12th Chemistry Chapter 8 Exercise Transition and Inner Transition Elements Solutions Maharashtra Board

Class 12 Chemistry Chapter 8

Balbharti Maharashtra State Board 12th Chemistry Textbook Solutions Chapter 8 Transition and Inner Transition Elements Textbook Exercise Questions and Answers.

Transition and Inner Transition Elements Class 12 Exercise Question Answers Solutions Maharashtra Board

Class 12 Chemistry Chapter 8 Exercise Solutions Maharashtra Board

Chemistry Class 12 Chapter 8 Exercise Solutions

1. Choose the most correct option.

Question i.
Which one of the following is diamagnetic
a. Cr3⊕
b. Fe3⊕
c. Cu2⊕
d. Sc3⊕
Answer:
d. Sc3⊕

Maharashtra Board Class 12 Chemistry Solutions Chapter 8 Transition and Inner Transition Elements

Question ii.
Most stable oxidation state of Titanium is
a. +2
b. +3
c. +4
d. +5
Answer:
c. +4

Question iii.
Components of Nichrome alloy are
a. Ni, Cr, Fe
b. Ni, Cr, Fe, C
c. Ni, Cr
d. Cu, Fe
Answer:
(c) Ni, Cr

Question iv.
Most stable oxidation state of Ruthenium is
a. +2
b. +4
c. +8
d. +6
Answer:
(b) +4

Question v.
Stable oxidation states for chromium are
a. +2, +3
b. +3, +4
c. +4, +5
d. +3, +6
Answer:
d. +3, +6

Question vi.
Electronic configuration of Cu and Cu+1
a. 3d10, 4s0; 3d9, 4s0
b. 3d9, 4s1; 3d94s0
c. 3d10, 4s1; 3d10, 4s0
d. 3d8, 4s1; 3d10, 4s0
Answer:
c. 3d10, 4s1; 3d10, 4s°

Question vii.
Which of the following have d0s0 configuration
a. Sc3⊕
b. Ti4⊕
c. V5⊕
d. all of the above
Answer:
d. All of the above

Question viii.
Magnetic moment of a metal complex is 5.9 B.M. Number of unpaired electrons in the complex is
a. 2
b. 3
c. 4
d. 5
Answer:
d. 5

Question ix.
In which of the following series all the elements are radioactive in nature
a. Lanthanoids
b. Actinoids
c. d-block elements
d. s-block elements
Answer:
b. Actinides

Question x.
Which of the following sets of ions contain only paramagnetic ions
a. Sm3⊕, Ho3⊕, Lu3⊕
b. La3⊕, Ce3⊕, Sm3⊕
c. La3⊕, Eu3⊕, Gd3⊕
d. Ce3⊕, Eu3⊕, Yb3⊕
Answer:
d. Ce3⊕, Eu3⊕, Yb3⊕

Maharashtra Board Class 12 Chemistry Solutions Chapter 8 Transition and Inner Transition Elements

Question xi.
Which actinoid, other than uranium, occur in a significant amount naturally?
a. Thorium
b. Actinium
c. Protactinium
d. Plutonium
Answer:
a. Thorium

Question xii.
The flux added during extraction of Iron from hematite are its?
a. Silica
b. Calcium carbonate
c. Sodium carbonate
d. Alumina
Answer:
b. Calcium carbonate

2. Answer the following

Question i.
What is the oxidation state of Manganese in
\(\text { (i) } \mathrm{MnO}_{4}^{2-}(\mathrm{ii}) \mathrm{MnO}_{4}^{-} \text {? }\)
Answer:
Oxidation state of Manganese in
\((i) \mathrm{MnO}_{4}^{2-} is +6
(ii) \mathrm{MnO}_{4}^{-}is +7\)

*Question ii.
Give uses of KMnO4

Question iii.
Why salts of Sc3⊕, Ti4⊕, V5⊕ are colorless?
Answer:
(i) Sc3+ salts are colourless :

  • The electronic configuration of 21Sc [Ar| 3d1 4s2 and Sc3+ [Ar] d°.
  • Since there are no unpaired electrons in 3d subshell, d → d transition is not possible.
  • Therefore, Sc3+ ions do not absorb the radiations in the visible region. Hence salts of Sc3+ are colourless (or white).

(ii) Ti4+ salts are colourless :

  • The electronic configuration of 22Ti [Ar] 3d24s2 and Ti4+ : [Ar] d°
  • Since there are no unpaired electrons in 3d subshell, d-*d transition is not possible.
  • Therefore, Ti3+ ions do not absorb the radiation in visible region. Hence salts of Ti3+ are colourless.

(iii) Vs5+ salts are eolourless :

  • The electronic configuration of 23V : [Ar] 3d34s2 and V5+ : [Ar] 3d°
  • Since there are no unpaired electrons in 3d-subshell, d – d transition is not possible.
  • Therefore, V5+ ions do not absorb the radiations in the visible region. Hence, V5+ salts are colourless, a

Question iv.
Which steps are involved in the manufacture of potassium dichromate from chromite ore?
Answer:
Steps in the manufacture of potassium dichromate from chromite ore are :

  • Concentration of chromite ore.
  • Conversion of chromite ore into sodium chromate (Na2CrO4).
  • Conversion of Na2CrO4 into sodium dichromate (Na2Cr2O7).
  • Conversion of Na2Cr2O7 into K2Cr2O7.

Question v.
Balance the following equation
(i) KMnO4 + H2C2O4 + H2SO4 → MnSO4 + K2SO4 + H2O + O2
(ii) K2Cr2O7 + KI + H2SO4 → K2SO4 + Cr2(SO4)3 + 7H2O + 3I2
Answer:
(i) 2KMnO4 + 3H2SO4 + 5H2C2O4 → K2SO4 + 2MnSO4 + 8H2O + 10CO2
(ii) Acidified potassium dichromate oxidises potassium iodide (KI) to iodine (I2). Potassium dichromate is reduced to chromic sulphate. Liberated iodine turns the solution brown K2Cr2O7 + 6KI + 7H2SO4 → 4K2SO4 + Cr2(SO4)3 + 7H2O + 3I2 [Oxidation state of iodine increases from – 1 to zero]

Maharashtra Board Class 12 Chemistry Solutions Chapter 8 Transition and Inner Transition Elements

Question vi.
What are the stable oxidation states of plutonium, cerium, manganese, Europium?
Answer:
Stable oxidation states :
Plutonium + 3 to + 7
Cerium + 3, + 4
Manganese + 2, + 4, + 6, + 7
Europium +2, +3

Question vii.
Write the electronic configuration of chromium and copper.
Answer:
Chromium (24Cr) has electronic configuration,
24Cr (Expected) : Is2 2s2 2p6 3s2 3p6 3d4 4s2
(Observed) : Is2 2s2 2p6 3s2 3p6 3d5 4s1

Explanation :

  • The energy difference between 3d- and 45-orbitals is very low.
  • d-orbitals being degenerate, they acquire more stability when they are half-filled (3d5).
  • Therefore, there arises a transfer of one electron from 45-orbital to 3d-orbital in Cr giving more stable half-filled orbital. Hence, the configuration of Cr is [Ar] 3d5 4s1 and not [Ar] 3d4 4s2.

Copper (29CU) has electronic configuration,
29Cu (Expected) : Is2 2s3 2p6 3s3 3p6 3d9 4s2
(Observed) : Is2 2s2 2p6 3s2 3p6 3d10 4s1

Explanation :

  • The energy difference between 3d- and 45-orbitals is very low.
  • d-orbitals being degenerate, they acquire more stability when they are completely filled.
  • Therefore, there arises a transfer of one electron from 45-orbital to 3d-orbital in Cu giving completely filled more stable d-orbital.

Hence, the configuration of Cu is [Ar] 3d10 4s1 and not [Ar] 3d9 4s2.

Question viii.
Why nobelium is the only actinoid with +2 oxidation state?
Answer:

  • Nobelium has the electronic configuration 102NO : [Rn] 5f146d°7s2
  • No2+ : [Rn] 5f146d°
  • Since the 4f subshell is completely filled and 6d° empty, + 2 oxidation state is the stable oxidation state.
  • Other actinoids in + 2 oxidation state are not as stable due to incomplete 4f subshell.

Question ix.
Explain with the help of balanced chemical equation, why the solution of Ce(IV) is prepared in acidic medium.
Answer:
Ce4+ undergoes hydrolysis as, Ce4++ 2H2O → Ce(OH)4 + 4H+.
Due to the presence of H+ in the solution, the solution is acidic.

Question x.
What is meant by ‘shielding of electrons’ in an atom?
Answer:
The inner shell electrons in an atom screen or shield the outermost electron from the nuclear attraction. This effect is called the shielding effect.

The magnitude of the shielding effect depends upon the number of inner electrons.

Question xi.
The atomic number of an element is 90. Is this element diamagnetic or paramagnetic?
Answer:
The 90th element thorium has an electronic configuration, [Rn] 6d27s2. Since it has 2 unpaired electrons it is paramagnetic.

3. Answer the following

Question i.
Explain the trends in atomic radii of d-block elements
Answer:

  1. The atomic or ionic radii of 3-d series transition elements are smaller than those of representative elements, with the same oxidation states.
  2. For the same oxidation state, there is an increase in nuclear charge and a gradual decrease in ionic radii of 3d-series elements is observed. Thus ionic radii of ions with oxidation state + 2 decreases with increase in atomic number.
  3. There is slight increase is observed in Zn2+ ions. With the higher oxidation states, effective nuclear charge increases. Therefore ionic radii decrease with increase in oxidation state of the same element. For example, Fe2+ ion has ionic radius 77 pm whereas Fe3+ has 65 pm.

Maharashtra Board Class 12 Chemistry Solutions Chapter 8 Transition and Inner Transition Elements

Question ii.
Name different zones in the Blast furnace. Write the reactions taking place in them.
Answer:
(i) Zone of combustion : The hot air oxidises coke to CO which is an exothermic reaction, due to which the temperature of furnace rises.
C + 1/2 O2 → CO ΔH= – 220kJ
Some part of CO dissociates to give finely divided carbon and O2.
2CO → 2C + O2
The hot gases with CO rise up in the furnace and heats the charge coming down. CO acts as a fuel as well as a reducing agent.

(ii) Zone of reduction : At about 900 °C, CO reduces Fe2O3 to spongy (or porous) iron.
Fe2O3 + 3CO → 2Fe + 3CO2
Carbon also reduces partially Fe203 to Fe.
Fe2O3 + 3C → 2Fe + 3CO

(iii) Zone of slag formation : At 1200 K limestone, CaCO3 in the charge, decomposes and forms a basic flux CaO which further reacts at 1500 K with gangue (SiO2, Al2O3) and forms a slag of CaSiO3 and Ca3AlO3.
CaCO3 + CaO + CO2.
CaO + SiO2 → CaSiO3
12CaO + 2Al2O3 → 4Ca3AlO3 + 3O2

The slag is removed from the bottom of the furnace through an outlet.

(iv) Zone of fusion : The impurities in ore like MnO2 and Ca3(PO4)2 are reduced to Mn and P while SiO2 is reduced in Si. The spongy iron moving down in the furnace melts in the fusion zone and dissolves the impurities like C, Si, Mn, phosphorus and sulphure. The molten iron collects at the bottom of furnace. The lighter slag floats on the molten iron and prevents its oxidation.

The molten iron is removed and cooled in moulds. It is called pig iron or cast iron. It contains about 4% carbon.

Question iii.
What are the differences between cast iron, wrought iron and steel?
Answer:

Cast iron Wrought iron Steel
(1) Hard and brittle
(2) Contains 4% carbon.
(3) Used for making pipes, manu­facturing automotive parts, pots, pans, utensils
(1) Very soft
(2) Contains less than 0.2% carbon.
(3) Used for making pipes, bars for stay bolts, engine bolts and rivets.
(1) Neither too hard nor too soft.
(2) Contains 0.2 to 2% carbon
(3) Used in buildings infrastruc­ture, tools, ships, automobiles, weapons etc.

Question iv.
Iron exhibits +2 and +3 oxidation states. Write their electronic configuration. Which will be more stable? Why?
Answer:
The electronic configuration of Fe2 + and Fe3+ :
Fe2+ : Is2 2s2 2p6 3s2 3p6 3d6
Fe3+ : Is2 2s2 2p6 3s2 3p6 3d5

Due to loss of two electrons from the 4.v-orbital and one electron from the 3d-orbital, iron attains 3+ oxidation state. Since in Fe3+, the 3d-orbital is half-filled, it gets extra stability, hence Fe3+ is more stable than Fe2+.

Question v.
Give the similarities and differences in elements of 3d, 4d and 5d series.
Answer:
Similarity :

  • They are placed between .s-block and p-block of the periodic table.
  • All elements are metals showing metallic characters.
  • Some are paramagnetic.
  • Most of them give coloured compounds.
  • They have catalytic properties.
  • They form complexes.
  • They have variable oxidation states.

Differences :

  • In 4d and 5d series lanthanide and actinoid contraction is observed. In 3d series atomic size changes are less marked.
  • 4d and 5d elements have high coordination numbers compared to 3d elements.
  • 4d and 5d series have similar properties whereas 3d series have different properties.

Question vi.
Explain trends in ionisation enthalpies of d-block elements.
Answer:

  1. The ionisation enthalpies of transition elements are quite high and lie between those of 5-block and p-block elements. This is because the nuclear charge and atomic radii of transition elements lie between those of 5-block and p-block elements.
    Maharashtra Board Class 12 Chemistry Solutions Chapter 8 Transition and Inner Transition Elements 12
  2. As atomic number of transition elements increases along the period and along the group, first ionisation enthalpy increases even though the increase is not regular.
  3. If IE1; IE2 and IE3 are the first, second and third ionisation enthalpies of the transition elements, then IE1 < IE2 < IE3.
  4. In the transition elements, the added last differentiating electron enters into (n – 1) d-orbital and shields the valence electrons from the nuclear attraction. This gives rise to the screening effect of (n – 1) d-electrons.
  5. Due to this screening effect of (n – 1) d electrons, the ionisation enthalpy increases slowly and the increase is not very regular.

Maharashtra Board Class 12 Chemistry Solutions Chapter 8 Transition and Inner Transition Elements

Question vii.
What is meant by diamagnetic and paramagnetic metal? Give one example of diamagnetic and paramagnetic transition metal and lanthanoid metal.
Answer:

  1. Paramagnetic substances : When a magnetic field is applied, substances which are attracted towards the applied magnetic field are called paramagnetic substances. Example : Ni2+, Pr4+
  2. Diamagnetic substances : When a magnetic field is applied, substances which are repelled by the magnetic fields are called diamagnetic substances. Example : Zn2+, La3+
  3. Ferromagnetic substances : When a magnetic field is applied, substances which are attracted very strongly are called ferromagnetic substances. These substances can be magnetised. For example, Fe, Co, Ni are ferromagnetic.

Question viii.
Why the ground-state electronic configurations of gadolinium and lawrencium are different than expected?

Question ix.
Write steps involved in the metallurgical process
Answer:
The various steps and principles involved in the extraction of pure metals from their ores are as follows.:

  • Concentration of ores in which impurities (gangue) are removed.
  • Conversion of ores into oxides or other reducible compounds of metals.
  • Reduction of ores to obtain crude metals.
  • Refining of metals giving pure metals.

Question x.
Cerium and Terbium behaves as good oxidising agents in +4 oxidation state. Explain.
Answer:

  • The most stable oxidation state of lanthanoids is +3.
  • Hence, Ce4+ (cerium) and Tb4+ (terbium) tend to get + 3 oxidation state which is more stable.
  • Since they get reduced by accepting electron, they are good oxidising agents in + 4 oxidation state.

Question xi.
Europium and Ytterbium behave as good reducing agents in +2 oxidation state explain.
Answer:

  • The most stable oxidation state of lanthanoids is + 3.
  • Hence, Eu2+ and Yb2+ tend to get + 3 oxidation states by losing one electron.
  • Since they get oxidised, they are good reducing agents in + 2 oxidation state.

Activity :
Make groups and each group prepare a PowerPoint presentation on the properties and applications of one element. You can use your imagination to create some innovative ways of presenting data.

You can use pictures, images, flow charts, etc. to make the presentation easier to understand. Don’t forget to cite the reference(s) from where data for the presentation is collected (including figures and charts). Have fun!

12th Chemistry Digest Chapter 8 Transition and Inner Transition Elements Intext Questions and Answers

Do you know? (Textbook Page No 165)

Question 1.
In which block of the modern periodic table are the transition and inner transition elements placed?
Answer:
The transition elements are placed in d-block and inner transition elements are placed in f-block of the modern periodic table.

Use your brain power! (Textbook Page No 167)

Question 1.
Fill in the blanks with correct outer electronic configurations.
Answer:
Answers are given in bold.
Maharashtra Board Class 12 Chemistry Solutions Chapter 8 Transition and Inner Transition Elements 6

Maharashtra Board Class 12 Chemistry Solutions Chapter 8 Transition and Inner Transition Elements

Try this….. (Textbook Page No 168)

Question 1.
Write the electronic configuration of Cr and Cu.
Answer:
24Cr : [Ar] 3d54s1 30Cu : [Ar] 3d104s1

Can you tell? (Textbook Page No 168)

Question 1.
Which of the first transition series element shows the maximum number of oxidation states and why?
Answer:

  • 25Mn shows the maximum number of oxidation states, + 2 to +7.
  • 25Mn : [Ar] 3d54s3
  • Mn has incompletely filled J-subshell.
  • Due to small difference in energy between 3d and 4s -orbitals, Mn can lose (or share) electrons from both the orbitals.
  • Hence Mn shows oxidation states from + 2 to +7.

Question 2.
Which elements in the 4d and 5d-series will show maximum number of oxidation states?
Answer:
In 4d-series maximum number of oxidation states are for Ruthenium Ru ( + 2, +3, + 4„ +6, +7, + 8). In 5d-series, maximum number of oxidation states are for Osmium, Os ( + 2 to + 8).

Try this ….. (Textbook Page No 168)

Question 1.
Write the electronic configuration of Mn6+, Mn4+, Fe4+, Co5+, Ni2+.
Answer:

Ions Electronic configuration of valence shell
Mn6+ [Ar] 3d1
Mn4+ [Ar] 3d3
Fe4+ [Ar] 3d4
Co5+ [Ar] 3d4
Ni2+ [Ar] 3d8

Try this ….. (Textbook Page No 171)

Question 1.
Pick up the paramagnetic species from the following : Cu1+, Fe3+, Ni2+, Zn2+, Cd2+, Pd2+.
Answer:
The following ions are paramagnetic : Fe3+, Ni2+, Pd2+

Try this ….. (Textbook Page No 171)

Question 1.
What will be the magnetic moment of transition metal having 3 unpaired electrons?
(a) equal to 1.73 B.M.?
(b) less than 1.73 BM.
(c) more than 1.73 B.M.?
Answer:
By spin-only formula, \(\mu=\sqrt{n(n+2)}\) where n is number of unpaired electrons.
\(\mu=\sqrt{3(3+2)}=\sqrt{3(5)}=3.87 \mathrm{~B} . \mathrm{M}\)
Thus the value is more than 1.73 B.M.

Use your brain power! (Textbook Page No 171)

Question 1.
A metal ion from the first transition series has two unpaired electrons. Calculate the magnetic moment.
Answer:
\(\)\begin{aligned}
\mu &=\sqrt{n(n+2)} \\
&=\sqrt{2(2+2)} \\
&=\sqrt{8} \\
&=2.84 \text { B.M. }
\end{aligned}\(\)

Maharashtra Board Class 12 Chemistry Solutions Chapter 8 Transition and Inner Transition Elements

Problem (Textbook Page No 172)

Question 1.
Calculate the spin-only magnetic moment of divalent cation of a transition metal with atomic number 25.
Answer:
For element with atomic number 25. electronic configuration of its divalent cation will be : [Ar] 3d5.
Maharashtra Board Class 12 Chemistry Solutions Chapter 8 Transition and Inner Transition Elements 16

Try this….. (Textbook Page No 172)

Question 1.
Calculate the spin-only magnetic moment of a divalent cation of element Slaving atomic number 27.
Answer:
Electronic configuration of divalent ion of an element with atomic number 27 : [Ar] 3d7;
Maharashtra Board Class 12 Chemistry Solutions Chapter 8 Transition and Inner Transition Elements 17

Can you tell? (Textbook Page No 172)

Question 1.
Compounds of s and p-block elements are almost white. What could be the absorbed radiation? (uv or visible)?
Answer:
The white colour of a compound indicates the absorption of uv radiation.

Can you tell? (Textbook Page No 181)

Question 1.
Why f-block elements are called inner transition metals?
Answer:
f-block elements are called inner transition elements since f-orbital lies much inside the f-orbital in relation to the transition metals, These elements have 1 to 14 electrons in their f-orbital.

Question 2.
Are there an similarities between transition and inner transition metals?
Answer:
There are some properties similarity between transition and inner transition metals.

  • They are placed between s and p-block elements.
  • They are metals with filling of inner suhshells in their electronic configuration.
  • They show variable oxidation slates.
  •  They show magnetism.
  • They form coloured compounds.
  • They have catalytic property.

Problem (Textbook Page No 184)

Question 1.
Which of the following will have highest fourth ionisation enthalpy, La4+, Gd4+, Lu4+.
Answer:
La : 4f°5d16s2
Gd : 4f15d16s2
Lu : 4f145d16s2
Lu will have the highest fourth ionisation enthalpy since Lu3+ has the most stable configuration of 4f14.

Use your brain power! (Textbook Page No 185)

Question 1.
Do you think that lanthanoid complex would show magnetism?
Answer:
Lanthanoid complexes may show magnetism.

Question 2.
Can you calculate the spin only magnetic moment of lanthanoid complexes using the same formula that you used for transition metal complexes?
Answer:
You cannot calculate magnetic moment of lanthanoid complexes using spin only formula as you have to consider orbital momentum also.

Maharashtra Board Class 12 Chemistry Solutions Chapter 8 Transition and Inner Transition Elements

Question 3.
Calculate the spin only magnetic moment of La3+. Compare the value with that given in the table.
Answer:
La3+ ion has no unpaired electron.
Maharashtra Board Class 12 Chemistry Solutions Chapter 8 Transition and Inner Transition Elements 56
La3+ ion has zero value of magnetic moment same as given in the table.

Maharashtra Board Class 12 Chemistry Solutions Chapter 8 Transition and Inner Transition Elements 51

12th Std Chemistry Questions And Answers:

12th Chemistry Chapter 12 Exercise Aldehydes, Ketones and Carboxylic Acids Solutions Maharashtra Board

Class 12 Chemistry Chapter 12

Balbharti Maharashtra State Board 12th Chemistry Textbook Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids Textbook Exercise Questions and Answers.

Aldehydes, Ketones and Carboxylic Acids Class 12 Exercise Question Answers Solutions Maharashtra Board

Class 12 Chemistry Chapter 12 Exercise Solutions Maharashtra Board

Chemistry Class 12 Chapter 12 Exercise Solutions

1. Choose the most correct option.

Question i.
In the following resonating structures A and B, the number of unshared electrons in valence shell present on oxygen respectively are
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 1
Answer:
c. 4, 6

Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids

Question ii.
In the Wolf -Kishner reduction, alkyl aryl ketones are reduced to alkylbenzenes. During this change, ketones are first converted into
a. acids
b. alcohols
c. hydrazones
d. alkenes
Answer:
c. hydrazones

Question iii.
Aldol condensation is
a. electrophilic substitution reaction
b. nucleophilic substitution reaction
c. elimination reaction
d. addition – elimination reaction
Answer:
d. addition-elimination reaction

Question iv.
Which one of the following has the lowest acidity?
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 2
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 3
Answer:
(c)

Question v.
Diborane reduces
a. ester group
b. nitro group
c. halo group
d. acid group
Answer:
d. acid group

Question vi.
Benzaldehyde does NOT show positive test with
a. Schiff reagent
b. Tollens’ ragent
c. Sodium bisulphite solution
d. Fehling solution
Answer:
d. Fehling solution

2. Answer the following in one sentence

Question i.
What are aromatic ketones?
Answer:
The compounds in which Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 23 group is attached to either two aryl groups or one aryl and one alkyl group are called aromatic ketones.

For example :
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 24

Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids

Question ii.
Is phenylacetic acid an aromatic carboxylic acid?
Answer:
Phenylacetic acid is not an aromatic carboxylic acid.

Question iii.
Write reaction showing conversion of ethanenitrile into ethanol.
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 311

Question iv.
Predict the product of the following reaction:
\(\mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{COOCH}_{3} \frac{\mathrm{i} \cdot \mathrm{AlH}(\mathrm{i}-\mathrm{Bu})_{2}}{\text { ii. } \mathrm{H}_{3} \mathrm{O}^{\oplus}}{\longrightarrow} ?\)
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 313

Question v.
Name the product obtained by reacting toluene with carbon monoxide and hydrogen chloride in presence of anhydrous aluminium chloride.
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 314

Question vi.
Write reaction showing conversion of Benzonitrile into benzoic acid.
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 315

Question vii.
Name the product obtained by the oxidation of 1,2,3,4-tetrahydronaphthalene with acidified potassium permanganate.
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 316

Question viii.
What is formalin?
Answer:
The aqueous solution of formaldehyde (40%) is known as formalin.

Question ix.
Arrange the following compounds in the increasing order of their boiling points : Formaldehyde, ethane, methyl alcohol.
Answer:
Ethane, formaldehyde, methyl alcohol.

Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids

Question x.
Acetic acid is prepared from methyl magnesium bromide and dry ice in presence of dry ether. Name the compound which serves not only reagent but also as cooling agent in the reaction.
Answer:
The cooling agent used in the above reaction is dry ice (O = C = O).

3. Answer in brief.

Question i.
Observe the following equation of reaction of Tollens’ reagent with aldehyde. How do we know that a redox reaction has taken place. Explain.
\(\begin{array}{r}
\mathrm{R}-\mathrm{CHO}+2 \mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}+\mathrm{OH}^{-} \stackrel{\Delta}{\longrightarrow} \\
\mathrm{R}-\mathrm{COO}^{-}+2 \mathrm{Ag} \downarrow+4 \mathrm{NH}_{3}+2 \mathrm{H}_{2} \mathrm{O}
\end{array}\)
Answer:
Tollen’s reagent oxidises acetaldehyde to acetic acid (carboxylate ion) and Ag in Tollen’s reagent complex are reduced to silver. In this reaction, oxidation and reduction takes place simultaneously hence, it is a redox reaction.

Question ii.
Formic acid is stronger than acetic acid. Explain.
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 146
In acetic acid, the methyl group is an electron-donating group. The acetate ion formed gets destabilized due to the electron releasing effect of methyl group ( +1 effect) which is higher than that of H-atom in the corresponding formic acid. As a result, acetic acid dissociates to a lesser extent. Thus decreasing the acidity of acetic acid.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 147
Formic acid having lower pKa value than acetic acid. Hence, formic acid is a stronger acid than acetic acid.

Question iii.
What is the action of hydrazine on cyclopentanone in presence of —. KOH in ethylene glycol?
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 308

Question iv.
Write reaction showing conversion of Acetaldehyde into acetaldehyde dimethyl acetal.
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 184

Question v.
Aldehydes are more reactive toward nucleophilic addition reactions than ketones. Explain.
Answer:
The reactivity of aldehydes and keones is due to the polarity of carbonyl group which results in electrophilicity of carbon. The reactivity is further explained on the basis of electronic effect and steric effects.

(1) Influence of electronic effects: A ketone has two electron-donating aJ.kyl groups ( + I effect) bonded to carbonyl carbon which are responsible for decreasing its positive polarity and electrophilicity. In contrast. aldehydes have only electron-donating group bonded to carbonyl carbon. This shows aldehydes are more electrophilic than ketones.

(2) Steric effects : Two bulky alkyl groups in ketone come in the way of the incoming nucleophile. This is called steric hindrance to nucleophilic attack.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 160
On the other hand. nucleophile can easily attack the carbonyl carbon in aldehyde because has one alkyl group and is less crowded or sterically less hindered.

Hence aldehydes are more reactive and can easily be attacked by nucleophiles.

Question vi.
Write reaction showing the action of the following reagent on propane nitrile
a. Dilute NaOH
b. Dilute HCl ?
Answer:
(1) Action of dil NaOH:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 115
(2) Action of dil HCl:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 116

Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids

Question vi.
Arrange the following carboxylic acids with increasing order of their acidic strength and justify your answer.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 4
Answer:
The increasing order of acidity will be 1 <3 <2. Acidity depends on mainly two factors : (1) ease of proton release (2) stability of conjugate base formed. In example (3) the ether O exerts a I effect and is closer to COOH group than in 2 (1 effect diminishes). Also the conjugate base formed will be stabilized by the same – I effect by delocalization of charge.

4. Answer the following

Question i.
Write a note on
a. Cannizaro reaction
b. Stephen reaction.
Answer:
(1) The carbon atom adjacent to carbonyl carbon atom is called a-carbon atom (α – C) and the hydrogen atom attached to a-carbon atom is called α-hydrogen atom (α – H).
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 226
(2) The α-hydrogen of aldehydes and ketones is acidic in nature due to (i) the strong-I effect of carbonyl group (ii) resonance stabilization of the carbanion.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 227
(3) Aldol condensation reaction is characteristic reaction of aldehydes and ketones containing active α-hydrogen atom.

(4) When aldehydes or ketones containing α – H atoms are warmed with a dilute base or dilute acid, two molecules of them undergo self condensation to give β-hydroxy aldehyde (aldol) or β-hydroxy ketone (ketol) respectively. The reaction is known as Aldol addition Reaction.

(5) In aldol condensation, the product is formed by the nucleophilic addition of α-carbon atom of a second molecule which gets attached to carbonyl carbon atom of the first molecule and α-hydrogen atom of the second molecule gets attached to carbonyl oxygen atom of the first molecule forming (- OH) group to give β-hydroxy aldehyde or ketone.

(6) This is a reversible reaction, establishing an equilibrium favouring aldol formation to a greater extent than ketol formation.

(7) For aldehyde :
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 228
Acetaldol on heating undergoes subsequent elimination of water giving rise to α, β unsaturated aldehyde.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 229
The overall reaction is called aldol condensation. It is a nucleophilic addition-elimination reaction. For ketone :
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 230
Diacetone alcohol on dehydration by heating forms α, β unsaturated ketone.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 231

Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids

Question ii.
What is the action of the following reagents on toluene ?
a. Alkaline KMnO4, dil. HCl and heat
b. CrO2Cl2 in CS2
c. Acetyl chloride in presence of anhydrous AlCl3.
Answer:
(1) Action of alkaline KMnO4 : When toluene is heated with alkaline KMnO4. (methyl group gets oxidised to earboxy lic group) benzoic acid is obtained
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 304

(2) Action of CrO2Cl2 in C2:
Answer:
When Loluenc is ircated with soluion of chromyl chloride (CrO2Cl2) in Cs2, brown chromium complex is obtained, which on acid hydrolysis gives benzaldehyde.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 305

(3) Action of acetyl chloride in presence of anhyd. AlCl3.
Answer:
When toluene is treated with acetyl chloride in presence of anhydrous AlCl3 4-methyl acetophenone is obtained.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 306

Question iii.
Write the IUPAC names of the following structures :
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 5

Question iv.
Write reaction showing conversion of p- bromoisopropyl benzene into p-Isopropyl benzoic acid (3 steps).
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 117

Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids

Question v.
Write reaction showing aldol condensation of cyclohexanone.
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 247

Activity :
Draw and complete the following reaction scheme which starts with acetaldehyde. In each empty box, write the structural formula of the organic compound that would be formed.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 323

12th Chemistry Digest Chapter 12 Aldehydes, Ketones and Carboxylic Acids Intext Questions and Answers

Use your brain power! (Textbook Page No 254)

Question 1.
Classify the following as aliphatic and aromatic aldehydes.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 18
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 19

Use your brain power! (Textbook Page No 255)

Question 1.
Classify the following as simple and mixed ketones. Benzoplienone, acetoneq hutanoneq acetophenone.

Compoun
Benzophenone ……………………………………………..
Acetone ……………………………………………..
Butanone ……………………………………………..
Acetophenone ……………………………………………..

Answer:

Compound
Benzophenone Simple ketone
Acetone Simple ketone
Butanone Mixed ketone
Acetophenone Mixed ketone

Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids

Use your brain power! (Textbook Page No 264)

Write IUPAC names for the following compounds.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 49

Try this ….. (Textbook Page No 260)

Question 1.
Draw structures for the following :
(1) 2-Methylpentanal
(2) Hexan-2-one
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 50

Can you tell? (Textbook Page No 260)

Question 1.
Which is the reagent that oxidizes primary alcohols to only aldehydes and does not oxidize aldehydes further into carboxylic acid ?
Answer:
The reagent that is used to make only aldehydes is-heated Cu at 573 K.

Use your brain power! (Textbook Page No 261)

Question 1.
Write the structure of the product formed on Rosenmund reduction of ethanoyl chloride and benzoyl chloride.
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 84

Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids

Can you think? (Textbook Page No 261)

Question 1.
What is the alcohol formed when benzoyl chloride is reduced with pure palladium as the catalyst ?
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 85

Use your brain power! (Textbook Page No 262)

Question 1.
Name the compounds which are used for the preparation of benzophenone by Friedel-Crafts acylation reaction. Draw their structures.
Answer:
The compounds which are used in preparation of benzophenone by Friedel – Crafts reaction are :
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 106

Use your brain power! (Textbook Page No 263)

Identify the reagents necessary to achieve each of the following transformations:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 108
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 109

Use your brain power! (TextBook Page No 264)

Predict the products (name and structure) in the following reactions.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 133
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 134

Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids

Problem 12.1 : (Textbook Page No 276)

Question 1.
Alcohols (R – OH), phenols (Ar – OH) and carboxylic acids (R – COOH) can undergo ionization of O – H bond to give away proton H+; yet they have different pKa values, which are 16, 10 and 4.5 respectively. Explain.
Solution :
pKa value is indicative of acid strength. Lower of pKa value stronger the acid. Alcohols, phenols and carboxylic acids, all involve ionization of an O – H bond. But their different pKa values indicate that their acid strengths are different. This is because the resulting conjugate bases are stabilized to different extents.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 150
As the conjugate base of carboxylic acid is best stabilized, among the three, carboxylic acids are strongest and have the lowest pKa value. As conjugate base of alcohols is destabilized, alcohols are weakest acids and have highest pIQ value. As conjugate base of phenols is moderately stabilized, phenols are moderately acidic and have intermediate pBQ value.

Try this….. (Textbook Page No 277)

Question 1.
Compare the following two conjugate bases and answer.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 151
(1) Indicate the inductive effects of CH3 – group in (a) and Cl-group in (b) by putting arrowheads in the middle of appropriate covalent bonds.
(2) Which species is stabilized by inductive effect, (a) or (b) ?
(3) Which species is destabilized by inductive effect, (a) or (b) ?
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 152
(2) The monochloroacetate ion formed gets stabilised due to electron-withdrawing of Cl atom (- I effect).
(3) Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 153 The acetate ion formed gets destabilised due to electron releasing effect of methyl group

Use your brain power! (Textbook Page No 277)

Question 1.
(1) Compare the pKa values and arrange the following in an increasing order of acid strength. CI3CCOOH, ClCH2COOH, CH3COOH, Cl2CHCOOH
(2) Draw structures of conjugate bases of monochloroacetic acid and dichloroacetic acid. Which one is more stabilized by – 1 effect?
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 154

Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 155 The dichloroacetate ion formed gets stabilised due electron-withdrawing effect of two chlorine atoms. (- 1 effect)

Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids

Try this….. (Textbook Page No 277)

Question 1.
Arrange the following acids in order of their decreasing acidity.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 156
Answer:
Acidity in the decreasing order

Try this ….. (Textbook Page No 267)

Question 1.
Draw the structure of propanone and indicate its polarity.
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 161

Can you tell? (Textbook Page No 268)

Question 1.
Simple hydrocarbons, ethers, ketones and alcohols do not get oxidized by Tollen’s reagent. Explain, Why?
Answer:
(1) Due to the presence of hydrogen atom in aldehyde group Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 167, an aldehyde is oxidised to carboxylic acid Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 168 which is not possible in case of ethers, ketones, alcohols and hydrocarbons.
(2) In ketones, carbonyl atom is attached to C-atom, hence C – C bond in Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 169 can’t be broken easily.
(3) H atom attached to carbonyl carbon can be oxidised to – OH group giving carboxylic group Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 168 Therefore, aldehyde reduces Tollen’s reagent, whereas simple hydrocarbons, ethers, ketones and alcohols do not reduce Tollen’s reagent.

Use your brain power! (Textbook Page No 269)

Question 1.
Sodium bisulfite is sodium salt of sulfurous acid, write down its detailed bond structure.
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 179
Bond structure of sodium bisulfite

Use your brain power! (Textbook Page No 270)

Predict the product of the following reaction:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 185
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 186

Use your brain power! (Textbook Page No 271)

Question 1.
Draw the structures of
(1) The semicarbazone of cyclohexanone
(2) The imine formed in the reaction between 2-methylhexanal and ethyl amine
(3) 2, 4-dinitrophenyl hydrazone of acetaldehyde.
Answer:
(1) The semi carbazone of cyclohexanone.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 217
(2) The imine formed between 2-methyl hexanal and ethyl amine.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 218
(3) 2, 4-dinitrophenylhydrazone of acetaldehyde.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 219

Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids

Try this….. (Textbook Page No 272)

Question 1.
Write chemical reactions taking place when propan-2-ol is treated with iodine and sodium hydroxide.
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 224

Question 2.
When acetaldehyde Is treated with dilute NaOH, the following reaction is observed.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 225
(1) What are the functional groups in the product?
(2) Can another product be formed during the same reaction? (Deduce the answer by doing atomic audit of reactant and product)
(3) Is this an addition reaction or condensation reaction?
Answer:
(1) There arc two functiona’ groups in the product: -CRO and -OH
(2) No other product can be formed in the same reaction.
(3) This is an addition reaction.

Use your brain power! (Textbook Page No 273)

Question 1.
Observe the following reaction.
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 234

Question 2.
Will this reaction give a mixture of products like a cross aldol reaction ?
Answer:
No, since benzaldehyde, Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 235 does not have a-hydrogen atom, it will not undergo self aldol condensation.

Use your brain power! (Textbook Page No 274)

Question 1.
Can isobutyraldehyde undergo a Cannizzaro reaction? Explain.
Answer:
Since isobutyraldehydeMaharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 252contains a-carbon atom, it cannot undergo Cannizzaro reaction.

Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids

Can you tell? (Textbook Page No 279)

What is the term used for elimination of water molecule ?
Answer:
Dehydration.

Use your brain power! (Textbook Page No 278)

Question 1.
Fill in the blanks and rewrite the balanced equations.
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 302
Answer:
Maharashtra Board Class 12 Chemistry Solutions Chapter 12 Aldehydes, Ketones and Carboxylic Acids 303

12th Std Chemistry Questions And Answers:

Maharashtra Board Class 12 BK & Accountancy Previous Year Question Papers with Solutions Pdf

Maharashtra Board Class 12th BK & Accountancy Previous Year Question Paper, BK & Accountancy Question Paper Class 12 Maharashtra State Board Pdf 2024, Maharashtra Board 12th BK & Accountancy Question Paper Pdf Download, BK & Accountancy Previous Year Question Paper Class 12 Maharashtra State Board Pdf, 12th BK & Accountancy Paper Pattern 2024 Maharashtra Board, Maharashtra Board Model Paper Class 12th BK & Accountancy Pdf with Solutions, BK & Accountancy Class 12 Maharashtra Board Question Paper.

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Maharashtra Board Class 12 BK & Accountancy Previous Year Question Paper, Previous Year BK & Accountancy Question Paper Class 12 Maharashtra State Board Pdf, Class 12 Maharashtra Board BK & Accountancy Previous Year Question Paper, BK & Accountancy Paper Pattern Class 12 Maharashtra Board 2024, 12th BK & Accountancy Question Paper Maharashtra State Board Pdf with Answers, Class 12 BK & Accountancy Question Paper 2024 Maharashtra Board.

Maharashtra Board Class 12th Previous Year Question Paper

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7

Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 Questions and Answers.

Maharashtra State Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7

(I) Choose the correct alternatives:

Question 1.
Area of the region bounded by the curve x2 = y, the X-axis and the lines x = 1 and x = 3 is
(a) \(\frac{26}{3}\) sq units
(b) \(\frac{3}{26}\) sq units
(c) 26 sq units
(d) 3 sq units
Answer:
(a) \(\frac{26}{3}\) sq units

Question 2.
The area of the region bounded by y2 = 4x, the X-axis and the lines x = 1 and x = 4 is
(a) 28 sq units
(b) 3 sq unit
(c) \(\frac{28}{3}\) sq units
(d) \(\frac{3}{28}\) sq units
Answer:
(c) \(\frac{28}{3}\) sq units

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7

Question 3.
Area of the region bounded by x2 = 16y, y = 1 and y = 4 and the Y-axis, lying in the first quadrant is
(a) 63 sq units
(b) \(\frac{3}{56}\) sq units
(c) \(\frac{56}{3}\) sq units
(d) \(\frac{63}{7}\) sq units
Answer:
(c) \(\frac{56}{3}\) sq units

Question 4.
Area of the region bounded by y = x4, x = 1, x = 5 and the X-axis is
(a) \(\frac{3142}{5}\) sq units
(b) \(\frac{3124}{5}\) sq units
(c) \(\frac{3142}{3}\) sq units
(d) \(\frac{3124}{3}\) sq units
Answer:
(b) \(\frac{3124}{5}\) sq units

Question 5.
Using definite integration area of circle x2 + y2 = 25 is
(a) 5π sq units
(b) 4π sq units
(c) 25π sq units
(d) 25 sq units
Answer:
(c) 25π sq units

(II) Fill in the blanks:

Question 1.
Area of the region bounded by y = x4, x = 1, x = 5 and the X-axis is _________
Answer:
\(\frac{3124}{5}\) sq units

Question 2.
Using definite integration area of the circle x2 + y2 = 49 is ___________
Answer:
49π sq units

Question 3.
Area of the region bounded by x2 = 16y, y = 1, y = 4 and the Y-axis lying in the first quadrant is _________
Answer:
\(\frac{56}{3}\) sq units

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7

Question 4.
The area of the region bounded by the curve x2 = y, the X-axis and the lines x = 3 and x = 9 is _________
Answer:
234 sq units

Question 5.
The area of the region bounded by y2 = 4x, the X-axis and the lines x = 1 and x = 4 is __________
Answer:
\(\frac{28}{3}\) sq units

(III) State whether each of the following is True or False.

Question 1.
The area bounded by the curve x = g(y), Y-axis and bounded between the lines y = c and y = d is given by \(\int_{c}^{d} x d y=\int_{y=c}^{y=d} g(y) d y\)
Answer:
True

Question 2.
The area bounded by two curves y = f(x), y = g(x) and X-axis is \(\left|\int_{a}^{b} f(x) d x-\int_{b}^{a} g(x) d x\right|\)
Answer:
False

Question 3.
The area bounded by the curve y = f(x), X-axis and lines x = a and x = b is \(\left|\int_{a}^{b} f(x) d x\right|\)
Answer:
True

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7

Question 4.
If the curve, under consideration, is below the X-axis, then the area bounded by curve, X-axis, and lines x = a, x = b is positive.
Answer:
False

Question 5.
The area of the portion lying above the X-axis is positive.
Answer:
True

(IV) Solve the following:

Question 1.
Find the area of the region bounded by the curve xy = c2, the X-axis, and the lines x = c, x = 2c.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q1
= c2 log(\(\frac{2 c}{c}\))
= c2 . log 2 sq units.

Question 2.
Find the area between the parabolas y2 = 7x and x2 = 7y.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q2
For finding the points of intersection of the two parabolas,
we equate the values of y2 from their equations.
From the equation x2 = 7y, y2 = \(\frac{x^{4}}{49}\)
∴ \(\frac{x^{4}}{49}\) = 7x
∴ x4 = 343x
∴ x4 – 343x = 0
∴ x(x3 – 343) = 0
∴ x = 0 or x3 = 343, i.e. x = 7
When x = 0, y = 0
When x = 7, 7y = 49
∴ y = 7
∴ the points of intersection are O(0, 0) and A(7, 7)
Required area = area of the region OBACO
= (area of the region ODACO) – (area of the region ODABO)
Now, area of the region ODACO = area under the parabola y2 = 7x
i.e. y = √7 √x
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q2.1
Area of the region ODABO = Area under the parabola
x2 = 7y
i.e. y = \(\frac{x^{2}}{7}\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q2.2
∴ required area = \(\frac{98}{3}-\frac{49}{3}=\frac{49}{3}\) sq units.

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7

Question 3.
Find the area of the region bounded by the curve y = x2 and the line y = 10.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q3
By the symmetry of the parabola,
the required area is twice the area of the region OABCO
Now, the area of the region OABCO
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q3.1

Question 4.
Find the area of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}\) = 1.
Solution:
By the symmetry of the ellipse, the required area of the ellipse is 4 times the area of the region OPQO.
For the region OPQO, the limits of integration are x = 0 and x = 4.
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q4
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q4.1

Question 5.
Find the area of the region bounded by y = x2, the X-axis and x = 1, x = 4.
Solution:
Required area = \(\int_{1}^{4} y d x\), where y = x2
= \(\int_{1}^{4} x^{2} d x\)
= \(\left[\frac{x^{3}}{3}\right]_{1}^{4}=\frac{4^{3}}{3}-\frac{1}{3}=\frac{64-1}{3}\)
= 21 sq units.

Question 6.
Find the area of the region bounded by the curve x2 = 25y, y = 1, y = 4, and the Y-axis.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q6

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7

Question 7.
Find the area of the region bounded by the parabola y2 = 25x and the line x = 5.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q7
Given the equation of the parabola is y2 = 25x
∴ y = 5√x …… [∵ IIn first quadrant, y > 0]
Required area = area of the region OQRPO
= 2(area of the region ORPO)
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q7.1

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1

Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 7 Application of Definite Integration Ex 7.1 Questions and Answers.

Maharashtra State Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1

Question 1.
Find the area of the region bounded by the following curves, the X-axis, and the given lines:
(i) y = x4, x = 1, x = 5
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q1(i)

(ii) y = \(\sqrt{6 x+4}\), x = 0, x = 2
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q1(ii)

(iii) \(\sqrt{16-x^{2}}\), x = 0, x = 4
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q1(iii)

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1

(iv) 2y = 5x + 7, x = 2, x = 8
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q1(iv)

(v) 2y + x = 8, x = 2, x = 4
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q1(v)

(vi) y = x2 + 1, x = 0, x = 3
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q1(vi)

(vii) y = 2 – x2, x = -1, x = 1
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q1(vii)

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1

Question 2.
Find the area of the region bounded by the parabola y2 = 4x and the line x = 3.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q2
Required area = area of the region OABO
= 2(area of the region OACO)
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q2.1

Question 3.
Find the area of the circle x2 + y2 = 25.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q3
By the symmetry of the circle, its area is equal to 4 times the area of the region OABO.
Clearly, for this region, the limits of integration are 0 and 5.
From the equation of the circle, y2 = 25 – x2.
In the first quadrant y > 0
∴ y = \(\sqrt{25-x^{2}}\)
∴ area of the circle = 4(area of region OABO)
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q3.1

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1

Question 4.
Find the area of the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{25}\) = 1
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q4
By the symmetry of the ellipse, its area is equal to 4 times the area of the region OABO.
Clearly, for this region, the limits of integration are 0 and 2.
From the equation of the ellipse,
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q4.1
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q4.2

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 6 Definite Integration Miscellaneous Exercise 6 Questions and Answers.

Maharashtra State Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

(I) Choose the correct alternative:

Question 1.
\(\int_{-9}^{9} \frac{x^{3}}{4-x^{2}} d x\) = ________
(a) 0
(b) 3
(c) 9
(d) -9
Answer:
(a) 0

Question 2.
\(\int_{-2}^{3} \frac{d x}{x+5}\) = _________
(a) -log(\(\frac{8}{3}\))
(b) log(\(\frac{8}{3}\))
(c) log(\(\frac{3}{8}\))
(d) -log(\(\frac{3}{8}\))
Answer:
(b) log(\(\frac{8}{3}\))

Question 3.
\(\int_{2}^{3} \frac{x}{x^{2}-1} d x\) = _________
(a) log(\(\frac{8}{3}\))
(b) -log(\(\frac{8}{3}\))
(c) \(\frac{1}{2}\) log(\(\frac{8}{3}\))
(d) \(\frac{-1}{2}\) log(\(\frac{8}{3}\))
Answer:
(c) \(\frac{1}{2}\) log(\(\frac{8}{3}\))

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 4.
\(\int_{4}^{9} \frac{d x}{\sqrt{x}}\) = ___________
(a) 9
(b) 4
(c) 2
(d) 0
Answer:
(c) 2

Question 5.
If \(\int_{0}^{a} 3 x^{2} d x=8\), then a = __________
(a) 2
(b) 0
(c) \(\frac{8}{3}\)
(d) a
Answer:
(a) 2

Question 6.
\(\int_{2}^{3} x^{4}\) dx = ________
(a) \(\frac{1}{2}\)
(b) \(\frac{5}{2}\)
(c) \(\frac{5}{211}\)
(d) \(\frac{211}{5}\)
Answer:
(d) \(\frac{211}{5}\)

Question 7.
\(\int_{0}^{2} e^{x}\) dx = _______
(a) e – 1
(b) 1 – e
(c) 1 – e2
(d) e2 – 1
Answer:
(d) e2 – 1

Question 8.
\(\int_{a}^{b} f(x) d x\) = ________
(a) \(\int_{b}^{a} f(x) d x\)
(b) –\(\int_{a}^{b} f(x) d x\)
(c) –\(\int_{b}^{a} f(x) d x\)
(d) \(\int_{0}^{a} f(x) d x\)
Answer:
(c) –\(\int_{b}^{a} f(x) d x\)

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 9.
\(\int_{-7}^{7} \frac{x^{3}}{x^{2}+7} d x\) = _________
(a) 7
(b) 49
(c) 0
(d) \(\frac{7}{2}\)
Answer:
(c) 0

Question 10.
\(\int_{2}^{7} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{9-x}} d x\) = _________
(a) \(\frac{7}{2}\)
(b) \(\frac{5}{2}\)
(c) 7
(d) 2
Answer:
(b) \(\frac{5}{2}\)

(II) Fill in the blanks:

Question 1.
\(\int_{0}^{2} e^{x} d x\) = ________
Answer:
e2 – 1

Question 2.
\(\int_{2}^{3} x^{4} d x\) = __________
Answer:
\(\frac{211}{5}\)

Question 3.
\(\int_{0}^{1} \frac{d x}{2 x+5}\) = ____________
Answer:
\(\frac{1}{2} \log \left(\frac{7}{5}\right)\)

Question 4.
If \(\int_{0}^{a} 3 x^{2} d x\) = 8, then a = _________
Answer:
2

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 5.
\(\int_{4}^{9} \frac{1}{\sqrt{x}} d x\) = _________
Answer:
2

Question 6.
\(\int_{2}^{3} \frac{x}{x^{2}-1} d x\) = _________
Answer:
\(\frac{1}{2} \log \left(\frac{8}{3}\right)\)

Question 7.
\(\int_{-2}^{3} \frac{d x}{x+5}\) = _________
Answer:
\(\log \left(\frac{8}{3}\right)\)

Question 8.
\(\int_{-9}^{9} \frac{x^{3}}{4-x^{2}} d x\) = _____________
Answer:
o

(III) State whether each of the following is True or False:

Question 1.
\(\int_{a}^{b} f(x) d x=\int_{-b}^{-a} f(x) d x\)
Answer:
True

Question 2.
\(\int_{a}^{b} f(x) d x=\int_{a}^{b} f(t) d t\)
Answer:
True

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 3.
\(\int_{0}^{a} f(x) d x=\int_{a}^{0} f(a-x) d x\)
Answer:
False

Question 4.
\(\int_{a}^{b} f(x) d x=\int_{a}^{b} f(x-a-b) d x\)
Answer:
False

Question 5.
\(\int_{-5}^{5} \frac{x^{3}}{x^{2}+7} d x=0\)
Answer:
True

Question 6.
\(\int_{1}^{2} \frac{\sqrt{x}}{\sqrt{3-x}+\sqrt{x}} d x=\frac{1}{2}\)
Answer:
True

Question 7.
\(\int_{2}^{7} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{9-x}} d x=\frac{9}{2}\)
Answer:
False

Question 8.
\(\int_{4}^{7} \frac{(11-x)^{2}}{(11-x)^{2}+x^{2}} d x=\frac{3}{2}\)
Answer:
True

(IV) Solve the following:

Question 1.
\(\int_{2}^{3} \frac{x}{(x+2)(x+3)} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q1
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q1.1

Question 2.
\(\int_{1}^{2} \frac{x+3}{x(x+2)} d x\)
Solution:
Let I = \(\int_{1}^{2} \frac{x+3}{x(x+2)} d x\)
Let \(\frac{x+3}{x(x+2)}=\frac{A}{x}+\frac{B}{x+2}\)
∴ x + 3 = A(x + 2) + Bx
Put x = 0, we get
3 = A(2) + B(0)
∴ A = \(\frac{3}{2}\)
Put x + 2 = 0, i.e. x = -2, we get
-2 + 3 = A(0) + B(-2)
∴ 1 = -2B
∴ B = \(-\frac{1}{2}\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q2

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 3.
\(\int_{1}^{3} x^{2} \log x d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q3

Question 4.
\(\int_{0}^{1} e^{x^{2}} \cdot x^{3} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q4
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q4.1

Question 5.
\(\int_{1}^{2} e^{2 x}\left(\frac{1}{x}-\frac{1}{2 x^{2}}\right) d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q5

Question 6.
\(\int_{4}^{9} \frac{1}{\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q6

Question 7.
\(\int_{-2}^{3} \frac{1}{x+5} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q7

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 8.
\(\int_{2}^{3} \frac{x}{x^{2}-1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q8

Question 9.
\(\int_{0}^{1} \frac{x^{2}+3 x+2}{\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q9

Question 10.
\(\int_{3}^{5} \frac{d x}{\sqrt{x+4}+\sqrt{x-2}}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q10

Question 11.
\(\int_{2}^{3} \frac{x}{x^{2}+1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q11
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q11.1

Question 12.
\(\int_{1}^{2} x^{2} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q12

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 13.
\(\int_{-4}^{-1} \frac{1}{x} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q13

Question 14.
\(\int_{0}^{1} \frac{1}{\sqrt{1+x}+\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q14
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q14.1

Question 15.
\(\int_{0}^{4} \frac{1}{\sqrt{x^{2}+2 x+3}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q15
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q15.1

Question 16.
\(\int_{2}^{4} \frac{x}{x^{2}+1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q16

Question 17.
\(\int_{0}^{1} \frac{1}{2 x-3} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q17

Question 18.
\(\int_{1}^{2} \frac{5 x^{2}}{x^{2}+4 x+3} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q18
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q18.1

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 19.
\(\int_{1}^{2} \frac{d x}{x(1+\log x)^{2}}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q19

Question 20.
\(\int_{0}^{9} \frac{1}{1+\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q20
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q20.1

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2

Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 6 Definite Integration Ex 6.2 Questions and Answers.

Maharashtra State Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2

Evaluate the following integrals:

Question 1.
\(\int_{-9}^{9} \frac{x^{3}}{4-x^{2}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q1

Question 2.
\(\int_{0}^{a} x^{2}(a-x)^{3 / 2} d x\)
Solution:
We use the property
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q2

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2

Question 3.
\(\int_{1}^{3} \frac{\sqrt[3]{x+5}}{\sqrt[3]{x+5}+\sqrt[3]{9-x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q3

Question 4.
\(\int_{2}^{5} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{7-x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q4

Question 5.
\(\int_{1}^{2} \frac{\sqrt{x}}{\sqrt{3-x}+\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q5

Question 6.
\(\int_{2}^{7} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{9-x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q6

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2

Question 7.
\(\int_{0}^{1} \log \left(\frac{1}{x}-1\right) d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q7
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q7.1

Question 8.
\(\int_{0}^{1} x(1-x)^{5} d x\)
Solution:
We use the property
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q8

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1

Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 6 Definite Integration Ex 6.1 Questions and Answers.

Maharashtra State Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1

Evaluate the following definite integrals:

Question 1.
\(\int_{4}^{9} \frac{1}{\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q1

Question 2.
\(\int_{-2}^{3} \frac{1}{x+5} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q2

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1

Question 3.
\(\int_{2}^{3} \frac{x}{x^{2}-1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q3
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q3.1

Question 4.
\(\int_{0}^{1} \frac{x^{2}+3 x+2}{\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q4

Question 5.
\(\int_{2}^{3} \frac{x}{(x+2)(x+3)} d x\)
Solution:
Let I = \(\int_{2}^{3} \frac{x}{(x+2)(x+3)} d x\)
Let \(\frac{x}{(x+2)(x+3)}=\frac{A}{x+3}+\frac{B}{x+2}\)
∴ x = A(x + 2) + B(x + 3)
Put x + 3 = 0, i.e. x = -3, we get
-3 = A(-1) + B(0)
∴ A = 3
Put x + 2 = 0, i.e. x = -2, we get
-2 = A(0) + B(1)
∴ B = -2
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q5
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q5.1

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1

Question 6.
\(\int_{1}^{2} \frac{d x}{x^{2}+6 x+5}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q6

Question 7.
If \(\int_{0}^{a}(2 x+1) d x\) = 2, find the real values of ‘a’.
Solution:
Let I = \(\int_{0}^{a}(2 x+1) d x\)
= \(\left[2 \cdot \frac{x^{2}}{2}+x\right]_{0}^{a}\)
= a2 + a – 0
= a2 + a
∴ I = 2 gives a2 + a = 2
∴ a2 + a – 2 = 0
∴ (a + 2)(a – 1) = 0 1
∴ a + 2 = 0 or a – 1 = 0
∴ a = -2 or a = 1.

Question 8.
If \(\int_{1}^{a}\left(3 x^{2}+2 x+1\right) d x\) = 11, find ‘a’.
Solution:
Let I = \(\int_{1}^{a}\left(3 x^{2}+2 x+1\right) d x\)
= \(\left[3\left(\frac{x^{3}}{3}\right)+2\left(\frac{x^{2}}{2}\right)+x\right]_{1}^{a}\)
= \(\left[x^{3}+x^{2}+x\right]_{1}^{a}\)
= (a3 + a2 + a) – (1 + 1 + 1)
= a3 + a2 + a – 3
∴ I = 11 gives a3 + a2 + a – 3 = 11
∴ a3 + a2 + a – 14 = 0
∴ (a3 – 8) + (a2 + a – 6) = 0
∴ (a – 2)(a2 + 2a + 4) + (a + 3)(a – 2) = 0
∴ (a – 2)(a2 + 2a + 4 + a + 3) = 0
∴ (a – 2)(a2 + 3a + 7) = 0
∴ a – 2 = 0 or a2 + 3a + 7 = 0
∴ a = 2 or a = \(\frac{-3 \pm \sqrt{9-28}}{2}\)
The latter two roots are not real.
∴ they are rejected.
∴ a = 2.

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1

Question 9.
\(\int_{0}^{1} \frac{1}{\sqrt{1+x}+\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q9

Question 10.
\(\int_{1}^{2} \frac{3 x}{9 x^{2}-1} d x\)
Solution:
Let I = \(\int_{1}^{2} \frac{3 x}{9 x^{2}-1} d x\) = \(\int_{1}^{2} \frac{3 x}{(3 x)^{2}-1} d x\)
Put 3x = t
∴ 3 dx = dt
∴ dx = \(\frac{d t}{3}\)
When x = 1, t = 3 × 1 = 3
When x = 2, t = 3 × 2 = 6
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q10
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q10.1

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1

Question 11.
\(\int_{1}^{3} \log x d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q11

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 5 Integration Miscellaneous Exercise 5 Questions and Answers.

Maharashtra State Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

(I) Choose the correct alternative from the following:

Question 1.
The value of \(\int \frac{d x}{\sqrt{1-x}}\) is
(a) 20\(\sqrt{1-x}\) + c
(b) -2\(\sqrt{1-x}\) + c
(c) √x + c
(d) x + c
Answer:
(b) -2\(\sqrt{1-x}\) + c

Question 2.
\(\int \sqrt{1+x^{2}} d x\) =
(a) \(\frac{x}{2} \sqrt{1+x^{2}}+\frac{1}{2} \log \left(x+\sqrt{1+x^{2}}\right)+c\)
(b) \(\frac{2}{3}\left(1+x^{2}\right)^{3 / 2}+c\)
(c) \(\frac{1}{3}\left(1+x^{2}\right)+c\)
(d) \(\frac{(x)}{\sqrt{1+x^{2}}}+c\)
Answer:
(a) \(\frac{x}{2} \sqrt{1+x^{2}}+\frac{1}{2} \log \left(x+\sqrt{1+x^{2}}\right)+c\)

Question 3.
\(\int x^{2}(3)^{x^{3}} d x\) =
(a) \(\text { (3) }^{x^{3}}+c\)
(b) \(\frac{(3)^{x^{3}}}{3 \cdot \log 3}+c\)
(c) \(\log 3(3)^{x^{3}}+c\)
(d) \(x^{2}(3)^{x 3}\)
Answer:
(b) \(\frac{(3)^{x^{3}}}{3 \cdot \log 3}+c\)
Hint:
Put x3 = t

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

Question 4.
\(\int \frac{x+2}{2 x^{2}+6 x+5} d x=p \int \frac{4 x+6}{2 x^{2}+6 x+5} d x\) + \(\frac{1}{2} \int \frac{d x}{2 x^{2}+6 x+5}\), then p = __________
(a) \(\frac{1}{3}\)
(b) \(\frac{1}{2}\)
(c) \(\frac{1}{4}\)
(d) 2
Answer:
(c) \(\frac{1}{4}\)
Hint:
\(\int \frac{x+2}{2 x^{2}+6 x+5} d x=\int \frac{\frac{1}{4}(4 x+6)+\frac{1}{2}}{2 x^{2}+6 x+5} d x\)

Question 5.
\(\int \frac{d x}{\left(x-x^{2}\right)}\) = ________
(a) log x – log(1 – x) + c
(b) log(1 – x2) + c
(c) -log x + log(1 – x) + c
(d) log(x – x2) + c
Answer:
(a) log x – log(1 – x) + c

Question 6.
\(\int \frac{d x}{(x-8)(x+7)}\) = __________
(a) \(\frac{1}{15} \log \left|\frac{x+2}{x-1}\right|+c\)
(b) \(\frac{1}{15} \log \left|\frac{x+8}{x+7}\right|+c\)
(c) \(\frac{1}{15} \log \left|\frac{x-8}{x+7}\right|+c\)
(d) (x – 8)(x – 7) + c
Answer:
(c) \(\frac{1}{15} \log \left|\frac{x-8}{x+7}\right|+c\)

Question 7.
\(\int\left(x+\frac{1}{x}\right)^{3} d x\) = _________
(a) \(\frac{1}{4}\left(x+\frac{1}{x}\right)^{4}+c\)
(b) \(\frac{x^{4}}{4}+\frac{3 x^{2}}{2}+3 \log x-\frac{1}{2 x^{2}}+c\)
(c) \(\frac{x^{4}}{4}+\frac{3 x^{2}}{2}+3 \log x+\frac{1}{x^{2}}+c\)
(d) \(\left(x-x^{-1}\right)^{3}+c\)
Answer:
(b) \(\frac{x^{4}}{4}+\frac{3 x^{2}}{2}+3 \log x-\frac{1}{2 x^{2}}+c\)
Hint:
\(\left(x+\frac{1}{x}\right)^{3}=x^{3}+3 x+\frac{3}{x}+\frac{1}{x^{3}}\)

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

Question 8.
\(\int\left(\frac{e^{2 x}+e^{-2 x}}{e^{x}}\right) d x\)
(a) \(e^{x}-\frac{1}{3 e^{3 x}}+c\)
(b) \(e^{x}+\frac{1}{3 e^{3 x}}+c\)
(c) \(e^{-x}+\frac{1}{3 e^{3 x}}+c\)
(d) \(e^{-x}-\frac{1}{3 e^{3 x}}+c\)
Answer:
(a) \(e^{x}-\frac{1}{3 e^{3 x}}+c\)

Question 9.
∫(1 – x)-2 dx = ___________
(a) (1 + x)-1 + c
(b) (1 – x)-1 + c
(c) (1 – x)-1 – 1 + c
(d) (1 – x)-1 + 1 + c
Answer:
(b) (1 – x)-1 + c

Question 10.
\(\int \frac{\left(x^{3}+3 x^{2}+3 x+1\right)}{(x+1)^{5}} d x\) = _______
(a) \(\frac{-1}{x+1}+c\)
(b) \(\left(\frac{-1}{x+1}\right)^{5}+c\)
(c) log(x + 1) + c
(d) log|x + 1|5 + c
Answer:
(a) \(\frac{-1}{x+1}+c\)
Hint:
x3 + 3x2 + 3x + 1 = (x + 1)3

(II) Fill in the blanks.

Question 1.
\(\int \frac{5\left(x^{6}+1\right)}{x^{2}+1}\)dx = x4 + ___x3 + 5x + c.
Answer:
\(-\frac{5}{3}\)
Hint:
x6 + 1 = (x2 + 1)(x4 – x2 + 1)

Question 2.
\(\int \frac{x^{2}+x-6}{(x-2)(x-1)} d x\) = x + ______ + c
Answer:
4 log|x – 1|
Hint:
x2 + x – 6 = (x + 3)(x – 2)

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

Question 3.
If f'(x) = \(\frac{1}{x}\) + x and f(1) = \(\frac{5}{2}\) then f(x) = log x + \(\frac{x^{2}}{2}\) + _______
Answer:
2
Hint:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 II Q3

Question 4.
To find the value of \(\int \frac{(1+\log x) d x}{x}\) the proper substitution is __________
Answer:
1 + log x = t

Question 5.
\(\int \frac{1}{x^{3}}\left[\log x^{x}\right]^{2} d x\) = p(log x)3 + c, then p = _______
Answer:
\(\frac{1}{3}\)
Hint:
\(\frac{1}{x^{3}}\left(\log x^{x}\right)^{2}=\frac{1}{x^{3}}(x \log x)^{2}=\frac{(\log x)^{2}}{x}\)

(III) State whether each of the following is True or False:

Question 1.
The proper substitution for \(\int x\left(x^{x}\right)^{x}(2 \log x+1) d x \text { is }\left(x^{x}\right)^{x}=t\)
Answer:
True

Question 2.
If ∫x e2x dx is equal to e2x f(x) + c where c is constant of integration, then f(x) is \(\frac{(2 x-1)}{2}\).
Answer:
False

Question 3.
If ∫x f(x) dx = \(\frac{f(x)}{2}\), then f(x) = \(e^{x^{2}}\).
Answer:
True

Question 4.
If \(\int \frac{(x-1) d x}{(x+1)(x-2)}\) = A log|x + 1| + B log|x – 2|, then A + B = 1.
Answer:
True

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

Question 5.
For \(\int \frac{x-1}{(x+1)^{3}} e^{x} d x\) = ex f(x) + c, f(x) = (x + 1)2.
Answer:
False

(IV) Solve the following:

1. Evaluate:

(i) \(\int \frac{5 x^{2}-6 x+3}{2 x-3} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q1(i)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q1(i).1

(ii) \(\int(5 x+1)^{\frac{4}{9}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q1(ii)

(iii) \(\int \frac{1}{2 x+3} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q1(iii)

(iv) \(\int \frac{x-1}{\sqrt{x+4}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q1(iv)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q1(iv).1

(v) If f'(x) = √x and f(1) = 2, then find the value of f(x).
Solution:
By the definition of integral
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q1(v)

(vi) ∫|x| dx if x < 0
Solution:
∫|x| dx = ∫-x dx …..[∵ x < 0]
= -∫x dx
= \(-\frac{x^{2}}{2}\) + c

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

2. Evaluate:

(i) Find the primitive of \(\frac{1}{1+e^{x}}\)
Solution:
Let I be the primitive of \(\frac{1}{1+e^{x}}\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q2(i)

(ii) \(\int \frac{a e^{x}+b e^{-x}}{\left(a e^{x}-b e^{-x}\right)^{2}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q2(ii)

(iii) \(\int \frac{1}{2 x+3 x \log x} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q2(iii)

(iv) \(\int \frac{1}{\sqrt{x}+x} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q2(iv)

(v) \(\int \frac{2 e^{x}-3}{4 e^{x}+1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q2(v)

3. Evaluate:

(i) \(\int \frac{d x}{\sqrt{4 x^{2}-5}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(i)

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

(ii) \(\int \frac{d x}{3-2 x-x^{2}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(ii)

(iii) \(\int \frac{d x}{9 x^{2}-25}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(iii)

(iv) \(\int \frac{e^{x}}{\sqrt{e^{2 x}+4 e^{x}+13}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(iv)

(v) \(\int \frac{d x}{x\left[(\log x)^{2}+4 \log x-1\right]}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(v)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(v).1

(vi) \(\int \frac{d x}{5-16 x^{2}}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(vi)

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

(vii) \(\int \frac{d x}{25 x-x(\log x)^{2}}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(vii)

(viii) \(\int \frac{e^{x}}{4 e^{2 x}-1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(viii)

4. Evaluate:

(i) ∫(log x)2 dx
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q4(i)

(ii) \(\int e^{x} \frac{1+x}{(2+x)^{2}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q4(ii)

(iii) ∫x e2x dx
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q4(iii)

(iv) ∫log(x2 + x) dx
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q4(iv)

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

(v) \(\int e^{\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q4(v)

(vi) \(\int \sqrt{x^{2}+2 x+5} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q4(vi)

(vii) \(\int \sqrt{x^{2}-8 x+7} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q4(vii)

5. Evaluate:

(i) \(\int \frac{3 x-1}{2 x^{2}-x-1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(i)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(i).1

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

(ii) \(\int \frac{2 x^{3}-3 x^{2}-9 x+1}{2 x^{2}-x-10} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(ii)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(ii).1
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(ii).2

(iii) \(\int \frac{1+\log x}{x(3+\log x)(2+3 \log x)} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(iii)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(iii).1
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(iii).23

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6

Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 5 Integration Ex 5.6 Questions and Answers.

Maharashtra State Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6

Evaluate:

Question 1.
\(\int \frac{2 x+1}{(x+1)(x-2)} d x\)
Solution:
Let I = \(\int \frac{2 x+1}{(x+1)(x-2)} d x\)
Let \(\frac{2 x+1}{(x+1)(x-2)}=\frac{A}{x+1}+\frac{B}{x-2}\)
∴ 2x + 1 = A(x – 2) + B(x + 1)
Put x + 1 = 0, i.e. x = -1, we get
2(-1) + 1 = A(-3) + B(0)
∴ A = \(\frac{1}{3}\)
Put x – 2 = 0, i.e. x = 2, we get
2(2) + 1 = A(0) + B(3)
∴ B = \(\frac{5}{3}\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6 Q1
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6 Q1.1

Question 2.
\(\int \frac{2 x+1}{x(x-1)(x-4)} d x\)
Solution:
Let I = \(\int \frac{2 x+1}{x(x-1)(x-4)} d x\)
Let \(\int \frac{2 x+1}{x(x-1)(x-4)}=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x-4}\)
∴ 2x + 1 = A(x – 1)(x – 4) + Bx(x – 4) + Cx(x – 1)
Put x = 0, we get
2(0) + 1 = A(-1)(-4) + B(0)(-4) + C(0)(-1)
∴ 1 = 4A
∴ A = \(\frac{1}{4}\)
Put x – 1 = 0, i.e. x = 1, we get
2(1) + 1 = A(0)(-3) + B(1)(-3) + C(1)(0)
∴ 3 = -3B
∴ B = -1
Put x – 4 = 0, i.e x = 4, we get
2(4) + 1 = A(3)(0) + B(4)(0) + C(4)(3)
∴ 9 = 12C
∴ C = \(\frac{3}{4}\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6 Q2

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6

Question 3.
\(\int \frac{x^{2}+x-1}{x^{2}+x-6} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6 Q3
∴ 1 = A(x – 2) + B(x + 3)
Put x + 3 = 0, i.e. x = -3, we get
1 = A(-5) + B (0)
∴ A = \(\frac{-1}{5}\)
Put x – 2 = 0, i.e. x = 2, we get
1 = A(0) + B(5)
∴ B = \(\frac{1}{5}\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6 Q3.1

Question 4.
\(\int \frac{x}{(x-1)^{2}(x+2)} d x\)
Solution:
Let I = \(\int \frac{x}{(x-1)^{2}(x+2)} d x\)
Let \(\frac{x}{(x-1)^{2}(x+2)}=\frac{A}{x-1}+\frac{B}{(x-1)^{2}}+\frac{C}{x+2}\)
∴ x = A(x – 1)(x + 2) + B(x + 2) + C(x – 1)2
Put x – 1 = 0, i.e. x = 1, we get
1 = A(0)(3) + B(3) + C(0)
∴ B = \(\frac{1}{3}\)
Put x + 2 = 0, i.e. x = -2, we get
-2 = A (-3)(0) + B(0) + C(9)
∴ C = \(-\frac{2}{9}\)
Put x = -1, we get,
-1 = A(-2)(1) + B(1) + C(4)
But B = \(\frac{1}{3}\) and C = \(-\frac{2}{9}\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6 Q4

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6

Question 5.
\(\int \frac{3 x-2}{(x+1)^{2}(x+3)} d x\)
Solution:
Let I = \(\int \frac{3 x-2}{(x+1)^{2}(x+3)} d x\)
Let \(\frac{3 x-2}{(x+1)^{2}(x+3)}=\frac{A}{x+1}+\frac{B}{(x+1)^{2}}+\frac{C}{x+3}\)
∴ 3x – 2 = A(x + 1)(x + 3) + B(x + 3) + C(x + 1)2
Put x + 1 = 0, i.e. x = -1, we get
3(-1) – 2 = A(0)(2) + B(2) + C(0)
∴ -5 = 2B
∴ B = \(-\frac{5}{2}\)
Put x + 3 = 0, i.e. x = -3, we get
3(-3) – 2 = A(-2)(0) + B(0) + C(4)
∴ -11 = 4C
∴ C = \(-\frac{11}{4}\)
Put x = 0, we get
3(0) – 2 = A(1)(3) + B(3) + C(1)
∴ -2 = 3A + 3B + C
But B = \(-\frac{5}{2}\) and C = \(-\frac{11}{4}\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6 Q5

Question 6.
\(\int \frac{1}{x\left(x^{5}+1\right)} d x\)
Solution:
Let I = \(\int \frac{1}{x\left(x^{5}+1\right)} d x\)
= \(\int \frac{x^{4}}{x^{5}\left(x^{5}+1\right)} d x\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6 Q6

Question 7.
\(\int \frac{1}{x\left(x^{n}+1\right)} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6 Q7

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6

Question 8.
\(\int \frac{5 x^{2}+20 x+6}{x^{3}+2 x^{2}+x} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6 Q8
Let \(\frac{5 x^{2}+20 x+6}{x(x+1)^{2}}=\frac{A}{x}+\frac{B}{x+1}+\frac{C}{(x+1)^{2}}\)
∴ 5x2 + 20x + 6 = A(x + 1)2 + Bx(x + 1) + Cx
Put x = 0, we get
0 + 0 + 6 = A(1) + B(0)(1) + C(0)
∴ A = 6
Put x + 1 = 0, i.e. x = -1, we get
5(1) + 20(-1) + 6 = A(0) + B(-1)(0) + C(-1)
∴ -9 = -C
∴ C = 9
Put x = 1, we get
5(1) + 20(1) + 6 = A(4) + B(1)(2) + C(1)
But A = 6 and C = 9
∴ 31 = 24 + 2B + 9
∴ B = -1
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.6 Q8.1