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CHEMISTRY UNIT-1 ATOMIC STRUCTURE BLUE PRINT Part I

Part II

Part III

Part IV

Total

2x1=2

1x 3 = 3

1x5=5

NIL

10

PART-II THREE MARKS QUESTIONS For questions number: 31 1. Distinguish between a particle and a wave.

2. State Heisenberg’s Uncertainty principle 3. What is the significance of negative electronic energy?

4. What are the significances of de-Broglie wave?

5. Why is He2 not formed?

6. What are the significances of bond order?

7.What are bonding and antibonding molecular orbitals?

8. What are the consequence of intra and inter molecular hydrogen bonding?

9. Give any three importance of Hydrogen bonding?

10. Calculate the de-Broglie’s wavelength of a particle of mass 3.313 x 10-31kg moving with a velocity of 107ms-1 PART-III FIVE MARKS QUESTIONS SECTION – A For question number: 52 1. Discuss the shape of s,p and d orbitals 2. Explain the formation of oxygen molecule by molecular orbital theory.

3. Give any five postulates of molecular orbital theory.

4. Explain the formation of N2 molecule by using molecular orbital theory.

5. Write short note on hydrogen bonding.

6. Discuss Davisson and Germer’s experiment.

7. Derive de Broglie’s equation.

UNIT-2 PERIODIC CLASSIFICATION

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Part III Part IV

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1 x 5 = 5 09

PART-II THREE MARKS QUESTIONS For question number: 32

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1 Define atomic radius and ionic radius 2. Define electron affinity or electron gain enthalpy.

3. What is screening effect or shielding effect?

4. Define Electro negativity.

5. Define electron affinity 6. Define Ionization energy.

7. Compare the ionisation energy of nitrogen with that of oxygen.

8. Why is the first ionisation energy of Beryllium greater than that of Lithium?

9. Why is electron affinity of fluorine less than that of chlorine?

10. Why first ionisation energy is much greater than second ionisation energy?

11. Electron affinity of noble gasses Be, Mg, N are zero why?

12. If the d (C-Cl) is 1.76 and r(Cl) is 0.99, find the radius of carbon atom 13. Mention the advantages and disadvantages of Pauling’s and Milikan’s electronegativity scale.

14. Define second and third Ionization potentials.

15. Calculate the effective nuclear charge experienced by the 4s electron in K atom.

PART-IV FIVE MARKS QUESTIONS For question number : 64a 1. Explain Pauling’s method to determine ionic radii.

2. Explain the Pauling scale for the determination of electronegativity. Give the disadvantage of Pauling scale.

3. Explain the various factors that affect electron affinity.

4. Explain any three factors which affect the ionisation energy.

5. How do electro negativity values help to find out the nature of bonding between atoms?

UNIT-3 p-BLOCK ELEMENTS BLUE PRINT Part I

Part II

Part III

Part IV

Total

1x1=2

2x3=6

NIL

1x5=5

12

PART-II THREE MARKS QUESTIONS For question number: 33 and 34 1. What is inert pair effect?

2. What is burnt alum?

3. Prove that P2O5 is a powerful dehydrating agent.

4. Prove that H3PO3 is triprotic or bibasic. Explain 5. Write the structure of phosphoric acid 6. How is potash alum prepared?

7. Write about the Holme’s signal.

8. Write the use of silicones 9. Write a note on plumbo solvency.

10. What is meant by smoke screen?

11. Write the uses of Neon.

12. Write the uses of Helium.

13. Write the uses of Fluorine.

14. Illustrate the oxidizing power of fluorine.

15. Why is HF not stored in silica or glass bottles?

Write its equation.

16. Draw the electron dot formula of Phosphorous acid and Phosphorous pentachloride.

17. Prove that PH3 is a powerful reducing agent.

18. Write the use of Phosphoric acid.

19. How is PCl5 hydrolyzed?

20. What are interhalogen compounds? How are they named? Give one example. Give the preparation of any one.

18. Why He-O2 mixture is used by deep sea divers?

19. What do you mean by inert pair effect?

20. Give the preparation of XeF4. What is its shape?

21. What is the action of heat on phosphine?

22. What are silicones?

24.Give the electron dot formula of i. PCl3 ii.

PCl5 iii. H3PO4 iv.H4P2O7 PART-IV FIVE MARKS QUESTIONS For question number : 64b 1. How is lead extracted from its ore?

2. How is fluorine isolated from their fluorides by

5

Dennis method?

3. Discuss the structure of interhalogen compounds of AX3 and AX5 type 4. Write note on halogen hydrides.

5. What are silicones? Mention the uses of silicones.

6. Illustrate: (i) tribasic nature of orthophosphoric acid (ii) reducing property of phosphorus acid 7. How noble gases are isolated from air by Ramsay Rayleighs method? Describe in detail.

8. How does Fluorine differ from other halogens?

9.Describe in detail how noble gases are isolated by dewar’s method.

UNIT-4 d-BLOCK ELEMENTS

BLUE PRINT Part I

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2x1=2 2x3=6

Part III

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1 x 5 = 5 Problem = 5 18

PART-II THREE MARKS QUESTIONS For question number: 35 and 36 1. Why are Zn2+ salts white while Ni2+ salts are coloured?

2. Why transition metal ions are coloured?

3. Why do transition elements form alloys?

4. Why do d-block elements have variable oxidation states?4 5. Most of the transition metals and their compounds have catalytic activity why?

6. Why do transition elements form complexes?

7. A substance is found to have a magnetic moment of 3.9 BM. How many unpaired electrons does it contain.

8. How is chrome plating done.

9. What is the action of zinc on hot NaOH solution?

10. What happen when KI solutions is added to an aqueous solution of copper sulphate?

11. Explain why Mn2+ is more stable than Mn3+ 12. What is the action of heat on copper ?

13. What is blue vitriol? How is it prepared?

14. Show that K2Cr2O7 is a powerful oxidizing agent.

15. What is aqua regia? Give the reaction of Au with aqua regia.

16. What is the reaction of CuSO4 with KCN? 17. What is the action of heat on copper sulphate

crystals? Write the equation.

18. What is spitting of silver and how is it prevented?

19. Give any two evidences for the oxidizing nature of potassium dichromate.

20. Write the reaction of Ag with a) Dil HNO3 b) Conc. HNO3 21. What is purple of Cassius? How is it prepared?

22. What is Lunar caustic? How is it prepared?

23. Write a short note on Blister copper.

24. What is philosophers wool?

PART-III FIVE MARKS QUESTIONS SECTION – A For question number: 53 1. Explain the extraction of silver from its chief ore (Argentite) 2. Explain aluminothermic process involved in the extraction of chromium.

3. Explain how potassium dichromate is extracted from chromite ore.

4. How is zinc extracted from its chief ore?

5. How is gold extracted from its ore?

Problem 1. A compound of chromium, in which chromium exists in +6 oxidation state. Its chief ore (A) on roasting with molten alkali gives compound (B).This compound on acidification gave compound C. Compound C on treatment with KCl gave compound D. Identify the compounds A,B,C and D. Explain with proper chemical reactions.

2. An element (A) belong to group no 11 under periodic no 4. (A) is reddish brown metal (A) reacts with HCl in the presence of air and gives compound (B) . (A) also reacts with concentrated HNO3 to give compound (C) with the liberation of NO2 identify (A) , (B) and (C). Explain the reaction.

UNIT-5 f-BLOCK ELEMENTS

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2 x 1 = 2 NIL

Part III

Part IV

1 x 5 = 5 NIL

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PART-III FIVE MARKS QUESTIONS SECTION – A For question number: 54 1. Mention the oxidation state and any three uses of lanthanides.

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2. Describe the extraction of lanthanides from monazite sand.

3. Compare the points of similarities and differences between lanthanides and actinides 4. What is lanthanide contraction? Discuss its causes and any two consequences.

UNIT-6 CO-ORDINATION AND BIO - COORDINATION COMPOUNDS

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PART-III FIVE MARKS QUESTIONS SECTION – A For questions number: 55 1. Give the postulates of valence bond theory.

2. Write note on (a) Hemoglobin (b) Chlorophyll 3. Give the postulates of Warner’s Theory of Coordination.

4. For the complex K4 [Fe(CN)6], [Cu(NH3)4] SO4 mention. (i) Name (ii) Central metal iron (iii) Ligands (iv) Coordination number (v) IUPAC name 5. [Ni(CN)6]2- is square planar where as [Ni(NH3)4]2+ is tetrahedral. Explain.

6. Explain the coordination and Ionization isomerism with suitable example.

7. Write note on (a) Chelates (b) optical isomerism.

PART-IV FIVE MARKS QUESTIONS For question number: 65a 1. For the complexes K4[Fe(CN)6], [Cu(NH3)4] SO4 mention a) IUPAC name b) Central metal ion c) Geometry of the complex d) Ligand e) Co-ordination number 2. How is chlorophyll important in environmental chemistry? Mention its function.

3. Explain the co-ordination isomerism and ionisation isomerism with example.

4. [Ni(CN)4]2- is diamagnetic whereas

[Ni(NH3)4]2+ is paramagnetic. Explain 5. [Fe(CN)6]4- is diamagnetic whereas [FeF6]4 is paramagnetic. Explain 6. Apply VB theory to find out the geometry of [Ni(NH3)4]2+ and calculate its magnetic moment.

7. Dive the functioning of Heamoglobin.

UNIT-7 NUCLEAR CHEMISTRY

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1 x 1 = 1 1x 3 = 3 NIL

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1 x 5 = 5 09

PART-II THREE MARKS QUESTIONS For question number: 37 1. What is the Q value of a nuclear reaction?

2. What is nuclear fusion?

3. What is spallation reaction?

4. Give any three differences between chemical reactions and nuclear reactions.

5. Calculate the number of α and β particles emitted when 90Th 232 nucleus is converted into 82Pb 208 6. Explain the principle behind the Hydrogen bomb.

7. What is binding energy of nucleus? How will you calculate give example.

8. The decay constant for 6C14 is 2.31 x 10-4 year-1 calculate the half-life period.

9. What is radio carbon dating?

10. Calculate the decay constant of U238 the t½ = 140 days?

11. Calculate the disintegration constant in seconds if half-life period is 100 seconds.

12. Define radioactivity and half life period.

13. Half-life period of 79Au198 nucleus is 150 days. Calculate its average life.

14. What are the nuclear reactions taking place in sun and stars?

15. How the mechanism of photosynthesis in plants is studied with the help of radioactive isotopes?

16. Write any five radioactive isotopes and its uses.

17. What is mass defect?

PART-IV FIVE MARKS QUESTIONS For question number: 65b

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1. Explain the principle underlying the function of hydrogen bomb.

2. Explain the nuclear reactions that take place in sun.

3. Write a note on radio carbon dating.

4. Explain Nuclear fusion reaction.

5. How are radioactive isotopes useful in medicine?

6. Explain nuclear fission reaction with an example.

7. How do Nuclear fusion reaction differ from nuclear fission reaction?

UNIT-8 SOLID STATE BLUE PRINT Part I

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1x1=1

1x 3 = 3 NIL

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1 x 5 = 5 09

PART-II THREE MARKS QUESTIONS For question number: 38 1. Calculate the number of atoms in i. simple cube ii. FCC iii.BCC iv. edge centred 2. Write any three properties of crystals.

3. State Bragg’s law.

4. Write the significance of Bragg’s equation 5. What is vitreous state?

6. What are superconductors?

7. What is Schottky defect?

8. Write a note on Frenkel defect.

9. What is meant by superconducting transition temperature?

10. Write a note on metal excess defect.

11. Give two applications of super conductor.

12. Sketch the a) simple b) face-centered and c) Body centered cubic.

13. Write a note on molecular and metallic crystals.

14. Write the structure of Rutile.

15. what are ionic crystals write its characteristics.

PART-IV FIVE MARKS QUESTIONS For question number: 66a 1. Explain Bragg’s spectrometer method.

2. Write the properties of ionic crystals.

3. Explain the nature of glass.

4. What are superconductors? Write their uses.

5. Explain Schottky and Frenkel defects.

UNIT-9 THERMODYNAMICS-II

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2 x 1 = 1 1x 3 = 3 1 x 5 = 5 NIL

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PART-II THREE MARKS QUESTIONS For question number: 39 1. What is entropy? What is its unit?

2. When does entropy increase in a process.

3. State Trouton’s rule.

4. Give the Kelvin – Planck statement of second law of thermodynamics.

5. State Clausius statement of second law of thermodynamics.

6. What types of liquids or substances deviate from Trouton’s Rule?

7. What is Gibb’s free energy?

8. Write about the spontaneity of reaction based on entropy.

9. Give the entropy statement of second law of thermodynamics. Mention the unit of entropy also.

10. What standard entropy change of formation.11. How is efficiency of a machine determined?

12. Calculate the change of entropy for the process, water (liquid) water (vapour 373K) inbolving ∆H vap = 40850J Mol-1 at 373k.

13. What is entropy change of an engine that operates 1000 C when 453.6K cal of it supplied to it.

14. Calculate the maximum % efficiency possible from a thermal engine. Operating between 1100 C and 250C. PART-III FIVE MARKS QUESTIONS SECTION – B For question number: 56 1. What are characteristics of entropy?

2. Write the various statements of second law of thermodynamics.

3. Prove that ∆G = ∆H - T∆S. 4. Mention the characteristics of free energy.

UNIT-10 CHEMICAL EQUILIBRIUM-II

BLUE PRINT

8

Part I

Part II

2 x 1 = 2 1x 3 = 3

Part III

Part IV Total

1 x 5 = 5 NIL

10

PART-II THREE MARKS QUESTIONS For question number: 40 1. Define rate constant.

2. Dissociation of PCl5 decreases in presence of increase in Cl2 why?.

3. What is the role of catalyst in chemical equilibrium?

4. What is the effect of change of temperature in equilibrium?

5. Why do equilibrium reactions referred to as dynamic equilibrium?

6. What is the relationship between formation equilibrium constant and dissociation constant?

7. What is equilibrium constant?

8. What is reaction quotient? How is it related to equilibrium constant?

9. Give the KP and KC values for the formation of HI and PCl3?

10. State Le Chatelier’s principle.

11. The equilibrium constant KC for A(g) | B(g) is 2.5 x10-2. The rate constant of the forward reaction is 0.05 sec-1. Calculate the rate constant of the reverse reaction.

12. Write the equilibrium constants KC for the following reactions: (i) H2O2(g) H 2O(g) + ½ O2(g) (ii) CO(g) + H2O(g) CO2 (g) + H2(g) 13. In the equilibrium H2 + I2 2HI, the number of moles of H2, I2 and HI are 1, 2, 3 moles respectively. Total pressure of the reaction mixture is 60 atm. Calculate the partial pressures of H2 and HI in the mixture PART-III FIVE MARKS QUESTIONS SECTION – B For question number: 57 1. Derive the relation KP = KC (RT)ng for a general chemical equilibrium reaction.

2. Derive the expressions for KC and KP for the decomposition of PCl5.

3. Apply Le Chatelier’s principle to contact process for the manufacture of SO3.

4. Apply Le Chatelier principle of Haber’s pro-

cess for the manufacture of ammonia.

5. Derive the expression for KC and KP for the formation of HI. 6. The dissociation equilibrium constant of HI is 2.06 x 10-2 458K. At equilibrium the concentrations of HI and I2 are .36 M and .15 M respectively . What is the equilibrium concentrations of H2 at 458K.

UNIT-11 CHEMICAL KINETICS-II

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1x1=1 2x3=6 1x5=5

Part IV Total NIL

11

PART-II THREE MARKS QUESTIONS For question number: 41 & 42 1. What is half-life period?

2. Define order of a reaction.

3. What is Pseudo first order reaction? Give an example 4. Derive an equation for the half-life period of a first order reaction.

5. What are simple and complex reactions?

6. Write the Arrhenius equation and explain the terms.

7. What are opposing reactions? Give an example.

8. What is activation energy?

9. What are parallel reactions? Give an example.

10. What is zero order reaction?

11. Draw the potential energy diagram for activation energy.

12. Write a note on consecutive reactions.

13. Show that for a first order reaction time required for 99% completion is twice the time required.

14. The initial rate of a first order reaction is 5.2 x 10-6 mol lit-1 s-1 at 298 K, when the initial concentration of the reactant is 2.6 x 10-3 mol lit-1. Calculate the first order rate constant of the reaction at the same temperature.

15. The half-life period of a first order reaction is 20 mins. Calculate the rate constant.

16. The rate constant of a reaction is 1.54 x 10-3 sec. Calculate t½?

PART-III FIVE MARKS QUESTIONS

SECTION – B

For question number: 58

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1. State the characteristics of First order reaction.

2. Explain the experimental determination of rate constant for decomposition of H2O2 in aqueous solution.

3. Explain the experimental determination of rate constant of acid hydrolysis of methyl acetate.

4. State the differences between simple and complex reactions.

5. If 30% of first order reaction is completed in 12 minutes. What percentage will be completed in 65.33 minutes?

6. If first order reaction is 75% complete in 100 minutes. What are the rate constant and Half life period of the reaction

UNIT-12 SURFACE CHEMISTRY

BLUE PRINT BLUE PRINT Part I

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3 x 1 = 3 1x 3 = 3 NIL

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1 x 5 = 5 11

PART-II THREE MARKS QUESTIONS For question number: 43 1. What is homogeneous catalysis? Give an example.

2. What is heterogeneous catalysis? Give an example.

3. What are negative catalyist?

4. What are active centres?

5. What is catalytic poison? Give an example.

6. What are promoters? Give an example.

7. Write a note on auto-catalyst.

8. What are emulsions?

9. What is electro dialysis?

10. What is induced catalysis?

11. Why a colloidal system of gas in gas does not exist?

12. What is electrophoresis?

13. Who is electro osmosis?

14. What is Tyndall effect?

15. What is the reason for the stability of colloids?

16. What are lyophilic and lyophobic colloids?

PART-IV FIVE MARKS QUESTIONS For question number: 66b 1. Write any three methods for the preparation of colloids by dispersion methods.

2. Write briefly on intermediate compound formation theory of catalysis with an example.

3. Write note on purification of colloids by differ-

ent methods.

4. Write briefly about the adsorption theory of catalysis.

5. Write any two chemical methods for the preparation of colloids.

6. What is electro-osmosis? Explain.

7. Distinguish physical adsorption and chemical adsorptions.

UNIT-13 ELECTROCHEMISTRY-I

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1 x 1 = 1 1x 3 = 3 NIL

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1x5=5

14

70th Problem = 5

PART-II THREE MARKS QUESTIONS For question number: 44 1. State Ostwald dilution law.

2. State Faraday’s first and second laws of electrolysis.

3. Define equivalent conductance. Give the equation for it.

4. Define molar conductance.

5. What is Kohlrausch’s law?

6. What is meant by common ion effect?

7.Define ionic product of water.

8. Why phenolphthalein is not a suitable indication for the titration of strong acid and weak bases?

9. What are pH indicants?

10. What are buffer solutions?

11. What are intrinsic and extrinsic semiconductors?

12. The mass of the substance deposited by the passage of 10 ampere of current for 2 hours 40 minutes and 50 seconds is 9.65 g. Calculate the electrochemical equivalent.

PART-IV FIVE MARKS QUESTIONS For question number: 67a 1. Write the postulates of Arrhenius theory of electro lytic dissociation.

2. Explain the Buffer action with an example. 3 .

Explain quinonoid theory of indicators.

4. How will you select the indicators for titration? Explain with titration curve.

5. Derive and Explain Ostwald’s dilution law. 6 .

10

Derive Henderson equation.

7. Explain Ostwald’s theory of indicators.

8. Whatis the pH of a solution containing .5M propionic acid and .5M sodium propionate?

The Ka of propionic acid is 1.34 x 10-5.

9. Calculate the pH of .1M acetic acid solution dissociation constant of a acetic acid is 1.8 x 10 -5.

10. 0.1978g of copper is deposit by a current of 0.2 ampere in 15minutes. What is the electro chemical equivalent of copper?

UNIT-14 ELECTRO CHEMISTRY-II

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NIL

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1x5=5

1x5=5

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PART-III FIVE MARKS QUESTIONS AND PART-IV FIVE MARKS QUESTIONS For question number: 59 and 67b 1. Write an account on cell terminology.

2. How is sytandard hydrogen eltrode construc ed? Explain its function.

3. Derive the Nernst equation. or) Establish a re lation between free energy and e.m.f.

4. Write a note on IUPAC convention of repre sentation of cell.

5. How will you determine the emf of a half cell?

6. Describe Daniel cell.

7. What is the potential of a half – cell consisting zinc electrode in 0.01 M ZnSO4, solution at 250C and E0 = 0.7862?

8. Calculate the equilibrium constant for 2 Ag+ + Zn Zn2+ = 2Agr (E0 = 1.56 V) 9. Calculate the e.m.f of zinc-silver cell at 250C when [Zn2+ ] = 0.10 M and [ Ag+] = 10M. 10. Calculate the standard e.m.f and standard free energy change of the following cell Zn | Zn2+ || Cu2+ | + Cu 11. The emf of the half cell Cu2+ (aq) / Cu (S) containinig ) 0.1 M Cu2+ solution is + 0.30 1V. Calculate the statdard emf of the half cell

UNIT-15 ISOMERISM IN ORGANIC CHEMISTRY

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PART-II THREE MARKS QUESTIONS For question number: 45 1. Give the structure of cis-trans isomers of 2-Pentane.

2. How will you differentiate Cis-trans isomers?

3. Differentiate diastereometer from enantiomer.

4. Label the following as E, Z isomers.

5. What are types of isomerism in organic compounds?

6. What are optical isomers? Give example.

7. What is racemic mixture? Give example.

8. Mesotartaric acid is optically inactive. Why? 9. What are the conditions for a compound to be optically active?

10. Distinguish between racemic and meso forms.

11. Draw the possible isomers of formula C6H4Cl2.

12. What are enantiomers? Give example.

PART-IV FIVE MARKS QUESTIONS For question number: 68a 1. Discuss the optical isomerism in tartaric acid.

2. Write a short account on cis-trans isomerism.

3. Distinguish enantiomers from diastereomers.

Give an example each.

4. Describe the conformations of cyclohexanol.

Comment on their stability.

5. Explain internal and external compensation with suitable examples.

6. Distinguish racemic form from Meso form with suitable example.

7. Out line isomerism in 1,3 – Butadiene.

UNIT-16 HYDROXY DERIVATIVES

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70th problem = 5 12

PART-II THREE MARKS QUESTIONS

11

Total

For question number: 46 & 47 1. Write a note on Kolbe’s reaction.

2. Explain Dow’s process.

3. Alcohols cannot be used as a solvent for Grignard reagents. Why? 4. How can the consumption of alcohol by a person be detected?

5. How is TNG prepared from glycerol?

6. What happens when glycerol reacts with KHSO4?

7. Why is glycol more viscous than ethanol?

8. Why Glycol is hygroscopic in nature?.

9. How will you convert C2H5OH to C2H5OC2H5?

10. How is glycerol synthesized from propylene?

11. How is terylene prepared from ethylene glycol?

12. Write the dye test for phenol.

13. How will you convert phenol to phenolphthalein?

14. Give chemical test to distinguish between propan-2-ol and 2-methyl propan-2-ol.

15. Give a brief account on coupling reaction of phenol with benzene diazonium chloride.

16. Give the reaction, where Glycerol is acting as a catalyst in the preparation of Formic acid.

17. Give the uses of Glycol.

18. What is saponification reaction?

19. How will you get benzene from phenol?

20. Give brief account of coupling reaction.

21. How is allyl alcohol obtained from glycerol?

22. Give the uses of Glycerol.

23. What kind of Nucleophilic substitution taking place, when SOCl2 reacts with alcohol?

Give the reaction.

24. How Bakelite is prepared from Phenol?

25. Phenol is insoluble in NaHCO3 solution but acetic acid is soluble. Give reason.

26. Give any three uses of benzyl alcohol 27. What happens when ethylene reacts with alkaline KMnO4?

28. How is ethylene glycol converted into dioxan?

29. Write Lederer’Manasse reaction.

30. Write a note on sehotten’boumann reaction.

UNIT-17 ETHERS

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2 x 1 = 2 NIL 1 x 5 = 5 NIL 07 PART-III FIVE MARKS QUESTIONS For question number: 60 1. Discuss the isomerism exhibited by ethers.

2. Give any three methods of preparing diethyl ether.

3. Give any two methods of preparation of anisole and explain the reaction of HI with anisole.

4. How do ethers react with HI? Give the significance of the reaction.

5. Distinguish between the anisole and diethyl ether.

6. Explain the Reaction of ether with PCl5, one mole of HI, Excess of HI. 7. How do ethers react with i.Cl2 in the absence of light ii. concentrated acids.

UNIT-18 CARBONYL COMPOUNDS

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1 x 1 = 1 1 x 3 = 3 1 x 5 = 5 70th problem = 5 14 PART-II THREE MARKS QUESTIONS Question number: 48 1. Give the IUPAC names for the following: a) Crotonaldehyde b) Methyl n-propyl ketone c) Phenyl acetaldehyde 2. Explain haloform reaction with an example.

3. Write two tests to identify aldehydes.

4. How is acetophenone prepared by Friedel – Crafts method?

5. How is urotropine prepared? Mention its use.

6. What is Rosenmund’s reduction? What is the purpose of adding BaSO4 in it?

7. What is formalin? Write its use.

8. What is Popott’s Rule?

9. How Mesitylene is prepared?

10. Write note Clemmension Reduction?

11. How will you prepared Benzhydrol?

12. What is the reaction of acetaldehyde with conc. H2SO4? Give the use of the product formed.

13. Write the reaction and acetaldehyde with Tollen’s reagent and Fehling’s solution.

12 |

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Blue Print

14. Benzaldehyde gives Cannizaro reaction but acetaldehyde does not. Why? 15. How will you prepare Malachite green from benzaldehyde?

16. Write note on Wolff-Kishner reduction.

17. Give the reaction of acetone with CHCl3?

18. What is the reaction of acetone with dry HCl? 19. Give the test to identify salicylic acid?

20. Give any five reactions of acetone which are different from aldehydes.

21. Give any two methods of preparation of benzo phenone.

PART-III FIVE MARKS QUESTIONS For question number: 61 1. How is acetone converted to (i) Mesitylene (ii) Mesityl oxide 2. How is acetone converted to (i) Phorone and (ii) Isopropyl alcohol?

3. Clauisen – Schmith reaction Mechanism 4. Write the mechanism of crossed aldol condensation of aceton.

5. Write note on (i) Perkins reaction (ii) Stephen’s reaction.

6. Write note on Cannizaro reaction mechanism.

7. How will you prepare a) acetophenone from benzene and b) benzoin from benzaldehyde.

8. Illustrate the reducing properties of acetaldehyde with example.

9. Explain clemmenson reduction.

PART-IV FIVE MARKS QUESTIONS 1. An organic compound (A) of molecular formula C7H6O is not reduced by fehling’s

solution but will undergo Cannizaro reaction.

Compound (A) reacts with aniline to give (B). Compoud (A) also reacts with Cl2 in the presence of caltalyst to give compound (C). Identify (A) ,(B) (C) and explain the reaction.

2. An organic compound (A) of molecular formula C2H4O is reduces Tollen’s reagent.

(A) treatment with HCN gives Compound (B).

Compound (B) an hydrolysis with an acid gives (C) With molecular formula C3H6O3. Compoud (C) is optically active.

Compound (C) on treatment with fentons reagent gives compound (D) with molecular formula C3H4O3. Compound. (C) and compound (D) give effervescence

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with NaHCO3. Identify (A) ,(B) ,(C) and (D) and the explain the reaction.

UNIT-19 CARBOXYLIC ACIDS

BLUE PRINT Part I

Part II

Part III

Part IV

Total

1x1=1

1x3=3

1x5=5

1x5=5

14

PART-II THREE MARKS QUESTIONS For question number: 49 1. What is aspirin? How is it prepared?

2. Formic acid reduces Tollen’s reagent but acetic acid does not. Give reasons.

3. Write a note on HVZ reaction.

4. What is meant by esterification reaction? Write the equation.

5. What is the reaction of lactic acid with H2SO4?

6. What are the uses of benzoic acid?

7. Mention the uses of oxalic acid.

8. How is methyl salicylate prepared?

9. What are the uses of Lactic acid?

10. Give the uses of formic acid.

11. Give the source and trivial name of a) HCOOH b) C3H7COOH 12. Account for the reducing property of formic acid.

13. Write two test to identify corboxylic acids.

PART-III FIVE MARKS AND PART-IV FIVE MARKS QUESTIONS For question number: 62 and 68b 1. Give the account for the reducing property of formic acid.

2. Discuss the mechanism of bromination of salicyclic acid.

3. Distinguish between formic acid and acetic acid 4. How is latic acid manufactured in large scale?

How can it be converted into cyclic diester.

5. How is benzoic acid obtained from a) C6H5CH2CH3 b) phenyl cyanide c) Carbon dioxide 6. How are the following conversion carried out a) Lactic acid to lactide b) Succinic acid to succinimide c) Salicylic acid to aspirin 7. How is oxalic acid manufactured from sodium formate.

8. Give the mechanism involved in the esterification of a carboxylic acid with alcohol.

9. Bring the conversion of Benzoic acid to ben-

13

zyl alcohol.

10. Explain the mechanisms of i. Kolbe’s reaction. ii. Trans-esterification.

11. What happens when lactic acid is (i) treated with dilute H2SO4 (ii) treated with PCl5 (iii) oxidised with acidified KMnO4 12. What happens when (1) Oxalic acid is treated with NH3 (2) Benzoic acid is treated with PCl5 13. Explain the reactions of CH3CONH2 with (i) P2O5 (ii) Br2 / NaOH (iii) Hydrolysis by an acid 14. What happens when Lactic acid is (i) Treated with H2SO4 (ii) Oxidised with Fenton’s reagent (iii) Added to PCl5 15. Compare the acidic nature of acetic acid with chloro substituted acetic acid.

UNIT-20 ORGANIC NITROGEN COMPOUNDS

BLUE PRINT Part I

Part II

Part III Part IV

3 x 1 = 3 1 x 3 = 3 NIL

Total

1 x 5 = 5 11

PART-II THREE MARKS QUESTIONS For question number: 50 1. How does Phosgene react with aniline?

2. Write a note on Gattermann reaction.

3. Write note on chloropicrin.

4. What is mustard oil reaction?.

5. How will you convert acetamide to methyl amine? Give equation.

6. What is Gabriel phthalimide synthesis?

7. Write a note on Gomberg Bechmann reaction.

8. Comment on the basic nature of aniline.

9. Explain diazotisation with a suitable example 10. Why is amide more acidic than an amine?

11. Mention the uses of diazonium salt.

12. How nitrobenzene is reduced in alkaline medium?

13. Write a note on electrolytic reduction of nitrobenzene.

14. Compound A is yellow coloured liquid and it is called oil of mirbane A on reduction with Sn/HCl gives B. B answers to carbonylamine test? What are A and B 15. An aromatic organic compound (A) having molecular formula C2H7N is treated with

nitrous acid to give (B) of molecular formula Mù£‚èœ Þ«î£ ñŸÁ‹ Blue Print & ð°F 3 | 3.2.2014 | 13

C2H6O which answers iodoform test.

Identify (A) and (B) and explain the reaction.

16. When benzamide is treated with bromine and alkali gives compound A. Also when benzamide is reduced by LiAlH 4. Compound B is formed. Find A and B. Write the equations 17. An organic compound (A) having molecular formula C2H5 ON is treated with Bromine and KOH gives (B) of molecular formula CH5N. Identify (A) and (B) and Write the equation involved.

18. An organic compound (A) having molecular formula C6H7N gives (B) with HNO2 /HCL at 273K the aqua solution of (B) on heating gives (C) which gives violet colour with nutral FeCl3.

Identify (A) ,(B) and(C).

PART-IV FIVE MARKS QUESTIONS For question number: 69a 1. How does nitrous acid react with primary, secondary and tertiary amines?

2. Distinguish between primary, secondary and tertiary amines.

3. How are the following conversions carried out? (i) Nitro methane to methyl amine (ii) methyl amine to methyl isocyanides (iii) benzene diazonium chloride to biphenyl 4. How is (i) phenol, (ii) chlorobenzene (iii) biphenyl prepared by using benzene diazonium chloride?

5. Write a note on reduction of nitrobenzene under different conditions.

6. Explain the following reactions in aniline i.

Coupling ii.schotten-baumann iii. carbile amine.

UNIT-21 BIO MOLECULES

BLUE PRINT Part I

Part II

Part III

Part IV

Total

2x1=2

NIL

NIL

1x5=5

07

PART-IV FIVE MARKS QUESTIONS For question number: 69b 1. Explain biological importants of lipids.

2. Write a note on amino acids.

3. Discuss the structure of fructose in detail.

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4. What is peptide bond? Illustrate the formation of peptide bond in glycylalanine.

Draw the structure.

6. Prove the structure of glucose.

5. Outline the classification of carbohydrates giving an example for each.

UNIT-22 CHEMISTRY IN ACTION

BLUE PRINT Part I

Part II

NIL

1 x 3 = 3 NIL

Part III

Part IV

Total

1 x 5 = 5 08

PART-II THREE MARKS QUESTIONS For question number: 51 1. What are anti protozoals.?

2. What are anesthetics? Give one example 3. What are antibiotics?

4. In what way are antacids important?

5. Write a brief note on ‘Antiseptic’.

6. What are Chromospheres? Give two examples.

7. Define Chemotherapy. Give two examples of antimalarials.

8. Give the characteristics of dyes.

9. How is Dacron prepared? Give any one of its uses.

10. How is nylon 66 prepared? Give its use.

11. Write note on antioxidants.

12. What are artificial sweetening agents? Give two examples.

13. Write a note on polystyrene.

14. Why the Iodoform and Phenolic solutions are called antiseptic?

15. How does potassium meta bisulphate preserve the food materials PART-III FIVE MARKS QUESTIONS For question number: 63 1. Write a note on anesthetics.

2. Explain briefly on Rocket propellants.

3. Write briefly on Buna rubbers.

4. Write a note on analgesics and antipyretics.

5. What are chromophores and auxochromes?

Give two examples for each.

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ECONOMICS BLUE PRINT

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Part-B

Chapter-8: 3 Marks 10x3=30

Chapter-1 1. What are the main divisions of Economics?

2. Distinguish between free goods and economics goods?

3. Write Adam Smith’s Wealth definition?

4. Define Consumption.

5. Write the properties of wealth?

6. How to calculate Per Capita Income?

Chapter-2 1. Name the important general economic systems?

2. List the basic features of Socialism.

3. What is Opportunity Cost?

4. Is india a mixed Economy?

Chapter-3 1. Define the law of Diminishing Marginal Utility?

2. Define ‘‘Consumer’s Surplus” in the words of marshall?

3. What is meant by Alternatives?

4. Write two approaches of Consumer Demand Theory.

5. Define Law of Equi-Marginal Utility?

Chapter-4 1.

2.

3.

4.

5.

What is demand?

Write a note on Giffen Paradon?

What are the types of Elasticity of demand.

Enumerate the determinants of demand.

Define Supply?

Chapter-5 1. What is Equilibrium Price.

2. Distinguish between change in demand and shift in demand.

3. Differentiate the short period from the long period.

4. Describe Fixed Input?

Chapter-6 1. Define Labour.

2. What is Production Function? and what are its classification?

3. What are the Forms of Capital?

4. What is Land?

5. Who is Enterpreneur?

Chapter-7 1. Define Opportunity Cost?

2. Define Marginal Cost?

3. Mention the relationship between MC and AC. 4. What is money cost?

5. What are Factors affecting Production Cost?

6. Distinguish between long period and short period?

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1. Write notes on Market time?

2. Write the meaning and definition of market?

3. Mention any three benefits of Perfect Competition?

4. What is Monopoly?

5. What are the conditions of Price Descrimination?

6. When the Firm gets Maximum Profits?

Chapter-9 1. What are the assumptions of Marginal Productivity Theory of Distribution?

2. Distinguish between real wages and money wages?

3. What are the three motivities of Liquidity preference?

4. Define Modern Theory of Rent?

5. What is Standard of living theory wages?

6. What are theory of Interest?

7. Distinguish Normal Profit and Super Normal Profit?

8. Write the Marginal Productivity Theory of Profit.

Chapter-10 1. What are the assumptions of Say’s law of markets?

2. What are 3 motivities of liquidity preference theory.

3. Write a note on multiplier?

4. Classify the total expenditure of an economy?

5. What is mean by Consumption Function.

6. Write the specification of the ‘‘Keynes Law of Consumption.

7. What is Investment.

Chapter-11 1. Define Money.

2. Define Monetary Policy.

3. What are the Four Components of Money Supply in India?

4. What is Inflation.

5. What is Inflationary Spiral?

6. What are Dear Money and Cheap Money?

Chapter-12 1.

2.

3.

4.

5.

6.

7.

What is the subject matter of Public Finance?

What are the Kinds of Tax? What are the Cannons of Taxation?

What are the Kinds of Tax. What is Zero based Budget?

What is mean by Budget?

What is Government Non-Tax Revenue?

Part-C

10 Marks

Lesson-2

1. Write a note on Traditional Economy.

2. Explain the salient features of Capitalism.

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3. What are the merits and demerits of a mixed economy?

4. Explain ‘Opportunity Cost” with an example.

Lesson-5

1. Explain the shift in supply with diagram.

2. Explain with a help of diagram how demand and supply excert influence on price in the short period.

3. Describe the flatter long run supply curve.

Money Policy 3. Explain the equation of exchange.

Lesson-12

1. What are main source of tax and non-tax revenue of the state government?

2. Define Budget. Explain the balanced and unbalanced budget.

3. What are the Limitations of Fiscal Policy?

Part-D

Lesson-6

1. Explain the merits and demerits of division of Labour.

2. What are the function of Entrepreneur?

3. Distinguish between laws of returns and returns to scale.

4. Explain different Kinds of Production Function.

5. Write Assumptions of the Law of Variable Proportions.

Lesson-7

1. Explain the meaning of Fixed and Variable factors and costs.

2. Explain the relationship between AR and MR curve.

3. Explain the short run average cost curves.

4. Explain the relationship between SAC and SMC.

Lesson-8

1. Briefly explain the Classification of Markets.

2. What are the methods of Controlling monopoly.

3. Write the advantages of perfect competition.

4. Describe the wastes of monopolistic competition.

Lesson-9

1. Discuss Marshall’s theory of Quasi-rent.

2. Write a note on the subsistence theory of wages.

3. Discuss the Abstinence or waiting theory of interest.

4. Briefly describe the dynamic theory of profits.

5. Explain Schumpeter’s theory of profits.

6. Explain the Wage Fund Theory and the Residual Claimant.

7. Explain the Keynesian theory of Interest.

Lesson-10

1. What are the criticisms of Say’s Law? 2. Describe the consumption function with a diagram.

3. What are the assumptions of Keynes Simple Income determination?

4. Explain Investment Function.

Chapter-1

3x20=60

1. Discuss the relationship betweeen economics and other social science.

2. Examine Marshall’s definition of economics.

3. Discuss the nature and importance of economic laws.

4. Discuss the nature and scope of economics.

Chapter-3

1. Explain the characteristics of human wants.

2. Explain consumer’s surplus with the help of a diagram and bring out its importance and its criticism.

3. Explain the Indifference Curve Approach.

Chapter- 4

1. Discuss the law of demand.

2.Explain the methods of measurement of Price Elasticity of demand in detail.

Chapter-8

1.Define Monopoly? Explain the Price and output determinantion under monopoly.

2.How is the price and output determined in the short run under perfect competition?

3.Explain the advantages and disadvantages of Monopoly?

4.Differentiate the Perfect Competition from Monopoly.

Chapter-9

1.Discuss the Marginal Productivity Theory of Distribution.

2.Examine Recardian Theory of Rent.

3.Explain the Marginal Productivity Theory of Wages.

Chapter-11

1.Describe the Functions of Money.

2.Discuss the Objectives and instruments of Monetary Policy.

3.Discuss the Causes, Effects, and Remedies for Inflation.

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Lesson-11

1. Explain the difficulties of barter system.

2. Explain a) Dear Money Policy b) Cheap

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MATHS UNIT 1 Applications of Matrices and Determinants Part A

Part B

Part C

2 1

Total Marks:26 Section B (6marks) 1. Find the adjoint of the matrix ( 3 -1)

( 2 - 4) 2. Find the adjoint of the matrix A = ( 1 2 ) and verify that result A (adj A) = (adj A ) A =(A) I

3 -5 -1 -2 -2 3. Show that the adjoint of A =

2 1 -2 is 3AT

2 -2 1

(

4

(

1 2 -2 4. Find the inverse of -1 3 0

0 -2 1

(

5 5. If A = 7

(

6.

7.

)

)

)

( )

2 2 -1 T T T 3 and B = -1 1 verify that (AB) =B A

)

-1 2 -2

For A= 4 -3 4 show that A =A-1

4 -4 5 Solve by matrix inverse method. 7x +3y =-1, 2x+y =0 0 1 2 1 8. Find the rank of 2 -3 0 -1

1 1 -1 0

) )

( (

1 2 -1 3 9. Find the rank of 2 4 1 -2

3 6 3 -7 10. Solve the non-homogeneous systems of linear equation by determinants method 4x+5y=9, 8x+10y=18 11. Examine the consistency of system of equations if it is consistant then solve.

X+y+Z= 7, x+2y+3Z=18 y+2Z =6

Examples

-1 2 1. Find the inverse of 1 -4

( )

(

)

( )

1 2 0 -1 2. If A = 1 1 and B= 1 2 verify that (AB )-1 =B-1A-1.

3.

4.

( (

)

Solve by matrix inverse method x+y = 3, 2x +3y =8

1 1 -1 Find the rank of 2 -3 4

3 -2 3

)

3 1 -5 -1 5. Find the rank of 1 -2 1 -5 1 5 -7 2 6.

7.

8.

Solve by determinant method x+2y=3, 2x=4y=8 Examine the consistency of the equations 2x-3y+7z=5, 3x+y-3z=13, 2x+19y-47z=32 State and prove reversal law of inverse by matrix method.

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18

)

Section –C 10 marks

3 -3 4 1. Find the adjoint of the matrix A= 2 -3 4 and verify the result (adj A) = (adj A) A= 1A1I

0 -1 1

3 -3 4 2. Find the inverse of the matrix A = 2 -3 4 and verify that A3= A-1 0 -1 1 3. Solve by matrix inversion method of the system of linear equation x-3y-8z+10=0, 3x+y=4, 2x+5y+6z=13 4. Solve by determinant method 2x-y+z=2, 6x-3y+3z=6, 4x-2y+2z=4 5. Solve 1/x+2/y-1/z=1, 2/x+4/y+1/z=5, 3/x-2/y-2/z=0 6. A small seminar hall can hold 100 chairs. Three different colours (red, blue and green ) of chairs are available. The cost of a red chair is rs.240, cost of a blue chair is rs.260 and cost of a green chair is rs.300. The total cost of chair is rs.25,000 Find atleast 3 different solution if the number of chairs in each colour to be purchased.

7. Examine the consistency of system of equations. If it is consistant then solve x-3y-8z=-10, 3x+y-4z=0, 2x+5y+6z-13=0 8. Discuss the solutions of the system of equations for all values of X+y+z=2, 2x+y-2z=2, x+y+4z=2 9 For what values of k, the system of equations of kx+y+z=1, x+ky+z=1, x+y+ kz=1 have i unique solution ii more than one solution iii No solution Examples 1. Solve by matrix inversion method 2x-y+3z=9, X+y+z=6, x-y+z=6 2. Solve by determinant method X+y+2z=6, 3x+y-z=2, 4x+2y+z=8 3. Show that the equation x+y+z=6, x+2y+3z=14, x+4y+7z=30 are consistant and solve them 4. Investigate for what values of λ,µ the simultaneous equations x+y+z=6, x+2y+2z=10, x+2y+λz =14 hare (i) No solution (ii) a unique solutions and (iii) an infinite number of solution 5. For what value of µ the equations x+y+3z=0, 4x+3y+µ = 0, 2x+y+2z=0 hare a (i) trivial solution (ii) Non trivial solution.

( (

)

UNIT 2. VECTORS Blue print Total Marks:38 Section A 6

Section B Section C 2

2

Section B (6 marks)

r r r r 01. i. If a = i + j + 2k

r r r r r r r r and b = 3i + 2 j − k find a + 3b . 2a − b r r r ii. A force of magnitude 5 units acting parallel to 2i − 2 j + k displaces

(

)(

)

the point of application from (1,2,3) to (5, 3,7). Find the work done.

02. If a and b and unit vectors inclined at an angle θ, then prove that θ aˆ − bˆ (i) cos θ = 1 aˆ + bˆ (ii) tan = ˆ 2

2

2

aˆ + b

r

r

r

r

a = 3i + j − 4k 03. Find the unit vector whose length 5 and which are perpendicular to the vectors r r r r and b = 6ri + 5 j − r2k . r r r r r r r r r r 04. i. If and show that and are parallel. ii. Find the a−d b −c a ×b = c ×d a×c = b ×d area of the parallelogram whose diagonals are represented by and 2i+3j+6K and 3i-6j-2k 05. Find the area of the triangle whose vertices are (3,-1,2), (1,-1,-3) and (4,-3,1) r r r 06. Show that torque about the points A(3, -1, 3) of a force through the point B(5, 2, 4) 4i + 2 j + k r r r is i + 2 j − 8k .

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07. i. Show that vectors a,b,c are coplanar if and only if a+b, b+c, c+a are coplanar. r r r r r r r ii. The volume of a parallelopiped whose edges are represented by-12 12 i + λk ,3 j − k ,2i + j −15 15 k - is 546. Find the value of λ.

08. i. Can a vector have direction angles 30°, 45°, 60° . ii. If the points (λ, 0, 3), (1, 3, -1) and (-5, -3, 7) are collinear then find l 09. Find the angle between the following lines x − 1 y + 1 z − 4 and x + λ = y + 2 z − 4 = = = 2

3

2

6

2

x − 2 y +1 z − 3 10. Find the angle between the line and the plane 3x + 4y + z + 5 = 0. = = 3 −1 −2 11. If A(-1, 4, -3) is one end of a diameter AB of the sphere. x2 + y2 + z2 - 3x - 2y + 2z - 15 = 0, then find the coordinates of B.

13 a. For any vector r,prove that r = ( r . i ) i + ( r . j ) j + ( r . k ) k b. If │a │= 13, │b│= 5, a • b = 60, then find │a x b│.

14 Prove that twice the area of a parallelogram is equal to the area of another parallelogram formed by taking as its adjacent sides the diagonals of the former parallelogram.

15. Show that the vectors 3i – 2j +k, i – 3j + 5k and 2i + j – 4k form a right angled triangle.

16. Find the vector equation of a sphere with centre having position vector 2i – j + 3k and radius 4 units. Also find the equation in cartesian form.

17. (a) Show that the vector i + j + k is equally inclined with the co-ordinate axes. (b) If the sum of two unit vectors is a unit vector, prove that the magnitude of their difference is √3.

18. Prove by vector method the mid point of the hypotenus of a right angled triangle is equidistant from its vertices.

19. Find the vector and Cartesian equation of the line joining the points (1,-2,1) and (0,-2,3).

20 Find the equation of the sphere on the joint of the pairs A and B having positions vectors 2i + 6j –7k and - 2i + 4j - 3k respectively as a diameter.

21 (a)If the points (λ, 0, 3), (1,3,-1) and (-5, -3, 7) are collinear then find λ. (b)Find the angle between 2x - y + z = 4 and x + y + 2z = 4.

22. Forces of magnitutes 3 and 4 units acting in the directions 6i + 2j + 3k and 3i – 2j + 6k respectively act on a particle which is displaced from the point (2,2,-1) to (4, 3, 1).

Find the work done by the forces.

23. Force 2i + 7j, 2i – 5j + 6k, -i +2j –k act at a point P whose position vector is 4i – 3j –2k. Find the moment of the resultant of three forces acting at P about the point Q whose position vector lis 6i + j –3k.

24 Find the vector and cartesian equation of the sphere on the join of the points A and B having position vectors 2i + 6j – 7k and –2i + 4j – 3k respectively as a diameter. Find the centre and radius of the sphere.

Section C (10 marks) 1. Prove that cos ( A + B ) = cosA cosB - sinA sinB by vector method.

2. Prove that sin(A -B) = sinA cos B - cos A sin B by vector method.

3. If a = 2i + 3j – 5k, b = -i + j + 2k and c = 4i – 2j + 3k Verify that (a x b) x c ≠ a x (b x c ) 4. If a = 2 i + 3 j −k, b = −2 i + 5 k, c = j −3 k Verify that a×(b×c ) =( a. c) b − (a. b) c 5. If a = i + j + k ,b = 2 i + k,, c = 2 i + j + k, d = i + j + 2 k Verify (a× b) × (c × d) = [ab d] c − [ab c] d x −1

y +1

z

6. Show that the lines 1 = − 1 = 3 and x − 2 = y − 1 = − z − 1 intersect 1 2 1 and find their point of intersect.

7. Find the vector and cartesian equation to the plane through the point (-1, 3, 2) and perpendicular to the planes x + 2y + 2z = 5 and 3x + y + 2z = 8 8. Find the vector and cartesian equations of the plane passing through the points (1,-2,3) and B(-1,2, -1) and its parallel to the line . x − 2 = y + 1 = z − 1 2

3

4

9. Find the Vector and Cartesian equations of the plane through the points (1,2,3)and (2,3,1) perpendicular to the plane 3x -2y + 4z – 5 = 0.

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10. Derive the equation of the plane in the intercept form.

Examples 11. Prove by vector method Altitudes of a triangle are concurrent.

12. Prove that cos ( A -B ) = cosA cosB + sinA sinB by vector method.

13. Prove that sin(A + B) = sinA cos B + cos A sin B by vector method.

14. Find vector and cartesian equation of the plane passing through the points (2,2,-1) (3,4,2) and (7,0,6).

UNIT 3 Complex Numbers PART A

PART B

PART C

4

2

1

Total Marks: 26

Section B 6 Marks 1+ i n 1. i Find the least positive integern such that 1 -1 =1 ii If (1+i) (1+2i) (1+3i)-------(1+n i) = x+iy show that 2,5,10……. (1+n2) = x2+y2

( )

2. i. Prove that if w3 =1, then ( -1+i√3 )5 +(-1 i√3 )5 =-1 2 2 ii. cos 2θ + sin 2θ)2 (cos3θ- sin 3θ) -3 -6 8 cos 4θ + sin 4θ) (cosθ + i sinθ) 3. Find the real values of x and y for (1+i) x-2i +(2-3i) (y+i) =i 3+1 3-i 4. For what values of x and y, the numbers -3+ix2y and x2+y+4i are complex conjucate of each other 5. Find the square root of -8-6i 6. Prove that the points representing the complex numbers (7+5i), (5+2i), (4+7i) and (2+4i) term a parallelogram.

7. If arg (z-1) =π /6 and arg (z+1) =2π/3 Then prove that 1z1=1 8. P represents the variable complex numbers z Find the locus of p. If 1z -5i = 1Z +5i 9. Solve 6x4-25x3+32x2+3x-10=0. Given that one of the roots of 2-i.

(cos∝+ sin∝)3 10 simplify cos (sinβ+ i consβ)4 11 If cos∝+cosβ+cosγ =0 sin∝ +sinβ+sinγ) prove that i. Cos3∝+ cosβ+3γ =3 cos (∝+B+3y ) ii. Sin 3∝+sin 3β +sin 3γ =3 sin (∝+β+γ) 12. Prove that (1+1√3 )n +(n-i √3 )n = 2 n+1 - cos π5/3 13. If x1/x=2 cosθ prove that xx+1/xx = 2cosnθ and xx-1/xx = zi sinnθ 15. Solve x4+4=0 14. Prove that if co3=1 then 1/1+2w - 1/1+w + 1/2+w = 0 Examples 16. If ( a1+ib1) (a2+ib2)......=An+ibn=A+iB prove that i (a2+b2) (a22+b22)..... (ax2+bn2) = A2+b2 ii tan b1 b2 bn B + tan-1 + tan-1 = KTT+tan-1 a1

a2

an

A

17. Prove that complex numbers 3+3i, -3-3i, -3√3+3√3i are the vertices of an equilateral triangle in the complex plane 18. Find the square root of -7+24i 19. Solve the equation x4-4x2+8x+35=0 If one of its roots is 2+√3i 20. If n is a positive integer , prove that (√3+i )n+( √3-i)n = 2n+1 cos nπ/6 21. State and prove the triangle inequality of complex numbers.

Section –c (10 marks)

z-1 1. P represents the variable complex numbers Find the locus of p, if arg z+3 = π/2 2. If ∝ and β are the roots of the equations x2-2px+ (p2+q2)=0 and tan θ q/y+p show that (y+ ∝)n - (y+b) n = qn-1 = sin θ ∝-β sinn θ 3. If ∝ and β are the roots of x2-2x+4=0 prove that ∝n βn=i2n+1 sin nπ/3 and deduct ∝9-β9 4. If a=cosz∝+isin2∝, b =cos 2β+i sin zβ and c=cos zγ+isinz γ prove that Mù£‚èœ Þ«î£ ñŸÁ‹ Blue Print & ð°F 3 | 3.2.2014 | 21

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i. √abc + 1/√abc = z cos (∝+β+γ) ii. a2b2+c2 /abc = z cos2 (∝+β-γ)

5. Find all the values of (-3-i)2/3 6. Find all the values of ( 1/2 - i √3/2 ) ¾ and hence prove that the product of the values is 1 Examples (z+1)1 7. P represents the variable complex numbers z, find the locus of p is = Re= z+1 =1 8. If ∝ and β are the roots of x2-2x+2=0 and cot θ=y+1 show that ( y+α)x - (y+β)n singθ 9. Solve the equations x4+x5-x4-1=0

sinnθ 10. Find all the values of ( √3+i )2/3

UNIT 4 Analytical geometry PART A PARTB PARTC 4

1

3

Total Marks: 40

PART B (6 marks) 1. The focus of a parabolic mirror is at a distance of 8cm from its centre (vertex). If the mirror is 25cm deep, find the diameter of the mirror.

2. Find the equation of the ellipse if the foci are (=3, 0) and the vertices are (=5,0) 3. Find the equation of ellipse if the centre is (3,1), one of the foci is (6,-1) and passing through the point (8,-1) 4. Find the locus of a point which moves so that the sum of its distances from (3,0) and (-3,0) is 9 5. Find the equation of the hyperbola if centre centre : (1,-2): length of the transverse axis is 8, e=5/4 and the transverse axis is parallel to x-axis 6. Find the equation of the hyperbola if centre: (2,5); the distance between the directrices is 15, the distance between the foci is 20 and the transverse axis is parallel to y-axis 7. Show that the locus of a point which moves so that the difference if its distances from the points (5,0) and (-5,0) is 8 is 9x2-16y2=144 8. Find the equations of the tangent and normal (i) To the ellipse 2x2+3y2=6 at (3,0) (ii) To the hyperbola 9x2-5y2=31 at (2,1) 9. Find the equations of the tangents (i) To the parabola y2=16x, perpendicular to the line 3x-y+8=0 (ii) To the hyperbola 4x2-y2=64, which are parallel to 10x-3y+9=0 10. Find the equation of the two tangents that can be drawn from the point (1,2) to hyperbola 2x2-3y2=6 11. Show that the line x-y+4=0 is a tangents to the ellipse x2+3y2=12 12. Find the equation to the chord of a contact of tangents to the ellipse 2x2+5y2=20 13. Find the angle between the asymptotes of the hyperbola 4x2-5y2-16x+10y+31=0 14. Show that the tangent to a rectangular hyperbola terminated by its asymptotes is bisected at the point of contact 15. Find the equations of the asymptotes of the following rectangular hyperbola xy-kx-hy=0 16. Prove that the tangent at any point to the rectangular hyperbola forms with the asymptotes a triangle of constant area.

Examples 17. Find the equation of the parabola if the curve is open rightward, vertex is (2,1) and passing through point (6,5) 18. The headlight of a motor vehicle is a parabolic reflector of diameter 12cm and depth 4cm. Find the position of bulb on the axis of the reflector for effective functioning of the headlight 19. Find the equation of the ellipse whose one of the loci is (2,0) and the corresponding directrix in x=8 and centricity 1/2 20. The centre of the ellipse is (2,3). One of the foci is ( 8,3). Find the other focus 21. Find the equation of a point which moves so that the sum of its distances from (-4,0) and (4,0) is 10 22. Find the equation of hyperbola whose direction is 2x+y=1 focus (1,2) and eccentricity 3 23. Find the equation of the hyperbola whose transverse axis is parallel to y-axis centre (0,0) length of semi-conjugate axis is 4 and eccentricity is 2 24. Find the equation of the hyperbola whose centre is (2,1) one of the foci is (8,1) and the corresponding directix is x=4 25. Find the equations of the tangents to the parabola y2=5x from the point (5,13) . Also find the points of contact PART C (10 Marks) 1. Find the equations of directrices, latus rectum and lengths of latus rectums of the ellipse x2+4y28x-16y-68=0 Mù£‚èœ Þ«î£ ñŸÁ‹ Blue Print & ð°F 3 | 3.2.2014 22 |

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2. Find the eccentricity , centre, foci, vertices of the following ellipse and draw the diagram : 16x2+9y2+32x-36y=92 3. A kho-kho player in a practice session while running realises that the sum of the distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him if the distance between the poles is 6m 4. A satellite is travelling around the earth in an elliptical orbit having the earth at a focus and of eccentricity ½. The shortest distance that the satellite gets to the earth is 400kms. Find the longest distance that the satellite gets from the earth.

5. The orbit of the planet mercury around the sun is in elliptical shape with sun at a focus. The semi major axis is of length 36 million miles and the eccentricity of the orbit is 0.206. Find (i) how close the mercury gets to the sun ? (ii) the greatest possible distance between mercury and sun 6. Find the eccentricity, centre, foci and vertices of the following hyperbolas and draw their diagrams x2-3y2+6x+6y+18=0 7. Find the equation of the hyperbola if its asymptotes are parallel to x+2y-12=0 and x+2y+8=0,(2,4) is the centre of the hyperbola and it passes through (2,0) 8. Find the equation of the rectangular hyperbola which has for one of its asymptotes the line x+2y-5=0 and passes through the points (6,0) and (-3,0).

9. Find the equation of the rectangular hyperbola which has its centre at (2,1), one of its asymptotes 3x-y-5=0 and which passes through the point (1,-1) Examples 10. Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus rectum for the following parabolas and hence draw their graphs. X2-2x+8y+17=0 11. The girder of a railway bridge is in the parabolic form with span 100ft and the highest point on the arch is 10ft, above the bridge, find the height of the bridge at 10ft to the left or right from the midpoint of the bridge 12. On lighting a rocket cracker its gets projected in a parabolic path and reaches a maximum heights of 4 mts when it is 6 mts away from the point of projection. Finally it reaches the ground 12 mts away from the starting point.

13. Find the angle of projection, Assume that water issuing from the end of a horizontal pipe, 7.5m above the ground , describes a parabolic path. The vertex of the parabolic path is at the end of the pipe, At a position 2.5m below the line of the pipe, the flow of water has curved outward 3m beyond the vertical line through the end of the pipe. How far beyond this vertical line will the water strike the ground ?

14. A comet is moving in a parabolic orbit around the sun which is at the focus of a parabola. When the comet is 80 million kms from the sun, the line segment from the sun to the comet makes an angle or π/3 radians with the axis of the orbit. Find (i) the equation of the comets orbit (ii)how close does the comet come nearer to the sun ? (Take the orbit as open rightward) 15. A cable of a suspension bridge hangs in the form of a parabola when the load is uniformly distributed horizontally. The distance between two towers is 1500 ft, the points of support of the cable on the towers are 200ft above the road way and the lowest point on the cable is 70ft above the roadway.

Find the vertical distance to the cable (parallel to the roadway) from a pole whose height is 122ft.

16. Find the eccentricity, centre, foci, vertices of the ellipse 36x2+4y2-72x+32y-44=0 17. The ceiling in a hallway 20ft wide is in the shape of a semi ellipse and 18ft high at the centre. Find the height of the ceiling 4 feet from either wall if the height of the side walls is 12ft 18. The orbit of the earth around the sun is elliptical in shape with sun at a focus. The semi major axis is of length 92.9 million miles and eccentricity is 0.017. Find how close the earth gets to the sun and the greatest possible distance between the earth and the sun 19. Find the equations and length of transverse and conjugate axes of the hyperbola 9x2-36x-4y2-16y+56=0 20. Find the eccentricity , centre, foci, and vertices the hyperbola 9x2-16y2-18x-64y-199=0 and also trace the curve 21. Find the equation of the hyperbola which passes through the point (2,3) and has the asymptotes 4x+3y-7=0 and x-2y=1.

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UNIT 5. DIFFERENTIAL CALCULUS APPLICATION - I Blue print Total Marks:36 Section A

Section B

Section C

4

2

2

Section B (6 marks)

1. Prove that the curve 2x2 + 4y2 = 1 and 6x2 – 12y2 = 1 cut each other at right angles.

2. Find at what points on the circle x2 + y2 = 13, the tangent is parallel to the line 2x + 3y = 7.

2 3 2 +1, −2 ≤ x ≤ 2 3. Lagrange’s Using Rolle’s theorem find the points on the curve y = x 4.4.Verify functions: Verify Lagrange’slaw lawofofmean meanfor forthe thefollowing following functions:f(x) f(x)= =x x-3 -5x5x-2 -3x, 3x,[1,[1 where the tangent is parallel to x − axis. x x 3 5.5.Obtain the Obtain theMaclaurin’s Maclaurin’sseries seriesfor fore e - 5x2 - 3x, [1, 3] 4. Verify Lagrange’s law of mean for the following functions: f(x) = x x 5. the Obtain the Maclaurin’s series for e 6.6.Obtain Obtain theMaclaurin’s Maclaurin’sSeries Seriesexpansion expansionfor: for:tan tanx,x,−π/2 −π/2< <x <π/2 x <π/2. .

6. Obtain the Maclaurin’s Series expansion for: tan x, −π/2 < x <π/2 .

7.7.Obtain the Obtain theMaclaurin’s Maclaurin’sseries seriesfor forlog log(1(1+ +x).x).

7. Obtain the Maclaurin’s series for log (1 + x). x x 8.8.(i)(i)Determine the y y= =exe(ii) Determine thedomain domainofofconvexity convexityofofthe thefunction function (ii)State StateRolle’s Rolle’stheore theor (ii) State Rolle’s theorem.

8. (i) Determine the domain of convexity of the function y = e lim ⁡���� −�

lim ⁡���� −� 9.9.Evaluate the Evaluate thelimit limitfor for�→�→ 9. Evaluate the limit for 0 0�−���� �−����

1 1

(�()��)−1� −1 10. the 10.Evaluate Evaluate thelimit limitfor forlim lim �→1 10. Evaluate the limit for lim �→1 2 11.

11.Find Findthe theintervals intervalsononwhich whichf is f isincreasing increasingorordecreasing. decreasing.f(x) f(x)= =2020- -x x- -x x2 11. Find the intervals on which f is increasing or decreasing. f(x) = 20 - x - x2 3 3 12. the f(x) 12.Find Find theintervals intervalsononwhich whichf is f isincreasing increasingorordecreasing. decreasing. f(x)= =x x-3 -3x3x+ +1 1 12. Find the intervals on which f is increasing or decreasing. f(x) = x - 3x + 1 13. Find the intervals of concavity and the points of inflection of the following functions 13. the ofofconcavity 13.Find Find theintervals intervals concavityand andthe thepoints pointsofofinflection inflectionofofthe thefollowing followingfunction functi 3 2 f(x) = 2x 3 3 +25x2 - 4x f(x) f(x)= =2x2x+ +5x5x- -4x4x Examples 14. Find the equations of the tangent and normal to the curve y = x Examples Examples 3 at the point (1, 1).

2

15. Find the equation of the tangent at the point (a,b) to the curve xy = c .

16. Obtain the Maclaurin’s series for loge(1 + x) 3 3 14. the 17. Evaluate : lim log(sin x) 14.Find Find theequations equationsofofthe thetangent tangentand andnormal normaltotothe thecurve curvey y= =x xat atthe thepoint point(1,(1,1).1) 2 2 2 X →π/2 (π − 2x) 15. ofofthe 15.Find Findthe theequation equation thetangent tangentatatthe thepoint point(a,b) (a,b)totothe thecurve curvexyxy= =c c. .

18. Find the absolute maximum and minimum values of the function.

16. the series 16.Obtain Obtain theMaclaurin’s seriesfor forloge(1 loge(1+ +x)x) 3Maclaurin’s - 3x2 + 1, -1/2 ≤ x ≤ 4 f(x) = x 3 log(sin 17. : lim - 3x + 2 19. Determine the points of inflection if any, of the function y = x log(sinx)x) 17.Evaluate Evaluate : lim 2 2 2 , −1 ≤ x ≤ 1 20. Using Rolle’s theorem find the value(s) of c. f(x) = √1 - x X →π/2 (π − 2x) X →π/2 (π − 2x) 21. Verify Rolle’s theorem for f(x) = ex sin x, 0 ≤ x ≤ π 18. minimum 18.Find Findthe theabsolute absolutemaximum maximumand and minimumvalues valuesofofthe thefunction. function.

Section – C (10 marks) 3 3 2 2 f(x) + +1, -1/2 ≤ y≤x=≤xx ≤and 4 4suppose that the tangent line at P intersect the curve again at f(x) =x xP-be-3x 1.= Let a3x point on1, the-1/2 curve Q. Prove that the slope at Q is four times the slope P. function y = x3 -3 3x + 2 19. if if any, ofatofthe 19.Determine Determinethe thepoints pointsofofinflection inflection any, the function y = x - 3x + 2 2. Show that the equation of the normal to the curve x = a cos3 θ ; y = a sin3 θ at ‘θ’ is 1 1−−�2�,2 ,−1−1≤ ≤x ≤x ≤1 1 20. theorem the 20.Using UsingRolle’s Rolle’s theorem thevalue(s) value(s)ofofc.c.f(x) f(x)= =√√ x cos θ - y sin θ = a cos 2find θ.find 2 2 3. Rolle’s Find the angle between thef(x) curves and atπ the point of intersection 21. theorem forfor x, x,0y≤0= ≤(xx -≤x2)≤π 21.Verify Verify Rolle’s theorem f(x)= =eyxe=sin xxsin 2 4. If the curves y = x and xy = k are orthogonal then prove that 8k2 = 1 Evaluate : lim (cot x) sin x Section Section– –CC(10 (10marks) marks) x →0 5. Find the intervals of concavity and the points of inflection of the following functions 4 xpoint - 6x2onon 1.1.Let the LetP Pbef(x) bea =apoint thecurve curvey y= =x xand andsuppose supposethat thatthe thetangent tangentline lineatatP Pintersect intersectthe t 6. Find the intervals of concavity and the points of inflection of the following functions curve again at Q. Prove that the slope at Q is four times the slope at P. curve again at Q. Prove that the slope at Q is four times the slope at P. y = 12x2 - 2x3 - x4 3 3 2.2.Show the Showthat thatthe theequation equationofofthe thenormal normaltoto thecurve curvex x= =a acos cos�3 �; y; y= =a asin sin�3 �atat‘�’ ‘�’isi Examples 7.cos A�car travelling west at�.

50 km/hr. and car B is travelling towards north x xcos 2 2�. �- A-yisysin sin� �= =afrom acos cos

2 2 2 2 y y= =(x(x- -2)2) atatthe 3.3.Find between y= thepoint pointofofintersection intersectio Findthe theangle angle between thecurves curves =x xand Mù£‚èœ Þ«î£the ñŸÁ‹ Blue Print y & ð°F 3 and | 3.2.2014 24 | 24 2 2 2 2 4.4.IfIfthe thecurves curvesy y= =x xand andxyxy= =k kare areorthogonal orthogonalthen thenprove provethat that8k8k= =1 1 sinsin x x Evaluate : lim (cot x) Evaluate : lim (cot x) x x→0 →0

at 60 km/hr. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 kilometers and car B is 0.4 kilometers from the intersection?

8. A water tank has the shape of an inverted circular cone with base 2 metres and height 4 metres. If water is being pumped into the tank at a 2m3/min, find the rate at which the water level is rising when the water is 3m.

9. Prove that the sum of the intercepts on the coordinate axes of any tangent to the curve x = a cos4 θ, y = a sin4 θ, 0 ≤ x ≤ π/2 is a constant.

10. Find the angle between the curves y = x2 and y = (x - 2)2 at the point of intersection. Find the condition for the curves ax 2 + by2 = 1, a1x2 + b1y2 = 1 to intersect orthogonally.

11. The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cms. If the area of the printed material on the poster is fixed at 384 cm2, find the dimension of the poster with the smallest area.

12. Show that the volume of the largest right circular cone that can be inscribed in a sphere of radius a is 8/27 (volume of the sphere).

13. Show that of all the rectangles with a given perimeter the one with the greatest area is square.

14. A man is at a point P on a bank of a straight river, 3 km wide, and wants to reach point Q, 8 km downstream on the opposite bank, as quickly as possible. He could row his boat directly across the river to point R and then run to Q, or he could row directly to Q, or he could row to some point S between Q and R and then run to Q. If he can row at 6 km/h and run at 8 km/h where should he land to reach Q as soon as possible ?

UNIT 6 DIFFERENTIAL CALCULUS –APPLICATIONS II PART A

PART B

PART C

2

1

1

Total Marks-18 Section-B ( 6 marks) 1. Use differentials to find an approximate value for 1/10-1 2. The radius of a circular disc is given as 24 cm with a maximum error in measurement of 0.02cm (i) use differentials to estimate the maximum error in the calculated area of the disc (ii) compute the relative error ∂10 and ∂W 3. Find if w = X2+Y2 where ∂u ∂r x=u2-r2, y=2ur 4. Using Euler’s theorem if u is homogenous function of x and y of degree n prove that x∂24 y∂2u ∂u + (n-1) x ∂x2 ∂x∂y ∂x 5. Using Euler’s theorem if r= zeax+by and z is a homogeneous functions of degree n in x and y prove that ∂r +y∂r x = (ax+by+n)V ∂x ∂y Examples 6. Use differentials to find an approximate value for 3 √65 7. Show that the percentage error in the nth root of a number is approximately 1/n times the percentage error in the number ∂u 8. If u= log (tanx+tany+tanz ) prove that = z sin zx x =2 ∂ 9. If u = (x-y) (y-z) (z-x) then show that ux+uy+uz=0 10. If z= yex2 where x=2f and y=1-t then find dz/dt 11. The time of swingT of a pendulum is given by T= k √1 where k is a constant , Determine the percentage error in the time of swing if the length of the pendulum changes from 32.1cm to 32 cm

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Section c -10 marks Use the differentials to find the approximate value for y= 3 √ 1.02 + 4 √ 1.02 Trace the curve y=x3+1 Trace the curve y=2x3 Trace the curve y=x3 Using Euler’s theorem prove that ∂u ∂u x-y x +y 1/2 tan u if u = sin-1 √x+ √ y ∂x ∂y

1.

2.

3.

4.

5.

(

)

∂2u ∂zu 6. Verify ∂x∂y = ∂y∂x + for the functions u= tan-1 ( x/y) 7. Using Euler’s theorem x3+y3 if u= tan-1 x-y prove that ∂u ∂u +y x = sin zu ∂x ∂y

(

)

UNIT 7 INTEGRAL CALCULUS AND ITS APPLICATIONS

dx x2+5x+6 -1 3 2 (sin x ) 2 1-x √

∫ Evaluate ∫

PARTB

PARTC

4

1

2

0

3. Evaluate (i) 4. Evaluate (i)

SECTION B (6 marks)

dx

∫

∫ ∫

1

log 3-x dx 3+x π4 cos82xdx

-1

0

(ii) -π /4

∫

3

0

∫

Log (tanx) dx

8.

0

x3cos3 x dx

x5 e-4x dx

√x √x+√3-x

∫ log (1/x-1) dx x Evaluate ∫ sin /4 dx

6. Evaluate by using properties era π2

0

π4

∫∞

(ii)

5. Evaluate by using properties

7.

Total Marks-30

2

1. Evaluate 1 2.

PART A

1

0

2π

a

0

π/3

∫

dx 1+ √cotx π/3 dx 10. Evaluate by using properties π/6

1+ √tanx 11. Find the area enclosed by the centre y2=x and y=x-2 12. Find the area of the region bounded by y=2x+4, y=1 and y=3 and y-axis 13. Find the area included between the parabola y2=4ax and its latus rectums 14. Find the area of the circle whose radius is a.

9. Evaluate

π/6

∫

15. Evaluate

π/2

∫ f(sinx)f(sinx) + f (cos x)

0

dx

Section –c (10 marks) 1. Find the area of the region bounded by the curve y=3x2-x and the x - axis between x= -1 and x=1 2 2 2. Find the area of the region bounded by the ellipse x /a + y /5=1 between the two latus rectum.

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3.

4.

5.

6.

Find the area of the region bounded by the parabola y2=4x and the line 2x-y=4 Desire the formula for the volume of right circular cone with radius r and height h Find the perimeter of the circle with radius a Find the surface area of the solid generated by revolving the arc of the parabola y2=4ax, bounded by its latus rectum about x-axis 7. Prove that the curved surface area of a sphere of radius r intercepted between two parallel planes at a distance a and b from the centre of the sphere is 2πr (b-a) and hence deduct the surface area of the sphere. (b>a) Examples 8. Find the area of the region common to the circle x2+y2=16 and the parabola y2=6x 9. Compute the area between the curve y= sinx and y=cosx and the lines x=0 and x=π 10. Find the area of the curve y2= (x-5)2 (x-6) i. between x=5 and x=6 i i. between x=6 and x=7 11. Find the length of the curve (x/a)z/3 +(y/a)z/3=1 12. Show that the surface area of the solid obtained by revolving the arc of the curve y = sinx from x=0 to x=π about x-axis is 2π (√2+log (1+√2 ) 13. Find the surface area of the solid generated by revolving the cycloid x= a(t=sin t ), y=a(1+cost) about its base (x-axis)

UNIT 8 DIFFERENTIAL EQUATIONS Blue print

Total Marks:30 Section A

Section B

Section C

4

1

2

Section B (6 marks) 1. Form the differential equation from the following equations y = Ae2x + Be−5x {A, B} 2. Find the differential equation that will represent the family of all circles having centres on the x-axis and the radius is unity.

3. solve :cos2xdy + yetanxdx = 0 4. solve : dy/(dx ) = sin(x + y) 5. solve :dy/dx = y(x − 2y) / x(x − 3y) 6. Find the equation of the curve passing through (1, 0) and which has slope 1 + y/x at (x, y).

7. solve : (1 + x2)dy/dx + 2xy = cosx 8. solve : (y − x)dy/dx = a2 9. solve : (D2 + 14D + 49)y = e−7x + 4 10. solve : (D2 − 2D − 3)y = sinx cosx Examples 11. Form the differential equation from the following equations. y = ex (A cos 3x + B sin 3x) 12. Solve 3ex tan y dx + (1 + ex) sec2y dy = 0 13. Solve : ex 1 − y2 dx + y x dy = 0 14. Solve: (x2− y)dx + (y2 − x) dy = 0, if it passes through the origin.

15. The normal lines to a given curve at each point (x, y) on the curve pass through the point (2, 0). The curve passes through the point (2, 3). Formulate the differential equation representing the problem and hence find the equation of the curve.

16. Solve : xdy − ydx = x2 + y2 dx 17. Solve : (x + 1)dy/dx − y = ex(x + 1)2 18. Solve : (D2 − 6D + 9)y = e3x 19. Solve : (D2 + 4D + 13)y = cos 3x 20. Solve : (D2 − 3D + 2)y = x

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Section C (10 marks) 1. Solve (x + y)2 dy/(dx ) = 1 2. Solve : dy/dx +y/x x = sin(x2) 3. Show that the equation of the curve whose slope at any point is equal to y + 2x and which passes through the origin is y = 2(ex − x − 1) 4. Solve :d2y/ dx2 − 3dy/dx + 2y = 2e3x when x = log2, y = 0 and when x = 0, y = 0 5. Solve : (D2 − 6D + 9) y = x + e2x 6. Solve : (D2 − 1)y = cos2x − 2 sin 2x 7. The sum of Rs. 1000 is compounded continuously, the nominal rate of interest being four percent per annum. In how many years will the amount be twice the original principal? (loge2 = 0.6931).

8. The rate at which the population of a city increases at any time is proportional to the population at that time. If there were 1,30,000 people in the city in 1960 and 1,60,000 in 1990 what population may be anticipated in 2020. log e(16/13) = .2070 ; e.42 = 1.52 9. A radioactive substance disintegrates at a rate proportional to its mass When its mass is 10 mgm, the rate of disintegration is 0.051 mgm per day. How long will it take for the mass to be reduced from 10 mgm to 5 mgm. [loge2 = 0.6931] Examples 10. Find the cubic polynomial in x which attains its maximum value 4 and minimum value 0 at x = − 1 and 1 respectively.

11. Solve : (x3 + 3xy2)dx + (y3 + 3x2y)dy = 0 12. Solve : (1 + ex/y)dx + ex/y(1 − x/y) dy = 0 given that y = 1, where x = 0 13. Solve : (1 − x2) dy/dx + 2xy = x (1 − x2) 14. Solve : (1 + y2)dx = (tan−1y − x)dy 15. Solve : (D2 − 4D + 1)y = x2 16. In a certain chemical reaction the rate of conversion of a substance at time t is proportional to the quantity of the substance still untransformed at that instant. At the end of one hour, 60 grams remain and at the end of 4 hours 21 grams. How many grams of the substance was there initially 17. The temperature T of a cooling object drops at a rate proportional to the difference T − S, where S is constant temperature of surrounding medium. If initially T = 150°C, find the temperature of the cooling object at any time t.

18. For a postmortem report, a doctor requires to know approximately the time of death of the deceased.

He records the first temperature at 10.00 a.m. to be 93.4°F. After 2 hours he finds the temperature to be 91.4°F. If the room temperature (which is constant) is 72°F, estimate the time of death. (Assume normal temperature of a human body to be 98.6°F).

19. The number of bacteria in a yeast culture grows at a rate which is proportional to the number present.

If the population of a colony of yeast bacteria triples in 1 hour. Show that the number of bacteria at the end of five hours will be 35 times of the population at initial time.

UNIT 9 DISCRETE MATHEMATICS SECTION B (6 MARKS) Construct the truth table for the statement (pvq) ^(∼q) Use the truth table to establish the statement is tautology or contradiction (p^(∼p) ^ (∼q)^ p) Show that p↔q ≡ (p →q)^(q↔p) Show that p↔q ≡ (-p)vq)^(-qvp) Show that (p^q) (pvq) isa atautology Show that ∼ (p v q ) = (∼p) ^ (∼q) Use the truth table to determine whether the statement (∼p v q ) v (p^(∼q) is a tautology Prove that the set of all 4th roots of unity forms an abelian group under ultiplication Show that the sets of all 2X2 non singular matrices forms a non-abelian infinite group under matrix Mù£‚èœ Þ«î£ ñŸÁ‹ Blue Print & ð°F 3 | 3.2.2014 28 |

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multiplication. (where the entries belong to R) 1 0 -1 0 1 0 -1 0 10. Show that the set of four matrix 0 1 0 1 0 -1 0 -1 multiplication of matrices 11. (i) prove that identity elements of a groups is unique (ii)Prove that the inverse element of an element of a group is unique 12. State and prove cancellation laws on groups 13. State and prove reversal law on inverse of a group

, form an abelian group under

Section –c (10 marks) 1. Show that (Z*) is an infinite abelian group where * is defined as a*b= a+b+2 2. Show that the set G of all positive rationals forms a group under the composition * defined by a*b=ab/3 for all a,b ∈ G 3. Prove that (c,+) is an infinite abelian group 4. Show that the set G of all rational numbers except -1 forms an baelian group with respect to the operation * given by a*b =a+b+ab for all a, b ∈ G 5. Show that the set {[1],[3],[4],[5],[9]} forms an abelian group under multiplication modulo 11 6. Show that the set of all matrices of the form 1 0 w 0 w2 0 0 1 0 w2 0 w 2 2 0 1 0 w 0 w 1 0 w 0 w 0 where w3=1, w1 form a group under matrix multiplications 7. Show that the set G of all matrices of the form ( x/x x/x ) where x ∈ R-{0}is a group matrix multiplications 8. Let G be the set of all rational numbers except 1 and * be defined on G by a*b=a+b-ab for all a, b∈G. Show that (G,*) is an infinite abelian group 9. Prove that the set of four functions f1,f2,f3,f4 on the set of non-zero complex c-{0} defined z by f1(Z) = f2(Z) =-2 , f3(z)=1/z and f4(z)= 1/-z z∈ C –{0} forms an abelian group with respect to the composition of functions 10. Show that (Zn+n) forms group. 10 show that (Z7{0)},.. 7) forms a group 11. Show that the set G = {2n/nz is an abelian group under multiplication.

Section b (6 marks) 1. Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of queens 2. A discrete random variable x has the following probability distributions X 0 1 2 3 4 5 6 7 8 P(x) a 3a 5a 7a 9a 11a 13a 15a 17a 3. In an entrance examination a student has to answer all the 120 questions. Each question has four options and only one option is correct. A student gets 1 mark for a correct answer and loses half mark for a wrong answer. What is the expectations of the mark scored by a students if he chooses the answer to each questions at random?

4. The probability distribution of a random variable x is given below; X 0 1 2 3 P(X=x) 0.1 0.3 0.5 0.1 If y= x2+2x find the mean and variance of y 5. Find the mean and variance for the following variability density functions f(x) = xe-x , if x0 otherwise 6. Four coins are tossed simultaneously, what is the probability of getting (a) exactly 2 heads (b) at least two heads (c) at most two heads 7. The overall percentage of passes in a certain examination is 80. If 6 candidates appear in the examination what is the probability that atleast 5 pass the examinations 8. Suppose that the amount of cosmic radiation to which a person is exposed when flying by jet across the United States is a random variable having a normal distribution with a mean of 4.5m rem and a standard Mù£‚èœ Þ«î£ ñŸÁ‹ Blue Print & ð°F 3 | 3.2.2014 | 29

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deviation of 0.59m rem, What is the probability that a person will be exposed to more than 5.20m rem of cosmic radiation of such a flight.

9. Find the probability mass function, and the cumulative distribution function for getting 3 when two dice are thrown 10. A continuous random variable x follows that the probability law f(x)=kx (1-x)10, 0<x< 0 elsewhere Find k 11. Verify that the following are probability density functions (a) F(x) = 2/9 x, 0 ≤x≤ 3 0 elsewhere (b) f (x) = {1/π 1/1+x2, - ∞ < x< 2 12. An urn contains 4 white and 3 red balls, Find the probability distribution of numbers of red balls in three draws one by one from the urn (i) with replacement (ii) without replacement 13. In a continuous distribution that p.d. of X is F(x)=3/4x(2-x, 0<x<2 0 otherwise Find the mean and the variance of the distribution 14. A game is played with a single fair die, a player wins Rs. 20 if a 2 turns ups, rs.40 if a 4 turns up loses rs. 30 if a 6 turns up. While he neither wins nor loses if any other face turns up. Find the expected sum of money he can win 15. Let x be a binomially distributed variable with mean 2 and standard deviation 3 2 . find the corresponding probability function 16. In a Poisson distribution if P(X=2)=P(x=3) find P(x=5) {given e-3=0.050}.

17. The difference between the mean and the variance of a Binomial distribution is 1 and the difference between their squares is 11. Find n 18. (i) find the expected value of the number on a die when thrown (ii)In a binimisl Fidytibuyion ig n = 5 and P (x=3)=2 P(x=2) find P 19. (i) if the sum of mean and variance of a binomial distribution is 4.8 for 5 trials, find the distribution (ii) The mean of a binomial distribution is 6 and its standard deviation is 3. Is this statement true or false? Comment 20. (i) A die is thrown 120 times and getting 1 or 5 is considered a success. Find the mean and variance of the number of success (ii) let x have a Poisson distribution with mean 4, find (a) P(X<3) (b) P(2<X<5) Section C 10 marks 1. The probability density function of a random variable x is F(x) =kxα-1 e-B xa,x,αβ>0 elsewhere Find (i) K, (ii) P(x>10) 2. The number of accidents in a year involving taxi drivers in a city follows a Poisson Distribution with mean equal to 3. Out of 100 taxi drivers find approximately the number of drivers with (I) no accident in a year 3. The mean weight of 500 males student in a certain college in 151 pounds and the standards deviation is 15 pounds. Assuming the weights are normally distributed find how many students weigh (i)between 120 and 155 pounds(ii)more than 185 pounds 4. Find c, µ and σ2 of the normal distribution whose probability function is given by f(x) =c e –x2+3x-∞ <x<∞ 5. A random variable X has the following probability mass function X 0 1 2 3 4 5 6 P(x=x) k 3k 5k 7k 9k 11k 13k (a) Find k (b) evaluate p(x<4), p(x<5) and p(3<x<6) (c) What is the smallest value of x for which p(X<x)>1/2 6. An urn contains 4 white and 3 red balls. Find the probability distribution of number of red balls in three draws one by one from the urn. (i)with replacement (ii) without replacement 7. Obtain k, µ and σ2 of the normal distribution whose probability distribution function is given by f (x) =k e -2x2+4x-∞<x<∞ 8. The air pressure in a randomly selected tyre put on a certain model new car is normally distributed with mean value 31psi and standard deviation 0.2psi (i) what is the probability that the pressure for a randomly selected tyre (a) between 30.5 and 31.5 psi (b) between 30 and 32psi (ii) what is probability that the pressure for a randomly selected tyre exceeds 30.5 psi?

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