STRATEGIC ACTION PLAN FOR SLOW LEARNERS MATHEMATICS CLASS XII

Transcription

1 STRATEGIC ACTION PLAN FOR SLOW LEARNERS MATHEMATICS CLASS XII SL NO TOPIC AREAS IDENTIFIED MARKS. INVERSE TRIGONOMETRIC FNS Applications 4. MATRICES & DETERMINANTS PROPERTIES OF DETERMINANTS SOLVING EQUATIONSUSING MATRIX METHOD DIFFERENTIATION Using Logarithms 4 4. APPLICATION OF DERIVATIVES INCREASING AND DECREASING FNS 5. APPLICATION OF INTEGRALS AREA BETWEEN TWO CURVES USING FORMULAE 4 6 OR AREA OF TRIANGLE 6. DIFFERENTIAL EQUATIONS LINEAR DIFFERENTIAL 4 EQN. 7 VECTORS DOT PRODUCT 4 VECTOR PRODUCT 8. THREE DIMENSIONAL GEOMETRY SHORTEST DISTANCE 4 BETWEEN TWO LINES 9 LINEAR PROGRAMMING GRAPH 6 0 PROBABILITY BAYE S THEOREM 4 TOTAL 50

2 Inverse Trigonometric Functions Areas to be revised:. Principal value branch table.. Properties of Inverse Trigonometric functions. Properties: tan ifxy<. tan +tan tan if x > 0, y > 0, xy> tan ifx < 0, y < 0, xy>.tan tan Problems tan ifxy>- tan ifx > 0, y < 0, xy<- tan ifx < 0, y> 0, xy<-. Prove that Sol: L.H.S = tan tan tan =tan. - tan tan tan tan = tan tan = tan ( xy> -) = tan tan = R.H.S.

3 . If,then find the value of x. Sol.: We have sinsin cos, = sinsin cos sin = sin cos = x= 3. Write the value of. Sol:tan sin x since cos = tan sin tan x tan Prove that = tan tan =tan tan tan tan =tan tan =tan tan tan tan, =tan Let tan tan sin sin tan =sin = R.H.S 3

4 5. Find the value of xy< Sol: tan sin cos x <,y>0, =tan [tan tan tan sin =tan x [ tan tan =tan [ tan cos 6. Prove that Sol. L.H.S tan tan tan tan = tan tan = tan tan = tan tan = tan x =tan tan 7. Prove that Sol. LHS =cot 7cot 8 cot 8 =tan tan tan =tan tan since x since cot tan tan 3 tan 3 8 tan 8 3 x 8 =tan tan cot 3RHS 4

5 8. Solve Sol.=tan tan 3 =tan tan => 6x +5x-=0 => (6x-)(x+) = 0 x= or sincex= - doesn t satisfy the equation, x=/6 is the only solution of the given equation. 9. Solve for x, Sol. Given tan tan =tan = tan = = If, then solve the following for x Sol. Given tan tan tan = tan tan = tan tan = = & 4 As 0 <x <,x -8 therefore x=/4 5

6 Matrices & Determinants. Let A= express A as a sum of two matrices such that one is symmetric and other is skew symmetric. Sol. A can be expressed as A= [A+A ] + ½ [A-A ] () Where A+A and A-A are symmetric and skew symmetric matrices respectively A+A = A- A = Putting the values of A+A and A - A in equation () we get A= ½ Using properties of determinants, prove that a ab ac Sol.LHS let ba b bc ca cb c takinga, b, c common from R, R and R 3 respectively a b c bc a b c a b c Now taking a, b, c common from C, C and C 3 respectively applying R R + R 0 0 expanding along first row, we get 00 = 4 3. Using properties of determinants, show that Sol. LHS = let applying R R +R + R 3 we get 6

7 Taking common from R applyingc C C,C 3 C 3 C 0 0 expanding along R = = 4. Using properties of determinants, show that Sol. LHS Multiplying C, C and C 3 by a, b and c respectively, we get taking a, b, c common from R, R and R 3 respectively applying C C + C + C 3 Taking common ( from C Applying R R R, R 3 R 3 R expanding along C, we get 0 00 RHS 7

8 5. Prove that Sol. let applying C C +C +C 3 Taking common from C, we get applying R R -R, R 3 R 3 -R 0 0 expanding along C we get = = RHS 6. Using properties of determinants prove that Sol. LHS let applying C C +C +C 3 Taking ( common from C, we get applying R R -R, R 3 R 3 -R and expanding along C we get 0 RHS 7. Prove that Sol.LHS= let 8

9 Taking a,b,c common from R,R and R 3 respectively taking applying R R +R +R 3 common from R applying C C C, C 3 C 3 C = 0 0 0expanding along R, we get 0 0 RHS 8. If a, b, c are real numbers and show that either a+b+c=0 ora=b=c Sol. Let applying C C +C +C 3 = applying R R -R, R 3 R 3 -R and expanding along C and on simplification we get given 0 either 0 or 0 or 0 or 0 Either a+b+c=0 or a=b=c 9

10 . Two schools decided to award some of their selected students for the values honesty, regularity and hardwork at the rate of Rs. X,Rs. y and Rs.z respectively per student the first school allotted a total of Rs.5,000 for its,and students for the respective values, while the second school kept Rs.9000 for theses values for 3,and students respectively. If the sum of three awards per students is Rs.0,000 then find the values of x,yand z using matrices. Suggest one more value which should also be included for the awards. Sol. We can represent given information, by the system of equation x y z x y z 9000 x y z 0,000 Rewriting the above equations in matrix form AXB Where A3 X B A 0, so A exists and have unique solutions 3 3 adja A XA B Hence the award for honesty Rs 000, award for regularity Rs 3000 and award for handworkrs 5000 Value: Any one value like sincerity or helpfulness etc can be awarded.. There are 3 families A,B and C. The no. of men, women and children in these families are as under Men Women Children Family A 3 Family B 3 Family C 4 6 Daily expenses of men, women and children are Rs00, Rs50 and Rs00 respectively only men and women earn and children do not. Using matrix multiplication, calculate the daily expenses of each family. what impact does more children in the family create on the society? Sol. The No. of men, women and children in families A,B and C can be represented by 3 x 3 matrix as 0

11 family A 3 Xfamily B 3 and daily expenses of men, women and children can be family C 4 6 Men 00 represented by 3 x matrix as Y women 50Daily expense of each family is given children 00 by the product XY family A 3 00 family A 050 XY family B 350 family B 50 Hence daily expense of family C family C 300 i Family A Rs 050 ii Family B Rs 50 iii Family C Rs 300 VALUE: More children in the family will increase the expenses of family, which will affect the economy of society. 3. For the matrix A show that A 5A4I0 hence find A Sol. Given A A AA A 5A4I A 5A4I0 Premultiplying by A both sides, we get A A 5A A4A IA 0 A 5I4A 0 4A 5I A A 5I A A Try These. If A and I is the identity matrix of order then show that A 4A3I0 hence find A Ans To raise money for an Orphanage, students of three schools A,B and C organized an exhibition in their locality, where they sold paper bags, scrap book and pastel sheets made by them using recycled paper, at the rate of Rs 0, Rs 5 and Rs 5 per unit respectively. School A sold 5 paper bags, scrap books and 34 pastel

12 sheets. School B sold paper bags, 5 scrap books and 8 pastel sheets while school C sold 6 paper bags 8 scrap books and 36 pastel sheets. Using matrices find the total amount raised by each school. By such exhibition, which values are inculcated in the students? Ans: School ARs 850 B Rs 805 C Rs 970 Values: helping the orphans, use of recycle paper. 3. Find non zero values of x satisfying the matrix equation Librarian Mr.Ajeet Kumar has purchased 0 dozen autobiography of great person, 8 dozen historical books, 0 dozen story books related to moral teaching the cost prices are Rs.80, Rs.60 and Rs.40 respectively. Find the total amount of money that he invested for library using matrix algebra. Which type of books is more useful for students and why? Ans: Rs.060, autobiography of great person is more useful for students as it educate a lesson to them for being a great person If A0 prove that A 3 6A 7AI Using matrix, solve 3x y3z8, xy z, 4x 3yz4 Ans: x, y, z Find A, if A hence solve the following system of linear equation 3 xy5z0, x y z, x3y z 8. Solve using matrix, 3 3 0, 0, Using properties of Determinants, show that 4 Sol: L.H.S Let Applying R > R +R +R 3 Taking common from R Applying R > R R, R 3 > R 3 R 0 0

13 Applying R >R +R +R Expanding along R, we get =RHS 0. If x, y,z are all different and Sol: Let = 0, then show that 0. + (Taking common x,y,z from R,R and R 3 respectively) = Applying R >R R, R 3 >R 3 R 0 0 Expanding along C and simplifying, we get Since 0 and x,y,z are all different 0,0, we get 0. TRY THESE I. Using properties of determinants, prove the following

14 = II. Using properties of determinants solve for x ,0 5. Using properties of det. Prove that i. 9 ii. [Ans:,, [Ans: x=4] [Ans: x=, ] [Ans: x= ] Solutions of Linear Equations using Matrices. Solve 7;3455;3 using Matrix method. Sol: The given system of equations can be expressed in Matrix from A x =B, where ,, exists and given system has unique solution X=A B 4

15 7 3 9, X=A B= =>,, If 3 4, find A, using A solve the system of equations 3 5, 3 4 5, Sol: 3 4=0 6+5= 0 A is a non singular Matrix, so A exists adj A= 9 3, The given system of equations can be expressed as 3 5 A x =B Where A= 3 4, X=, B=5 3 0 AX=B => A B => X= = => x=, y=, z= Determine the product 7 3 and using it solve the equations., 4, Sol: Let A=, CA= 7 3 = => A = [

16 The given system of equations can be written inmatrix form as PX=B Where P=, X=, B= PX=B = > But P= = X= = 4,, 4. Solve 4,, Sol: Rewriting the given equations in Matrix form, we get by using Matrix method. AX=B Where A= 4 6 5, 6 9 0, = 00 0 A is non singular so A exists and X= A B adj A= , A = X=A B => => = => x=, y=3, z=5 6

17 5. The management committee of a residential colony decided to award some of its members (say x) for honesty, some(say y) for helping other and some others (say z) for supervising the workers to kepp the colony neat and clean. The sum of all the awardees is. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards. Sol: According to the question, the system of values is, , 0. The above system of equations can be written in matrix for AX=B as 3 3=33 where A= 3 3,, A =9+ 7=3 0, So A exists. AX=B = > A B adj A= 0, X= 0 33= =4=> x=3, y=4, z=5 5 Number of awards for honesty = 3 Number of awards for helping others= 4 Number of awards for supervising = 5 Value: The management can include cleanliness for awarding the members. Or the management can also include the persons, who work in the field of health and hygiene. Or any other relevant answer Given A4 4 B 3 4 find BA and use this to solve the 5 0 system of equations yz7, x y3, x3y4z7 7

18 0. Sum of three numbers is 0. If we multiply the first by and add the second number and subtract the third we get 3. If we multiply the first by 3 and add second and third to it we get 46. Find the numbers. Ans: 3,, 5 3. If A and B 3 0 then find AB 5 0 use AB B A 4. Express the matrix A3 5 7 as the sum of a symmetric and a skew symmetric matrix. 3. Find a matrix X such that ABX0, when A, B A trust has fund Rs.50,000 that is to be invested in two different types of bonds. The first bond pays 0%P.A interest which will be given to adult education and second bond pays % interest P.A which will be given to financial benefits of the trust using matrix multiplication, determine how to divide Rs.50,000 among two types of bonds, if the trust fund obtains an annual total interest of Rs.800. what are the values reflected in the question. 5. An agriculture firm possesses 00 acre cultivated land that must be cultivated in two different mode of cultivations : organic and inorganic. The yield for organic and inorganic system of cultivation is 5 quintals/acre and 0 quintals/acre respectively.using matrix method determine how to divide 00 acre land among two modes of cultivation to obtain yields of 600 quintals. Which mode of cultivation do you prefer most and why? 8

19 LOGARITHMIC DIFFERENTIATION : Rules of logarithmic function = / = =n Change of base rule = loge =, log = 0, PRACTICE QUESTIONS:. Differentiate Solution: DIFFERENTIABILITY 9

20 . If find 3. If,, 0

21 4. Differentiate with respect to x: 5. If,

22 6. If 7. find

23 8. Differentiate ( PRACTICE QUESTIONS: Find for the following :. If If prove that. Differentiate with respect to x:. If, 3

24 APPLICATIONS OF DERIVATIVES INCREASING AND DECREASING FUNCTIONS:. Steps for working rule : i) Find f (x) in factor form. ii) Solve f (x) = 0 and find the roots. iii) If there are n roots,then divide the real number line R into (n+ ) disjoint open intervals. iv) Find the sign of f (x) in each of the above intervals. v) f(x) is increasing or decreasing in the intervals when f (x) is positive or negative respectively Tips and Techniques :. If the coefficient of the highest power is +ve then the rightmost interval in the Real Line is +ve & the other intervals from right to left get alternatively signed. The given function is increasing in +ve signed intervals and decreasing in the ve signed intervals.. If the coefficient of the highest power is ve then the rightmost interval in the Real Line is ve & the other intervals from right to left get alternatively signed. The given function is increasing in +ve signed intervals and decreasing in the ve signed intervals. 4

25 SAMPLE QUESTIONS AND SOLUTION:. Find the intervals in which the function f given by is a) strictly increasing b) strictly decreasing 5

26 . Find the intervals in which the function f given by,, is strictly increasing or strictly decreasing. SOLUTION: 3. Find the intervals in which the function f given by is a) strictly increasing b) strictly decreasing 6

27 4. Find the intervals in which the function f given by is a) strictly increasing b) strictly decreasing.5. Find the intervals in which the function f given by is a) strictly increasing b) strictly decreasing 7

28 5. Show that log, x> is an increasing function throughout its domain. 8

29 PRACTICE QUESTIONS: Find the intervals in which the following functions are increasing and decreasing Prove that the function f given by,. Prove that the function f given by,. Show that the function is neither increasing nor decreasing on (0,) TANGENTS AND NORMALS :. Find the equation of the tangent to the curve, at 9

30 . Find the equation of the tangent to the curve, which is parallel to the line 3. Find the equation of the tangent to the curve, which is parallel to the line 30

31 4. Find the points on the curve at which the slope of the tangent is equal to the y coordinate of the point. 3

32 5. Find the equations of the tangent and the normal to the curve, 3

33 6. Find the equation of the normal at the point ( am,am 3 ) for the curve a y = x 3 7. Find the equation of the tangent and normal to the curve, 33

34 PRACTICE QUESTIONS:. Find the points at which the tangent to the curve Is parallel to the x axis.. Find the slope of the normal to the curve, 3. Find the points on the curve at which the tangent has the equation 4. Prove that the tangents to the curve,, Are at right angles. 5. Find the equation of the tangent to the curve, which is parallel to the line 6. Find the equation of the tangent and normal to the curve 7. Find the equations of the normals to the curve which is parallel to the line 8. Find the equation of the tangent to the curve,. 9. Find the equation of the tangent to the curve, That are parallel to the line x+y = 0 0. Find the equations of the normal at a point on the curve x = y which passes through The point (, ). Also find the equation of the tangent 34

35 INTEGRATION & APPLICATION OF INTEGRALS POINTS TO REMEMBER: A: Integration of standard functions 35

36 ; where m B: Integration by substitution /. /. / Where C: Integration using trigonometric identities

37 D: Integration of special functions Every quadratic polynomial can be expressed in one of the three forms, or by completing square method Example: [see that x coefficient is ] [Add and subtract square of (half the x coefficient)i.e. ] It is in the form of where and E: Integration of quadratic equations. Express Q. E as and use, and 3 formulae. Find A, B Such that L.E= A.. +B, separate integrals and proceed 3. Express Q. E as and use 4,5 and 6 formulae 4. Find A, B Such that L.E= A.. +B, separate integrals and proceed 5. Express Q. E as and use 7,8 and 9 formulae A rational expression is called proper if degree of is smaller than degree of If is proper and the polynomial can be expressed as product of linear/quadratic factors, then it can be decomposed into small fractions called partial fractions. (All non repeated linear factors only) (Repeated but linear factors only ) (involve non repeated quadratic factors) 37

38 6... Find A, B Such that L.E= A.. +B, separate integrals and proceed F: Integration of a rational expression If it is proper, decompose it into partial fractions and then integrate. If it is not proper, divide by Q(x) And can be split into partial fractions as it is proper G: Integration by parts. / ( learn ILATE rule). / H: Definite Integrals I: Definite integral by limit Sum method. To find ; follow the steps mentioned below.. Write (constant) 3. Find 4. Substitute the in the formula 5. Use values :,,, J: Application of Integrals denote the area under the curve bounded by three lines, and y=0. To find the area bounded by a curve and a curve/straight line, a. First find the points of intersection where the curve intersects the curve/line. b. Draw the rough sketch of the curve and curve/straight line c. Write the required area using definite integrals and then solve.. Evaluate. Evaluate Questions for practice 3. Evaluate 4. Evaluate 38

39 5. Evaluate 6. Evaluate 7. Evaluate 8. Evaluate 9. Evaluate 0. Evaluate. Evaluate. Evaluate 3. Evaluate 4. Evaluate 5. Evaluate 6. Evaluate 39

40 7. Evaluate 8. Evaluate 9. Evaluate 0. Evaluate. Evaluate. Evaluate 3. Evaluate 4. Evaluate 5. Evaluate 6. Evaluate 7. Evaluate 8. Evaluate 9. Evaluate 30. Evaluate 3. Evaluate 3. Evaluate 33. Evaluate by the method of limit of sums. 34. Evaluate by the method of limit of sums 35. Evaluate by the method of limit of sums 36. Using integration, find the area of the region bounded by the curves,,, 37. Using the integration, find the area of the region bounded by the curve and the line 38. Sketch the graph of and evaluate the area under the curve above x axis and between to. 39. Using the integration, find the area of the region bounded by the curve and 40. Find the area of the circle which is interior to the parabola. 4. Using integration find the area of the triangle ABC, coordinates of whose vertices are A(4, ), B(6, 6) and C(8, 4). 4. Using the integration find the area of the triangular region whose sides have equations, and 43. Using the integration find the area of the triangular region whose sides have equations, and 44. Using the integration, find the area of the region enclosed between the two circles and 40

41 HINTS/SOLUTIONS Write, expand Ans: Put sin x = t, continue as in problem 3 Ans: Divide Nr and Dr by cos x, replace by Ans: Use identities Ans: Write Ans: Multiply and divide by, and put Obtain the form, let Obtain the form = Write use formula D5 Ans: Write Sol: 9 Evaluate 0 Evaluate Evaluate Similar to previous problem given for practice Similar to previous problem given for practice Similar to previous problem given for practice Evaluate Similar to previous problem given for practice 3 Evaluate Similar to previous problem given for practice 4 Evaluate Similar to previous problem given for practice 5 Evaluate Write = and integrate Ans: 6 Evaluate Write integrate then 4

42 7 Evaluate 8 9 Ans: (by parts) Evaluate Take and use by parts formula Ans: Evaluate 0 Evaluate x=t) Evaluate (put Evaluate 3 Evaluate 4 Evaluate 5 6 Evaluate Evaluate = Ans: = Ans: +C, For 4, ; and For, ; For 4, ; For 4, = Use rule H4, add both integrals,. Put cosx =t, = Use rule H4, add both integrals, Use rule H7,, Use rule H4 again add integrals Ans. Ans: = where Sin x Cos x = t, after substitution,it becomes = =.. 4

43 7 Evaluate Evaluate Evaluate Evaluate 3 Evaluate 3 Evaluate where Sin x Cos x = t, after substitution, it becomes Ans: Put x = Tan t, with that Use rule H4, Use rule H5, Add Refer problem 7 Use Rule H4, simplify to get let Given for practice use H4 Use rule H4, simplify to get Apply H7, Ans: 43

44 34. Evaluate by the method of limit of sums. Here, and = = Evaluate by the method of limit of sums Here, and = = Evaluate by the method of limit of sums Here, and = = 37. Using integration, find the area of the region bounded by the curves,,, Draw the lines y=x+, y= x, x= 3, x=3, y=0 Area of Shaded region: =. 38. Using the integration, find the area of the region bounded by the curve and the line The curves and intersect at (, ), (,) Area of Shaded region: = 44

45 39. Sketch the graph of and evaluate the area under the curve above x axis and between to. Area of Shaded region: =. =9 40. Using the integration, find the area of the region bounded by the curve and Points of intersection (0,0), (,) Area of Shaded region:= 4. Find the area of the circle which is interior to the parabola. Circle and parabola intersect at,,, Area of Shaded region: = 4. Using integration find the area of the triangle ABC, coordinates of whose vertices are A(4, ), B(6, 6) and C(8, 4). Eq to AB :, Eq to BC :, Eq to AC : Required area = = = =7 43. Using the integration find the area of the triangular region whose sides have equations, and Given for practice do Same as above 44. Using the integration find the area of the triangular region whose sides have equations, and Given for practice do Same as above 45

46 45. Using the integration, find the area of the region enclosed between the two circles and Given for practice 46. Find the particular solution of the differential equation = 0 for x=, y=. Given for practice ALL THE BEST 46

47 DIFFERENTIAL EQUATIONS POINTS TO REMEMBER:. An equation involving derivative (derivatives) of the dependent variable with respect to independent variable (variables) is called a differential equation. Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation 3. The highest power (positive integral index) of the highest order derivative involved in the given differential equation is defined as the degree of the differential equation. 4. The curve y = φ (x) is called the solution curve (integral curve) of the given differential equation if the derivatives of y, satisfy the differential equation. 5. The solution which contains arbitrary constants and satisfy the given differential equation is called the general solution (primitive) of the differential equation 6. The solution of a differential equation independent from arbitrary constants is called a particular solution of the differential equation 7. To obtain differential equation when the general solution(family of curves) is given (i). Identify the number of arbitrary constants involved in general solution, say n (ii). Derivate the general solution for n times let,,,. (iii). Eliminate the arbitrary constants by using,,,. (iv). The equation then obtained involve differentials in place of arbitrary constants, and it is required differential equation. 8. To obtain general/particular solution when the differential equation is given (i). (VARIABLE SEPARABLE)If the given differential equation can be expressed as OR then it can be solved by separating the variables and integrating both sides. OR (ii). (HOMOGENEOUS)If the given equation is expressed as, such that both, the functions and are homogeneous then; substitute and. Then the differential equation can be solved by using method discussed in type (i). (iii). (LINEAR D E)If the given differential equation is in the form are functions of, OR where where are functions of ; then it is called linear differential equation(lde). It can be solved by multiplying both sides of the LDE by integrating factor I.F., and integrating both sides. and write. 47

48 Questions for Practice 48

49 45. Solve the differential equation: Solve the differential equation 0, given that 0, 47. Find the particular solution of the differential equation, given that 0 when 48. If is a solution of the differential equation and 0, then find the value of 49. Find the particular solution of the differential equation 0, given that when Find the general solution of the differential equation 5. Solve the differential equation, given that. 5. Solve the differential equation 53. Solve the differential equation ; 54. Solve the differential equation 55. Solve the differential equation Find the particular solution of the differential equation 3 = 0 for x=, y=. 57. Obtain the differential equation of all circles of radius r 58. Show that the differential equation 0 is a homogeneous. Find the particular solution of this differential equation, given that x=0 when y=. HINTS/SOLUTIONS. given differential equation: 0 implies 0 integrating both sides 0implies 0 log. Given differential equation 0, given that 0, 0implies General Solution is tan tan 0, implies C = Particular Solution: tan tan 3. Given differential equation implies General Solution: log given that 0 when implies Particular Solution: log implies 4. Given differential equation log log log General Solution : and0 implies C = 4, Particular solution: 4 the value of is 5. given differential equation 0, 0integrating we get General Solution log 49

50 given that when 0 that implies log log P.S. 6. Given differential equation OR put implies integrate both sides : log 3 tan 3 7. Given differential equation given that integrating factor is, Multiply both sides by x integrating both sides G.S. C=0, PS.= 8. Given differential equation integrating factor is, Multiply both sides by integrating both sides GS. 9. Given differential equation integrating factor is, Multiply both sides by integrating both sides G.S. log 0. Given differential equation (Homogeneous) Put y = Vx Ans G.S. =. Given differential equation 3 can be written as 3 (LinearDE) Ans: General Solution 3. Find the particular solution of the differential equation 3 = 0 for x=, y=. Given for practice 3. Obtain the differential equation of all circles of radius r Equation to a circle with radius r is given by..() where a, b are arbitrary constants Derivating () both sides with respect x we get. 0.() Derivating () both sides with respect x we get. 0.(3) So,. ;. substituting these in eq, we get OR 4. Show that the differential equation 0 is a homogeneous. Find the particular solution of this differential equation, given that x=0 when y=. (Put x = Vy ) 50

51 5

52 =================================================================== 5

53 ====================================================================== Hint: 3 and 3 3 : and 9 d = Hint: Convert in to Vector form and proceed. The vector form of the lines are = 7 6 ) = ) 53

54 ============================================================================ ============================================================================= ===============================================================================. Find the distance between two parallel lines Hint :, ( = 93, 7 54

55 STRATEGIC ACTION PLAN FOR SLOW LEARNERS SOME IMPORTANT RESULTS/CONCEPTS LINEAR PROGRAMMING ** Solving linear programming problem using Corner Point Method. The method comprises of the following steps:. Find the feasible region of the linear programming problem and determine its corner points. Evaluate the objective function Z = ax + by at each corner point. Let M and m, respectively denote the largest and smallest values of these points. 3. (i) When the feasible region is bounded, M and m are the maximum and minimum values of Z. (ii) In case, the feasible region is unbounded, we have: 4.(a) M is the maximum value of Z, if the open half plane determined by ax + by >M has no point in common with the feasible region. Otherwise, Z has no maximum value. (b) Similarly, m is the minimum value of Z, if the open half plane determined by ax + by < m has no point in common with the feasible region. Otherwise, Z has no minimum value. 55

56 SOLVED PROBLEMS: ) A Shopkeeper sells only tables and chairs. He has only Rs 6,000 to invest and has a space for at most 0 items. A Table costs him Rs 400 and a chairs costs him Rs 50. He can sell a table at a profit of Rs 40 and a chair of Rs 30. Supposing he can sell whatever he buys, formulate the problem as a LPP and solve it graphically for maximum profit. Sol: Let x tables and y chairs are bought Y Then LPP is 5 To maximize Z = 40x + 30y Subject to constraints, C(0, 0) B(, ) 3 3 X+y 0 400x + 50y A(5, 0), =>8x +5y 0, X X 0, y Possible points for maximum Z are A(5, 0), B(, ), C(0, 0) 3 3 POINT Z = 40x + 30y VALUE A(5, 0) [MAXIMUM VALUE] B(, ) C(0, 0) Z is maximum for B(, ), ie., 6 tables and 3 chairs must be purchased and sold for a 3 3 maximum profit of Rs

57 ) One kind of cake requires 00 g of flour and 5 g of fat and another kind of cake requires 00 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and kg of fat. Assuming there is no shortage of other ingredients used in making the cake. Sol: Let x cakes of type and y cakes of type are made. Y Then LPP is 50 To maximize Z = x + y Subject to constraints, C(0,0) X 0, y 0 0 B(0,0) 00x + 00y 5000 => x + y 50 0 A(5,0) X And 5x + 50y 000 =>x + y POINT Z = x+ y VALUE A(5, 0) B(0, 0) maximum C(0, 0) Z is maximum at B (0, 0) Hence, 0 cakes of type and 0 cakes of type must be made for maximum number of 30 cakes. ) A Manufacturing company makes two models A and B of a product. Each piece of model A requires 9 labour hours for fabricating and labour hour for finishing. Each piece of model B requires labour hours for fabricating and 3 labour hour for finishing. For Fabricating and finishing, the maximum labour hours available are 80 and 30 respectively. The company makes a profit of Rs 8000 on each piece of model A and Rs 000 on each piece of Model B. How many pieces of model A and Model B should be manufactured per week to realise a maximum profit? What is the maximum profit per week? Sol: Suppose x is the number of pieces of model A and y is the number of pieces of Model B. Then Total profit (in Rs ) = 8000x + 000y Y Maximize Z = 8000x + 000y Subject to the constraints: 9x + y 80 C(0,0) B(,6) => 3x + 4y 60 X + 3y 30 0 A(0,0) X X 0, y 0 57

58 CORNER POINT Z = 8000x + 000y 0(0, 0) 0 A(0, 0) B(, 6) C(0, 0) 0000 We find that maximum value of Z is 6000 at B(,6). Hence, the company should produce pieces of Model A and 6 pieces of Model B to realize maximum profit and maximum profit then will be Rs ) A Dealer in rural area wishes to purchase a number of sewing machines. He has only Rs 5760 to invest and has space for atmost 0 items for storage.an electronic sewing machine cost him rs 360 and a manually operated sewing machine rs 40.He can sell an electronic sewing machine at a profit of Rs and a manually operated sewing machine at a profit of Rs 8. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profot? Make it as LPP and solve it graphically. Sol: Let dealer purchased x electronic sewing machines and y manually operated sewing machines. Our problem is to maximize, Z = x + 8y (i) Subject to constraints x + y 0 (ii) 360x + 40y 5760 or 3x + y 48 (iii) X o, y 0 (IV) On solving Equations we get x = 8 and y = So, the point of intersection of the lines is B(8,). Graphical representation of the lines is given below Y C(0,0) 3X + Y = 48 B(8, ) X + Y = 0 A(6,0) X 0 58

59 : Feasible region is OABCA The corner points of the feasible region are O(0,0), A(6,0), B(8,), C(0,0). The value of Z at these points is as follows CORNER POINTS Z = x + 8y 0(0,0) Z = (0) + 8(0) = 0 A(6,0) Z = x = 35 B(8,) Z = x x = 39 C(0,0) Z = x x 0 =360 The maximum value of Z= Rs 39 at point B(8,). Hence, dealer should purchased 8 electronic and manually operated sewing machines to get maximum profit. 59

60 PRACTICE PROBLEMS )An aeroplane can carry a maximum of 00 passengers.a profit of Rs 000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket.the airlines reserves at least 0 seats for executive class.however, atleast 4 times as many passengers prefer to travel by economy class than by the executive class.determine how many tickets of each type must be sold inorder to maximize the profit for the airline.what is the maximum profit? )There are two types of fertilizers,f and F. F consists of 0% nitrogen and 6% phosphoric acid and F consists of 5% nitrogen and 0% phosphoric acid. After testing the soil condition a farmer finds that she needs atleast 4 kg of nitrogen and 4kg of phosphoric acid for her crop. If F cost Rs 6/kg and F cost Rs5/kg, determine how much of each type of fertilizer should be used, so that nutrient requirements are met at a minimum cost? 3)A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 7 units of workers (male and female) and capital respectively, which he uses to produces two types of goods A and B. To produce one unit of A, two workers and three units of capital are required while three and one unit of capital is required to produce one unit of B. If A and B are priced at Rs.00 and 0 per unit respectively, how should he uses his resources to maximize the total revenue? From the above as an LPP and solve graphically. Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate. 4)A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and,400 calories. Two foods X and Y are available at a cost of Rs. 4 and Rs. 3 per unit respectively. One unit of the food X contains 00 units of vitamins, unit of mineral and 40 calories, whereas one unit of food Y contains 00 units of vitamins, units of minerals and 40 calories. Find what combination of X and Y should be used to have least cost? Also find the least cost. 60

61 PROBABILITY BAYE S THEOREM If E, E,..., En are n non empty events which constitute a partition of sample space S, i.e. E, E,..., En are pairwise disjoint and E E... En = S anda is any event of nonzero probability, then P (Ei).P (A/Ei) P(Ei A) = for any i =,, 3,..., n n P(Ej) P(A/Ej) j= Problems with solutions Q.Bag contains 3 red and 4 black balls and Bag contains 4 red and 5 black balls. One Ball is transferred from Bag to Bag and then two balls are drawn at random (without replacement) from Bag. The balls so drawn are found to be both red in color. Find the probability that the transferred ball is red. ANS: Total No. of balls in st bag = 3+4= 7 And total No. of balls in nd bag = 4+5= 9 Let, E : transferred ball is red E : transferred ball is black. A: Getting both red from nd bag (after transfer) 3 4 P (E ) = and P (E ) = 7 7 P (A/E ) =P (getting both red balls from nd bag, when transfer ball is red) 0 = 5 C / 0 C = = 45 9 P (A/E ) =P (getting both red balls from nd bag, when transfer ball is black) 6 = 4 C / 0 C = = 45 5 Therefore, by Baye s theorem P (E /A) = P (E).P (A/E) P (E).P (A/E) + P (E).P (A/E) = = 5 9 6

62 Q.Given three identical boxes,and 3, each containing two coins. In box, both coins are gold coins, in box, both are silver coins and in box 3, there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is probability tha5t the other coin in the box is also of gold? Solution: let E,E,and E3, be the events that boxes,and 3 are chosen respectively. Then, P (E ) = P (E ) = P (E 3 ) = 3 Also, let A be the event that the coin drawn is of gold Thus, P (A/E ) = P (a gold coin from bag) = = P (A/E ) = P (a gold coin from bag) = 0 P (A/E 3 ) = P (a gold coin from bag3) = Now the probability that the other coin in the box is of gold box = The probability that the gold coin is drawn from the = P (E /A) By Baye s theorem, we know that P (E /A) = P (E)P (A/E) P (E)P (A/E) + P (E) P (A/E) + P (E3) P (A/E3) = 3 X 3 X+ X 0 + X 3 3 = 3 Q3.A man is known to speak truth 3 out of 4 times. He throws a die and report that it is a six. Find the probability that it is actually a six. Solution:P (E ) = Probability that six occurs = 6 P (E ) = Probability that six does not occurs = 6 5 P (A/E ) = Probability that the man reports that six occurs when six has actually occurred on the die. 6

63 Probability that the man speaks the truth = 4 3 P (A/E ) = Probability that the man reports that six occurs when six has not actually occurred on the die. Probability that the man speaks the truth = Thus by Baye s theorem, we get 3 = 4 4 P (E /A) = Probability that the report of the man that six has occurred actually a six. P (E /A) = P (E).P (A/E) P (E).P (A/E) + P (E).P (A/E) = 6 3 X X + X = 8 X 4 8 = 3 8 Q4.There are three coins. One is two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin? SOLUTION: Let E,E and E3 be the events that coins I,II and III are chosen respectively. Let A be the event of getting a head. P(E)=P(E)=P(E3)= 3 P(A/E)=, P(A/E)=75%= 4 3, P(A/E3)= P (E).P (A/E) Required probability= p(e/a) = P (E).P (A/E) + P (E).P (A/E) + P(E3).P(A/E3) = 4 9 Q5. An insurance company insured 000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.0, 0.03 and 0.5 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver? SOLUTION: Insured scooter drivers=000 Car drivers = 4000 Truck drivers=6000 Total drivers=

64 Let E,E and E3 be the events that scooter driver, car driver and truck driver are selecting respectively Let A be the event of meeting with an accident P(E)= =, P(E)= =,P(E3)= = P(A/E)=0.0 and P(A/E)=0.03 and P(A/E3)=0.5 P (E).P (A/E) By Bayes theorem P(E/A) = P (E).P (A/E) + P (E).P (A/E) + P(E3).P(A/E3) = 5 PROBABILITY DISTRIBUTION MEAN & VARIANCE OF RANDOM VARIABLE The probability distribution of a random variable X is the system of numbers X : x x... xn P(X): p p... pn Where, pi > 0, Pi = n i= Problem with Solutions. From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs. Sol: Let X denotes the number of defective bulbs X= 0,,,3 or 4 Probability of getting a defective bulb= 6 = 30 Probability of getting a non defective bulb= = P(X=0)=P(no defective bulb)=p( all 4 good ones)= ( ) 4 = P(x=)= 4c ( 5 ) ( 5 4 ) 3 P(x=)= 4c ( 5 ) ( 5 4 ) 56 = = P(x=3)= 4c 3 ( ) 3 ( ) = P(x=4)= ( 5 ) 4 = 65 Probability distribution of X is X P(X)

65 PRACTICE QUESTIONS. Suppose a girl throws a die if she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets,,3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw,,3 or 4 with the die?. In answering a question on amultiple choice test, a student either knows the answer or guesses. Let ¾ be the probability that he knows the answer and ¼ be the probability that he guesses. Assuming that a student who guesses the answer will be correct with probability ¼. What is the probability that a student knows the answer given that he answered it correctly? 3. Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the expectation of X. 4. Two cards are drawn simultaneously(or successively without replacement) from a well shuffled pack of 5 cards. Find the mean, variance and standard deviation of the number of kings. TOPIC: RELATIONS, FUNCTIONS & INVERSE TRIGONOMETRIC FUNCTIONS. Show that f: R R defined by f (x) = [ x ] is neither one one nor onto.. Find fοg for f(x) = e x ; g(x) = log x 3. Check for commutative property for the operation : * : R x R R defined by a * b = a + 3b a + b 4. Find, if the binary operation, *, given by a * b =, in the set of real numbers is associative. 5. Let S = {,,3}. Find whether the function f : S S defined as f = {(,3), (3,), (,)} has inverse. If yes, find f. 6. For θ = sin, find the value of θ 7. Find the Principal value of tan ( 3 ) 8. Evaluate: sin(cot x) 3 9. Evaluate: cos(tan 4 )X 0. Express in simplest form: sin [3x 4x 3 ]. Prove: cos x cos y = cos [ xy + x y ] x π. Prove: tan tan = + x x x 3. Show that the function f : R R given by f(x) = 3x 4 is a bijection. 4. Find fοg and gοf if f(x) = x + and g(x) = x 5. Let f:r R defined by f(x) = x 3 and g : R R defined by g(x) = Show that fοg = I R = gοf. x

66 6. Let * be a binary operation on N, given by a * b = lcm (a,b) for a,b N. Find a) * 4 b) 3 * 5 c) Is * associative. x + x 7. Solve: tan + tan = tan ( 7) x x cos cos + x + x π x π 8. Prove that: tan = +,0 < x < + cos x cos x 4 9. Let R 0 denote the set of all non zero real numbers and let A = R 0 x R 0. If * is a binary operation on A defined by : (a,b) * (c,d) = (ac, bd) for all (a,b), (c,d) A. a) Show that * is both commutative and associative on A. b) Find the identity element in A c) Find the invertible element in A. 0. Show that the function f: R R given by f(x) = x 3 + x is a bijection. Find the inverse. 66

67 MATRICES & DETERMINANTS i 0 0 i. If A = and B =, show that AB BA 0 i Find a matrix X, for which X = If A =, prove that A A T is a skew symmetric matrix If A = find k for A = ka I 4 5. If A and B are symmetric matrices, show that AB is symmetric, if AB = BA. 6. Find the equation of the line joining (, ) and (3, 6) using determinants. k 7. For what value of k the matrix has no inverse. 3 4 a b 8. For, find determinant {A(adj A)} c d 4 x 4 9. Evaluate x if = 5 6 x 0. Vertices of a triangle ABC are A(,3), B(0,0) and C(k,0). Find the value of k such that the area of the triangle ABC is square units Express the matrix A = 5 4 as a sum of symmetric and skewsymmetric matrices. 3 3 cosθ isinθ. If A =, then prove by principle of Mathematical induction isinθ cosθ that A n cos nθ i sin nθ = isin nθ cos nθ 3 3. If A =, evaluate A 3 4 A + A cos x sin x 0 4. If f(x) = sin x cos x 0, show that f(x) f(y) = f(x + y) By using elementary transformations, find the inverse of A =

68 68 Show that the matrix A = 3 satisfies the equation A 5A + 7I = 0. Hence find A. 6. Using properties of determinants, show that c b b a a c a c b a c b = (a + b+ c) (a c) 7. Show that c x x x b x x x a x x x = 0, where a, b, c are in A.P. 8. Prove that ) )( )( ( α γ γ β β α αβ γα βγ γ β α = 9. Without expanding prove that x x x y x x x y x x x y x = If A = and B = verify that (AB) = B A. If x y z and, = z z z y y y x x x show that xyz =. If A =, prove that A 4A 5I = 0. Hence find A 3. Using matrix method, solve the system: x + y + z = 3 ; x y + z =, x y + 3z = 4. Using matrix method, solve the system: x + y z = ; 3x + y z = 3 ; x y z = 5. Solve the system using matrices: = + z y x ; 0 = + + z y x ; 3 3 = + z y x 6. Given A = and B = , compute (AB)

69 69 7. If A = , prove that A = A 6A + I 8. Show that x = is a root of the equation x x x x x = 0 and solve it completely.

70 INVERSE TRIGONOMETRIC FUNCTIONS. Evaluate: 3. Prove that : 3 sin x = sin (3x 4x 3 ) 3. Evaluate: 4. Find the Principal value of 5. Evaluate: sin(cot x) 6. Express in simplest form: 7. Prove: x tan tan = + x x x π 8. Evaluate: Solve for x: 0. Evaluate: x x x x +. Solve: tan + tan = tan ( 7). Prove that: tan + cos x + + cos x cos x π = + cos x 4 x,0 π < x < 3. Solve for x: 4. Prove that : 5. Solve for x: tan (x + ) + tan (x ) = 6. Prove that : 7. Solve for x : 8. Prove that : tan 9. Write in simplest form: 0. Prove that : 70

71 7

72 TOPIC : DIFFERENTIATION Questions 0 to 0 carry 0 mark each Questions to carry 04 marks each Questions 3 to 9 carry 06 marks each dy. Find for y = log [x + + x ] dx dy x. Find for y = tan dx x dy 3. Find for y = x sec x dx 4. Find y for y = cos x 5. Find y for y = x sinx cos x dy 6. For y = log, show that = cosec x + cos x dx 7. Find the interval at which f(x) = x x 4 is increasing. π 8. Show that f(x) = tan x 4x is decreasing in < x < The cost function of a firm is given by C = 4x x Find the marginal cost when x = The radius of a spherical bubble is increasing at the rate of 0.5 cm / sec. At what rate is the volume of the bubble increasing when its radius is cm? dy. Find if x 6 + y 6 + 6x y = 6. dx dy. If y = x + x + x +..., prove that = dx y 3. If y = e x tan x, then prove that : ( + x d y ) ( x +x dy ) + ( x )y = dx dx 0 4. If y = (log x), then prove that x y + x y = d y π 5. If x = cos t cos t, y = sint sin t, find at t = dx + x 6. Differentiate: tan x with respect to tan x 7. Find the equations of tangents to the curve y = x 3 + x + 6 which are perpendicular to the line x + 4 y + 4 = Using differentials find the approximate value of 4 8 7

73 9. Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long. 0. Find the local maximum and local minimum values, if any for f(x) = sinx + π cos x for 0 < x <. Find two positive numbers whose sum is 6 and sum of whose cubes is maximum.. Find the equation of the tangent to the curve x + 3y 3 = 0 which is parallel to the line 4x y 5 = 0 3. Find all the points of local maxima and minima and the corresponding maximum and minimum values of the function: f(x) = x 4 8x 45 x Find all the points of local maxima and minima and the corresponding maximum and minimum values of the function: f(x) = sin x + cos x where π 0 < x < 5. Show that the rectangle of maximum perimeter which can be inscribed in a circle of radius a is a square of side a 6. A figure consists of a semi circle with a rectangle on its diameter. Given the perimeter of the figure, find its dimensions in order that the area may be maximum. 7. Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm. 8. Show that the semi vertical angle of a right circular cone of given surface area and maximum volume is sin 3 9. Show that the volume of the greatest cylinder which can be inscribed in 4 a cone of height h and semi vertical angle α is π h 3 tan α 7 73

74 WEIGHTAGE TO DIFFERENT TOPICS S.No. Name of the Topic Marks allotted 0 RELATIONS AND FUNCTIONS 0 0 ALGEBRA 3 03 CALCULUS VECTORS & 3 D GEOMETRY 7 05 LINEAR PROGRAMMING PROBABILITY 0 TOTAL 00 74

75 INTEGRATION AND APPLICATION x dx. Evaluate: + 3 x 4x ax e 3x. If ( e + bx) dx = +, find the values of a and b 4 3. Evaluate: sin 4x cos7xdx x 4. Evaluate: sin dx x 5dx 5. Evaluate: x + 3 x 3 x x + x 6. Evaluate: dx x x 7. Evaluate: dx + x sin x 8. Evaluate: dx + cos x 0 π 0 9. Evaluate using properties of definite integrals: cot x + 0. Evaluate using properties of definite integrals: x. Evaluate: x log( + x) dx dx. Evaluate: 3 x + x x + 3. Evaluate : dx 4 x + x + 4. Evaluate : e x [tan x + log sec x] dx 5. Evaluate : tan xdx 4 6. Evaluate: cos xdx dx x x ) 7. Evaluate: ( ) ( π 0 8. Evaluate: cos xdx a a π cot x 0 tan x π π 3 cos xdx 9. Prove that: f ( x) dx = f ( a x) dx. Use it to evaluate: x xdx

76 x sin x + cos x 0 0. Evaluate: dx π 3. Evaluate as limit of a sum: (x + x + 5) dx. Evaluate as limit of a sum: (x + x) dx log( + x) 3. Evaluate: dx + x 0 π 0 4 x 4. Evaluate: dx + sin x + cos x 0 5. Find the area of the region: {(x,y) : y 4x, 4x + 4y 9} x y 6. Find the area of the smaller region bounded by the ellipse + = b a x y and the line + = a b 7. Using integration find the area of the region given by: {(x,y) : 0 y x, 0 y x+, 0 x } 8. Using integration, find the area of the triangular region whose vertices are (,0), (,) and (3,) 9. Using integration, find the area bounded by the lines: x + y =, y x = and x + y = 7. 76

77 DIFFERENTIAL EQUATIONS. Solve: dy + y = x dx. Solve: dy + y cot x = sec x. dx 3. Solve: ( +y )dx + x dy = 0 given that y() =. dy 4. Solve: = y tanx given that y(0) =. dx dy y 5. Solve : + dx x = 0 dy 6. Solve: (x ) = xy, given that y() =. dx 7. Solve: dy y = x e x dx 8. Solve: (x xy)dy = y dx. 9. Show that y = e x +ax +b is a solution of the differential equation: e x y = 0. Show that y = A cosx B sin x is a solution of the differential equation y + y = 0. Form the differential equation representing the family of curves y = e x (A+Bx),where A and B are constants.. Form the differential equation corresponding to y = a(b x ) by eliminating a and b. 3. Show that y = x sin3x is the solution of the differential equation: y +9y 6cos3x = 0 4. Solve: x y dx (x 3 + y 3 )dy = Solve: y dx + (x xy) dy = Solve: x y = y x + y dy 7. Solve: = (y/x) + tan(y/x) dx 8. x y dx + y x dy = Determine the order and degree of the equation. ds dt d s + 3s dt 0. Solve (y + xy)dx + (x xy )dy = 0 = 0, Order -, Degree 77