Sec: OSR.

IIT CO-SC Date: 23-05-21

Time: 3 Hrs Max. Marks: 300

Name of the Student: ___________________ H.T. NO:

23-05-21_ Sr.IIT CO-SC _Jee-Main_GTM-38_Syllabus

PHYSICS: Total Syllabus
CHEMISTRY: Total Syllabus
MATHEMATICS: Total Syllabus
MISTAKES
SUBJECT JEE JEE TOTAL
SYLLABUS Q'S EXTRA SYLLABUS Q'S Q'S

MATHS

PHYISCS

CHEMISTRY
Narayana IIT Academy 23-05-21_OSR.IIT CO SC_ JEE-MAIN_GTM-38_Q’P
PHYSICS MAX.MARKS: 100
SECTION – I
(SINGLE CORRECT ANSWER TYPE)
This section contains 20 multiple choice questions. Each question has 4 options (1), (2), (3) and (4) for its
answer, out of which ONLY ONE option can be correct.
Marking scheme: +4 for correct answer, 0 if not attempted and -1 if not correct.
1. Current versus time and voltage versus time graphs of a circuit element are shown
below. The type of circuit element is

(A) Capacitance of 2F (B) Resistance of 2 

(C) capacitance of 1 F (D) a voltage source of emf 1 V

2. What will be reading of the AC voltmeter across the resistance, and AC ammeter in the
circuit shown in the figure?

(A) 300 V, 2A (B) 800 V, 2A (C) 100 V, 2A (D) 200 V, 2A

3. A plane mirror is placed at origin in yz-plane, facing the positive x - axis. An object
starts from (2, 0, 0)m with velocity  2iˆ  2ˆj ms 1 . The relative velocity of image with

respect to object is along

(A) Positive x - axis (B) negative x - axis

(C) Positive y - axis (D) negative y - axis

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Narayana IIT Academy 23-05-21_OSR.IIT CO SC_ JEE-MAIN_GTM-38_Q’P
4. The fundamental frequency of sonometer wire of length  is f0 . A bridge is now
introduced at a distance of  from the centre of the wire      . The number of beats
heard if both sides of the bridge are set into vibration in their fundamental modes are:

8f 0  f 0  2f 0  4f 0 
(A) (B) (C) (D)
   

5. The angle of incidence at which reflected light is totally polarized for a ray travelling
from air to glass (Absolute refractive Index of glass is  )

1 1 1
(A) Sin 1 (B) cos 1   (C) cos1 2  1 (D) cot 1  
  

6. The electric field inside a thin long cylindrical shell of uniform charge density is

(A) zero

(B) constant and radially inward

(C) constant

(D) directly proportional to the distance from the center

7. A converging beam of light is incident on a spherical surface of radius R as shown. The

refracted beam is diverging in nature. The angle of the converging beam is the same as
angle of the diverging beam. What is the distance of the image from the surface.

1  
2  1

 

  1    1  2   2 
(A)  R (B)  R (C)  R (D)  R
  1   1    1    1

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Narayana IIT Academy 23-05-21_OSR.IIT CO SC_ JEE-MAIN_GTM-38_Q’P
8. In case of induced electric field, due to a time dependent magnetic field in a cylindrical
region, which of the statements is correct?
(A) It obeys the Gauss’ law for electric flux.
(B) The work done by induced electric field while moving a charged particle in a closed
loop is zero.
(C) The induced electric field and time-varying magnetic field are parallel to each other
at any instant of time.
(D) Electrostatic potential can be defined for the induced electric filed.
9. A thin conducting rod OA of length l is rotated about end O with constant angular
frequency . The magnetic field in the quadrants are B1 = 0; B2 = B0; B3 = 2B0;
B4 =4B0. B2 and B4 are into the plane of the paper whereas B3 is out of the plane. The
average emf induced in one cycle is

B2  B0 B1  0
O
A
B3  2B0 B4  4B0

(A) 13 B0l 2 (B) 3

B 0 l 2 (C) 1
B0l 2 (D) 3
B0l 2
16 8 6 4

10. In Bohr’s model of hydrogen –like species, which of the following is true?

(A) KE of electron in ground state of hydrogen atom is equal to KE of electron in n = 3

state of Li2+

(B) KE of electron in n = 3 state of hydrogen is equal to KE of electron is ground state

of Li2+

(C) Energy of hydrogen atom in ground state of hydrogen atom is equal to energy of
Li2+ in n = 2 state

(D) None of these

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11. Three materials are placed in a magnetic field of 10–3 Tesla. The resulting fields inside
each material are found to be 0.8 × 10–3 Tesla, 0.98 × 10–3 Tesla and 0.6 Tesla. These
materials are :
(A) two diamagnetic and one ferromagnetic
(B) two diamagnetic and one paramagnetic
(C) two paramagnetic and one ferromagnetic
(D) all diamagnetic
12. Sketch the voltage across central 2.2 k resistor as a function of time.

(A) (B) (C) (D)

13. A vertical ring of radius r and resistance R falls vertically. It is in contact with two
vertical rails which are joined at the top with a conducting wire of zero resistance. The
rails are conducting, frictionless and have negligible resistance. Horizontal uniform
magnetic field of magnitude B perpendicular to the plane of ring and rails exists. When
the speed of ring is V, the current in the section PQ is:

2BrV 4BrV 8BrV

(A) zero (B) (C) (D)
R R R

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Narayana IIT Academy 23-05-21_OSR.IIT CO SC_ JEE-MAIN_GTM-38_Q’P
14. A ray of light enters a diamond symmetrical shape (n = 2) from air and is being
internally reflected near the bottom as shown in the figure. Find maximum value of
angle of  possible?

 3 1 
(A) sin 1   (B) sin 1  3 1 
 2 
 2 1   2 1 
(C) sin 1   (D) sin 1  
 2   3 
15. A car is moving towards a huge wall with a speed  c 10 , where c  speed of sound in still
air. A wind is also blowing parallel to the velocity of the car in the same direction and
with the same speed as car. If the car sounds a horn of frequency f ,
then what is the frequency of the reflected sound of the horn heard by driver of the
car?

10 f 11 f 11 f 10 f
(A) (B) (C) (D)
9 10 9 11

16. The temperature at the bottom of a pond of depth L is maintained at 4 C . The

temperature of air, just above the layer of ice frozen at the pond’s surface, is 2 C for the
past many days. Assume that the thermal conductivity of ice is three times that of water.
The thickness of the frozen layer of the ice must be (Neglect the expansion due to
freezing of water)

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

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Narayana IIT Academy 23-05-21_OSR.IIT CO SC_ JEE-MAIN_GTM-38_Q’P
17. In the figure shown, the two identical balls of mass M and radius R each, are placed in
contact with each other on the frictionless horizontal surface. The third ball of mass
M and radius R 2 , is coming down vertically and has a velocity  v0 just before it strikes
simultaneously these two balls and itself comes to rest just after collision. Then, each of
the two bigger balls will move after collision with speed 

v0 v0 2 v0 v0
(A) (B) (C) (D)
2 5 5 3
18. Energy liberated in the de-excitation of hydrogen atoms from 3 rd level to 1 st level falls
on a photo – cathode. Later when the same cathode is exposed to a spectrum of some
unknown hydrogen like gas, excited to 2nd energy level it its found that the de-Broglie
wave length of the fastest photo electrons now ejected has decreased by a factor of
‘3’.For this new gas, difference of the energy gap of 3rd energy level and 1st energy level
And (energy gap of 3rd energy level and 2nd energy level) is found to be 3 times the
ionization energy of hydrogen atom. Then

(A) The gas is Helium (B) Work function of photo cathode is 8.5eV

(C) Both 1 and 2 are true (D) Both 1 and 2 are wrong

19. When tow soap bubbles of different radii coalesce, some portions of their surfaces make
a common surface. At any point on the circumference of the common surface, the three
surfaces meet at angles   and  . What should relation should these angles bear?

(A)    (B)      (C)      (D)     

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Narayana IIT Academy 23-05-21_OSR.IIT CO SC_ JEE-MAIN_GTM-38_Q’P
20. The Young’s double-slit experiment is performed with blue light and green light of
wavelengths 4360 Ao and 5460 Ao respectively. If X is the distance of 4th maximum
from the central one, then:

(A) X(Blue)=X(Green) (B) X(Blue) < X(Green)

X  Blue  5460
(C) X(Blue)>X(Green) (D) 
X  Green  4360

SECTION-II
(NUMERICAL VALUE ANSWER TYPE)
This section contains 10 questions. The answer to each question is a Numerical value. If the numerical value has
more than two decimal places, Truncate/Round-off the value of Two decimal places. Have to Answer any 5
only out of 10 questions and question will be evaluated according to the following marking scheme:
Marking scheme: +4 for correct answer, 0 in all other cases.
1
21. A planet of radius R    radius of Earth  has the same mass density as Earth. Scientists
10

R
dig a well of depth on it and lower a wire of the same length and of linear mass
5

density 103 kgm1 into it. If the wire is not touching anywhere, the force applied at the

top of the wire by a person holding it in place is_____N (take the radius of Earth =

6 106 m and the acceleration due to gravity on Earth is 10ms2 )

22. The amplitude of a simple pendulum, oscillating in air with a small spherical bob.

decreases from 10 cm to 8 cm in 40 seconds. Assuming that stokes law is valid, and ratio

of the coefficient of viscosity of air to that of carbon dioxide is 1.3, the time in which

amplitude of this pendulum will reduce from 10 cm to 5 cm in carbondioxide will be

close to____(In sec)(In 5 = 1.6, In 2=0.7)

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23. In the arrangement shown in the figure when the switch S2 is open, the galvanometer
shows no deflection for l = L/2. When the switch S2 is closed, the galvanometer
5
shows no deflection for l = L . The internal resistance (r) of 6 V cell, ____ In Ω 
12

24. Figure shows a meter bridge. If there is no current through galvanometer then l1 is equal
to__ (in cm)

25. A resistance R = 12  is connected across a source of varying emf as shown in figure.

Its emf changes with times shown in the graph. What is the heat dissipated in the
resistance in the first four seconds___(In joule)

26. Figure shows the variation of force acting on a particle of mass 400 g executing simple
harmonic motion. The frequency of oscillation of the particle is__ (in Hez)(Round off
the answer to two decimal places)
F
(N)

2.0
1.5
1.0
0.5
x(cm)
–20 –15 –10 –5 5 10 15 20
–0.5
–1.0
–1.5
–2.0

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Narayana IIT Academy 23-05-21_OSR.IIT CO SC_ JEE-MAIN_GTM-38_Q’P
27. In the circuit shown in the figure, potential difference between points A and B is 16V.
The current passing through resistance 2 will be____(In ampere)

28. A point charge q is placed inside a neutral conducting spherical shell of inner radius 2R
and outer radius 3R at a distance of R from the centre of the shell. The electric potential
1 aq
at the centre of shell will be ,then a+b is_____
40 bR

 E0 ,0  X  1
29. The potential energy of a particle of mass m is given by V  x   
0 , X  1

1 and 2 are the de-Broglie wavelengths of the particle, when 0  X  1 and X  1

respectively. If the total energy of particle is 2E0 , then 1 / 2 is

30. On end of an unstretched spring of force constant K1 is attached to the ceiling of

an elevator. A block of mass 1.5 kg is attached to the other end. Another spring of
force constant K 2 is attached to the bottom of the mass and to the floor of the elevator
as shown in figure. At equilibrium, the deformation in both the springs are equal and
is 40cm each. Now if the elevator starts to move with constant acceleration upwards,
the additional deformation in each spring (in steady state) is 8 cm. Find the elevator's
acceleration.______(in ms-2 ) [ g  10ms2 ]

k1

k2

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The important thing in life is not the triumph but the struggle.
Narayana IIT Academy 23-05-21_OSR.IIT CO SC_ JEE-MAIN_GTM-38_Q’P
CHEMISTRY MAX.MARKS: 100
SECTION – I
(SINGLE CORRECT ANSWER TYPE)
This section contains 20 multiple choice questions. Each question has 4 options (1), (2), (3) and (4) for its
answer, out of which ONLY ONE option can be correct.
Marking scheme: +4 for correct answer, 0 if not attempted and -1 if not correct.
31. For the redox reaction

xMnO 4  yH 2 C 2 O 4 +zH +  mMn 2+ +nCO 2 +pH 2O

The value of x, y, m and n are

(A)10,2,5,2 (B)2,5,2,10 (C)6,4,2,5 (D)3,5,2,10

32. Which of the following oxides shows electrical properties like metals?

(A) SiO 2 (B)MgO (C) SO2  s  (D) CrO2

33. Which of the following aqueous solutions should have the highest boiling point?

(A) 1.0 M NaOH (B) 1.0 M Na 2SO4

(C) 1.0 M NH 4 NO3 (D) 1.0 M KNO3

34. The charge required for the reduction of 1 mole of MnO 4 to MnO 2 is

(A)1 F (B)3 F (C)5F (D)7F

35. For the reaction

2SO 2 +O 2  2SO3 ,

The rate of disappearance of O2 is 2 104 mol L-1 s-1 .The rate of appearance of SO3 is

(A) 2×10-4 mol L-1 s-1 (B) 4×10-4 mol L-1 s-1

(C) 1×10-1 mol L-1 s-1 (D) 6×10-4 mol L-1 s-1

36. Which of the following electrolyte will have the maximum coagulating value of AgI+Ag 
sol?
(A) Na 2S (B) Na 3PO 4 (C) Na 2SO 4 (D) NaCl

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Narayana IIT Academy 23-05-21_OSR.IIT CO SC_ JEE-MAIN_GTM-38_Q’P
37. Which of the following is an amphoteric oxide?

(A) V2O5 , Cr2O3 (B) Mn 2O7 , Cr2O3

(C) CrO,V2O5 (D) V2 O5 ,V2O 4

38. The IUPAC name of

 Co  NH 3  4 Cl  NO 2   Cl is

(A)Tetraamminechloridonitrito-N-cobalt(III) chloride

(B) Tetraammnechloridonitrocobalt(II) chloride

(C)Tetraamminechloridonitrocobalt(I) chloride

(D)Tetraamminechloridodinitrocobalt(III) chloride

39. Which of the following is a polymide polymer?

(A)Terylene (B)Nylon-6,6 (C)Buna-S (D)Bakelite

40. At a particular temperature , the ratio of molar conductance to specific conductance of

0.01 M NaCl solution is
(A) 105 cm3 mol1 (B) 103 cm3 mol1
(C) 10 cm3 mol1 (D) 105 cm 2 mol1
41 The common impurity present in bauxide is
(A) CuO (B) ZnO (C) Fe2O3 (D) Cr2O3

42. When the vapours of tertiary butylalcohol are passed through heated copper at 573 K,
the product formed is

(A)But 2-ene (B)2-Butanone (C)2-Methyl propene (D)Butanal

43. Among the following the branched chain polymer is

(A)Polyvinyl chloride (B)Bakelite

(C)Low density polythene (D)High density polythene

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0
44. 0
For a cell reaction involving two electron changes Ecell  0.3V at 25 C the equilibrium
constant of the reaction is

(A) 1010 (B) 3102 (C) 10 (D) 1010

45. The value of rate constant of a pseudo first order reaction

(A) Depends only on temperature

(B)Depends on the concentration of reactants present in small amounts

(C) Depends on the concentration of reactants present in excess

(D) Is independent of the concentration of reactants

46. CH 3 3 SiCl Is used during polymerization of organosilicons because

(A) The chain length of organosilicon polymer can be controlled by adding CH 3 3 SiCl

(B) CH 3 3 SiCl Improves the quality and yield of the polymer

(C) CH 3 3 SiCl Does not block the end terminal of silicone polymer

(D) CH 3 3 SiCl Acts as a catalyst during polymerization

47. When PbO2 reacts with concentrated HNO3 the gas evolved is

(A) NO2 (B) O2 (C) N 2 (D) N 2O

48.  Fe  NO  Cl  and  Fe O  NO  Cl  show

 2 3 3   3 3 

(A) Linkage isomerism

(B) Geometrical isomerism

(C) Optical isomerism

(D) Hydrate isomerism

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Narayana IIT Academy 23-05-21_OSR.IIT CO SC_ JEE-MAIN_GTM-38_Q’P
49. The products X and Z in the following reaction sequences are

(A) Isopropyl benzene and acetone

(B) Cumene peroxide and acetone

(C) Isopropyl benzene and isopropyl alcohol

(D) Phenol and acetone

50. The reaction of Benzenediazonium chloride with aniline yields yellow dye. The name of
the yellow dye is

(A) p-Hydroxyazobenzene (B) p- Aminoazobenzene

(C) p- Nitroazobenzene (D) o-Nitroazobenzene

SECTION-II
(NUMERICAL VALUE ANSWER TYPE)
This section contains 10 questions. The answer to each question is a Numerical value. If the numerical value has
more than two decimal places, Truncate/Round-off the value of Two decimal places. Have to Answer any 5
only out of 10 questions and question will be evaluated according to the following marking scheme:
Marking scheme: +4 for correct answer, 0 in all other cases.
2h
51. In H sample, e is present in the orbit in which angular momentum of e is . Electron

make transition upto ground state. If 3rd highest energy photon has energy of X eV then

approximate numerical value of   is

2x
 5 

52. Calculate the percentage of hydrolysis in 106 M aqueous solution of NaOCN.

 K b for OCN   1010 

 

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53. How much water (in L) should be added to 10g CH 3COOH to give pOH equal to _____

11.K a  CH3COOH   1.8  105

54. The weight of glucose dissolved in 100 grm of water produce same lowering of vapor

pressure as

1 gm urea dissolved in 50gm of water at same temperature is ____

k 3

 B 
A 
 C
k 2
55.  1M  k 1 1M  1M 

If all the reactions in the above system give first order kinetics what is the rate of

disappearance of ‘B’ in the given instant ______

56. 92 U
238
disintegrates to give 4 and 6 particles. The group number of the daughter element

in modern periodic table is _____

57. The ion An  is oxidized to AO3 by MnO 4 changing to Mn 2 in acidic medium. Given that

2.68 103 mole of An  requires 1.61103 mole of MnO 4 .Thus, n is

58. The ratio of the inversion temperature of a gas to its Boyle temperature is _____

59. Magnetic moment of a 3-d a series element ‘A’ is 4.92BM if magnetic moment of A2 is

also 4.92BM then what is the group number of A in modern periodic table ____

60. How many of the followings cations have non-inert gas configuration

TI  , Pb 4 , Hg 2 , Bi 3 , Ag  , Cs  , Be 2

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Narayana IIT Academy 23-05-21_OSR.IIT CO SC_ JEE-MAIN_GTM-38_Q’P
MATHEMATICS MAX.MARKS: 100
SECTION – I
(SINGLE CORRECT ANSWER TYPE)
This section contains 20 multiple choice questions. Each question has 4 options (1), (2), (3) and (4) for its
answer, out of which ONLY ONE option can be correct.
Marking scheme: +4 for correct answer, 0 if not attempted and -1 if not correct.
61. Let z1 ,z2 ,z3 be there distinct complex numbers lying on a circle with center at the origin
such that z1  z2 z3 , z2  z3 z1 and z3  z1 z2 are real numbers, then z1 z2 z3 equals

(A) -1 (B) 1 (C) 0 (D) None of these

62. If three distinct real number a, b and c satisfy a 2 a  p   b 2 b  p   c 2 c  p 

Where p  R, then the value of bc+ ca+ ab is

(A) –p (B) p (C)0 (D) p 2 2

63. If a,b,c are three unequal numbers such that a,b,c are in A.P. and b  a,c  b,a are in
G.P., then ratio a : b : c is equal to

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

x3
Solution set of the inequality 3x 0.333...  1 27 is
x
64.

(A) 3 2 ,5 (B)  ,3 2] (C) 0, (D) none of these

65. An eight digit number divisible by 9 is to be formed by using 8 digits out of the digits
0,1,2,3,4,5,6,7,8.9 without replacement. The number of ways in which this can be done
is

(A)9! (B) 2(7!) (C) 4(7!) (D) (36) (7!)

66. The reminder when 22003 is divided by 17 is

(A) 1 (B) 2 (C)8 (D) None of these

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67. The system of homogeneous equations

tx  t  1 y  t 1 z  0

t  1 x  ty  t  2 z  0

t 1 x  t  2 y  tz  0
Has non-trivial solution for

(A) Exactly three real values of t (B) Exactly two real values of t

(C)Exactly one real values of t (D) Infinite number of values of t

68. Nine identical balls are numbers 1,2,…9 are put in a bag. Person A draws a ball and
gets the number a. The ball is put back the bag. Next person B draws a ball gets the
number b. the probability that a and b satisfies the inequality a- 2b 10  0 is

52 55 61 62
(A) (B) (C) (D)
81 81 81 81

1
69. If 3  4     , then 2cot  is equal to
sin 2

(A)1  cot (B) 1  cot (C)1  cot (D) 1  cot

70. The angle A of the  ABC is obtuse. Let x  2635  tan BtanC,if  x denotes the greatest

integer function, the value of  x  is

(A)2364 (B) 2634 (C)2364 (D)None

71. The value of x between 0 and 2 which satisfy the equation sin x 8 cos 2 x  1 are in
A.P. The common difference of the A.P. is

(A)  8 (B)  4 (C) 3  8 (D) 5  8

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72. If p,x1 ,x2 ...xi ,... and q, y1 , y2 ..., yi ,... are in A.P., with common difference a and b

respectively, then locus of the center of mean position of the points Ai  xi , yi  ,i  1,2...n
is

(A) ax  by  aq  bp (B) bx  ay  ap  bq

(C) bx  ay  bp  aq (D) ax  by  bq  ap

73. If one of the lines given by the equation 2 x 2  axy  3 y 2  0 coincide with one of those

given by 2 x 2  bxy  3 y 2  0 and the other lines represented by them be perpendicular

then

(A) a  5, b  1 (B) a  5, b  1

(C) a  5, b  1 (D)None of these

74. If 1and 2 be the angles which the lines

 x 2  y 2 cos2 sin 2  sin 2    x tan  ysin  2 make with the axis of x, then if
   6 , tanθ1  tanθ 2 is equal to

(A) 8 3sin 2 (B) 8 3 cosec 2

(C)  8 3 cos ec 2 (D)  4 cos ec 2

75. on the line joining the points A (0,4) and B(3,0), a square ABCD is constructed on the
side of the line away from the origin Equation of the circle having centre at C and
touching the axis of x is

(A) x 2  y 2  14 x  6 y  49  0 (B) x 2  y 2  14 x  6 y  9  0

(C) x 2  y 2  6 x  14 y  49  0 (D) x 2  y 2  6 x  14 y  9  0

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a3 x 2 a 2 x
76. The locus of the vertices of the family of parabolas y    2a is
3 2

105 3 35 64
(A) xy  (B) xy  (C) xy  (D) xy 
64 4 16 105

77. Tangent are drawn from any point on the hyperbola x 2 9  y 2 4  1 to the circle

x 2  y 2  9 . If the locus of the mid-point of the chord of the contact is

a  x 2  y 2   bx 2  cy 2 then the value of a 2  b 2  c 2 is equal to

(A)7873 (B)8773 (C)3877 (D)None

78. The relation R given by  x, y  : x 2  3xy  2 y 2  0, x, y  R is

(A) Reflexive but not symmetric (B) symmetric but not transitive

(C) Symmetric and transitive (D) An equivalence relation

79. Let x1 , x2 , x3 , ...xk be k observations and wi  axi  b for i  1,2 ,3,...k , where a and b are
constants. If mean of xi is 52 and their standard deviation is 12 and means of wi is 60
and their standard deviation is 15 then the value of a and b should be

(A) a  1.25,b  5 (B) a  1.25,b  5

(C) a  2.5,b  5 (D) a  2.5,b  5

80. The negation of P  q  r  is

(A) ~ p  q  r  (B) p  q  r 

(C) p  q  r  (D) ~ p  q  r 

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SECTION-II
(NUMERICAL VALUE ANSWER TYPE)
This section contains 10 questions. The answer to each question is a Numerical value. If the numerical value has
more than two decimal places, Truncate/Round-off the value of Two decimal places. Have to Answer any 5
only out of 10 questions and question will be evaluated according to the following marking scheme:
Marking scheme: +4 for correct answer, 0 in all other cases.
x
81. If the function f  x  x 3
e 2 and g  x  f 1  x then the value of g' 1 is _____

dy
82. If xe xy  y  sin 2 x, then at x=0, is equal to
dx

83. If z1  z2  z3  0 and z1  z2  z3  1, then z12  z22  z32 equals

2 2 2
84. If z1  z2  z3  1 and S  z2  z3  z3  z1  z1  z2

Let  = minimum value of S and  = minimum value of S then    

85. Find the number of rational roots of p  x  2 x98  3x97  2 x96  ....  2 x  3  0.

86. If x, y,z  R,x  y  z  4 and x 2  y 2  z 2  6 find the minimum possible value of z

87. Let p(x) be a polynomial with integral coefficients. Let a, b, c be three distinct integers
such that p a   p b  p c   1. find the number of integral roots of p(x)___

Coefficient of x10 in the expansion of 1  x 2  x3  is___

8
88.

89. The number of natural numbers which are smaller than 2 108 and which can be written
by means of the digit 1 and 2 is_________ .

90. The number of ordered pairs (m, n), m, n  1,2 ,....100 such that 7m  7n is divisible by
5 is___

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