DIRECTORATE OF EDUCATION GOVT. OF NCT OF DELHI

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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 NOTE Questions requiring Higher Order thinking skills (HOTS) have been added in every chapter Such questions are marked with a star, and to help the students, hints to their solutions are given along with the answers 6 XII Maths

CHAPTER RELATIONS AND FUNCTIONS POINTS TO REMEMBER Empty relation is the relation R in X given by R = φ 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 function 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, ab, A 5 An operation * on A is associative if (a * b) * c = a * (b * c) abc,, 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 7 XII Maths

VERY SHORT ANSWER TYPE QUESTIONS If A is the set of students of some boys school then write, which types 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 weights of students and a b > 0} R 4 = {(a, b) : a, b are students studying in same class} R 5 = {(a, b) : age of a is greater than age of b} If A = {,, 4, 5} then write whether each of the following relations on set A is a function or not? Give reasons also (i) {(, ), (, 4), (4, 5), (5, )} (ii) {(, 4), (, 4), (5, 4), (4, 4)} (iii) {(, ), (, 4), (5, 4)} (iv) {(, ), (, 5), (4, 5)} (v) {(, ), (, ), (4, 4), (4, 5)} * If f : R R, g : R R defined by 7 8 + 7 f ( ) =, g ( ) = then 8 find (i) (fog) (7) = (ii) (gof) (7) = 4 If f, g are the functions, given by f = {(, ), (, ), (, 7), (4, 6)} find fog g = {(0, 4), (, ), (, )} 5 If f ( ) = + write (fof) () 6 If f : R R defined by f ( ) =,find f ( ) =? 5 8 XII Maths

7 Check the following functions for one-one Also, give the reason for your answer (i) f : R R st f() = + R (ii) f : R {0} R {0} such that f() = (iii) f : R R such that f() = (iv) f : R R such that f() = (v) f : R R such that f() = ( ) ( ) ( ) (vi) f : R R such that f() = [] R where [ ] denotes the greatest integer function (vii) f : R R, f() = sin R (viii) f : [0, π] [, ], f() = cos [ 0, π] π π (i) f :, R, f ( ) = tan 8 Check whether the following functions are onto or not Give one reason for your Answer π π f :,,, f = sin (i) [ ] ( ) π π (ii) f : R,, f ( ) = tan (iii) f : R R, f () = (iv) f : R { 0 } R, f ( ) = (v) f : N N, f() = (vi) f : (0 ) R, f() = (vii) f : R { } R, f ( ) = + sin π [ ] f : R,, f = where + [ ] denotes the greatest integer function (viii) [ ] ( ) 9 If * is a Binary operation defined on R then if (i) a * b = a b, write 8 * ( * ) ab * write 4 * * 6 (ii) a b = ( ) 9 XII Maths

a b (iii) a * b = then write 0 * ( 7 * ) (iv) a * b = a find 5 * ( * ) b SHORT ANSWER TYPE QUESTIONS 0 Check the following relations for each of (i) Refleivity; (ii) Symmetricity; (iii) Transitivity; (iv) Equivalence Relation (a) R = {(A, B); A = B, A, B are line segments in the same plane} (b) R = {(a, b), (b, b), (c, c), (a, c), (b, c)} in the set A = {a, b, c} (c) R = {(a, b) : a b, a, b, R} (d) R 4 = {(a, b) : a divides b, a, b Î A } where A = {,, 4, 5} (e) R 5 = {a, b), (b, a), (a, a)} in {a, b, c} (f) R 6 = {(a, b) : a b, a, b N} *(g) R 7 = {(a, b) : a, b R, a b } (h) R 8 = {(a, b) : a b is multiple of 5, a, b, R} (i) R 9 = {(a, b) : b = a and a, b R} (j) R 0 = {(a, b) : a b is an integer, a, b R} Check the injectivity and surjectivity of the following functions (i) f : R R, f ( ) = 7 5 (ii) f : N N, defined by f() = N (iii) f : R R, defined by f() = (iv) f : R R, defined by f() = *(v) f : N N, defined by f ( ) = +, if is odd, if is even (vi) f : R R, defined by f() = ( + ) + 4 π π (vii) f : R, defined by f ( ) = tan ` 4 4 (viii) [ ] ( ) f : R, defined by f = sin 0 XII Maths

*(i) f : R R, defined by f ( ) = + () f : R Z, defined by f() = [], where [ ] denotes the greatest integer function :, defined by = + (i) f R { } R f ( ) *(ii) f : R R, defined by f() = ( ) ( ) * If A = N N and a Binary operation * is defined on A as * : A A A such that (a, b) * (c, d) = (ac, bd) Check whether * is commutative and Associative Find the identity element for * on A if any Let * is a Binary operation defined on R by a * b = a b, then (i) (ii) Is * commutative? Show that * is not associative by giving one eample for it 4 If * is a Binary operation defined on R {0} defined by (i) Is * Commutative? a a * b =, then b (ii) Is * Associative? 5 Let * be a binary operation on Q {} such that a * b = a + b ab (i) (ii) Prove that * is commutative and associative Also find the identity element in Q {} (if any) 6 If f : R R, defined by f ( ) = find f () if after checking f() for one-one and onto 4 7 If f : R {, }, defined by f() = sin invertible? If not, give reason If yes, find f () *8 If f : R R, f ( ) =, then find (fog) () and (gof) () Are they equal? 9 If f() = sin, g() = then find fog and gof Are they equal? XII Maths

CHAPTER INVERSE TRIGONOMETRIC FUNCTIONS POINTS TO REMEMBER Principal value branches of the branches of the inverse trigonometric function with their domains and Ranges : Function : Domain Range sin : [ ] cos π π,, : [, ] [0, π] π π,, 0 cosec : R ( ) { } π, 0, π sec : R ( ) [ ] tan : R cot : R (0, π) π π, Note : ( sin ) =, sin ( sin ) etc sin VERY SHORT ANSWER TYPE QUESTIONS ( Mark for Each Part) Write the principal value of (i) sin ( ) (ii) sin ( ) (iii) cos ( ) (iv) ( ) (v) tan (vi) cos tan (vii) cosec ( ) (viii) cosec () (i) cot () cot (i) sec ( ) (ii) sec () XII Maths

(iii) sin + cos + tan ( ) Simplify each of following using principal value : (i) tan sec (ii) sin cos (iii) tan () cot ( ) (iv) cos + sin (v) tan ( ) + cot (vi) ( ) ( ) (vii) tan () + cot () + sin () (viii) ( ) cosec + sec cot sin (i) 4π sin sin 5 () π cos cos 5 7 (i) π tan tan 6 5 (ii) π cosec cosec 4 SHORT ANSWER TYPE QUESTIONS (4 Marks for Each Part) Show that 4 Prove 5 Prove + cos + cos π tan = + + cos cos 4 sin + sin y = sin y + y sin sin y = sin y + y 6 Prove + y = y ( ) ( y ) cos cos cos *7 Prove a cos b sin a a tan = tan, tan + > 0 b cos + a sin b b cos + cos π tan cot = 0, π sin cos 4 8 Prove ( ) 9 Prove 0 Prove a tan = sin = cos a a a 8 8 00 cot tan cos + tan tan sin = tan 7 7 6 XII Maths

Prove Solve Solve + + π tan = + cos + 4 π cot + cot = 4 π tan + tan = 4 4 Prove ( ) ( ) cot tan + tan + cos + cos = π 5 (i) Prove a b b c c a tan + tan + tan = 0, abc,, > 0 ab + bc + ac (ii) *6 Prove that Find the value of π cot sin sin sin 4,, = (i) ( ) cos = cos 4,, (ii) ( ) 4 XII Maths

CHAPTER and 4 MATRICES AND DETERMINANTS POINTS TO REMEMBER Matri : A matri is an ordered rectangular array of numbers or functions The numbers of functions are called the elements of the matri Order of Matri : A matri having m rows and n coloumns 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 The diagonal elements may or may not be 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 Skew symmetric Matri : A square matri A = [a ij ] is said to be a skew symmetric matri if A = A Inverse of a Matri : Inverse of matri A = Adj A 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 5 XII Maths

(AB) = B A (A ) = (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 if y y y = 0 VERY SHORT ANSWER TYPE QUESTIONS ( Mark Each) 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 = find A 0 4 If A = [ ] and 6 B =, find AB 5 What is the element a in the matri A = λ[a ij ] where λ 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 If a + b =, 5 a b 5 5 find the value of a 9 Write the matri X if 8 7 5 X = 6 0 0 0 6 XII Maths

* 0 How many matrices of order are possible with each entry 0 or? Give an eample of two non zero matrices A and B such that AB = 0 If 0 A =, then find (A) 4 5 If A = = P + Q where P is symmetric and Q is skew-symmetric matri, then find the 0 matri P 4 If A = cos 0 sin 0, what is A? sin 70 cos 70 a + ib c + id 5 Find the value of the determinants c + id a ib 6 Find the value of y if 8 4 4y = 4 7 Write the cofactor of the element 5 in the determinants 6 6 0 4 5 7 a d g 8 Write the minor of the element b in the determinant b e h c f i * 9 If =, find the values (s) of 5 5 π * 0 If R, 0, and sin = 0, then find the values of sin 4 sin 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? * If A is a square matri of order and A =, find the value of A Find the area of triangle with vertices A(0, ), B(0, 4), C(, ) 4 For what value(s) of λ, the points (λ, 0), (, 0) and (4, 0) are colinear? 5 If A = [a ij ] is a matri and M ij s denotes the minors of the corresponding elements a ij s then, write the epression for the value of A by epanding A by third column 7 XII Maths

6 If 0 π < < and the matri sin is singular, find the value of sin 7 For what value of λ, the matri 5 λ λ + has no inverse? *8 If A is a square matri of order such that adj A = 5, find A 9 In the system of educations A = B, write the condition that the given system of educations has infinite solutions 0 If 5 A =, find adj (adj A) 6 8 It A = B, where A and B are square matrices of order and B = 5 What is A? If the matri A = sin α cosα cos α, find AA sinα If B = 0, and C = Find B C 0 * 4 Let A be a non singular matri of order such that A = 5 What is adj A? 5 Find a matri B such that 6 5 0 B = 5 6 0 + y + y + 6 If =,find and y 0 y + 5y 0 6 7 If 0 0 A = 0 0 For what value of, A will be a scalar matri 0 0 a b b c c a * 8 Find if = b c c a a b c a a b b c 9 Determine the value of for which the matri 4 A = is singular? 6 40 If 5 A =, write the matri A(adj A) 4 Write the adjoint of the matri A = 4 8 XII Maths

4 Find the value of P 0 0 a q 0 b c r 4 If A is a matri and A ( A) 0 adj =, what is A 0 ( * 44 If A = 0 0, Find A 0? 0 sin A + B + C) sin B cos C * 45 If A, B, C are angles of triangle Find the value of if = sinb O tan A cos ( A + B) tan A O + y y + z z + 46 If A = z y Write the value of det A 47 If A = 4 + + is symmetric matri, then find 48 If 49 If 50 If A = 4, find 5A A = 4 Find A sin cos π A =,0< < and A + A = I where I is unity matri, find the value of cos sin SHORT ANSWER TYPE QUESTIONS (4 Marks Each) 5 Construct a matri A = [a ij ] 4 whose entries are given by a ij = i i + j j 5 Find, y, z and w if y + z 5 = y + w 0 5 Find A and B if 0 A + 8 = and A B = 0 6 54 Let 0 4 0 0 A =, B = and C = 0, verify that (AB)C = A(BC) 6 9 XII Maths

55 Find the matri X so that 7 8 9 X = 4 5 6 4 6 = and = 4, verify that (AB) = B A 56 If A B [ ] 57 Epress the matri A = 4 as the sum of a symmetric and a skew symmetric matri 58 Epress the matri matri = P + Q where P is a symmetric and Q is a skew symmetric 4 5 59 Find the inverse of the following matri by using elementary transformations 7 6 60 Find the inverse of the matri by using elementary transformations θ 0 tan * 6 If A = θ tan 0 I + A = ( I A) = and I is the identity matri of order, show that cos θ sin θ sin θ cos θ 6 Find the value of such that [ ] 5 = 0 5 6 If 4 A =, find and y such that A A + yi = 0 5 * 64 If cos θ sin θ A =, sinθ cosθ then prove that n cos nθ sin nθ A = n N sinnθ cosnθ 65 If 4 A =, then prove that n + n 4n A =, n n where n is any positive integer 66 Find A (adj A) without finding (adj A) if A = 0 0 XII Maths

67 Given that A = Compute A and show that 9I A = A 4 7 68 Given that matri A = Show that A 4A + 7I = 0 Hence find A 69 If 8 A =, verify that 4 A I = A 70 Show that A = satisfies the equation 6 + 7 = 0 Hence find A 4 7 Prove that the product of two materices cos φ cos φsin φ cos θ cos θsin θ cos θsin θ sin θ and cos φsin φ sin φ is zero when θ and φ differ by an π odd multiple of 7 If A is any square matri Then show that (A A ) is a skew symmetric matri log 0 * 7 If, y, z are the 0th, th and 5th terms of a GP find the value of D if = log y log z 5 74 Show that : sin α sin β cos α cos β sin ( α + δ) sin ( β + δ ) = 0 sin γ cos γ sin γ + δ ( ) 75 Using the properties of determinant, prove the following questions (75 to 79) b + c a a b c + a b = 4 a b c c c a + c b + c c + a a + b a b c 76 q + r r + p p + q = p q r y + z z + + y y z XII Maths

77 a bc ac + c a + ab b ac = 4 a b c ab b + bc c + a b c 78 a + b c = ( + a + b + c) a b + c 79 Show that : y z ( ) ( ) ( ) ( ) y z = y z z y yz + z + y yz z y 80 (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 Given 0 0 A = and B = 0 0 Find the product AB and also find (AB) 8 (i) Using determinants find the area of the triangle whose vertices are (, ), (, 4) and (5, ) (ii) If =, find the value of 8 Solve the following equations for a + a a a a + a = a a a + 0 84 Verify that (AB) = B A for the matrices 4 5 A = and B = 5 4 85 Show that tanθ tanθ cos θ sinθ = tan θ tan θ sin θ cos θ 86 Using matri method to solve the following system of equations : 5 7y =, 7 5y = XII Maths

LONG ANSWER TYPE QUESTIONS (6 Marks Each) * 87 Let A = and f() = 4 + 7 Show that f(a) = 0 Use this result to find A 5 88 If cos α sin α 0 A = sin α cos α 0, find adj A and verify that A (adj A) = (adj A) A = A I 0 0 * 89 Find the matri X for which 90 If X = 7 5 0 4 0 A = 0 Show that A A I = 0 Hence find A 0 9 Using elementary transformations, find the inverse of each of the matrices in Question 9 to 9 9 9 0 0 0 0 4 5 4 7 94 Show that ( + ) y z y z = y ( + z ) yz = yz + y + z ( + ) z yz y 95 By using properties of determinants prove that + a + b ab b ( ) 96 Solve the system of linear equations by using matri in equation 96 to 98 y + 4z = z = XII Maths ( ) ab a + b a = + a + b b a a b

y z = 97 y z = 7 + y z = 7 + y z = 98 + y 5z = 6 + y + z = 4 + y + 6z = 9 * 99 If A = 4 find A and hence solve the system of linear equations 7 + 4y + 7z = 4, y + z = 4, + y z = 0 00 Find A, where A =, hence solve the system of linear equations : 4 + y z = 4 + y + z = y 4z = 0 Solve by matri method the following system of linear equations : y = 0 + y + z = 8 y + z = 7 * 0 The sun 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 0 Compute the inverse of the matri A = 5 6 5 5 5 and verify that A A = I 0 * 04 If the matri A = 0 and B = 0, 4 0 then compute (AB) 4 XII Maths

05 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 = * 06 Solve the following system of equations using matri method 0 + + = y z 4 4 6 5 + = y z 6 9 0 + = y z 07 For the matri A = Show that A 6A + 5A + I = 0 and hence find A 5 XII Maths

CHAPTER 5 DIFFERENTIATION POINTS TO REMEMBER Continuity of a Function : A function f() is said to be continuous at = c if lim f ( ) = f ( c) ie, LHL at = c = RHL at = c = f(c) f() is continuous in [a, b] iff : f() is continuous at = c c [ a b], c If f and g are two continuous function then f + g, f g, f g, cf, f are all continuous function f g is continuous at = a provided g(a) 0 Every polynomial function is a continuous function f() is said to be derivable at = c iff by f(c) f ( ) f ( c) lim c c eists and value of this limit is denoted du dv v u d dv du d u ( u v ) = u + v, = d d d d d d v v If y is a function of u and u is function of them, dy dy du = [chain rule] d du d If = φ (t), y = φ (t) then dy d φ ( ) t = = φ ( t ) g ( t ) d y say then ( ) dt = g t d 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 atleast one point c Î (a, b) st ( ) f ( b) f ( a) f c = b a VERY SHORT ANSWER TYPE QUESTIONS ( Mark Each) At what point f() = 5 is not differentiable 6 XII Maths

* What is derivative of f() if f() =, 0 At what point f() = is not differentiable 4 Write the points of discontinuity of f ( ) = + + 5 + 6 5 Write all the points of discoutinuity of f() = [], where [] is the greatest integer function * 6 At what point, f(n) is discontinuous where, f() is signum function defined as 0 f ( ) = 0 = 0 7 Write the interval in which f() is continuous where f() = e log 8 For what value of λ, f ( ) = λ + < is continuous on R * 9 Write the interval in which f() is continuous where ( ) log f = 9 sin * 0 Write the value of K given that f ( ) = 0 K = 0 * What is the derivative of 6 with respect to * What is the Derivative of f(log ) is f() = log If Mean value theorem holds for f() = e, [0, ], then for what value of, M V T is verified * 4 What is d ( ) sin + cos if d 5 Given g(0) = and f() = g() f () and g () eist then what is f (0) 6 Write the derivative of the following function wrt (a) a > 0 tan, + a a > 0 (b) sec (c) log ( ) 5 5 * (d) tan + (e) tan (f) e tan (g) cos sin 5 (h) e 7 XII Maths

(i) sin ( ) 0 (j) ( ) sin cos SHORT ANSWER TYPE QUESTION (4 Marks) Discuss the continuity of the following functions at indicated points 0 7 f ( ) = at = 0 0 = 0 0 8 f ( ) = at = 0 = 0 e 0 9 f ( ) = log ( + ) at = 0 5 = 0 sin 0 0 f ( ) = at = 0 0 = 0 f() = + + at =, = sin < 0 f ( ) = at = 0 + 0 + 0 f ( ) = δm at = 0 = 0 sin 0 4 f ( ) = at = 0 = 0 5 For what value of K, f ( ) = 6 For what values of a and b 0 < < 4 k < 5 is continuous in it s domain + + a if < + f ( ) = a + b if = is continuous at = + + b if > + 8 XII Maths

* 7 If f ( ) = then find the point of discontinuity if any of f [f{f()] 8 Prove that f() = is continuous at = but not differentiable at = tan < 0 5 sin 9 For what value of K, f ( ) = K =0 4 + < 0 is continuous at = 0 0 Show that f() = [] is discontinuous at = Also discuss the countinuity at [ ] represents greatest integer function Check the differentiability of f() = + at = 5 =, where p sin * If f ( ) = 0 0 = 0 is differentiable at = 0, then find value of p 5 if n For what value of a and b f ( ) = a + b if < < 0 if 0 is continuous 4 If y = (log ) + log dy then find d 5 If y dy = tan + tan find d 6 If y = dy sin tan then find + d 7 If / + y / = a / then show that dy d y = 8 If y = tan, show that ( ) 9 If f() = log ( + sec ), find f() d y dy + + = 0 d d 40 If + y + y + = 0 then prove that dy d = ( ), y + 9 XII Maths

4 If y + y = a then prove that dy d = y 4 If ( + y) m + n = m y n dy y then prove that = d 4 If α + α y = α + y where α > 0 then prove that dy y d +α = 0 *44 If y = + sin + sin cot + sin sin dy where π/ < < π, find d *45 If dy y = sin a a find when a d 46 Find the derivative of tan wrt sin + 47 If = sin log y a then show that ( ) y y a y = 0 48 If y + = tan + tan + dy then show that 0 d = 49 If y = f + and f () = sin dy, then find d 50 If y + y + = a b dy, then find d 5 If a = θ + θ y = a θ θ 5 if = a cos θ dy then find d y = a sin θ then find d y d 5 If = ae θ (sin θ cos θ) y = ae θ dy π (sin θ + cos θ), then show that at = is d 4 54 Find dy t t if =, y = d + t + t 40 XII Maths

* 55 If dy y = sin then find d 56 If y = ( ) tan dy,find e sin d t dy 57 If y = a sin t, = a cos t + log tan, find d 58 If y = sin 4cos dy cos, find 5 d 59 If y = + 5 dy sin, find d y dy 60 If y =, find d 6 If sin y = sin (a + y) then show that dy d = sin ( a + y) sin a 6 If y = cos, find d y d in terms of y 6 If sin y sin, m = then prove that ( ) d y dy + m y = 0 d d 64 If a + y = then show that b 4 d y d = b a y 65 If y a + d y a = 0 then prove that + = 5 d v 0 66 Find d y d when y = log e 4 XII Maths

CHAPTER 6 APPLICATIONS OF DERIVATIVES POINTS TO REMEMBER Rate of Change : If and y are connected by y = f() then dy d of y wrt 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 of intersection P m m tan θ =, + mm where m, m are slopes of tangent at the point 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 < Þ 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 δ), (c + δ) of c, st f() < f(c) (c δ, c + δ) 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, 4 XII Maths

Some useful results : Figure Curved Total SA Volume Surface area SA Sphere 4πr 4πr 4 π r Cone πr πr + πr π r h Cylinder πrh πrh + πr πr h VERY SHORT ANSWER TYPE QUESTIONS ( Mark Each) Write minimum value of f() = + + in [0, ] * If Rolle s theorem in applicable for the function f() = + in [, 4] then find the real no c verifying Rolle s theorem Find the interval where f() = cos defined in [0, π] is decreasing 4 Find the interval where f() =, (, ) is decreasing * 5 For what value (s) of λ, the function, f() = sin λ is always strictly increasing * 6 Write the interval in which f() = is increasing (where > 0) 7 Eamine if f() = 9 + 5 + + is increasing or decreasing (0, ) * 8 Write the least value of ( ) f = +, ( > 0 ) 9 Write the maimum value of f ( ) = [ ] in 0, + * 0 Find the maimum and minimum value of f() = sin + * On the curve f ( ) =, find the points at which tangent is parallel to the chord joining the points A, and B(, 6) dy * If the tangent to the curve at a point P is perpendicular to ais, then what is the value of d (if it eists) at the point P dy * If normal to the curve at a point P on y = f() is parallel to y ais, then write the value of d at P 4 What is the slope of the tangent to the curve y = at (, ) 4 XII Maths

*5 If the tangent to the curve y = at any point P is parallel to the line y = 0, then find the coordinates of P 6 If the tangent to the curve = at, y = at is perpendicular to -ais then write the coordinates of the point of contact of tangent *7 If curves y = e and y = be cut each other orthogonally, then find b *8 At which point on y = 4, the tangent makes an angle of 45 with the positive direction of *9 If k + y = P is normal to the curve y = at (, 6) then what is value of k 0 How many etreme values [maimum or minimum] are there of f() = What is equation of normal to the curve y = sin at origin SHORT ANSWER TYPE QUESTIONS (4 Marks Each) Sand is pouring out from a pipe at the rate of Cu cm/s The falling sand forms a cone one 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 A particle moves along the curve y = 5 + Find the points on the curve at which y co-ordinate is changing 5 times as fast as the co-ordinate 4 Find points of local maima/minima for f() If f() = sin cos where 0 < < π Also find the local maimum or minimum values 4 5 Find the intervals in which the function f ( ) = is increasing or decreasing *6 If f() = + then using differentials, find the approimate value of f(9) 7 Find the value (s) of a for which : (i) f() = a is increasing on R (ii) g() = sin + a is increasing on R 8 If radius of right circular cone is increasing at the rate of 0π cm /sec, find the rate at which the height of the cone is hanging at the instant when radius 5 cm and height 4 cm b,, 0 * 9 Find the least value of the function f ( ) = a + ( a b > ) 0 For the curve y =, find all the points on the curve at which the tangent passes through the origin Prove that the function : π f() = 50 + sin is strictly increasing on π 44 XII Maths

Show that the normal at any point θ to the curve = a cos θ + aθ sin θ, y = a sin θ aθ cos θ is at a constant distance from the origin Using errors and approimations find the approimate value of the following (i) ( ) 00 ; (ii) 008; (iii) 0005 4 Find the interval in which f() = sin cos, 0 < < p, is increasing or decreasing * 5 If log = 000 and log 0 e = 044 find approimate value of log 0 () 6 Find the interval (s) in which y = ( ) increases 7 Find the interval (s) in which function f() = 5 + 6 + is strictly increasing or decreasing 8 Find the point of the curve y = where the tangent is parallel to ais? 9 Find the equation of the tangent to the curve y = e / at the point where curve cuts y ais 40 If 8k = then show that the curves y = and y = k cut at right angles 4 Determine the interval in which function, f() = sin + cos in [0, π] is strictly increasing or decreasing 4 Find the maimum value of f() = sin + 4 cos in [0, π/] 4 Find the two positive numbers and y such that their sum is 5 and product y is maimum 44 Find the least value of a such that the function f() = + a + is strictly increasing on (, 4) 45 Show that the acute angle of intersection between the curves y = 6 and y = is 46 Find approimate value of 007 using differentials tan π 47 Find all the points of the curve y = at which the tangent is 9 6 (i) Parallel to the ais; (ii) Parallel to y ais * 48 Find all values of a R such that the function f() = (a + ) a + 9a decreases for all R * 49 Find the condition that the line cos α + y sin α = p be a tangent to the curve y + a b = 50 Find equation of tangent at = π to the curve y = cot cot + 4 45 XII Maths

LONG ANSWER TYPE QUESTION (6 Marks Each) 5 Show that the point (, ) on y = + is nearest to the point (, ) 5 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 / ) / 5 If the length of three sides of a trapezium other than base are equal to 0cm, then find the area of trapezium when it is maimum 54 A given quantity of metal is to be cast into half cylinder with a rectangular base and semi-circular ends Show that when total surface areas is minimum, the ratio of length of cylinder to the diameter of its semi-circular ends is π : (π + ) π π π 55 Show that f() = sin 4 + cos 4, [0, π/] is increasing on, and decreasing on 0, 4 4 56 Find the interval in which f() = log ( ) + 4 + is increasing or decreasing, 57 Find the equation of tangent to the curve y = ) ( ) at the points where the curve cuts the ais 58 Show that the semi-verticle angle of a cone of maimum volume and given height is tan 59 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 60 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? 6 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? 6 For a given curved surface of a right circular cone when volume is maimum, prove that semivertical angle is sin 6 Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and semi-vertical angle α is 4 πh 7 tan α 64 Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8 7 of the volume of the sphere 65 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? 46 XII Maths

66 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 67 Cost of fuel for running a car is proportional to the square of speed generated in km/hr It costs Rs 48 per hour when the car is running the speed of 6 km/hr What is the most economical speed if the fied charges are Rs 00 per hour over and above the running cost 68 Two sides of a triangle are of lengths a and b and angle between them is θ What value of θ will maimize the area of triangle? Also find the maimum area 47 XII Maths

CHAPTER 7 INTEGRATION POINTS TO REMEMBER Integration is inverse process of Differentiation STANDARD FORMULAE 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 48 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 INTEGRATION BY SUBSTITUTION f ( ) d = log f ( ) + c f ( ) [ ] n+ n f ( ) [ f ( )] f ( ) d = + c n + n+ f ( ) ( f ( )) d = + c n f ( ) n + [ ] INTEGRATION BY PARTS b a b f ( ) g ( ) d = f ( ) g ( ) d f ( ) g ( ) d d PROPERTIES OF DEFINITE INTEGRALS f ( ) d = F ( b) F ( a), where F ( ) = f ( ) d f ( ) d = f ( ) d a a f ( ) d = f ( t) dt b b a b a 49 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 VERY SHORT ANSWER TYPE QUESTIONS ( Mark Each) Evaluate the following integrals b a b a (i) + d (ii) sin d (iii) cos cos d (iv) d (v) π 0 4 + sin log d 4 + cos (vii) ( + ) (i) cosec cosec cot d d ( ) sin cos Evaluate the following integrals (vi) (viii) () ( ) log log log d 8 8 8 + + + d 8 99 4 cos d (i) + (ii) a d a 50 XII Maths

(iii) 4 sin e + sec d (v) d + (iv) (vi) + cos d cos sec cosec d log tan (vii) cos + sin a log log a d (viii) ( + ) cos e e d π (i) sin d, < < π () ( + log ) d (i) sin d (ii) tan 6 d + (iii) log d (v) (vii) (i) (iv) a + b d a + b + c sin d (vi) ( c c + ) d a + b cos d + log e d + e (viii) () 6 + 5 d 9 4 d (i) 6 5 d (ii) d 4 9 (iii) a b d Evaluate the following definite integrals : π (i) sin sin + cos 0 d (ii) e d (iii) π π 7 sin d (iv) π sin 4 4 π sin + cos d 5 XII Maths

π (v) 0 + d (vi) 0 sin + cos (vii) 0 tan d + (viii) 0 e + e d *(i) e e 5 log 4 log e log log e d () 0 e d SHORT ANSWER TYPE QUESTIONS (4 Marks Each) 4 Evaluate the following integrals : (i) cosec tan + ( ) 4 d (ii) + d + + (iii) d sin ( a) sin ( b) (v) cos cos cos d (vii) (i) (i) 4 sin cos d 4 cot cosec d cos d cos ( + a) * (iii) ( ) 6 + d (iv) (vi) (viii) () (ii) (iv) cos ( + a) d cos ( a) 5 cos d 5 4 sin cos d sin cos d a sin + b cos 6 6 sin + cos d sin cos sin + cos d sin (v) (vii) d ( 4 * + ) (vi) d + sin sin d * sin (viii) 5 5 5 5 5 d 5 Evaluate : (i) d + + 4 *(ii) d 6 log + 7 log + ( ) 5 XII Maths

(iii) d + (iv) 9 + 8 d (v) d + (vi) ( a) ( b) d (vii) sin ( α) d sin ( + a) (viii) 5 d + + (i) (i) d + 6 + () + 4 + d (ii) ( ) *(iii) sec + d 6 Evaluate : (iv) d + + d sinθ cosθ 6 cos θ 4 sinθ d (i) d ( 7 + ) *(ii) ( ) ( ) a b d (iii) sin d ( + cos ) ( + cos ) (iv) d + 6 (v) + d ( + cos ) ( + cos ) (vi) d ( + ) ( ) ( + ) (vii) + + d ( ) ( ) (viii) ( ) ( + + ) ( ) ( ) + + 4 d 4 * (i) d () 4 6 d ( ) ( ) + + 4 d (i) sin ( cos ) * (ii) d cos ( sin ) (iii) sin d sin 4 (iv) d 4 + + (v) tan d (vi) + 9 d 4 + 8 log * (vii) ( ) d 5 XII Maths

* (viii) d, where [] is greatest integer funbction 7 Evaluate : (i) 5 sin d (iii) e cos ( b + c) d (ii) sec d a *(iv) (vi) n ( ) (v) cos d *(vii) *(i) (i) ( sin + cos ) 6 sin d + 9 log d (viii) ( ) sin d a + ( ) () + log + log 4 d (ii) sin cos d tan d + sin e d + cos (iii) (v) e e d ( ) + ( + ) d (iv) (vi) e e + d ( + sin ) d ( + cos ) (vii) ( ) d *(viii) ( ) log log (i) ( ) + ( ) sin log cos log d 8 Evaluate the following definite integrals : log log + d ( log ) (i) π 4 0 sin + cos d 9 + 6 sin (ii) Π 0 cos log sin d * (iii) d + (v) 0 π 0 cos ( + sin ) ( + sin ) ( + sin ) d (iv) (vi) π 0 0 sin sin ( ) sin 4 4 + cos d d 54 XII Maths

(vii) (i) π 0 4 sin d (viii) 5 d * () + 4 + 0 4 π d d + sin d + cos 9 Evaluate : (i) { + + d } * (ii) ( ) (iii) π 0 π π sin cos d d (iv) log ( + tan ) d + sin π 4 0 π (v) 0 log sin d (vi) π 0 sin ( ) + cos d < (vii) f ( ) d where f ( ) = + when < when when < π (viii) 0 sin cos d 4 4 sin + cos (i) π 0 d a cos + b sin 0 Evaluate the following integrals as limit of a sum (i) d (ii) e d (iii) 0 d (iv) e d 0 (v) ( + ) d (vi) ( 7 5 ) d 0 55 XII Maths

Evaluate the following integrals (i) tan + tan d + tan (ii) tan 4 4 sec d (iii) d (iv) 0 sin d + *(v) + sin log d sin *(vi) log + + d (vii) 8 0 + 0 d (viii) π 0 e e cos cos + e cos d *(i) + a d () d ( ) LONG ANSWER TYPE QUESTIONS (6 MARKS EACH) Evaluate the following integrals : (i) 5 5 + 4 d (ii) d ( ) ( ) + 4 d (iii) ( + ) ( ) d (iv) 4 d 4 6 π (v) ( ) 0 tan + cot d (vi) 4 d + (vii) tan d (viii) ( + ) ( ) 0 + 0 cot d *(i) 0 log ( + ) d ( + ) *() + cos ( ) 5 d cos *(i) d ( sin cos ) ( sin + cos ) (ii) sin sin cos + cos d 56 XII Maths

(iii) + d (iv) + Evaluate the following integrals as limit of sums : sin π d 4 (i) ( + ) d (ii) ( + ) 0 d 4 (iii) ( + 4 ) d (iv) ( + ) 5 0 e d (v) ( + ) d (vi) ( + + ) 0 5 d 57 XII Maths

CHAPTER 8 APPLICATIONS OF THE INTEGRALS POINTS TO REMEMBER AREA OF BOUNDED REGION Area bounded by the curve y = f(), the ais and between the ordinate a + = 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 absussa a + y = c and y = d is given by Area = d d c c ( ) dy = f y dy = f( y) O y d c y d c O = 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 58 XII Maths

y y = f( ) y = gm 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 c Area = f ( ) d f ( ) d a b c SHORT ANSWER TYPE QUESTIONS (4 Marks Each) Find 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 LONG ANSWER TYPE QUESTIONS (6 Marks Each) 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 (, ) 59 XII Maths

(b) (, ) (0, 5) and (, ) 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 Find the area of the region {(, y) : + y + y} Find the area of the region bounded by y = and y = * Using integration find the area enclosed by the curve y = cos, y = sin and ais in the interval (0, π/) * 4 Sketch the graph y = 5 Evaluate 6 0 5 d * 5 Find the area enclosed by the curve y = and y = 6 Find the area enclosed by the curve y = sin between = 0 and 7 Find the area bounded by semi circle 8 Find area of region given by {(, y) : y } y = 5 and -ais π = and -ais 9 Find area of smaller region bounded by ellipse y + = and straight line + y = 6 9 4 0 Find the area of region bounded by the curve = 4y and line = 4y * Find the area bounded by the ellipse = a ( e ) and e < a + y = and ordinate = ae and = 0, where b b Find the area enclosed by parabola y = 4a and the line y = m Find the area of region bounded by y = and the line y = 4 Using integration find the area of region in first quadrant enclosed by -ais the line = y and the circle + y = 4 5 Draw a sketch of the region {(, y) : + y 4 + y} and find its area 6 Find the area enclosed between the y-ais the line y = and the curve + y = 0 7 Find smaller of two areas bounded by the curve y = and + y = 8 60 XII Maths

CHAPTER 9 DIFFERENTIAL EQUATION 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 : 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) Homogenous differential equation : A differential equation of the form (, y) (, ) dy f = d 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 pd Its solution is y (I F) = q(i F)d where IF = e VERY SHORT ANSWER TYPE QUESTIONS 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 (iv) d d 5 5 d y dy + log = 0 5 d d 6 XII Maths

*(v) dy d y + = (vi) d d 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 + (vii) dy sec d = (viii) dy y = d log 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 y log d d tan y sec y dy + = = (vi) ( ) (vii) d + = sin y dy + y d (viii) = y + dy 4 (i) Verify that (ii) SHORT ANSWER TYPE QUESTIONS (4 Marks Each) y sin m = e is a solution of ( ) d y dy Show that y = sin (sin ) is a solution of diff equation (iii) Show that y = A + B d y dy + tan = y cos d d is a solution of m y = 0 d d 6 XII Maths

d y dy + 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 (v) Verity that y = 4a ( + a) is a solution of y dy dy = = d d (vi) Show that y = ae + be + ae + be is a solution of *(vii) Verify that y log ( a ) = + + satisfies the diff equation d y dy y = d 0 d ( ) a d + d y + dy = d 0 *(viii) (i) () (i) (ii) Find the differential equation of family of all circles having centres on -ais and radius units 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 Find the differential equation corresponding to the family of curve ay = ( c), where c is an arbitrary constant By eliminating the constants a and b obtain the differential equation for which y = ae + be + is a solution *(iii) Form the differential equation corresponding to the family of curves y = c( c) (iv) Form the differential equation representing the family of curves (y b) = 4( a) 5 Solve the following diff equations dy + = d dy (ii) y cot sin d + = * (i) ( y ) (iii) dy y log d + = *(iv) (tan y ) dy = ( + y )d dy + y + e y = 0 (vi) d tan (v) ( ) ( ) d sin + y = cos +, > 0 dy 6 XII Maths

(vii) (i) d y e dy + = + (viii) y e y d = ( + e y ) dy dy cos cos sin d + = 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) = (vi) dy y sin cos y e d = + (vii) tan tan y d + sec sec y dy = 0 dy tan sin sin d = + + (viii) y ( y) ( y) dy (i) d + y y = e + e () ( e ) dy ( y ) e d y ( ) + + + = 0, 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 = + + y y *(iv) + e d = e dy = 0 y (v) sin dy y y y d = sin y dy (vi) = + tan y y d (vii) dy y = d + y (viii) dy d + y y = e + e dy log log d = () dy y y d = + *(i) y ( y ) (i) d y d = e + sin (ii) dy = d y 64 XII Maths

8 (i) Form the differential equation of the family of circles touching y-ais at (0, 0) *(ii) Form the differential equation of family of parabolas having verte at (0, 0) and ais along the (i) positive y-ais (ii) +ve -ais (iii) Form the differential equation of the family of hyperbols centred at (0, 0) and aes along the coordinate ais *(iv) Form differential equation of all circles passing through origin and whose centre lie on -ais (v) Form the differential equation of family of curves, y = a sin (b + c), a, b, c are arbitrary constants cos sin y y 9 Show that the differential equation : y( dy + yd) = ( yd dy) is homogeneous and solve it 0 dy + y Show that the differential equation = d y 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 π sin + y cos = sin cos given that y = when = d LONG ANSWER TYPE QUESTIONS (6 Marks Each) Solve the following differential equations : (i) ( + y ) d = ( y + y )dy (ii) dy yd = + 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 65 XII Maths

CHAPTER 0 VECTORS AND THREE DIMENSIONAL GEOMETRY POINTS TO REMEMBER 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 λ R then λ a in a direction parallel to a λ a is vector of magnitude If a 0 then a is unit vector in direction a a Scalar Product : a b = a b cosθ where θ 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 θ n where n is a unit vector perpendicular to a and b, and θ is the angle between a and b Unit vector perpendicular to plane of Vector a and b are collinear if a b = 0 i j k a b = a a a b b b where a = a i + a j + a k and b = b i + b j + b k a b a and b is ± a b 66 XII Maths

Area of a triangle whose two sides are a and b = a b Area of a parallelogram whose adjacent sides are a and b = a b If a, b represents the two diagonals of a parallelogram, then area of parallelogram = a b THREE DIMENSIONAL GEOMETRY Distance between P(, y, z ) and Q(, y, z ) is PQ = + y y + z z ( ) ( ) ( ) The coordinates of point R which divides line segment PQ where P(, y, z ) and Q(, y, z ) m + n my + ny mz + nz in ratio m : n are,, m + n m + n m + n If α, β, γ are the angles made by any line with coordinate aes respectively then l, m, n Where l = cos α, m = cos β, n = cos γ 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, m = ± b, n = ± c 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 + λ b (ii) Passing through two points A ( a ) and B ( b ) is r = a + λ( 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 θ between two lines with DC s l, m, n and l, m, n is given by cos θ = l l + m m + n n OR with DR s a b c or a, b, c aa + bb + cc cos θ = a b c a b c + + + + 67 XII Maths

If lines are r = a + λ b and r = a + µ b then, b b cos θ = b b Equation of plane : (i) Passing through A ( a ) and perpendicular to ( n ) r n = d where a n = d (ii) Passing through three given points is r a n = 0 is ( ) y y z z y y z z = 0 y y z z Or (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 θ = 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 ) + λ(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 + λ n = d + λd is ( ) Angle between a plane r n = d m n and a line r = a + λ m is sin θ = m n VERY SHORT ANSWER TYPE QUESTIONS ( Mark Each) * 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 = i j + k? Write when + y = + y 4 What is the area of a parallelogram whose sides are given by i j and i + 5 k? * 5 What is the angle between a and b, If a b = and a b = 68 XII Maths

6 If A is the point (4, 5) and vector AB respectively then write point B has components and 6 along -ais and y-ais 7 What is the point of trisection of PQ nearer to P if position of P and Q are i + j 4 k and 9 i + 8 j 0 k 8 What is the vector in the direction of i + j + k, whose magnitude is 0 units? * 9 What are the direction cosines of a vector equiangular with co-ordinate aes? 0 What are the angles which i 6 j + k Write a unit vector perpendicular to both the vectors i j + k and i + j k What is the projection of the vector i j on the vector i + j? If a =, b = 4 For what value of λ, a = λ i + j + 4 k 5 What is a ( a b ) ( a b ) and a b, what is the value of a + b?,if + = is perpendicular to and b = a? b = i + 6j + k? 6 What is the angle between a and b, if a b = a + b? * 7 In a parallelogram ABCD, AB = i j + 4 k and side BC? AC = i + j + 4 k What is the length of 8 What is the area of a parallelogram whose diagonals are given by vectors i + j k i + k? 9 Find if for a unit vector a, ( a) ( + a) = 0 If a + b = a + b, then what is the angle between a and b If a and b are two unit vectors and a between a and b? + b is also a unit vector then what is the angle 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? ( ) ( ) ( ) What is the angle between and y if y = y? 4 Write a unit vector in y-plane, making an angle of 0 with the +ve direction of ais * 5 Two adjacent sides of a parallelogram are i 4 j + 5 k and i j k Find a unit vector parallel to the diagonal, which is coinitial with a and b 69 XII Maths