |A|. 31. Let A be a non-singular square matrix of order 3 × 3 find |adj A| if |A| = 10. 32.  2 1 If A    find 3 4  33.  3 If A   1 2 3 and B   4  find |AB|.    0   A 1 1 . SHORT ANSWER TYPE QUESTIONS (4 MARKS) 34. x  y Find x, y, z and w if   2x  y 2x  z   –1   0 3x  w   35. Construct a 3 × 3 matrix A = [a ij] whose elements are given by 1  i  j if i  j  aij =  i  2 j if i  j  2  XII – Maths 28 5 . 13  
  29. 29. 36.  1 2 3   3 0 1 Find A and B if 2A + 3B =   and A  2B   1 6 2 . 2 0 1   37.  1 If A   2 and B   2 1 4 , verify that (AB)´ = B´A´.    3   38. 3 1  3  2 2 1  P  Q where P is a symmetric and Q Express the matrix    4 5 2   is a skew-symmetric matrix. 39.  cos  If A =    sin  sin    cos n  sin n   n  , then prove that A    sin n  cos n  cos     where n is a natural number. 40.  2 1  5 2  2 5 Let A    , B  7 4 , C  3 8 , find a matrix D such that 3 4     CD – AB = O. 41.  1 3 2  1 1 x 1  2 5 1  2   0 Find the value of x such that     15 3 2   x     42. Prove that the product of the matrices cos 2   cos 2  cos  sin   cos  sin      and  2 2 cos  sin  sin   cos  sin  sin       is the null matrix, when  and  differ by an odd multiple of 43.  . 2  5 3 2 –1 If A    show that A – 12A – I = 0. Hence find A . 12 7  29 XII – Maths
  30. 30. 44. 2 3 2 If A    find f(A) where f(x) = x – 5x – 2. 4 7  45. 4 If A   2 46.  1 2 3 Find the matrix X so that X   4 5 6  47. 2 3  1 2 –1 –1 –1 If A    and B   1 3 then show that (AB) = B A .  1 4   48. 3 , find x and y such that A2 – xA + yI = 0. 5   7 8 9   2 4 6 .   Test the consistency of the following system of equations by matrix method : 3x – y = 5; 6x – 2y = 3 49. Using elementary row transformations, find the inverse of the matrix  6 3 , if possible. A 1  2  50. 3 1 By using elementary column transformation, find the inverse of A   . 5 2 51. cos  If A    sin   sin   and A + A´ = I, then find the general value of . cos    Using properties of determinants, prove the following : Q 52 to Q 59. 52. a b c 2a 2a 3 2b b c a 2b  a  b  c  2c 2c c a b 53. x  2 x  3 x  2a x  3 x  4 x  2b  0 if a, b, c are in A.P . x  4 x  5 x  2c 54. sin  cos  sin     sin  cos  sin      0 sin  cos  sin      XII – Maths 30
  31. 31. b 2 c b 55. c 56. a 2 c 59. 2 a  ab b 2 2 b 2 2 2 a b 2 2 2  4a b c . 2 a b a p q  2 p x  y x b ac  c 2 ac  bc b x b b c b q y c r . z 2 2 2 2  4a b c . 2 c 2 c  x x  a  b  c  . x c Show that : 2 x yz (i) (ii) 61. 2 bc x 60. a c a r  p z  x 2 x a a a 2 a c ab 58. 2 2 b c q r y  z a 57. 2 y z 2 2 z   y  z   z  x   x  y   yz  zx  xy  . xy y zx If the points (a, b) (a´, b´) and (a – a´, b – b´) are collinear. Show that ab´ = a´b. 2 If A   2 0 Given A   2 5 4  and B   2 1  1 2 3  verity that AB  A B . 5   0 2 and B   1  0  1  1 0  . Find the product AB and  1  also find (AB)–1. 62. Solve the following equation for x. a  x a  x a  x a  x a  x a  x a  x a  x  0. a  x 31 XII – Maths
  32. 32. 63.  0 If A    tan   2 that,  tan   2  and I is the identity matrix of order 2, show 0    cos  I  A  I  A   sin  64.  sin   cos    Use matrix method to solve the following system of equations : 5x – 7y = 2, 7x – 5y = 3. LONG ANSWER TYPE QUESTIONS (6 MARKS) 65. Obtain the inverse of the following matrix using elementary row operations 0 1 2 A   1 2 3 .   3 1 1   66. 1  1 1 2  2 0 0   9 2 3 2 3  Use product   to solve the system of equations 3 2 4  6 1 2    x – y + 2z = 1, 2y – 3z = 1, 3x – 2y + 4z = 2. 67. Solve the following system of equations by matrix method, where x  0, y  0, z  0 2 3 3 1 1 1 3 1 2    10,    10,    13. x y z x y z x y z 68. 1 where A   2  3  equations : Fi nd A–1, 2 3 3 3  2  , hence solve the system of linear  –4   x + 2y – 3z = – 4 2x + 3y + 2z = 2 3x – 3y – 4z = 11 XII – Maths 32
  33. 33. 69. The sum of three numbers is 2. If we subtract the second number from twice the first number, we get 3. By adding double the second number and the third number we get 0. Represent it algebraically and find the numbers using matrix method. 70. Compute the inverse of the matrix.  3 A   15   5  71. If the matrix 1 6 2 1 A  0  3  1 5  and verify that A–1 A = I3.   5  1 2 2 2 1  and B –1   0 3   1 4    2 3 0 0 –1 ,  2  then compute (AB)–1. 72. Using matrix method, solve the following system of linear equations : 2x – y = 4, 2y + z = 5, z + 2x = 7. 1 0 1 1 A 2  3I . if A   1 0 1 . Also show that A 1    2  1 1 0   73. Find A 74.  1  Find the inverse of the matrix A   1  0  column transformations. 75.  2 3 Let A   and f(x) = x2 – 4x + 7. Show that f (A) = 0. Use this result 1 2   2 3 2 2  0  by using elementary   1 to find A5. 76. cos   sin  0 If A   sin  cos  0 , verify that A . (adj A) = (adj A) . A = |A| I3.    0 0 1   33 XII – Maths
  34. 34. 77.  2 1 1 For the matrix A   1 2 1 , verify that A3 – 6A2 + 9A – 4I = 0, hence    1 1 2   find A–1. 78. Find the matrix X for which 3 2  7 5  . X   79.  1 1 2 1 .    2 1 0 4  By using properties of determinants prove the following : 1 a2  b 2 2ab 2b 2ab 1 a2  b 2 2a 2b 2a 1 a2  b 2  y  z 2 xy zx 2 81. xy x  z  yz xz 80. yz 3  2xyz  x  y  z  . x  y 2 a ab a b c 2a 3a  2b 4a  3b  2c  a 3 . 3a 6a  3b 10a  6b  3c x 82. If x, y, z are different and y z 83. 3  1  a 2  b 2  . x 2 1 x 3 y 2 1  y 3  0. Show that xyz = – 1. z2 1 z 3 If x, y, z are the 10th, 13th and 15th terms of a G.P. find the value of log x   log y log z XII – Maths 10 1 13 1 . 15 1 34
  35. 35. 84. Using the properties of determinants, show that : 1 a 1 1 1 1 1  1 1 b 1  abc  1      abc  bc  ca  ab  a b c 1 1 1 c 85. Using properties of determinants prove that bc b 2  bc c 2  bc a 2  ac ac c 2  ac  ab  bc  ca  a 2  ab b 2  ab 86. I f 3 ab 3 2 1  A  4 1 2  , find A–1 and hence solve the system of equations   7 3 3    3x + 4y + 7z = 14, 2x – y + 3z = 4, x + 2y – 3z = 0. ANSWERS 1. x = 2, y = 7 2. 0 1 1 0    3. 11. 4. 4 5. 9 6 0 29 .   6.  3 5   3 1 .   7. AB = [26]. 8. x = 5 9. x = – 5 10.  0  1  11. a2 + b2 + c2 + d2. 12. x = – 13 13. k 14. |A| = 1. 15. 46 16. –4 3 2 35 1 . 0  XII – Maths
  36. 36. 5 . 3 17. P = – 8 18. x  19. k 10 . 3 20. 54. 21. 40. 22. 729 23. 9 sq. units 24. x = ± 2 25. 0 26. 0 27. 9 28.  8 3  6 5  .   29. 1728 30. |A| = 9 31. 100 32. 11 33. |AB| = – 11 34. x = 1, y = 2, z = 3, w = 4 35. 3 3 2 5 2 4 5 2 .   5 6 7    36. 9 11 7 7 A  1 18  7  7 9 2 1  5    7 7 7 7 , B    4  4  12  5   7 7   7 7  40.  191 D    77 110  . 44   43. 45. 41. x = – 2 or – 14 3  7 A 1   .  12 5  44. f(A) = 0 x = 9, y = 14 46.  1 2 x  . 2 0 XII – Maths 36
  37. 37. 48. Inconsistent 49. Inverse does not exist. 50.  2 1 A –1   .  5 3 51.   2n   61.  1 AB    2 62 0, 3a 2 1 2 –1  ,  AB    2 6 2 64. 2  . 1   x  11 24 65. 67. A –1  1   2   4   5   2   1 2 3 3 2 1  2 1 .  1  2 1 1 1 x  , y  , z  2 3 5 71.  AB 1  1  21 11 7  .  19  73. A 1  1 1 1 1   1 1 1 .  2  1 1 1   . 24 x = 0, y = 5, z = 3  6 17 13  1   14 5 8   67   15 9 1   A 70. x = 1, y = – 2, z = 2 10 2  1 , y  1 A 1 72. x = 3, y = 2, z = 1. 68. 69. 16 12 66.  , n z 3  2 0 1  5 1 0   0 1 3   1 3  74. 37 A 1 3 2 6   1 1 2    2 2 5   XII – Maths
  38. 38. 75. 93   118 A5   .  31 118  77. A 78.  16 3  X  .  24 5  83. 0 86. x = 1, y = 1, z = 1. XII – Maths 38 1 3 1  1 4   1 1 3 1 1 1.   3
  39. 39. CHAPTER 5 CONTINUITY AND DIFFERENTIATION POINTS TO REMEMBER  A function f(x) is said to be continuous at x = c iff lim f  x   f c  x c i.e., lim – f  x   lim f  x   f  c  x c x c  f(x) is continuous in (a, b) iff it is continuous at x  c  c  a, b  .  f(x) is continuous in [a, b] iff (i) (ii) (iii) f(x) is continuous in (a, b) lim f  x   f  a , x a  lim f  x   f  b  x b –  Trigonometric functions are continuous in their respective domains.  Every polynomial function is continuous on R.  If f (x) and g (x) are two continuous functions and c  R then at x = a (i) (ii) g (x) . f (x), f (x) + c, cf (x), | f (x)| are also continuous at x = a. (iii)  f (x) ± g (x) are also continuous functions at x = a. f x  is continuous at x = a provided g(a)  0. g x  f (x) is derivable at x = c in its domain iff 39 XII – Maths
  40. 40. lim  x c f  x   f c  f  x   f c   lim , and is finite x c x c x c The value of above limit is denoted by f´(c) and is called the derivative of f(x) at x = c.  d dv du u · v   u ·  v · dx dx dx  d u    dx  v   If y = f(u) and u = g(t) then v· du dv u· dx dx v2 dy dt  If y = f(u),  dy du  du  f ´ u  .g´ t   Chain Rule  dt x = g(u) then, dy dy du f ´u     . dx du dx g ´u   d  1  sin x  dx 1 1 x 2 , d  1  1 tan x  , dx 1  x2 d  1 sec 1 x   , dx x x2  1 d  x e dx   ex, d  1 cos x dx   1 1 x 2 d  1  1 cot x  dx 1 x 2 d  1 cosec 1x   dx x x2  1 d 1  log x   dx x  f (x) = [x] is discontinuous at all integral points and continuous for all x  R – Z.  Rolle’s theorem : If f (x) is continuous in [ a, b ], derivable in (a, b) and f (a) = f (b) then there exists atleast one real number c  (a, b) such that f´ (c) = 0. XII – Maths 40
  41. 41.  Mean Value Theorem : If f (x) is continuous in [a, b] and derivable in (a, b) then there exists atleast one real number c  (a, b) such that f ´ c   f b  – f a  . b a  f (x) = logex, (x > 0) is continuous function. VERY SHORT ANSWER TYPE QUESTIONS (1 MARK) 1. For what value of x, f(x) = |2x – 7| is not derivable. 2. Write the set of points of continuity of g(x) = |x – 1| + |x + 1|. 3. What is derivative of |x – 3| at x = – 1. 4. What are the points of discontinuity of f  x   5. Write the number of points of discontinuity of f(x) = [x] in [3, 7]. 6. x  3 if x  2  f x    4 if x  2 is a continuous function for all The function,  2x if x  2   x  1   x  1 . x  7 x  6 x  R, find .  tan3x , x 0  2K ,  7.  For what value of K, f  x    sin2x x 0 is continuous  x  R . 8. Write derivative of sin x w.r.t. cos x. 9. If f(x) = x2g(x) and g(1) = 6, g´(1) = 3 find value of f´(1). 10. Write the derivative of the following functions : (i) (iii) log3 (3x + 5) e 6 loge  x 1 (ii) e log 2 x ,x 1 41 XII – Maths
  42. 42. (iv) sec 1 x  cosec 1 x , x  1. (v) sin1 x 7 2  (vi) logx 5, x > 0. SHORT ANSWER TYPE QUESTIONS (4 MARKS) 11. Discuss the continuity of following functions at the indicated points. (i) (ii)  sin 2x  3x , x  0  g x    at x  0.  3 x  0  2  (iii)  x 2 cos 1 x  x  0  f x    0 x  0   (iv) f(x) = |x| + |x – 1| at x = 1. (v) 12. x  x , x  0  f x    x at x  0.  2, x  0  x   x  , x  1 f x    at x  1. 0 x  1  at x  0.  3x 2  kx  5, For what value of k, f  x    1  3x   0  x  2 is continuous 2  x  3  x   0, 3. 13. For what values of a and b     f x        XII – Maths x  2 a x  2 a b x  2  2b x  2 if x  –2 if x  –2 is continuous at x = 2. if x  –2 42
  43. 43. 14. Prove that f(x) = |x + 1| is continuous at x = –1, but not derivable at x = –1. 15. For what value of p,  x p sin 1 x  x  0  f x    is derivable at x = 0. 0 x 0   1 2  2x  dy  1  1   2 tan1    , 0  x  1 find , .  tan   x  1  x2   dx  16. If y  17.  1 x  dy  ? If y  sin  2 tan1  then 1 x  dx  18. If 5x + 5y = 5x+y then prove that 19. If x 1  y 2  y 1  x 2  a then show that 20. If 21. If (x + y)m + 22.  2x   2x  w.r.t. sin1  . Find the derivative of tan1   1 x 2    1 x 2   23. Find the derivative of loge(sin x) w.r.t. loga(cos x). 24. If xy + yx + xx = mn, then find the value of 25. If x = a cos3, y = a sin3 then find dy  5y  x  0 . dx dy 1 y 2  . dx 1 x 2 1  x 2  1  y 2  a  x  y  then show that n = xm . yn then prove that 43 dy 1 y 2  . dx 1 x 2 dy y  . dx x dy . dx d 2y . dx 2 XII – Maths
  44. 44. 26. x = aet (sint – cos t) If y = aet (sint + cost) then show that dy  at x  is 1. dx 4 dy . dx 27. 1 2   If y  sin  x 1  x  x 1  x  then find – 28. If y  x loge x  loge x  29. x Differentiate x 30. Find 31.  1  sin x  If y  tan1   1  sin x  32. 1 If x  sin  loge a 33. Differentiate log x log x , x  1 w .r .t . x 34. If sin y = x sin (a + y) then show that 35. If y = sin–1x, find 36. If 37. a cos If y  e 38. If y3 = 3ax2 – x3 then prove that 39. Verify Rolle's theorem for the function, y = x2 + 2 in the interval [a, b] where a = –2, b = 2. 40. Verify Mean Value Theorem for the function, f(x) = x2 in [2, 4] x x then find dy . dx w.r.t. x. dy y x , if  cos x    cos y  dx 1  sin x   dy  where 2  x   find dx . 1  sin x   y  then show that (1 – x2) y´´ – xy´ – a2y = 0.  2 dy sin a  y   . dx sin a d 2y in terms of y. dx 2 d 2y b 4 x2 y2  2 3.  2  1, then show that dx 2 a y a2 b XII – Maths 1 x ,  1  x  1 show that 1  x 2  , 44 d 2y dy x  a2 y  0 2 dx dx d 2 y 2a 2 x 2  . dx 2 y5
  45. 45. ANSWERS 1. x = –7/2 2. R 3. –1 4. x = 6, 7 5. Points of discontinuity of f(x) are 4, 5, 6, 7 i.e. four points. Note : At x = 3, f(x) = [x] is continuous. because lim f  x   3  f 3 . x 3 3 . 4 6.  7 . 2 7. k  8. –cot x 9. 15 10. (i) (ii) e log2 (iii) 6 (x – 1)5 (iv) 0 7 x2 x . 2 1 x 7 (vi) (i) Discontinuous (ii) Discontinuous (iv) continuous (v) 11. 3 log3 e 3x  5 (iii) Continuous x 1 .log2 e. x  loge 5 2 x loge x  . (v) Discontinuous 12. k = 11 13. a = 0, b = –1. 15. p > 1. 16. 0 22. 1 17. x 1 x 2 . 23. –cot2x logea 24. x y 1  y x log y dy  x 1  log x   yx  . dx x y log x  xy x 1 45 XII – Maths
  46. 46. 25. d 2y 1  cosec  sec 4 . dx 2 3a 28. x log x 29. x dy 1    x x . x x log x 1 log x  . dx  x logx   30. dy y tan x  logcos y  dx x tan y  logcos x 31. dy 1  . dx 2 27. 2log x x  1   logx    log logx  . x  logx  Hint. : sin x x    cos for x   ,   . 2  2 2  1 log log x    , x  1 x x  33. log x log x  35. sec2y tany. XII – Maths dy 1 1   . dx 1  x 2 2 x 1 x 46
  47. 47. CHAPTER 6 APPLICATIONS OF DERIVATIVES POINTS TO REMEMBER  Rate of Change : Let y = f(x) be a function then the rate of change of y with respect to x is given by dy  f ´  x  where a quantity y varies with dx another quantity x. dy  or f ´ x 0  represents the rate of change of y w.r.t. x at x = x0. dx  x  x 0   If x = f(t) and y = g (t) By chain rule dy dy  dx dt  (i) dx dx if  0. dt dt A function f(x) is said to be increasing (non-decreasing) on an interval (a, b) if x1  x2 in (a, b)  f(x1)  f (x2)  x1, x2  (a, b). Alternatively if f ´ x 0   0  x  a, b  , then f(x) is increasing function in (a, b). (ii)  A function f(x) is said to be decreasing (non-increasing) on an interval (a, b). If x1  x2 in (a, b)  f(x1)  f(x2)  x1, x2  (a, b). Alternatively if f´(x)  0  x  (a, b), then f(x) is decreasing function in (a, b). The equation of tangent at the point (x0, y0) to a curve y = f(x) is given by dy  y  y0  x  x 0  .  dx  x 0 , y 0  47 XII – Maths
  48. 48. where dy   slope of the tangent at the point  x 0, y 0 . dx  x 0 ,y 0   dy  does not exist then tangent is parallel to y-axis at dx  x 0 , y 0   (x0, y0) and its equation is x = x0. (i) (ii)   If If tangent at x = x0 is parallel to x-axis then dy  0 dx  x  x 0  1 Slope of the normal to the curve at the point (x0, y0) is given by  dy  dx  x  x 0  Equation of the normal to the curve y = f(x) at a point (x0, y0) is given by y  y0   1  x  x 0 . dy  dx  x 0 ,y 0   dy   0. then equation of the normal is x = x0. dx  x 0 ,y 0    If  dy  If dx   x 0 ,y 0  does not exit, then the normal is parallel to x-axis and the equation of the normal is y = y0.  Let y = f(x) x = the small increment in x and y be the increment in y corresponding to the increment in x Then approximate change in y is given by dy  dy   x  dx  or dy = f ´(x) x The approximate change in the value of f is given by f  x   x  XII – Maths  x  ´ f x  x f  48
  49. 49.  Let f be a function. Let point c be in the domain of the function f at which either f´ (x) = 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) f´(x) changes sign from negative to positive as x increases through c, then c is a point of local minima. (iii)  f´(x) changes sign from positive to negative as x increases through c, then c is called the point of the local maxima. f´(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. Such a point is called a point of inflexion. Second Derivative Test : Let f be a function defined on an interval I and let c  I. Then (i) x = c is a point of local maxima if f´(c) = 0 and f´´(c) < 0. f (c) is local maximum value of f. (ii) x = 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 (1 MARK) 1. The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate of increase of perimeter of the square. 2. The radius of the circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference? 3. If the radius of a soap bubble is increasing at the rate of 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 10 cm, how fast is the enclosed area increasing? 1 cm sec. At 2 what rate its volume is increasing when the radius is 1 cm. 49 XII – Maths
  50. 50. 5. The total revenue in rupees received from the sale of x units of a product is given by R(x) = 13x2 + 26x + 15. Find the marginal revenue when x = 7. 6. Find the maximum and minimum values of function f(x) = sin 2x + 5. 7. Find the maximum and minimum values (if any) of the function f(x) = – |x – 1| + 7  x  R . 8. Find the value of a for which the function f(x) = x2 – 2ax + 6, x > 0 is strictly increasing. 9. Write the interval for which the function f(x) = cos x, 0  x  2 is decreasing. 10. What is the interval on which the function f  x   increasing? log x , x  0,   is x 4 3 x is increasing? 3 11. For which values of x, the functions y  x 4  12. Write the interval for which the function f  x   13. Find the sub-interval of the interval (0, /2) in which the function f(x) = sin 3x is increasing. 14. Without using derivatives, find the maximum and minimum value of y = |3 sin x + 1|. 15. If f (x) = ax + cos x is strictly increasing on R, find a. 16. Write the interval in which the function f(x) = x9 + 3x7 + 64 is increasing. 17. What is the slope of the tangent to the curve f = x3 – 5x + 3 at the point whose x co-ordinate is 2? 18. At what point on the curve y = x2 does the tangent make an angle of 45° with positive direction of the x-axis? 19. Find the point on the curve y = 3x2 – 12x + 9 at which the tangent is parallel to x-axis. XII – Maths 50 1 is strictly decreasing. x
  51. 51. 20. What is the slope of the normal to the curve y = 5x2 – 4 sin x at x = 0. 21. Find the point on the curve y = 3x2 + 4 at which the tangent is perpendicular 1 to the line with slope  . 6 22. Find the point on the curve y = x2 where the slope of the tangent is equal to the y – co-ordinate. 23. If the curves y = 2ex and y = ae–x intersect orthogonally (cut at right angles), what is the value of a? 24. Find the slope of the normal to the curve y = 8x2 – 3 at x  25. 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. 26. 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 3 cm? 27. For the curve y = (2x + 1)3 find the rate of change of slope at x = 1. 28. Find the slope of the normal to the curve x = 1 – a sin  ; y = b cos2 at   1 . 4  2 29. If a manufacturer’s total cost function is C(x) = 1000 + 40x + x2, where x is the out put, find the marginal cost for producing 20 units. 30. Find ‘a’ for which f (x) = a (x + sin x) is strictly increasing on R. SHORT ANSWER TYPE QUESTIONS (4 MARKS) 31. A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which the y co-ordinate is changing 8 times as fast as the x coordinate. 32. 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 2 cm/sec. How fast is its height on the wall decreasing when the foot of the ladder is 4 metres away from the wall? 51 XII – Maths
  52. 52. 33. 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 15 cm. 34. A man 2 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. 35. Water is running out of a conical funnel at the rate of 5 cm3/sec. If the radius of the base of the funnel is 10 cm and altitude is 20 cm, find the rate at which the water level is dropping when it is 5 cm from the top. 36. The length x 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 x = 8 cm and y = 6 cm. Find the rate of change of (a) Perimeter (b) Area of the rectangle. 37. Sand is pouring from a pipe at the rate of 12c.c/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when height is 4 cm? 38. The area of an expanding rectangle is increasing at the rate of 48 cm2/ 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? 39. Find a point on the curve y = (x – 3)2 where the tangent is parallel to the line joining the points (4, 1) and (3, 0). 40. Find the equation of all lines having slope zero which are tangents to the curve y  1 . x  2x  3 2 41. Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1. 42. Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3. 43. Show that the curves 4x = y2 and 4xy = k cut as right angles if k2 = 512. 44. Find the equation of the tangent to the curve y  3x  2 which is parallel to the line 4x – y + 5 = 0. XII – Maths 52
  53. 53. 45. Find the equation of the tangent to the curve x  y  a at the point  a2 a2   , .  4 4 16 . 3 46. Find the points on the curve 4y = x3 where slope of the tangent is 47. Show that 48. Find the equation of the tangent to the curve given by x = a sin3t, x y   1 touches the curve y = be–x/a at the point where the a b curve crosses the y-axis. y = b cos3 t at a point where t  49.  . 2 Find the intervals in which the function f(x) = log (1 + x) – x , x  1 1 x is increasing or decreasing. 50. Find the intervals in which the function f(x) = x3 – 12x2 + 36x + 17 is (a) Increasing (b) Decreasing. 51. Prove that the function f(x) = x2 – x + 1 is neither increasing nor decreasing in [0, 1]. 52. Find the intervals on which the function f  x   53. Prove that f  x   54. Find the intervals in which the function f(x) = sin4x + cos4x, 0  x  x is decreasing. x 1 2 x3  x 2  9 x, x  1, 2 is strictly increasing. Hence find 3 the minimum value of f (x).  is 2 increasing or decreasing. 55. Find the least value of 'a' such that the function f(x) = x2 + ax + 1 is strictly increasing on (1, 2). 53 XII – Maths
  54. 54. 3 56. 5 Find the interval in which the function f x   5x 2  3x 2 , x  0 is strictly decreasing. 57. Show that the function f(x) = tan–1 (sin x + cos x), is strictly increasing on   the interval  0,  .  4 58. 59.   Show that the function f x   cos  2x   is strictly increasing on  4 3 7    .  ,  8 8  Show that the function f  x     sin x is strictly decreasing on  0,  .  2 x Using differentials, find the approximate value of (Q. No. 60 to 64). 1 1 60. 0.009  3 . 62. 0.0037  2 . 61. 255  4 . 63. 0.037. 1 64. 25.3 . 65. Find the approximate value of f (5.001) where f(x) = x3 – 7x2 + 15. 66. Find the approximate value of f (3.02) where f (x) = 3x2 + 5x + 3. LONG ANSWER TYPE QUESTIONS (6 MARKS) 67. Show that of all rectangles inscribed in a given fixed circle, the square has the maximum area. 68. Find two positive numbers x and y such that their sum is 35 and the product x2y5 is maximum. 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 XII – Maths 54 2 times the radium of the base.
  55. 55. 71. Show that the semi vertical angle of right circular cone of given surface  1 area and maximum volume is sin1   .  3 72. 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  2 a3 3 2 2 b3 . 73. 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. 27 74. Find the interval in which the function f given by f(x) = sin x + cos x, 0  x  2 is strictly increasing or strictly decreasing. 75. Find the intervals in which the function f(x) = (x + 1)3 (x – 3)3 is strictly increasing or strictly decreasing. 76. Find the local maximum and local minimum of f(x) = sin 2x – x,    x  . 2 2 77. Find the intervals in which the function f(x) = 2x3 – 15x2 + 36x + 1 is strictly increasing or decreasing. Also find the points on which the tangents are parallel to x-axis. 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 1 metre min. How 2 fast must h (height) be changing when r and h are 10 metres. is constant but radius r is increasing at the rate of 79. Find the equation of the normal to the curve x = 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 = 4x3 – 2x5, find all the points at which the tangent passes through the origin. 81. Find the equation of the normal to the curve x2 = 4y which passes through the point (1, 2). 55 XII – Maths
  56. 56. 82. Find the equation of the tangents at the points where the curve 2y = 3x2 – 2x – 8 cuts the x-axis and show that they make supplementary angles with the x-axis. 83. Find the equations of the tangent and normal to the hyperbola at the point (x0, y0). x2 y2  1 a2 b 2 84. A window is in the form of a rectangle surmounted by an equilateral triangle. Given that the perimeter is 16 metres. Find the width of the window in order that the maximum amount of light may be admitted. 85. A jet of an enemy is flying along the curve y = x2 + 2. A soldier is placed at the point (3, 2). What is the nearest distance between the soldier and the jet? 86. Find a point on the parabola y2 = 4x which is nearest to the point (2, –8). 87. A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each cover and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum. 88. A window in the form of a rectangle is surmounted by a semi circular opening. The total perimeter of the window is 30 metres. Find the dimensions of the rectangular part of the window to admit maximum light through the whole opening. 89. An open box with square base is to be made out of a given iron sheet of area 27 sq. meter, show that the maximum value of the box is 13.5 cubic metres. 90. A wire of length 28 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? 91. Show that the height of the cylinder of maximum volume which can be inscribed in a sphere of radius R is 92. 2R 3 . Also find the maximum volume. Show that the altitude of the right circular cone of maximum volume that 4r can be inscribed is a sphere of radius r is . 3 XII – Maths 56
  57. 57. 93. 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 h 3 tan2 . 27 95. Show that the right triangle of maximum 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 box 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  : ( + 2). ANSWERS 1. 0.8 cm/sec. 2. 4.4 cm/sec. 3. 2 cm3/sec. 4. 80 cm2/sec. 5. Rs. 208. 6. Minimum value = 4, maximum value = 6. 7. Maximum value = 7, minimum value does not exist. 8. a  0. 9. [0, ] 10. (0, e] 11. x  1 12. (– , 0) U (0, ) 13.    0,  .  6 14. Maximum value = 4, minimum valve = 0. 15. a > 1. 16. R 17. 7 18.  1 1  , .  2 4 19. (2, – 3) 20. 1 4 21. (1, 7) 57 XII – Maths
  58. 58. 1 . 2 22. (0, 0), (2, 4) 23. 24. 1 – . 4 25. 6r 26. 2  cm2/cm 27. 72 28.  30. a > 0. 31. 4, 11 33. a . 2b 29. Rs. 80. 32. 8  cm sec. 3 1 cm sec.  34. 2.5 km/hr. 35. 4 cm sec. 45 36. (a) –2 cm/min, (b) 2 cm2/min 37. 1 cm sec. 48 38. 7.11 cm/sec. 39.  7 1  , .  2 4 40. y  42. 2x + 3my = am2 (2 + 3m2) 44. 48x – 24y = 23 45. 2x + 2y = a2 46. 128   8 128   8 , ,   , .  3 27   3   27  48. y = 0 49. Increasing in (0, ), decreasing in (–1, 0). 50. Increasing in (– , 2)  (6, ), Decreasing in (2, 6). XII – Maths 31  and  4,   .  3 58 1 . 2
  59. 59. 25 . 3 52. (– , –1) and (1, ). 53. 54.     Increasing in  ,  Decreasing in  0,  4 2  55. a = – 2. 56. Strictly decreasing in (1, ). 60. 0.2083 61. 3.9961 62. 0.06083 63. 0.1925 64. 5.03 65. –34.995 66. 45.46 68. 25, 10 74.     5  Strictly increasing in 0,    , 2  4  4    . 4   5  Strictly decreasing in  , . 4 4   75. Strictly increasing in (1, 3)  (3, ) Strictly decreasing in (–, –1)  (–1, 1). 76. Local maxima at x  Local max. value   6 3   2 6 Local minima at x    6 Local minimum value  77.  3   2 6 Strictly increasing in (–, 2]  [3, ) Strictly decreasing in (2, 3). 59 XII – Maths
  60. 60. Points are (2, 29) and (3, 28). 78. 3  metres min.  79. x + y tan – a sec = 0. 80. (0, 0), (–1, –2) and (1, 2). 81. x + y = 3 82. 5x – y – 10 = 0 and 15x + 3y + 20 = 0 83. xx 0 a 2  yy 0 b 2  1, y  y0 2 a y0  x  x0 b 2x 0  0. 16 84. 6 3 85. 5 86. (4, –4) 87. 3cm 88. 60 30 , . 4 4 90. 112 28 cm, cm. 4 4 XII – Maths 60
  61. 61. CHAPTER 7 INTEGRALS POINTS TO REMEMBER  Integration is the reverse process of Differentiation.  Let  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-axis. d F  x   f  x  then we write dx  f x dx  F x   c . STANDARD FORMULAE 1. 2.    x n 1  c x dx   n  1   log x  c n n  1 n  –1   ax  b n  1 c    n  1 a n  ax  b  dx   1  log ax  b  c a 3.  sin x dx 5.  tan x . dx 4.  – cos x  c. n  1 n  1  cos x dx  sin x  c.  – log cos x  c  log sec x  c . 61 XII – Maths
  62. 62. 6.  cot x dx 8.  cosec x . dx 2  cosec x cot x dx 11.  sec x dx 12.  cosec x dx 13. x x  e dx  e  c . 15.  1 1 16. 1 x 17. x 18. 19. 20. 2 1 x 2 2 x 2 a 2 14. –1 –1  x 2 a 2 1  x 2 dx  1 x  c. –1 log 2a dx  1 x  c , x  1. 1 a a  x  c. a  x log 2a dx  x  a dx  x  c, x  1 . 1 1 XII – Maths  log cosec x – cot x  c . dx  sec 1 a  – cosec x  c . dx  sin 1 x  sec x . tan x . dx x . dx  tan x  c.  log sec x  tan x  c . dx  tan 2  sec 9.  – cot x  c. 10. 2 7.  log sin x  c . tan x a  c. x a –1 x  c. a 62 ax c log a  sec x  c .
  63. 63. 21. 22. 23.    1 a 2 2 – x 1 2 2 a  x 1 x 2 – a 2 2 dx  sin  c. 2 dx  log x  dx  log x  x 2  25.  a  x dx  x 2 a x 2 2  a 2 x 2 2  x 2  a dx  x sin a 2 2 2 2 a  x  c. 2 a 2 2 2 a x a  x dx   x a 24. 26. –1  c. –1 x  c. a 2 2 log x  a  x log x  x 2  c. 2 x 2 a 2  2 a 2 2 a 2  c. 2 RULES OF INTEGRATION 1.  k .f x dx  k  f x dx . 2.  k f x   g x dx  k  f x dx  k  g x dx . INTEGRATION BY SUBSTITUTION 1. 2. f ´ x   f  x  dx  log f  x   c . n  f  x   f ´x  dx   f  x  n  1  c. n 1 63 XII – Maths
  64. 64. 3. f ´ x   f  x   n  f  x  n  1 dx   c. –n  1 INTEGRATION BY PARTS  f  x  . g  x  dx  f x .    g  x  dx  –  f ´ x  .   g  x  dx  dx . DEFINITE INTEGRALS b  f  x  dx  F  b   F  a  , where F  x    f  x  dx . a DEFINITE INTEGRAL AS A LIMIT OF SUMS. b  f  x  dx a where  f  a   f  a  h   f  a  2h    lim h    .....  f  a  n  1 h   h0    b  a h  b . or h  a  n  f  x  dx  lim  h f  a  rh   h0    r 1   PROPERTIES OF DEFINITE INTEGRAL b 1.   a c  b f  x  dx  a  f t  dt . a b  f  x dx   f  x  dx   f  x  dx. a a b 4. 2. b b 3. b a f  x  dx  – f  x  dx . (i)  a XII – Maths c b f  x dx   a f  a  b  x  dx . a (ii)  0 64 a f  x dx   f  a  x  dx. 0
  65. 65. a 5.  f  x   0; if f  x  is odd function. –a a 6. a  f x dx  20 f x dx , if f(x) is even function. a 2a 7.  0  2a f  x  dx , if f  2a  x   f  x   f  x  dx    0  0, if f  2a  x   f  x   VERY SHORT ANSWER TYPE QUESTIONS (1 MARK) Evaluate the following integrals 1 1. 3. 1  sin x  cos 1 x  dx . 2. e x dx . 1 1  x  x8  8  6.  x log x log log x  dx .  e 1 5. x 99 cos4 x dx . 1 7. 0  4  3 sin x  log  dx .  4  3 cos x   8. 9.  cos 2x  2sin2 x   dx .    cos 2 x 10. 2 11.  x c  c x  dx . 8 x  dx . x 8 4.  1 sin2 x dx . 1 a log x  e x log a  dx . 2 7  sin x dx .   2 12. 65 d  f  x  dx  .  dx   XII – Maths
  66. 66. 1  sin2 x cos2 x dx . 15.  log e dx . e 17. 2 x x e x dx . x dx . 2 19.  x  1 21.  cos 23.  sec x .log sec x  tan x  dx . 2 1  dx . 14.  16.  dx . ex  a x dx . 18. 13. x  x 1 x x 1 e dx . x dx . 20.  22.  x cos   1 dx . 24.  cos   x sin  dx . 26. 1    x  2  dx .   28.  x  cos x dx . 30.  32. 1     ax  ax  dx .   x 1 1 3 25.  cot x .logsin x dx . 27.  x 2  3 log x  dx . 29.  1 1  cos x dx . sin x x  1  33. 0 34. 0 x dx  x e 1  e x 1 dx . x e  ex 2 x  log x dx . 31. x 1  sin x cos x dx . 2 XII – Maths where [ ] is greatest integer function. 66
  67. 67. 35. 2 2 0 x dx where [ ] is greatest integer function. f x  b 36. a f x   f a  b  x  dx . 38. If 40. x 1 1x x dx . 39. 37. a f x  dx  b f  x  dx . 2 x dx . 1 a 1 0 1 x 2  , then what is value of a. 4  b a SHORT ANSWER TYPE QUESTIONS (4 MARKS) 41. (i)  x cosec  tan–1x 2  1 x 4 dx . (ii) 1  x 1 x 1 x 1 x 1 dx . cos  x  a  (iii)  sin  x  a  sin  x  b  dx . (iv)  cos  x  a  dx . (v)  cos x cos 2x cos 3x dx . (vi)  cos (viii)  cot 2 (vii)  sin (ix)  4 x cos x dx . sin x cos x 2 2 2  dx . (x) a sin x  b cos x 6 (xi) 2  6 sin x  cos x 2 2 dx . (xii) sin x cos x 67  5 3 x dx. 4 x cosec x dx . 1 dx . 3 cos x cos  x  a  sin x  cos x dx . sin 2x XII – Maths
  68. 68. 42. Evaluate : (i) x x 4  x 2 dx . 1 *(ii)  x  6  log x  (iii) 1 x  x (v) (vii) (ix) (xi) 43. dx   7 log x  2   (iv) x  a  x  b 5x  2 2 dx . . 2 1  3x  1 2 dx . dx . (viii)  2x  1 x  2 4x  x (x) dx . 2   3x  2  x 2  x  1 dx . Evaluate : dx (i)  x x (ii)  1  cos x   2  3 cos x  dx . (iii)  cos XII – Maths (vi) 7  1 . sin x sin  cos  2 d .   cos   2 68 (xii)   x x  1 9  8x  x 2 sin  x    dx . sin  x    x 2 dx . 2 dx .  6x  12 2 1  x – x dx . sec x  1 dx .
  69. 69. (iv) (v) (vii) (ix) (xi) 44. x 1   x  1  x  2   x  3  dx . x 2  x  2   x  2  x  1 dx . (vi) . (viii) dx   2x  1  x sin x  sin 4x  2  4 (x) dx . (xii) tan x dx .  x 2  1  x 2  2    x 3  3   x 2  4  dx . dx  sin x 1  2 cos x  . x x 4 x x 2 1  x 2 2  9  dx . 1 dx .  81 Evaluate : (i) x 5 (iii) e ax (v) 3 (ii) cos bx  c  dx .  cos  sin (vi)  1  sin 2x  dx .   1  cos 2x   e 2x (ix)  2ax  x dx . (xi) e x (xii)   log  log x    log x    2 x 3 (viii) x dx. (vii)   sec (iv) sin x dx . e x  x (x) e 3 x dx . 6x –1 1  9x –1 tan 2 dx . x dx .  x  1 dx .   2x 2   x 2  1  x  12 dx .  2  sin 2x  dx . 1  cos 2x  1   dx . 2   69 XII – Maths
  70. 70. (xiii) (xiv)  x  2 (xv)   2x  5  (xvi) 45.   6x  5   x 2 2 6  x  x dx . x  3 dx . x  3 x 2  4x  3 dx .  4x  8 dx . Evaluate the following definite integrals :  4 (i)  sin x  cos x  9  16 sin 2x 2 dx . (ii) 0 0 1 (iii)  cos 2x log sin x dx . x 0 1 x 1 x 1 2 2 2  (iv) dx . sin 1 x 1  x 2 3 2 0 dx .  2 (v)  sin 0 2 sin 2x 4 4 dx . (vi) x  cos x x 5x 2 1 2 dx .  4x  3  2 (vii) x  sin x  1  cos x dx . 0 46. Evaluate : 3 (i)    x  1  x  2  x  3  dx . (ii) 1 XII – Maths 0 70 x  1  sin x dx .
  71. 71.   4 (iii) 2  log 1  tan x  dx .  log sin x dx . (iv) 0 0  (v) x sin x  1  cos 2 0 dx . x 3  2x  x   3 f  x  dx where f  x    x  3 x  2  3x  2  2  when  2  x  1 2 (vi)  when  1  x  1 when 1  x  2.  2 (vii) x sin x cos x  sin 0 4 4  (viii) a x 2 0 47. dx . x  cos x 2 2 dx . 2 cos x  b sin x Evaluate the following integrals (i)  3 6 1 (iii)  1  (v) 1 dx 1 (ii) tan x  1  sin x log   1  sin x x tan x  sin 0    dx .   sec x cosec x (iv) e 0 a dx . (iv) 0  a 71 1  2x  dx .  1  x2   e cos x cos x e a  x  cos x dx . dx . a  x XII – Maths
  72. 72. 1 48.  2x dx where [ ] is greatest integer function. 0 log x  log sin x 49. e 51. dx .  sin2x dx . 52. sin x e  sin x sin2x dx . b 4 53.  log x 1 log x 50. sin x dx . 54. dx . a  f x dx  f a  b  x  dx .   4 a 55. 1  sec x  tanx dx . 56. sin2 x  1 cos x dx . 57. 1  tan x  1 tan x dx . 58. ax  b x  c x dx . 59. Evaluate sin1 x  cos 1 x b dx , x   0, 1 (i)  sin (ii)  1 x 1 x  x 2  1  log  x 2  1  2 log x    dx 4 x (iii) (iv) (v) XII – Maths 1 x  cos 1 x dx x2   x sin x  cos x  2 1  sin dx  3 x dx a  x (vi)   6 72 sin x  cos x sin 2x dx
  73. 73.  2 (vii)  sin  – x  cos x  dx  2 2 (viii)  x 2  dx , where [x] is greatest integer function 1 3 2 (ix)  x sin x dx . 1 LONG ANSWER TYPE QUESTIONS (6 MARKS) 60. Evaluate the following integrals : (i) (iii) x x 5 5  4 dx . (ii)  x 2x dx  x  1  x 3   x  1  x  3  2 dx (iv) x x 4 x 4  4 2 dx 4 dx – 16  2 (v)  cot x  dx . tan x  (vi) 0  (vii) –1  1  x 0 61. x tan x 2 2  1 dx . 1 dx . Evaluate the following integrals as limit of sums : 4 (i)  2  2x  1 dx . (ii) 2  x 2  3  dx . 0 73 XII – Maths
  74. 74. 3 2   3x (iii) 4  2x  4  dx .   3x (iv) e 2x  dx . 0 1 5  x (v) 2 2  3 x  dx . 2 62. Evaluate 1  cot (i) 1 1  x  x 2  dx 0 dx   sin x  2 cos x  2 sin x  cos x  (ii) 1  (iii) log 1  x  0 1  x2 63. 1  sin x  sin2x dx . 65.  sec 3 x dx .  2 dx   2 log sin x (iv) – log sin 2x  dx . 0 3 sin   2 cos  64.  5  cos2   4 sin  d . 66. e 2x cos 3x dx . ANSWERS 1.  2. x  c. 2e – 2 2 3. tan x + c. 4. 8x x9 x2   8log x   c. log8 9 16 5. 0 6. log | log (log x) | + c XII – Maths 74
  75. 75. 8. x a 1 a x  c a  1 loga 7. 0 9. tan x + c 10. 0 11. x c 1 c x  c c  1 log c 12. f(x) + c 13. tan x – cot x + c 14. 2 32 2 32 x   x  1  c 3 3 15. log |x| + c 16. e    a 17. 2x e x c log 2e  18. 2  x  13 2  2  x  11 2  c . 3 19. log x  1  20. 2e 21. x cos2  + c 22. log x cos   1  c. cos  23. log sec x  tan x 2  c 24. log cos   x sin  c sin  25. logsin x 2  c 26. x4 1 3x 2  2  3 log x  c. 4 2x 2 1  c. x 1 2 2 x x log e a   c c 27. 1 log 2  3log x  c . 3 28. log |x + cos x| c 29. 2 log |sec x/2| + c. 30. 1 log x e  e x  c . e 75 XII – Maths
  76. 76. 31. x  log x 2  c 32. a 33. 0 34. 1 35.  36. b a 2 37. –1 38. 0 39. 1 40. 0 2 2  1 x 2 log ax   2x  c . 2 a (i) 1 1  log  cosec  tan1 x 2   2   c . 2  x  (ii) 41. 1 2 1 x  x x 2  1  log x  x 2  1  c. 2 2   1 (iii) (iv) sin  a  b  sin  x  a  sin  x  b  c x cos 2a – sin 2a log |sec (x – a)| + c. 1 (v) log 12x  6 sin 2x  3 sin 4x  2 sin 6x   c . 48 2 3 sin x  1 5 (vi) sin x  (vii) 1  1 1 1   2x  sin 2x  sin 4x  sin 6x   c . 32  2 2 6  (viii) 4  cot6 x cot x      c.  6 4  3 sin x  c . 5 1 (ix)  a2  b 2  2 2 2 2  c. a sin x  b cos x [Hint : put a2 sin2 x + b2 cos2 x = t] XII – Maths 76
  77. 77. (x) –2 cosec a cos a  tan x . sin a  c . [Hint. : Take sec2 x as numerator] (xi) (xii) 42. tan x – cot x – 3x + c. sin–1 (sin x – cos x) + c. (i) 1 tan 3 (ii) (iii) log –1  2x 2    3 2 log x  1 1   c.  [Hint : put x2 = t] C [Hint : put log x = t] 3 log x  2 1 log 5 5 – 1  2x c 5  1  2x (iv)  x  4 sin1   c.  5   (v) 2 log x  a  x b c (vi)  cos  sin 1  cos x   sin . log sin x    cos      Hint :   5 (vii) (viii) (ix) 2 log 3 x  2 x  1  6  4x  x sin  x    sin  x     11 tan  1 3 2 x  3 log x 2 2 2 1    2 2 sin x  sin    sin  x    3 x  1   c  2   6x  12  2 3 tan  4 sin 2 sin x  sin   c 1  x  3  c  3   x  2  c   2   77 XII – Maths
  78. 78. (x) 1  1 x  x 3 2 2  1  3  2x  1 1  x  x 2 8  5 sin 1 16 1   2  x   x  x  1    7 2  c    1 2 2 3  x  x 1   log x  2  8  3 2 (xi) x 2 (xii)  log cos x   x  1  2 x  1    c  5  1 2 cos x  cos x  c  2 [Hint : Multiply and divide by 43. (i) 1 log 7 (ii) (iii) x x 7 7 c 1 1  cos x log c 2  3 cos x 2 1 log cos   2  3 (iv) (v) 9 log 1  cos   c . 3 log x  3  10 4 log x  2  15 x  4 log  x  2 2 1 x 1 c 6 c x 1 (vi) x  2 3 tan x  1  x     3 tan    c 2   3 1  [Hint : put x2 = t] (vii) XII – Maths 2 17 log 2x  1  1 log x 17 2  4  1 34 78 tan 1 x 2 c sec x  1 ]
  79. 79. (viii) 1  log 1  cos x  2 1 log 1  cos x  6 2 log 1  2 cos x  c 3 [Hint : Multiply Nr and Dr by sin x and put cos x = t] (ix) 1 1  sin x log 8 (x) 1 1  sin x log 2 (xi) x x 1 2 1 tan 44. (i) 2 sin x c c 1 tan x   tan x  1 log     2 tan x  2 2 tan x  1 3 2 2 sin x  x 1 1 tan 1 1 log 4 2  x 1 2 2 (xii) 1  2 tan x  1 c 2 tan x  1  x2  9   c  3 2  1 3 3 3   x cos x  sin x   c  3 1  sec x tan x  log sec x  tan x   c 2 [Hint : Write sec3x = sec x . sec2 x and take sec x as first function] (ii) (iii) e ax 2 a b (iv) 2x tan 2 1  a cos  bx 3x  1  c   b sin  bx  c    c1 log 1  9x 2 c [Hint : put 3x = tan ] 3 (v) (vi) 2  x sin x  cos x   c 3  x 4  1 x x 1 tan x –   c.    4  12 4 79 XII – Maths
  80. 80. 1 (vii) (ix) (x) e 2x tan x  c . 2 x a 2ax  x e (viii) 2  2 a 2 1 sin 2 x  c. 2x  x  a c   a   x  x  1 e   c.  x  1  (xi) ex tan x + c. (xii) x log log x – x  c . [Hint : put log x = t  x = et ] log x (xiii) 2  6  x  x 2 32  25  2x  1 2 1  2x  1   8 6  x  x  sin  c  5   8  4 (xiv) (xv) 1  x  2  x 2  9  3 log x  2 x 2  9  c 2 2 x 2  4x  3  3 1  log x  2  3 2  x  2    2   x 2 x 2  4x  3  4x  3  c 2 (xvi)  x  2   2   1 45. (i) XII – Maths x 2  4 x  8  2 log  x  2   log 3. (ii) 20 80 –  4 x 2  4x  8  c
  81. 81. (iii) (v)  – 4 1 . 2  [Hint : put x2 = t] (vii) 4 15 (vi) (viii) 47. (i) (iii) log 2. 2 6 log   . 5 2 x sin x       Hint :   dx .   1  cos x 1  cos x    8.  (ii) log 2. (iv) 8  – log 2. 2 2  . 4 95/12.   Hint :   (vii)  8 1 (v) 1 25 /2. (iii) – . 5 – 10 log (i)  2 (vi) 46. (iv)  2  1 f  x  dx  2  1 f  x  dx  2  2 f  x  dx   f  x  dx    1  1 2 . 16    Hint : Use   2 . 2ab  . (ii) 12 0. (iv) 81  a  a f x   0  f a  x    0    log 2. 2 /2. XII – Maths
  82. 82. (v)  2 (vi) 4 48. 1 2 49. –x cos x + sin x + c. 50. x + log x + c. 51. 1 log sec x  tan x  c . 2 52. 1  sin3x     sin x   3  2 53. 2 2 54. 0 55. log |1 + sin x| + c 56. x – sin x + c 57. log |cos x + sin x| + c 58. a. a c x  b c x  C. log a c  log b c  (i) 2  2x  1 2 x  x2 sin1 x   x c   (ii) –2 1  x  cos 1 x  (iii) 59.  XII – Maths 1 1  1  2  3 x  3 2 x  x2  c  1  2  log  1  2     c  x  3  82
  83. 83. (iv) sin x  x cos x c x sin x  cos x (v)  x  a  tan1 x  ax  c (vi) 2 sin1 (vii) 0 (viii)  2  (ix) 60. (i) a 3 1 2 3  5 3 1  2 .   x  4 log x  5 log x  1  4 3 log x  1 4  log x 2 1  1 tan –1 x  c. 2 1 x  log 2 (ii) (iii) 1 x 1 – x 1 log x  1  5 1 1 1 2x  1 x  (vi) x x  log x 2 log x 1 4 – 10 81 log x  3  8 log x  2 1  tan x  2 2 4 tan 10 log x  1  2 (v) –1 2 8 (iv) tan 1  c. 1 –1 x    c. 2  27 2  x  3  c.  x    c.  2 2.  1 2 2 tan 1  x2  1 2x  1 4 2 83 log x x 2 2  2x  1  2x  1 c XII – Maths
  84. 84. (vii) /8. (i) 14. (iii) 26. (iv) 1 26 61. (ii) 127  e . 2 (i)   log 2 2 (ii)  (iii)  log 2. 8 (iv) 62. . 2 141 (v) 8 . 3   1 log   .  2 2 1 tan x  x log c 5 2 tan x  1 63. 1 1 2 log 1  cos x  log 1  cos x   log 1  2cos x  c. 6 2 3 64. 3log  2  sin   65. 1 1 sec x  tan x  log sec x  tan x  c. 2 2 66. e 2x 2cos 3x  3 sin3x   c . 13 XII – Maths 4 c. 2  sin  84
  85. 85. CHAPTER 8 APPLICATIONS OF INTEGRALS POINTS TO REMEMBER AREA OF BOUNDED REGION  Area bounded by the curve y = f(x), the x axis and between the ordinates, x = a and x = b is given by b Area =  f  x  dx a y y y = f(x) O a a x b b O x y = f(x)  Area bounded by the curve x = f(y) the y-axis and between abscissas, y = c and y = d is given by d Area =  f y  dy c y y d d x = f(y) x = f(y) c c x O 85 O x XII – Maths
  86. 86.  Area bounded by two curves y = f(x) and y = g(x) such that 0  g(x)  f(x) for all x  [a, b] and between the ordinate at x = a and x = b is given by Y y = f(x) B A y = g(x) a O X b b Area =  f  x  – g  x   dx a  Required Area k  b  f x  dx   f  x  dx. a k Y y = f(x) A2 O A y 1 A1 X B(k, 0) x = b x=a LONG ANSWER TYPE QUESTIONS (6 MARKS) 1. Find the area enclosed by circle x2 + y2 = a2. 2.  Find the area of region bounded by   x , y  : x  1  y   3. Find the area enclosed by the ellipse x a XII – Maths 86 2 2  y b 2 2  1 2 25  x  . 
  87. 87. 4. Find the area of region in the first quadrant enclosed by x–axis, the line y = x and the circle x2 + y2 = 32. 5. Find the area of region {(x, y) : y2  4x, 4x2 + 4y 2  9} 6. Prove that the curve y = x2 and, x = y2 divide the square bounded by x = 0, y = 0, x = 1, y = 1 into three equal parts. 7. Find smaller of the two areas enclosed between the ellipse x a and the line 2 2  y b 2 2  1 bx + ay = ab. 8. Find the common area bounded by the circles x 2 + y 2 = 4 and (x – 2)2 + y2 = 4. 9. Using integration, find the area of the region bounded by the triangle whose vertices are (a) 10. (–1, 0), (1, 3) and (3, 2) (b) (–2, 2) (0, 5) and (3, 2) Using integration, find the area bounded by the lines. (i) x + 2y = 2, y – x = 1 and 2x + y – 7 = 0 (ii) y = 4x + 5, y = 5 – x and 4y – x = 5. 11. Find the area of the region {(x, y) : x2 + y2  1  x + y}. 12. Find the area of the region bounded by y = |x – 1| and y = 1. 13. Find the area enclosed by the curve y = sin x between x = 0 and x  3 2 and x-axis. 14. Find the area bounded by semi circle y  15. 16. Find area of smaller region bounded by ellipse 2 Find area of region given by {(x, y) : x2  y  |x|}. 25  x x 2 9  y and x-axis. 2  1 and straight 4 line 2x + 3y = 6. 87 XII – Maths
  88. 88. 17. Find the area of region bounded by the curve x 2 = 4y and line x = 4y – 2. 18. Using integration find the area of region in first quadrant enclosed by 3 y and the circle x2 + y2 = 4. x-axis, the line x  19. Find smaller of two areas bounded by the curve y = |x| and x2 + y2 = 8. 20. Find the area lying above x-axis and included between the circle x2 + y2 = 8x and the parabola y2 = 4x. 21. Using integration, find the area enclosed by the curve y = cos x,   y = sin x and x-axis in the interval  0,  .  2 22. Sketch the graph y = |x – 5|. Evaluate  6 x  5 dx . 0 23. Find area enclosed between the curves, y = 4x and x2 = 6y. 24. Using integration, find the area of the following region :  x , y  : x 1  y  5  x2  ANSWERS 1. a2 sq. units. 2.   1  25 –  sq. units.  4 2 3. ab sq. units 5. 8. 10. 2  6 9 8  9 8 4. 1 sin    1  sq. units 3  8   2 3  sq. units   3  (a) 6 sq. unit XII – Maths 7. (4 – 8) sq. units    2  ab sq. units 4 9. (a) 4 sq. units (b) 2 sq. units [Hint. Coordinate of vertices are (0, 1) (2, 3) (4, – 1)] 88
  89. 89. (b) 15 sq. [ Hint : Coordinate of vertices are (– 1, 1) (0, 5) (3, 2)] 2 11. 1     sq. units 4 2 12. 13. 3 sq. units 14. 1 sq. units 25  sq. units 2 1 15. 17. 3 9 sq. units 16. 3    2  sq. units 2 sq. units 18. 8  sq. unit 3 19. 2 sq. unit. 20. 4  8  3  sq. units 3 21. 2  22. 13 sq. units. 23. 8 sq. units. 24. 1  5   sq. units   4 2 2  sq. units. 89 XII – Maths
  90. 90. CHAPTER 9 DIFFERENTIAL EQUATIONS POINTS TO REMEMBER  Differential Equation : Equation containing derivatives of a dependant variable with respect to an independent variable is called differential equation.  Order of a Differential Equation : The order of a differential equation is defined to be the order of the highest order derivative occurring in the differential equation.  Degree of a Differential Equation : Highest power of highest order derivative involved in the equation is called degree of differential equation where equation is a polynomial equation in differential coefficients.  Formation of a Differential Equation : We differentiate the family of curves as many times as the number of arbitrary constant in the given family of curves. Now eliminate the arbitrary constants from these equations. After elimination the equation obtained is differential equation.  Solution of Differential Equation (i) Variable Separable Method dy  f  x, y  dx We separate the variables and get f(x)dx = g(y)dy Then (ii)  f  x  dx  g  y  dy  c is the required solutions. Homogenous Differential Equation : A differential equation of the form XII – Maths  dy dx  f x, y  g x , y  90 where f(x, y) and g(x, y) are both
  91. 91. homogeneous functions of the same degree in x and y i.e., of the form dy y  F   is called a homogeneous differential equation. x dx For solving this type of equations we substitute y = vx and then dy dv . The equation can be solved by variable dx dx separable method. (iii)  v  x Linear Differential Equation : An equation of the from dy  Py  Q where P and Q are constant or functions of x only dx is called a linear differential equation. For finding solution of this type of equations, we find integrating factor (I.F .)  e Solution is y I .F .    Q. I.F .  dx  P dx . c VERY SHORT ANSWER TYPE QUESTIONS (1 MARK) 1. Write the order and degree of the following differential equations. (i) dy (ii)  cos y  0. dx 5 4 (iii) d y dx (v) (vii) 4  d 2y   sin x   2  .  dx   d 2y  1   2  dx  dx  d 3y   3  dx  2  d 2y    2  dx  5 (iv) d y dx 13 dy  dy    dx   5 2 2  3 d y dx 2  4.  dy   log   0.  dx   32 . (vi) 2   dy   1     dx       2  k d y dx 2 . 3  sin x . 91 (viii) dy  dy   tan   0  dx   dx XII – Maths
  92. 92. 2. Write the general solution of following differential equations. (i) dy  x 5  x 2 2 . (ii)  x . (iv)  dx (iii) dy  x 3 e x e dx (v) dy  1  cos 2x (vi) . dy dx 1  cos 2y  5 x y .  1  2y . 3x  1 Write integrating factor of the following differential equations (i) dy  y cos x  sin x dx (ii) dy 2  y sec x  sec x  tan x dx (iii) x (v) x (vii) 4. dy dx dx 3. (ex + e–x) dy = (ex – e–x)dx x 2 dy 4  y  x . dx dy  3y  x dx dy 1  dx 1 x 2 dy  y log x  x  y dx (iv) (vi) 3 x dy  y tan x  sec x dx y  sin x Write order of the differential equation of the family of following curves (i) (iii) (v) y = Aex + Bex + c (ii) Ay = Bx2 (x – a)2 + (y – b)2 = 9 (iv) Ax + By2 = Bx2 – Ay (vi) y = a cos (x + b) x a (vii) XII – Maths 2 2  y b 2 2  0. y = a + bex+c 92
  93. 93. SHORT ANSWER TYPE QUESTIONS (4 MARKS) 5. (i) Show that y  e 1  x 2  d 2 dx (ii) y 2 m sin  x 1 x is a solution of dy 2  m y  0. dx Show that y = sin(sin x) is a solution of differential equation 2 d y dx 2   tan x  dy 2  y cos x  0. dx 2 2 (iii) x d y dy B  x  y  0. Show that y  Ax  is a solution of 2 dx x dx (iv) Show that y = a cos (log x) + b sin (log x) is a solution of x 2 2 d y dx (v) dy  y  0. dx  x  Verify that y  log x  equation : a2 (vi) 2  x2  x 2  a2 d 2y dx 2 x  satisfies the differential dy  0. dx Find the differential equation of the family of curves y = ex (A cos x + B sin x), where A and B are arbitrary constants. (vii) (viii) 6. Find the differential equation of an ellipse with major and minor axes 2a and 2b respectively. Form the differential equation representing the family of curves (y – b)2 = 4(x – a). Solve the following differential equations. (i) dy  y cot x  sin 2x . (ii) x dy 2  2y  x log x . dx dx 93 XII – Maths
  94. 94. (iii) dx  dy (iv) 1 sin x . y  cos x  x 3 cos x , x  0. x dy  cos x  sin x . dx  (v) (vi) 7. ydx  x  y ye dx  y 3  y3  dy  0  2 xe y  dy Solve each of the following differential equations : dy dy   2  2y  .  dx dx  (i) y  x (ii) cos y dx + (1 + 2e–x) sin y dy = 0. (iii) x 1  y dy  y 1  x dx  0. 2 1  x 2   1 – (iv) (v) (vi) 2 y 2  dy  xy dx  0. (xy2 + x) dx + (yx2 + y) dy = 0; y(0) = 1. dy 3 3 x  y sin x cos x  xy e . dx (vii) 8. tan x tan y dx + sec2 x sec2 y dy = 0 Solve the following differential equations : (i) x2 y dx – (x3 + y3) dy = 0. (ii) x 2 dy  x 2 2  xy  y . dx (iii) XII – Maths  x 2  y 2  dx  2 xy dy  0, 94 y  1  1.
  95. 95. x x    y sin dx  x sin  y dy .        y y (iv) dy (vi) x dy (viii) 2xy  dx 9. 2  y 1– y  dx 1 x  3xy (ix) (v)  y 2 (vii) 2 dy dx dy  y  y  tan   .  x x  e x y 2 y  x e . dx 2 2 .  dx   x 2   xy dy  0 Form the differential equation of the family of circles touching y-axis at (0, 0). (ii) Form the differential equation of family of parabolas having vertex at (0, 0) and axis along the (i) positive y-axis (ii) positive x-axis. (iii) Form differential equation of family of circles passing through origin and whose centre lie on x-axis. (iv) 10. (i) Form the differential equation of the family of circles in the first quadrant and touching the coordinate axes. Show that the differential equation dy  dx x  2y x  2y is homogeneous and solve it. 11. Show that the differential equation : (x2 + 2xy – y2) dx + (y2 + 2xy – x2) dy = 0 is homogeneous and solve it. 12. Solve the following differential equations : (i) dy  2y  cos 3x . dx dy 2    y cos x  2 sin x cos x if y    1  2 dx (ii) sin x (iii) 3e tan y dx   1  e x x  sec2 y dy 95  0 XII – Maths
  96. 96. 13. Solve the following differential equations : (i) (x3 + y3) dx = (x2y + xy2)dy. (ii) x dy – y dx  (iii)  y  y y  x cos    y sin    dx x  x  x 2 2  y dx .  y  y – x  y sin    x cos    dy  0. x  x  (iv) x2dy + y(x + y) dx = 0 given that y = 1 when x = 1. y (v) xe x  y  x dy  0 if y(e) = 0 dx (vi) (vii) 16. (x3 – 3xy2) dx = (y3 – 3x2y)dy. dy  dx y  y  cosec    0 given that y  0 when x  1  x x Solve the following differential equations : 2 dy  tan x  y . dx (i) cos x (ii) x cos x dy  y  x sin x  cos x   1. dx  (iii) x x  1  e y dx  e y  1  x  dy  0.  y   (iv) (y – sin x) dx + tan x dy = 0, y(0) = 0. XII – Maths 96

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