NCERT Chemistry Question Paper (Class - 12)

:: Chapter 1 - The Solid State ::

INTEXT QUESTIONS

Question 1.1: Why are solids rigid?

Question 1.2: Why do solids have a definite volume?

Question 1.3: Classify the following as amorphous or crystalline solids: Polyurethane, naphthalene, benzoic acid, teflon, potassium nitrate, cellophane, olyvinylchloride, fibre glass, copper.

Question 1.4: Why is glass considered a super cooled liquid?

Question 1.5: Refractive index of a solid is observed to have the same value along all directions. Comment on the nature of this solid. Would it show cleavage property?

Question 1.6: Classify the following solids in different categories based on the nature of intermolecular forces operating in them:
Potassium sulphate, tin, benzene, urea, ammonia, water, zinc sulphide, graphite, rubidium, argon, silicon carbide.

Question 1.7: Solid A is a very hard electrical insulator in solid as well as in molten state and melts at extremely high temperature. What type of solid is it?

Question 1.8: Ionic solids conduct electricity in molten state but not in solid state. Explain.

Question 1.9: What type of solids are electrical conductors, malleable and ductile?

Question 1.10: Give the significance of a ‘lattice point’.

Question 1.11: Name the parameters that characterize a unit cell.

Question 1.12: Distinguish between (i) Hexagonal and monoclinic unit cells (ii) Face−centred and end−centred unit cells.

Question 1.13: Explain how much portion of an atom located at (i) corner and (ii) body−centre of a cubic unit cell is part of its neighboring unit cell.

Question 1.14: What is the two dimensional coordination number of a molecule in square close packed layer?

Question 1.15: A compound forms hexagonal close−packed structure. What is the total number of voids in 0.5 mol of it? How many of these are tetrahedral voids?

Question 1.16: A compound is formed by two elements M and N. The element N forms ccp and atoms of M occupy 1/3rd of tetrahedral voids. What is the formula of the compound?

Question 1.17: Which of the following lattices has the highest packing efficiency (i) simple cubic (ii) body−centred cubic and (iii) hexagonal close−packed lattice?

Question 1.18: An element with molar mass 2.7 × 10−2 kg mol−1 forms a cubic unit cell with edge length 405 pm. If its density is 2.7 × 103 kg m−3, what is the nature of the cubic unit cell?

Question 1.19: What type of defect can arise when a solid is heated? Which physical property is effected by it and in what way?

Question 1.20: What type of stoichiometric defect is shown by: (i) ZnS (ii) AgBr

Question 1.21: Explain how vacancies are introduced in an ionic solid when a cation of higher valence is added as an impurity in it.

Question 1.22: Ionic solids, which have anionic vacancies due to metal excess defect, develop colour.Explain with the help of a suitable example.

Question 1.23: A group 14 element is to be converted into n−type semiconductor by doping it with a suitable impurity. To which group should this impurity belong?

Question 1.24:What type of substances would make better permanent magnets, ferromagnetic or ferrimagnetic. Justify your Solution:.

EXERCISE

Question 1:Define the term 'amorphous'. Give a few examples of amorphous solids.

Question 2. What makes a glass different from a solid such as quartz?

Question 3. Classify each of the following solids as ionic, metallic, molecular, network (covalent) or amorphous.

Question 4. (i) What is meant by the term 'coordination number' ? (ii) What is the coordination number of atoms (a) in a cubic close packed structure? (b) in a body–centered cubic structure?

Question 5. How can you determine the atomic mass of an unknown metal if you know its density?

Question 6. 'Stability of a crystal is reflected in the magnitude of its melting points'. Comment.
Collect melting points of solid water, ethyl alcohol, diethyl ether and methane from a data book. What can you say about the intermolecular forces between these molecules?

Question 7. How will you distinguish between the following pairs of terms (i) Hexagonal close packing and cubic close packing (ii) Crystal lattice and unit cell (iii) Tetrahedral void and octahedral void.

Question 8 How many lattice points are there in one unit cell of each of the following lattice? (i) Face–centred cubic (ii) Face–centred tetragonal (iii) Body–centred

Question 9. Explain (i) The basis of similarities and differences between metallic and ionic crystals. (ii) Ionic solids are hard and brittle.

Question 10 Calculate the efficiency of packing in case of a metal crystal for (i) simple cubic (ii) body–centred cubic (iii) face–centred cubic (with the assumptions that atoms are touching each other).

Question 11 Silver crystallises in fcc lattice. If edge length of the cell is 4.07 × 10-8 cm and density is 0.5 g cm3, calculate the atomic mass of silver.

Question 12 A cubic solid is made of two elements P and Q. Atoms of Q are at the corners of the cube and P at the body–centre. What is the formula of the compound? What are the coordination numbers of P and Q?

Question 13 Niobium crystallises in body–centred cubic structure. If density is 8.55 g cm–3, calculate atomic radius of niobium using its atomic mass 93 u.

Question 14 If the radius of the octahedral void is r and radius of the atoms in closepacking is R, derive relation between r and R.

Question 15 Copper crystallises into a fcc lattice with edge length 3.61 × 10–8 cm. Show that the calculated density is in agreement with its measured value of 8.92 g cm-3.

Question 16 Analysis shows that nickel oxide has the formula Ni0.98O1.00. What fractions of nickel exist as Ni2+ and Ni3+ ions?

Question 17 What is a semiconductor? Describe the two main types of semiconductors and contrast their conduction mechanism.

Question 18: Non–stoichiometric cuprous oxide, Cu2O can be prepared in laboratory. In this oxide, copper to oxygen ratio is slightly less than 2:1. Can you account for the fact that this substance is a p–type semiconductor?

Question 19: Ferric oxide crystallises in a hexagonal close–packed array of oxide ions with two out of every three octahedral holes occupied by ferric ions. Derive the formula of the ferric oxide.

Question 20: Classify each of the following as being either a p–type or an n–type semiconductor: (i) Ge doped with In (ii) B doped with Si.

Question 21: Gold (atomic radius = 0.144 nm) crystallises in a face–centred unit cell. What is the length of a side of the cell?

Question 22: In terms of band theory, what is the difference (i) Between a conductor and an insulator (ii) Between a conductor and a semiconductor

Question 23: Explain the following terms with suitable examples: (i) Schottky defect (ii) Frenkel defect (iii) Interstitials and (iv) F–centres

Question 24: Aluminium crystallises in a cubic close–packed structure. Its metallic radius is 125 pm. (i) What is the length of the side of the unit cell? (ii) How many unit cells are there in 1.00 cm3 of aluminum?

Question 25: If NaCl is doped with 10−3 mol % of SrCl2, what is the concentration of cation vacancies?

Question 26: Explain the following with suitable examples: (i) Ferromagnetism (ii)Paramagnetism (iii)Ferrimagnetism (iv)Antiferromagnetism (v)12–16 and 13–15 group compounds.

:: Chapter 2 - Solutions ::

EXERCISE

2.2 Give an example of a solid solution in which the solute is a gas.

2.3 Define the following terms:
(i) Mole fraction

2.3 Define the following terms:
(ii) Molality

2.3 Define the following terms:
(iii) Molarity

2.3 Define the following terms:
(iv) Mass percentage.

2.4 Concentrated nitric acid used in laboratory work is 68% nitric acid by mass in aqueous solution. What should be the molarity of such a sample of the acid if the density of the solution is 1.504 g mL–1?

2.5 A solution of glucose in water is labelled as 10% w/w, what would be the molality and mole fraction of each component in the solution? If the density of solution is 1.2 g mL–1, then what shall be the molarity of the solution?

2.6 How many mL of 0.1 M HCl are required to react completely with 1 g mixture of Na2CO3 and NaHCO3 containing equimolar amounts of both?

2.7 A solution is obtained by mixing 300 g of 25% solution and 400 g of 40% solution by mass. Calculate the mass percentage of the resulting solution.

2.8 An antifreeze solution is prepared from 222.6 g of ethylene glycol (C2H6O2) and 200 g of water. Calculate the molality of the solution. If the density of the solution is 1.072 g mL–1, then what shall be the molarity of the solution?

:: Chapter 3 - Electrochemistry ::

CONCEPT

 Electrochemistry and It's Uses

construction and functioning of Daniell cell

electrode potential, cell potential & Represent a galvanic cell

Structure and Working of Standard Hydrogen Electrode

 measure the standard potential of Cu²⁺

 Use of platinum or gold in standard hydrogen electrode

What is the Nernst Equation

 Relation between EӨcell and Kc

Nernst Equation for the given chemical reaction

Gibbs energy of reaction taking place in an electrochemical cell

Conductance of Electrolytic Solutions

What is cell constant

What is a superconductor

Factor effecting conductance & ionic conductance

What are Electronically conducting polymers and there advantages

Problems takes place in measuring of conductivity

What is a conductivity cell

Method to measure conductance using Wheatstone bridge

Concept of Molar Conductivity

Kohlrausch law of independent migration of ions

How to measure equilibrium constant and limiting molar conductivity of week electrolytite

Faraday’s Laws of Electrolysis

Explain type of cells

What is corrosion explain how corrosion works as a cell

What is hydrogen economy

EXERCISE

Question 1:Arrange the following metals in the order in which they displace each other from the solution of their salts.
Al, Cu, Fe, Mg and Zn

Question 2:Given the standard electrode potentials,
K⁺/K = −2.93V, Ag⁺/Ag = 0.80V,
Hg²⁺/Hg = 0.79V
Mg²⁺/Mg = −2.37 V, Cr³⁺/Cr = − 0.74V Arrange these metals in their increasing order of reducing power.

Question 3:Depict the galvanic cell in which the reaction Zn(s) + 2Ag⁺(aq) → Zn²⁺(aq) + 2Ag(s) takes place. Further show:

(i) Which of the electrode is negatively charged?
(ii) The carriers of the current in the cell.
(iii) Individual reaction at each electrode.

Question 4:Calculate the standard cell potentials of galvanic cells in which the following reactions take place:

(i) 2Cr(s) + 3Cd²⁺(aq) → 2Cr³⁺(aq) + 3Cd
(ii) Fe²⁺(aq) + Ag⁺(aq) → Fe³⁺(aq) + Ag(s)

Calculate the =∆rGθ and equilibrium constant of the reactions.

Question 5:Write the Nernst equation and emf of the following cells at 298 K:

(i) Mg(s) | Mg²⁺(0.001M) || Cu²⁺(0.0001 M) | Cu(s)
(ii) Fe(s) | Fe²⁺(0.001M) || H⁺(1M)|H₂(g)(1bar) | Pt(s)
(iii) Sn(s) | Sn²⁺(0.050 M) || H⁺(0.020 M) | H₂(g) (1 bar) | Pt(s)
(iv) Pt(s) | Br₂(l) | Br−(0.010 M) || H

Question 6:In the button cells widely used in watches and other devices the following reaction takesplace:

Zn(s) + Ag₂O(s) + H₂O(l) → Zn²+(aq) + 2Ag(s) + 2OH⁻(aq)  Determine and for the reaction.

Question 7:Define conductivity and molar conductivity for the solution of an electrolyte. Discuss their variation with concentration.

Question 8:The conductivity of 0.20 M solution of KCl at 298 K is 0.0248 Scm−1. Calculate its molar conductivity.

Question 9:The resistance of a conductivity cell containing 0.001M KCl solution at 298 K is 500M. What is the cell constant if conductivity of 0.001M KCl solution at 298 K is 0.146 × 10−3S cm^−1.

Question 10:The conductivity of sodium chloride at 298 K has been determined at different concentrations and the results are given below:

Question 11: Conductivity of 0.00241 M acetic acid is 7.896 × 10⁻⁵ S cm⁻¹. Calculate its molar conductivity and if for acetic acid is 390.5 S cm² mol⁻¹, what is its dissociation constant?

Question 12:How much charge is required for the following reductions:

(i) 1 mol of Al³⁺ to Al.
(ii) 1 mol of Cu²⁺ to Cu.
(iii) 1 mol of MnO^4– to Mn²⁺.

Question 13:How much electricity in terms of Faraday is required to produce

(i) 20.0 g of Ca from molten CaCl₂.
(ii) 40.0 g of Al from molten Al₂O₃.

Question 14: How much electricity is required in coulomb for the oxidation of (i) 1 mol of H2O to O2. (ii) 1 mol of FeO to Fe₂O₃.

Question 15: A solution of Ni(NO₃)2 is electrolysed between platinum electrodes using a current of 5 amperes for 20 minutes. What mass of Ni is deposited at the cathode?

Question 16:Three electrolytic cells A,B,C containing solutions of ZnSO₄, AgNO3 and CuSO₄, respectively are connected in series. A steady current of 1.5 amperes was passed through them until 1.4 g of silver deposited at the cathode of cell B. How long did the current flow? What mass of copper and zinc were deposited?

Question 17: Using the standard electrode potentials given in Table 3.1, predict if the reaction between the following is feasible:

(i) Fe₃₊(aq) and I⁻(aq)
(ii) A^g+ (aq) and Cu(s)
(iii) Fe³⁺ (aq) and Br−(aq)
(iv) Ag(s) and Fe³⁺(aq)
(v) Br² (aq) and Fe²⁺ (aq).

Question 18: Predict the products of electrolysis in each of the following:

(i) An aqueous solution of AgNO₃ with silver electrodes.
(ii) An aqueous solution of AgNO₃with platinum electrodes.
(iii) A dilute solution of H₂SO₄with platinum electrodes.
(iv) An aqueous solution of CuCl₂ with platinum electrodes.

IN TEXT SOLUTION

Question 3.1: How would you determine the standard electrode potential of the systemMg2+ | Mg?
Can you store copper sulphate solutions in a zinc pot?

Question 3.3:Consult the table of standard electrode potentials and suggest three substances that an oxidise ferrous ions under suitable conditions.

Question 3.4:Calculate the potential of hydrogen electrode in contact with a solution whose pH is 10.

Question 3.5: Calculate the emf of the cell in which the following reaction takes place:

Question 3.6: The cell in which the following reactions occurs:

Question 3.7: Why does the conductivity of a solution decrease with dilution?

Question 3.8:Suggest a way to determine the Λ°m value of water.

Question 3.9:The molar conductivity of 0.025 mol L−1 methanoic acid is 46.1 S cm2 mol−1. Calculate its degree of dissociation and dissociation constant. Given λ0(H+)= 349.6 S cm2 mol–1 and λ0(HCOO–) = 54.6 S cm2 mol–1

Question 3.10:If a current of 0.5 ampere flows through a metallic wire for 2 hours, then how many electrons would flow through the wire?

Question 3.11:Suggest a list of metals that are extracted electrolytically.

Question 3.12:Consider the reaction:
Cr2O72– + 14H+ + 6e– → 2Cr3+ + 8H2O
What is the quantity of electricity in coulombs needed to reduce 1 mol of Cr2O72– ?

Question 3.14:Suggest two materials other than hydrogen that can be used as fuels in fuel cells.

Question 3.15: Explain how rusting of iron is envisaged as setting up of an electrochemical cell.

:: Chapter 4 - Chemical Kinetics ::

Question 4.11 The following results have been obtained during the kinetic studies of the reaction: 2A + B → C + D Determine the rate law and the rate constant for the reaction.

Question 4.12 The reaction between A and B is first order with respect to A and zero order with respect to B. Fill in the blanks in the following table:

Question 4.13 Calculate the half-life of a first order reaction from their rate constants given below:

(i) 200 s–1
(ii) 2 min–1
(iii) 4 years–1

Question 4.14 The half-life for radioactive decay of 14C is 5730 years. An archaeological artifact containing wood had only 80% of the 14C found in a living tree. Estimate the age of the sample.

Question 4.15 The experimental data for decomposition of N2O5 [2N2O5 → 4NO2 + O2] in gas phase at 318K are given below:

(i) Plot [N2O5] against t.
(ii) Find the half-life period for the reaction.
(iii) Draw a graph between log[N2O5] and t.
(iv) What is the rate law ?
(v) Calculate the rate constant.
(vi) Calculate the half-life period from k and compare it with (ii).

Question 4.16 The rate constant for a first order reaction is 60 s–1. How much time will it take to reduce the initial concentration of the reactant to its 1/16th value?

Question 4.17 During nuclear explosion, one of the products is 90Sr with half-life of 28.1 years. If 1μg of 90Sr was absorbed in the bones of a newly born baby instead of calcium, how much of it will remain after 10 years and 60 years if it is not lost metabolically.

Question 4.18 For a first order reaction, show that time required for 99% completion is twice the time required for the completion of 90% of reaction.

Question 4.19 A first order reaction takes 40 min for 30% decomposition. Calculate t1/2.

Question 4.20 For the decomposition of azoisopropane to hexane and nitrogen at 543 K, the following data are obtained. Calculate the rate constant.

Question 4.21 The following data were obtained during the first order thermal decomposition of SO2Cl2 at a constant volume. SO2Cl2 (g) → SO2 (g) + Cl2 (g)

Question 4.22 The rate constant for the decomposition of N2O5 at various temperatures is given below: Draw a graph between ln k and 1/T and calculate the values of A and Ea. Predict the rate constant at 30° and 50°C.

Question 4.23 The rate constant for the decomposition of hydrocarbons is 2.418 × 10–5s–1 at 546 K. If the energy of activation is 179.9 kJ/mol, what will be the value of pre-exponential factor.

Question 4.24 Consider a certain reaction A → Products with k = 2.0 × 10–2s–1. Calculate the concentration of A remaining after 100 s if the initial concentration of A is 1.0 mol L–1.

Question 4.25 Sucrose decomposes in acid solution into glucose and fructose according to the first order rate law, with t1/2 = 3.00 hours. What fraction of sample of sucrose remains after 8 hours ?

Question 4.26 The decomposition of hydrocarbon follows the equation k = ( 4.5 × 1011s–1) e-28000K/T Calculate Ea.

Question 4.27 The rate constant for the first order decomposition of H2O2 is given by the following equation: log k = 14.34 – 1.25 × 104K/T Calculate Ea for this reaction and at what temperature will its half-period be 256 minutes?

Question 4.28 The decomposition of A into product has value of k as4.5 × 103 s–1 at 10°C and energy of activation 60 kJ mol–1. At what temperature would k be 1.5 × 104s–1?

Question 4.29 The time required for 10% completion of a first order reaction at 298K is equal to that required for its 25% completion at 308K. If the value of A is 4 × 1010s–1. Calculate k at 318K and Ea.

Question 4.30 The rate of a reaction quadruples when the temperature changes from 293 K to 313 K. Calculate the energy of activation of the reaction assuming that it does not change with temperature. 

:: Chapter 5 - Surface Chemistry ::

Question 5.1 Distinguish between the meaning of the terms adsorption and absorption. Give one example of each.

Question 5.2 What is the difference between physisorption and chemisorption?

Question 5.3 Give reason why a finely divided substance is more effective as an adsorbent.

Question 5.4 What are the factors which influence the adsorption of a gas on a solid?

Question 5.5 What is an adsorption isotherm? Describe Freundlich adsorption isotherm.

Question 5.6 What do you understand by activation of adsorbent? How is it achieved?

Question 5.7 What role does adsorption play in heterogeneous catalysis?

Question 5.8 Why is adsorption always exothermic ?

Question 5.9 How are the colloidal solutions classified on the basis of physical states of the dispersed phase and dispersion medium?

Question 5.10 Discuss the effect of pressure and temperature on the adsorption of gases on solids.

Question 5.11 What are lyophilic and lyophobic sols? Give one example of each type. Why are hydrophobic sols easily coagulated ?

Question 5.12 What is the difference between multimolecular and macromolecular colloids? Give one example of each. How are associated colloids different from these two types of colloids

Question 5.13 What are enzymes ? Write in brief the mechanism of enzyme catalysis.

Question 5.14 How are colloids classified on the basis of (i) physical states of components (ii) nature of dispersion medium and (iii) interaction between dispersed phase and dispersion medium

Question 5.15 Explain what is observed

(i) when a beam of light is passed through a colloidal sol.
(ii) an electrolyte, NaCl is added to hydrated ferric oxide sol.
(iii) electric current is passed through a colloidal sol?

Question 5.16 What are emulsions? What are their different types? Give example of each type.

Question 5.17 What is demulsification? Name two demulsifiers.

Question 5.18 Action of soap is due to emulsification and micelle formation. Comment.

Question 5.19 Give four examples of heterogeneous catalysis.

Question 5.20 What do you mean by activity and selectivity of catalysts?

Question 5.21 Describe some features of catalysis by zeolites.

Question 5.22 What is shape selective catalysis?

Question 5.23 Explain the following terms:

(i) Electrophoresis
(ii) Coagulation
(iii) Dialysis
(iv) Tyndall effect.

Question 5.24 Give four uses of emulsions.

Question 5.25 What are micelles? Give an example of a micellers system.

Question 5.26 Explain the terms with suitable examples:

(i) Alcosol
(ii) Aerosol
(iii) Hydrosol.

Question 5.27 Comment on the statement that “colloid is not a substance but a state of substance”

:: Chapter 6 - General Principles and Processes of Isolation of Elements ::

Question 6.1 Copper can be extracted by hydrometallurgy but not zinc. Explain.

Question 6.2 What is the role of depressant in froth floatation process?

Question 6.3 Why is the extraction of copper from pyrites more difficult than that from its oxide ore through reduction?

Question 6.4 Explain:
(i) Zone refining
(ii) Column chromatography.

Question 6.5 Out of C and CO, which is a better reducing agent at 673 K ?

Question 6.6 Name the common elements present in the anode mud in electrolytic refining of copper. Why are they so present ?

Question 6.7 Write down the reactions taking place in different zones in the blast furnace during the extraction of iron.

Question 6.8 Write chemical reactions taking place in the extraction of zinc from zinc blende.

Question 6.9 State the role of silica in the metallurgy of copper.

Question 6.10 What is meant by the term “chromatography”?

Question 6.11 What criterion is followed for the selection of the stationary phase in chromatography?

Question 6.12 Describe a method for refining nickel.

Question 6.13 How can you separate alumina from silica in a bauxite ore associated with silica? Give equations, if any.

Question 6.14 Giving examples, differentiate between ‘roasting’ and ‘calcination’.

Question 6.15 How is ‘cast iron’ different from ‘pig iron”?

Question 6.16 Differentiate between “minerals” and “ores”.

Question 6.17 Why copper matte is put in silica lined converter?

Question 6.18 What is the role of cryolite in the metallurgy of aluminium ?

Question 6.19 How is leaching carried out in case of low grade copper ores?

Question 6.20 Why is zinc not extracted from zinc oxide through reduction using CO?

Question 6.21 The value of ΔfG0 for formation of Cr2 O3 is – 540 kJmol−1and that of Al2 O3 is – 827 kJmol−1. Is the reduction of Cr2 O3 possible with Al ?

Question 6.22 Out of C and CO, which is a better reducing agent for ZnO ?

Question 6.23 The choice of a reducing agent in a particular case depends on thermodynamic factor. How far do you agree with this statement? Support your opinion with two examples.

Question 6.24 Name the processes from which chlorine is obtained as a by-product. What will happen if an aqueous solution of NaCl is subjected to electrolysis?

Question 6.25 What is the role of graphite rod in the electrometallurgy of aluminium?

Question 6.27 Outline the principles of refining of metals by the following methods:

(i) Zone refining
(ii) Electrolytic refining
(iii) Vapour phase refining

Question 6.28 Predict conditions under which Al might be expected to reduce MgO. (Hint: See Intext question 6.4) 

:: Chapter 7 - The p-Block Elements ::

Question 7.1 Discuss the general characteristics of Group 15 elements with reference to their electronic configuration, oxidation state, atomic size, ionisation enthalpy and electronegativity.

Question 7.2 Why does the reactivity of nitrogen differ from phosphorus?

Question 7.3 Discuss the trends in chemical reactivity of group 15 elements.

Question 7.4 Why does NH3 form hydrogen bond but PH3 does not?

Question 7.5 How is nitrogen prepared in the laboratory? Write the chemical equations of the reactions involved.

Question 7.6 How is ammonia manufactured industrially?

Question 7.7 Illustrate how copper metal can give different products on reaction with HNO3.

Question 7.8 Give the resonating structures of NO2 and N2O5.

Question 7.9 The HNH angle value is higher than HPH, HAsH and HSbH angles. Why? [Hint: Can be explained on the basis of sp3 hybridisation in NH3 and only s–p bonding between hydrogen and other elements of the group].

Question 7.10 Why does R3P = O exist but R3N = O does not (R = alkyl group)?

Question 7.11 Explain why NH3 is basic while BiH3 is only feebly basic.

Question 7.12 Nitrogen exists as diatomic molecule and phosphorus as P4. Why?

Question 7.13 Write main differences between the properties of white phosphorus and red phosphorus.

Question 7.14 Why does nitrogen show catenation properties less than phosphorus?

Question 7.15 Give the disproportionation reaction of H3PO3.

Question 7.16 Can PCl5 act as an oxidising as well as a reducing agent? Justify.

Question 7.17 Justify the placement of O, S, Se, Te and Po in the same group of the periodic table in terms of electronic configuration, oxidation state and hydride formation.

Question 7.18 Why is dioxygen a gas but sulphur a solid?

Question 7.19 Knowing the electron gain enthalpy values for O → O– and O → O2– as –141 and 702 kJ mol–1 respectively, how can you account for the formation of a large number of oxides having O2– species and not O–? (Hint: Consider lattice energy factor in the formation of compounds).

Question 7.20 Which aerosols deplete ozone?

Question 7.21 Describe the manufacture of H2SO4 by contact process?

Question 7.22 How is SO2 an air pollutant?

Question 7.23 Why are halogens strong oxidising agents?

Question 7.24 Explain why fluorine forms only one oxoacid, HOF.

Question 7.25 Explain why inspite of nearly the same electronegativity, oxygen forms hydrogen bonding while chlorine does not.

Question 7.26 Write two uses of ClO2.

Question 7.27 Why are halogens coloured?

Question 7.28 Write the reactions of F2 and Cl2 with water.

Question 7.29 How can you prepare Cl2 from HCl and HCl from Cl2? Write reactions only.

Question 7.30 What inspired N. Bartlett for carrying out reaction between Xe and PtF6?

Question 7.31 What are the oxidation states of phosphorus in the following:

(i) H3PO3
(ii) PCl3
(iii) Ca3P2
(iv) Na3PO4
(v) POF3? Exercises Chemistry 208

Question 7.32 Write balanced equations for the following:

(i) NaCl is heated with sulphuric acid in the presence of MnO2.
(ii) Chlorine gas is passed into a solution of NaI in water.

Question 7.33 How are xenon fluorides XeF2, XeF4 and XeF6 obtained?

Question 7.34 With what neutral molecule is ClO– isoelectronic? Is that molecule a Lewis base?

Question 7.35 How are XeO3 and XeOF4 prepared?

Question 7.36 Arrange the following in the order of property indicated for each set:

(i) F2, Cl2, Br2, I2 - increasing bond dissociation enthalpy.
(ii) HF, HCl, HBr, HI - increasing acid strength.
(iii) NH3, PH3, AsH3, SbH3, BiH3 – increasing base strength.

Question 7.37 Which one of the following does not exist?

(i) XeOF4
(ii) NeF2
(iii) XeF2
(iv) XeF6

Question 7.38 Give the formula and describe the structure of a noble gas species which is isostructural with:

(i) ICl4 –
(ii) IBr2 –
(iii) BrO3 –

Question 7.39 Why do noble gases have comparatively large atomic sizes?

Question 7.40 List the uses of neon and argon gases.

:: Chapter 8 - The d- and f- Block Elements ::

Question 8.1 Write down the electronic configuration of:

(i) Cr3+
(iii) Cu+
(v) Co2 +
(vii) Mn2+
(ii) Pm3+
(iv) Ce4+
(vi) Lu2+
(viii) Th4+

Question 8.2 Why are Mn2+ compounds more stable than Fe2+ towards oxidation to their +3 state?

Question 8.3 Explain briefly how +2 state becomes more and more stable in the first half of the first row transition elements with increasing atomic number?

Question 8.4 To what extent do the electronic configurations decide the stability of oxidation states in the first series of the transition elements? Illustrate your answer with examples.

Question 8.5 What may be the stable oxidation state of the transition element with the following d electron configurations in the ground state of their atoms : 3d3, 3d5, 3d8 and 3d4?

Question 8.6 Name the oxometal anions of the first series of the transition metals in which the metal exhibits the oxidation state equal to its group number.

Question 8.7 What is lanthanoid contraction? What are the consequences of lanthanoid contraction?

Question 8.8 What are the characteristics of the transition elements and why are they called transition elements? Which of the d-block elements may not be regarded as the transition elements?

Question 8.9 In what way is the electronic configuration of the transition elements different from that of the non transition elements?

Question 8.10 What are the different oxidation states exhibited by the lanthanoids?

Question 8.11 Explain giving reasons:

(i) Transition metals and many of their compounds show paramagnetic behaviour.
(ii) The enthalpies of atomisation of the transition metals are high.
(iii) The transition metals generally form coloured compounds.
(iv) Transition metals and their many compounds act as good catalyst

Question 8.12 What are interstitial compounds? Why are such compounds well known for transition metals?

Question 8.13 How is the variability in oxidation states of transition metals different from that of the non transition metals? Illustrate with examples.

Question 8.14 Describe the preparation of potassium dichromate from iron chromite ore. What is the effect of increasing pH on a solution of potassium dichromate?

Question 8.15 Describe the oxidising action of potassium dichromate and write the ionic equations for its reaction with: (i) iodide (ii) iron(II) solution and (iii) H2S Exercises 235 The d- and f- Block Elements

Question 8.16 Describe the preparation of potassium permanganate. How does the acidified permanganate solution react with (i) iron(II) ions (ii) SO2 and (iii) oxalic acid? Write the ionic equations for the reactions.

Question 8.17 For M2+/M and M3+/M2+ systems the EV values for some metals are as follows: Cr2+/Cr -0.9V Cr3/Cr2+ -0.4 V Mn2+/Mn -1.2V Mn3+/Mn2+ +1.5 V Fe2+/Fe -0.4V Fe3+/Fe2+ +0.8 V Use this data to comment upon: (i) the stability of Fe3+ in acid solution as compared to that of Cr3+ or Mn3+ and (ii) the ease with which iron can be oxidised as compared to a similar process for either chromium or manganese metal.

Question 8.18 Predict which of the following will be coloured in aqueous solution? Ti3+, V3+, Cu+, Sc3+, Mn2+, Fe3+ and Co2+. Give reasons for each.

Question 8.19 Compare the stability of +2 oxidation state for the elements of the first transition series.

Question 8.20 Compare the chemistry of actinoids with that of the lanthanoids with special reference to: (i) electronic configuration (iii) oxidation state (ii) atomic and ionic sizes and (iv) chemical reactivity.

Question 8.21 How would you account for the following:

(i) Of the d4 species, Cr2+ is strongly reducing while manganese(III) is strongly oxidising.
(ii) Cobalt(II) is stable in aqueous solution but in the presence of complexing reagents it is easily oxidised.
(iii) The d1 configuration is very unstable in ions.

Question 8.22 What is meant by ‘disproportionation’? Give two examples of disproportionation reaction in aqueous solution.

Question 8.23 Which metal in the first series of transition metals exhibits +1 oxidation state most frequently and why?

Question 8.24 Calculate the number of unpaired electrons in the following gaseous ions: Mn3+, Cr3+, V3+ and Ti3+. Which one of these is the most stable in aqueous solution?

Question 8.25 Give examples and suggest reasons for the following features of the transition metal chemistry

(i) The lowest oxide of transition metal is basic, the highest is amphoteric/acidic.
(ii) A transition metal exhibits highest oxidation state in oxides and fluorides.
(iii) The highest oxidation state is exhibited in oxoanions of a metal.

Question 8.26 Indicate the steps in the preparation of:

(i) K2Cr2O7 from chromite ore.
(ii) KMnO4 from pyrolusite ore.

Question 8.27 What are alloys? Name an important alloy which contains some of the lanthanoid metals. Mention its uses.

Question 8.28 What are inner transition elements? Decide which of the following atomic numbers are the atomic numbers of the inner transition elements : 29, 59, 74, 95, 102, 104.

Question 8.29 The chemistry of the actinoid elements is not so smooth as that of the lanthanoids. Justify this statement by giving some examples from the oxidation state of these elements.

Question 8.30 Which is the last element in the series of the actinoids? Write the electronic configuration of this element. Comment on the possible oxidation state of this element.

:: Chapter 9 - Coordination Compounds ::

Question  9.1 Explain the bonding in coordination compounds in terms of Werner’s postulates.

Question  9.2 FeSO4 solution mixed with (NH4)2SO4 solution in 1:1 molar ratio gives the test of Fe2+ ion but CuSO4 solution mixed with aqueous ammonia in 1:4 molar ratio does not give the test of Cu2+ ion. Explain why?

Question  9.3 Explain with two examples each of the following: coordination entity, ligand, coordination number, coordination polyhedron, homoleptic and heteroleptic.

Question  9.4 What is meant by unidentate, didentate and ambidentate ligands? Give two examples for each.

Question  9.5 Specify the oxidation numbers of the metals in the following coordination entities:
(i) [Co(H2O)(CN)(en)2]2+
(ii) [CoBr2(en)2]+
(iii) [PtCl4]2–
(iv) K3[Fe(CN)6]
(v) [Cr(NH3)3Cl3 ]

Question  9.6 Using IUPAC norms write the formulas for the following:
(i) Tetrahydroxozincate(II)
(ii) Potassium tetrachloridopalladate(II)
(iii) Diamminedichloridoplatinum(II)
(iv) Potassium tetracyanonickelate(II)
(v) Pentaamminenitrito-O-cobalt(III)
(vi) Hexaamminecobalt(III) sulphate
(vii) Potassium tri(oxalato)chromate(III)
(viii) Hexaammineplatinum(IV)
(ix) Tetrabromidocuprate(II)
(x) Pentaamminenitrito-N-cobalt(III)

Question  9.7 Using IUPAC norms write the systematic names of the following:
(i) [Co(NH3)6]Cl3
(ii) [Pt(NH3)2Cl(NH2CH3)]Cl
(iii) [Ti(H2O)6]3+
(iv) [Co(NH3)4Cl(NO2)]Cl
(v) [Mn(H2O)6]2+
(vi) [NiCl4]2–
(vii) [Ni(NH3)6]Cl2
(viii) [Co(en)3]3+
(ix) [Ni(CO)4]

Question  9.8 List various types of isomerism possible for coordination compounds, giving an example of each.

Question  9.9 How many geometrical isomers are possible in the following coordination entities?
(i) [Cr(C2O4)3]3–
(ii) [Co(NH3)3Cl3]

Question  9.10 Draw the structures of optical isomers of:
(i) [Cr(C2O4)3]3– (ii) [PtCl2(en)2]2+
(iii) [Cr(NH3)2Cl2(en)]+ 259 Coordination Compounds

Question  9.11 Draw all the isomers (geometrical and optical) of:
(i) [CoCl2(en)2]+
(ii) [Co(NH3)Cl(en)2]2+
(iii) [Co(NH3)2Cl2(en)]+

Question  9.12 Write all the geometrical isomers of [Pt(NH3)(Br)(Cl)(py)] and how many of these will exhibit optical isomers?

Question  9.13 Aqueous copper sulphate solution (blue in colour) gives: (i) a green precipitate with aqueous potassium fluoride and (ii) a bright green solution with aqueous potassium chloride. Explain these experimental results.

Question  9.14 What is the coordination entity formed when excess of aqueous KCN is added to an aqueous solution of copper sulphate? Why is it that no precipitate of copper sulphide is obtained when H2S(g) is passed through this solution?

Question  9.15 Discuss the nature of bonding in the following coordination entities on the basis of valence bond theory:
(i) [Fe(CN)6]4–
(ii) [FeF6]3–
(iii) [Co(C2O4)3]3–
(iv) [CoF6]3–

Question  9.16 Draw figure to show the splitting of d orbitals in an octahedral crystal field.

Question  9.17 What is spectrochemical series? Explain the difference between a weak field ligand and a strong field ligand.

Question  9.18 What is crystal field splitting energy? How does the magnitude of Δo decide the actual configuration of d orbitals in a coordination entity?

Question  9.19 [Cr(NH3)6]3+ is paramagnetic while [Ni(CN)4]2– is diamagnetic. Explain why?

Question  9.20 A solution of [Ni(H2O)6]2+ is green but a solution of [Ni(CN)4]2– is colourless. Explain.

Question  9.21 [Fe(CN)6]4– and [Fe(H2O)6]2+ are of different colours in dilute solutions. Why?

Question  9.22 Discuss the nature of bonding in metal carbonyls.

Question  9.23 Give the oxidation state, d orbital occupation and coordination number of the central metal ion in the following complexes:
(i) K3[Co(C2O4)3]
(iii) (NH4)2[CoF4]
(ii) cis-[Cr(en)2Cl2]C l
(iv) [Mn(H2O)6]SO4

Question  9.24 Write down the IUPAC name for each of the following complexes and indicate the oxidation state, electronic configuration and coordination number. Also give stereochemistry and magnetic moment of the complex:
(i) K[Cr(H2O)2(C2O4)2].3H2O
(ii) [Co(NH3)5Cl-]Cl2
(iii) CrCl3(py)3
(iv) Cs[FeCl4]
(v) K4[Mn(CN)6]

Question  9.25 What is meant by stability of a coordination compound in solution? State the factors which govern stability of complexes.

Question  9.26 What is meant by the chelate effect? Give an example. 9

Question  9.27 Discuss briefly giving an example in each case the role of coordination compounds in:
(i) biological systems
(iii) analytical chemistry
(ii) medicinal chemistry
(iv) extraction/metallurgy of metals.

Question  9.28 How many ions are produced from the complex Co(NH3)6Cl2 in solution?
(i) 6
(ii) 4
(iii) 3
(iv) 2

Question  9.29 Amongst the following ions which one has the highest magnetic moment value?
(i) [Cr(H2O)6]3+
(ii) [Fe(H2O)6]2+
(iii) [Zn(H2O)6]2+

Question  9.30 The oxidation number of cobalt in K[Co(CO)4] is
(i) +1
(ii) +3
(iii) –1
(iv) –3

:: Chapter 10 - Haloalkanes and Haloarenes ::

NCERT Mathematics Question Paper (Class - 12)

:: Chapter 1 - Number System ::

Q1. Determine whether each of the following relations are reflexive, symmetric and transitive:

(ii) Relation R in the set N of natural numbers defined as
R = {(x, y) : y = x + 5 and x < 4}

(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
R = {(x, y) : y is divisible by x}

(iv) Relation R in the set Z of all integers defined as
R = {(x, y) : x – y is an integer}

Q 1. Determine whether each of the following relations are reflexive, symmetric and transitive:

(i) Relation R in the set A = {1, 2, 3, ..., 13, 14} defined as
R = {(x, y) : 3x – y = 0}

Q 1.Determine whether each of the following relations are reflexive, symmetric and transitive:

(v) Relation R in the set A of human beings in a town at a particular time given by

(a) R = {(x, y) : x and y work at the same place}
(b) R = {(x, y) : x and y live in the same locality}
(c) R = {(x, y) : x is exactly 7 cm taller than y}
(d) R = {(x, y) : x is wife of y}
(e) R = {(x, y) : x is father of y}

2. Show that the relation R in the set R of real numbers, defined as
R = {(a, b) : a ? b2} is neither reflexive nor symmetric nor transitive.

3. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

4. Show that the relation R in R defined as R = {(a, b) : a ? b}, is reflexive and transitive but not symmetric.

5. Check whether the relation R in R defined by R = {(a, b) : a ? b3} is reflexive, symmetric or transitive.

6. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

7. Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y) : x and y have same number of pages} is an equivalence relation.

8. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by
R = {(a, b) : |a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

9. Show that each of the relation R in the set A = {x ? Z : 0 ? x ? 12}, given by
(i) R = {(a, b) : |a – b| is a multiple of 4}
(ii) R = {(a, b) : a = b}

:: Chapter 2 - Inverse Trigonometric Functions ::

EXERCISE

Question 1. Find the principal values of the following:

Question 6. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Question 7. Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y) : x and y have same number of pages} is an equivalence relation.

Question 8. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) :
|a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Question 9. Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by (i) R = {(a, b) : |a – b| is a multiple of 4} (ii) R = {(a, b) : a = b} is an equivalence relation. Find the set of all elements related to 1 in each case.

Question 10. Give an example of a relation. Which is
(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Question 11. Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

Question 12. Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T2}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?

Question 13. Show that the relation R defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Question 14. Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2) : L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Question 15. Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4,4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.
(A) R is reflexive and symmetric but not transitive.
(B) R is reflexive and transitive but not symmetric.
(C) R is symmetric and transitive but not reflexive.
(D) R is an equivalence relation.

Question 16. Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b > 6}. Choose the correct answer.
(A) (2, 4) ∈ R
(B) (3, 8) ∈ R
(C) (6, 8) ∈ R
(D) (8, 7) ∈ R

EXERCISE

Question 1. Show that the function f : R → R defined by f (x) = 1 x is one-one and onto, where R is the set of all non-zero real numbers. Is the result true, if the domain R is replaced by N with co-domain being same as R?

Question 2. Check the injectivity and surjectivity of the following functions:

(i) f : N → N given by f (x) = x2
(ii) f : Z → Z given by f (x) = x2
(iii) f : R → R given by f (x) = x2
(iv) f : N → N given by f (x) = x3
(v) f : Z → Z given by f (x) = x3


Question 3. Prove that the Greatest Integer Function f :
R→R, given by f (x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Question 4. Show that the Modulus Function f : R→R, given by f (x) = | x |, is neither oneone nor onto, where | x | is x, if x is positive or 0 and | x | is – x, if x is negative. 5. Show that the Signum Function f : R→R, given by

is neither one-one nor onto.

State whether the function f is bijective. Justify your answer.

Question 10. Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by

Question 11. Let f : R → R be defined as f(x) = x4. Choose the correct answer.
(A) f is one-one onto (B) f is many-one onto
(C) f is one-one but not onto
(D) f is neither one-one nor onto.

Question 12. Let f : R → R be defined as f (x) = 3x. Choose the correct answer.
(A) f is one-one onto
(B) f is many-one onto
(C) f is one-one but not onto
(D) f is neither one-one nor onto.
 

EXERCISE

EXERCISE

Show that is commutative and associative. Find the identity element for on A, if any.

Question 12. State whether the following statements are true or false. Justify.

(i) For an arbitrary binary operation on a set N, a a = a ∀ a ∈ N.
(ii) If is a commutative binary operation on N, then a (b c) = (c b) a

Question 13. Consider a binary operation on N defined as a b = a3 + b3. Choose the correct answer.

(A) Is both associative and commutative?
(B) Is commutative but not associative?
(C) Is associative but not commutative?
(D) Is neither commutative nor associative?

Define the relation R in P(X) as follows: For subsets A, B in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify your answer.

Question 9. Given a non-empty set X, consider the binary operation :

P(X) × P(X) → P(X) given by A B = A ∩ B ∀ A, B in P(X), where P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation

Question 10. Find the number of all onto functions from the set {1, 2, 3, ... , n} to itself.

Question 11. Let S = {a, b, c} and T = {1, 2, 3}. Find F–1 of the following functions F from S to T, if it exists.
(i) F = {(a, 3), (b, 2), (c, 1)}
(ii) F = {(a, 2), (b, 1), (c, 1)}

Question 12. Consider the binary operations : R × R → R and o : R × R → R defined as a b = |a – b| and a o b = a, ∀ a, b ∈ R. Show that is commutative but not associative, o is associative but not commutative. Further, show that ∀ a, b, c ∈ R, a (b o c) = (a b) o (a b). [If it is so, we say that the operation distributes over the operation o]. Does o distribute over ? Justify your answer.

Question 13. Given a non-empty set X, let : P(X) × P(X) → P(X) be defined as A * B = (A – B) ∪ (B – A), ∀ A, B ∈ P(X). Show that the empty set φ is the identity for the operation and all the elements A of P(X) are invertible with A–1 = A. (Hint : (A – φ) ∪ (φ – A) = A and (A – A) ∪ (A – A) = A A = φ).

Question 14. Define a binary operation on the set {0, 1, 2, 3, 4, 5} as

:: Chapter 3 - Matrix ::

EXERCISE

Question 1. In the matrix , write

(i) The order of the matrix, (ii) The number of elements,
(iii) Write the elements a13, a21, a33, a24, a23.

Question 2. If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

Question 3. If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements ?

Question 4. Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
(i)aij=(i + j)2/2

Question 4. Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
(ii)aij=i/j

Question 4. Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
(iii)aij=(i + 2j)2/2

Question 5. Construct a 3 × 4 matrix, whose elements are given by:
(i)aij=1/2| -3i + j |

Question 5. Construct a 3 × 4 matrix, whose elements are given by:
(ii)aij=2i - j

Question 6. Find the values of x, y and z from the following equations:

(i)

Question 6. Find the values of x, y and z from the following equations:

(ii)

Question 6. Find the values of x, y and z from the following equations:

(iii)

Question 7. Find the value of a, b, c and d from the equation

Question 8 .is a square matrix, if
(A) m < n (B) m > n (C) m = n (D) None of these

Question 9.Which of the given values of x and y make the following pair of matrices equal

Question 10. The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:

EXERCISE

=Question 1. Let Find each of the following:
(i) A + B (ii) A – B (iii) 3A – C

=Question 1. Let Find each of the following:
(iv) AB (v) BA

=Question 2. Compute the following:

=Question 3. Compute the indicated products.

=Question 4. If , then compute (A+B) and (B – C). Also, verify that A + (B – C) = (A + B) – C.

=Question 5. If , then compute 3A – 5B.

=Question 6. Simplify

:: Chapter 4 - Determinants ::

Exercise

Question 1. Evaluate the determinants in Exercises 1 and 2.

Question 2. Evaluate the determinants in Exercises 1 and 2.

Question 3. If , then show that | 2A | = 4 | A |

Question 4. If , then show that | 3 A | = 27 | A |

Question 5. Evaluate the determinants

Question 6. If , find | A |

Question 7. Find values of x, if 

Question 8. If , then x is equal to

Exercise

Using the property of determinants and without expanding in Exercises 1 to 7, prove That 


Using the property of determinants and without expanding in Exercises 1 to 7, prove That


Using the property of determinants and without expanding in Exercises 1 to 7, prove That


Using the property of determinants and without expanding in Exercises 1 to 7, prove That


Using the property of determinants and without expanding in Exercises 1 to 7, prove That 

By using properties of determinants, in Exercises 8 to 14, show that:

 

Question 15.Let A be a square matrix of order 3 × 3, then | kA| is equal to

(A) k| A|
(B) k2 | A|
(C) k3 | A|
(D) 3k | A |

Question 16. Which of the following is correct?

(A) Determinant is a square matrix.
(B) Determinant is a number associated to a matrix.
(C) Determinant is a number associated to a square matrix.
(D) None of these

Exercise

Question 1. Find area of the triangle with vertices at the point given in each of the following :

(i) (1, 0), (6, 0), (4, 3)
(ii) (2, 7), (1, 1), (10, 8)
(iii) (–2, –3), (3, 2), (–1, –8)

Question 2. Show that points A (a, b + c), B (b, c + a), C (c, a + b) are collinear.

Question 3. Find values of k if area of triangle is 4 sq. units and vertices are
(i) (k, 0), (4, 0), (0, 2)
(ii) (–2, 0), (0, 4), (0, k)

Question 4.
(i) Find equation of line joining (1, 2) and (3, 6) using determinants.
(ii) Find equation of line joining (3, 1) and (9, 3) using determinants.

Question 5. If area of triangle is 35 sq units with vertices (2, – 6), (5, 4) and (k, 4). Then k is

(A) 12 (B) –2 (C) –12, –2 (D) 12, –2

Exercise

Write Minors and Cofactors of the elements of following determinants:

Exercise

=Find adjoint of each of the matrices in Exercises 1 and 2.

=Verify A (adj A) = (adj A) A = |A| I in Exercises 3 and 4

=Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.

17. Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to

(A) |A|
(B) |A|2
(C) |A|3
(D) 3|A|

18. If A is an invertible matrix of order 2, then det (A–1) is equal to

(A) det
(A) (B)1/det (A)
(C) 1
(D) 0

Exercise

Question 1. x + 2y = 2 and 2x + 3y = 3

Question 2. 2x – y = 5 and x + y = 4

Question 3. x + 3y = 5 and 2x + 6y = 8

Question 4. x + y + z = 1 , 2x + 3y + 2z = 2 and ax + ay + 2az = 4

Question 3x–y – 2z = 2, 2y – z =-1 and –3x – 5y = 3

Question 6. 5x – y + 4z = 5,2x + 3y + 5z = 2 and 5x – 2y + 6z = –1

Solve system of linear equations, using matrix method, in Exercises 7 to 14.

Question 7. 5x + 2y = 4 and 7x + 3y = 5

Question 8. 2x – y = –2 and 3x + 4y = 3

Question 9. 4x – 3y = 3 and 3x – 5y = 7

Question 10. 5x + 2y = 3 and 3x + 2y = 5

Question 11. 2x + y + z = 1, x – 2y – z =3/2 and 3y – 5z = 9

Question 12. x – y + z = 4, 2x + y – 3z = 0 and x + y + z = 2

Question 13. 2x + 3y +3 z = 5, x – 2y + z = – 4 and 3x – y – 2z = 3

Question 14. x – y + 2z = 7,3x + 4y – 5z = – 5 and 2x – y + 3z = 12

Question 15. If , find A–1. Using A–1 solve the system of equations 2x – 3y + 5z = 11 3x + 2y – 4z = – 5 x + y – 2z = – 3

Question 16. The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.

:: Chapter 6 - Application of Derivatives ::

EXERCISE

Question 1. Find the rate of change of the area of a circle with respect to its radius r when (a) r = 3 cm (b) r = 4 cm

Question 2. The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm?

Question 3. The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

Question 4. An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

Question 5. A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing? 6. The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

Question 7. The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8cm and y = 6cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

Question 8. A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.

Question 9. A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm. 10. A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall

Question 11. A particle moves along the curve 6y = x3 +2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.

Question 12. The radius of an air bubble is increasing at the rate of 1 2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

Question 13. A balloon, which always remains spherical, has a variable diameter 3 (2 1) 2 x + . Find the rate of change of its volume with respect to x.

Question 14. Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

Question 15. The total cost C(x) in Rupees associated with the production of x units of an item is given by C(x) = 0.007x3 – 0.003x2 + 15x + 4000. Find the marginal cost when 17 units are produced.

Question 16. The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 13x2 + 26x + 15. Find the marginal revenue when x = 7.

Question 17. The rate of change of the area of a circle with respect to its radius r at r = 6 cm is

(A) 10π
(B) 12π
(C) 8π
(D) 11π

Question 18. The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5. The marginal revenue, when x = 15 is

(A) 116
(B) 96
(C) 90
(D) 126

EXERCISE

Question 1. Show that the function given by f (x) = 3x + 17 is strictly increasing on R.

Question 2. Show that the function given by f (x) = e2x is strictly increasing on R.

Question 3. Show that the function given by f (x) = sin x is (a) strictly increasing in 0, 2 (b) strictly decreasing in , 2 (c) neither increasing nor decreasing in (0, π)

Question 4. Find the intervals in which the function f given by f (x) = 2x2 – 3x is (a) strictly increasing (b) strictly decreasing

Question 5. Find the intervals in which the function f given by f (x) = 2x3 – 3x2 – 36x + 7 is (a) strictly increasing (b) strictly decreasing

Question 6. Find the intervals in which the following functions are strictly increasing or decreasing:

(a) x2 + 2x – 5
(b) 10 – 6x – 2x2
(c) –2x3 – 9x2 – 12x + 1
(d) 6 – 9x – x2
(e) (x + 1)3 (x – 3)3

Question 7. Show that log(1 ) 2 2 y x x x = + −+ , x > – 1, is an increasing function of x throughout its domain.

Question 8. Find the values of x for which y = [x(x – 2)]2 is an increasing function.

Question 9. Prove that 4sin (2 cos ) y θ = −θ + θ is an increasing function of θ in 0, 2

Question 10. Prove that the logarithmic function is strictly increasing on (0, ∞).

Question 11. Prove that the function f given by f (x) = x2 – x + 1 is neither strictly increasing nor strictly decreasing on (– 1, 1)

Question 12. Which of the following functions are strictly decreasing on 0, 2?

(A) cos x
(B) cos 2x
(C) cos 3x
(D) tan x

Question 13. On which of the following intervals is the function f given by f (x) = x100 + sin x –1 strictly decreasing ? (A) (0,1) (B) , 2 (D) None of these

Question 14. Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on (1, 2).

Question 15. Let I be any interval disjoint from (–1, 1). Prove that the function f given by f (x) x 1 x = + is strictly increasing on I.

Question 16. Prove that the function f given by f (x) = log sin x is strictly increasing on 0, 2and strictly decreasing on .

Question 17. Prove that the function f given by f (x) = log cos x is strictly decreasing on 0, 2 ⎠ and strictly increasing on , 2 .

Question 18. Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.

Question 19. The interval in which y = x2 e–x is increasing is

(A) (– ∞, ∞)
(B) (– 2, 0)
(C) (2, ∞)
(D) (0, 2)

EXERCISE

Question 1. Find the slope of the tangent to the curve y = 3x4 – 4x at x = 4.

Question 2. Find the slope of the tangent to the curve 1, 2 2 y x x x − = ≠ − at x = 10.

Question 3. Find the slope of the tangent to curve y = x3 – x + 1 at the point whose x-coordinate is 2.

Question 4. Find the slope of the tangent to the curve y = x3 –3x + 2 at the point whose x-coordinate is 3.

Question 5. Find the slope of the normal to the curve x = acos3 θ, y = asin3 θ at . 4 π θ =

Question 6. Find the slope of the normal to the curve x = 1− asinθ, y = bcos2 θ at . 2 π θ =

Question 7. Find points at which the tangent to the curve y = x3 – 3x2 – 9x + 7 is parallel to the x-axis.

Question 8. Find a point on the curve y = (x – 2)2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

Question 9. Find the point on the curve y = x3 – 11x + 5 at which the tangent is y = x – 11.

Question 10. Find the equation of all lines having slope – 1 that are tangents to the curve 1 1 y x = − , x ≠ 1.

Question 11. Find the equation of all lines having slope 2 which are tangents to the curve 1 3 y x = − , x ≠ 3.

Question 12. Find the equations of all lines having slope 0 which are tangent to the curve 2 1 . 2 3 y x x = − +

Question 13. Find points on the curve 2 2 1 9 16 x + y = at which the tangents are (i) parallel to x-axis (ii) parallel to y-axis.

Question 14. Find the equations of the tangent and normal to the given curves at the indicated points:

(i) y = x4 – 6x3 + 13x2 – 10x + 5 at (0, 5)
(ii) y = x4 – 6x3 + 13x2 – 10x + 5 at (1, 3)
(iii) y = x3 at (1, 1)
(iv) y = x2 at (0, 0)
(v) x = cos t, y = sin t at 4 t π =1

Question 15. Find the equation of the tangent line to the curve y = x2 – 2x +7 which is (a) parallel to the line 2x – y + 9 = 0 (b) perpendicular to the line 5y – 15x = 13.

Question 16. Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and x = – 2 are parallel.

Question 17. Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.

Question 18. For the curve y = 4x3 – 2x5, find all the points at which the tangent passes through the origin.

Question 19. Find the points on the curve x2 + y2 – 2x – 3 = 0 at which the tangents are parallel to the x-axis.

Question 20. Find the equation of the normal at the point (am2,am3) for the curve ay2 = x3.

Question 21. Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.

Question 22. Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).

Question 23. Prove that the curves x = y2 and xy = k cut at right angles* if 8k2 = 1.

Question 24. Find the equations of the tangent and normal to the hyperbola 2 2 2 2 1 x y a b − = at the point (x0, y0).

Question 25. Find the equation of the tangent to the curve y = 3x − 2 which is parallel to the line 4x − 2y + 5 = 0 . Choose the correct answer in Exercises 26 and 27.

Question 26. The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is

(A) 3
(B) 1 3
(C) –3
(D) 1 3 −

Question 27. The line y = x + 1 is a tangent to the curve y2 = 4x at the point

(A) (1, 2)
(B) (2, 1)
(C) (1, – 2)
(D) (– 1, 2)

EXERCISE

Question 1. Using differentials, find the approximate value of each of the following up to 3 places of decimal.

(i) 25.3
(ii) 49.5
(iii) 0.6
(iv) 1 (0.009)3
(v) 1 (0.999)10
(vi) 1 (15)4
(vii) 1 (26)3
(viii) 1 (255)4
(ix) 1 (82)4
(x) 1 (401)2
(xi) 1 (0.0037)2
(xii) 1 (26.57)3
(xiii) 1 (81.5)4
(xiv) 3 (3.968)2
(xv) 1 (32.15)5

Question 2. Find the approximate value of f (2.01), where f (x) = 4x2 + 5x + 2.

Question 3. Find the approximate value of f (5.001), where f (x) = x3 – 7x2 + 15.

Question 4. Find the approximate change in the volume V of a cube of side x metres caused by increasing the side by 1%.

Question 5. Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%.

Question 6. If the radius of a sphere is measured as 7 m with an error of 0.02 m, then find the approximate error in calculating its volume.

Question 7. If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find the approximate error in calculating its surface area.

Question 8. If f(x) = 3x2 + 15x + 5, then the approximate value of f (3.02) is (A) 47.66 (B) 57.66 (C) 67.66 (D) 77.66 9. The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is

(A) 0.06 x3 m3
(B) 0.6 x3 m3
(C) 0.09 x3 m3
(D) 0.9 x3 m3

EXERCISE

Question 1. Find the maximum and minimum values, if any, of the following functions given by

(i) f
(x) = (2x – 1)2 + 3
(ii) f (x) = 9x2 + 12x + 2
(iii) f
(x) = –
(x – 1)2 + 10
(iv) g
(x) = x3 +

Question 1. It is given that at x = 1, the function x4 – 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.

Question 2. Find the maximum and minimum values of x + sin 2x on [0, 2π].

Question 3. Find two numbers whose sum is 24 and whose product is as large as possible.

Question 4. Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.

Question 5. Find two positive numbers x and y such that their sum is 35 and the product x2 y5 is a maximum.

Question 6. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

Question 7. A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible.

Question 8. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum ?

Question 9. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Question 10. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.

Question 11. Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?

Question 12. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

Question 13. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8 27 of the volume of the sphere.

Question 14. Show that the right circular cone of least curved surface and given volume has an altitude equal to 2 time the radius of the base.

Question 15. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan−1 2 .

Question 16. Show that semi-vertical angle of right circular cone of given surface area and maximum volume is sin 1 1

Miscellaneous Exercise on Chapter 6

Question 1. Using differentials, find the approximate value of each of the following: (a) 1 17 4 81 (b) ( ) 1 33 5 −

Question 2. Show that the function given by f (x) log x x = has maximum at x = e.

Question 3. The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base

Question 4. Find the equation of the normal to curve x2 = 4y which passes through the point (1, 2).

Question 5. Show that the normal at any point θ to the curve x = a cosθ + a θ sin θ, y = a sinθ – aθ cosθ is at a constant distance from the origin.

Question 6. Find the intervals in which the function f given by ( ) 4sin 2 cos 2 cos f x x x x x x − − = + is

(i) increasing
(ii) decreasing.

Question 7. Find the intervals in which the function f given by 3 3 f (x) x 1 , x 0 x = + ≠ is

(i) increasing
(ii) decreasing.

Question 8. Find the maximum area of an isosceles triangle inscribed in the ellipse 2 2 2 2 1 x y a b + = with its vertex at one end of the major axis.

Question 9. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?

Question 10. The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

Question 11. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is

Question 12. m. Find the dimensions of the window to admit maximum light through the whole opening.

Question 13. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the maximum length of the hypotenuse is 2 2 3 (a3 + b3 )2 .

Question 14. Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has

(i) local maxima
(ii) local minima
(iii) point of inflexion

14. Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π]

Question 15. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4 3 r .

Question 16. Let f be a function defined on [a, b] such that f ′(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).

Question 17. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R 3 . Also find the maximum volume.

Question 18. Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle α is one-third that of the cone and the greatest volume of cylinder is 4 3 tan2 27 πh α . Choose the correct answer in the Exercises from 19 to 24.

Question 19. A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of

(A) 1 m3/h
(B) 0.1 m3/h
(C) 1.1 m3/h
(D) 0.5 m3/h

Question 20. The slope of the tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2,– 1) is

(A) 22 7
(B) 6 7
(C) 7 6
(D) 6 7

Question 21. The line y = mx + 1 is a tangent to the curve y2 = 4x if the value of m is

(A) 1
(B) 2
(C) 3
(D) 1 2

Question 22. The normal at the point (1,1) on the curve 2y + x2 = 3 is

(A) x + y = 0
(B) x – y = 0
(C) x + y +1 = 0
(D) x – y = 0

Question 23. The normal to the curve x2 = 4y passing (1,2) is

(A) x + y = 3
(B) x – y = 3
(C) x + y = 1
( D) x – y = 1

Question 24. The points on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with the axes are

(A) 4, 8 3
(B) 4, 8 3
(C) 4, 3 8
(D) 4, 3 8

:: Chapter 7 - Integral ::

EXERCISE

Question 1. Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.

Question 2. Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in the first quadrant. Fig

Question 3. Find the area of the region bounded by x2 = 4y, y = 2, y = 4 and the y-axis in the first quadrant.

Question 4. Find the area of the region bounded by the ellipse 2 2 1 16 9 x y + = .

Question 5. Find the area of the region bounded by the ellipse 2 2 1 4 9 x y + = .

Question 6. Find the area of the region in the first quadrant enclosed by x-axis, line x = 3 y and the circle x2 + y2 = 4.

Question 7. Find the area of the smaller part of the circle x2 + y2 = a2 cut off by the line 2 x= a .

Question 8. The area between x = y2 and x = 4 is divided into two equal parts by the line x = a, find the value of a.

Question 9. Find the area of the region bounded by the parabola y = x2 and y = x .

Question 10. Find the area bounded by the curve x2 = 4y and the line x = 4y – 2.

Question 11. Find the area of the region bounded by the curve y2 = 4x and the line x = 3. Choose the correct answer in the following Exercises 12 and 13.

Question 12. Area lying in the first quadrant and bounded by the circle x2 + y2 = 4 and the lines x = 0 and x = 2 is

(A) π
(B) 2 π
(C) 3 π
(D) 4 π

Question 13. Area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3 is

(A) 2
(B) 9 4
(C) 9 3
(D) 9 2

EXERCISE

Question 1. Find the area of the circle 4x2 + 4y2 = 9 which is interior to the parabola x2 = 4y.

Question 2. Find the area bounded by curves (x – 1)2 + y2 = 1 and x2 + y2 = 1.

Question 3. Find the area of the region bounded by the curves y = x2 + 2, y = x, x = 0 and x = 3.

Question 4. Using integration find the area of region bounded by the triangle whose vertices are (– 1, 0), (1, 3) and (3, 2).

Question 5. Using integration find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x =

Question 4.Choose the correct answer in the following exercises 6 and 7.

Question 6. Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is

(A) 2 (π – 2)
(B) π – 2
(C) 2π – 1
(D) 2 (π + 2)

Question 7. Area lying between the curves y2 = 4x and y = 2x is

(A) 2 3
(B) 1 3
(C) 1 4
(D) 3 4

Miscellaneous Exercise on Chapter

Question 1. Find the area under the given curves and given lines:

(i) y = x2, x = 1, x = 2 and x-axis
(ii) y = x4, x = 1, x = 5 and x-axis

Question 2. Find the area between the curves y = x and y = x2.

Question 3. Find the area of the region lying in the first quadrant and bounded by y = 4x2, x = 0, y = 1 and y = 4.

Question 4. Sketch the graph of y = x + 3 and evaluate 0 6 3 − ∫ x + dx .

Question 5. Find the area bounded by the curve y = sin x between x = 0 and x = 2π.

Question 6. Find the area enclosed between the parabola y2 = 4ax and the line y = mx.

Question 7. Find the area enclosed by the parabola 4y = 3x2 and the line 2y = 3x + 12.

Question 8. Find the area of the smaller region bounded by the ellipse 2 2 1 9 4 x + y = and the line 1 3 2 x y + = .

Question 9. Find the area of the smaller region bounded by the ellipse 2 2 2 2 x y 1 a b + = and the line 1 x y a b + = .

Question 10. Find the area of the region enclosed by the parabola x2 = y, the line y = x + 2 and the x-axis.

Question 11. Using the method of integration find the area bounded by the curve x + y = 1 . [Hint: The required region is bounded by lines x + y = 1, x– y = 1, – x + y = 1 and – x – y = 1].

Question 12. Find the area bounded by curves {(x, y) : y ≥ x2 and y = | x |}.

Question 13. Using the method of integration find the area of the triangle ABC, coordinates of whose vertices are A(2, 0), B (4, 5) and C (6, 3).

Question 14. Using the method of integration find the area of the region bounded by lines: 2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0

Question 15. Find the area of the region {(x, y) : y2 ≤ 4x, 4x2 + 4y2 ≤ 9} Choose the correct answer in the following Exercises from 16 to 20.

Question 16. Area bounded by the curve y = x3, the x-axis and the ordinates x = – 2 and x = 1 is

(A) – 9
(B) 15 4 −
(C) 15 4
(D) 17 4

Question 17. The area bounded by the curve y = x | x | , x-axis and the ordinates x = – 1 and x = 1 is given by

(A) 0
(B) 1 3
(C) 2 3
(D) 4 3 [Hint : y = x2 if x > 0 and y = – x2 if x < 0].

Question 18. The area of the circle x2 + y2 = 16 exterior to the parabola y2 = 6x is

(A) 4 (4 3) 3 π −
(B) 4 (4 3) 3 π +
(C) 4 (8 3) 3 π −
(D) 4 (8 3) 3 π +

Question 19. The area bounded by the y-axis, y = cos x and y = sin x when 0 2 x π ≤ ≤ is

(A) 2 ( 2 −1)
(B) 2 −1
(C) 2 +1
(D) 2

:: Chapter 8 - Application Of Integrals ::

EXERCISE

1. Find the area of the region bounded by the curve y^2 =x and the lines x = 1 , x = 4 and the x axis

2.Find the area of the region bounded by y^2 = 9x, x=2, x =4 and the x axis in the first quadrant.

3. Find the area of the region bounded by x^2 = 4y, y = 2, y = 4 and the y-axis in the first quadrant.

Find the equation of the region bounded by the ellipse x^2/16 + y^2/9 =1

Find the equation of the region bounded by the ellipse x^2/4 + y^2/9 =1

6. find the area of the region in the first quadrant enclosed by x axis, line x =root 3 y and the circle x^2 + y^2 = 4

7.Area between x=y2 and x=4 is divided in two equal parts by the line x = a, find the value of a

8. The area between x^2 = y and x = 4 is divided into two equal parts by the line x = a, find the value of a.

9. Find the area of the region bounded by the parabola y = x^2 and y= |x|\

10. Find the area bounded by the curve x^2 =4y and the line x = 4y- 2

11. Find the area of the region bounded by the curve y2 = 4x and the line x = 3.

12. Area lying in the first quadrant and bounded by the circle x2 + y = 4 and the lines x = 0 and x = 2 is 2

13. Area of the region bounded by the curve y^2 =4x , y axis and the line y=3 is

EXERCISE

1. Find the area of the circle 4x^2 + 4y^2 = 9 which is interior to the parabola x^2 =4y

2. Find the area bounded by curves (x – 1)^2 + y^2 = 1 and x^2 + y^2 = 1

3.Find the area of the region bounded by the curves y= x^2 + 2 , y=x , x =0 and x = 3

4.Using integration find the area of region bounded by the triangle whose vertices are (– 1, 0), (1, 3) and (3, 2).

5. Using integration find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x = 4.

6. Smaller area enclosed by the circle x^2 +y^2 = 4 and the lines x + y = 2 is (A) 2 (π – 2) (B) π – 2 (C) 2π – 1 (D) 2 (π + 2)

7. Area lying between the curves y^2 = 4x and y = 2x is

Miscellaneous Solutions

1. Find the area under the given curves and given lines (i) y = x^2 , x=1 , x= 2 and x axis (ii) y = x^4 , x=1 , x= 5 and x axis

2. Find the area between the curves y = x and y = x^2

3.Find the area of the region lying in the first quadrant and bounded by y = 4x^2, x=0, y=1 and y= 4

4. Sketch the graph of y = x + 3 and evaluate integration limits 6 to 0 of x + 3 dx

5. Find the area bounded by the curve y = sin x between x = 0 and x = 2π.

6. Find the area enclosed between the parabola y^2 = 4ax and the line y =mx

7. Find the area enclosed by the parabola 4y = 3x^2 and the line 2y = 3x + 12 using integration to find area,

8. Find the area of the smaller region bounded by the ellipse x^2/9 + y^2/4 = 1 and the line x/3 + y/2 =1

:: Chapter 9 - Differential Equations ::

EXERCISE 9.2

In each of the Exercises 1 to 10 verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:

Question 1. y = ex + 1 : y″ – y′ = 0

Question 2. y = x² + 2x + C : y′ – 2x – 2 = 0

Question 3. y = cos x + C : y′ + sin x = 0

Question 4. y = 1+ x² : y′ = 1 2 xy + x

Question 5. y = Ax : xy′ = y (x ≠ 0)

Question 6. y = x sin x : xy′ = y + x x² − y² (x ≠ 0 and x > y or x < – y)

Question 7. xy = log y + C : y′ = 2 1 y − xy (xy ≠ 1)

Question 8. y – cos y = x : (y sin y + cos y + x) y′ = y

Question 9. x + y = tan–1y : y² y′ + y² + 1 = 0

Question 10. y = a²

 − x2 x ∈ (–a, a) : x + y dy dx = 0 (y ≠ 0)

Question 11. The number of arbitrary constants in the general solution of a differential equation of fourth order are:

(A) 0
(B) 2
(C) 3
(D) 4

Question 12. The number of arbitrary constants in the particular solution of a differential equation of third order are:

(A) 3
(B) 2
(C) 1
(D) 0

EXERCISE 9.3

In each of the Exercises 1 to 5, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

Question 1. x y 1 a b + =

Question 2. y² = a (b² – x²)

Question 3. y = a e3x + b e– 2x

Question 4. y = e²x (a + bx)

Question 5. y = ex (a cos x + b sin x)

Question 6. Form the differential equation of the family of circles touching the y-axis at origin.

Question 7. Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.

Question 8. Form the differential equation of the family of ellipses having foci on y-axis and centre at origin.

Question 9. Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.

Question 10. Form the differential equation of the family of circles having centre on y-axis and radius 3 units.

Question 11. Which of the following differential equations has y = c1 ex + c2 e–x as the general solution?
(A) 2 2 d y y 0 dx + =
(B) 2 2 d y y 0 dx − =
(C) 2 2 d y 1 0 dx + =
(D) 2 2 d y 1 0 dx − =

Question 12. Which of the following differential equations has y = x as one of its particular solution?
(A) 2 2 2 d y x dy xy x dx dx − + =
(B) 2 2 d y x dy xy x dx dx + + =
(C) 2 2 2 d y x dy xy 0 dx dx − + =
(D) 2 2 d y x dy xy 0 dx dx |
 

EXERCISE 9.4

EXERCISE 9.3

Question 17. Find the equation of a curve passing through the point (0, –2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.

Question 18. At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (– 4, –3). Find the equation of the curve given that it passes through (–2, 1).

Question 19. The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.

Question 20. In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 double itself in 10 years (loge2 = 0.6931).

Question 21. In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e0.5 = 1.648).

Question 22. In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?

Question 23. The general solution of the differential equation dy ex y dx = + is (A) ex + e–y = C (B) ex + ey = C (C) e–x + ey = C (D) e–x + e–y = C

Question 16. A homogeneous differential equation of the from dx h x dy y = can be solved by making the substitution.
(A) y = vx
(B) v = yx
(C) x = vy
(D) x = v

Question 17. Which of the following is a homogeneous differential equation?
(A) (4x + 6y + 5) dy – (3y + 2x + 4) dx = 0
(B) (xy) dx – (x3 + y3) dy = 0
(C) (x3 + 2y2) dx + 2xy dy = 0
(D) y2 dx + (x2 – xy – y2) dy = 0

Miscellaneous Exercise on Chapter 9

Question 1. For each of the differential equations given below, indicate its order and degree (if defined).
(i) 2 2 2 d y 5x dy 6y log x dx dx +
(ii) 3 2 dy 4 dy 7 y sin x dx dx
(iii) 4 3 4 3 d y sin d y 0 dx dx

Question 2. For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
(i) y = a ex + b e–x + x2 : 2 2 2 x d y 2 dy xy x 2 0 dx dx + − + − =
(ii) y = ex (a cos x + b sin x) : 2 2 d y 2 dy 2y 0 dx dx − + =
(iii) y = x sin 3x : 2 2 d y 9y 6cos3x 0 dx + − =
(iv) x2 = 2y2 log y : (x2 y2 ) dy xy 0 dx + − =

Question 3. Form the differential equation representing the family of curves given by (x – a)2 + 2y2 = a2, where a is an arbitrary constant.

Question 4. Prove that x2 – y2 = c (x2 + y2)2 is the general solution of differential equation (x3 – 3x y2) dx = (y3 – 3x2y) dy, where c is a parameter.

Question 5. Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

Question 6. Find the general solution of the differential equation 2 2 1 0 1 dy y dx x − + = − .

Question 7. Show that the general solution of the differential equation 2 2 1 0 1 dy y y dx x x + + + = + + is given by (x + y + 1) = A (1 – x – y – 2xy), where A is parameter.

Question 8. Find the equation of the curve passing through the point 0, 4 whose differential equation is sin x cos y dx + cos x sin y dy = 0.

Question 9. Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0.

Question 10. Solve the differential equation 2 ( 0) x x y e ydx x e y y dy y ≠ .

Question 11. Find a particular solution of the differential equation (x – y) (dx + dy) = dx – dy, given that y = –1, when x = 0. (Hint: put x – y = t)

Question 12. Solve the differential equation 2 1( 0) x e y dxx x x dy.

Question 13. Find a particular solution of the differential equation cot dy y x dx + = 4x cosec x (x ≠ 0), given that y = 0 when 2 x π = .

Question 14. Find a particular solution of the differential equation (x + 1) dy dx = 2 e–y – 1, given that y = 0 when x = 0.

Question 15. The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?

Question 16. The general solution of the differential equation y dx x dy 0 y − = is
(A) xy = C
(B) x = Cy2
(C) y = Cx
(D) y = Cx2

Question 17. The general solution of a differential equation of the type P1 Q1 dx x dy + = is
(A) P1 ( P1 ) Q1 C dy dy y e∫ = ∫ e∫ dy +
(B) P1 ( P1 ) . Q1 C dx dx y e∫ = ∫ e∫ dx +
(C) P1 ( P1 ) Q1 C dy dy x e∫ = ∫ e∫ dy +
(D) P1 ( P1 ) Q1 C dx dx x e∫ = ∫ e∫ dx +

Question 18. The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
(A) x ey + x2 = C
(B) x ey + y2 = C
(C) y ex + x2 = C
(D) y ey + x2 = C