3. Total number of functions from the set A to set B is n. 4. Total number of one-one functions from the set A to set B is n Pm

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1 ASSIGNMENT CLASS XII RELATIONS AND FUNCTIONS Important Formulas If A and B are finite sets containing m and n elements, then Total number of relations from the set A to set B is mn Total number of relations on the set A is m m Total number of functions from the set A to set B is n Total number of one-one functions from the set A to set B is n Pm if n m, otherwise 0 5 Total number of onto functions from set A to set B is n nr n m ( ) Cr r if m n, otherwise 0 6 Total number of bijective functions from the set A to set B is m!, if m n, otherwise 0 7 Total number of binary operations on A is m m Q Prove that the relation R on the set Z of all integers defined by a, b, c, d N N is divisible by n is an equivalence relation on Z Q Prove that the relation R on the set N N defined by a, b, c, d N N is an equivalence relation Q Let N be the set of all natural numbers and let R be a relation on N N, defined by a, b Rc, d ad bc for all a, b, c, d N N Q Let a relation Show that R is refleive and symmetric but not transitive R r a, b R c, d a d b c for all Show that R is an equivalence relation on N N R on the set R of real numbers be defined as Q5 Let R be a relation defined on the set of natural numbers N as Show that R is neither refleive nor symmetric but it is transitive Q6 Let S be a relation on the set R of all the real numbers defined by a, b R ab 0 for all a, b R R, y :, y N, y S a b R R a b, : Prove that S is not an equivalence relation on R Q7 If a b(mod m) means that m divides a b Show that is an equivalence relation on Z Q8 Le N be the set of all natural numbers and let R be a relation in N, defined by R a, b : a is a factor of b Show that R is refleive and transitive but not symmetric y Q9 Prove that the relation R on the set Z of all integers defined by R y y for all, y Z is an equivalence relation Q0 Test whether the following relations R, R, R are (i) refleive (ii) symmetric (iii) transitive: (i) R on Q0 defined by( a, b) R a / b (ii) R on Z defined by( a, b) R a b 5 (iii) R on R defined by ( a, b) R a ab b 0 Q Classify the following functions as injection, surjection or bijection: (i) f : R R, f ( ) (ii) f : Z Z, f ( ) (iii) f : Z Z, f ( ) 5 (iv) f : R R, f ( ) sin (v) (vii) (i) f : R R, f ( ) (vi) f : R R, f ( ) sin cos (viii) f : R R, f ( ) f : Q {} Q, f ( ) f : Q Q, f ( ) () f : R R, f ( ) 5 Q Prove that the function f : N N, f ( ) is one-one but not onto Q Let f : R {} R {} is a mapping defined by f ( ), show that f is bijective Q Let f : N {} N f ( n) the highest prime factor of n Show that f is neither one-one nor onto Find the range of f also

2 Q5 Find gof and fog when and f : R R and g : R R are defined by: (i) (iii) f ( ) and g( ) 5 (ii) f ( ) and g( ) f ( ) 8 and g( ) (iv) f ( ) and g( ) and f : A B and g : B C be defined as Q6 If A,,,,5, B,5,7,9, C 7,, 7,79 f ( ) and g( ) Epress as the sets of ordered pairs and verify ( gof ) f og Q7 Let f be the greatest integer function and g be the modulus function, then prove that: (i) gof fog 0 (ii) gof fog (iii) fog gof Q8 If f ( ) where Show that fof is an identity function Q9 Let f : R R: f ( ) and g : R R: g( ) ( ) Show that ( fog) I R ( gof ) Q0 Show that the function f : R R defined by f ( ) is invertible Also find the inverse of f Q Determine whether or not each of the definition of given below gives a binary operation In the event that is not a binary operation, give justification: a b (i) On N,defined by ab (ii) On R, defined by ab a ab (iii) On N, defined by ab a b a (iv) On Q, defined by a b b Q For each of binary operations defined below, determine whether is commutative and associative: a b (i) On Q,defined by ab (ii) On R, defined by ab a b (iii) On Z, defined by ab a b ab (iv) On Z, defined by ab a b ab Q Let A be a set of all real numbers ie A R { } Let be defined on A as ab a b ab (a) Prove that (i) is a binary operation on A (ii) is commutative and associative (iii) 0 is the identity element a (iv) is the inverse of a a Q Show that the operation on Z, defined by ab a b, satisfies (i) closure property (ii) associative property (iii) commutative property Also find the identity element and inverse of an element a A Q5 Let S Ro R A binary operation is defined on S as follows: ( a, b) ( c, d) ( ac, bc d) ( a, b),( c, d) R0 R Find: (i) identity element in S (ii) invertible element in S ANSWERS 0 (i) R is symmetric but neither refleive nor transitive (ii) R is refleive and symmetric but not transitive (iii) f : Z Z f ( ) 5 is refleive but neither symmetric nor transitive (i) Neither an injecton nor a surjection (ii) Neither an injecton nor a surjection (iii) bijective (iv) Neither an injecton nor a surjection (v) bijective (vi) surjective but not injective (vii) Neither an injecton nor a surjection (viii) injective but not surjective (i)injective () bijective set of all prime numbers 5(i), (ii), 6 (iii) 7,,,, 7,, 79, 0 8, (iv), 6 f ( ) ( ) (i) No (ii) Yes (iii)no (iv) No (i) commutative but not associative (ii) commutative but not associative (iii) commutative and associative (iv) neither commutative nor associative identity is and inverse of a is ( a) 5 (i) (,0) b (ii), a a

3 ASSIGNMENT CLASS XII INVERSE TRIGONOMETRY Q Find the principle value of the following: (a) sin ( ) (b) cos tan ( ) sec ( ) (e) cos ec ( ) (f) sin sin 5 (g) cos cos (h) tan tan 8 (i) cos cos 7 (j) 5 tan tan 6 Q Evaluate the following: (a) cos sin 5 (b) cos ec cos 5 cos tan 8 tan cos 7 (e) tan tan 5 (f) sin cos 5 Q Prove the following: 5 6 (a) tan tan (b) sin sin sin tan sec tan 5 7 tan tan tan tan (e) cot 5 cot 7 cot 8 (f) sin tan (g) sin cos tan (h) cos tan tan (i) tan cos sin (j) sin tan cos Q Write the following in the simplest form: sin (a) tan (b) tan cos tan cos sin tan a a (e) cos tan sin cot (f) cot Q5 Solve the following equations: (a) tan tan (b) tan 6 tan tan 7 sin sin cos sin 6 (e) sin cos (f) sin cos 6 ANSWERS (a) (b) 5 (e) (f) (g) (h) (i) 6 (j) (a) 5 (b) (e) 7 (f) (a) (b) cos cos 7 5 a e) (f) tan 5(a) (b) 7 (e) (f)

4 ASSIGNMENT CLASS XII MATRICES ( i j) Construct a matri A, matri B, whose elements are given by aij 5 5 If A 5 and B 5, show that AB BA O 5 If A 0 and I 0, then (i) find, so that A A I (ii) prove A A A O cos sin cos sin (a) If A, then show that A sin cos sin cos cos i sin n cos n i sin n (b) If A i sin cos, then prove by mathematical induction that A, nn i sin n cos n n( n ) n n If A 0, prove by mathematical induction that A 0 n, n N Epress the following matrices as the sum of symmetric and a skew-symmetric matri: (a) (b) (a) Find the matri C, such that A B C is a zero matri, where 0 A, B 0 0 (b) Find a matri X such that A B X 0, where 7 From the following equation, find the values of and y : A, B 5 (a) y 5 (b) 0 y y 0 0 y 5y 8 If A, B and C, 0 verify that : ' ' ' ' ' ( i)( A B ) A B ( ii)( AB) B A' ( iii) ( AB) C A( BC) ( iv) A( B C) AB AC 9 (a) If A,find f ( A ) if f ( ) (b) A,find f ( A ) if f ( ) 5 7 If A, find a matri B such that AB I If A,prove that A A A (e) If A, find k such that A 8A ki 0 7 (f) If A, prove that A 5 A I 0

5 0 5 5 Let A, B, C Find a matri D such that CD AB (a) If A and B 0, verify that ( AB)' B ' A' (b) If A and B, verify that ( AB )' B ' A ' 5 Using elementary transformations, find the inverse of the following matrices: (a) A 7 (b) A A A 0 7 ANSWERS , 0, 5 (a) (b) (a) (b) X 7 7 (a), y 9 (b), y (a) f ( A) 6 (b) f ( A) 0 5 B (e) k (a) A (b) A 7 A A 6

6 ASSIGNMENT CLASS XII DETERMINANTS Find the values of, if (i) (ii) (iii) Using properties of determinants, show that: a b c a a (i) b b c a b a b c a b c (ii) a b c a b bc c a ab bc ca c c c a b bc ca ab b c c a a b (iii) c a a b b c a b cab bc ca a b c a b b c c a y z (v) y z yz y y z z y z y z (vi) (vii) y z y y z z y z (iv) a a bc b b ca 0 c c ab b c a a b c a b abc a b c c c a b y z z y (i) y z y z y y z z y y z z (i) b ca b a bb ca ca b c (iii) a bc a c ab c a b b c c a a b c abc a a a (viii) a a a a a b a b () a b a a b 9b a b a b a b a (ii) a b c b c a b b c c a a b a b b c (iv) c a c a a b ca c (v) a b abc abc bc ca ab a b c c (vi) 0 b a a b 0 c b a b c a c b c c a 0 Without epanding the determinants, show that: a bc a a a a bc a a (i) b ca b b (ii) c ab c c b b ca b b (iii) b c c a a b b c a c c ab c c a b b c c a a b c c a a b b c c a b Using properties of determinants, solve for : a a a 8 (i) a a a 0 (ii) 8 0 a a a 8 (iii) 6 0

7 5 Using determinants, find the area of the triangle whose vertices are,,, 6and 5, Are the given points collinear? 6 Find the equation of the line joining A(,)and B(0,0) using determinants and find k if D( k, 0) is a point such that ar( ABD)issqunits 7 Find the value of if the area of the triangle with vertices (, ),(, 6) and (5, ) be 70sqcm 8 Find the value of so that the points (, ),(, ) and (,6 ) are collinear? 9 Let A, verify that: ( i) A( adja) ( adja) A A I ( ii)( adja) adj( A ) 5 ( )( T T iv A ) ( A ) ( ) T T v adja ( adja) ( )( ) iii A A 0 If A, B, verify that: ( i) ( AB) B A ( ii) adj( AB) ( adjb)( adja) 5 0 Given A and B 9 7 Compute( AB) Ans: ( AB) If A, prove that A A5I 0Hence find A Ans: A Find A, so that ( i ) A ( ii) A ( iii) 0 A Ans: A 5 5 A 5 7 A 0 Find whether the following system of equations is consistent or not, find the solution of the system also: y z 5 7y z y z 6 (i) y z (ii) 6 8y z 5 y z 5 y 6z 7 y 7z 0 Ans: inconsistent consistent, y, z consistent, k, y 8 k, z k 5 Using matri method, solve the following system of linear equations: y z y z 7 y z y z (i) y z (ii) z (iii) y 0 (iii) (iv) y z y z y z 5 y y z y z y z 7 Ans:, y, z, y, z, y, z, y, z, y, z 5 6 Find the product of matrices A 7 5, B and use it for solving the equations y z, y z 7, y z Ans: AB I,, y, z 7 If 5 A, find (v) y y z 7 A Hence solve the following: y 5z 0, y z, y z Ans:, y, z

8 ASSIGNMENT CLASS XII CONTINUITY AND DIFFERENTIABILITY Important Formulas A function f ( ) is continuous at a iff lim f ( ) lim f ( ) f ( a ) a a f ( ) f ( a) A function f ( ) is differentiable at a iff lim eists finitely ie a a f ( a h) f ( a) f ( a h) f ( a) lim lim h0 h h0 h Show that the function sin, 0 f ( ) 0 0 is continuous at 0 Show that the function sin cos, 0 f ( ) 0 is continuous at 0 5 when 0 Show that the function f ( ) when is continuous at Show that the function f ( ) is continuous at 0 5 Show that the function, 0 f ( ) 0 is discontinuous at 0, 6 If f is defined as f ( ) 0 sin, 0 7 Show that the function f ( ) 0 Show that f is everywhere continous ecept at is discontinuous at 0 8 Find the value of k so that the function f is continous at the indicated point: cos, 0 (a) f ( ) k 0 9 Show that the function at 0 if f ( ) if (b) 5, 5 f ( ) 5 k 5 is not differentiable at at 5 0 Discuss the continuity and differentiability of f ( ) ANSWERS 8(a) (b) 0 0 continous but not differentiable at,

9 ASSIGNMENT CLASS XII DIFFERENTIATION Find dy for the following: d (a) y a (b) 5 y sin ( ) cos sin y cos sin y log cos e (e) y log( ) (f) y sin sin Show that d a sin a a d a If y, prove that dy ( ) y 0 d If y a, prove that 5 Find dy for the following: d n dy ny d a (a) sin (cos ) cos (sin ) (b) cos tan sin cos sin tan cos sin sin tan sin (e) sin sin tan sin sin 6 Find dy for the following: d (a) cos ( ) (b) cot tan 5 (e) sin tan 7 If y a y, prove that dy d y dy 8 If y, prove that y d 9 If y log, show that dy y 0 d

10 0 If y log y, prove that dy log d log If log y y tan, prove that dy y d y If sin y, prove that dy y d If cos sin y cos, find dy d a b If a b y y, prove that dy y d 5 If f ( ) ', find 0 f 6 Differentiate tan wrt sin 7 If asin t cos t, y b cos t cos t, show that dy d at t b a 8 If t t a, y t t, show that dy t d at 9 If cos cos and y sin sin, find d y n y 0 d d y 0 If y Acos n Bsin n, prove that d d y dy If y e sin cos, prove that y 0 d d If y tan, show that d y dy 0 d d ANSWERS (a) ( a ) (f) (b) 55 ( ) sec ( ) 5 (a) (e) sin( 6) sec ( ) (b) (e) 6(a) cos 5 7log 6 9 sin cos e cos e (b) (e) ( ) cos sin log sin cos sin tan cos log cos

11 ASSIGNMENT CLASS XII APPLICATION OF DERIVATIVES Questions based on Mean Value Theorem Verify Rolle s Theorem for each of the following functions: f ( ) 6 in, f ( ) 7 6 in, (a) (b) f ( ) in, f ( ) sin cos in 0, Discuss the applicability of the Rolle s Theorem for the following functions on the indicated intervals: f ( ) on, f ( ) on, (a) (b) when 0 when y 6,,,where the tangent is parallel f ( ) [ ] on, f ( ) on 0, Using Rolle s Theorem, find the points on the curve to -ais Verify Lagrange s Mean Value Theorem for each of the following functions: (a) f ( ) in, (b) f ( ) in 0, f ( ) on,5 f ( ) 6 on, 5 Find a point on the parabola y,where the tangent is parallel to chord joining,0 and, 6 Use Lagrange s Mean Value Theorem to determine a point P on the curve f ( ) definded on,, where the tangent is parallel to the chord joining the end points of the curve Questions based on Rate of Change of Quantities 7 Find the points on the curve y 8 for which abscissa and ordinate change at the same rate 8 A partical moves along the curve y Find the points on the curve at which y-coordinate is changing twice as fast as -coordinate 9 The volume of spherical balloon is increasing at the rate of 5 cm / sec Find the rate of change of its surface area at the instant when its radius is 5 cm 0 The surface area of a spherical bubble is increasing at the rate of cm / sec Find the rate at which the volume of the bubble is increasing at the instant its radius is 6 cm Water is leaking from a conical funnel at the rate of 5 cm / sec If the radius of the base of funnel is 5 cm and its altitude is 0 cm, find the rate at which water level is dropping when it is 5 cm from top Water is running into a conical vessel, 5 cm deep and 5 cm in radius, at the rate of 0 cm / sec When the water is 6 cm deep, find at what rate is: (a) water level rising? (b) water surface area increasing? wetted surface of vessel increasing? Questions based on Increasing and Decreasing Functions Show that the function f ( ) 6 8 is an increasing function on R Show that the function f ( ) cos 5 Find the intervals on which the following functions are (i) increasing (ii) decreasing: (a) f ( ) (b) f ( ) 56 is strictly decreasing on 0, f ( ), 0 f ( ) (f) f ( ) (g) f ( ), 0 (h) (e) ( ) f e f ( ) sin cos in 0,

12 Questions based on Tangent and Normal 6 Prove that the tangents to the curves y 5 6 at the points (,0)and (,0) are at right angles y 7 Find points on the curve at which the tangents are parallel to (a) -ais (b)y-ais Find the equation of the tangent to the curve y 5, which is parallel to the line y 0 9 Find the equation of the normals to the curve 0 Find the equation of the tangent to the curve At what points on the curve y y, which is parallel to the line y y 8 y, which is parallel to the line y 5 0 0, is the tangent parallel to the y-ais Show that the curves y a and y a touch each other Find the equation of the tangent to the curve sin, y cos at Find the points on the curve 9y, where the tangents are perpendicular to the line y 0 5 If the tangent to the curve at (, 6) is parallel to the line y 5 0, find a and b y a b 6 Find equation of the tangent to the curve y, which are perpendicular to 9y 0 7 Show that the curves y and y k cut at right angles if k 8 8 Find the equations of the tangent and the normal to the following curves at the indicated points: (a) y at (, ) (b) y at y at (, ) y at ( a cos, bsin ) a b Questions based on Approimations 9 Using differentials, find the approimate value of the following: (a) 007 (b) If y and if chnges from to 99, what is the approimate change in the y? Find approimate change in volume V of a cube of side meters caused by increasing the side by % If the radius of a sphere is measured as 9 cm with an error of 00 cm, find approimating error in calculating its volume Questions based on Maima and Minima Find the points of local maima or local minima and the corresponding local maimum and minimum values of each of the following functions: (a) f ( ) 6 0 (b) f ( ) f ( ) ( )( ) f ( ) 8 05 (e) f ( ) sin, where (f) f ( ) cos, where 0 Find the absolute maimum value and absolute minimum value of the following functions: (a) f ( ) in, 5 (b) f ( ) 8 8 in, 5 Find the maimum and minimum values of the following functions: (a) f ( ) sin on 0, (b) f ( ) sin cos on 0 6 Show that f ( ) sin cos is maimum at in the interval 0,

13 Questions based on Maima and Minima( Word Problems) 7 Show that of all the rectangles of the given area, the square has the smallest perimeter 8 Find the point on the curve y which is nearest to the point (,) 9 A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semi-circle Find the dimensions of the rectangle, so that its area is maimum Also find the maimum area 0 A right circular cylinder is inscribed in a given cone Show that the curved surface area of the cylinder is maimum when diameter of cylinder is equal to the radius of the base of cone An open tank with the square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water Show that the cost of the material will be least when the depth of the tank is half of its width Show that a closed right circular cylinder of given surface area and maimum volume is such that its height is equal to the diameter of the base Show that the height of the right circular cylinder of maimum volume that can be inscribed in a given h right circular cone of height h is Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long 5 A closed circular cylinder has a volume of 56 cu Cm What will be the radius of its base so that its total surface area is minimum? 6 Of all the rectangles each of which has a perimeter 0 m, find one which has maimum area Find the maimum area also 7 A wire of length 0 m is to be cut into two pieces One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle Where the wire should be cut so thet the sum of the areas of the square and triangle is minimum? 8 An open bo with a square base id to be made out of a given quantity of sheet of area a Show that the maimum volume of the bo is a 6 9 Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and semivertical angle 0 is 0 8 h 50 Show that a right triangle of given hypotenuse has maimum area when it is an isosceles triangle 5 A window is in the form of an rectangle above which there is a semi-circle If the perimeter of the window is p cm, show that the window will allow the maimum possible light only when the radius of the semi-circle is p 5 Show that the rectangle of maimum area that can be inscribed in a circle of radius r is a square of side r 5 Two sides of a triangle have lengths a and b and the angle between them is What value of will maimize the area of the triangle? Find the maimum area of the triangle also 5 A rectangular window is surmounted by an equilateral triangle Given that the perimeter is 6 m, find the width of the window so that the maimum amount of light may enter 55 Show that a right circular cylinder which is open at top, and has a given surface area, will have greatest volume if its height is equal to the radius of its base 56 Show that the maimum volume of the cylinder which can be inscribed in the sphere of radius 5 cm is 500 cm 57 Show that the right circular cylinder of given volume open at the top has minimum total surface area, provided its height is equal to the radius of its base 58 Prove that the surface area of the solid cuboid, of square base and given volume, is minimum when it is a cube 59 Show that the height of the cone of maimum volume that can be inscribed in a sphere of radius cm is 6cm 60 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?

14 ANSWERS (APPLICATION OF DERIVATIVES) (a) c (b) 8 c (a) c 6 (b) c c 5 8, ;, (b) / sec 9 0 cm (b) inc on inc on, 0 cm / sec 0, dec on,, c 7 9 c 5, 6, 6 6 cm / sec cm / sec (a) 5 Not applicable in any case 0,6 7, cm / sec 0 0 cm 0 / sec 5 (a) inc on,,, inc on,,, dec on,,,,,,, dec on (e) inc on, dec on 8 8 (f) inc on,,, dec on, (g) inc on,,,0 0, 5 5 (h) inc on,, dec on 0, 7 (a) No point (b) (, 0) and (, 0) y y y 0 (, )and (, ), dec on y, and, 5 a, b y 0 and 9 y 0 8 (a) y 0, y 0 (b) 0 y 8 0, 0 y 0 y y 0, y 6 0 cos sin, a sec by cos ec a b a b 9 (a) 095 (b) m 97 cm (a)local ma value is at and Local min value is 8at 6 (b)local ma value is 68at and Local min value is 67 at 6and 6at 5 Local ma value is 0 at and Local min value is at 0 95 Local ma value is 05at 0and at 5 and Local min value is at (e) Local ma value is at and Local min value is at (a)abs ma is 8at,Abs min is 0at (b)abs ma is 57at, Abs min is 6at (a) ma value is at and minvalue is at r (b) ma value is at and minvalueis at 8 (,) 9 units, r units; r units 6 5 cm 5 7 cm 6 square, 00m m , , ab

CLASS XII MATHEMATICS. Weightage (Marks) (i) Relations and Functions 10. (ii) Algebra 13. (iii) Calculus 44

CLASS XII MATHEMATICS. Weightage (Marks) (i) Relations and Functions 10. (ii) Algebra 13. (iii) Calculus 44 CLASS XII MATHEMATICS Units Weightage (Marks) (i) Relations and Functions 0 (ii) Algebra (iii) Calculus 44 (iv) Vector and Three Dimensional Geometry 7 (v) Linear Programming 06 (vi) Probability 0 Total