mock test qp

..Part - A : Mathematics..

SECTION – I - Single Correct Choice TypeThis section contains 8 multiple choice questions. Each questionhas 4 choices (a), (b), (c) and (d) for its answer, out of which ONLYONE is correct.

1. The complex numbers 1 + i and 1 + 2i are both roots of theequation x5 – 6x4 + Ax3 + Bx2 + Cx + D = 0,where A, B, C, D ∈ R. The value of D is(a) – 20 (b) 20(c) 10 (d) 2

2. The set of values of x for which the identity

1 1 21cos cos 3 3

2 2 3xx x− − π + + − =

holds good, is

(a) [0, 1] (b)

21,0 (c)

1,21 (d) –1,0,1

3. Let q

2008 rdx 1 xln C

px x 1 x

= +

+ + ∫ , where p, q, r ∈ N and

need not be distinct, then the value of (p + q + r) equals(a) 6024 (b) 6022 (c) 6021 (d) 6020

IIT-JEE MOCK TEST - 1

Space for Rough Work

* GENERAL INSTRUCTIONS :

1. Section I : Q. No. 1 to 8, Q. No. 21 to 28 and Q. No. 41 to 48 are Single Correct Choice Type questions. For this section, 3 markswill be awarded for correct answer and zero marks for no answer. In all other cases, –1 mark will be awarded.

2. Section II : Q. No. 9 to 12, Q. No. 29 to 32 and Q. No. 49 to 52 are Multiple Correct Choice Type questions. For this section, 4marks will be awarded for correct answer and zero marks for no answer. In all other cases, –1 mark will be awarded.

3. Section III : Q. No. 13 & 14, Q. No. 33 & 34 and Q. No. 53 & 54 are Multiple Matrix-Match Type questions. Each questioncontains statements given in two columns, which have to be matched. The statements in Column-I are labelled A, B, C and D,while the statements in Column-II are labelled p, q, r, s and t. Any given statement in Column -I can have correct matching withONE OR MORE statement(s) in Column-II. 2 marks each will be awarded for darkening the correct bubble in each row. Hencethe maximum marks of each question is 8 marks. No negative mark will be awarded for an incorrectly bubbled answer.

4. Section IV : Q. No. 15 to 20, Q. No. 35 to 40 and Q. No. 55 to 60 are Comprehension Type questions. For this section, 4 markswill be awarded for correct answer and zero marks for no answer. In all other cases, –1 mark will be awarded.

Time : 180 minutes Max. Marks : 240

PAPER - 1


2 IIT-JEE Mock Test - 1

4.

x1

x

2 tan xlim−

→∞

π

equals eL then L is equal to

(a)2π

(b)2

−π

(c)2π

− (d) 1

5. Let C1 and C2 are circles defined by x2 + y2 – 20x + 64 = 0 andx2 + y2 + 30x + 144 = 0. The length of the shortest linesegment PQ that is tangent to C1 at P and to C2 at Q is(a) 15 (b) 18 (c) 20 (d) 24

6. The sum of all positive integral values of ‘a’, a ∈ [1, 500] forwhich the equation [x]3 + x – a = 0 has solution is (where [ ]denote the greatest integer function)(a) 462 (b) 512 (c) 784 (d) 812

7. Area enclosed by the graph of the function y = ln2x – 1 lyingin the 4th quadrant is

(a)2e

(b)4e

(c)1

2 ee

+ (d)

14 e

e −

8. Equation of the plane containing the lines

r (1,1,0) t (1, 1,2)= + −r , r (2,0, 2) s ( 1,1,0)= + −

r is(a) x + 3y + z = 4 (b) x + y – 2 = 0(c) 5x – 3y – 4z = 2 (d) None of these

SECTION – II - Multiple Correct Choice TypeThis section contains 4 multiple choice questions. Each questionhas 4 choices (a), (b), (c) and (d) for its answer, out of which ONEOR MORE is/are correct.

9. Let f (x) = x3 + px2 + qx + 6, where p, q ∈ R and f ' (x) < 0 in

largest possible interval 5

, 13

− − then (p + q) is greater

than(a) –2 (b) 2 (c) 4 (d) 6

10. Choose the correct options

(a) The sum 12

1

2are tan

n

r r−

=

∑ equals

34π

(b) 2nlim n sin (2 1 n 2n ) (n N)→∞

π + − π ∈ equals π

(c) Period of the function f (x) = sin22x + cos42x + 2, is 4π

(d)1

1/2 3/2

1

(1 x) (1 x) dx−

+ −∫ equals 2π

11. Choose the incorrect statement(a) If a.b a.c b c (a 0)= ⇒ = ≠

r rr r r r r

(b) If a b a c b c (a 0)× = × ⇒ = ≠r rr r r r r

(c) If a.b a.c and a b a c b c (a 0)= × = × ⇒ = ≠r r rr r r r r r r r

(d) If 1 2 3v , v , vr r r are non-coplanar vectors and

2 31

1 2 3

v vk

v .(v v )×

=×

r rrr r r ; 3 1

21 2 3

v vk

v .(v v )×

=×

r rrr r r and

1 23

1 2 3

v vk

v .(v v )×

=×

r rrr r r then 1 2 3

1 2 3

1k .(k k )v .(v v )

× =×

r r rr r r

12. Choose the correct options(a) Locus of the feet of the perpendiculars drawn from the

foci on a variable tangent to the hyperbola16y2 – 9x2 = 1 is x2 + y2 = 1/16

(b) A line passing through the point (21, 30) and normal to

the curve y 2 x= can have the slope equal to –5(c) The magnitude of the gradient of the tangent at an

extremity of latus rectum of the hyperbola 2 2

2 2x y 1a b

− =

is equal to e (where e is the eccentricity of thehyperbola)

(d) TP and TQ are tangents to the parabola, y2 = 4ax at Pand Q. If the chord PQ passes through the fixed point(–a, b) then the locus of T is y = 2a (x – a)


IIT-JEE Mock Test - 1 3

SECTION – III - Matrix-Match TypeThis section contains 2 questions. Each question contains statements given in two columns, which have to bematched. The statements in Column-I are labelled A, B, C and D, while the statements in Column-II arelabelled p, q, r, s and t. Any given statement in Column -I can have correct matching with ONE OR MOREstatement(s) in Column-II. The appropriate bubbles corresponding to the answers to these questions have to bedarkened as illustrated in the following example:

p q r s tppp

qqq

rrr

sss

ttt

p q r s tABCD

If the correct matches are A–p, s and t; B–q and r; C–p and q; and D–s and t; then the correct darkeningof bubbles will look like the given :

13. Column-I Column-II

(A) The values of 2 4cos sinθ + θ for all θ p. belong to (0, 1]

(B) In a ∆ABC if tan A < 0 then values of tan B tan C q. belong to 3,1

4

(C) For any real ,n n Iθ ≠ π ∈ the values of 2

2cos 1

cos cosθ −

θ + θr. are less than 0

or greater than 2

(D) If 0, 0 and3

A B A Bπ

> > + = then the values of 3 tan tanA B s. belong to (0, 1)

14. Column-I Column-II

(A) The roots of cubic equation (z + αβ)3 = α3 ( )0, Rα ≠ α ∈ represent the vertices of a p. | tan α |

triangle of area equal to

(B) If α is a complex number then the radius of the circle zz

− α− α

= 2 is equal to q. 2||4

33α

(C) If arg z = α and 1z − =1 then z 2

z−

is equal to r. ||32

α−α

(D) Let A and B represent complex numbers z1 and z2, which are roots of the equation s.2

cos4 2 α

z2 + pz + q = 0. If ∠AOB = α ≠ 0 and OA = OB, where O is the origin then 2p

q is equal to



SECTION – IV - Comprehension TypeThis section contains 2 groups of questions. Each group has 3multiple choice questions based on a paragraph. Each questionhas 4 choices (a), (b), (c) and (d) for its answer, out of which ONLYONE is correct.

Paragraph for question nos. 15 to 17

Let E : denotes the event that a student does his homeworkwith P (E) = p and F : denotes the event that he answer the questioncorrectly.

15. If p = 0.75, then the value of P (E/F) equals

(a)8

16(b)

1016

(c)1216

(d)1516

16. The relation P (E/F) ≥ P (E) holds good for(a) all values of p in [0, 1](b) all values of p in (0, 1) only(c) all values of p in [0.5, 1] only(d) p = 0, 1

17. Suppose that each question has n alternative answers ofwhich only one is correct, and p is fixed but not equal to 0 or1 then P (E/F)(a) decreases as n increases for all p ∈ (0, 1)(b) increases as n increases for all p ∈ (0, 1)(c) remains constant for all p ∈ (0, 1)(d) decreases if p ∈ (0, 0.5) and increases if p ∈ (0.5, 1) as

n increases.

Paragraph for question nos. 18 to 20

Consider a polygon of sides n which satisfies the equation3.nP4 = n–1P5.

18. Rajdhani express travelling from Delhi to Mumbai has 10stations enroute. Number of ways in which a train can bestopped at 3 stations if no two of the stopping stations areconsecutive, is(a) 20 (b) 35(c) 56 (d) 85

19. Number of quadrilaterals that can be made using the verticesof the polygon of sides 10 if exactly two adjacent sides ofthe quadrilateral are common to the sides of the n-gon, is(a) 50 (b) 60(c) 70 (d) None of these

20. Number of quadrilaterals that can be formed using thevertices of a polygon of sides 10 if exactly 1 side of thequadrilateral is common with the side of the n-gon, is(a) 150 (b) 100(c) 96 (d) None of these

..Part - B : Physics..


21. Two blocks each of mass m lie on a smooth table. They areattached to two other masses as shown in the figure. Thepulleys and strings are light. An object O is kept at rest onthe table. The sides AB and CD of the two blocks are madereflecting. Find the acceleration of two images formed inthose two reflecting surfaces w.r.t. each other.



3m 2m

m m

B D

A C

O

(a) 17g/6 (b) 7g/6(c) 11g/6 (d) 5g/6

22. A small quantity of solution containing Na24 radionuclide(half life 15 hours) of activity 1.0 microcurie is injected intothe blood of a person. A sample of the blood of volume1cm3 taken after 5 hours shows an activity of 296disintegrations per minute. Determine the total volume ofblood in the body of the person. Assume that the radioactivesolution mixes uniformly in the blood of the person.(1 curie = 3.7 × 1010 disintegrations per second).(a) 5.91 litres (b) 0.91 litres(c) 3.21 litres (d) 4.12 litres

23. Consider the potentiometer circuit arranged as in figure.The potentiometer wire (resistance 15r) is 600 cm long. If thejockey touches the wire at a distance of 560 cm from A, whatwill be the current in the galvanometer ?

N

r

A

E/2

E

Gr

(a) 3E/22r (b) 5E/22r(c) 3E/11r (d) 5E/11r

24. Three identical positive charges Q are arranged at the verticesof an equilateral triangle. The side of the triangle is a. Findthe intensity of the field at the vertex of a regular tetrahedronof which the triangle is the base.

(a) 2KQ

6a

(b)2

KQ2

a(c) 2

KQ3

a(d) None of these

25. A point source is emitting sound in all directions. Find theratio of distance of two points from the point source wherethe difference in loudness levels is 3 dB. (log10 2 = 0.3)(a) 2 (b) 3

(c) 1/2 (d) 1/ 226. 0.5 mole of an ideal gas at constant temperature 27°C kept

inside a cylinder of length L and cross-section area A closedby a massless piston.

The cylinder is attached with a conducting rod of length L ,cross-section area (1/9) m

2 and thermal conductivity K,whose other end is maintained at 0°C. If piston is movedsuch that rate of heat flow through the conducing rod isconstant then velocity of piston when it is at height L/2from the bottom of cylinder is :[Neglect any kind of heat loss from system]

(a)K

m / secR

(b)

Km / sec

10R

(c)K

m / sec100R

(d) K

m / sec1000R



27. A U-tube of base length l filled with same volume of twoliquids of densities ρ and 2ρ is moving with an accelerationa on the horizontal plane. If the height difference betweenthe two surfaces (open to atmosphere) becomes zero, thenthe height h is given by –

h a

(a)a

2gl (b)

3a2g

l

(c)ag

l (d)2a3g

l

28. An object is moving in the xy plane with the position as a

function of time given by ˆ ˆr x(t)i y(t) j= +r . Point O is at

r 0=r . The distance of object from O is definitely decreasingwhen(a) vx > 0, vy > 0 (b) vx < 0, vy < 0(c) xvx + yvy < 0 (d) xvx + yvy > 0


29. If the first minima in a Young’s double slit experiment occursdirectly in front of one of the slits, (distance between slitand screen D=12cm and distance between slits d=5cm.) thenthe wavelength of the radiation used can be

(a) 2 cm (b) 4 cm(c) 2/3 cm (d) 4/3 cm

30. A small sphere of mass m suspended by a thread is firsttaken aside so that the thread forms the right angle with thevertical and then released, then(a) total acceleration of sphere as a function of θ is

2g 1 3cos+ q(b) thread tension as a function of θ is T =3mg cos θ(c) the angle θ between the thread and the vertical at the

moment when the total acceleration vector of the sphere

is directed horizontally is 1cos 1 / 3-

(d) the thread tension at the moment when the verticalcomponent of the sphere's velocity is maximum will bemg.

31. A horizontal plank has a rectangular block placed on it. Theplank starts oscillating vertically and simple harmonicallywith an amplitude of 40 cm. The block just loses contactwith the plank when the latter is at momentary rest. Then(a) the period of oscillation is (2π/5)(b) the block weighs double its weight, when the plank is

at one of the positions of momentary rest(c) the block weighs 0.5 times its weight on the plank

halfway up(d) the block weighs 1.5 times its weight on the plank

halfway down extreme.32. A nonconducting disc having uniform positive charge Q, is

rotating about its axis with uniform angular velocity ω. Themagnetic field at the centre of the disc is

O

RQ

(a) directed outward

(b) having magnitude 0Q4 R

µ ωπ

(c) directed inwards

(d) having magnitude 0Q2 R

µ ωπ




p q r s tppp

qqq

rrr

sss

ttt

p q r s tABCD


33. Each of the properties of sound listed in the column - I primarily depends on one of the quantities in column - II. Write down thematching pairs from the two columns.

Column I Column II(A) pitch p. waveform(B) quality q. frequency(C) loudness r. intensity(D) Sound level s. wavelength

34. According to Bohr’s model of H-atom, if Vn denotes the potential energy of electron, Kn the kinetic energy, En the totalenergy and rn the radius of nth orbit, then match the following :

Column I Column II

(A) ?n

n

VK

= p. 0

(B) xnn Er ∝ , x = ? q. – 2

(C)1 , ?yn Z y

r∝ = r. – 1

(D) Angular momentum in lowest orbital s. 1


Paragraph for question nos. 35 to 37The figure shows the interference pattern obtained in

double-slit experiment using light of wavelength 600nm. 1, 2, 3, 4and 5 are marked on the five fringes.

1 2 3 4 5

Central bright fringe



35. The third order bright fringe is(a) 2 (b) 5(c) 4 (d) 3

36. Which fringe results from a phase difference of 4π betweenthe light waves emanating from two slits(a) 2 (b) 3(c) 5 (d) 4

37. Let ∆XA and ∆XC represent path differences between wavesinterfering at 1 and 3 respectively then( | ∆XC | – | ∆XA | ) is equal to(a) 0 (b) 300nm(c) 600 nm (d) 900nm

Paragraph for question nos. 38 to 40Superposition of waves results in maximum and minimum of

intensities such as in case of standing waves. This phenomenonis called as interference. Another type of superposition result ininterference in time which is called as beats. In this case waves areanalyzed at a fixed point as a function of time. If the two waves areof nearby same frequency are superimposed, at a particular point,intensity of combined waves gives a periodic peak and fall. Thisphenomenon is beats. If ω1 and ω2 are the frequencies of twowaves then by superimposed y = y1 + y2, we get at

x = 0, 1 2 1 2y 2A cos .t sin .t2 2

ω − ω ω + ω =

Thus amplitude frequency is small and fluctuates slowly. Abeat i.e., a maximum of intensity occurs, also intensity dependson square of amplitude. The beat frequency is given by

beat 1 2| |ω = ω − ω .

Number of beats per second is called as beat frequency.A normal ear can detect only upto 15 Hz of frequency because ofpersistence of ear.

38. If two sound sources of frequency difference 25 Hz aresounded together. Then which of the following is correct ?(a) A normal human ear will hear 25 Hz beat frequency(b) A normal human ear will hear only 10 Hz beat frequency(c) A normal human ear cannot hear this frequency(d) A normal human ear cannot hear this beat frequency

39. Two sources of frequency 50 Hz and 52 Hz vibrated togetherto produce beats, frequency of resultant wave –(a) 51 Hz (b) 1 Hz(c) 2 Hz (d) 52 Hz

40. The frequency of beats produced when two sources ofsound are activated, one emitting wavelength 32 cm, other32.2 cm is (Take vsound = 350 m/s)(a) 14 (b) 18(c) 7 (d) 10

..Part - C : Chemistry..


41. An evacuated glass vessel weighs 50.0 g when empty, 148.0g when filled with a liquid of density 0.98 g mL–1 and 50.5 gwhen filled with an ideal gas at 760 mmHg at 300 K. Determinethe molar mass or the gas.(a) 123 g mol–1 (b) 133 g mol–1

(c) 128 g mol–1 (d) 111 g mol–1

42. The energy of the electron in the second and third Bohrorbits of the hydrogen atom is – 5.42 × 10–19 J and– 2.41 × 10–19 J, respectively. Calculate the wavelength ofthe emitted radiation when the electron drops from third tosecond orbit.(a) 6.104 × 10–6 m. (b) 6.604 × 10–7 m.(c) 2.604 × 10–7 m. (d) 4.604 × 10–7 m.



43. The densities of graphite and diamond are 2.25 and 3.51 gm/cc. The ∆Gºr value are 0 J mol–1 and 2900 J mol–1 for graphiteand diamond respectively. Calculate the equilibrium pressurefor the conversion of graphite into diamond at 25ºC.(a) 0.12 × 109 Pa (b) 3.12 × 109 Pa(c) 5.32 × 109 Pa (d) 1.52 × 109 Pa

44. A solution has 0.05 M Mg2+ and 0.05 NH3. Calculate theapproximate concentration of NH4Cl required to preventthe formation of Mg(OH)2 in solution.

sp[Mg(OH) ]2K = 9.0 × 10–12 and ionization constant of NH3

is 1.8 × 10–5.[Given : log 1.8 = 0.2552, log 1.34 = 0.1271](a) 0.067 M (b) 0.67 M(c) 1.67 M (d) 1.27 M

45. – I effect is shown by(a) Aryloxy (b) Ethyl(c) Isopropyl (d) tert-butyl

46. Consider the cell Ag(s) | AgBr(s)|Br– (aq)|| AgCl(s) | Cl– (aq)| Ag(s) at 25ºC. The solubility product constants of AgBr &AgCl are respectively 5 × 10–13 & 1 × 10–10. For what ratioof the concentrations of Br– & Cl– ions would the emf of thecell be zero ? [Given : 0.059 log 5 × 10–13 = –0.7257](a) 1 : 200 (b) 1 : 100(c) 1 : 500 (d) 200 : 1

47. Which of the following complex will show geometrical aswell as optical isomerism. [en = ethylene diamine](a) [Pt(NH3)2Cl2] (b) [Pt(NH3)Cl4](c) [Pt(en3)]4+ (d) [Pt(en)2Cl2]

48.O

Ph CH2CH Br2

O NaOCH3Product

Product is/are

(a)O

Ph CH CH OCH2 2 3

O (b)O

Ph CO CH3O

(c)O

PhC – OCH3

O

(d)O

Ph COCH3


49. Which of the following statement is false regarding follow-ing reaction ?

Cl

Cl

NO21

2

34

5

6+ NH3

heatpressure

(a) No reaction is possible because —Cl is present onbenzene ring.

(b) A nucleophilic substitution will take place in whichboth —Cl will be replaced by two —NH2 groups.

(c) A nucleophilic substitution will take place in whichonly —Cl attached on C1 will be replaced by —NH2.

(d) A nucleophilic substitution will take place in whichonly —Cl attached on C4 will be replaced by —NH2.



50. Which of the following statement(s) is/are true ?

(a) ionisation energy ∝ 1Screening effect

(b) The first ionisation energies of Be and Mg are morethan ionisation energies of B and Al respectively

(c) Atomic and ionic radii of Niobium and Tantalum arealmost same

(d) Metallic and covalent radii of potassium are 2.3 Å and2.03 Å respectively.

51. Two elements A and B form compounds having molecularformula AB2 and AB4. When dissolved in 20g of C6H6. 1gof AB2 lowers the freezing point by 2.3 K, whereas 1.0g of

AB4 lowers it by 1.3 K. The molar depression constant forbenzene is 5.1 K kg mol–1. Then –(a) Atomic mass of A = 25.58 u(b) Atomic mass of B = 42.64 u(c) Atomic mass of A = 42.64 u(d) Atomic mass of B = 25.58 u

52. Choose the correct options for inferences drawn.(a) Canary yellow precipitate with ammonium molybdate

→ PO43–

(b) Brown ring test with dil. H2SO4 → NO2–

(c) Yellow ppt. with HgCl2 solution → SO42–

(d) Yellow ppt. with HgCl2 solution → NO3–


p q r s tppp

qqq

rrr

sss

ttt

p q r s tABCD


53. Column I Column II(A) Definite dipole moment p. CH3Cl(B) CH4 + Cl2 (excess) q. CHCl3(C) CH3COCH2CH3 + alkaline Cl2 r. CCl4(D) Reacts with ammonium silver nitrate s. CH2Cl2

54. Column I Column II(A) CH2 = CHCN 3 2(CH ) NH+ → p. Transition state involves

pentavalent carbon

(B) 2CH CHCN= catalyst→ q. Nucleophilic substitution

(C) 3

O||

CH C Cl− − 3 2(CH ) NH+ → r. Nucleophilic addition

(D) ClCH2CH = CHCN + (CH3)2NH → s. Free radical addition




Paragraph for question nos. 55 to 57The shapes of molecules can be predicted by VSEPR theory,

hybridization and dipole moment. Total number of hybrid orbitals(H) on the central atom of a molecule can be calculated by usingthe following relation :

H = [Total no. of valence electron pairs (P)–3 × (no. of atomssurrounding the central atom, excluding Hydrogen atoms)]

One can also calculate total no. of bond pairs (n) aroundcentral atom as n = total number of atoms surrounding the centralatom also, total no. of lone pairs (m) = H – n

Thus, VSEPR notation of a molecule can be written asAXnEm. Where, A denotes central atom of the molecule.

X denotes bond pairs on central atom of the molecule.E denotes lone pairs on central atom of the molecule.In a polar molecule, the net dipole moment of the molecule

∝ m55. VSEPR notation of chlorine trifluoride molecule is

(a) AX5 (b) AX3(c) AX2E3 (d) AX3E2

56. Some molecules are given belowCO2, SO2, H2O I II IIIThe incorrect increasing order of dipole moment of givenspecies is –(a) I < II < III (b) II < I < III(c) III < II < I (d) III < I < II

57. Total number of hybrid orbitals on central iodine in tri-iodideion, are(a) 2 (b) 3(c) 4 (d) 5

Paragraph for question nos. 58 to 60Consider the inter conversion of nitrosotriacetoamine into

nitrogen phorone and water.

(aq) N2(g) + H2O(l)

+ (aq) – 20 kJ/ mol

The reaction is 1st order in each direction, with an equilibriumconstant of 104, the activation energy for the forward reaction is57.45 kJ/ mol. Assuming arrhenius preexponetial factor of 1012 s–1.58. What is the expected forward rate constant at 300K , if we

initiate this reaction starting with only reactant(a) 102 (b) 106

(c) 108 (d) 104

59. If the change in entropy of the reaction is 0.07 kJ. K–1 mol–1 at 1 atm pressure. Calculate up to which temperature thereaction would not be spontaneous. (For forward reaction)(a) T < 285.7 K (b) T > 250 K(c) T < 340.2 K (d) T > 200 K

60. Calculate Kp of the reaction at 300 K(a) 2.4 × 104 atm–1 (b) 104 atm(c) 24.6 × 104 atm (d) 2.82 × 102 atm–1



..Part - A : Mathematics..

SECTION – I - Single Correct Choice Type

This section contains 4 multiple choice questions. Each questionhas 4 choices (a), (b), (c) and (d) for its answer, out of which ONLYONE is correct.

1. f(x) = sin4 x6π +

+ cos4 x

6π +

is

(a) not a periodic function

(b) a periodic function with period 2π

(c) a periodic function with period π

(d) a period function with period π/2

2. If ω is an imaginary cube root of unity, then a root of the

equation

2

2

2

x 1

x 1

1 x 2

+ ω ω

ω + ω

ω + = 0 is

(a) x = 1 (b) x = ω (c) x = ω2 (d) x = 03. Let f (x, y) = 0 be the equation of a circle such that

f (0, y) = 0 has equal real roots and f (x, 0) = 0 has two distinctreal roots. Let g (x, y) = 0 be the locus of point P from wheretangents to circle f (x, y) = 0 make angle π/3 between themand g (x, y) = x2 + y2 – 5x – 4y + c, c ∈ R.Let Q be a point from where tangents drawn to circleg (x, y) = 0 are mutually perpendicular. If A, B are the pointsof contact of tangents drawn from Q to circle g (x, y) = 0,then area of triangle QAB is

(a)2512

(b)258

(c)254

(d)252

* GENERAL INSTRUCTIONS :

1. Section I : Q. No. 1 to 4, Q. No. 20 to 23 and Q. No. 39 to 42 are Single Correct Choice Type questions. For this section, 3 markswill be awarded for correct answer and zero marks for no answer. In all other cases, –1 mark will be awarded.

2. Section II : Q. No. 5 to 9, Q. No. 24 to 28 and Q. No. 43 to 47 are Multiple Correct Choice Type questions. For this section, 4marks will be awarded for correct answer and zero marks for no answer. In all other cases, –1 mark will be awarded.

3. Section III : Q. No. 10 & 11, Q. No. 29 & 30 and Q. No. 48 & 49 are Multiple Matrix-Match Type questions. Each questioncontains statements given in two columns, which have to be matched. The statements in Column-I are labelled A, B, C and D,while the statements in Column-II are labelled p, q, r, s and t. Any given statement in Column -I can have correct matching withONE OR MORE statement(s) in Column-II. 2 marks each will be awarded for darkening the correct bubble in each row. Hencethe maximum marks of each question is 8 marks. No negative mark will be awarded for an incorrectly bubbled answer.

4. Section IV : Q. No. 12 to 19, Q. No. 31 to 38 and Q. No. 50 to 57 are Integer Answer Type questions. In this section, 4 markswill be awarded for correct answer and –1 mark for each wrong answer. Marks will be awarded only if you have darken theappropriate bubble.

Time : 180 minutes Max. Marks : 240

PAPER - 2



4. If b1 , b2 and b3 (b1 > 0) are three successive terms of a G.P.with common ratio r, the value of r for which the inequalityb3 > 4b2 – 3b1 holds, is given by(a) r > 3 (b) r < 1 (c) 1 < r < 2 (d) 2 < r < 3


5. Let f : A → B and g : B → C be functions and gof : A → C.Which of the following statement is true(a) If gof is one-one then f and g both are one-one(b) If gof is one-one then if is also one-one(c) If gof is bijection then f is one-one and g is onto(d) If f and g are both one-one then gof is one-one.

6. In the expansion of (x + y + z)20

(a) coefficient of x7y8z7 is zero(b) total number of distinct terms is 231

(c) every term is of the form 20 r r k k20!x y z

(20 r)!(r k)!k!

- -

- -(d) sum of coefficient is 320

7. The locus of the mid point of the focal radii of a variablepoint moving on the parabola, y2 = 4ax is a parabola whose(a) latus rectum is half the latus rectum of the original parabola(b) vertex is (a/2, 0)(c) directrix is y-axis(d) focus has the coordinate (a, 0)

8. For the hyperbola 9x2 – 16y2 – 18x + 32y – 151= 0(a) one of the directrix is x = 21/5(b) length of the latus rectum = 9/2(c) focii are (6, 1) and (– 4, 1)(d) eccentricity is 5/4

9. Let 1 1 1 2E .....3 50 3 50

= + + + + upto 50 terms, then

(a) E is divisible by exactly 2 primes(b) E is prime(c) E ≥ 30 (d) E < 35


p q r s tppp

qqq

rrr

sss

ttt

p q r s tABCD

If the correct matches are A–p, s and t; B–q and r; C–p and q; and D–s and t; then the correct darkeningof bubbles will look like the given :10. Column-I Column-II

(A) If a,b,c are unequal positive numbers and b is A.M of a and c then the roots of p. of opposite signsax2 + 2bx + c = 0 are

(B) If a ∈ R, then the roots of the equation x2 – (a + 1)x – a2 – 4 = 0 are q. rational numbers(C) If a, b, c are unequal positive numbers and b is H.M of a and c then the roots of r. real and unequal

ax2 + 2bx + c = 0 are

(D) If ±a b < c and a = 0 then the roots of a2x2 + (b2 + a2 – c2) x + b2 = 0 are s. imaginary

t. of same sign



11. Column-I Column-II(A) a, b, c, d are four distinct real numbers and they are in A.P. If p. A rational number

2 (a – b) + x (b – c)2 + (c – a)3 = 2 (a – d) + (b – d)2 + (c – d)3 then the value of x can be(B) If roots of equation ax2 + bx + c = 0, 0a ≠ are α and β and the roots of the equation q. An irrational number

a5x2 + ba2c2x + c5 = 0 are 4 and 8 then | αβ | is

(C) If log3 (log5 x) + log1/3 (log1/5 y) = 1 and x2y = 1 ( , )x y R∈ then the value of 5x + y is r. 2

(D) Let a, b, c are rational numbers 1a ≠ − and x1, x2 are the real roots of the equation s. 26x3 – ax2 + bx – c = 0 then x1 x2 is

SECTION – IV - Integer Answer TypeThis section contains 8 questions. The answer to each of thequestions is a single-digit integer, ranging from 0 to 9. Theappropriate bubbles below the respective question numbers in theORS have to be darkened. For example, if the correct answers toquestion numbers X, Y, Z and W (say) are 6, 0, 9 and 2, respectively,then the correct darkening of bubbles will look like the following:

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12. If n > 3 and n N∈ , then find the value of ( ).nr rC C=

0 1 2( 1)( 1)( 1) ( 2)( 2)( 2)C abc C a b c C a b c− − − − + − − − +

( 1) ( )( )( )nnC a n b n c n…+ − − − − .

13. Find the principle value of the argument of the complexnumber ii .

14. If sin cosq q≠ and x, y, z satisfy the equations

cos sin cos 1x p y p z q− + = +

sin cos 1 sinx p y p z q+ + = −

( ) ( )cos sin 2x p q y p q z+ − + + =

then find the value of x2 + y2 + z2.15. A person has 6 friends and during a certain vacation he met

them during several dinners. He found that he dinned withall the 6 exactly on one day, with every 5 of them on 2 days,with every 4 of them on 3 days, with every 3 on 4 days; withevery 2 on 5 days. Furthers every friend was present at 7dinners and every friend was absent at 7 dinners. Then findthe number of dinner(s) he had alone.

16. A straight line cuts the x-axis at point A (1, 0), and y-axis at

point B, such that .4

OAB Cπ ∠ = α α >

is a middle point

of AB, if B' is a mirror image of point B with respect to line OCand C' is a mirror image of point C with respect to line BB',then find the ratio of the areas of triangles 'and ' 'ABB BB C .



17. If a point P on the parabola 2 4y x= is taken such that thepoint is at shortest distance from the circle

2 2 2 2 2 2 0,x y x y+ + − + = tangents are drawn to the

circle and the parabola, then the area of the triangle PAB is

a where A and B are the points of contact on two differentlines on circle and parabola respectively, then find the valueof a.

18. P (x, y) is a point, which moves in the xy plane such that2 [ y ]=3 [ x ], where [ . ] denotes the greatest integer functionand 2 5x− ≤ ≤ , 3 6y− ≤ ≤ . Then find the area of theregion containing the point P (x, y) .

19. If 2 3

sin cos

( ) tan2 sin 2 5

x x x

f x x x xx x x

= − − , then find the value of

0

( )Limx

f xx→

′ .

..Part - B : Physics..

SECTION – I - Single Correct Choice TypeThis section contains 4 multiple choice questions. Each questionhas 4 choices (a), (b), (c) and (d) for its answer, out of which ONLYONE is correct.20. In the figure shown S0 is a monochromatic source of light

emitting light of wavelength λ (in air). Light falls on slit Sfrom S0 and then reach the slits S1 and S2 through a mediumof refractive index µ1. Light from slits S1 and S2 reach thescreen through medium of refractive index µ3.A thin transparent film of refractive index µ2 and thicknesst is placed infront of S2 . Point P is symmetrical w.r.t. S1 andS2. Using the values d = 1 mm, D = 1m, µ1 = 4/3, µ2 = 3/2,µ3 = 9/5 and t = (4/9) × 10–5 m. The distance of centralmaxima from P is –

d P

Screen

D 2D

S

S2

S1dS0 µ1

µ1

µ3

µ3µ ,t2

(a) 1mm (b) 0(c) 2mm (d) 0.5mm

21. When a galvanometer is shunted with a 4Ω resistance, thedeflection is reduced to one-fifth. If the galvanometer isfurther shunted with a 2Ω wire, the further reduction in thedeflection will be (the main current remains the same).(a) (8/13) of the deflection when shunted with 4Ω only(b) (5/14) of the deflection when shunted with 4Ω only(c) (3/4) of the deflection when shunted with 4Ω only(d) (3/13) of the deflection when shunted with 4Ω only

22. Two vibrating strings of the same material but length L and2L have radii 2r and r respectively. They are stretched underthe same tension. Both the strings vibrate in theirfundamental modes, the one of length L with frequency 1νand the other with frequency 2ν . The ratio 1 2/ν ν is givenby :(a) 2 (b) 4(c) 8 (d) 1

23. A steel wire of length 4 m and diameter 5 mm is stretched by5 kg-wt. Find the increase in its length, if the Young’smodulus of steel of wire is 2.4 × 1012 dyne/cm2.(a) 1.0041 cm. (b) 0.0041 cm.(c) 4.1 cm. (d) 1.2 cm.




24. A parallel beam of light (λ = 5000 Å) is incident at an angle α= 30° with the normal to the slit plane in a Young’s doubleslit experiment. Assume that the intensity due to each slit atany point on the screen is I0. Point O is equidistant from S1and S2. The distance between slits is 1mm.

O

S1

S2

3 m

(a) the intensity at O is 4I0(b) the intensity at O is zero(c) the intensity at a point on the screen 1m below O is 4I0(d) the intensity at a point on the screen 1m below O is

zero25. A double star is a system of two stars of masses m and 2m,

rotating about their centre of mass only under their mutualgravitational attraction. If r is the separation between thesetwo stars then their time period of rotation about their centreof mass will be proportional to(a) r3/2 (b) r(c) m1/2 (d) m–1/2

26. In the figure shown, the plates of a parallel plate capacitorhave unequal charges. Its capacitance is C. P is a pointoutside the capacitor and close to the plate of charge –Q.The distance between the plates is ‘d’.

2Q – Q

P

(a) A point charge at point 'P' will experience electric forcedue to capacitor

(b) The potential difference between the plates will be3Q/2C

(c) The energy stored in the electric field on the regionbetween the plates is 9Q2/8C

(d) The force on one plate due to the other plate is2

20

Q2 dπε

27. Suppose the potential energy between electron and proton

at a distance r is given by 2

3Ke3r

− . Application of Bohr’ss

theory to hydrogen atom in this case shows that(a) energy in the nth orbit is proportional to n6

(b) energy is proportional to m–3 (m : mass of electron)(c) energy in the nth orbit is proportional to n–2

(d) energy is proportional to m3 (m = mass of electron)28. During an experiment, an ideal gas is found to obey a

condition 2P

constant=ρ

[ρ = density of the gas]. The gas

is initially at temperature T, pressure P and density ρ. Thegas expands such that density changes to ρ /2

(a) The pressure of the gas changes to 2 P

(b) The temperature of the gas changes to 2 T(c) The graph of the above process on the P-T diagram is

parabola(d) The graph of the above process on the P-T diagram is

hyperbola




p q r s tppp

qqq

rrr

sss

ttt

p q r s tABCD


29. Column I Column II

(A) Electron jumps from lower to higher orbit p. Potential Energy changes

(B) A body is lifted in gravitational force q. Kinetic Energy changes

(C) Spring is compressed r. Negative work is done

(D) A body is pulled on a rough surface s. Conservative force acts

t. Non-conservative force acts

30. Column I Column II

A. Wattless current p. When circuit contains only C

B. Oscillating current q. When circuit contains only L

C. Heating effect r. When both L and C are used

D. Phase difference of 2π

s. When all L, C & R are used




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31. The masses of 10 kg and 20 kg respectively are connectedby a massless spring in fig. A force of 200 newton acts onthe 20 kg mass. At the instant shown, the 10 kg mass hasacceleration 12 m/sec2. What is the acceleration (in m/s2) of20 kg mass?

10 kg 20 kg200 newton

32. A particle is moving on x-axis has potential energyU = 2 – 20x + 5x2 joules along x-axis. The particle is released atx = – 3. If the mass of the particle is 0.1 kg, then find how manytimes of ten is the maximum velocity (in m/s) of the particle.

33. The length of a wire between the two ends of a sonometer is105cm. If the sum of the distances of the positions of the twobridges from one end is expressed as (182 – x) cm so that thefundamental frequencies of the three segments are in theratio of 1 : 3 : 15. Find the value of x.

34. A brass rod of length 50 cm and diameter 3.0 mm is joined to asteel rod of the same length and diameter. If the change inlength of the combined rod at 250 °C is (A × 10–2) cm, given theoriginal lengths are at 40.0 °C, what is the sum of digits of A?The ends of the rod are free to expand (Co-efficient of linearexpansion of brass = 2.0 × 10–5 K–1, steel = 1.2 × 10–5 K–1 ).

35. Two long parallel wires carrying current 2.5 amperes and1 ampere in the same direction (directed into the plane ofthe paper ) are held at P and Q respectively such that theyare perpendicular to the plane of paper. The points P and Qare located at a distance of 5 metres and 2 metres respectivelyfrom a collinear point R (see figure). An electron movingwith a velocity of 4 × 105 m/s along the positive x- directionexperiences a force of magnitude 3.2 × 10–20 N at the pointR. Find the value of I (in ampere).

→−−−−−−−−−−−−

mm

52

1AA5.2x.............OO

RQPX X

36. A thin half ring of radius R = 20 cm is uniformly charged witha total charge q = 0.70 millicoloumb (mC). If the magnitudeof electric field strength at the curvature centre of this halfring is (100 k) V/m, find the value of k.



37. A hydrogen like atom (atomic number Z) is in a higher excitedstate of quantum number n. The excited atom can make atransition to the first excited state by successively emittingtwo photons of energy 10.2 eV and 17.0 eV respectively.Alternately, the atom from the same excited state can makea transition to the second excited state by successivelyemitting two photons of energies 4.25 eV and 5.95 eVrespectively. Determine the value of n.(Ionization energy of H-atom = 13.6 eV)

38. A plano convex lens has a thickness of 4 cm . When placedon a horizontal table, with the curved surface in contact withit, the apparent depth of the bottom point of the lens is foundto be 3 cm. If the lens is inverted such that the plane face is incontact with the table, the apparent depth of the centre of theplane face is found to be 25/8 cm. If the focal length (in cm) ofthe lens is 25y, find the value of y.

..Part - C : Chemistry..


39. The bond dissociation enthalpies of H2(g) and N2(g) are+ 435.95 kJ mol–1 and + 941.8 kJ mol–1 and enthalpy offormation of NH3(g) is – 46.024 kJ mol–1. What is theenthalpy of atomization of NH3(g)(a) 0.170 MJ mol–1 (b) 1.70 MJ mol–1

(c) 1.170 MJ mol–1 (d) 2.130 MJ mol–1

40. In Ostwald’s process for the manufacture of nitric acid, thefirst step involves the oxidation of ammonia gas by oxygengas to give nitric oxide gas and steam. What is the maximumweight of nitric oxide that can be obtained starting onlywith 10.00 g of ammonia and 20.00 g of oxygen ?(a) 15g (b) 20g(c) 10g (d) 25g

41. The molar volume of liquid benzene (density=0.877 g ml–1)increases by a factor of 2750 as it vapourizes at 20ºC. At27ºC when a non-volatile solute (that does not dissociate)is dissolved in 54.6 cm3

of benzene, vapour pressure of thissolution, is found to be 98.88 mm Hg. Calculate the freezingpoint of the solution.

Given : Enthalpy of vapourization of benzene (l) = 394.57Jg–1. Enthalpy of fusion of benzene (l) = 10.06 kJ mol–1

Molal depression constant for benzene = 5.0 K kg mol–1.

[Given : log 100.2 = 2.00086, log 74.63 = 1.8729]

(a) 177.65 K (b) 277.65 K

(c) 517.65 K (d) 237.15 K

42. An unknown compound (A) with the M.F. C9H12O doesnot decolorize Br2 in CCl4 and is oxidised by hot KMnO4 togive PhCO2H. The compound reacts with Na to give acolourless and odourless gas. From the following resultsdeduce the correct structure for (A).

(i) The colour of Cr2O72- changes from orange to blue-green.

(ii) The compound can be resolved.

(iii) No precipitate of CHI3 is observed with I2 / OH-.

(iv) Oxidation with CrO3 / pyridine gives a chiral compound.

(a) Ph

CH3H

OH

(b) CH3

H

OHPh

(c) Ph

CH3OH

CH3

(d) Ph OH




43. A solution of 0.2g of a compound containing Cu2+ andC2O4

2– ions on titration with 0.02M KMnO4 in presence ofH2SO4 consumes 22.6 mL of the oxidant. The resultantsolution is neutralized with Na2CO3, acidified with dil. aceticacid and treated with excess KI. The liberated iodine requires11.3 mL of 0.05 M Na2S2O3 solution for complete reduction.Then the correct options are(a) Amount of C2O4

2– in the solution = 1.13 × 10–3 mol(b) Amount of Cu2+ in the solution = 0.56 × 10–3 mol

(c) Amount of C2O42– in the solution = 0.56 × 10–3 mol

(d) Amount of Cu2+ in the solution = 1.13 × 10–3 mol44. Choose the correct statements

(a) BeO is insoluble but BeSO4 is soluble in water.(b) The carbon hydride of the type CnH2n+2 act as Lewis

acid or base(c) Due to its high bond enthalpy, dihydrogen is not

particularly reactive at room temperature(d) The s-block elements are very reactive

45. Which of the following is (are) correct statement(s)(assuming oxidation number of metal does not affect crystalfield energy)(a) Considering H2O to be a weak ligand then on the basis

of CFSE only, we can say that [Co(H2O)6]2+ is morestable than [Co(H2O)6]3+

(b) On the basis of CFSE only [Fe(NH3)6]2+ is more stablethan [Fe(NH3)6]

+3

(c) All octahedral complexes of Ni(ll) are bound to be outerd-complex

(d) The type of d-orbital involved in the hybridisation fora square planar complex (CN = 4) is 2z

d .

46. An aromatic chiral compound G (C8H8Br2) on treatmentwith aqueous NaOH gives H (C8H9BrO). On heating ‘G’with potassium tert-butoxide I (C8H7Br) is formed. With oneequivalent of methyl magnesium bromide in ether p-bromoisopropylbenzene is formed which observation/s is/are correct about these reactions.

(a) (I) is an optically inactive aromatic alkene

(b) H is an optically active phenol

(c) G =

(d) Benzylic halide gives nucleophilic substitution withfaster rate than aryl halide

47. Choose the correct options for

CH3

CH314

H OHNa

aMe – Cl

I

IIRed PI2

bMeONa

(a) I and II are identical

(b) I and II are different

(c) Mechanism of formation of I and II are same

(d) Mechanism of formation of I and II are different



SECTION – III - Matrix - Match Type

This section contains 2 questions. Each question contains statements given in two columns, which haveto be matched. The statements in Column-I are labelled A, B, C and D, while the statements in Column-II are labelled p, q, r, s and t. Any given statement in Column-I can have correct matching with ONE ORMORE statement(s) in Column-II. The appropriate bubbles corresponding to the answers to thesequestions have to be darkened as illustrated in the following example:

p q r s tp

pp

q

qq

r

rr

s

ss

t

tt

p q r s tA

BCD

If the correct matches are A—p, s and t; B—q and r; C—p and q; and D—s and t; then the correctdarkening of bubbles will look like the given.

48. Column - 1 Column - II

(A) Left behind as waste in Kipp’s apparatus p. Mohr’s salt

(B) It is green in colour q. Green vitriol

(C) On heating it leaves a residue that is brown in colour r. Basic copper carbonate

(D) On heating it leaves a residue that is black in colour s. Hydrated cupric chloride

49. Column - I Column - II

(A) CH3CH2 CH2CN p. Reduction with Pd–C/H2

(B) CH3 CH2 OCOCH3 q. Reduction with SnCl2/HCl

(C) CH3–CH= CH–CH2OH r. Development of foul smell on

treatment with chloroform and

alcoholic KOH

(D) CH3CH2CH2CH2NH2 s. Reduction with diisobutylaluminium

hydride(DIBAL-H)

t. Alkaline hydrolysis




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50. The dipole moment of KCl is 3.332 × 10–29 Coulomb meterswhich indicates that it is a highly polar molecule. Theinteratomic distance between K+ and Cl– in this molecule is2.6 ×10–10 m. The percentage ionic character of KCl is 10xthen find the value of x.

51. At 27ºC, hydrogen is leaked through a tiny hole into a vesselfor 20 minutes. Another unknown gas at the sametemperature and pressure as that of H2 is leaked throughthe same hole for 20 minutes. After the effusion of the gasesthe mixture exerts a pressure of 6 atmosphere. The hydrogencontent of the mixture is 0.7 mole in a container of volume3 litres. If the molecular weight of the unknown gas isexpressed as (1040 – A), then what is the value of A.

52. How many dipeptides are possible from two molecules ofa typical α-amino acid ?

53. The edge length of unit cell of LiCl having rock salt typelattice is 5.14Å. If Li+ ions precisely fit into the octahedralwoids of closed packed structure of Cl– ions. If the ionicradius (in pm) of Cl– ions is expressed as 121x then what isthe value of x?

54. The vapour pressure of pure benzene at a certain temperatureis 640 mm Hg. A non-volatile non-electrolyte solid weighing2.176 g is added to 39.0 g of benzene. The vapour pressure ofthe solution is 600 mm Hg. If the molecular weight of the solid

substance is expressed as 130

A then find the value of A.

55. We have taken a saturated solution of AgBr. Ksp of AgBr is12 × 10–14. When 10–7 mole of AgNO3 are added to 1 litre ofthis solution then, conductivity (specific conductance) ofthis solution is find as 11x × 10–7 (S m–1 units). Find thevalue of x. Given, molar conductance of Ag+, Br– and NO3

–

are 6×10–3 Sm2mol–1, 8×10–3 Sm2mol–1 and 7×10–3 Sm2mol–1.

56. Consider a reaction aG + bH → Products. Whenconcentration of both the reactants G and H is doubled, therate increases by eight times. However, when concentrationof G is doubled keeping the concentration of H fixed, therate is doubled. What will be the overall order of the reaction?

57. Reductive ozonolysis of myrcene, a terpenoid

gives how many different products?

mock test qp

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