CLASS XII MATHEMATICS Units Weightage (Marks) (i) Relations and Functions 0 (ii) Algebra (Matrices and Determinants) (iii) Calculus 44 (iv) Vector and Three dimensional Geometry 7 (v) Linear Programming 06 (vi) Probability 0 Total : 00 Design Type of Questions Weightage of Number of Total Marks each question questions (i) Very short answer (VSA) 0 0 0 (ii) Short Answer (SA) 04 48 (iii) Long Answer (LA) 06 07 4 Internal Choice There will be internal choice in 4 questions of short answer type and in questions of Long answer type Questions requiring Higher Order thinking skills (HOTS) have been added in every chapter XII Maths
CHAPTER RELATIONS AND FUNCTIONS Empty relation is the relation R in X given by R = X X Universal relation is the relation R in X given by R = X X Refleive relation R in X is a relation with (a, a) R, a X 4 Symmetric relation R in X is a relation satisfying (a, b) R (b, a) R 5 Transitive relation R in X is a relation satisfying (a, b) R and (b, c) R (a, c) R 6 Equivalence relation R in X is a relation which is refleive, symmetric and transitive 7 A function f : X Y is one-one (or injective) if f( ) = f( ) =,, X 8 A function f : X Y is onto (or surjective) if given any y Y, X such that f() = y 9 A function f : X Y is called bijective if it is one-one and onto 0 For f : A B and g : B C, the functions gof : A C is given by (gof) () = g[f()] A A function f : X Y is invertible if g : Y X such that go f = I and fog = I y A function f : X Y is invertible if and only if f is one-one and onto A binary operation * on a set A is a function * : A A A 4 An operation * on A is commutative if a * b = b * a, a, b A 5 An operation * on A is associative if (a * b) * c = a * (b * c) a, b, c A 6 An element e A, is the identity element for * : A A A if a * e = a = e * a, a, A 7 An element a A is invertible for * : A A A if there eists b A such that a * b = e = b * a, where e is the identity for * The element b is called inverse of a and is denoted by a XII Maths
If A is the set of students of a school then write, which of following relations are (Universal, Empty or neither of the two) R = {(a, b) : a, b are ages of students and a b 0} R = {(a, b) : a, b are weights of students, and a b < 0} R = {(a, b) : a, b are students studying in same class} R 4 = {(a, b) : a, b are age of students and a > b} Is the relation R in the set A = {,,, 4, 5} defined as R = {(a, b) : b = a + } refleive? If R, be a relation in set N given by R = {(a, b) : a = b, b > 5} Does elements (5, 7) R? 4 If f : {, } {,, 5} and g : {,, 5} {,,, 4} be given by f = {(,,), (, 5)}, g = {(, ), (, ), (5, )} Write down gof 5 Let g, f : R R be defined by g, f Write fog 6 If f : R R defined by f 5 be an invertible function, write f () 7 If f, Write fo f() 8 Let * is a Binary operation defined on R, then if (i) a * b = a + b + ab, write * (ii) a * b a b, Write ( * ) * 4 (iii) a * b = 4a 9b, Write ( * ) * XII Maths
9 What is the number of bijective function from a set A to B, when A and B have same number of elements 0 If f, g : R R be defined by 7 8 7 f, g, then 8 What is fog (7) Determine whether each of the following relations are (i) Refleive (ii) symmetric (iii) Transitive (iv) Equivalence relation (a) R = {(a, b) : a b, a, b R} (b) R = {(a, b) : (a b) is a multiple of, a, b R} (c) R = {(a, b) : (a b) is an even integer, a, b R} (d) R 4 = {(a, b) : a b = 0, a, b R} (e) R 5 = {(a, b) : a b, a, b R} (f) R 6 = {(a, b) : a = b +, a, b R} Check the following functions for one-one and onto (a) f : R R, f 5 (b) f : R R, f 4 (c) f : R R, f (d) f : A R, f, where A = R { } Let f : R R be a function defined by f Show that f is invertible and hence find 7 f 4 Let 4 4 f : R R be a function given by f 4 4 Show that f is invertible with f 4 4 4 XII Maths
5 Show that function f : A B defined as f is invertible and hence find f 4 5 7 where 7 A R, B R 5 5 6 Let * be a binary operation on Q Such that a * b = a + b ab (i) (ii) Prove that * is Commutative and associative Find identify element of * in Q (if eists) 7 If * is a binary operation defined on R {0} defined by a * b a, b and associativity then check * for cummutativity 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 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 (ii) Every non zero element of Q is invertible with 6 a, a Q 0 a Let f : A B defined as 4 4 f a where A, B R, what is fof () A relation R in the set R of real numbers is defined as R = {(a, b) : a b } Is R refleive? Consider f : R + [ 5, ] as f() = 9 + 6 5 Show that f is invertible and Hence find f 5 XII Maths
R Universal relation R Empty relation R Neither empty nor universal R 4 Neither empty nor universal Not refleive (5, 7) R 4 gof = {(, ), (, )} 5 fog = I R ie fog () = 5 6 f 7 fof 8 (i) (ii) 9 n! 0 7 69 7 (iii) 09 (a) Refleive and Transitive but not symmetric (b) (c) (d) (e) (f) Equivalence relation Equivalence relation Neither refleive nor symmetric nor transitive Neither refleive nor symmetric nor transitive Neither refleive nor symmetri nor transitive (a) Bijective function (ie, one-one and onto) (b) (c) (d) Neither one-one nor onto Neither one-one nor onto one-one but not onto 7 f 6 XII Maths
7y 4 5 f 5y 6 (ii) 0 is the identity element of * in Q 7 (i) * is not commutative (ii) * is not associative 8 (i) * is commutative as well as associative (ii) (iii) is identity element It is not refleive 6 f 7 XII Maths
CHAPTER INVERSE TRIGONOMETRIC FUNCTIONS sin, cos, etc, are angles,,,, sin (sin ) = and sin (sin y) y y cosec sin ; sec cos ; cot tan 4 sin ( ) = sin ; tan ( ) = tan ; cosec ( ) = cosec 5 cos ( ) = cos ; sec ( ) = sec ; cot ( ) = cot 6 sin cos or cos sin 7 tan cot or cot tan 8 sec cosec or cosec sec 9 y tan tan y tan ; y y 0 y tan tan y tan ; y y Infact every formula in trigonometry can be written in the language of inverse trigonometric functions Function Domain Range/Principal Value Branch Graph Inverse sine function [, +], sin = y = sin y Inverse cosine function [, +] [0, ] cos = y = cos y 8 XII Maths
Y Inverse tangent function R, tan = y Inverse cosecant function,,, 0 0, y = cosec = cosec y iff Inverse secant function ], ] [+, +[ 0, y = sec = sec y Inverse cotangent function R ]0, [ y = cot = cot y Write the principal value of (i) sin (ii) sin (iii) cos (iv) (v) tan (vi) cos tan (vii) cosec ( ) (viii) cosec () (i) cot () cot 9 XII Maths
(i) sec ( ) (ii) sec () (iii) sin cos tan What is value of the following functions (using principal value) (i) tan sec (ii) sin cos (iii) tan () cot ( ) (iv) cos sin tan cot (v) (vi) cosec sec (vii) tan () + cot () + sin () (viii) cot sin (i) 4 sin sin 5 () 7 cos cos 5 (i) 5 tan tan 6 (ii) cosec cosec 4 Show that cos cos tan cos cos 4 cos cos tan cot 0, sin cos 4 4 Prove 5 Prove 6 Prove a tan sin cos a a a 8 8 00 cot tan cos tan tan sin tan 7 7 6 7 Prove 8 Solve tan cos 4 cot cot 4 0 XII Maths
9 Solve 0 Prove Prove tan tan 4 tan tan m m n tan tan n m n 4 Prove y y tan sin cos y y Solve for cos tan 4 tan a cos b sin a tan b cos a sin b a if tan 0 b cot tan tan cos cos 5 Prove 6 Prove a b b c c a tan tan tan 0 ab bc ac If a, b, c > 0 7 Find the value of cot sin 8 Find value of for tan (cos ) = tan ( cosec ) 9 Epress the following in simplest form sin 0 If tan a + tan b + tan c = then prove that a + b + c = abc XII Maths
What is value of if sin sin cos 5 If f() = cos (log ) then what is value of f() + f(e) What is range of y = sin [], where [ ] is greatest integer function 4 What is value of cot sec sin (i) (ii) (iii) 5 6 (iv) 6 (v) (i) 6 (iii) (vi) () (i) 0 (ii) (v) (i) 5 (vi) () 6 5 (vii) (i) (iii) 6 (viii) (ii) (iv) (vii) (viii) (i) 6 6 (ii) 4 8 = 9 7 4 8 0 or 9 sin sin 4 =, 0, 4 0 6 XII Maths
CHAPTER and 4 MATRICES AND DETERMINANTS Matri : A matri is an ordered rectangular array of numbers or functions The numbers or functions are called the elements of the matri Order of Matri : A matri having m rows and n columns is called the matri of order mn Zero Matri : A matri having all the elements zero is called zero matri or null matri Diagonal Matri : A square matri is called a diagonal matri if all its non diagonal elements are zero Scalar Matri : A diagonal matri in which all diagonal elements are equal is called a scalar matri Identity Matri : A scalar matri in which each diagonal element is I, is called an identity matri or a unit matri A = [a ij ] n n a ij = 0 when i j = when i = j is a identity matri Transpose of a Matri : If A = [a ij ] 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 If A = [a ij ] m n Then transpose A = A = [a ij ] n m Transpose of A is denoted by A or A T Symmetric Matri : A square matri A = [a ij ] is said by symmetric if A = A or a ij = a ji i & j Skew Symmetric Matri : A square matri A = [a ij ] is said to be a skew symmetric matri if A = A or a ij = a ji i & j Inverse of a Matri : Inverse of a square matri A Adj A, provided A 0 A where (Adj A) is the adjoint matri which is the transpose of the cofactor matri Singular Matri : A square matri is called singular if A = 0, otherwise it will be called a nonsingular matri 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 If A is a nonsingular matri then its inverse eists and A is called invertible matri (AB) = B A Adj (AB) = (Adj B) (Adj A) XII Maths
(AB) = B A (A ) = (A ) Adj A = (Ad j A) If A is any non singular matri of order n, then adj A = A n If A be any given square matri of order n Then A (adj A) = (adj A) A = A I Where I is the identity matri of order n A B = A B where A and B are square matrices of same order Area of triangle with vertices (, y ), (, y ) and (, y ) = y y y The points (, y ), (, y ), (, ) are collinear y y y 0 What is the matri of order whose general element a ij is given by a ij i j if i j i j if i j If the matri P is the order and the matri Q is of order m, then what is the order of the matri PQ? If A 0 find A 4 If A = [ ] and B 6, find AB 5 What is the element a in the matri A = [a ij ] where R and a ij i j if i j i j if i j 6 Let P and Q be two different matrices of order n and n p then what is the order of the matri 4Q P, if it is defined 7 Let A be a 5 7 type matri, then what is the number of elements in the second column 8 Write the matri X if 8 7 5 X 6 0 0 0 9 Give an eample of two non zero matrices A and B such that AB = 0 4 XII Maths
0 If A P Q 0 matri P where P is symmetric and Q is skew-symmetric matri, then find the If A cos 0 sin 0 sin 70 cos 70, what is A? a ib c id Find the value of the determinants c id a ib Find the value of y if 8 4 4y 4 4 Write the cofactor of the element 5 in the determinant 6 6 0 4 5 7 a d g 5 Write the minor of the element b in the determinant b e h 6 If, find the values (s) of 5 5 c f i 7 If A = [a ij ] is matri and A ij is denote the co-factors of the corresponding elements a ij s, then what is the value of a A + a A + a A? 8 If A is a square matri of order and A =, find the value of A 9 For what value(s) of, the points (, 0), (, 0) and (4, 0) are colinear? 0 If 0 and the matri sin sin is singular, find the value of For what value of, the matri 5 has no inverse? If A 5, 6 8 find adj (adj A) It A = B, where A and B are square matrices of order and B = 5 What is A? 4 If the matri A sin cos, find AA cos sin 5 XII Maths
5 If B 0, and C 0 Find B C 6 Let A be a non singular matri of order such that A = 5 What is adj A? 7 Find a matri B such that 6 5 0 B 5 6 0 8 y y If, find 0 y 5y 0 6 and y 9 If A 0 0 0 0 0 0 For what value of, A will be a scalar matri a b b c c a 0 Find if b c c a a b c a a b b c Determine the value of for which the matri A 4 6 is singular? If A 5, write the matri A(adj A) Find the value of P 0 0 a q 0 b c r 4 If A is a matri and A A 0 adj, 0 what is A y y z z 5 If A z y Write the value of det A 6 If A 4 is symmetric matri, then find 7 Find, y, z and w if y z 5 y w 0 6 XII Maths
8 Find A and B if 0 A 8 and A B 0 6 9 Let 0 4 0 0 A, B and C 0, 6 verify that (AB)C = A(BC) 40 Find the matri X so that X 7 8 9 4 5 6 4 6 and 4, 4 If A B verify that (AB) = B A 4 Epress the matri matri P Q 4 5 where P is a symmetric and Q is a skew symmetric 4 Find the inverse of the following matri by using elementary transformations 7 6 44 Find the value of such that 5 0 5 45 If A 4, 5 find and y such that A A + yi = 0 46 Find A (adj A) without finding (adj A) if A 0 47 Given that A 4 7 Compute A and show that 9I A = A 48 Given that matri A Show that A 4A + 7I = 0 Hence find A 49 Show that A 4 satisfies the equation 6 + 7 = 0 Hence find A 7 XII Maths
50 Prove that the product of two matrices cos cos sin cos cos sin and cos sin sin cos sin sin odd multiple of is zero when and differ by an 5 Show that : sin cos sin sin cos sin 0 sin cos sin 5 Using the properties of determinant, prove the following questions 5 to 56 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 5 q r r p p q p q r y z z y y z 54 a bc ac c a ab b ac 4 a b c ab b bc c a b c 55 a b c a b c a b c 56 Show that : y z y z y z z y yz z y yz z y 57 (i) If the points (a, b) (a, b ) and (a a, b b ) are collinear Show that ab = a b (ii) If 5 4 A and B verity that AB A B 5 8 XII Maths
58 Given A 0 0 and B 0 0 59 Solve the following equations for Find the product AB and also find (AB) a a a a a a a a a 0 60 Verify that (AB) = B A for the matrices A 4 5 and B 5 4 6 Using matri method to solve the following system of equations : 5 7y =, 7 5y = 6 Let A and f() = 4 + 7 Show that f(a) = 0 Use this result to find A 5 6 If A cos sin 0 sin cos 0, 0 0 64 Find the matri X for which find adj A and verify that A (adj A) = (adj A) A = A I X 7 5 0 4 65 Using elementary transformations, find the inverse of the matrices 66 By using properties of determinants prove that a b ab b ab a b a a b b a a b 67 Solve the system of linear equations by using matri in equations y + 4z = z = y z = 9 XII Maths
68 Find A, where + y z = 4 + y + z = y 4z = A, hence solve the system of linear equations : 4 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 0 0 and B 0, 4 0 then compute (AB) 7 Determine the product equations 4 4 4 7 5 and use it to solve the system of y + z = 4, y z = 9, + y + z = 7 Solve the following system of equations using matri method 0 y z 4 4 6 5 y z 6 9 0 y z 74 For the matri A Show that A 6A + 5A + I = 0 and hence find A 0 XII Maths
75 How many matrices of order are possible with each entry as 0 or 76 If sin - 0 R, 0, and sin 4 sin Then find the value of 77 If A is a square matri of order such that adj A = 5, find A 78 If 0 0 A, find A 0 0 79 If 0 tan A and I is the identity matri of order, show that tan 0 I + A = (I A) cos sin sin cos 80 Using properties of determinants, show that b c a bc c a b ca a b b c c a a b c a b c a b c ab 8 If, y, z are the 0 th, th and 5 th terms of a GP find the value of log 0 log y log z 5 8 Show that y z y z y z yz yz y z z yz y XII Maths
8 If A 4, find A and hence solve the system of equations 7 + 4y + 7z = 4, y + z = 4, + y z = 0 0 0 0 m 4 [8] 5 0 6 7 5 8 5 0 9 A 0 0 0, B 0 0 0 0 0 0 a + b + c + d ( ) / 4 8 5 id fg 6 [Hint : 5 = 7] 7 0 8 54 [Hint : order A = ( ) A ] 9 = any real number 0 8 5 6 8 40 4 I 5 7 9 6 6 5 5 6 6 5 8 =, y = 9 0 0; [Hint : [R R + R + R ]] XII Maths
= 4 4 0 0 4 pqr 4 5 0 6 5 7 =, y =, z =, w = 4 8 A 9 9 5 7 7 7 7 7 7, B 8 4 4 5 7 7 7 7 7 7 40 X 0 4 7 44 = or = 4 45 = 9, y = 4 46 4I 47 7 A 4 48 A 7 49 4 A 7 58 AB, AB 6 6, y 4 4 6 5 8 9 A 8 [Hint : A 4A + 7I = 0, A = 4A 7I, A = 4 (4A 7I 7A) 64 X 6 4 5 [Hint : if A B = P, X = A P B ] 65 0 0 5 67 0 8, y, z 9 9 9 68 A 6 7 4 5 8 67 5 9 =, y =, z = XII Maths
69 =, y =, z = [Hint : Suppose three numbers as, y, z] 70 A 0 5 0 0 7 6 AB 7 9 0 7 =, y =, z = [Hint : (AB) = B A ] 7 =, y =, z = 5 [Hint : Let u, v, w ] y z 74 A 4 5 9 4 5 75 64 76, 6 77 +55 78 0 8 0 8 =, y =, z = 4 XII Maths
CHAPTER 5 CONTINUITY AND DIFFERENTIATION Continuity of a Function : A function f() is said to be continuous at = c if lim f f c ie, Lt f Lt f f c c c f() is continuous in ]a, b[ if it is continuous at c c a, b f() is continuous in [a, b] iff it is continuous in (a, b) and lim f f a lim b f f b a f() and g() are continuous functions at = c and d is constant then, f + g, f g, df, f g, f + d, f are all continuous at = c c f g is continuous at = c provided g(c) 0 Every polynomial function is continuous on R Every trigonometric function, Eponential function and logarithmic function are continuous in their respective domain f() is derivable at = c iff f f c lim c c denoted by f (c) eists and value of this limit is called the derivative at = c and is d u v u dv v du d d d dv dv v u d v v d u d d If y = f(u), = g(u) then dy f u d g u 5 XII Maths
If y is a function of t and t is a function of then, dy dy dt d dt d If = (t), y = (t) then dy t d y g t say then d t d dt g t d Rolle s theorem : If f() is continuous in [a, b] and derivable in (a, b) and f(a) = f(b) then there eists atleast one real no c (a, b) stf (c) = 0 LMVT : If f() is continuous in [a, b] and derivable in (a, b) then these eists atleast one point c (a, b) such that f b f a f c b a Write for what value of, f() = + is not derivable Write the set of points of discontinuity of the function, g() = What is derivative of f() = at 4 What are the points of discontinuity of the function f 5 Write all the points of discontinuity of the function f() = [] in [, ], where [] denotes the greatest integer function, 0 6 At what point sgn f 0, 0 7 Is the function e sin is continuous on R? is discontinuous, 8 If f then for which value of, f() is continuous on R 9 Write the value of k, for which f 0 What is the derivative of 6 wrt sin, 0 k, 0 is continuous R Given that g(0) = 7 and f() = g() Also f () and g () eist, then write value of f (0) Write the derivatives of the following functions (i) log ( ) (ii) e log 6 XII Maths
(iii) (iv) tan cot (v) sin 0 Discuss the continuity of following functions at the indicated points (i) f, 0, 0 at 0 (ii) f tan, 0 5 5 0 at 0 (iii) sin 0 f at 0 0 0 (iv) f() = + at = sin, 0 (v) f at 0 0 sin (vi), 0 f at 0 0, 0 (vii) f at 0 0 4 For what value of K, f 5 For what values of a and b 0 4 k 5 is continuous in it s domain a if f a b if b if is continuous at = 6 Prove that f() = + is continuous at = but not derivable at = 7 XII Maths
p sin, 0 7 If f is derivable at = 0 the find the value of p 0 0 dy 8 If y = (log ) + log then find? d 9 If y dy tan tan, 0, find d 0 If y dy sin tan then? d If + y = + y dy y then Prove that d If y = tan then show that d y dy 0 d d If y y a then show that dy d y 4 If y a y then show that dy d y dy y 5 If ( + y) r + s = r + y s then prove that d 6 If y sin sin tan where sin sin dy find d 7 Find the derivative of tan 8 Find derivative of log (sin ) wrt log with respect to sin 9 If sin log y then show that ( a ) y y a y = 0 dy 0 If y + y + = a b then find? d If = a cos, y = a sin then find d y d If = ae (sin cos ) dy y = ae (sin + cos ) then show that at is d 4 8 XII Maths
If 4 If sin dy y then find d y e 4 sin tan, dy find d 5 If 5 dy y sin,? d 6 If y = y dy, find d 7 If sin y = sin (a + y) then show that 8 If y = sin, find d y d in terms of y dy d sin a y n, n z sin a 9 If a y b then show that 4 b d y d a y 40 If y a + = 0 then Prove that d y d a 5 y 4 If a + by + by = the prove that d y h ab d h by 4 Write the points of discontinuity of f() = [] in [, 9] 4 Write the critical points of f log 44 Evaluate lim, where [ ] denotes the greatest integer function 45 If for a function f(), f () = ( ) ( + ) then write the interval in which f() is increasing or decreasing a 7, 46 If f b 5 0 is continuous for all values of, then find the value of a and b = / Derivative does not eist 4, 0, 9 XII Maths
5 0,,, 6 = 0 7 Yes 8 = 9 k = 0 4 f (0) = 7 (i) log e (ii) (iii) log e (iv) 0 (v) (i) Discontinuous (ii) Discontinuous (iii) Continuous (iv) continuous (v) Discontinuous (vi) Continuous (vii) Discontinuous 4 7 K 5 a = 0, b = 6 7 p > 8 log e log log log log log e e e e e 9 0 0 6 / 7 8 cot 0 y dy y y log y log d y log y d y d dy sec 4 cosec a d 4 4 y cosec sin 5 6 y y log y y log 8 sec y tan y 4 4, 5, 6, 7, 8, 9 4, 0, 44 Limit Does not eist 45 decreasing in (, ] and increasing in {, ) 46 a 5, b 0 XII Maths
CHAPTER 6 APPLICATIONS OF DERIVATIVES Rate of Change : If and y are connected by y = f() then dy of y wrt d represents the rate of change Equation of tangent to the curve y = f() at the point P(, y ) is given by y y dy Similarly equation of normal is y y d P dy d P The angle of intersection between two curves is the angle between the tangents to the curves at the point of intersection intersection P m m tan, m m where m, m are slopes of tangent at the point of A function f() is said to be strictly monotonic in (a, b) if it is either increasing or decreasing in (a, b) A function f() is said to be strictly increasing in (a, b) if, in (a, b) st < f( ) < f( ) Alternatively, f() is increasing in (a, b) if f () > 0 a b A function f() is said to be strictly decreasing in (a, b) if, in (a, b) st < f( ) > f( ) Alternatively, f() is strictly decreasing in (a, b) if f () < 0 (a, b) A function f() is said to have local maimum value at = c, if there eists a neighbourhood (c, c + ) of c, st f() < f(c) (c, c + ), c Similarly, local minimum value can be defined Local maimum and local minimum values of f() may not be maimum and minimum value of f() Critical Point : A point c is called critical point of y = f() if either f (c) = 0 or f (c) does not eist If f() is defined in [a, b] and f () = 0 gives =,,, n then Ma {f(a), f( ), f( ), f( n ), f(b)} is called global maimum value Similarly, Global minimum value can be defined, XII Maths
If f() = cos, [0, ] then write the interval in which f() is decreasing Write the interval in which f() = is decreasing For what value of, f() = sin is strictly increasing 4 Write the maimum value of f in [0, ] 5 Find the ma and min value of f sin 5 6 Write the slope of the normal to the curve f() = 7 + at = 7 If normal to the curve at a point P on y = f() is parallel to y ais, then write the value of f () at P 8 On the curve f, find the points at which tangent is parallel to the chord joining the points A, and B, 6 9 Write the least value of f, 0 0 y = is normal to the curve y = at which point? If y = e and y = b e cut each other at right angles, find the value of b If the tangent to y = is parallel to the line y = 0 then find the point of contact of tangent with the curve At which point on y 5, tangent makes an angle of 0 with the +ve directions of -ais 4 In which interval f() = tan is increasing? 5 If the radius of the circle is decreasing at the rate of cm/sec then, write the rate at which area is changing when r = 5 cm 6 If length and breadth of a rectangle are increasing at the rate of 5 cm/sec and cm/sec respectively Find the rate at which area is increasing if length = cm and breadth = 0 cm 7 Sand is pouring out from a pipe at the rate of cm /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 sand cone increasing when the height is 4 cm XII Maths
8 Find the points of local maima/minima for f() If f() = sin cos where 0 < < Also find the local maimum and minimum values 9 Find the interval(s) in which 4 f is increasing or decreasing? 0 Find the interval in which f() = log ( ) + 4 +, ( > ) is increasing or decreasing? For the curve y =, find the points on the curve at which the tangent passes through (0, 0) Prove that the function : f() = 50 + sin is strictly increasing on, Show that f cos 4 is an increasing function in 5, 8 8 4 Find the intervals in which f() = + 05 + 5 is increasing or decreasing? 5 Separate the interval [0, ] into the interval in which f() = sin is increasing or decreasing? 6 Find the point on the curve y = + 9 + 6 at which slope of the tangent to the curve is minimum Also, find the minimum slope 7 Find the absolute maimum value of f() = + sin in [0, ] 8 Show that the surface area of a closed cuboid with a square base and given volume is minimum when it is a cube y yy at, is a b a b 9 Show that the equation of the tangent to P y a 0 Find the equation of tangent to y = 4a at m a, m y Show that touches the curve y-ais y e at the point where the curve crosses If = y and y = r cut each other at right angles then find the value of r A point on the hypotenuse of a right 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 / ) / 4 If the length of three sides of a trapezium other than base are equal to 0 cm, then find the area of trapezium when it is maimum 5 Show that f() = sin 4 + cos 4, [0, /] is increasing on, 4 and decreasing on [0, /4] XII Maths
6 Find the equation of tangent to the curve y = ( ) ( ) at the points where the curve cuts the ais 7 Show that the semi-vertical angle of a cone of maimum volume and given height is tan 8 Prove that the radius of the right circular cylinder of greatest curved surface which can be inscribed in a given cone is half of that of the cone 9 A rectangular sheet of tin 45 cm 4 cm is to be made into a bo without top by cutting off square from each corner and folding up the flaps What should be the side of the square to be cut off so that the volume of the bo is maimum? 40 A wire of length 8 m is to be cut into two pieces One of the pieces is to be made into a square and the other into a circle What should be the lengths of the two pieces so that the combined area of the square and the circle is minimum? 4 For a given curved surface of a right circular cone when volume is maimum, prove that semivertical angle is sin 4 Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and semi-vertical angle is 4 h 7 tan 4 Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is of the volume of the sphere 8 7 44 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? 45 Show that f is neither increasing nor decreasing in (, ) 46 Find the least value of a such that f() = + a + is increasing on [, ] 47 In which interval f() = 5 / 5/, 0 is decreasing? 48 Find the point on curve y = ( + ) where tangent is parallel to the chord joining the points A(, 7) and B(, 5) 49 Verify mean value theorem for f in [0, 4] (if applicable) 7 50 Using differentials, find the approimate value of 007 4 XII Maths
5 If y = sin and changes from to 4 Find the approimate change in the value of y 5 A rectangular window is surmounted by an equilateral triangle Given that the perimeter is 6m, find the width of the window so that the maimum amount of light may enter 5 A particular moves along the curve 6y = + Find the points on the curve at which y-coordinate is changing 8 times as fast as -coordinate [0, ] (, 0], 4 / 5 Ma value = 8, Min Value = 6 7 0 8 9 0, 8, 4 6, 5 9, 4 4 (, ) 5 Decreasing at the rate of 0 cm /sec 6 74 cm /sec 7 cm sec 48 8 Local ma value = at Local min value = at 4 7 4 9 Decreasing in, 0, 8 8 Increasing in, 0, 8 8 0 Increasing in (, ) and decreasing in (, ) 5 XII Maths
7 4,, 0, 0 4 Increasing in (, 7) (5, ), Decreasing in ( 7, 5) 5 5 Decreasing in 0,, Increasing in 6 6 6 (, ), Minimum slope = 6 7 5, 6 6 0 y a m /8 m 4 75 sq units 6 + y = 0, 7 y 4 = 0 9 5 cm 40 8 m, m 4 4 44 5 46 47 [, ) 48 6, 5 5 49 MV Theorem Not applicable at =, because f() is not derivable 50 095 5 0 5 6 6 m 5 4,, 4, 6 XII Maths
CHAPTER 7 INTEGRATION Integration is inverse process of Differentiation n d n c n n log c n n a b c n n a b d n 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 cot d log sin c 7 8 sec d tan c 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 4 6 d tan c 5 a d log c 7 a a a d sin c, d sec c, a d log c a a a 8 d tan c 9 a a a a d sin c a 7 XII Maths
0 d log a c a a d log a c a a d a sin c a 4 a a d a log a c a a d a log a c 5 e d e c 6 a d a c log a f d log f c f n f f f d c n n f f n d n f n c f g d f g d f g d d b a f d F b F a, where F f d b f d f d a a f d f t dt b b a b a 8 XII Maths
b c b 4 f d f a b d f d f d f d a a c a 5 f 0; if f is odd function a a a 6 f d f d, if f a f 0 0 = 0 if f(a ) = f() Integral as limit of sum : b h 0 a lim f d h f a f a h f a h f a n h b a where h h b a b a Evaluate the following integrals (i) d (ii) sin d (iii) cos cos d (iv) d (v) (vii) 0 4 sin log d 4 cos d sin cos (vi) cosec cosec cot d Evaluate the following integrals (i) (ii) a d a (iii) 4 sin e sec d (iv) cos d cos 9 XII Maths
(v) d (vi) sec cosec d log tan (vii) cos sin a log log a d (viii) cos e e d (i) sin d, () log d (i) sin d (ii) log d (iii) a b d a b c (v) c d (vii) c 6 5 d (iv) (vi) (viii) sin d a b cos d log d 9 4 (i) 6 5 d () d 4 9 Evaluate the following definite integrals : (i) sin sin cos 0 d (ii) 7 sin d (iii) 0 d (iv) sin cos 0 d (v) 0 tan d (vi) 0 e e d 4 Evaluate the following integrals : (i) cosec tan 4 d (ii) d 40 XII Maths
(iii) sin a sin b (v) cos cos cos d (vii) 4 sin cos d d (iv) (vi) (viii) cos cos 5 cos d a d a 4 cot cosec d (i) sin cos a sin b cos d () cos d cos a (i) 6 6 sin cos d sin cos (ii) sin cos d sin 5 Evaluate : (i) 4 d *(ii) d 6 log 7 log (iii) d (iv) 9 8 d (v) a b d (vi) sin sin d a (vii) 5 d (viii) d 6 (i) 4 d () d (ii) d (iii) sec d 6 Evaluate : (i) d (ii) 7 sin d cos cos (iii) d cos cos (iv) d (v) d (vi) 4 d (vii) d 4 (viii) d sin cos 4 XII Maths
(i) sin d sin 4 () d 4 (i) tan d 7 Evaluate : (i) 5 a sin d (iii) e cos b c d (v) cos d (vii) (i) sin e d cos e d (ii) (ii) (iv) (vi) (viii) () 9 d 8 sec d 6 sin d 9 e e tan d d d (i) sin d cos e (ii) 8 Evaluate the following definite integrals : (i) 4 0 sin cos d 9 6 sin (ii) 0 log log d log cos log sin d (iii) (v) 0 0 sin sin d 4 4 cos d (iv) (vi) 0 sin d 5 d 4 (vii) 0 9 Evaluate : sin d cos (i) d (ii) 0 sin d 4 XII Maths
4 (iii) log tan d 0 (iv) 0 log sin d (v) 0 sin cos d when (vi) f d where f when when (vii) 0 sin cos sin 4 4 cos d (viii) 0 a cos b sin d 0 Evaluate the following integrals as limit of a sum (i) d (ii) e d (iii) e d (iv) 0 0 Evaluate the following integrals d (i) d (ii) 0 sin d (iii) sin log d sin (iv) 0 e e cos cos e cos d Evaluate the following integrals : (i) 5 5 4 d (ii) d 4 d (iii) d (iv) 4 d 4 6 4 XII Maths
(v) 0 tan cot d (vi) 4 d (vii) 0 tan d Evaluate the following integrals as limit of sums : 4 (i) d (ii) 0 d (iii) 4 d (iv) 5 (v) d 4 0 e d 4 Evaluate the following integrals : d (ii) tan cot d (i) sin cos d (iv) e (iii) cos log d (v) e d (vi) 4 4 4 sin d (vii) 0 tan d 44 XII Maths
5 Evaluate sin cos d cos (i), 0, sin (ii) d (iii) log log 4 d (iv) sin cos d (v) sin a d (vi) 6 sin cos d sin d sin cos d (viii) a cos b sin 0 (vii) d, where [] is greatest integer function (i) () sin d 6 Evaluate cot d d (ii) sin cos sin cos (i) 0 (iii) log d (iv) 0 0 log sin log sin d (i) c (ii) tan + c (iii) cosec + c (iv) 45 XII Maths
(v) 0; (vii) cot cosec + C (i) (i) sin + c 5 c (ii) 5 a log c a (iii) cos 4 e tan c (iv) cosec + c 4 (v) c (vi) log [log(tan )] + c (vii) tan + c (viii) a a a log a c (i) cos sin c log log c () (i) cos c (ii) (log ) + c (iii) log a b c c (iv) log a b cos c b (v) c c log c c c (vi) log + log + c (vii) 5 tan 0 4 c (viii) log c (i) 4 5 log c 40 4 5 () log 4 9 c (i) 4 (iii) 4 (ii) 0; (iv) 4 (v) (vi) log cosec tan c 4 (i) log c (ii) tan e 4 46 XII Maths
(iii) sin a log sin a b sin b c (iv) cos a sin a log sec ( a) + c (v) 6 sin sin 4 sin 6 c 48 (vi) (vii) (viii) 5 sin sin sin c 5 sin sin 4 sin 6 c 6 6 4 cot cot c 6 4 (i) a b a sin b cos c [Hint : put a sin + b cos = t] () cosec a cos a tan sin a c [Hint : Take sec as numerator] (i) tan cot + c (ii) sin (sin cos ) + c 5 (i) tan c [Hint : put = t] (ii) log log log C [Hint : put log = t] (iii) 5 log 5 5 c (iv) 4 sin c 5 (v) log a b c (vi) cos cos sin sin log sin sin sin c cos Hint : sin sin sin sin sin (vii) 5 log tan c 6 47 XII Maths
(viii) (i) log 6 tan c 4 4 sin c () 5 sin c 8 6 5 7 log c 8 (i) (ii) log cos cos cos c [Hint : Multiply and divide by sec ] 6 (i) 7 log 7 7 c (ii) cos log cos c (iii) (iv) 5 log log c 8 8 9 4 log log c 0 5 6 (v) 4 log c (vi) tan tan c [Hint : put = t] (vii) (viii) log log 4 tan c 7 7 4 log cos log cos log cos c 6 [Hint : multiply N r and D r by sin and put cos = t] (i) sin sin log log 8 sin 4 sin c () log c 48 XII Maths
(i) tan tan tan tan log tan tan tan c (ii) 9 tan c 7 (i) cos sin c sec tan log sec tan c [Hint : Write sec = sec sec and take sec as first function] (ii) a e (iii) a cos b c b sin b c c a b (iv) tan log 9 c [Hint : put = tan ] (v) sin cos c (vi) (vii) 4 tan c 4 4 e tan c e (viii) c e (i) c () e c (i) e tan + c (ii) log log c log [Hint : put log = t = et ] 8 (i) log 0 (ii) / (iii) 4 [Hint : put = t] (iv) log 4 49 XII Maths
(v) (vi) 5 5 6 5 0 log log 8 5 (vii) / sin Hint : d cos cos 9 (i) 8 (ii) (iii) log 8 (iv) log (v) 4 (vi) 95/ Hint : f d f d f d f d (vii) 6 (viii) ab Hint : Use a a f f a 0 0 0 (i) 6 (ii) e e (iii) (iv) 0 e (i) (ii) log (iii) 0 (iv) / (i) (ii) 5 4 log log log log tan c 4 4 5 4 4 Hint : 5 log log 4 tan c 5 5 5 50 Hint : 4 A B c D 4 50 XII Maths
(iii) 8 7 log log c 8 8 (iv) log tan c (v) (vi) tan log 4 c (vii) /8 (i) 4 (ii) 6 7 8 (iii) 6 (iv) e (v) 4 4 (i) c (ii) c (iv) c e (iii) tan log c (v) e (vi) 0 (vii) 4 5 (i) sin c (ii) cos c (iii) log c (iv) sin cos sin cos c (v) a tan a c (vi) a sin 5 XII Maths
(vii) 0 (viii) ab (i) 5 () 6 (i) log (ii) tan log 5 tan c (iii) log 8 (iv) log 5 XII Maths
CHAPTER 8 APPLICATIONS OF THE INTEGRALS Area bounded by the curve y = f(), the ais and between the ordinats, = a and = b is given by Area = b f d a y y = f( ) y O a b O a b y = f( ) Area bounded by the curve = f(y) the y-ais and between abcissae, y = c and y = d is given by Area = d c d dy f y dy c = f( y ) O y d c d c O y = f( y ) Area bounded by two curves y = f() and y = g() such that 0 g() f() for all [ab] and between the ordinate at = a and = b is given by b Area = f g d a 5 XII Maths
y y = f( ) y = g( ) O d c 4 If the curve y = f() interest the ais (-ais) then the area of shaded region is given by Area = A A a A c A b c Area = f d f d a b c F ind the area enclosed by circle + y = a Find the area of region bounded by y = 4, =, = 4 and ais in first quadrant Find the area enclosed by the ellipse a y a b b 4 Find the area of region in the first quadrant enclosed by ais the line y = and the circle + y = 5 Find the area of region {(, y) : y 4, 4 + 4y 9} 6 Prove that the curve y = and, = y divide the square bounded by = 0, y = 0, =, y = into three equal parts 7 Find smaller of the two areas enclosed between the ellipse b + ay = ab a y and the line b 8 Find the common area bounded by the circles + y = 4 and ( ) + y = 4 9 Using integration find area of region bounded by the triangle whose vertices are (a) (, 0), (, ) and (, ) (b) (, ) (0, 5) and (, ) 54 XII Maths
0 Using integration find the area bounded by the lines (i) + y =, y = and + y 7 = 0 (ii) y = 4 + 5, y = 5 and 4y = 5 F ind the area of the region {(, y) : + y + y} Find the area of the region bounded by y = and y = Find the area enclosed by the curve y = sin between = 0 and 4 Find the area bounded by semi circle y 5 Find area of region given by {(, y) : y } 5 and -ais and -ais 6 Find area of smaller region bounded by ellipse y and straight line + y = 6 9 4 7 Find the area of region bounded by the curve = 4y and line = 4y 8 Using integration find the area of region in first quadrant enclosed by -ais the line y and the circle + y = 4 9 Draw a sketch of the region {(, y) : + y 4 + y} and find its area 0 Find smaller of two areas bounded by the curve y = and + y = 8 Find the area lying above -ais and included between the circle + y = 8 and the parabola y = 4 Using integration find the area enclosed by the curve y = cos, y = sin and -ais in the interval 0, Sketch the graph y = 5 Evaluate 6 0 5 d a sq units 8 sq units ab sq units 4 4 8 sq units 55 XII Maths
5 9 9 sin 6 8 8 sq units 7 ab 4 sq units 8 8 sq units 9 (a) 4 sq units (b) sq units 0 (a) 6 sq unit [Hint Coordinate of verties are (0, ) (, ) (4, )] (b) 5 sq [Hint Coordinate of verties are (, ) (0, 5) (, )] sq units 4 sq units sq units 4 5 7 5 sq units sq units 6 sq units 9 sq units 8 8 sq unit 9 sq unit 0 sq unit 4 8 sq units 5 sq units sq units 56 XII Maths
CHAPTER 9 DIFFERENTIAL EQUATION 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 : The degree of differential equation is defined to be the degree of highest order derivative occurring in it after the equation has been made free from radicals and fractions Solving a differential equation dy (i) Type f g y d d = h(y) dy The : Variable separable method separate the variables and get f() f d h y dy c is the required solution (ii) dy Homogenous differential equation : A differential equation of the form d f, y g, y where f(, y) and g(, y) are both homogeneous functions of the same degree in and y ie, of the form dy F y is called a homogeneous differential equation Substituting d dy dv y = v and then v, we get variable separable form d d (ii) Linear differential equation : Type I : dy py q where p and q are functions of d p d Its solution is y (I F) = q(i F)d where I F e Write the order and degree of the following differential equations dy (i) cos y 0 d (ii) dy d d y 4 d (iii) 4 d y d y sin 4 d d 5 (iv) 5 d y d 5 dy log 0 d 57 XII Maths
*(v) dy d y d d (vi) dy d y K d d (vii) d y d y d d sin Write the general solution of following differential equations (i) dy d 5 (ii) (e + e ) dy = (e e )d (iii) dy e e d (iv) dy 5 y d (v) dy d cos cos y (vi) dy d y What is the integrating factor in each of the following linear differential equations dy (i) y cos sin d (ii) dy y d cos (iii) sin cos dy dy y cos (iv) log y tan e d d (v) dy d y log d dy (vi) tan y sec y 4 (i) Verify that (ii) y sin m e is a solution of d y dy Show that y = sin (sin ) is a solution of diff equation d y d (iii) Show that y A B dy tan y cos d is a solution of d y dy y 0 d d m y 0 d d (iv) Show that function y = a cos (log ) + b sin (log ) is the solution of d y dy y 0 d d 58 XII Maths
(v) (vi) Find the differential equation of the family of curves y = e (A cos + B sin ), where A and B are arbitrary constants Find the differential equation of an ellipse with major and minor aes a and b respectively (vii) Form the differential equation corresponding to the family of curves y = c( c) (viii) Form the differential equation representing the family of curves (y b) = 4( a) 5 Solve the following diff equations dy (i) y cot sin d (ii) dy y log d (iii) d sin y cos, 0 dy dy (iv) cos cos sin d (v) y ey d = ( + e y ) dy 6 Solve each of the following differential equations : (i) (iii) dy dy y y (ii) cos y d + ( + e ) sin y dy = 0 d d (iv) y dy y d 0 y dy y d 0 (v) (y + ) d + (y + y) dy = 0; y(0) = dy (vi) d y sin cos y e (vii) tan tan y d + sec sec y dy = 0 7 Solve the following differential equations : (i) y d ( + y ) dy = 0 (ii) (iii) y d y dy y 0, dy y y d (iv) sin y d sin y dy y y dy y y (v) tan d (vi) dy y d y (vii) dy d e y y e (ii) dy d y 59 XII Maths
8 (i) Form the differential equation of the family of circles touching y-ais at (0, 0) (ii) (iii) Form the differential equation of family of parabolas having verte at (0, 0) and ais along the (i) positive y-ais (ii) +ve -ais Form differential equation of all circles passing through origin and whose centre lie on -ais 9 dy y Show that the differential equation d y 0 Show that the differential equation : is homogeneous and solve it ( + y y ) d + (y + y ) dy = 0 is homogeneous and solve it Solve the following differential equations : dy (i) y cos d (ii) dy sin y cos sin cos given that y = when d Solve the following differential equations : (i) ( + y ) d = ( y + y )dy (ii) dy y d y d (iii) y y y y y cos y sin d y sin cos dy 0 (iv) dy + y( + y) d = 0 given that y = when = y dy (v) e y 0 given that y = 0 when = e d (vi) ( y ) d = (y y)dy dy dy (i) Write the order and degree of the differential equation tan 0 d d 60 XII Maths
(ii) What will be the order of the differential equation, corresponding to the family of curves y = a cos ( + b), where a is arbitrary constant (iii) What will be the order of the differential education y = a + be + c where a, b, c are arbitrary constant dy (iv) Find the integrating factor for solving the differential education y tan cos d (v) Find the integrating factor for solving the differential equation d dy y sin y 4 (i) Form the differential equation of the family of circles in the first quadrant and touching the coordinate aes (ii) Verify that y log a satisfies the differential education (iii) d y dy a d 0 d dy y y Show that the general solution of the differential equation d by ( + y + ) = A( y y) Write A is parameter 0 is given 5 Solving the following differential equation (i) (tan y ) d = ( + y ) d dy d (ii) y (iii) y y e d e dy 0 y (iv) ( sin y) dy + tan y d = 0, y(0) = 0 6 Solve the following differential equation y y dy y d y y d dy cos (i) sin (ii) e tan y d + ( e ) sec y dy = 0 given that y, when = 4 (iii) dy d y cot cot given that y(0) = 0 6 XII Maths
(i) order =, degree = (ii) order =, degree = (iii) order = 4, degree = (iv) order = 5, degree not defined (v) order =, degree = (vi) order =, degree = (vii) order =, degree = 6 (i) y log c (ii) 6 6 y log e e c e (iii) 4 e y e c 4 e (iv) 5 + 5 y = c (v) (y ) + sin y + sin = c (vi) log + + log e y = c (i) e sin (ii) e tan (iii) e / (iv) e log (v) 4 (vi) sec y d y dy (v) y 0 d d [Hint : find dy d d y, and eliminate A and B ] d (vi) (vii) (viii) 5 dy d y = dy y y d d d dy dy 4y y d d d y dy 0 d d dy [ Hint : divide y by and find c] d sin (i) y sin c (ii) y 4 loge c 6 6 XII Maths
c (iii) y sin, 0 (v) = y e y + cy (iv) y = tan + ce tan 6 cy y (ii) (i) (iii) y c e sec y c (iv) y log y c y (v) y (vi) log y cos cos e e 5 c (vii) log tan y cos y c 7 (i) y log y c (ii) y tan log c [Hint : Homogeneous Equation] (iii) + y = cos y (iv) y ce [Hint : Put y (vi) c y y v ] (v) sin y c (vii) (viii) y e e c [Hint : Factorise RHS] sin y sin c 8 (i) dy y y 0 [Hint : The family of circles is, y d g 0 ] 6 XII Maths
dy (ii) y, y d dy d dy (iii) y y 0 d 9 log y y y tan c c y y 0 sin cos (i) y (ii) y sin cosec ce (i) log y c y (ii) c y y (iii) y cos y c [Hint : Put y = v] (iv) y y [Hint : Put y = v] y log log, 0 (vi) c y y (v) (i) Order =, Degree = not define (ii) Order = (iii) Order = (iv) sec (v) e tan y 4 (i) ( y) { + y } = ( + y y ) 5 (i) = tan y + c e tan y (ii) = y + ce y (iii) y ye c (iv) y = sin 6 (i) C y sec y (ii) ( e) tan y = ( e ) (iii) y = 64 XII Maths
CHAPTER 0 VECTORS AND THREE DIMENSIONAL GEOMETRY Vector : A directed line segment represents a vector Addition of vectors : If two vectors are taken as two sides of a triangle taken in order then their sum is the vector represented by the third side of triangle taken in opposite order (triangle law) Multiple of a vector by a scalar : a is any vector and R then a in a direction parallel to a a is vector of magnitude If a 0 then a is unit vector in direction a a Scalar Product : a b a b cos where is the angle between a and b Projection of a along b is a a a a b b Vectors a and b are perpendicular iff a b 0 Cross Product : a b a b sin n b, and is the angle between a and b a b Unit vector perpendicular to plane of a and b is a b Vector a and b are collinear if a b 0 i j k a b a a a where b b b a a i a j a k and b b i b j b k where n is a unit vector perpendicular to a and 65 XII Maths
Area of a triangle whose two sides are a and b is a b Area of a parallelogram whose adjacent sides are a and b is a b If a, b represents the two diagonals of a parallelogram, then area of parallelogram a b Distance between P(, y, z ) and Q(, y, z ) is P Q y y z z The coordinates of point R which divides line segment PQ where P(, y, z ) and Q(, y, z ) in ratio m : n are m n my ny mz nz,, m n m n m n If are the angles made by any line with coordinate aes respectively then l, m, n Where l = cos, m = cos, n = cos are called the, direction cosines of the line and l + m + n = If a, b, c are the direction ratios then direction cosines are a b c l, m, n a b c a b c a b c Direction ratios of a line joining (, y, z ) and (, y, z ) are : y y : z z Vector equation of straight line : (i) Through a point A a and parallel to vector b is r a b (ii) Passing through two points A a and B b is r a b a (iii) Line passing through two given points (, y, z ) and (, y, z ) is y y z z, in cartesian form y y z z Angle between two lines with DC s l, m, n and l, m, n is given by cos = l l + m m + n n with DR s <a, b, c > or <a, b, c > OR a a b b c c cos a b c a b c 66 XII Maths
If lines are r a b and r a µ b then, b b cos b b Equation of plane : (i) Passing through A a and perpendicular to n where a n d (ii) Passing through three given points is is r a n 0 Or r n d y y z z y y z z 0 y y z z (iii) Having intercepts a, b, c on coordinate aes is y z a b c n n Angle between two planes r n d and r n d is cos n n Distance of a point (, y, z ) from a plane a + by + cz + d = 0 is a by cz d a b c Equation of plane passing through intersection of two planes a + b y + c z + d = 0 and a + b y + c z + d = 0 is (a + b y + c z + d ) + (a + b y + c z + d ) = 0 Equation of plane passing through intersection of two planes r n d and r n d r n n d d m n Angle between a plane r n d and a line r a m is sin m n is What is the horizontal and vertical components of a vector a of magnitude 5 making an angle of 50 with the direction of -ais What is a R such that a, where Write when y y i j k? 4 What is the area of a parallelogram whose sides are given by i j and i 5 k? 5 If A is the point (4, 5) and vector AB respectively then write point B has components and 6 along -ais and y-ais 67 XII Maths
6 What is the point of trisection of PQ nearer to P if position of P and Q are i j 4k and 9i 8 j 0 k 7 What is the vector in the direction of i j k, whose magnitude is 0 units? 8 What are the angles which i 6 j k makes with coordinate aes 9 Write a unit vector perpendicular to both the vectors i j k and i j k 0 What is the projection of the vector i j on the vector i j? If a, b and a b, what is the value of a b? For what value of, a i j 4k is perpendicular to b i 6j k?, if What is a a b a b and b a? 4 What is the angle between a and b, if a b a b? 5 What is the area of a parallelogram whose diagonals are given by vectors i j k i k? 6 Find if for a unit vector a, a a 7 If a b a b, 8 If a and b are two unit vectors and a between a and b? then what is the angle between a and b b is also a unit vector then what is the angle 9 If i, j, k are the usual three mutually perpendicular unit vectors then what is the value of i j k j i k k j i? 0 What is the angle between and y if y y? Write a unit vector in y-plane, making an angle of 0 with the +ve direction of ais If a, b and c are unit vectors with a b c 0, a b b c c a? If a and b are unit vectors such that a what is the angle between a and b? b then what is the value of is perpendicular to 5 a 4 b, then 4 Write a unit vector which makes an angle of 4 with ais and with z-ais and an acute angle with y-ais 68 XII Maths
5 What is the ratio in which y plane divides the line segment joining the points (,, 4) and (, 5, 6)? 6 If coordinate of the point P on the join of Q(,, ) and R( 5,, ) is 4, then in what ratio P divides QR 7 What is the distance of a point P(a, b, c) from -ais? 8 Write the equation of a line passing through (,, ) and perpendicular to plane y + 4z = 7 9 What is the angle between the lines = y = z and 6 = y = 4z? 0 What is the perpendicular distance of plane y + z = 0 from origin? What is the y-intercept of the plane 5y + 7z = 0? Write the value of, so that the lines given below are perpendicular to each other y z y 5 z and 4 4 5 A (,, 0), B(5,, ) and C(5, 8, 0) are the vertices of ABC D and E are mid points of AB and AC respectively What are the direction cosines of DE? 4 What is the equation of the line, which passes through the point (, 4, 5) and parallel to y 4 z 8? 5 5 6 5 What is the angle between the straight lines : y z y z,? 4 6 If the direction ratios of a line are proportional to,, then what are the direction cosines of the line? 7 If a line makes angles and with -ais and y-ais respectively then what is the acute angle 4 made by the line with z ais? 8 What is the acute angle between the planes + y z + = 0 and 4 + 4y z + 5 = 0? 9 What is the distance between the planes + y z + = 0 and 4 + 4y z + 5 = 0 40 What is the equation of the plane which cuts off equal intercepts of unit length on the coordinate aes 4 What is the equation of the plane through the point (, 4, ) and parallel to the plane + y z = 7? 4 Write the vector equation of the plane which is at a distance of 8 units from the origin and is normal to the vector i j k 69 XII Maths
4 What is equation of the plane if the foot of perpendicular from origin to this plane is (,, 4)? 44 What is the angle between the line 4 = 0? y z and the plane + y z + 4 4 45 If O is origin OP = with direction ratios proportional to,, then what are the coordinates of P? 46 What is the distance between the line r i j k i j 4k from the plane r i 5 j k 5 0 47 If ABCDEF is a regular heagon then using triangle law of addition prove that : AB AC AD AE AF AD 6 AO O being the centre of heagon 48 Points L, M, N divides the sides BC, CA, AB of a ABC in the ratios : 4, :, : 7 respectively Prove that AL BM CN is a vector parallel to CK where K divides AB in ratio : 49 If PQR and P Q R are two triangles and G, G are their centroids, then prove that PP QQ RR GG 50 PQRS is parallelogram L and M are mid points of QR and RS Epress PL and PM in terms of PQ and PS Also prove that PL PM PR 5 The scalar product of vector i j k with a unit vector along the sum of the vectors i 4 j 5k and i j k is equal to Find the value of 5 a, b and c are three mutually perpendicular vectors of equal magnitude Show that a b + c makes equal angles with a, b and c with each angle as cos 5 If i j and b i j k then epress in the form of, where is parallel to and is perpendicular to 54 If a, b, c are three vectors such that a b c 0 then prove that a b b c c a 55 If a, b 5, c 7 and a b c 0, find the angle between a and b 70 XII Maths