Maths I & Ii Hsc

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Theory of Attributes Ex. No. 1 1. Find the missing frequencies in the following data of two attributes A and B. N= 800, AB= 120, B= 500, A= 300. 2. For a data for 2 attributes, it is given that N=500, A= 150, B= 100, AB= 60, find the other calss frequencies. 3. In a population of 10,000 adults, 1290 are literate, 1390 are unemployed and 820 are literate unemployed. Find the number of (i) literate employed. (ii) literates, (iii) employed. 4. In a co – educational school of 200 students contained 150 boys. An examination was conducted in which 120 passed. If 10 girls failed, find the number of (i) boys who failed, (ii) girls who passed. 5. In a sample of 240 persons, 40 were graduates and 5 were graduates employed. If 40 non – graduates were employed, find the number of unemployed non – graduates and the number of unemployed persons. 6. If for 3 attributes A, B and C, it is given that (ABC) = 210, αBC= 280, ABγ= 180 αBγ= 240, AβC=250, αβC=160, Aβγ=360, αβγ=32, find (A), (B), (C), (AB), (AC) and (BC). 7. If for 3 attributes A, B, C, it is given that (ABC) = 370, αBC= 1140, ABγ= 230, αBγ= 960, AβC= 260, αβC=870, Aβγ=140, αβγ=1030, find Bγ, A, B, C. 8. If N = 800, (A)=224, (B) = 301, (C) = 150, (AB) = 125, (AC) = 72, (NC) = 60 and (ABC) = 32, find AβCand ABγ.

Ex. No. 2 Check the consistency of the following data. 1. 2. 3. 4. 5. 6. 7. 8.

A= 100, B=150, AB= 60, N=500. A= 100, B= 150, AB= 140, N=500. A= 300, β= 400, Aβ= 200, N=1000. A- 150, β= 45, AB= 125, N=200. AB= 40, αβ= 70, α= 160, N=200. AB= 75, αβ= 50, α= 55, N=300. AB= 50, Aβ= 79, αB=89, αβ= 782. AB= 200, A=300, B= 300, N=1000.

Ex. No. 3 1. Discuss the association of A and B if 1

OMTEX CLASSES “THE HOME OF SUCCESS” i. N = 100, (A) = 50, (B) = 40, (AB) = 20. ii.(AB) = 25, Aβ= 30, αβ= 25, αB= 20. 2. Discuss the association between attributes A and B if i. N = 100, (A) = 40, (B) = 60, (AB) = 30. ii.N = 1000, (A) = 470, (B) = 620, (AB) = 320. iii.N = 500, α= 300, β= 350, AB= 60. iv.N = 1500, α= 1117, B= 360, AB= 35. 3. Find the association between literacy and unemployment in the following data. Total No. Of adults 100 0 No. Of literate

130

No. Of unemployed

140

No. Of literate unemployed

80

4. Find the association between literacy and employment from the following data. Total Adults 10000 Unemployed 1390 Literates

1290

Literate unemployed

820

Comment on the result. 5. Show that there is very little association between the eye colour of husband s and wives from the following data. Husband with light eyes and wives with light eyes = 309 Husband with light eyes and wives with dark eyes = 214 Husband with dark eyes and wives with light eyes = 132 Husband with dark eyes and wives with dark eyes = 119 6. 88 persons are classified according to their smoking and tea drinking habits. Find Yule’s coefficient and draw your conclusion. Smokers Non – smokers Tea Drinkers

40

33

Non Tea Drinkers

3

12

7. Show that there is no association between sex and success in examination from the following data. Boys Girls Passed examination

120

40

Failed examination

30

10

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OMTEX CLASSES “THE HOME OF SUCCESS” 8. Find Yule’s coefficient to determine if there is association between the heights of spouses Tall Husbands Short Husbands Tall Wives

60

10

Short Wives

10

50

9. 300 students appeared for an examination and of these, 200 passed. 130 had attended a coaching class and 75 of these passed. Find the number of unsuccessful students who did not attend the coaching class. Also find Q. 10.Calculate Yule’s coefficient of association between smokers and coffee drinkers, from the following data. Coffee Drinkers Non – coffee Drinkers Smokers

90

65

Non – smokers

260

110

11.Out of 700 literates in town, 5 were criminals. Out of 9,300 literates in the same town, 150 were criminals. Find Q. 12.Examine the consistency of the following data and if so, find Q. N = 200, (AB) = 24, α= 160, αβ= 70. 13.Find Yule’s coefficient of association for the following data. Intelligent husbands with intelligent wives 40 Intelligent husbands with dull wives 100 Dull husbands with intelligent wives 160 Dull husbands with dull wives 190

1. i.

LOGIC

2. Express the following in the symbolic form Hari is either intelligent or hard working.

ii. a+b2=a2+b2if and only if ab=0. 3. Given p ≡ x is an irrational number. q ≡ x is the square of an integer. Write the verbal statement for the following.

i. p~q ii. ~p →q 4. Using the statements:

P: Kiran passed the examination. S: Kiran is sad. And assuming that ‘not sad’ is happy, represent the following statement in symbolic form.

“Kiran failed or Kiran passed as well as he is happy” 5. Write the following statements in symbolic form. i. Bangalore is a garden city and Mumbai is a metropolitan city. ii. Ram is tall or Shyam is intelligent. 6. Write the following statements in symbolically. i. If a man is happy, then he is rich.

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OMTEX CLASSES “THE HOME OF SUCCESS” ii. If a man is not rich, then he is not happy. 7. Write the following statements in symbolic form. i. Akhila likes mathematics but not chemistry. ii. IF the question paper is not easy then we shall not pass. 8. Let p : Riyaz passes B.M.S. q : Riyaz gets a job. r : Riyaz is happy. Write a verbal sentence to describe the following.

i. p→q r ii.p q~r

9. Using appropriate symbols, translate the following statements into symbolic form. “A person is successful only if he is a politician or he has good connections”. 10.Express the following statements in verbal form:

i.p⋀q ii. ~p ⋁q where p:Sacgube is smart. q:Sachin is healthy. 11.Let p: Rohit is tall. q: Rohit is handsome. Write the following statements in verbal form using p & q.

a. ~p⋀(~q) b. p (~pq) 12.Give a verbal statement for a. p→~q b. ~p↔ ~q p :Rama is young. q :Rama is intellignet. 13.Construct the truth table and determine whether the statement is tautology, contradiction or neither. i. ( p → q) ⋀ (q ⋀ ~q) ii. [ p ⋁ (~ q ⋀ p)] → p iii. ~( p ⋀ q) iv. p → (q → p) v. p ⋁ (~q ⋀ p). vi. ~ ( p ↔ q). vii. [ p ⋁ (~ q ⋀ ~p)] → p viii.( p → ~q) → (q ⋀ ~q) ix. [q ⋀ ( p → q)] → p x. ~( ~p ⋀ ~q ) xi. [~(p ⋁ q) ⋀ p] 14.Do as directed. i. Prove that the following statements are logically equivalent: p → q ≡ ~q → ~p

ii. Show that the statements p → q and ~( p ⋀ ~q) are equivalent. iii. Write the truth table for “Disjunction”. Write the disjunction of the statements: India is a democratic country. France is in India. iv. Using the truth table, Prove that p ⋀ (~p ⋁ q) ≡ p ⋀ q. v. Show that p ↔ q ≡ ( p → q ) ⋀ ( q → p ). vi. Using truth table show that, p → q ≡ (~p ⋁ q) vii. Using truth table prove that, p → q ≡ (~q) → (~p) viii.Prove that the statement pattern ( p ⋀ q) ⋀ (~p⋁~q) is a contradiction. ix. Show that the following pairs of statements are equivalent: p ⋀ q and ~ (p → ~q). 15.Represent the following statements by Venn Diagrams: i. No politician is honest. ii. Some students are hard working. iii. No poet is intelligent. iv. Some poets are intelligent. v. Some mathematicians are wealthy. Some poets are mathematicians. Can you conclude that some poets are wealthy? vi. Some parallelograms are rectangles. vii. If a quadrilateral is a rhombus, then it is a parallelogram. viii.No quadrilateral is a triangle. ix. Sunday implies a holiday. x. If U = set of all animals. D = Set of dogs. W = Set of all wild animals; Observe the diagram and state

whether the following statements are true or false a. All wild animals are dogs. b. Some dogs are wild. i. Some students are obedient. ii. No artist is cruel. iii. All students are lazy. iv. Some students are lazy. v. All students are intelligent. vi. Some students are intelligent.

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OMTEX CLASSES “THE HOME OF SUCCESS” vii. All triangles are polygons. viii.Some right angled triangles are isosceles.

ix. All doctors are honest. x. Some doctors are honest.

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LIMIT Ex. No. 1. [Algebraic Limits] 1. limx→4x3-x-7x2+ 3x-25326 2. limx→-2x2+x+1x2-x+3 13 3. limx→2x3+x2-12x3-x2-x-2167 4. limx→3x3-x-24x3+x2-362633 5. limx→13x3+4x2-6x-12x3-x-1115 6. limx→1x4-3x+23x4-x-2111 7. limx→2x3-2x2-4x+8x3-5x2+8x-441 8. limx→3x3-x-24x3-6x2+9x[ined] 9. limx→4x3-8x2+16xx3-x-160 10.limx→1x3-x2-x+1x3+x2-5x+312 11.limx→128x3-14x3-x(3) 12.limx→ 2x2+2x-4x2-32x+4(-3) 13.limx→ 3x2+x3-6x3-3x32

14.limx→-3x3+6x2+9xx3+5x2+3x-934 15.limx→1x2+x-2x2-4x+3-32 16.limx→3x5-2432x2-91354 17.limy→3y2-4y+32y2-3y-929 18.limx→12x2-x-1x-172 19.limx→4x3-64x3-15x-41611 20.limx→3x2-9x3-6x2+11x-6(3) 21.limx→1x2+2xx-3x-1(5) 22.limx→3x2+33x-12x4-9512 23.limx→2x4-16x2-5x+685 24.limx→2x4-16x2-5x+6(-32) 25.limx→1x7+x4-2x3-2x+1(11)

Ex No 2. [Algebraic Limits] 1. 2. 3. 4. 5. 6. 7. 8.

limx→21x-2-2x2-2x12 limx→51x-5-5x2-5x15 limx→31x-3-9xx3-27(0) limx→21x-2-2xx2-3x+232 limx→21x2-5x+6-12x2-7x+6-3 limx→41x23x-4-1x2-13x+36(-225) limx→-31x2+4x+3-1x2+8x+15-12 limx→a1x2-3ax+2a2-12x2-3ax+a2(3a2)

9. limx→11x-1-1x2-x(1) 10.limx→13xx2+x-2-4x2+2x-31112 11.limx→3x2-5x+6x2-9-x3-27x2+x1216942 12.limx→21y-2-4y3-2y2(1) 13.limx→21x-2-1x2-3x+2(1) 14.limx→31y-3-27y4-3y3(1)

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Ex No 3 limx→axn-anx-a=nan-1

1. 2. limx→ax3-a3x10-a10(310a7) 3. limx→ax25-a25x15-a15(5a103a) 4. limy→by5-b5 y9-b9(59b4) 5. limy→by15-b15y20-b20(34b5) 6. limx→2x7-128x6-6473 7. limx→2x6-64x10-1024380 8. limx→3x8-38x12-3122243 9. limx→5x7-57x10-51071250 10.limx→ax-6-a-6x-8-a-8(3a24) 11.limx→ax-5-a-5x-7-a-75a27 12.limh→0 a+h6-a6h6a5 13.limh→0 ha+h8-a818a7 14.limx→2 x12-212x13-2133256 15.limx→a3x-3ax-a23a16 16.limx→2x2-4xx-22423 17.limx→2 x2-22x32-232423 18.limx→1x+x2+x3+∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙+xn-n x-1 Note: Sn=n22a+n-1d Ans. n2[1+n] 19.limx→2310-x-2x-2 -112

Ex No 4 [Rationalizing] 1. limx→3x+6-3x2-9136 2. limx→42x+1-3x2-x-12121 3. limx→4x2+x-203x+4-4(24) 4. limx→5 x2-6x+514-2x-2(-8) 5. limx→8x2+17-9x2+x-728153 6. limx→3x3-5x-122x2-9-3(11) 7. limx→4x4-64xx2+9-5240 8. limx→4x2-16xx-883 9. limx→32+x-x2+x-79-x2125 10.limx→2x2-4xx-22423 11.limx→ -1x+1(x2+4x+5-x2+1)12 12.limx→2 x3-4-220-x2-4-6

13.limx→3x2+ x+6- 12x2-93736 14.limx→1x+3-22x-1-114 15.limx→0a+x-a-x4x14a 16.limx→ax+2a-3ax2-a2143a3 17.limh→0x+h-xh12x 18.limh→0x+h3-x3h3x2 19.limx→26+x-10-xx2-4182 20.limh→0a+h-aha+h12a 21.limx→43x+4-45x-4-435 22.limx→18+x-35-x-2-23 23.limx→2x3+x+2-10x2-44916

Ex. No. 5 Trigonometric Limits limx→0sinxx=1 & limx→0tanxx=1

1. 2. 3. 4. 5. 6. 7. 8.

limx→0sin25xx25 limx→0sin4xx(π) limx→0sinπxx4 limx→0sin5x4x54 limx→0sin3x2x32 limx→0tanxx1 limx→0xcotx1 7

OMTEX CLASSES “THE HOME OF SUCCESS” 9. limx→0sinxx0 10.limx→0tan3x2x32 11.limx→0sin2xtan3x23 12.limx→0sin25xx2(25) 13.limx→0sinx2+5xx(5) 14.limx→0Sin32xx3(8) 15.limx→0sin2x2x214 16.limx→0sin4xsin6x5x2245 17.limx→0sin3xsin5x7x2 757 18.limx→0xcosx+sinxx2+tanx2 19.limx→03sin2x+2x3x+2tan3x89 20.limx→07xcosx+3sinx3x2+tanx10 21.limθ→08sinθ-θcosθ3tanθ+θ273 22.limθ→02θ+3sinθ3θ+5tanθ58 23.limx→0x2+xsinx-1x2+x-223 24.limx→0x1-cosx(2) 25.limx→0sinx1-cosxx312 26.limx→01-cos3xx292 27.limx→01-cosmxx2m22 28.limx→01-cosmx1-cosnx(m2n2) 29.limx→0xtanx1-cosx(2)

Ex. 6. [Logarithmic Limits] limx→0 ax-1x=loga

1. 2. limx→0ea+x-eax 3. limx→0ax-bxx 4. limx→032x-23xsinx 5. limx→05x-3x4x-1 6. limx→03x-2x21-cos2x 7. limx→07x+8x+9x-3x+1x 8. limx→0a3x-a2x-ax+1xtanx 9. limx→0ax-bxtanx 10.limx→0e6x-e4x-e2x+1xtanx 11.limx→010x-2x-5x+1xsinx 12.limx→06x-3x-2x+1x2 13.limx→04x+14x-2x2 14.limx→05x+5-x-2x2 15.limx→032x-1sinx 16.limx→05x+15x-2xsinx 17.limx→032x-1sinx 18.limx→0ex+3x+4x-3x 19.limx→0ax+bx-2x+1x 20.limx→04x-11-cosxx3 21.limx→0ax-11-cosxx3 22.limx→0tanxe3x-ex 23.limx→02x-12sinxlogx+1 24.limx→04x-3x3sinx+sin4x 8

OMTEX CLASSES “THE HOME OF SUCCESS” 25.limx→1abx-axbx-1 26.limx→0e8x- e5x- e3x+1cos4x-cos10x

27.

Ex. 7. Exponential Limits limx→0 1+x1x=e 1. limx→01+2x5x=e10 2. limx→01+4x3x=e12 3. limx→01+5x34x=e203 4. limx→01-4x752x=1e107 5. limx→01+3x1-4x1x=e7 6. limx→04x+11-4x1x=e8 7. limx→02+x2-x1x=e 8. limx→04-8x4+5x1x=-1e134 9. limx→0log1+pxx=p 10.limx→0log1+3xx=3

11.limx→0log5+x-log5-xx=25 12.limx→0log10+log⁡(x+0.1)x = 10 13.limx→0log10+logx+1101x = 1 14.limx→0log7+x-log7-xx=27 15.limx→3logx-log3x-3=13 16.limx→2logx-log2x-2=12 17.limx→elogx-1x-e=1e 18.limx→1x1x-1=e 19.limx→2x-11x-2=e 20.limx→0x-31x-4=e

Ex. 8. Trigonometric Limits 1. 2. limx→02sinx-sin2xx3=(1) 3. limx→0tanx-sinxx3=12 4. limx→0sin3x-sin5xx=-2 5. limx→0cos3x-cosxx2=(-4) 6. limx→0x2cos4x-cos10x=142 7. limx→0cos4x-cos8x xtanx=(24) 8. limx→0x2cos14x-cos10x=-148 9. limx→0cos8x-cos2xcos12x-cos4x=1532 10.limx→0cosmx-cosnxcospx-cosqx=m2-n2p2-q2 11.limx→0secx-1x2secx+13=116 12.limh→0sinx+h-sinxh=(cosx) 13.limh→0tanx+h-tanxh=(sec2x) 14.limx→acosx-cosax-a=(sina) 15.limx→asecx-secax-a 16.limx→ax-acotx-cota 17.limx→asinx-sinax-a 18.limx→atanx-tanax-a 19.limx→π2secx-tanxπ2-x=12 20.limx→1sinπx1-x =(π) 21.limx→π2cosecx-1π2-x2=12 22.limx→11+cosπx1-x2=π22 23.limx→π41-tanxπ-4x=12 24.limx→π4cosx-sinxπ-4x=122 25.limx→π23cosx+cos3xπ-2x3= 26.limx→π5+cosx-2π-x2=18 9

OMTEX CLASSES “THE HOME OF SUCCESS” 27.limx→π33-tanxπ-3x=43 28.limx→π6cosx-3sinxπ-6x 29.limx→π41-tanx1-2sinx 30.limx→π42cosx-11-cotx-12

Ex. 9. Using first principle find

f'x

or Find

limh→0fx+h-f(x)h

1. 2. fx=x 3. fx=x2 4. fx= x3 5. fx=x4 6. fx=1x 7. fx=x 8. fx=c 9. fx=1x 10.fx=1xx 11.fx=2x+1 12.fx=xx 13.fx=1+x2 14.fx=1x+32 15.fx=sinx 16.fx=cosx 17.fx=cos5x 18.fx=sin2x 19.fx=sin2x 20.fx=cos2x 21.fx=xsinx 22.fx=x2sinx 23.fx=a2x 24.fx=log3x+2 25.fx=log⁡(2x-1)

Ex. 10. 1. 2. 1. 2. 3. 4. 5. 6.

If fx=3x+x2 find limh→0f2+h-f2h 7 If limh→0f3+h-f3h is to be find out, where fx=2x2-3x+5 9 Find limx→1fx-f1x2-1 where fx= x2+3 14 Find limh→0f1+h-f1h where fx= 7-2x -15 Evaluate limh→0 f3+h-f3h where fx= 7-2x -1 Find limh→0f-3+h-f-3h where fx=1x-5 -164 10

OMTEX CLASSES “THE HOME OF SUCCESS” 7. Evaluate limh→0f2+h-f(2)h where fx=1x2+2 -19 8. Find limh→0f1+h-f(1)h where fx=x+5x+1 (-1)

1.

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Continuity Ex. No. 1. I.

Discuss the continuity for the following functions and if the function discontinues, determine whether the discontinuity is removable. 1. fx=e3x-1sinxx2 for x≠0; =4 for x=0, at x=0. 2. fx=x+3-2 for x≠1; =2 for x=1, at x=1. 3. fx=3x-12sinxlog(1+x) for x≠0; =2log3 for x=0, at x=0. 4. fx=xcosx+3tanxx2+sinx for x≠0; =4 for x=0, at x=0. 5. fx=x+6-3x2-9 for x≠3; =12 for x=3, at x=3. 6. fx=e5x-e2xsin3x for x≠0; =1 for x=0, at x=0. 7. fx=3-tanxπ-3x for x≠π3; =43 for x=π3, at x=π3. 8. fx=5cosx-1π2-x for x≠π2; =2log5 for x=π2, at x=π2. 9. fx=5x-3x2x-1 for x≠0; =log53log2 for x=0, at x=0. 10.fx=x2-16x-4 for x≠4; =9 for x=4, at x=4. 11.fx=x2-x-1 for0≤x<2; =4x+1 for 2≤x≤4 ,at x=2. 12.fx=2x+3 for 0≤x<2; =4 for 2≤x≤5, at x=2. 13.fx=x2-x+5 for 0≤x<3; =2x+5 for 3≤x≤6, at x=3.

Ex. No. 2. 1. If fx=e3x-1ax, for x<0; =1, for x=0; =log⁡(1+bx)4x for x>0 is continuous at x=0, find a & b.

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OMTEX CLASSES “THE HOME OF SUCCESS” 2. If fx=3sinx-12xlog(1+x), for x≠0;is continuous at x=0, find f0. 3. If fx is continuous at x=0 where f(x) =ax-a-xx, for x≠0; =k for x=0 find k. 4. 5. 6. 7. 8.

If fx=7sinx-12xlog(1+5x) for x≠0;is continuous at x=0, find f0. If fx=cos3x-cosxx2 for x≠0;is continuous at x=0, find f0. If fx=1-coskxxtanx for x≠0;is continuous at x=0, find k, if f0=3. If fx=15x-3x-5x+1xtanx, for x≠0;is continuous at x=0, find f0. If fx is continuous at x=0 where fx =x2+α, for x>0; =2x2+1+β for x<0 find α&β if f0=2.

9. If fx is continuous at x=0 where fx =x2+α, for x≥0; =2x2+1+β for x<0 find α&β if f2=4. 10.Discuss the continuity of the following. fx=3x2-2x-12x2-x-15 in the interval or domain 0,5 11.Discuss the continuity of the following. fx=3x2-2x-1x2+7x+12 in the domain -2,-7

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ASSIGNMENT PROBLEMS AND SEQUENCING Ex. No. 1 1. 2. Solve the following minimal assignment problem. in Division I, he cannot be assigned A B C D the corrections of that division. 1 1 1 6 1 If the time required in days, for every teacher to asses the papers of the 2 6 1 0 1 various divisions is listed below find 3 2 0 2 1 the allocation of the work so as to 4 5 2 1 0 minimize the time required to 1 5 4 1 complete the assessment. 0 7 4 A B C D 1 1 I - 5 2 6 5 0 II 4 5 3 8 3. A Departmental Store has 4 wormers III 6 6 2 5 to pack their items. The timing in I 1 6 3 4 minutes required for each workers to V complete the packing per item sold is given below. How should the manager of the store assign the job to the workers, so as to minimize the total time of packing? Book Toy Crocker Catter s s y y A B C D

2 12 3 4

10 2 4 15

9 12 6 4

7 2 1 9

4. Solve the following minimal assignment problem. A B C D 1 2 3 4

3 5 1 4

4 6 2 1 0

6 1 0 3 6

5 9 2 4

5. For an examination, the answer papers of the divisions I, II, III and IV are to be distributed amongst 4 teachers A, B, C & D. It is a policy decision of the department that every teacher corrects the papers of exactly one division. Also, since Mr. A’s son is

6. Solve the following minimal assignment problem. A B C D I II III I V

1 2 3 3 2

1 1 1 4 1 3

1 1 1 0 6 1 1

5 8 1 7

7. A Departmental head has four subordinates and four task to be performed. The time each man would take to perform each task is given below. A B C D I II III I V

1 2 1 2 8 1 0

2 0 1 6 9 1 7

1 1 2 8 1 5

5 1 4 5 1

8. Minimise the following assignment problem.

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I II III I V

A

B

C D

2 9 1 0 7

1 3 1 2 2 6

3 6 4 1

M4

4 1 3 1 5 9

9. A team of 4 horses and 4 riders has entered the jumping show contest. The number of penalty points to be expected when each rider rides each horse is shown below. How should the horses be assigned to the riders so as to minimise the expected loss? Also find the minimum expected loss. HORS H H H H ES 1 2 3 4 RIDER S R1 R2 R3 R4

1 2 1 1 1 5

3 1 1 1 0 8

3 4 6 1

6 2 7 1

7 8 8 4

50 61 11 1

10 1 73 71

8 2 5 9

9 2 6 1

87

77

8 1 8 0

11. A Chartered Accountants’ firm has accepted ‘five’ new cases. The estimated number of days required by each of their ‘five’ employees for each case are given below, where ‘-‘means that the particular employee cannot be assigned the particular case. Determine he optimal assignment of cases to the employees so that the total number of days required completing these ‘five’ cases will be minimum. Also find the minimum number of days. CASES I I II I V EMPLOYE I I V ES E1 E2 E3 E4 E5

2 1 3 1 1 7

10. The owner of a small machine shop has ‘four’ machinists available to assign jobs for the day. ‘Five’ jobs are offered to be done on the day. The expected profits for each job done by each machinist are given below. Find the assignment of jobs to the machinists that will results in maximum profit. Also find the maximum profit. [One machinist can be assigned only ‘one’ job] JOBS A B C D E MACHINIS TS M1 M2 M3

8 7 4 8

5 3 6 4 3

2 4 3 2 6

4 4 2 4

2 5 1 3 7

6 7 2 5 3

12. The cost (in hundreds of Rs.) of sending material to ‘five’ terminals by ‘four’ trucks, incurred by a company is given below. Find the assignment of trucks to terminals which will minimize the cost. [‘One’ truck is assigned to only ‘one’ terminal] Which terminal will ‘not’ receive material from the truck company? What is the minimum cost? TRUCKS A B C D TERMINAL S T1 T2 T3 T4 T5

3 7 3 5 5

6 1 8 2 7

2 4 5 6 6

6 4 8 3 2

EX. NO. 2 15

OMTEX CLASSES “THE HOME OF SUCCESS”

1. 2. Find the sequence that minimises the total elapsed time, required to complete the following jobs on two machineries. 6. Solve the following problem for Jo A B C D E F G minimum elapsed time. Also state the b idling time for each machine. M 7 2 3 2 7 4 5 Job 1 2 3 4 5 6 1 Machine 8 3 7 2 5 1 M 4 6 5 4 3 1 4 A 2 Machine 3 4 5 2 1 6 B 3. Solve the following for minimum elapsed time and idling time for each Machine 8 7 6 9 1 9 machine. C 0 Jo A B C D E b M 1

5

1

9 3

1 0

M 2

2

6

7 8

4

7. Solve the following problem for minimum elapsed time. Also state the idling time for each machine. Job A B C D E F G

4. Solve the following problems for minimum elapsed time. Also state the idling time for the machine. Jo 1 2 3 4 5 6 7 8 9 b M 1

2 5 4 9 6 8 7 5

4

M 2

6 8 7 4 3 9 3 8

1 1

5. Solve the following problem for minimum elapsed time. Also state the idling time for each machine. Job 1 2 3 4 5

Machine A

2

7

6

3

8 7

9

Machine B

3

2

1

4

0 3

2

Machine C

5

6

4

1 0

4 5

1 1

8. Five jobs have to go through the machines A, B, C in order ABC. Following table shows the processing times in hours for the five jobs. Job J J J J J 1 2 3 4 5 Machine A

5

7

6

9

5

Machine A

8

1 0

6 7

1 1

Machine B

2

2

4

5

3

Machine B

5

6

2 3

4

Machine C

3

6

5

6

7

Machine C

4

9

8 6

5

Determine the sequence of jobs, which will minimise the total elapsed time. 9. Determine the eptimum sequence so as to minimize the total elapsed time.

16

OMTEX CLASSES “THE HOME OF SUCCESS” Type of Chairs

1

Number To be processed/d ay

4

Processing time on

Machi ne A

Machi ne B

2 3 4 5 6

6 5 2 4 3

4 12 14 20 8 10

8 6 16 22 10 2

17

OMTEX CLASSES “THE HOME OF SUCCESS”

BINOMIAL AND POISSON DISTRIBUTION Ex. 1 1.

2. An unbiased coin is tossed 6 times. Find the probability of getting 3 heads. (5/16) 3. Find the probability of getting atleast 4 heads, in 6 trials of a coin. (11/32) 4. An ordinary coin is tossed 4 times. Find the probability of getting a. No heads(1/16) b. Exactly 1 head(1/4) c. Exactly 3 tails(1/4) d. Two or more heads(11/16) 5. On an average ‘A’ can solve 40% of the problems. What is the probability of ‘A’ solving a. No problems out of 6. (729/15625) b. Exactly four problems out of 6. (432/3125) 6. The probability that a student is not a swimmer is 1/5. Out of five students considered, find the probability that a. 4 are swimmers. (256/625) b. Atleast 4 are swimmers/ (2304/3125) 7. In a certain tournament, the probability of A’s winning is 2/3. Find the probability of A’s winning atleast 4 games out of 5. (112/243) 8. A has won 20 out of 30 games of chess with B. In a new series of 6 games, what is the probability that A would win. a. 4 or more games. (496/729) b. Only 4 games. (80/243) 9. If the chances that any of the 5 telephone lines are busy at any instant are 0.1, find the probability that all the lines are busy. Also find the probability that not more than three lines are busy. (1/100000) (99954/100000) 10.It is noted that out of 5 T.V. programs, only one is popular. If 3 new programs are introduced, find the probability that a. None is popular. (64/125) b. At least one is popular. (61/125) 11.A marks man’s chance of hitting a target is 4/5. If he fires 5 shots, what is the probability of hitting the target a. Exactly twice (31/625) b. Atleast once. (3124/3125) 12.It is observed that on an average, 1 person out of 5 is a smoker. Find the probability that no person out of 3 is a smoker. Also find that atleast 1 person out of 3 is smoker. (64/125) (61/125). 13.A bag contains 7 white and 3 black balls. A ball drawn is always replaced in the bag. If a ball is drawn 5 times in this way, find the probability of we get 2 white and 3 black balls. (1323/100000)

Ex. 2. Binomial Distribution

For a binomial variate parameter means n, p and q. 18

OMTEX CLASSES “THE HOME OF SUCCESS” 1. A biased coin in which P(H) = 1/3 and P(T) = 2/3 is tossed 4 times. If getting a head is success then find the probability distribution. 2. An urn contains 2 white and 3 black balls. A ball is drawn, its colour noted and is replaced in the urn. If four balls are drawn in this manner, find the probability distribution if success denotes finding a white ball. 3. Find Mean and Variance of Binomial Distribution. If a. n = 12; p = 1/3 b. n = 10; p = 2/5 c. n = 100; p = 0.1 4. Find n and p for a binomial distribution, if a. Mean = 6; S.D. = 2. b. x=6, variance = 5 c. Mean=12, VAriance=10.2 d. x=10, σ=3.

Ex. 3. Poisson distribution Note: For a random variable x with a Poisson distribution with the parameterλ, the probability of success is given by. Px=λxe-λx!

Note: - For a Poisson distribution Mean = Variance = λ.

For a Poisson variate parameter means

λ

and

λ=np.

If

n≥100 & λ≤10.

1. 2. For a Poisson distribution with λ=0.7, find p(2). 3. For a Poisson distribution with λ=0.7, find p(x≤2). 4. If a random variable x follows Poisson distribution such that p(1) = p(2), find its mean and variance. 5. The probability that an individual will have a reaction after a particular drug is injected is 0.0001. If 20000 individuals are given the injection find the probability that more than 2 having reaction. 6. The average number of incoming telephone calls at a switch board per minute is 2. Find the probability that during a given period 2 or more telephone calls are received. 7. In the following situations of a Binomial variate x, can they be approximated to a Poisson Variate? a. n = 150 p = 0.05 b. n = 400 p = 0.25 8. For a Poisson distribution with λ=3, find p(2) , px≤3. 9. The average customers, who appear at the counter of a bank in 1 minute is 2. Find the probability that in a given minute a. No customer appears. b. At most 2 customers appear. 10.The probability that a person will react to a drug is 0.001 out of 2000 individuals checked, find the probability that a. Exactly 3 b. More than 2 individuals get a reaction. 11.A machine producing bolts is known to produce 2% defective bolts. What is the probability that a consignment of 400 bolts will have exactly 5 defective bolts? 19

OMTEX CLASSES “THE HOME OF SUCCESS” 12.The probability that a car passing through a particular junction will make an accident is 0.00005. Among 10000 can that pass the junction on a given day, find the probability that two car meet with an accident. 13.The number of complaints received in a super market per day is a random variable, having a Poisson distribution with λ= 3.3. Find the probability of exactly 2 complaints received on a given day. 14.For a Poisson distribution if p(1) = p(2), find p(3). 15.In a manufacturing process 0.5% of the goods produced are defective. In a sample of 400 goods. Find the probability that at most 2 items are defective. 16.In a Poisson distribution, if p(2) = p(3), find mean. 17.In a Poisson distribution the probability of 0 successes is 10%. Find its mean.

20

OMTEX CLASSES “THE HOME OF SUCCESS”

APPLICATION OF DERIVATIVES Ex. No. 1 Approx. 1. 2. 3. 4. 5. 6. 7. 8.

Find approximately, the value of 100.1, 64.1 Find approximately, the value of 328 to three decimal place. Find approximately, the value of326.96 to four decimal places. Find approximately, the value of 3997, 363 Find approximately, the value of 4.14& 3.074 Find approximately, the value of tan(45030') given 10=0.0175c Find approximately, the value of sin310, given 10=0.0175c, cos300=0.0866, sin300=0.5 9. Find approximately, the value of cos(89030’), given 10=0.0175c 10.Find approximately, the value of cos(30030’), given 10=0.0175ccos300=0.0866,sin300=0.5 11.Find approximately, the value of tan-11.001,tan-10.999 12.Find approximately, the value of e2.1given e2=7.389 13.Find approximately, the value of e1.002,given e=2.71828. 14.Find approximately, the value of log101016given log10e=2.3026. 15.Find approximately the value of loge101 given loge10=203026 16.Find approximately, the value of log9.01 given log3=1.0986. 17.Find approximately, the value of 51113, 80.714 18.Find approximately, the value of fx=2x3+7x+1 at x=2.001 19.Find approximately, the value of 5x2+80x at x=5.083 20.Find approximately, the value of 32.01,log3=1.0986 and 531.5

Ex. No. 2 Error 1. Radius of the sphere is measured as 12 cm with an error of 0.06cm. Find a. Approximate error b. Relative error c. Percentage error in calculating the volume. 2. Radius of a sphere is measures as 25 cm with an error of 0.01cm. Find

a. Approximate error b. Relative error c. Percentage error in calculating the volume. 3. Radius of a sphere is found to be 24cm with the possible error of 0.01cm. Find approximately a. Consequent error b. Relative error

21

OMTEX CLASSES “THE HOME OF SUCCESS” c. Percentage error in the surface area of the sphere. 4. The side of a square is 5 meter is incorrectly measured as 5.11 meters. Find up to one decimal place the resulting error in the calculation of the area of sphere. 5. If an edge of a cube is measured as 2m with an possible error of 0.5 cm. Find the corresponding error in calculating the volume of the cube. 6. Find the approx error in the surface area of the cube having an edge of 3m. If an error of 2cm is made in measuring the edge. Also find the percentage error. 7. The volume of a cone is found by measuring its height and diameter of base as 7 cm and 5 cm respectively. It is found that the diameter is measured incorrectly to the extent of 0.06 cm. Find the consequent error in the volume. 8. The diameter of a spherical ball is found to be 2cm with a possible error of 0.082mm. Find approximately the possible error in the calculated value of the volume of the ball. 9. Side of an equilateral triangle is measured as 6cm with a possible error of 0.4mm. Find approximate error in the calculated value of its area. 10.Find the approximate % error in calculating the volume of a

sphere, if an error of 2% is made in measuring its radius. 11.If an error of 0.3% in the measurement of the radius of spherical balloon, find the %error in its volume. 12.If the radius of a spherical balloon increases 0.1%. Find the approximate % increase in its volume. 13.Under ideal conditions a perfect gas satisfies the equation PV = K; where P = Pressure, V = Volume and K = Constant. If K = 60 and Pressure is found by measurement to be 1.5 unit with error of 0.05 per unit. Find approximately the error in calculating the volume. 14.In ∆ABC, ∠B is measured using the formula cosB=a2+c2-b22ac. Find the error in calculation of ∠B if an error of 2% is made in the measurement of side b. 15.Area of the triangle is calculated by the formula 12bcsinA. If ∠A is measured as 300 with 1% error. Find the % error in the area. 16.Time (T) for completing certain length (L) is given by the equation T=2πlg where g is a constant. Find the % error in the measure of period, if the error in the measurement of length (L) is 1.2%.

Ex. No. 3. MAXIMA AND MINIMA 1. 1. Examine each of the function for Maximum and Minimum.

i. x3-9x2+24x ii.2x3-15x2+36x+10 22

OMTEX CLASSES “THE HOME OF SUCCESS” 2.

3.

4.

5.

6.

7.

8.

9.

iii.8x3-75x2+150x Output 'Q', is given by Q=10+60x+7x22-x33. Where x is the input. Find Input for which output ‘Q’ is maximum. Find the position of the point P on seg AB of length 12cm, so that AP2+BP2 is minimum. Find two Natural Number whose sum is i. 30 and product is maximum. ii.18 and the sum of the square is minimum. iii.16 and the sum of the cube is minimum. Find two Natural numbers x and y such that i. x+y=6 and x2y is maximum. ii.x+y=60 and xy3is maximum. Product of two natural numbers is 36. Find them when their sum is minimum. Product of two Natural Number is 144. Find them when their sum is minimum. Divide 70 in two part, such that i. Their product is maximum ii.The sum of their square is minimum. Divide 100 in two part, such that the sum of their squares is minimum.

10.Divide 12 in two part, so that the product of their square of one part and fourth power of the other is maximum. 11.Divide 10 in two part, such that sum of twice of one part and square of the other is minimum. 12.The perimeter of a rectangle is 100 cm. Find the length of sides when its area is maximum. 13.Perimeter of a rectangle is 48cm. Find the length of its sides when its area is maximum. 14.A metal wire 36cm long is bent to form a rectangle. Find its dimensions when its areas is maximum. 15.A box with a square base and open top is to be made from a material of area 192 sq. cm. Find its dimensions so as to have the largest volume. 16.An open tank with a square base is to be constructed so as to hold 4000 cu.mt. of water. Find its dimensions so as to use the minimum area of sheet metal. 17.Find the maximum volume of a right circular cylinder if the sum of its radius and height is 6 mts.

23

OMTEX CLASSES “THE HOME OF SUCCESS”

INDEFINITE INTEGRATION Ex. No. 1 Integrate the following functions 1. 1. 4x3 2. 3x2 3. 3x2 4. 12x3 5. 1x 6. 4x3 7. 2x+5 8. 13x-2 9. 143x-23 10.142-3x3 11.2x+1+13x+5 12.13x-23 13.17-3x3 14.7x 15.53x 16.3x 2x 17.e5x+3-5x 18.4x3x 19.ex2x

20.4x52x 21.x-12x 22.x2+2x+3x 23.x+2x+3x 24.2x3+5x2+4x 25.x3-2x2+5x-7+x2axx 26.x2+3x-2+x7xx 27.13xx-13+x+2 28.x+1(x+2)2 29.1x1+1x 30.x+1x2 31.1x-x-1 32.1a+x-a 33.13x+10- 3x-7 34.x+1x-2 35.x+2x+3 36.x+1-x2x1-x2 37.sinxsec2x 38.cosxsin2x

39.tan23x-sin4x+3 40.11-x2-cosec2x2 41.cot2x-sin5x+3+1x. 42.14-9x2 43.15-3x2 44.15-4x2 45.19+x2 46.13x2+5 47.19x2+25 48.13x2+4 49.15x2+4 50.14x2+25 51.13x2+2 52.125-9x2 53.14-9x2

Ex. No. 2. Integrate the following functions 1. 1. sin2x 2. cos2x 3. sin2x2 4. cos2x2 5. cos23x 6. tan2x 7. cot2x 8. sin3x 9. cos3x 10.sinx.cosx 11.sin2xcos2x 12.sin3xcos3x

13.sinxcosxsin2x 14.1sin2xcos2x 15.11+cosx 16.11-cosx 17.11-cos2x 18.1+cosx1-cosx 19.1+cos2x 20.1-cos2x 21.1+cosx 22.1-cosx 23.1+sin2x 24.1+sinx

25.1-sinx 26.11-sinx 27.sinx1+sinx 28.sinx1-sinx 29.tanxsecx+tanx 30.cosx1-cos2x 31.cos2xcos2xsin2x 32.sinx-cosx2 33.sec2 3x-1 34.sin2x1+cosx 35.sin2x1+cosx2 36.1-2cosxsin2x 24

OMTEX CLASSES “THE HOME OF SUCCESS” 37.3cosx-41-cos2x 38.sin3xcos4x 39.sin5xcos3x 40.cos3xcos4x

41.cos5xcos7x 42.sin3xsin4x 43.sin5xsin7x 44.sin-1cosx

45.tan-1cosx1+sinx 46.tan-1sinx1-cosx 47.tan-1sin2x1+cos2x

Ex. No. 3. Integrate the following functions 1. Note: - Whenever the degree (Highest Power of a polynomial equation) of the numerator is greater than or equal to the degree of the denominator then divide the numerator by denominator. 1. 2. xx+1 3. x+3x-3 4. 2x+3x+1 5. 2x+1x+2 6. 5x+4x+2 7. 2x+13x-2 8. 2+x2-x 9. x2+1x-1 10.2x2+xx-1 11.x3+5x2+2x+32x-1 12.5x2-6x+32x+1 13.5x2+3x+12x-1 14.5x2+x-1x-1 15.x2-1x2+1 16.xx+12 17.x2-2x+3x-12

Ex. No. 4. Integration by Substitution 1. 1. 2. 3. 4. 5. 6. 7. 8.

xsinx2 2xex2 sinxx cosxx xn-1cosxn xn-1sinxn x2sec2x3 3x2tan2x3

9. exe2x-4 10.exe2x+1 11.ex4-e2x 12.sinx25-cos2x 13.cosx4-sin2x 14.log1+1xxx+1 15.log⁡(tanx2) sinx 16.cotxsinx

17.cos3x sinx 18.2xsinx2cosx 19.sin3x cosx 20.[tan-1x)21+x2 21.cosx1+sinx32 22.2+logxx2 23.ex1+xcos2xex 24.1xcos2logx 25

OMTEX CLASSES “THE HOME OF SUCCESS” 25.cosx1+sin2x 26.secx1-tanx22 27.1 x.logx 28.1xlogx.loglogx 29.logx3x 30.etanxsec2x 31.x+1x+logx22x 32.(cos-1x) 21-x2 33.esin-1x1-x2 34.cosxsin4x 35.(sin-1x) 31-x2 36.cos-1x1-x2 37.etan-1x1+x2 38.sin⁡(tan-1x)1+x2 39.1xcos2logx 40.x31+x2 41.1x2+3logx2 42.sinxsec2x 43.1xsin2logx 44.exx+1sin2xex 45.xtan-1x21+x4 46.tanxsecx+cosx

47.1a-1xn1x2 48.2x+1x2+x+5 49.4x+10x2+5x-1 50.5x2x2+3 51.2x+6x2+6x+10 52.2-x6+4x-x2 53.3x1-x2 54.cosxx+3sinx 55.secx.cosecxlogtanx 56.13tanx+1cos2x 57.11+ex1-e-x 58.xe-1+ex-1xe+ex 59.1+sin2xx+sin2x 60.cotxlogsinx 61.12x+xlogx 62.1ex+e-x2 63.x1+x1-x 64.x33+x23-x2 65.sin2xacos2x+bsin2x 66.11+e-x 67.e2x-1e2x+1 68.ex+1ex-1

69.1ex+1 70.exex2-1 71.1x+x 72.1+tanx1-tanx 73.1-tanx1+tanx 74.11+tanx 75.sinxsinx+cosx 76.cos2xsinx+cosx2 77.xx2+1 78.xx2-a2 79.x1-x2 80.2x+1x62+x-5 81.2x+3x2+3x-1 82.exex+1 83.1xa+blogx 84.sinxcosx2-3sin2x 85.a-x2ax-x2 86.ex-sinxex+cosx 87.sin2xa2sin2x+b2cos 2x

26

OMTEX CLASSES “THE HOME OF SUCCESS”

Ex. No. 5. Integration of the type fxax+bdx

or

fxax+bdx

1. 2. 2x+1x+1 3. 3x+4x-3 4. x2x+1 5. x2x+1 6. x2-x+1x+1 7. x2+x+3x-1 8. x22x-1 9. (2sin2x+sinx-3)cosxsinx-1 10.2e2x+9ex+5)exex+1 11.tan1+xx 12.cot1+xx 13.tan2+3logxx 14.secxx 15.seclogxx 16.secxx 17.11-cosx 18.11+sinx 19.11-sinx 20.sinxsinx+a 21.cosxcosx-a 22.sinxcosx-a 23.sinx-asinx+a 24.cosxcosx-a 25.cosx+acosx-a 26.sinx-asinx-b 27.cosx-acosx-b 28.1sinx-asinx-b 29.1cosx-acosx-b 30.1sinx-acosx-b 31.1cosx-asinx-b 32.1+tan2x1-tan2x 33.1sinx.cos2x 34.ex16-e2x 35.x21+x6 27

OMTEX CLASSES “THE HOME OF SUCCESS” 36.xx4+25 37.axa2x-9 38.sec2x3tan2x+2 39.sinx25-cos2x 40.sec2x4tan2x-9 41.1xlogx2+9x 42.cosx4sin2x-3 43.cosx4+sin2x 44.sinx4sin2x+5 45.a+xa-x 46.x+1x-1

28

OMTEX CLASSES “THE HOME OF SUCCESS”

Ex. No. 6. Integration of the type 1ax2+bx+cdx

OR

1ax2+bx+cdx

1. 2. 1x2+6x+10 3. 54x2+4x-15 4. 13x2-4x+5 5. 19x62+6x+5 6. 1x2+x+1 7. 115+4x-4x2 8. 54-2x-x2 9. 14+4x-3x2 10.1x2+4x+3 11.13x2-4x+2 12.1x2+4x+5 13.14x2-4x+3 14.13x2-4x-3 15.13+4x-4x2 16.19+8x-x2 17.exe2x+4ex+13

Ex. No. 7. Integration of the type mx+nax2+bx+cdx

OR

mx+nax2+bx+cdx

1. 2. 3. 4. 5. 6. 7. 8. 9.

2x+3x2+3x+1 2x-5x2-5x+2 x-3x2-6x+4 1-x3+2x-x2 2x+1x2+3x+5 3x+72x2+3x-2 x+3x2+4 2x+19-4x2 29

OMTEX CLASSES “THE HOME OF SUCCESS” 10.3x+5x2+4x+5 11.x3 +5x2+12x+10x2+4x+5 12.x-13+4x-3x2 13.2x-1x2-x+3 14.2x-3x2-3x+4 15.2x+1x2+3x-4 16.x+43-x 17.2x3+8x2-3x-6x2+6x+10

Ex. No. 8. [Important] Integration of the type 1a+bsinxdx

OR

1a+bcosxdx

Or

1asinx+bcosx+cdx

1. 2. 13-2sinx 3. 15+4cosx 4. 15-3cosx 5. 13+2sinx 6. 11+3cosx 7. 14+9sinx 8. 15-4cosx 9. 1sinx+cosx 10.1cosx-sinx 11.1cosα+sinx 12.11+cosαcosx 13.113+3cosx+sinx 14.13+2cos2x 15.11+sin2x 16.14+5sin2x 17.15+3cos2x 18.11-2cos2x

Ex. No. 9. [Important] Integration of the type 1a+bsin2xdx

OR

1a+bcos2xdx

Or

1asin2x+bcos2x+cdx

19. 30

OMTEX CLASSES “THE HOME OF SUCCESS” 1. 2. 3. 4.

13-2sin2x 12+3sin2x 12-3cos2x 13-sin2x

5. 6. 7. 8.

11+7cos2x 15-cos2x 14+5sin2x 12sin2x+3cos2x

9. 14cos2x+3sin2x 10.1a2sin2x+b2cos2x

Ex. No. 10. [Important] Integration of the type 1. 2. sin2x 3. cos2x 4. sin3x 5. cos3x 6. cos4x 7. cos5x 8. sin6x 9. cos7x 10.sin7x 11.sin3xcos4x 12.sin2xcos3x 13.sin5x 14.sin3xcos3x 15.sin3xsinx 16.sin3xcos4x 17.sin5xcos2x 18.cos5xsinx 19.tan3x 20.cotxsinx 21.sinx.cos3x 22.sin5xcos3x 23.cot3x 24.sec4x 25.tan4x 26.cosec4x 27.tan5x 28.cosec6x 29.cosec8x 30.sin5xcos3x 31.cos3xcos4x 32.sinxcosxcos3x 33.4sin3xcos2x 31

OMTEX CLASSES “THE HOME OF SUCCESS” 34.cos5xcos3x

Ex. No. 11. [Important] Integrate the following . 1. 2. 1x+1x+2 3. xx-1x+2 4. x2+1x+1x+2x-3 5. 3x-2x2-3x+2 6. x+1xx2-4 7. 11+x+x2+x3 8. x-1x+12 9. x+1x-12 10.1x+12x2+1 11.x3+2x2+6x2+x-2

Ex. No. 12. [Important] Integrate the following. uvdx=uvdx-vdx.ddxudx

1. 2. xsinx 3. exsin2x 4. excosx 5. logx+4 6. loglogxx 7. e4xsin3x 8. tan-1x 9. logx2+4 10.logxx3 11.xsec2x 12.x1+cos2x 13.cosx 14.x3logx 15.xex 32

OMTEX CLASSES “THE HOME OF SUCCESS” 16.logx2 17.ex

Ex. No. 13. [Important] Integrate the following exfx+f'xdx=exfx+c

1. 2. 3. 4. 5. 6. 7.

exsinx+cosx ex1x+logx excotx+logsinxdx excosecx1-cotxdx sinlogx+coslogx tanlogx+sec2logx

DEFINITE INTEGRATION Ex. No. 1. 1. 2. 01xdx 3. 13x13dx 4. 491x3dx 5. 0∞e-xdx 6. 12dx3x-2dx 7. -11dx1+x2dx 8. 011-x21+x2dx 9. 02dxx-x-1dx 10.0π2sinxcosxdx 11.025xx2+4dx 12.0π4cosxdx 13.0π2sin5xcos3xdx 14.0π2sin3xdx 15.π3π21+cosxdx 16.0π4etanxsec2xdx 17.012(sin-1x)31-x2dx 18.0π2sin2x.cosxdx 19.a2aa2-x2x2dx 33

OMTEX CLASSES “THE HOME OF SUCCESS” 20.3636-x2x2dx 21.124-x2x2dx 22.0π2sinx1+cosx3dx 23.0πexsin2xdx 24.0π2dx5+3cosxdx 25.-111-x2 1+x2dx 26.012dx1-2x21-x2 27.0π41-tanx1+tanxdx 28.01logxdx

Ex. No. 2. [Important] PROPERTIES 1. 1. abfxdx= abftdt 2. abfxdx= -bafxdx 3. abfxdx= acfxdx+ bcfxdx 4. 0afxdx= 0afa-xdx 5. abfxdx=abfa+b-xdx 6. 02afxdx= 0afxdx +0af2a-xdx 7. –aafxdx =20afxdx only if fxis even and =0 if fxis odd.

1. 2. 3. 4. 5. 6. 7. 8. 9.

12xx+3-xdx 12x+2x+2+5-xdx 545-xx-4+5-xdx 02xx+ 2-xdx 01xx+1-xdx 054x+44x+4+49-xdx 03x+2x+2+5-xdx 0π2sinxsinx+cosxdx 34

OMTEX CLASSES “THE HOME OF SUCCESS” 10.0π2asinx+bcosxsinx+cosxdx 11.0π2tanxtanx+cotxdx 12.0π211+tanxdx 13.π6π311+cotxdx 14.0πxsinx1+sinxdx 15.04dxx+16-x2 16.0∞dx1+x1+x2 17.031x+9-x2dx 18.0adxx+a2-x2 19.0πxsinx1+cos2xdx 20.0π4log1+tanxdx 21.0πxtanxsecx+tanxdx 22.01x1-xdx 23.04x4-xdx

35

OMTEX CLASSES “THE HOME OF SUCCESS”

DIFFERENTIAL EQUATION EX. NO. 1. A. Form the differential equations by eliminating the arbitrary constant. 1. 2. y=x2a+x 3. y=A.ex 4. y=ae-x 5. y=ax2 6. y=c2+cx 7. x-a2+y2=a2 8. y2=4ax 9. y2=4ax+a 10.x2+y2-2ax=0 11.y=4x-c2 12.y=ax2+1 13.xy=a 14.y=eax 15.y=sinax 16.y=cosx+a 17.y=mx+c 18.y=ax2+b 19.xa+yb=1 20.x2a2+y2b2=1 21.y=Ae3x+Be-3x 22.y=AeBx 23.y=ax-a 24.y=A.e2x+B.e5x 25.y=ae2x+be-2x 26.y=Acoslogx+Bsinlogx 27.y=Acos7x-Bsin7x 28.ex+Cey=1 29.x2+cy2=4 30.y=C1x2+C2x 31.Ax3+By2=5 (Note: Important sum use the condition for consistency) 32.x2a2+y2b2=1 33.C1x2+C2y2=5

EX. NO. 2. 36

OMTEX CLASSES “THE HOME OF SUCCESS” 1. 2. 3. 4. 5. 6.

Solve x2dydx=x2+xy+y2 Solve the differential equation ydx-xdy=0 Solve the differential equation dydx=e-2ycosx Solve tanydydx=sinx+y- sin⁡(x-y) Solve dydx=sinx+y+cosx+yby putting x+y=u. Find the particular solution of the differential equation y1+logx-xlogxdydx=0 when x=e and y=e2. 7. Solve the differential equation xdydx-ysinyx=x2ex by substituting yx=v. 8. Solve dydx=4x+3y-12 by using substitution 4x+3y-1=u. 9. Solve x+2y+1dx-2x+4y+3dy=0. 10.Find the particular solution of the differential equation 1-xdy-1+ydx=0, if y=4 when x=2. 11.Solve dydx=1+y21+x2 12.Solve the D.E. dydx=x+y+12x+y-1 13.Solve the D.E. x2+y2dx-2xydy=0 14.Solve y-xdydx=y2+dydx. Hence find the particular solution if y=2 when x=1. 15.Solve the equation ey+1cosxdx+eysinxdy=0 16.Verity that y=ax2+b is a solution of xd2ydx2-dydx=0. 17.Verify that y=Asin3x+Bcos3x is the general solution of the differential equation d2ydx2+9y=0. 18.Find the particular solution of the differential equation: x+1dydx -1=2e-y when x=1 & y=0. 19.Solve the differential equation dydx=4x+6y-22x+3y+3, by taking 2x+3y=t. 20.Verify that x2+y2=r2 is a solution of the D.E. y=xdydx+ r 1+dydx2. 21.Find the order and degree of the D.E. dydx=131+dydx2. 22.Determine the order and degree of the differential equation. d2ydx2+31dydx2-y=0. 23.Determine the order and degree of the D.E. d2ydx2+1dydx2=y. 24.Determine the order and degree of the differential equation 5dydx2=10x1dydx.

37

OMTEX CLASSES “THE HOME OF SUCCESS”

VITAL STATISTICS, MORTALITY RATES AND LIFE TABLE Crude Death Rate (C.D.R.) 1. For the following data, find the crude death rate. Age group 02550Above 25 50 75 75 Population

500 7000 6000 0

No. of deaths

800

600

2000

500

100

2. Compare the crude death rate of the two given population. Age group 03060 & 30 60 above Population A Deaths in A Population B Deaths in B

400 8000 0 120 180

3000

700 9000 0 320 250

4000

200

230

3. Compare the crude death rate of the two given population. Age group 02550Above 25 50 75 75 Population A in thousands

60

70

40

30

250

120

180

200

20

40

30

10

120

100

160

170

Deaths in A Population B in thousands

38

OMTEX CLASSES “THE HOME OF SUCCESS”

Deaths in B 4. For the following data Age group

Populatio Deat n hs

0-35

4000

80

35-70

3000

120

Above 70

1000

x

Find x if the C.D.R. = 31.25 per thousand. 5. For the following data Age group Population in thousands Deaths

020

2040

4060

Above 60

58

71

41

30

195

130

X

245

Find x if the C.D.R. = 3.75 6. For the following data. Age group Population in thousands Deaths

025 25

2540

4070 X

Above 70

28

15

125 1000 1570 0

1680

Find x if the C.D.R. = 55

39

OMTEX CLASSES “THE HOME OF SUCCESS”

Specific Deaths Rates (S.D.R.) 1. Find the Age Specific deaths rates (S.D.R.) for the following data. Age Populati No. of group on deaths 0-15

6000

150

15-40

20000

180

40-60

1000

120

Above 60

4000

160

2. Find the age Specific deaths rates (S.D.R.) for population A and B of the following. Age – group 03060 and 30 60 above Population A in thousands

50

90

30

150

180

200

60

100

20

120

160

250

Deaths in A Population B in thousands Deaths in B 3. Find the Age specific deaths rates (S.D.R.) for population A and B for the following. Age – group 03060Above 30 60 80 80 Population A in thousands

30

60

50

20

150

120

200

400

50

100

90

70

200

140

270

350

Deaths in A Population B in thousands Deaths in B

40

OMTEX CLASSES “THE HOME OF SUCCESS”

Standard Deaths Rates (S.T.D.R.) 1. Find the Standard Deaths Rates for the following data: Age – group 030Above 30 60 60 Population A in thousands Deaths in A

60

90

50

240

270

250

20

30

20

Standard Population in thousands

2. Find the Standard Deaths Rates for the following data. Age – group 02550Over 25 50 75 75 Population A in thousands Deaths in A Population B in thousands Deaths in B Standard Population in thousands

66

54

55

25

132

108

88

100

34

58

52

16

102

116

78

80

40

60

80

20

3. Taking A, as the standard population. Compare the standardized death rates for the population A and B for the given data. Age – group 030Above 30 60 60 Population A in thousands

5

7

3

150

210

120

6

8

2.5

240

160

7.5

Deaths in A Population B in thousands Deaths in B 4. Taking A, as the standard population. Compare the standardized death rates for the population A and B for the given data. Age – group 02040Above 20 40 75 75 Population A in

7

15

10

8 41

OMTEX CLASSES “THE HOME OF SUCCESS”

thousands

140

150

110

240

9

13

12

6

270

260

300

150

Deaths in A Population B in thousands Deaths in B

Life Tables 1. Construct the life tables for the rabbits from the following data. x 0 1 2 3 4 5 6 lx

10

1

0

2. Construct the life tables for the following data. x 0 1 2 3 4

5

6

lx

2

0

50

9

36

7

21

5

12

2

6

3. Construct the life tables for the following data. x 0 1 2 3 4

5

lx

0

30

26

18

10

4

4. Fill in the blanks in the following tabled marked by ‘?’ sign. Age lx dx qx px Lx Tx e0x 50

60

?

?

?

?

240

?

51

50

-

-

-

-

?

?

5. Fill in the blanks in the following table marked by ‘?’ sign. Age lx dx qx px Lx Tx e0x 56

400

?

?

?

?

3200

?

57

250

?

?

?

?

?

?

58

120

-

-

-

-

?

?

42

OMTEX CLASSES “THE HOME OF SUCCESS”

MATRICES Ex: 1 A. 1. Consider the Matrices A=2-134, B=461, C=2331, D = 4-63-241652, E = a33b , F = [56-7], G = [57a6] Answer the following questions. a. State the orders of the matrices A, C, D, G. b. Which of these are row matrixes? c. If G is a triangular matrix. Find a. d. If e11 = e12. Find a. e. For D, state the values of d21, d32, d13. 1. A = [aij]2×3 such that aij=i+j. Write down A in full.

2. Find which of the following matrices are non – singular.

A=33-88, B=5-20-416, C=1233-12707, D=2-1382612012, E=65-42101063 3. If A=63-4a is a singular matrix, find a. 4. If A=6-5142-114-1k is a singular matrix, find k. A. 1. Consider the matrices. A=12-13, B=3-542, C=1-123, D=a-b2-1a+b, E=3ab-a2, F=213-124, G=2-11234, H=2sinπ23cosπ24 Answer the following questions. i. Are matrices A and C, F and H, F and G, F'and G are equal. ii. If A=D, find a and b. iii. If B=E, find a and b. 1. If a-4b56-a+b=1156-5, find a and b. 2. Find a, b and c if a+2b2-bb+ca-c=2312

43

OMTEX CLASSES “THE HOME OF SUCCESS”

Ex: 2 1. A=2-135, B=-3216-15, C=62-15, D=4322-21, F=52a4, F=-26-3-5 Answer the following. a. Can you find, i. A + B; ii. A + C; iii. B + D, iv. B’ +D; v. A + A’; vi. D + D’; vii. C + F’. b. If A + F = 0, find b. c. If C – E = I, Find a. 1. If A =3-1243-5, B=-1248-13, C=824-237 Verify the following. a. A + B = B + A b. A + (B + C) = (A + B) + C c. A – (B – C) = A – B + C. d. 3(A + B – C) = 3A +3B – 3C e. (A + B)’ = A’ + B’. 1. If A=6321, B=0-13-2 obtain the matrix A -3 B. 2. Find x if 4536+ x=1010-5 3. If A=1234, ind matrix B such that A+B=0. 4. If A=12-34 and 2A+3B=0, find the matrix B. 5. If A=3215 find the matrix ‘X’ such that A-2X=187-6. 6. If A=122-3-10, B=101213 Find the matrix C such that A + B + C is a zero matrix. 7. If A=2124, B=12-30 Find the matrix X such that 2X + 3A – 4B = 0. 8. Find the matrix ‘X’ such that 3X+451-3=711-89. 9. Find the values of x and y satisfying the matrix equation. a. 1x0y24+ 31243-2=422652 b. 2x+1-1134y4+-164303=4556127 1. Find x, y & z if x+yy-zz-2xy-x=3-111

44

OMTEX CLASSES “THE HOME OF SUCCESS”

Ex: 3 1. Find the following products: a. 23-14-13 b.34-21 c.-2430 d.65-12-22 e.4x030y f.2xx-1 g.8-452 h.abcabc i.secθtanθsecθ-tanθ 2. Find x in the following cases. a.-32-1x=1 b.4x13-2x=8 c.4x-xx=21 d.5x-12xx4=20 e.x2x3xx2-1=0 f.xsinθcosθxsinθcosθ=5

Ex: 4 A. 1. Find AB and BA whenever they exist in each of the following cases. 2. A=2525, B=3113 3. A=231-2, B=1-230-12 4. A=1021-10, B=-1301 5. A=23-15-10321, B=0201231-12 6. A=3-12, B=43-5 7. A=10-343-2124, B=16-69-187-105-23 2. If A=1-243, B=5678, C=[-2083] Then verify the following a.ABC=ABC b.BA-C=BA-BC c.A+BC=AC+BC d.AI=IA=A, Where I identity matrix 1. If A=2-133, B=257-3-21, C=[-164321] verify the following. a.AB+C=AB+AC. b.AB-C=AB-AC. 2. If A=122212221 show that A2-4A is a scalar matrix. 3. If A=033-304-3-40, B=xyz, B'=xyz S.T.B'AB is a Null Matrix. 6. (A) Find the values of a and b from the matrix equation: 3241 a15b =4535 6. (B) Find the values of x and y a. 1232x532y-5=5-3-777-1 b. 4120213-2131238201=xy 6. (C). Find x, y, z values in each of the following cases. i. 134-20652-3xyz=98-4 ii. 322123-1=xyz ii. xyz105237-421=-4-47 iii. 5100111-312-233121=xyz 7. Find x, y, z, a, b, c if

1232xy33-1z=707abc 45

OMTEX CLASSES “THE HOME OF SUCCESS” 8. If A=100111, B=12-2331 , C =21, X=xyz Find the values of x, y, z if 5A-3BC=X 9. If A =415234, B=1-64203 Find the Matrix AB and without computing the Matrix BA, show that AB ≠ BA. 10. If A=3-520, B =1-234 Verify that AB ≠BA. 11. i. If A=1232-10, B=1324-1-3, find AB ii. If A=2103, B=123-2 verify that AB=A.|B| 12. If A=-201123, B=01231-1 show that AB is a Non singular matrix. 13. If A=24-1-2, Show that A2 is a null matrix. 14. If A = 1-1-11 show that A2=2A. 15. a. If A = [2411] Show that A satisfies the Matrix Equation A2=3A+2I. b.If A=1234, show that A2-5A-2I is a Zero matrix. 16. If A∝=cos∝sin∝-sin∝cos∝ show that AαAβ=AβAα=Aα+β

Ex: 5 1. 2. 3. 4.

If A=32128 and B=84-12-6 show that AB = 0. If A=12-1-2 B=4353, C=2175 show that BA = CA. Show that AB = AC does not imply that B = C. a. If A=3443, B=5665 show that AB = BA. b. If A=-36-24 show that A2=A.

5. If A=31-13, B=25-52 show that A+BA-B= A2-B2 6. If A=32128 , B=618-9-27 show that a. A+B2=A2+BA+B2 b. A+BA-B=A2+AB-B2. 7. If A=84105, B=5-410-8show that a. A+B2=A2+AB+B2 b. A+BA-B=A2-AB-B2. 8. If A=2-2-4-1341-2-3 and B=-1241-2-4-124 show that A+B2=A2+B2 9. If A=1-12-1 and B=1a4b such that A+B2=A2+B2 than find a & b. 10. If A=12-1-2 , B=2a-1b and A+B2=A2+B2, find a and b

Ex: 6 I.

Write down the following equation in the Matrix Form and hence find values of x, y, z using Matrix method. 46

OMTEX CLASSES “THE HOME OF SUCCESS” 1. 2. 3. 4. 5. 6.

x+3y+3z=12; x+4y+4z=15 ; x+3y+4z=13. x+y+z=6; 3x –y+3z =10 ; 5x+5y-4z=3. x+y+z=3; 3x-2y+3z=4; 5x+5y+z=11. x+y-z+4=0; 3x-2y+3z=4; 5x+5y+z=11. 4x-3y+z=1; x+4y-2z=10; 2x-2y+3z=4. x+y+z=1; 2x+y+2z=10 ; 3x+3y+4z=21.

I. 1. 2. 3. 4. 5. 6.

Solve the following equation by the methods of reduction. x+y+2z=7; 3x+y-5z=6; 2x+2y-3z=7. x+y+4z=4; 2x+3y+6z=5; 3x-2y-z=4. x-y+z=1; 3x-y+2z=1; 2x-2y+3z=2. x+y+z=3; 7x+y+z=9; 2x-y+3z=4 4x+2y-z=3; x-2t+z=-8; 2x-y+z=-7 3x+3y-4z=2; x-y+z=1; 2x-y=1.

Ex: 7 A. Find the inverse of each of the following Matrices by using elementary transformations. 1. 1325 2.3124 3.cosθsinθ-sinθcosθ 4.secθtanθtanθsecθ 5.cosecθ-cotθcotθcosecθ 6.2010-12101 7.12-2-1300-21 8.7-3-3-110-101 9.cosθsinθ0-sinθcosθ0001 10.sinθcosθ0cosθ-sinθ0001 11.cosecθcotθ0cotθcosecθ0001 12.secθtanθ0tanθsecθ0001

B. 1.If A=31-12, show that A2-5A+7I=0, Hence find A-1. 2.If A=2411, show that A2-3A=2I, Hence find A-1. 3.If A=1303, show that A2-4A+3I=0, Hence find A-1. 4.If A=122212221, show that A2-4A=5I, Hence find A-1. C. 1.If A=31-12, B=7306 find Matrix X such that AX=B. 2.If A=10-11, B=123456, find Matrix X such that AX=B. 3.x+2y-2z=5; -x+3y=0; -2y+z=-3, by using inverse of a Matrix. 4.If A+I=134-113-2-31, find the matrix A+IA-I. 5.If A=cosθsinθ-sinθcosθ, then Show that A2=cos2θsin2θ-sin2θcos2θ 6.If A=-200-2-13 , B=01231-1 , show that AB-1exists.

47

OMTEX CLASSES “THE HOME OF SUCCESS” 7.If A=2-110, B=3121, find AB-1. 8.If A=1ωω2ωω21ω21ω, B=ωω21ω21ω1ωω2 then show that AB=0, where w is cuberoot of unity. 9.Verify AB-1=B-1A-1, where A=1101, B=2413

48

OMTEX CLASSES “THE HOME OF SUCCESS”

DIFFERENTIATION (DERIVATIVES) EX. NO. 1.

Differentiate the following with respect to x.

1. 2. ax2+bx+c 3. 4x3-5x2+8x-1 4. xa+ax+ex+aa 5. 7x3+4x+sinx-2cosx 6. axb+1x3-logx 7. 5x+1x-xn 8. 5ax+4x3+logx2 9. 13x2+sin-1x+cos-1x 10.tan-1x+sec-1x 11.x+1x2 12.x1+3x 13.1e-x+15-x

EX. NO. 2. 1. 2. 5xex 3. exx3 4. 4xx4 5. x-33x 6. exsinx 7. xsin-1x 8. xtan-1x 9. exsecx 10.secx.tanx 11.xtanx 12.x2logx. 13.(x−1)(x-2) 14.2x+1cosx. 15.x2+1(x2−x+1) 16.x+1(x−2)(x+3) 17.exsinx.cosx 18.x.5xcosx 49

OMTEX CLASSES “THE HOME OF SUCCESS” 19.xx+1 20.1+x1-x 21.1+x1-x 22.x2+1x2-1 23.3x-52x+3 24.x3-5x+22x+1 25.x2+x-1x2+x-3 26.xx-1x-2 27.sinxlogx 28.cosx1+x 29.exlogx 30.x2+x-11+sinx 31.x+xx+1 32.x1+logx 33.sinx1+cosx 34.sinx1+cosx 35.1+sinx1-sinx 36.1+tanx1-tanx 37.ex-1ex+1 38.3+sinx1+3sinx 39.1+cosxxsinx 40.cosx+sinxcosx-sinx 41.1-sin2x1+sin2x 42.1-cos2x1+cos2x 43.xsinx-cosxxcosx+sinx 44.x2sinx+3xtanx 45.If y=tanx.Prove thatdydx=sec2x by using the rule ofddxuv

50

OMTEX CLASSES “THE HOME OF SUCCESS”

INDEX NUMBER EX. NO. 1. SIMPLE AGGREGATIVE METHOD I. FIND THE INDEX NUMBER. 1. Find Index number. [Ans. 137.73] Commoditi es

I II III IV V

Prices in 2002 (P0)

21.3 55.9 100.2 60.5 70.6

Prices in 2003 (P1)

30.7 88.4 130 90.1 85.7

2. Find Index number. [Ans. 180] Commoditi Prices Prices es in in 1990 2002 (P0) (P1)

A B C D E

12 28 10 16 24

3. Find Index number. Commoditie Price Price s s s in in 2000 2003

38 42 24 30 46 [Ans. 107.1, 109.375] Price s in 2006

51

OMTEX CLASSES “THE HOME OF SUCCESS” Trucks Cars Three wheelers Two wheelers

800 176 100 44

830 200 127 43

850 215 115 43

4. Find Index number. [Ans. 64.06, 39.06] Commoditi 199 200 200 es 8 0 5 P0 Stereo T.V. Computer Mobile

10 30 80 8

P1 6 20 50 6

P1 5 15 25 5

5. Find the index number for the year 2003 and 2006 by taking the base year 2000. [Ans. 48, 75.4359] Security at 200 200 200 Stock 0 3 6 market

A B C D E

P0

P1

P1

160 240 0 800 350 0 150

180 35 550 200 0 600

210 8 850 400 0 220

6. Calculate Index Number. [Ans. 69.078, 238.15] Real 199 199 200 Estate 0 8 6 Area wise

A B C

100 35 5

65 22 7

250 75 12

52

OMTEX CLASSES “THE HOME OF SUCCESS” D

12

11

25

7. Calculate Index Number. [Ans. 113.0952] Items 200 200 0 5

Wheat Rice Dal Milk Clothi ng

500 400 700 20 60

600 430 770 32 68

8. Calculate the Index Number. [Ans. 412.19, 92.68] Security at 198 199 199 Stock 8 1 4 market

A B C D E

P0

P1

P1

650 120 0 530 270 145 0

350 0 135 0 470 0 505 0 230 0

700 130 0 200 100 150 0

9. Compute the Index Number. [Ans. 110.526, 126.31579] Food Items

Potato Onion Tomat o Eggs Banan a

Units

Kg Kg Kg Doze n Doze n

200 4

200 5

200 6

P0

P1

P1

10 12 12 24 18

12 25 25 2 20

14 16 16 26 24

53

OMTEX CLASSES “THE HOME OF SUCCESS”

I.

The Index number by the method of aggregates is given in each of the following example. Find the value of x in each case. 1. Index Number = 180

Commodi ty

A B C D E

Base year

Current Year

P0

P1

12 28 X 26 24

38 41 25 36 40

[Ans. X = 10] 2. Index Number = 112.5 Commodi ty

Base Year

Current Year

P0

P1

I

3

5

II III IV V

16

25

40

35

7

10

14

x

[Ans. X = 15] 3. Index Number = 120 Commodi Base Current ty Year Year

I II III IV V

P0

P1

40 80 50 X 30

60 90 70 110 30

54

OMTEX CLASSES “THE HOME OF SUCCESS” [Ans. X = 100]

Ex. No. 2. Weighted Aggregative Index Numbers. 1. For the following data find Laspeyre’s, Paasche’s, Dorbish – Bowley’s and Marshall – Edgeworth Index Numbers. [Ans. 134.2, 130, 132.1, 132.05] Commoditi Base Year Current Year es Pric e

Quanti ty

Pric e

Quanti ty

A

20

3

25

4

B

30

5

45

2

C

50

2

60

1

D

70

1

90

3

2. For the following data find Laspeyre’s, Paasche’s, Dorbish - Bowley’s and Marshall – Edgeworth Index Numbers. [Ans. 144.11, 149.2, 146.66, 147.422] Commoditi Base Year Current Year es Pric e

Quanti ty

Pric e

Quanti ty

1

10

3

20

3

2

40

4

60

9

3

30

1

50

4

4

60

2

70

2

3. Find Fisher’s Price Index Number. [Ans. 132.1] {using log table} Commoditi Base Year Current Year es Pric e

Quanti ty

Pric e

Quanti ty

A

20

3

25

4

B

30

5

45

2

55

OMTEX CLASSES “THE HOME OF SUCCESS” C

50

2

60

1

D

70

1

90

3

4. Find Walsch’s Price Index Number.[Ans. 116.21] Commoditi Base Year Current Year es Pric e

Quanti ty

Pric e

Quanti ty

I

10

4

20

9

II

40

5

3

5

III

30

1

50

4

IV

50

0.5

60

2

5. Calculate Price Index Number by using Walsch’s Method. [Ans. 126.83] Commoditi Base Year Current Year es Pric e

Quanti ty

Pric e

Quanti ty

A

5

4

7

1

B

2

6

3

6

C

10

9

12

4

6. The ratio of Laspeyre’s and Paasche’s Index number is 28:27. Find x. [Ans. x = 4] Commoditi 1960 1965 es Pric e

Quanti ty

Pric e

Quanti ty

A

1

10

2

5

B

1

5

X

2

7. For the following the Laspeyre’s and Paasche’s index number are equal, find K. Commodi ty

P

Q

P

Q

0

0

1

1

A

4

6

6

5 56

OMTEX CLASSES “THE HOME OF SUCCESS” 4 B

4

K

4

57

OMTEX CLASSES “THE HOME OF SUCCESS”

Ex. No. 3. Cost of Living Index number There are two methods to construct cost of living index number. 1. Aggregative Expenditure Method. 2. Family Budget Method. 1. Taking the base year as 1995, construct the cost of living index number for the year 2000 from the following data. Group

1995

200 0

Pri ce

Quant ity

Pric e

Food

23

4

25

Clothes

15

5

20

Fuel and Lighting

5

9

8

12

5

18

8

6

13

House Rent Miscellaneous [Ans. 137.5]

2. The price relatives I, for the current year and weights (W), for the base year are given below find the cost of living Index number. Group

Food

Clothes

Fuel & Lighting

House Rent

Miscellaneo us

I

320

140

270

160

210

W

20

15

18

22

25

Fuel & Lighting

House Rent

Miscellaneo us

[Ans. 221.3] 3. Find the cost of living Index number. Group

Food

Clothes

58

OMTEX CLASSES “THE HOME OF SUCCESS”

I W

200

150

140

100

120

6

4

3

3

4

[Ans. 150] 4. Find the cost of living index number. Group

1995

200 0

Pric e

Quanti ty

Pric e

Food

90

5

200

Clothes

25

4

80

Fuel and Lighting

40

3

50

30

1

70

50

6

90

House Rent Miscellaneous [Ans. 208]

5. Find the cost of living index number. Group

1995

200 0

Pric e

Quanti ty

Pric e

Food

30

15

25

Clothes

45

10

30

Fuel and Lighting

25

12

20

12

8

15

36

20

35

House Rent Miscellaneous [Ans. 86.06]

6. Find x if the cost of living index number is 150 59

OMTEX CLASSES “THE HOME OF SUCCESS” Commodi ty

Foo d

Cloth es

Fuel & Lighting

House Rent

I

200

150

140

100

120

6

4

X

3

4

W

Miscellaneo us

60

OMTEX CLASSES “THE HOME OF SUCCESS”

Numerical method Newton’s Forward Interpolation Formula. Ex. No. 1 x

1. Using Newton’s Interpolation formula, find f(5) from the following table. 2 4 6 8

f(x)

4

7

11

18

2. Given the following table find f(24)using an appropriate interpolation formula. X 20 30 40 50 f(x)

512

439

346

243

3. In an examination the number of candidates who scored marks between certain limits were as follows. Marks 0-19 20-39 40-59 60-79 80-99 No. Of Candidates

41

62

65

50

17

Estimate the number of candidates geting marks less than 70. 4. The population of a town for 4 year was as given below. Year 1980 1982 1984 1986 Population (in Thousand)

x

54

58

63

5. For a function f(x), f(0) = 1, f(1) = 3, f(2) = 11, f(3) = 31. Estimate f(1.5), using Newton’s Interpolation formula. 6. For a function f(x), f(1) = 0, f(3) = 25, f(5) = 86, f(7) = 201. Find f(2.5) using Forward Difference interpolation formula. 7. Construct a table of values of the function fx= x2 for x = 0,1,2,3,4,5. Find (2.5) and f(2.5)2 using Newton’s Forward Interpolatino Formula. 8. Estimated values of logarithms upto 1 decimal are given below find log(25) 10 20 30 40

logx x

52

1

1.3

1.4

1.6

9. Estimated values of sines upto 1 decimal are given below find sin(450) 00 300 600 900

sinx

0

0.5

0.87

10.Find f(x) if f(0) = 8, f(1) = 12, f(2) = 18. 11.f(x) is a polynomial in x. Given the following data, find f(x) X 1 2 3

1

4 61

OMTEX CLASSES “THE HOME OF SUCCESS” f(x)

7

18

35

58

Also find f(1.1)

62

OMTEX CLASSES “THE HOME OF SUCCESS”

Ex. No. 2 Lagrange’s Interpolation Formula. 1. By using suitable interpolation formula estimate f(2) from the following table. X - 0 3 1 f(x )

3

1

1 9

2. By suing Lagrange’s Interpolation formula, estimate f(x) when x = 3 from the following table. X 0 1 2 5 f(x )

2 3

1 0

14 7

3. A company started selling a new product x in the market. The profit of the company per year due to this product is as follows: Year 1s 2n 7t 8t Profit (Rs. In lakh)

t

d

h

h

4

5

5

5

Find the profit of the company in the 6th year by using Lagrange’s Interpolation formula. 4. Using the Lagrange’s Interpolation formula, determine the percentage number of criminals under 35 years. Age % number of criminals Under 25 years Under 30 years Under 49 years Under 50 years

52 67.3 84.1 94.4

5. The function y = f(x) is given by the points (7,3), (8,1), (9,1), (10, 9). Find the value of y at x = 9.5 using Lagrange’s formula. 63

OMTEX CLASSES “THE HOME OF SUCCESS” 6. Given log1010=1, log1012=1.1, log1015=1.2 and log1020=1.3. find log1013= ? [Values are approximate and rounded off to 1 decimal place].

Ex. No. 3 Forward difference table 1. Form the difference table for f(x) = x2 +5 taking values for x = 0, 1 , 2 , 3. 2. Write down the forward difference table of the following polynomials f(x) for x = 0(1)5 a. f(x) = 4x-3 b. f(x) = x2 – 4x – 4. 3. Obtain the difference table for the data. Also what can you say about f(x). From the table? x 0 1 2 3 4 5 f(x)

0

3

8

15

24

35

4. By constructing a difference table, obtain the 6th term of the series 7, 11, 18, 28, 41. 5. Estimate f(5) from the following table. X 0 1 2 3 4 f(x)

3

2

7

24

59

6. By constructing a difference table, find 6th and 7th term of the sequence 6, 11, 18, 27, 38. 7. By constructing a difference table, find 7th and 8th term of the sequence 8, 14, 22, 32, 44, 58. 8. Given u4 = 0, u5 = 3, u6 = 9 and the second difference are constant. Find u2. 9. Find u9, if u3 = 5, u4 = 12, u5 = 21, u6 = 32, u7 = 45.

Ex. No. 4 x

1. Estimate the missing term by using "∈"and "∆" from the following table. a. 0 1 2 3 4

y

1

3

9

-

81

x

b. 1

2

3

4

5

6

7

y

2

4

8

-

32

64

128

c. 64

OMTEX CLASSES “THE HOME OF SUCCESS” x

1

2

3

4

5

f(x)

2

5

7

-

32

2. Find ∆fxin each of the following case, assuming the interval of difference to be 1. i. ii.fx= xx-1x-2x-3. iii.fx= x2+x. iv.fx= x2-2x+4. v.fx= 2x+3. 3. 4. Given fx= x2+3x+5 taking the interval of differentiating equal to 1. Find ∆fxand ∆2fx. 5. Given fx= x2-8x+2, taking the interval of differentiating equal to 1. Find ∆fxand ∆2fx. 6. Find ∆2fx if fx= xx+1x+2. 7. Evaluate i. ii.∆2x2+5 iii.∆sinax+b iv.∆cos⁡(ax+b) 8. 9. Evaluate i. ii.∆35ex iii.∆4(abx) iv.∆2∈x3 v.∆2x3∈x3 vi.∆21x vii.∆ex 10.

65

OMTEX CLASSES “THE HOME OF SUCCESS” 11.Show that ∆2∈ex.∈ex∆2ex=ex 12.Show that ∆logfx=log1+∆fxfx 13.If fx= ex. Show that fx, ∆f, ∆2fx, …………….∆nfx are in geometric progression. 14.Given: u0 = 3, u1 = 12, u2 = 81, u4 =100, u5 = 8, find∆5u0. 15.Given: u2 = 13, u3 = 28, u4 = 49, find ∆2u2. 16.Given: u2 = 13, u3 = 28, u4 = 49, u5 = 76. Compute ∆3u2+∆2u3. 17.Prove the following: i. ∆4fx= fx+3h-3fx+2h+3fx+h-fx ii.∆4fx=fx+4h-4fx+3h+6fx+2h-4fx+h+fx. iii.fa+3h=fa+3∆fa+3∆2fa+∆3fa. iv.fa+5h=fa+5∆fa+10∆2fa+10∆3fa+5∆4fa+∆5fa. 18.Assuming that the difference interval h = 1, prove the following. i. f4= f3+∆f2+∆2f1+∆3f1. ii.f7= f6+∆f5+∆2f4+∆3f4. iii.f5= f4+∆f3+∆2f2+∆3f1+∆4f1. iv.f2= f1+∆f0+∆2f-1+∆3f-1.

66

OMTEX CLASSES “THE HOME OF SUCCESS”

Revision for Board exam Derivatives & Integration 1.Differentiate y=tan-11+sinx+1-sinx1+sinx-1-sinxw.r.t.x. 2.Finddydxif y=cotxsinx+tanxcosx 3.Evaluate 3x+2x2-5x+4dx. 4.Evaluate e2xsin3xdx. 5.Evaluate 0π2logcosx dx. 6.Evaluate 02dxx+4-x2 7.Evaluate 15x+4-5x+2dx 8.Evaluate sin2xdx7sin2x+3cos2x. 9.Evaluate 1+sinxdx 10. Evaluate 4x+32x3+3x+432dx 11.Find the derivative ofsin2x+3w.r.t.x using the first principle. 12.Differenciate xxsinxw.r.t.x 13.Finddydxif y=cos-11-2x2 14.Evaluate 15+4cosxdx 15. Evaluate d4xsin3x dx 16. Evaluate 043x+53x+5+39-xdx. 17. Evaluate0π4log1+tanxdx 18.Evaluate logxdx 19. Evaluate cosxdx 20.Differentiate sinx w.r.t.cosx 21.Evaluate 012(sin-1x)1-x2dx 22. Evaluate 2x-1x-54dx 23.Evaluate sinx1-sinxdx 24.Differentiatex2+1x2+4w.r.to x. 25.Finddydxif y= sinx 26.Differentiate y=tan-1cosx1+sinxw.r.to x. 27. Evaluate sin3xdx 28. Evaluate 54x2+4x-15dx 29.Evaluate exdxe2x1+4ex 30. Evaluate x3+2x2+6x2+x-2dx. 31.Evaluate 0πxsinx1+cos2xdx 32.Evaluate 23dxxx3-1. 33.Differentiatetan-1sinx1+cosxw.r.t.x 34.Finddydxif x2+y2+6x-4y+10=0 35.Differentiatecosx+sinxcosx-sinxw.r.t.x 36.Evaluate 0π2cosecxcosecx +secxdx 37.Prove that 012sin-1x1-x232dx=π4-12log2 38.Evaluate 012-x21+x2dx 39.Evaluate –π2π2sin4xsin4x+cos4xdx 67

OMTEX CLASSES “THE HOME OF SUCCESS” 40. Evaluate cosxdx1+sinx2+sinx3+sinx). 41. Evaluate sec34xdx 42. Evaluate 2x+1x+45dx 43.If y=cosxlogx+ logxx, finddydx. 44.Differenciate 1-x21+x2w.r.t.2x1+x2. 45.Differentiatecot-1xexw.r.t.x 46.If x2+3xy2+y2=x, finddydx. 47. Evaluate cos2x dx 48.Finddydxwhere x=aθ+sinθ; y=a1-cosθ 49.If y=tanlogx, finddydx. 50, If fx= 1+4x, findf'2by first principle. 51.If x=acost+tsint; y=asint-tcost, finddydxat t=π4 52.Evaluate dx4ex+9e-x 53. Evaluate logxx3dx 54.Evaluate 01x1-x52 dx 55.Evaluate -111-x2 1+x2dx 56. Evaluate x21-xdx 57. Evaluate excotx+logsinxdx 58. Evaluate 012dx1-2x21-x2. 59.Evaluate dx2-3sin2x 60. Evaluate x3dxx2+1 61.If x13+y13=a13, finddydx 62.If y=xlogx, finddydx. 63.If y=xx+xsinx, finddydx 64.y=tanxx1+x2, finddydx. 65.Evaluate 3x2-4x+252×3x-2dx 66. Evaluate ex(1+tanx+tan2x)dx 67.Find the derivative of tanx w.r.t.x using the first principle 68.If y=cosx+y, finddydx 69.If y=exsinx, finddydx 70.Evaluate 04dxx+16-x2 71.Evaluate dxsinx-asinx-b 72. Evaluate logx2+ 4dx 73. Evaluate 0π2sinx1+cosx3dx 74. Evaluate 0adxx+a2- x2 75. Evaluate sinx1+sinxdx 76. Evaluate x3logxdx 77. Evaluate If y=x3logcosx, finddydx 78.Diff.w.r.to x, 1-cos2x1+cos2x 79. Evaluate sinx-asinx-adx 80.Evaluate cos2x sinx dx. 81.Finddydxif y= 3-x3+xx2+2. 82.If y=sin-1(2x1-x2), finddydx. 68

OMTEX CLASSES “THE HOME OF SUCCESS” 83.Evaluate loglogxxdx 84.Evaluate ex2-sin2x1-cos2xdx 85.Evaluate 0πdxsin2xdx 86.Show that 0ax2a-x32dx=1635a92. 87. Evaluate 1x2-6x-7dx 88. Evaluate x3x2+1dx 89.If x2y3=x+y5finddydx. 90.If fx= cosx, find f'π6from the first principle. 91.Differenciatesin3xcos2x w.r.t.x 92.If y=5x3+4x-5, finddydx. 93.Evaluate x2+14x4+x2- 2dx 94. Evaluate dxsin2x+2cos2x+3 95.Finddudvif u=2bt1+t2, v=a(1-t2)1+t2, in terms of u & v. 96. Evaluate cos4xdx 97. Evaluate sin-1xdx 98. Evaluate 01x+4x2+5dx 99.Evaluate 0aa-xxdx 100. Evaluate sin2xdx

Matrices 1. Find k if k 2 secθ -2tanθk2secθ2tanθ=29 2. If A= 213321, B=123423Show that AB'=B'.A'. 3. Find k if the following matrix is singular. A=6-5142-114-1k 4.

Find the matrix X such that 3A – 2B +4X = 5C.

5. Find x, y, z if 1-4352-13=xyz 6. If A=31-12, find the matrix A2-I. 7. Find the inverse of the matrix 2312. 8. If A=63-4ais a singular matrix, find a. 9. If A=3125, B=5621, then show that A+Bis a non-singular matrix. 10. If x+2y2-1x-y=42-15, find x& y. 11. If A=132-1, B=2643, find the matrix X such A-2B+X is a unit matrix. 12.ix r e of teh hat 3A - 2Bs. the following statements are true or falseis in India. If A=5443, B=-3445Find A,B & AB. 13.A=3-1243-5, B=-1248-13 verify that A+B'= A'+ B' 14. Find the values of x and y from the matrix equation:

1332x532y-5=5-3-777-1 15. If A=2411, find A-1 16. If 3x-164272+ 54-32y+310= 13101782 17. If A=ab00, B b0-a0Show that (A+B)2 = A2 + B2 + BA.

18. Find the inverse of the matrix 1303 69

OMTEX CLASSES “THE HOME OF SUCCESS” 19. Solve the equation by reduction method.

2x+4y-1=0 , x+y=1. 20. If A=39-4-12, B 4386Show that (A+B)2 = A2 + AB + B2. 21. Solve the equation by using reduction method.

x+y=1, 4x-3y=18. 22. If A = 2-11-23-2-44-3show that A2=A. 23. Find the matrix X such that AX = B, Where

A=1-22-1, B=-31 24. If A = 12& B=2 5 find AB. Does (AB)-1 exists?

25. If A = 133313331, show that A2-5A is a scalar matrix. 26. If A=2103, B=123-2, verify , AB=AB 27. Find the value of a & b from the matrix equation: 3214a15b=45-35. 28. If A=2-103, B=-123-2, verify that AB=A.B. 29. Find a matrix X such that 2X – 3A = B where A=-10212-1211, B=1210-13416. 30. If A=100111, B=12-2331, C=21, X=xyz, Find x, y, z if (5A-3B)C=X.

DISCRETE PROBABILITY DISTRIBUTIONS 1. 2.

Six coins are tossed simultaneously. What is the probability of more than 2 heads? Find the mean and variance of the Binomial Distribution, if n = 7, p = ¼.

3. For a poisson distribution with parameter 0.3, find p x<1and px ≠0. e-0.3=0.7408 4. 5. 6.

Workers in a factory have 20% chance of suffering from a disease. What is the probability that out of 6 workers selected 4 or more suffer from the disease. Six coins are tossed simultaneously. What is the probability of getting 2 heads? Find the mean and variance of the binomial distribution with parameters n = 16, p = ½ .

7. The probability that a poisson variate x takes a positive value is (1 – e – 1.5 ), find the variance. 8. For a binomial distribution probability of 1 and 2 successes are 0.4096 and 0.2048. Find p. 9. Six coins are tossed simultaneously what is the probability of getting at least two heads. 10. Find the Mean and Variance of the binomial distribution if n = 7, p = ¼ .

11. A variate follows poisson distribution with parameter 0.3, find p(0), p(1). (e – 0.3 = 0.7408). 12. How many tosses of a coin are needed so that the probability of getting at least one head is 87.5%. 13. A biased coin for which head is thrice as likely as tail in a toss, is tossed five times. Find the probability that three heads occur in these five tosses.

14. For a binomial distribution, mean is 6 and the standard deviation is 2.Find the probability that the number of success is exactly equal to the number of trials.

15. Between 2 pm and 4 pm the average number of phone calls per minute coming into a switch board of a company is 2.35. Find the probability that during one particular minute there will be at most 2 phone calls. [ e - 2.35 = 0.095374] 16. On an average A can solve 40% of the problem. What is the probability of A solving exactly 4 problems out of 6. 17. An unbiased die is thrown 5 times and occurrence of 1 or 6 is considered as success. Find the probability of at least one success.

18. For a Binomial distribution mean is 4 and Standard Deviation is 125. Find the parameters of the distribution.

70

OMTEX CLASSES “THE HOME OF SUCCESS” 19. In a certain plant there are 4 accidents on an average per months. Find the probability that in a given year there will be less than 4 accidents. e-4=0.0183 20. On an average A can solve 40% of the problems. What is the probability of A solving 4 problems out of 6. 21. An unbiased die is thrown 5 times and the occurrence of 1 or 6 is considered as success. Find the probability of exactly one success. 22. Find the binomial distribution whose mean is 9 and variance is 2.25.

23. Assuming that the probability of fatal accident in a factory during the year is 11200 . Calculate the probability that in a factory employing 300 workers there will be at least 2 fatal accidents in a year.

(e-0.25=0.7788).

24. For binomial variate x, with n = 6, p = 2/3, find px≥2. 25. An unbiased dice is thrown 5 times and the occurrence of 1 or 6 is considered as success. Find the probability of at least 4 successes. 26. For a binomial distribution the number of independent Bernoulli trials was 12 and probability of failure was 5/6. Find the means the variance of the binomial distribution.

27. If 2% of electric bulbs are defective. Find the probability that in a sample of 200 bulbs less than 2 are defective. e-4=0.0183. 28. Assuming that half of the MBA’s are commerce graduates and that the investigators interview 10 MBA’s to see whether they are commerce graduates what is the probability that 2 or less number of MBA’s will be commerce graduates. 29. An unbiased coin is tossed 6 times. Find the probability of getting at most two heads.

30. Find the parameters of binomial distribution if mean = 7/4, SD = 214. 31. A factory produces on an average 5% defective item. Find the probability that a randomly selected sample contains 2 or more defective items. e-0.05=0.9512. 32. The overall percentage of failures in an examination is 40. What is the probability that out of a group of 6 candidates at least 4 passed the exam. 33. A fair dice rolled 5 times getting an even number is considered as success. Find the probability of no successes. 34. Find the parameters for binomial distribution if mean = 15/2, variance = 15/8.

35. 4% of the bolts produced in a factory are defective. Find the probability that a random sample of 100 bolts contain at least one defective bolts. e-4=0.0183. 36. The probability that a man hits the target is 1/5. If he fires 5 times, what is the probability of hitting the target at least twice. 37. A dice is tossed 5 times what is the probability that 5 shows up exactly thrice. 38. Find mean and variance of the Binomial distribution, n = 10, p = 3/5.

39. A variate follows Poisson distribution with parameter 0.3 find px≥1if e-0.3=0.7408. 40. A has won 20 out of 30 games in chess against B. In a new series of 6 games what is the probability that A would win four games. 41. If for a binomial distribution probability of success is ¼ and the mean is 12.5, find the remaining parameters of the distribution. 42. The probability that A wins a game of chess against B is 2/3. Find the probability that A wins at least ‘one’ game out of the 4 games he plays against B. 43. If X is a Poisson random variable such that P(x=3) = P(x = 4), find the mean find the standard deviation of the distribution.

44. If X is a Poisson variate with mean 3, find px≥2.[Given:e3 =0.0498]

Numerical Method 1. Using Newton’s Backward formula, find cos750 if 71

OMTEX CLASSES “THE HOME OF SUCCESS” x0

0

30

60

9 0

cos x0

1

0. 8

0. 5

0

2. Evaluate: -221+x+x2+x3dx; dividing the interval [-2,2] into 4 equal parts by trapezoidal rule.

3. Find Δ2 f(x) = x3 + 3x + 5 if h = 1. 4.

Given h = 1. F(x) = x(x-1)(x-2) find Δf(2).

5. Find the 19th term of the sequence of 2, 7, 14, 23,34, ……. 6.

Estimate the missing figure x

0

1

2

3

4

Y

1. 5

1. 1

-

0. 6

0. 2

7. Find Δ2 f(x) if f(x) = x2 + x by taking difference interval h = 1. 8. Show that Δ log f(x) = log 1+Δfxfx. 9. Find the sixth and seventh term of 6, 11, 18, 27, 38, …… 10. Estimate the share capital in 2006. Year 200 200 200 200 200 0 1 2 3 4 Share Capital In Thousand

55

70

98

135

Marks

30– 40

40– 50

50– 60

60 – 70

70–80

Number of students

31

42

51

35

31

180

11. The marks of the students are given below.

12. Using Simpson’s 1/3 rule. Calculate 0611+x2dx by taking 7 equidistant ordinates. 13. Find Δ2(x2+5) 14. If f(x) = x2 + 2x – 4 , h = 1 find Δf(x). 15. Find the 15th term of the sequence 8, 12, 19, 29, 42, … 16. Estimate the missing term x

1

2

3

4

5

F(x )

2

5

7

-

3 2

17. If f(x) is a polynomial of second degree and if f(1) = 7, f(2) = 5, f(7) = 5, f(8) = 7, find f(x).

18. Using Simpson’s 3/8 Rule evaluate 04.5ydx. X

0

0. 5

1

1. 5

2

2. 5

3

3. 5

4

4. 5

Y

0

6

1 0

30

5 0

63

7 0

74

8 0

82

72

OMTEX CLASSES “THE HOME OF SUCCESS” 19. Find Δf(x) if f(x) = x(x-1)(x-2)(x-3) by taking h=1.

20. Find Δ23ex. 21. By constructing the forward difference table find the sixth and seventh terms in the sequence 6, 11, 18, 27, 38, ……… 22. Using the data estimate f(5). X 0 1 2 3 4 F(x )

3

2

7

2 4

5 9

23. The profit of a company (in lacs) is given below. Estimate the profit in the 6th year using suitable interpolation. Year 1 2 7 Profit (in lacs)

4

5

5

9 5

24. Evaluate 271x2-1. using trapezoidal Rule by dividing the interval [2,7] into 5 equal parts. 25. Construct a backward difference table X 1 2 3 4 5 6 F(x )

2

7

1 8

2 6

3 5

4 7

26. Find Δ2f(x) if f(x) = 2x2 + 3. 27. Given u2 = 10, u3=18, u4=29, u5=52. Compute Δ2u2+Δ2u3. 28. Evaluate Δ2x3Ex3 29. Given sin450 = 0.70, sin 500 = 0.76, sin 550 = 0.81, sin 600 = 0.86, find sin520, using Newton’s method of interpolation.

30. Apply Simpson’s 38th Rule to evaluate 0611+x taking 6 equal parts in [0,6]. Hence find the value of log 7.

31. Prepare the difference table for y = x3 in [0,6] by taking the difference interval h = 1. 32. If f(x) = x4 find Δ2f(x). 33. Show that f7=f6+Δf5+Δ2f4+Δ3f3+Δ4f3. 34. By constructing forward difference table find 6th and 7th terms of the sequence 3, 11, 31, 67, 131, ……… 35. Using Newton’s Interpolation formula find log25(upto two decimal) given that 36. Newton’s Interpolation formula find log25(upto two decimal) given that X 1 20 30 40 50 0 Log x

1

1. 3

1. 4

1. 6

2. 2

37. Evaluate 021+x4dx using trapezoidal Rule by dividing [0,2] into 4 parts. 38. If f(x) = x2 + x + 1 . Construct a forward difference table with x = 0, 1, 2, 3, 4, 5. 39. If f(x) = 2x3 + 3 find Δ2f(x). 40. If f(x) = ex show that f(x), Δf(x), Δ2f(x), … are in g.p. 41. Find the missing term X 1 2 3 4 5 Y

2

4

8

-

3

73

OMTEX CLASSES “THE HOME OF SUCCESS” 2 42. Find f(4) using Lagrange’s formula. X 1 2 5 7 F(x )

6

1 1

3 8

6 0

43. Using Simpson’s 1/3rd rule evaluate 06ydx if X

0

1

2

3

4

5

6

Y

0.1 4

0.1 6

0.1 7

0.1 9

0. 2

0.2 1

0.2 3

44. Complete the difference table if y0 = 2, Δy0=-1, Δ2y0=0, Δ3y0=3, Δ4y0=-2, where y = f(x) is a polynomial of degree 4.

45. Evaluate Δ3 f(x) if f(x) = x3 – 3x2 if h = 1. 46. Show that f5=f4+Δf3+Δ2f2+Δ3f1+Δ4f1. 47. Given u2 = 13, u3 = 28, u4 = 49, u5 = 76, find Δ3u2 and Δ2u3 without constructing the difference table. 48. Find the number of students who obtained less then 45 marks if Marks 304050607040 50 60 70 80 No. of students

31

42

51

35

31

49. Using Trapezoidal Rule evaluate 03x2dx by dividing the interval [0,3] into 6 equal parts. 50. By constructing the difference table find Δ2 y3 and Δ2y2 if y2=13, y3=28, y4 = 49, y5 = 76. 51. The population of a town is given as Year 199 199 199 199 0 1 2 3 Populatio n

105

107

109

112

52. Estimate f(2) using Lagrange’s formula if X 1 F(x )

3

0

3

1

9

53. Evaluate 0π2f(x)dx by Simpson’s 3/8 Rule if f(0) = 1, f(π/6) = 0.9354, f(π/3) = 0.7906, f(π/2) = 0.7071.

54. Find Δ2 f(1) if f(x) = x(x+1)(x+2), h=1. 55. With usual notation show that ΔE≅Δ and ΔE – 1 ≅∇.

56. Show that Δ3f(x) = f(x+3h) – 3f(x + 2h) + 3f(x + h) – f(x). 57. Without constructing the difference table find Δ2y2, Δ3y2 if y2 = 13, y3 = 28, y4 = 49, y5 = 76. 58. Evaluate Δ3fxif fx= x3- 3x2 by taking h = 1. 59. Evaluate Δ2x3Ex3 60. Show that Δ4 ex = ex(eh – 1)4. 61. Find the missing figure

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OMTEX CLASSES “THE HOME OF SUCCESS” X

1

2

3

4

5

F(x )

2

5

7

X

3 2

62. Find the polynomial for y if X

0

1

2

3

Y

1

0

1

0

63. Evaluate 1101x2dx by dividing the interval [1,10] into 9 equal parts using Simpson’s 3/8th Rule. Also find the error.

64. Show that f6=f5+Δf4+Δ2f3+Δ3f3 65. Find the seventh terms of the sequence 3, 9, 20, 38, 65. … 66. If y(0) = 1, y(1) = -1, y(3) = 10 find the polynomial using Lagrange’s interpolation formula.

67. Evaluate 04x+1012 using trapezoidal Rule if X

0

1

2

3

4

x+101 2

3.162 3

3.316 6

3.464 1

3.605 5

3.741 7

68. Find f(3.5), using Newton’s Backward Interpolation formula from the following table: X 0 1 2 3 4 F(x )

3

6

1 1

1 8

2 7

69. Find f(6) , using Lagrange’s Interpolation formula, given that f(1) = 4, f(2) = 5 , f(7) =5, f(8) = 4.

70. Using Simpson’s (3/8)th Rule, evaluate 06ydx from the table given below. X

0

1

2

3

4

5

6

Y

1

0. 7

0.5 8

0. 5

0.4 5

0.4 1

0.3 8

71. If f(0) = 5, f(1) = 6, f(2) = 10, f(3) = 15, evaluate 03fxdx using Trapezoidal Rule. 72. Construct both the difference tables (i.e.) backward and forward, for the sequence 8, 3, 0, -1, 0, 3.

73. Prove that ΔEfx=E[Δfx]

Regression equation 1.

Obtain the two regression lines X 1 2 3 4 5 Y

2.

5

7

9

1 1

1 3

Obtain the two regression lines for X 6 2 1 4 8 0

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OMTEX CLASSES “THE HOME OF SUCCESS” Y 3.

2

1

8 2

7 8

8

8 6

6

1 6

7 2

4

1 4

4 0

9 1

3

1 2

3 8

8 0

9 5

7 2

8 9

7 4

7

5

9

1 1

1 5

1 4

1 7

3 6

3 5

3 9

3 7

4 1

Find the husband’s age if wife’s age is 20. Wife’s 1 2 2 2 2 2 3 Age 8 0 2 3 7 8 0 Husband ’s Age

9.

6

Find both the lines of regression X 4 4 4 4 4 4 4 6 2 4 0 3 1 5 Y

8.

7

Find the two regression lines. Also find x when y = 10. X 1 7 9 5 8 6 1 1 0 Y

7.

8

Find the two regression equations and hence estimate y when x = 13 & estimate x when y = 10. X 1 1 1 1 9 1 6 4 0 5 1 2 Y

6.

5

The marks in Algebra (x) and in Geometry (y) of 10 students are given. Find both the regression equations and hence estimate y when x = 78 and x if y = 94. X 7 8 9 6 8 7 9 6 8 7 5 0 3 5 7 1 8 8 9 7 Y

5.

1 1

For the following find the regression line of y on x. Also estimate y when x = 4. X 1 2 3 Y

4.

9

2 3

2 5

2 7

3 0

3 2

3 1

3 5

From the following find x when y = 200. X 25 24 29 33 46 39 0 8 8 8 3 3 Y

13 7

14 7

18 4

19 6

27 6

26 0

10. Find both the lines of regression X 1 2 3 4 5 6 Y

2

4

7

6

5

6

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OMTEX CLASSES “THE HOME OF SUCCESS”

DIFFERENTIAL EQUATION 1. Verify that x2+y2=r2 is a solution of the D.E. y=xdydx+ r 1+dydx2. 2. Solve y-3=x3x+1dydx and find the particular solution when x=1 and y=4. 3. 4.

The surface area of a balloon being inflated increases at a constant rate. Initially its radius is 3 units and after 2 seconds its is 5 units. Find the radius after 5 seconds. The rate at which radioactive nuclei decay is proportional to the number of such nuclei that are present in a given sample. Half life period of what substance is 1500 years. Find what percentage of original radioactive nuclei will remain after 4500 years?

5. Solve: x2+ y2dx-2xy dy=0. 6. The population of a town increases at a rate proportional to the population at that time. If the population increases from 40,000 to 60,000 in 40 years, what will be the population in another 20 years? Given 32=1.2247 7. Solve the D.E. x-1y+1dy=-y-1x+1dx. 8. Find the order and degree of the D.E. dydx=131+dydx2. 9. Form the D.E. from y=Acos3x+sin3x by elimination arbitrary constants. 10. Solve: dydx+b2-y2a2-x2=0 11. Solve x-y1-dydx= ex by putting x-y = u. 12. If x items are produced by a manufacturer and ‘c’ is their total cost; also the rate of change of ‘c’ w.r.to x is (24x + 4). If the fixed cost is Rs. 200/- find the cost of producing 10 items.

13. Solve ex.tany.dx-ex-1sec2ydy=0 14. Form the differential equation by eliminating the arbitrary constant from the equation y=acosnx+bsin nx 15. Form the differential equation by eliminating the arbitrary constants from the general solution given by y=Aex+Be-x+Ce3x. 16. The rate of decay of the mass of radioactive substance at any instant is α times its mass at that instant. If α = 10 – 4 , show that the mass of the substance will be less than half its value today, after 10,000 years.

17. The population of a city increases at a rate proportional to the population at that time. If the population of the city increased from 20 lakhs to 40 lakhs in a period of 30 years. Find the population after another 15 years. [Take

2=1.41]

18. Solve x2dydx=xy+y2. 19. Solve the differential equation sinxcosy+cosxsiny dydx=0. 20. Determine the order and degree of the differential equation. d2ydx2+31- dydx2-y=0. 21. Form the D.E. by eliminating the arbitrary constants from the relationy=Acoslogx+ Bsin (logx). 22. Solve the D.E. 2ex+2y dx-3dy=0. 23. Solve the differential equation dydx=4x+3y-12 by putting 4x+3y-1=u. 24. The rate of growth of bacteria is proportional to the bacteria present. If the original number N doubles in 3 hours, find the number of bacteria in 6 hours.

25. The money invested in a company is compounded continuously. If Rs. 100 invested today becomes Rs. 200 in 6 years, show that it will become Rs. 4524.44 at the end of 33 years. (2=1.4142)

26. Solve dydx=2x3y2when, x=3 & y=2. 27. Determine the order and degree of the D.E. d2ydx2+1dydx2=y. 28. Find the particular solution of the differential equation dydx-x2=x2y when x=0 & y=2.

77

OMTEX CLASSES “THE HOME OF SUCCESS” 29. Form the differential equation by eliminating the arbitrary constant from the equation y = ax + b.

30. Form the differential equation by eliminating the arbitrary constants from x2a2+y2b2=1. 31. A person’s assets start reducing in such a way that the rate of reduction of assets is proportional to the square root of the existing assets. If the assets at the beginning were Rs. 10 lakhs & they dwindle (decreases) down Rs. 10,000 after 2 years. Show that the person will

229years. 32. Verify that y=ax2+b is a solution of x.d2ydx2-dydx=0. 33. Solve D.E. ydx-xdy=0. 34. Solve dydx =x+y+12x+2y+1 be bankrupt in

35. Due to internal dispute, a company’s share prices are going down. If the rate of falling of the prices of shares is directly proportional to its price in the market and if the original price of Rs. 12 per unit reduces to Rs. 6 per unit in 4 days, find the price after another 4 days.

36. Solve dydx=3x+y. 37. Form the differential equation of y=ae-x. 38. Solve the differential equation: x2dy+y2-xydx=0. 39. The rate of growth of population of a country at any time is proportional to the sixe of the population at that time. For a certain country, it is found that the constant of proportionality is 0.04. Show that the population of that country will be more than doubled in 25 years.

40. Solve the differential equation dydx=e-2ycosx. 41. Form the differential equation by eliminating the arbitrary constant from the equation y = a(xa)

42. Solve the differential equation x.dydx-yeyx=x2cosx by putting y = ux. 43. The rate of reduction of a person’s assets is proportional to the square root of the existing assets. If the assets dwindle from 25 lakhs to 6.25 lakhs in 2 years, in how many years will the person be bankrupt?

44. Verify that ysecx=tanx+c is a solution of dydx+ y tanx=secx. State particular solution if x = y = 0.

45. Determine the order and degree of the differential equation 5dydx2=10x-1dydx. 46. Solve: dydx=4x-3y3x-2y. 47.x items are produced by a manufacturer and C is their total cost. The rate of change of C w.r.to x is 10x + 5 . If the fixed cost is Rs. 600, find the cost of producing 20 items.

48.Solve the differencial equation dydx=4x+6y-22x+37+3by taking 2x+3y=t. 49. The rate of increase of the population of a city varies as the population at that time. In a period of 40 years, the population increased from 4 lakhs to 6 lakhs. Show that in another 20 years, the population will be 7.3482 lakhs.

Given 32=1.2247

50. Form the differential equation by eliminating arbitrary constants a, b from y = a e bx. 51. Verify that y = A sin 3x + B cos 3x is the general solution of the differential equation d2ydx2+9y=0.

APPLICATION OF DERIVATIVES. 1. 2.

The edge of a cube is 8 cm. but it is wrongly measures as 8.15 cm. Find the consequent error, relative error and the percentage of error in calculating the volume of the cube. Dive 50 into two parts such that their product is maximum.

3. Find for what value of x, f(x) has a maximum, where f(x) = 2x3 – 15x2 + 36x + 10. 4. Find the approximate value of loge5.1 given that loge5=1.609. 5.

The side of a square of size 5 meters is incorrectly measured as 5.11 meters. Find the resulting error in calculation of the area of the square.

78

OMTEX CLASSES “THE HOME OF SUCCESS” 6.

Divide 100 into two parts such that the sum of their squares is minimum.

7. Find the value of x, for which the function f(x) = x3 – 12x + 5 is decreasing. 8. Find approximately e2.1, given e2 = 7.389. 9.

The length of the side of a cube is measured to be 4cm. with a possible error of 0.01 cm. Find the consequent error in the volume of the cube.

10. Show that fx= x+1x has a minimum value at x = 1. 11. Find the value of demand x for which supply y=2x3-12x2-30x+8 is increasing. 12. Find the approximate value of fx=2x3+7x+1 at x=2.001. 13. If there is an error of 0.3% in the measurement of the radius of a spherical balloon, find the percentage error in calculation of its volume.

14. Find the approximate value of 4.14 . 15. Find the radius for maximum volume of right circular cylinder, if sum of its radius & height is 6 m.

16. Find the approximate value of e1.002 where e = 2.71828. 17. The radius of a sphere is measured as 10 cm. with an error of 0.04 cm. Find the approximate error in calculating its volume.

18. Find the approximate value of 257 19. Examine the function f(x) = 2x3 – 9x2 + 12x + 5 for maxima. 20. The total revenue of a firm, when the demand for the goods is D, is given by R = 12 + 36D – D2. If demand is measured as 12 with an error of 0.6, find the approximate error in calculating the revenue. 21. An edge of a cube measures 2 meters with a possible error of 0.5 cm. Find the approximate error in the surface area of the cube.

22. Find approximate value of 417. 23. A manufacturer can sell x items at the rate of Rs. (330-x) each. The cost of producing x items is x2 + 10x +12. How many items must be sold so that his profit is maximum?

24. Find approximate value of (4.1)4. 25. The total cost ‘C’ of producing x items is given by C = x3 – 300x2 + 12x. Find ‘x’ for which the marginal cost is decreasing.

26. To total revenue ‘R’ & the total cost ‘C’ of a firm are given by R = 380x – 3x2 & C = 20x respectively. Where x is the quantity. If there is an error of 0.5% in measuring the quantity, find approximately the consequent error in the calculation of the profit, when the quantity is 10 units.

L.P.P. 1.

2.

3.

4.

The standard weight of a brick has to be at least 5 unit. The raw materials A and B used to manufacture cost Rs. 2 per unit and Rs. 3 per unit. Requirements or raw materials is that it should not contain more than 4 units of A and minimum of 1 units of B. Formulate the problem as L.P.P. to minimize the cost. A factory uses three raw materials A, B, C to manufacture two different products P and Q. The factory has 24 units of A, 12 units of B and 18 units of C. One unit of product P requires 2 units of A, 2 units of B and 4 units of C. One unit of Q requires 3 units of A, 1 units of B and 4 units of C. The profit of P per unit is Rs. 5 and on Product Q per unit is Rs. 7. Formulate it as the L.P.P. problem. Health food X contains 4 units of Vitamins A per gram, 7 units of Vitamin B per gram. Food Y contains 6 units of Vitamin A per gram and 11 units of Vitamin B per gram. They cost 25 paise per gram and 35 paise per gram. The daily requirement of Vitamins A and B is 90 and 130 units. Formulate the L.P.P. A firm manufacturers two products A & B. The product A requires 15 minutes in plant I and 25 minutes in plant II. The product B needs 40 minutes in plant I and 20 minutes in plant II. Plants I

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OMTEX CLASSES “THE HOME OF SUCCESS”

5.

6.

7.

and II can be operated for 660 m and 720 m per day respectively. The profit on product A is Rs. 40 and on Product B is Rs. 25. Formulate the L.P.P. to maximize the profit. Food x contains 6 units of Vitamin A per gram and 7 units of Vitamins B per gram and costs Rs. 2 per gram. Food Y contains 8 units of Vitamin B per gram and 12 units of Vitamin B per gram cost Rs. 3 per gram. The daily minimum requirement of Vitamin A and B are 100 units and 120 units respectively. Formulate the L.P.P. Two different kinds of food A and B are being considered for diet. The minimum weekly requirements of fats, carbohydrates and proteins are 16, 25 & 15 units respectively. One Kg. of food A has 5 units of fats, 15 units of carbohydrates and 8 units of protein. One Kg. of food B has 7 units of fats, 10 units of carbohydrates and 9 units of protein. The price of food A is Rs. 4 per kg. and that of food B is Rs. 3 per kg. Formulate the L.P.P. to minimize the case. The inputs and outputs of two processes are as follows Inputs(unit Outputs(unit s) s) Proce ss

A

B

X

Y

1

5

3

5

8

2

4

5

4

4

The maximum amounts available if A and B are 200 units and 150 units respectively. At least 100 units of X and 80 units of Y are required the profit from processes 1 and 2 are 3000 and 4000 respectively. Formulate the above L.P.P. 8. A manufacturer requires 2 types of radio I and II. I require 2 hrs. in plant A and 3 hrs. in plant B. II requires 3 hrs. in plant A & 1 hr. in plant B. Plant A can operate for almost 15 hrs. a day and B can operate for almost 12 hrs. a day. The manufacturer makes a profit of Rs. 15 on I and Rs. 22 on II. Formulate the L.P.P. 9. A company must product 1000 kg. of a special mixture. The mixture consists of two ingredients A and B. A costs Rs. 18 per kg. and B costs Rs. 20 per Kg. No more than 300 kg. of ingredient A can be used and at least 150 kg. of B is required. Formulate the L.P.P. 10. A toy manufacturer produces two types of toys, dolls and guns, each of which must be processed through two machines A and B. The maximum availability of machines A and B per day are 12 and 18 hrs. respectively. The manufacturing doll requires 4 hrs. in A and 2 hrs. in B. A gun requires 3 hrs. in A and 6 hrs. in B. If the profit per doll is Rs. 20 and that of gun is Rs. 50. Formulate the L.P.P. to maximize the profit. 11. Two different kinds of foods A and B are to be considered. The minimum requirements for fats, carbohydrates and proteins are 9, 24, 12 units respectively. 1kg of A has 2, 16 & 4 units of each. 1 kg of B has 6, 4 & 3 units of each. The cast of a is Rs. 75 per kg. and of B is 60 per kg. Formulate the L.P.P. 12. A manufacturer produces bicycles and tricycles which are processed through two machines A and B. A is available for 120 hrs and 180 hrs. Tricycle requires 5 hrs on A and 3 hrs on B. Profit for tricycle is Rs. 45 for a bicycle is Rs. 65. A bicycle requires 3 hrs on A and 10 hrs on B. Formulate the LPP.

13. A company manufactures two models of cars: model A model B. To stay in business it must produce at least 50 cars of model A per month. It has facilities to produce at most 200 cars of model A and 150 cars of model B per month. The total demand for both models together is at most 300 cars per month. The profit per car is Rs. 4000 for model A and Rs. 3000 for model b. It is required to determine the number of cars of each type to be manufactured so as to get maximum profit. Formulate this problem as a L.P.P.

LIMITS 1. Evaluate limx→π4[cosx-sinxπ-4x] 80

OMTEX CLASSES “THE HOME OF SUCCESS” 2. Evaluate limx→1[3x+1-2x-1] 3. Evaluate limx→3[x2-5x+6x2-9-x3-27x2+x-12] 4. Evaluate limx→0[tanx-sinxx3] 5. Evaluate limx→2[x4-16x2-5x+6] 6. Evaluate limθ→π2[3cosθ+cos3θπ-2θ3] 7. Evaluate limx→a[x8-a8x12- a12] 8. Evaluate limx→π3[3-tanxπ-3x] 9. Evaluate limx→e[logx-1x-e] 10.Evaluate limx→-2[3x2-5kx-11]= -29, find K. 11.Evaluate limx→0[cosec x-cotxx] 12.Evaluate limx→π3[sec3x-83-tan2x] 13.Evaluate limx→2[1x-2-1x2-3x+2] 14.Evaluate limx→0[2x+3x+4x-3x+1x] 15.Evaluate limx→3[logx-log3x-3] 16.Evaluate limx→0[5x-13sin5x.tan3x.log⁡(1+x)] 17.Evaluate limx→0[7x-1sinx] 18.Evaluate limx→2[sin4x-8x2-5x+6] 19.Evaluate limx→0[32x-53x43x- 72x] 20.Evaluate limy→3[1y-3-27y4-3y3] 21.Evaluate limx→0[log10+logx+0.1x] 22.Evaluate limx→0[a+5x-a-5xx] 23. If f(x) = x+2, find limx→2[fx- f(2)x-2] 24.Evaluate limx→7[logx-log7x2-49] 25.Evaluate limx→π2[1-sin3xcos2x] 26.Evaluate limx→0[e8x- e5x- e3x+1cos4x-cos10x]

81