Maths I & Ii Hsc

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LOGIC

1.

2. Express the following in the symbolic form i. 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 “A person is successful only if he an integer. is a politician or he has good Write the verbal statement for the connections”. following. 10.Express the following i. p~q statements in verbal form: ii.~p →q i.p⋀q ii. ~p ⋁q where 4. Using the statements: p:Sacgube is smart. q:Sachin is healthy. P: Kiran passed the examination. 11.Let p: Rohit is tall. q: S: Kiran is sad. Rohit is handsome. And assuming that ‘not sad’ is Write the following happy, represent the following statements in verbal form statement in symbolic form. using p & q. “Kiran failed or Kiran passed as a. ~p⋀(~q) well as he is happy” b. p (~pq) 5. Write the following 12.Give a verbal statement for statements in symbolic form. a. p→~q i. Bangalore is a garden city and b. ~p↔ ~q Mumbai is a metropolitan city. p :Rama is young. ii.Ram is tall or Shyam is q :Rama is intellignet. intelligent. 13.Construct the truth table and 6. Write the following determine whether the statements in symbolically. statement is tautology, i. If a man is happy, then he is rich. contradiction or neither. ii.If a man is not rich, then he is not i. ( p → q) ⋀ (q ⋀ ~q) happy. ii.[ p ⋁ (~ q ⋀ p)] → p 7. Write the following iii.~( p ⋀ q) statements in symbolic form. iv.p → (q → p) i. Akhila likes mathematics but not v.p ⋁ (~q ⋀ p). chemistry. vi.~ ( p ↔ q). ii.IF the question paper is not easy vii.[ p ⋁ (~ q ⋀ ~p)] → p then we shall not pass. viii.( p → ~q) → (q ⋀ ~q) 8. Let p : Riyaz passes B.M.S. ix.[q ⋀ ( p → q)] → p q : Riyaz gets a job. x.~( ~p ⋀ ~q ) r : Riyaz is happy. xi.[~(p ⋁ q) ⋀ p] Write a verbal sentence to 14.Do as directed. describe the following. i. Prove that the following i. p→q r ii.p q~r statements are logically 9. Using appropriate symbols, equivalent: p → q ≡ ~q → ~p translate the following ii.Show that the statements p → q statements into symbolic and ~( p ⋀ ~q) are equivalent. form. iii.Write the truth table for “Disjunction”. Write the 1

OMTEX CLASSES “THE HOME OF SUCCESS” 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. 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)

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 3

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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. limx→0sin25xx25 3. limx→0sin4xx(π) 4. limx→0sinπxx4 5. limx→0sin5x4x54 6. limx→0sin3x2x32 7. limx→0tanxx1 8. limx→0xcotx1 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)

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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 25.limx→1abx-axbx-1 26.limx→0e8x- e5x- e3x+1cos4x-cos10x

27.

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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 27.limx→π33-tanxπ-3x=43 28.limx→π6cosx-3sinxπ-6x 29.limx→π41-tanx1-2sinx 30.limx→π42cosx-11-cotx-12 6

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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. 7. 8.

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 Evaluate limh→0f2+h-f(2)h where fx=1x2+2 -19 Find limh→0f1+h-f(1)h where fx=x+5x+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. 2. If fx=3sinx-12xlog(1+x), for x≠0;is continuous at x=0, find f0. 8

OMTEX CLASSES “THE HOME OF SUCCESS” 3. If fx is continuous at x=0 where f(x) =ax-a-xx, for x≠0; =k for x=0 find k. 4. If fx=7sinx-12xlog(1+5x) for x≠0;is continuous at x=0, find f0. 5. If fx=cos3x-cosxx2 for x≠0;is continuous at x=0, find f0. 6. If fx=1-coskxxtanx for x≠0;is continuous at x=0, find k, if f0=3. 7. If fx=15x-3x-5x+1xtanx, for x≠0;is continuous at x=0, find f0. 8. 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. division. Also, since Mr. A’s son is in A B C D Division I, he cannot be assigned the 1 1 1 6 1 corrections of that division. If the time required in days, for every 2 6 1 0 1 teacher to asses the papers of the 3 2 0 2 1 various divisions is listed below find 4 5 2 1 0 the allocation of the work so as to 1 5 4 1 minimize the time required to 0 7 4 complete the assessment. 1 1 A B C D 5 0 I - 5 2 6 3. A Departmental Store has 4 wormers II 4 5 3 8 to pack their items. The timing in II 6 6 2 5 minutes required for each workers to I 1 6 3 4 complete the packing per item sold is I given below. How should the manager V of the store assign the job to the workers, so as to minimize the total 6. Solve the following minimal time of packing? assignment problem. Book Toy Crocke Catte A B C D s s ry ry 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

I II II I 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 II I I V

1 2 1 2 8

2 0 1 6 9

1 1 2 8 1 5

5 1 4 5 1

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1 7

8. Minimise the following assignment problem. A B C D I II II I I V

2 9 1 0 7

1 3 1 2 2 6

3 6 4 1

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

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

6

7

50

10

8

M2 M3 M4

2 7 1 8 7 4 8

8 8 4 9 2 6 1

61 11 1 87

1 73 71 77

2 5 9 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

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 TERMINA LS T1 T2 T3 T4

3 7 3 5

6 1 8 2

2 4 5 6

6 4 8 3

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5 7 6 2

EX. NO. 2 1. 2. Find the sequence that minimises the total elapsed time, required to complete the following jobs on two machineries. Machine 4 9 8 6 5 Jo A B C D E F G C b M 1

7

2 3 2

7 4 5

M 2

4

6 5 4

3 1 4

6. Solve the following problem for minimum elapsed time. Also state the idling time for each machine. Job 1 2 3 4 5 6

3. Solve the following for minimum elapsed time and idling time for each machine. Jo A B C D E b M 1

5

1 9 3

1 0

M 2

2

6 7 8

4

Machine A

8 3 7 2

5

1

Machine B

3 4 5 2

1

6

Machine C

8 7 6 9

1 0

9

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

8

1 0

6 7

1 1

Machine A

5

7

6

9

5

Machine B

5

6

2 3

4

Machine B

2

2

4

5

3

Machine C

3

6

5

6

7

15

OMTEX CLASSES “THE HOME OF SUCCESS” 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. Type of Chairs

Number to be processed/day

Processing time on Machin

Machin

eA

eB

1 2 3 4 5 6

4 6 5 2 4 3

4 12 14 20 8 10

8 6 16 22 10 2

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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. 17

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? 18

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.

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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

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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 21

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.

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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 23

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 24

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

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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 26

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

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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 28

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. 29

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 30

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 31

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 32

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 33

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

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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. 35

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.

36

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

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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

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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

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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 40

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

-

-

-

-

?

?

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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

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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

43

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.

DIFFERENTIATION (DERIVATIVES) EX. NO. 1. 1. 2. 3. 4. 5. 6. 7. 8. 9.

Differentiate the following with respect to x.

ax2+bx+c 4x3-5x2+8x-1 xa+ax+ex+aa 7x3+4x+sinx-2cosx axb+1x3-logx 5x+1x-xn 5ax+4x3+logx2 13x2+sin-1x+cos-1x 44

OMTEX CLASSES “THE HOME OF SUCCESS” 10.tan-1x+sec-1x 11.x+1x2 12.x1+3x 13.1e-x+15-x

45

OMTEX CLASSES “THE HOME OF SUCCESS”

INDEX NUMBER EX. NO. 1. SIMPLE AGGREGATIVE METHOD 1. Find Index number. 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

46

OMTEX CLASSES “THE HOME OF SUCCESS” 2. Find Index number. Commoditi Prices Prices es in in 1990 2002 (P0) (P1)

A B C D E

12 28 10 16 24

38 42 24 30 46

3. Find Index number. Commoditie Price Price Price s s s s in in in 2006 2000 2003

Trucks Cars Three wheelers Two wheelers

800 176 100 44

830 200 127 43

850 215 115 43

4. Find Index number. 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. Security at 200 200 200 Stock 0 3 6 market P0

P1

P1 47

OMTEX CLASSES “THE HOME OF SUCCESS” A B C D E

160 240 0 800 350 0 150

180 35 550 200 0 600

210 8 850 400 0 220

6. Calculate Index Number. Real 199 199 200 Estate 0 8 6 Area wise

A B C D

100 35 5 12

65 22 7 11

250 75 12 25

7. Calculate Index Number. 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. Security at 198 199 199 Stock 8 1 4 market

A B C D E

P0

P1

P1

650 120 0 530 270 145

350 0 135 0 470 0

700 130 0 200 100 150 48

OMTEX CLASSES “THE HOME OF SUCCESS” 0

505 0 230 0

0

9. Compute the Index Number. 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

49