advanced practice test-10

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APT 120 Advanced Practice Test-10

Advanced Practice Test-10

TIME : 3 hrs M.M. : 360

Read the following Instructions very carefully before you proceed.

The Test consists of 90 questions.

The question paper consists of 3 parts: Part I : Chemistry, Part II : Physics, Part III : Mathematics. Each question has 4 choices (A), (B), (C) and (D), out of which Only One choice is Correct.

For each question you will be given 4 Marks if you have darkened only the bubble corresponding to the correct answer and zero mark if no bubble is darkened. In all other cases, minus one (–1) Mark (NEGATIVE MARKING) will be given.

For answering a question, an ANSWER SHEET (OMR SHEET) is provided separately. Please fill your Name, Roll Number, Group, Batch and the PAPER CODE properly in the space provided in the ANSWER SHEET.

IT IS YOUR OWN RESPONSIBILITY TO FILL THE OMR SHEET CORRECTLY.

A blank space has been provided on each page for rough work. You will not be provided with any supplement or rough sheet. However some blank pages for rough work are given at the end of this paper.

The use of log tables, calculator, mobile or any other electronic device is strictly prohibited.

Please do not disturb the invigilator or any other student for any confusion(s) in the paper. Violating the examination room discipline will immediately result in the cancellation of your paper and no excuses will be entertained.

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PART - I (CHEMISTRY) 120 MARKS

LINK COMPREHENSION TYPE This section contains 4 multiple choice questions relating to 15 paragraphs with 30 questions. Each question has four choices A, B, C and D out of which ONLY ONE choice is correct.

Paragraph for Questions 1 - 2

Any reaction, in which the electrons are exchanged between atoms or ions, represents simultaneous process of oxidation and reduction and is called as a redox reactions.

In a redox reaction, the species that loses electron is known as reducing agent and the species which accepts electrons is known as oxidising agent. In a redox reaction, milliequivalents of oxidising and reducing agent are same. A reaction in which same species is oxidised and reduced at the same time is called as a disproportionation reaction. n-factor of any oxidising or reducing agent is equal to total number of electrons loosed or gained by one mole of substance.

1. Which reaction represents the acid base reaction as well as redox reaction ? (A) 3 4 3 3 2 2 2Pb O 4HNO 2Pb(NO ) PbO 2H O

(B) 3 4 2 2 2Pb O 8HCl 3PbCl 4H O Cl

(C) 24 6 2 3 6 2Zn 2K [Fe(CN) ] K Zn [Fe(CN) ] 6K

(D) 2 2 3 2 2 4 62Na S O I Na S O 2NaI

2. If one mole of 3IO ions are heated with excess of iodide ions in presence of acidic conditions

according to equation, 3 2IO I I . How many moles of hypo solution (acidified Na2S2O3 solution) will be required to react completely with iodine thus produced ?

(A) 2

3 (B)

3

2 (C) 3 (D) 6


Equivalent weight is a term which has been used in several contexts in stoichiometric calculations. In its most general usage, it is the mass of a given substance which will : Supply or react with one mole of H+ ions in an acid-base reaction.

Supply or react with one mole of electrons e in a redox reaction.

3. ‘a’ gram KHC2O4 reduces 200 ml of 0.02 M KMnO4 in acidic medium and ‘b’ grams of KHC2O4 neutralises 100 ml of 0.02 M Ca(OH)2. Then a and b are related as :

(A) 2 5a b (B) 2a b (C) 5 2a b (D) a b

4. Equivalent weight of H3PO2 when it disproportionate into PH3 and H3PO3 is : (A) M (B) M/2 (C) 4M/3 (D) 3M/4


The electronic configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals. Knowledge of the electron configuration of different atoms is useful in understanding the structure of the periodic table of elements. The concept is also useful for describing phenomena like magnetic properties of atom, energy of the electrons etc.

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5. Which of the following ions will have the highest paramagnetic character ? (A) Co+3 (B) Cu+2 (C) Cr+3 (D) Fe+3

6. Following are the sets of quantum numbers for four electrons e1, e2, e3 and e4. n m s

1e 3 0 0 1

2

2e 4 0 0 1

2

3e 3 2 2 1

2

4e 3 1 1 1

2

The correct order of decreasing energy of these electrons is : (A) 4 3 2 1e e e e (B) 2 3 4 1e e e e

(C) 3 2 4 1e e e e (D) 1 4 2 3e e e e Paragraph for Question 7 - 8

Catalytic hydrogenation is a best way to convert unsaturated hydrocarbons alkenes and alkynes into alkanes. Reactants passed over surface of the metals like Pd, Pt, Ni and a transition state develops between the reactants from same side and that is why it is a syn addition.

Mechanism in case of other metals H2 molecule will be adsorbed on the surface of metal

To compare rate : (i) More the stable alkene less will be the rate of hydrogenation. (ii) Steric factor will also affect rate of hydrogenation. If Pd/BaSO4 is used alkenes are not reduced to alkanes and major product of alkyne reduction is cis alkene.

7. Number of isomers of C5H12 that can be prepared by catalytic hydrogenation of any alkene (A) 1 (B) 2 (C) 3 (D) None

Catalyst with Hydrogen adsorbed

catalyst with hydrogen and alkene adsorbed

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8. Correct statement(s) are : I. Methane cannot be prepared by catalytic hydrogenation of alkene II. Only 3 isomeric alkenes on catalytic hydrogenation can give iso-pentane as product III. solution of product(s) obtained show no optical rotation.

IV. solution of product(s) obtained show no optical rotation

The correct choice is : (A) I, III, IV (B) I, II, III (C) I, II, IV (D) I, II, III, IV

Paragraph for Questions 9 - 10 The value of standard electrode potential (E ) is useful for the understanding of redox reactions and suitability of an oxidant can be established by it. The value of cellE for a redox reaction is also used to find out equilibrium constant value of the reactions. Based on the equilibrium constant, the amount of the product formed can also be determined. Some half cell reactions and their standard reduction potentials are given below as : I. 2

4 2MnO 8H 5e Mn 4H O ; E 1.51V

II. 2 32 7 2Cr O 14H 6e 2Cr 7H O ; E 1.38V

III. 3 2Fe e Fe ; E 0.77 V

IV. 2Cl 2e 2Cl ; E 1.40V

V. 2 2 2 42CO 2H 2e H C O ; E 0.49V

VI. 2 22 8 4S O 2e 2SO E 2.0 V

9. Which one of the following oxidant can be used for the quantitative estimation of aqueous Fe(NO3)2 ? I. K2Cr2O7 in aqueous HCl II. KMnO4 in aqueous HCl III. K2Cr2O7 in aqueous H2SO4 IV. KMnO4 in aqueous H2SO4 The correct choice is : (A) I, III, IV (B) I, II, III, IV (C) I, III (D) II, IV

10. The approximate value of eqlog K for the given reaction

24 2 2 4 2 22MnO 6H 5H C O 2Mn 8H O 10CO would be:

(A) 255 (B) 170 (C) 330 (D) 83


The CO in a 20 L sample of gas was converted to CO2 by passing the gas over iodine pentoxide heated to

150 C : 2 5 2 2I O (s) 5CO(g) 5CO (g) I (g) . The iodine distilled at this temperature and was collected in an absorber containing 10 mL of 0.011 M Na2S2O3. The excess hypo was back-titrated with 5 mL of 0.001 M I2 solution.

11. What must be the milligrams of CO in 1 L of the original gas sample? (A) 0.35 mg (B) 0.70 mg (C) 0.07 mg (D) 3.5 mg

12. What is the weight (in mg) of iodine produced due to the reaction with l2O5(s) when is reacted with CO(g) in 20 L sample of gas?

(A) 6.35 mg (B) 127 mg (C) 12.7 mg (D) 0.635 mg

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Physical and chemical equilibria respond stress, e.g. change in pressure, temperature and concentration of reactant and products. According to Le Chatelier’s principlea system at equilibrium, when subjected to a disturbance (stress), responds in a way that tends to minimize the effect of the disturbance.

Effect of pressure : The principle implies that if a system at equilibrium is compressed, the reaction will adjust so as to minimize the effect of increase in pressure.

Effect of temperature : Le-Chatelier’s principle predicts that a system at equilibrium will tend to shift in the endothermic direction when temperature is raised.

Effect of addition of reactants or products : On addition of one or more reactants, the equilibrium will shift to the products side so that the added reactants are consumed and increase in their concentrations is opposed.

13. Consider the equilibrium, (g) 2 (g) 2 (g)2CO O 2CO heat . If O2 is added and volume of the reaction vessel is reduced, the equilibrium will :

(A) Shift in forward direction (B) Shift in reverse direction (C) Remain unchanged (D) Be unpredictable

14. If the volume of two systems describing the respective equilibria :

I. 3(s) (s) 2(g)Heat MgCO MgO CO II. (s) 2(g) (g)2C O 2CO heat is decreased, the equilibria will shift :

(A) to the right in both I and II (B) to the left in both I and II (C) to the left in I and to the right in II (D) to the right in I and to the left in II


Polar covalent molecules exhibit dipole moment. Dipole moment is equal to the product of charge separation, q and the bond length d for the bond. Unit of dipole moment is debye. Dipole moment is a vector quantity. It has both magnitude and direction. Hence, dipole moment of a molecule depends upon the relative orientation of the bond dipoles, as well as on the polarity of bonds. A symmetrical structure shows zero dipole moment. 15. Arrange the following compounds in increasing order of dipole moments : Chlorobenzene (I),

o-dichlorobenzene (II), m-dichlorobenzene (III) and p-dichlorobenzene (IV) (A) (IV) < (I) < (II) < (III) (B) (I) < (IV) < (II) < (III) (C) (IV) < (III) = (I) < (II) (D) (IV) < (I) < (III) < (II)

16. Dipole moment of AX4 type of molecule is zero. The shape of molecule can be : (A) Tetrahedral (B) Square planar

(C) (A) or (B) (D) None of these

Paragraph for Questions 17 - 18 Concrete is produced from a mixture of cement, water, sand and small stones. Cement consists primarily of calcium silicates and calcium aluminates formed by heating and grinding of clay and limestone. In the later steps of cement production a small amount of gypsum, CaSO4.2H2O is added to improve subsequent hardening of the concrete. The use of elevated temperatures during the final production may lead to formation of

unwanted hemihydrate, CaSO4.1

2H2O. Consider the following reaction

CaSO4.2H2O(s) CaSO4. 1

2H2O(s) + 1

1

2H2O(g)

The following thermodynamic data apply at 25°C, standard pressure: 1 bar

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Compound H/(kJ mol1) (Hf°) S(JK1 mol1)

CaSO4.2H2O(s) 2021 194

CaSO4.½H2O(s) 1575 130.5

H2O(g) 241.8 188.6

Gas constant, R = 8.314 J mol1 K1 = 0.08314 L bar mol1 K1 (Atomic mass : H = 1, O = 16, S = 32, Ca = 40)

17. The reaction mentioned above is : (A) Exothermic (B) Endothermic (C) Sublimation (D) Disproportionation

18. The standard change in enthalpy for the transformation of 1 kg of CaSO4.2H2O(s) to CaSO4.1

2H2O(s)

is : (A) 189.6 kJ (B) 443.2 kJ (C) 214.5 kJ (D) 484 kJ


A chemist accidently mixed up three anions A , B , and C . To separate them, he used a method in which a solution of 0.1M AgNO3 was added dropwise into the flask containing the mixture of A , B , and C . On the basis of different Ksp values of AgA, AgB and AgC respectively, different layers of ppt were deposited at different titration points Given the Ksp of AgA = 1010 ; AgB = 1310 ; AgC = 1610 and initial concentration of A , B , and C = 510 M each.

19. If AgA is white ppt, AgB is red ppt and AgC is black ppt. then what is the colour of bottom most layer? (A) White (B) Red (C) Black (D) Grey

20. What will be the concentration (in M) of B and C respectively when AgA starts precipitating? (A) 8 1110 and 10 (B) 11 1110 and 10 (B) 11 810 and 10 (D) 8 810 and 10


A metal carbide A1 on reaction with water produces a mixture of hydrocarbons B1 and B2.

2 22 4 4

AmmonicalH O Isilver nitrate

1 H SO , HgSO NaOHwhite ppt. B (g) (D) yellow ppt.

H2 2

unstableB (g) H O (E) (D)

21. Metal carbide A1 is : (A) 4 3Al C (B) 2 3Mg C (C) 2CaC (D) SiC

22. Identify the correct statement : (A) B1 and B2 are optical isomers (B) B1 and B2 are geometrical isomers

(C) B1 and B2 are structural isomers (D) None of these


Two identical bulbs contain an excess of solid A (volatile) and B (non-volatile) respectively. At 25C equilibrium gets established as shown above at equilibrium; total pressure inside both bulbs is 10 mm whereas partial pressure of H2S inside bulb (I) and (II) is 4 mm and 5 mm respectively.

Bulb - I Bulb -II

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23. If some A(g) is removed bulb-1 at equilibrium then : (A) Only decomposition equilibrium will shift forward (B) Only sublimation equilibrium will shift forward (C) Both decomposition and sublimation equilibria will shift forward (D) No change will happen in the state of both the equilibria

24. If 1 mm of B(g) is removed from bulb (II) and equilibrium is allowed to re-stablish at 25C the inside pressure will:

(A) Remain 10 mm (B) Increase slightly (C) Decrease slightly (D) Be unpredictable


Three diatomic species that are biologically important because they either promote or inhibit life are (i) CO, (ii) NO, and (iii) CN . The first binds to haemoglobin, the second is a neurotransmitter, and the third intercepts the respiratory electron-transfer chain. Their biochemical action is a reflection of their orbital structure.

25. In a complex 2 5 4[Fe(H O) (NO)]SO , NO is present as NOx . What is value of x and bond order in an

isolated NOx species? (A) 0, 2.5 (B) +1, 2.5 (C) 0, 3 (D) +1, 3

26. Which of the following is paramagnetic? (A) CO (B) NO (C) CN (D) All of these


A sample of iron weighing 15.0 g was heated with potassium chlorate 3(KClO ) in an evacuated container. The oxygen generated from the decomposition of 3KClO converted some of the Fe to 2 3Fe O . The combined mass of Fe and 2 3Fe O was 17.9 g. [Atomic Masses are : K = 39, Cl = 35.5, Fe = 56, O = 16]

27. What is the mass of 2 3Fe O formed? (A) 9.6 g (B) 10.9 g (C) 2.9 g (D) 5.6 g

28. What is the mass of 3KClO decomposed? (A) 14.8 g (B) 7.4 g (C) 9.6 g (D) 5.6 g


Substitution and elimination reactions always compete with each other in case of alkyl halides. It depends upon both the nature of halides and the type of reagents. It is due to the fact that a nucleophile also has a basic character and vice-versa.

Isopropyl bromide was treated separately with sodium tert-butoxide and sodium ethoxide under two different conditions.

Reaction 1 : Treatment of isopropyl bromide with sodium tert-butoxide at 40 C gave almost exclusively compound A.

Reaction 2 : Treatment of isopropyl bromide with sodium ethoxide at 30 C yielded compound A along with small amount of a compound B.

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Compound A was readily oxidized by a neutral solution of cold dilute potassium permanganate to give a brown precipitate.

29. Formation of A and B can best be explained by (A) E2 reaction and SN1 reaction respectively (B) E2 reaction and SN2 reaction respectively (C) E1 reaction and SN1 reaction respectively (D) SN2 reaction and E2 reaction respectively

30. If instead of isopropyl bromide, we take tert. pentyl bromide and treat it with ethanol at room temperature, then which of the following is(are) true ?

I. The reaction involves the formation of carbocation.

II. The major product will be 3 3

3

CH C CHCH|

CH

via E2 reaction.

III. The reaction follows nearly 90 - 95% racemisation

IV. The major product will be

2 5

3 3

3

OC H|

CH C CHCH|

CH

via SN1 reaction.

The correct option is : (A) I, II (B) I, IV (C) I, III, IV (D) None of these

PART - II (PHYSICS) 120 MARKS

LINK COMPREHENSION TYPE

This section contains 4 multiple choice questions relating to 14 paragraphs with 30 questions. Each question has four choices A, B, C and D out of which ONLY ONE choice is correct.


The drawing shows box 1 resting on a table, with box 2 resting on top of box 1. A massless rope passes over massless, frictionless pulley. One end of the rope connected to box 2 and the other end is connected to box 3. The weights of the three boxes are

1 2 355 35 and 28W N W N W N .

1. Determine the magnitude of the normal force that the table exerts on box 1. (A) 55 N (B) 62 N (C) 48 (D) 90 N

2. If the pulley is pulled upward with an acceleration that increases with time, a = t/4 where t is the time in seconds, what is the time when the block 2 is lifted off ?

(A) 2.5 sec (B) 5 sec (C) 1.25 sec (D) 3.75 sec


In the shown figure, the spring and strings are ideal. The spring has stiffness of 100 N/m and m = 1 kg. Friction exists between mass 2m and surface with coefficient 0 8. . The system is released with spring from its relaxed position. Based on above data, answer the following questions : (Take g = 10 m/s2)

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3. Maximum extension in spring is : (A) 8 cm (B) 4 cm (C) 36 cm (D) 20 cm

4. Magnitude of work done by gravity during the motion of system is : (A) 0.8 J (B) 1.6 J (C) 0.4 J (D) 4 J


One mole of a diatomic gas is heated under a “Kibolinsky Process” in which gas pressure and temperature is

controlled according to law 3 245

/P CT . Where C is Kibolinsky constant. In this process gas is heated by

300 K temperature. Answer the following questions : 5. Under above process the molar specific heat of gas is : (A) R/2 (B) R (C) 2 R (D) 3 R

6. The amount of work done on gas under above heating is : (A) 600 R (B) 750 R (C) 150 R (D) 100 R


A container of large uniform cross-sectional area A resting on a horizontal surface, holds two immiscible, non-viscous and incompressible liquids of densities d and 2d, each of height H/2 as shown in the figure. The lower density liquid is open to the atmosphere having pressure 0P .

A homogenous solid cylinder of length 2L L H / , cross-sectional

area A/5 is immersed such that it floats with its axis vertical at the liquid-liquid interface with length L/4 in the denser liquid :

7. The density D of the solid is : (A) 3d / 2 (B) 9 d / 4 (C) 7 d / 4 (D) 5 d / 4

8. The total pressure at the bottom of the container is :

(A) 032 4H LP dg

(B) 0 2

LP H dg

(C) 0 4LP H dg

(D) 0 2 4

H LP dg


A jet of water is following from a water gun W. a rod ABCD is attached to the water gun W with the end A of the rod fixed to the point where water jet is coming out. Angle made by rod with horizontal is same as the angle of projection of water Jet. Strings are tied to the rod at equal distances so that AB = BC = CD. The length of the strings are adjusted so that the lower ends just touch the curved stream of water. If the length of the string at B is 5cm.

9. The length of the string at C is : (A) 10 cm (B) 20 cm (C) 25 cm (D) 30 cm

10. The length of the string at D is : (A) 15 cm (B) 20 cm (C) 45 cm (D) 30 cm

d

2d

H/2

H/2

A

B

C

D

W

Stream of water

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When a ball of density d is dropped on to a horizontal solid surface, it bounces elastically from the surface and returns to its original position in time t. Next the ball is released and falls through the same height before striking the surface of a nonviscous liquid of density dL. (Consider size of ball to be very small) L( )d d

11. Time taken by the ball to come to rest when it enters into the liquid is : (Assume depth of liquid is very large)

(A) 2 L

dtd d

(B) 4 L

dtd d

(C) 2 L

dtd d

(D) 4 L

dtd d

12. Time taken by the ball to come back to the position from where it was released is :

(A) L

L

tdd d

(B) L

L

tdd d

(C) 2

L

L

tdd d

(D) 2

L

L

tdd d


Three identical parallel large plates are placed as shown in figure. The outer two

plates are given charge Q each and middle plate is given charge 2Q. The outer

plates are connected by a conducting wire.

13. The charge appearing at the left surface of the middle plate is :

(A) 23Q

(B) 43Q (C) 3

4Q (D) 5

3Q

14. Now the middle plate is moved by distance L right maintaining the parallelism, the charge flown through the connecting wire is :

(A) 0 (B) 2Q (C) 2

3Q (D)

3Q


Refer to the adjacent figure. Two geometrically identical balls A and B are submerged in water of density by means of two strings. The string attached to A is attached to the base of the container and the one attached to B, has its other end attached to hoop which is rigidly attached to the container. The density of A is 0 while that of B is 0 . The setup is placed on a weighing machine.

15. Suppose the string attached to A breaks then immediately afterwards. The reading of the weighing machine.

(A) Will increase (B) Will decrease (C) Will remain unaffected (D) May increase or decrease depending on the value of 0

16. Suppose the string attached to ball B were to break then just after the string breaks. The reading of the weighing machine

(A) Will remain unaffected (B) Will increase (C) Will decrease (D) May increase or decrease depending on the value of 0

L 2L

Q Q 2Q

B A

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The current through 12Ω resistance is zero.

17. The value of E is : (A) 24V (B) 36 V (C) 42 V (D) 45 V

18. The resistance r is : (A) 1Ω (B) 2Ω (C) 3Ω (D) 4Ω

Paragraph for Questions 19 - 20 Two particles whose masses are m and M are connected by a string of length a and placed on a horizontal frictionless table. Initially the string is taut and mass m is projected with a velocity v at right angles to the string.

19. The distance of each particle from original positions when the string has made a complete rotation will be:

(A) amm M

(B) 2 amm M

(C) 3 am4(m M)

(D) 4 am(m M)

20. In the previous question, tension in the string while the particles are moving will be:

(A) Zero (B) 2mMV

a(m M) (C)

22mMVa(m M)

(D) 2mMV

2a(m M)

Paragraph for Question 21 - 23

A Capacitor and a Pendulum : We begin with an uncharged, isolated, parallel plate capacitor having its plates maintained at a fixed distance apart and with an isolated independent voltage source. By connecting the two plates of the uncharged capacitor momentarily to the independent voltage source and then disconnecting the source, we are left with a charged and isolated capacitor. A small ball of cork, covered with a conducting foil, is suspended by an insulating thread between the two plates of the capacitor as a simple pendulum.

If the ball is initially at rest and is closer to the positive plate, it will be slightly attracted to that plate because of induction. On contact with the positive plate, some of the plate's positive charge is transferred to the ball by charge sharing. The positively charged ball then is repelled by the positive plate and attracted to the negative plate. Upon reaching the negative plate, the kinetic energy of the ball is completely converted into thermodynamic internal energy of the negative plate. The positive charge on the ball neutralizes some of the negative charge on the negative plate. The ball also then becomes negatively charged by charge sharing and subsequently is repelled by the negative plate and attracted back to the positive plate.

conducting ball

Insulating thread

+|Q| –|Q|

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The process continues with the electric pendulum swinging back and forth between the two plates until essentially all of the charge on the capacitor is neutralized and the capacitor is discharged. We imagine positive charge transferred one way, negative charge the other way until the two plates are discharged. We observe that the force between the plates decreases with each swing of the pendulum, confirming our account of the neutralization or discharge of the two plates. Once discharged, the field between them is zero, they do not exert electric force on each other.

21. During the swinging of the charged ball (A) the current is from left to right (B) the current is from right to left (C) the current is from left to right during when ball moves to the left and the current is from right

to left when ball moves to the right (D) the current is from right to left during when ball moves to the left and the current is from left to

right when ball moves to the right

22. Consider the moment when the ball leaves the positive plate taking away a charge of 0.01µC, leaving a charge of 8.85µC on the positive plate. The tension in the string, when the ball reaches the lowest position for the first time is nearly. (Assume the distance between the plates is 1 cm and length of the thread is 1m, area of the plates is 1m2 and mass of ball is 1mg).

(A) 6 × 10–5 N (B) 3 × 10–5 N (C) 11 × 10–5 N (D) 10–5 N

23. If the initial charge on the capacitor plates is 10µC, and the capacitance of the capacitor is 10µF, the total change in thermodynamics internal energy of the left plate is :

(A) 5µJ (B) 2.5µJ (C) 10µJ (D) 7.5µJ

Paragraph Questions 24 to 26

A magnetic field B perpendicular to the X–Y plane is

restricted to a region of length l and width b as shown in

the figure. An electron (mass of electron = m) enters into

the region of B

at origin ‘O’ with a speed V0 at an angle

θ to the X-axis.

24. The range of θ for which the electron emerges out of the region of magnetic field on the side of the

X-axis (i.e. at a point (x, 0) such that 0 x l ) is0 0

13 12

elB ebBGiven that : ,mV mV

(A) 45 60 (B) 0 60 (C) 30 60 (D) 60 90

25. Now suppose that the electron was initially projected at an angle 2

with a speed V. It emerges out

of the region at a point (x, b) such that 0 < x < l. Given that : 0 6beB .mV

Then, x is given by

(A) 43b (B) 2

3b (C)

3b (D)

35b

26. In the previous question, assume the instant at which electron emerges out of (x, b); 0 < x < l, to be t = 0. At t = 0, an electric field 0

ˆE E j

is switched on in the entire region. Find the minimum value of E0 for which the electron again re-enters the region of B

on the same side on which it exited.

Given that : 0 6 1beB leB. ,mV mV

(A) BV (B) 1.2 BV (C) 1.6 BV (D) 1.8 BV

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A man of mass m clings to a rope slung over a fixed pulley. The opposite end of the rope is tried to a weight of mass M lying on a horizontal table. The coefficient of friction between the weight and the table is µ. Find the acceleration of weight and the tension of the rope for two cases

The man moves downwards with respect to the rope with an acceleration b.

27. The acceleration of weight is

(A) 2 ( )2

m g b MgM m

(B) ( )2( )

m g b MgM m

(C) ( ) 33

m g b MgM m

(D) ( )m g b MgM m

28. The tension of rope is

(A) ( )Mm g g bM m

(B) ( )Mm g g bM m

(C) ( )Mm g g bM m

(D) ( )Mm g g bM m


For equal positive charges, each of value Q, are arranged at the four corners of a square of diagonal 2a. A small body of mass m carrying a unit positive charge is placed at a height h above the centre of the square.

29. What should be the value of Q in order that this body may be in equilibrium?

(A) 2 2 3/ 2( 2 )2omg h a

h (B) 2 2 3/ 2( )o

mg h ah

(C) 2 2 3/ 22 ( 2 )2omg h ah

(D) 2 2 3/ 2( )2omg h a

h

30. The type of equilibrium of the point mass is (consider only vertical displacement) (A) Stable equilibrium (B) Unstable equilibrium (C) Neutral equilibrium (D) Cannot be determined

PART - III (MATHEMATICS) 120 MARKS

LINK COMPREHENSION TYPE This section contains 4 multiple choice questions relating to 13 paragraphs with 30 questions. Each question has four choices A, B, C and D out of which ONLY ONE choice is correct.

Paragraph for questions 1 - 2

A function ( )f x is monotonic increasing or decreasing according as '( ) 0f x or '( ) 0.f x Let us consider a

function 2

0

( ) {2 2 sin (2 2)sin 1} ,0 2π. x

f x t t dt x Then :

1. Interval of x in which ( )f x is increasing is given by :

(A) π 5π,6 6

(B) π0,6

(C) 5π 5π,6 4

(D) 7π , 2π4

2. ( )f x is decreasing in the region :

(A) π 5π,6 6

(B) 5π 5π,6 4

(C) 7π0,4

(D) 5π 7π,4 4

Vidyamandir Classes


Paragraph for Questions 3 – 5 A chemical manufacturing company has a 1000 kl holding tank which it uses to control the release of pollutants into a sewage system. Initially the tank has 360 kl of water containing 2 kg of pollutant per kl. Water containing 3 kg of pollutant per kl enters the tank at the rate 80 kl per hour and is uniformly mixed with the water already in the tank. Simultaneously water is released from the tank at the rate of 40 kl per hour. If P(t) is the amount of pollutant (in kg) in the tank at time t hours.

3. The value of P(10) is approximately (A) 2000 (B) 3100 (C) 2109 (D) 4321

4. The value of 1

0

P t dt is

(A) 1140 – log5/3 (B) 1140 - 3240 109

log

(C) 2140 - 1140 89

log (D) 2140 – 3240 109

log

5. 10t

dPdt

is approximately equal to

(A) 158 (B) 149 (C) 129 (D) 212

Paragraph for Questions 6 – 7 Let A be a square matrix of order n×n . A constant is said to be characteristic root of A if there exists a n×1 matrix X such that AX X X 0 6. If is a characteristic root of A, then (A) A I is singular (B) A I is singular (C) A I is non-singular (D) None of these

7. If is a characteristic root of A and n is a natural number then n is a characteristic root of

(A) nA (B) 1nA (C) nA (D) n nA A A Paragraph for Q. 8 - 10

It is known that 2 2

2 2 1 x ya b

is the equation of the conjugate hyperbola of the hyperbola 2 2

2 2 1x ya b

. Also,

the pair of lines 2 2

2 2 0x ya b

are the asymptotes of the hyperbola 2 2

2 2 1x ya b

.

Let the equation of a hyperbola be 2 23 2 8 0x y x 8. The equation of its conjugate hyperbola is : (A) 2 23 2 8 0x y x (B) 2 23 2 8 0x y y

(C) 2 23 2 10 0x y x (D) 2 23 8x y

9. The equation of the asymptotes of the hyperbola is : (A) 2 23 2 0x y x (B) 2 23 0x y

(C) 2 23 2 0x y y (D) 2 23 2 1 0x y x 10. If ( 1 2a ) is a point exterior to the hyperbola but interior to its conjugate hyperbola then the number

of positive integral values of a is : (A) 3 (B) 2 (C) 1 (D) 4

Vidyamandir Classes


Paragraph for Q. 11 - 13 At times the methods of coordinates becomes effective in solving problems of properties of triangles. Without loss of generality, we may choose one vertex of the triangle as origin and one side passing through this vertex as x-axis.

11. If in Δ , 3, 4,ABC AC BC medians AD and BE are perpendicular, then area of triangle ABC must be

(A) 7 (B) 11 (C) 2 2 (D) None of these

12. Suppose the bisector AD of the interior angle A of ΔABC divides side BC into segments 4, 2,BD DC then we must have

(A) 6b and 4c (B) 2 6b and c 1 (C) 2 6b and 4 12c (D) None of these

13. If altitudes 7, 6CD AE and E divides BC such that 3 ,4

BEEC

and 14a then c must be

(A) 2 3 (B) 5 3 (C) 3 (D) 6 3


Consider the three vectors , andp q r and a scalar c such that ˆ ˆˆ ˆ ˆ ˆand ; and · 2p i j k q i j k p r q cp p r .

14. If x is a vector such that , thenp q r x p q r x is :

(A) ˆˆ ˆ( 2 )c i j k (B) a unit vector (C) indeterminate (D) ˆˆ ˆ( 2 ) / 2i j k

15. If y is a vector satisfying 1 c y p q r , then the vectors , andx y r (A) are collinear (B) are coplanar (C) represent the coterminous edges of a tetrahedron whose volume is |c| cu units (D) represent the coterminous edges of a parallelepiped whose volume is |c| cu units


If a circle with centre ( , )C intersects, a rectangular hyperbola with centre ( , )L h k at four points.

1 1 2 2 3 3( , ), ( , ), ( , )P x y Q x y R x y and 4 4( , )S x y , then the mean of the four points P, Q, R, S is the mean of the points C and L, in other words, the mid points of CL coincides with the mean point of P, Q, R and S analytically

1 2 3 4 1 2 3 4;4 2 4 2

x x x x y y y yh k .

16. Five points are selected on a circle of radius a. The centres of the rectangular hyperbolas, each passing through four of five points, all lie on a circle of radius.

(A) a (B) 2a (C) 2

a (D) 2a

17. A, B, C, D are the points of intersection of a circle and a rectangular hyperbola which have different centres. If AB passes through the centre of the hyperbola, then CD passes through:

(A) Centre of the hyperbola (B) Centre of the circle

(C) Mid points of the centres of the circle and hyperbola (D) None of these

Vidyamandir Classes


18. A circle with fixed centre (3 , 3 )h k and of variable radius cuts the rectangular hyperbola 2 2 29x y a at the points A, B, C, D. The locus of the centroid of the triangle ABC is given by:

(A) 2 2 2( 2 ) ( 2 )x h y k a (B) 2 2 2( ) ( )x h y k a

(C) 2 2

22 2

x y ah k

(D) 2 2

22 2

x y ah k


If 3 2( ) 2 3( 3) 6 2f x x a x ax a is cubic equation in x. Then give the answer of the following :

19. The value of ‘a’ for which f (x) has exactly one point of local maxima and one point of local minima : (A) ( ,1) (9, ) (B) ( ,1] [9, ) (C) [1, 9] (D) (1, 9)

20. The value of ‘a’ for which f (x) has local minima at some negative real x (A) ( ,1) (9, ) (B) ( ,1] [9, ) (C) (0, 1) (D) (1, 9)


A variable lines L is drawn through (0,0)O to meet the lines 1 2andL L given by 10 0y x and 20 0y x at the points A and B, respectively.

21. A point P is taken on L such that 2 1 1OP OA OB

. Then locus of P is :

(A) 3 3 40x y (B) 3 3 40 0x y (C) 3 3 40x y (D) 3 3 40y x

22. Locus of P if 2OP OA OB is :

(A) 2( ) 100y x (B) 2( ) 50y x (C) 2( ) 200y x (D) 2 100y x


Let x1, x2, x3, x4 be the roots (real or complex) of the equation x4 + ax3 + bx2 + cx + d = 0. If x1 + x2 = x3 + x4 and a, b, c, d R then.

23. If a = 2 then the value of b c is : (A) 1 (B) 1 (C) 2 (D) 2

24. If b + c = 1 and 2 a then for real values of ‘a’, c belongs to :

(A) 14

, (B) 3, (C) 1, (D) 4,


A circle C whose radius is 1 unit touches X-axis at point A. the centre Q of C lies in 1st quadrant. The tangent from O to circle touches it at T and a point P lies on this tangent such that OAP is a right angled triangle at A and its perimeter is 8 unit.

25. The length of QP is :

(A) 12

(B) 43

(C) 53

(D) None of these

Vidyamandir Classes


26. Equation of circle C is :

(A) 2 22 1 1 x y (B) 2 21 1 1 x y

(C) 2 22 2 1 x y (D) 2 21 3 1 x y


Let f x be a cubic polynomial which has local maximum at 1 andx f x has a local minimum at x = 1.

If 1 10 and 0 5 f f .

27. The distance between two horizontal tangents is : (A) 12 (B) 22 (C) 32 (D) 10

28. If x1, x2, x3 are the real roots of 0f x then 2 2 21 2 3x x x is equal to :

[Note : [k] denotes the greatest integer less than or equal to k] (A) 20 (B) 6 (C) 22 (D) 25


The equation 4 3 22 3 4 1 0x x x x has four distinct real roots 1 2 3 4x , x , x , x such that 1 2 3 4x x x x and product of two roots is unity, then :

29. The value of 1 2 1 3 2 4 3 4x x x x x x x x is :

(A) 0 (B) 1 (C) 5 (D) 1

30. The value of 3 32 4x x is :

(A) 2 5 5

8

(B) 4 (C) 27 5 5

4

(D) 18

advanced practice test-10

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