XII NM ALPHA

MOCK TEST-9

Date: 25-12-2023
Time: 3HRS Max. Marks: 300

Name of the Student: ___________________

PHYSICS: TOTAL SYLLABUS

CHEMISTRY: TOTAL SYLLABUS

MATHEMATICS: TOTAL SYLLABUS

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. A particle is moving parallel to x-axis as shown in the figure such that at all instants the
y - component of its position vector is constant and is equal to ‘b’. The angular velocity
of the particle ‘P’ about the origin at the given instant is

v v v 2
(A) cos  (B) sin  (C) sin  (D) vb
b b b

2. An object is placed on the surface of a smooth inclined plane of inclination . It takes

time t to reach the bottom. If the same object is allowed to slide down a rough
inclined plane of same inclination  ,so as to move the same distance, it takes time
‘nt’ to reach the bottom where n is a number greater than 1. The coefficient of friction
 is given by:
1  1 
(A)   tan   1   (B)   cot   1  
 n2   n2 
1 1
1 2 1 2
(C)   tan  1  2  (D)   cot  1  2 
 n   n 

3. A system of two bodies of masses ‘m’ and ‘M’ being interconnected by a spring of
stiffness k,in its natural length, moves towards a rigid wall on a smooth horizontal
surface as shown in figure with a K.E. of system ‘E’. If the body M sticks to the wall
after the collision, the maximum compression of the spring will be:

k
m
M

mE 2mE 2m  M  E 2ME

(A) (B) (C ) (D)
Mk  M  m k mk  M  m k

Page. No. 2
4. A sphere of mass m1 = 2kg collides with a sphere of mass m2 = 3kg which is at rest.
Mass m1, after the collision will move at right angle to the line joining centers of two
spheres at the time of collision, assuming colliding surfaces are smooth, if the
coefficient of restitution is

4 1 2 2
(A) (B) (C) (D)
9 2 3 3

5. From a circular disc of radius and mass 9 , a small disc of radius /3 is removed
from the disc. The moment of inertia of the remaining disc about an axis perpendicular
to the plane of the disc and passing through is

A) B) C) D)
6. A planet of radius R has an acceleration due to gravity of g s on its surface. A deep
smooth tunnel is dug on this planet, radially inward, to reach a point P located at a
R
distance of from the centre of the planet. Assume that the planet has uniform density.
2
The kinetic energy required to be given to a small body of mass m, projected radially
outward from P, so that it gains a maximum altitude equal to the thrice the radius of the
planet from its surface, is equal to

O P

63 3 9 21
A) mg s R B) mg s R C) mg s R D) mgs R
16 8 8 8

Page. No. 3
7. Water rises to a height of 10cm in a capillary tube and mercury falls to a depth of 3.42
cm in the same capillary tube. If the density of mercury is 13.6g/cc and the angle of
contact of mercury and water are 135 and 0 respectively, the ratio of surface tension of
water to mercury is
(A)1 : 0.15 (B)1 : 3 (C)1 : 6.5 (D)1.5 : 1
8. Two moles of ideal helium gas are in a rubber balloon at 30℃. The balloon is fully
expandable and can be assumed to require no energy in its expansion. The temperature
of the gas in the balloon is slowly changed to 35℃. The amount of heat required in
raising the temperature is nearly (take �㕅=8.31 �㔽/�㕚�㕜�㕙.�㔾)
A) B) C) D)
9. Two rigid boxes containing different ideal gases are placed on a table. Box A contains
one mole of nitrogen at temperature while box B contains one mole of helium at
temperature The boxes are then put into thermal contact with each other and heat
flows between them until the gases reach a common final temperature (Ignore the heat
capacity of boxes and heat exchange will happen only between boxes). Then, the final
temperature of the gases, in terms of is

A) B) C) D)
10. A particle free to move along the x-axis has potential energy given by
U ( x )  k [1  exp(  x 2 )] for   x   , where k is a positive constant of appropriate

dimensions. Then

A) At point away from the origin, the particle is in unstable equilibrium

B) For any finite non-zero value of x, there is a force directed away from the origin

C) If its total mechanical energy is k/2, thenits kinetic energy at the origin is k.

D) For small displacements from x = 0, the motion is simple harmonic.

11. The electric potential in a medium of dielectric constant ‘unity’ is   x, y, z   ax2 where
‘a’ is a constant of suitable dimensions. The total charge contained in a cube of
dimensions L  x, y, z  L is

A) zero B) 2a 0 L3 C) 16a 0 L3 D) 4a 0 L3

Page. No. 4
12. Three large plates are arranged as shown. How much charge will flow through the key
when it is closed?

A) B) C) D) Q
13. Assertion : A current I flows along the length of an infinitely long straight and thin
walled pipe. Then, the magnetic field at any point inside the pipe is zero.
 
Reason :  B .d l  o I

Read the assertion and reason carefully to mark the correct option out of the options
given below:

A) Both Assertion and Reason are true and the reason is the correct explanation of the
assertion.

B) Both Assertion and Reason are true but reason is not the correct explanation of the
assertion.

C) Assertion is true but Reason is false.

D) Assertion and Reason both are false.

14. The graph for an alloy of paramagnetic nature is shown in fig. the curie constant
is nearly.

0.4
0.3
0.2
0.1
O
0 2 4 6 7
1/T(in 10–3K–1)

A) B) C) D)

Page. No. 5
15. The natural frequency of the circuit shown in the figure is (Assume that equal currents i,
i exist in the two branches, and the charges on the capacitors are equal)
q q
i+ – + – i
C C
L L L

1 1 1 1 1 1 1 4
A) B) C) D)
2 LC 2 3LC  LC  3LC

16. In the circuit shown in the figure, the ac source gives a voltage V  20cos(2000 t ).
Neglecting source resistance, the voltmeter and ammeter reading will be ( 2  1.4 )

A) 0V, 0.47A B)1.68V, 0.47A C) 0V, 1.4 A D) 5.6V, 1.4 A

17. A plane Electromagnetic Waves travelling along the positive -direction has a
wavelength of 3 mm. The variation in the electric field occurs in the -direction with an
amplitude66Vm-1. The equations for the electric and magnetic fields as a function of
and are respectively

A)

B)

C)

D)

18. In young’s double slit experiment, the distance between the slits varies with time as
d  t    2d 0  d 0 sin wt  , where d0 and ‘w’ are positive constants. The difference between
the largest and the smallest fringe width obtained over time is___
(D= distance between slits and screen  d &  = wavelength of light used)
D D 2 D D
A) B) C) D)
2d 0 3d 0 3d 0 6d 0

Page. No. 6
19. Match the following.
List – I List – II
A) Moderator e) Absorbs heat
B) Control rods f) Prevents radiation from getting exposed
to the outside
C) Radiation shielding g) Absorb neutrons
D) Coolant h) Slow down neutrons
A) A – h, B – g, C – f, D – e B) A – g, B – h, C – f, D – e
C) A – h, B – g, C – e, D – f D) A – h, B – f, C – e, D –g
20. A student performs an experiment to determine the Young’s modulus of a wire, exactly
2 m long, by Searle’s method. In a particular reading, the student measures the extension
in the length of the wire to be 0.8 mm with an uncertainty of at a load of
exactly 1.0 kg. The student also measures the diameter of the wire to be 0.4mm with an
uncertainty of
Take (exact). The Young’s modulus obtained from the reading is
A) B)
C) D)

SECTION-II
(NUMERICAL VALUE ANSWER TYPE)
This section contains 10 questions. The answer to each question is a Numerical value. If the Answer in the
decimals , Mark nearest Integer only. 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, -1 in all other cases.
21. In the system shown all the surfaces are frictionless while pulley and string are massless.
Mass of block �㔴 is 3�㕚 and that of block �㔵 is �㕚. If the acceleration of block �㔵with
respect to ground after system is released from rest is ‘a’, then the value of ‘10 a’ is (take
g=10 m/s2and 5 2  7.0 )

Page. No. 7
22. An arrow sign is made by cutting and re-joining a quarter part of a square plate of side
'L=1m' as shown. The distance OC, where 'C' is the centre of mass of the arrow, is
…..(in cm)

23. A uniform solid cylinder of mass and radius is placed on a rough horizontal board
of same mass, which in turn is placed on a smooth surface. The coefficient of friction
between the board and the cylinder is =0.3. If the board starts accelerating with
constant acceleration as shown in the figure, then the maximum value of , so that the
cylinder performs pure rolling is…...(in m/s2) given g=10m/s2

24. The variation of lengths of two metal rods and with change in temperature is shown
in Figure. If the coefficients of linear expansion for the metal = n 106 / 0C ,then
the value of ‘n’ will be, in nearest integer, given C)

25. Two strings and of a sitar produce a beat frequency of When the tension of the
string is slightly increased the beat frequency is found to be If the frequency of
is then the original frequency of was (in Hz)
26. The potential difference across 8 ohm resistance is 48 volt as shown in the figure. The
value of potential difference across X and Y points will be…..(in Volts)
X
3

20 30 60

24 8 48V

1
Y

Page. No. 8
27. Two thin symmetrical lens of different nature have equal radii of curvature of all faces
R = 20 cm. The lenses are put close together and immersed in water. The focal length of
the system is 24 cm. The difference between refractive indices of the two lenses is ……
1 4
× . Refractive index of water is .
9 3

4
=
1 3

2

28. When a monochromatic point source of light is at a distance 0.2 m from a photoelectric
cell, the saturation current and cut-off voltage are 12.0 mA and 0.5 V. If the same source
is placed at 0.4 m away from the same photoelectric cell, then the saturation current,
now, will be …..(in mA)

29. In the circuit given if the current through the Zener diode is I Z . Find the value of 6 I Z (in
mA)

30. In a circuit for finding the resistance of a galvanometer by half deflection method, a 6 V battery
and a high resistance of 11 kΩ are used. The figure of merit of the galvanometer is
60μA/division. In the absence of shunt resistance, the galvanometer produces a deflection
of θ=9 divisions when current flows in the circuit. The value of the shunt resistance (in  ) that
can cause the deflection of θ/2, is closest to____

Page. No. 9
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. If  0 is the threshold wavelength for photoelectric emission from a metal surface,  is
the wavelength of light falling on the surface of metal and m is the mass of electron, then
the maximum speed of ejected electrons is given by
1 1

 2h 2  2hc
  0   
2
(A)    0     (B) 
m   m 
1 1

 2hc   0     2  2h  1 1   2
(C)    (D)     
 m   0   m  0  

32. Given below are two statements: One is labelled as Assertion A and the other is labelled
as Reason R.
Assertion A: In TlI3 which is isomorphousto CsI3 , the metal is present in +1 oxidation
state.
Reason R: Tl metal has fourteen f electrons in its electronic configuration.
In the light of the above statements, choose the most appropriate answer from the
options given below.
(A) Both A and R are correct and R is the correct explanation of A

(B) A is not correct but R is correct

(C) Both A and R are correct R is NOT the correct explanation of A

(D) A is correct but R is not correct

33. Assertion (A): Bond dissociation energy of F2 is smaller than that of Cl2
Reason (R): Bond dissociation energy is in order F2  Cl2  Br2  I 2
A) Assertion is True, Reason is True; Reason is correct explanation for Assertion
B) Assertion is True, Reason is True; Reason is NOT a correct explanation for Assertion
C) Assertion is True, Reason is False
D) Assertion is False, Reason is True

Page. No. 10
Narayana IIT Academy 21-12-23_SR.IIT_*CO-SC(MODEL-A,B&C)_JEE-MAIN_GTM-1_Q’P
34. 5 g of urea is dissolved in one kg of water. If the solution is cooled to 0.200o C , then
how many grams of ice would separate? (Kf of water = 1.86)

(A) 200 (B) 225 (C) 325 (D) 175

35. A brown precipitate insoluble in excess of aqueous ammonia solution is formed by

(A) FeCl3  NH 4 Cl  (B) FeCl3  NH 4 OH 

(C) CrCl3  NH 4OH  (D) Hg 2 Cl2  NH 4 OH 

36. For the first-order reaction A g   2Bg   C g  , the total pressure after time t from the start

of reaction with A is P and after infinite time, it is P . Then the rate constant of the
reaction is

1 P 1 2P
(A) ln (B) .ln
t P t 3  P  P 

1 2P 1 2P
(C) .ln (D) ln
t 3P  P t P  3P

37. The order of E oM 3

/ M 2
value is

(A) Co >Mn> Fe (B)Mn> Co > Fe (C) Co > Fe >Mn (D) Mn> Fe > Co

38. A chromatography column packed with silica gel (used as stationary phase) is used for
the separation of

The column was then eluted with a mixture of Dichloro methane-nitromethane in 80:20
ratio, the sequence of elution of A, B and C is

A) A, B, C B) C, B, A C) B, C, A D) A, C, B

Page. No. 11
39. Consider the following reactions, What is A

A) Acetylene B) propyne C) ethylene D) 2-butyne

40. In azomethane, H3CNNCH3, what are the molecular geometries around the carbon and
nitrogen atoms, respectively?
A) Tetrahedral at carbon, bent at nitrogen

B) Tetrahedral at carbon, linear at nitrogen

C) Square planar at carbon, bent at nitrogen

D) Square planar at carbon, linear at nitrogen

41. An organic compound contains only carbon, hydrogen, nitrogen, and oxygen. It is
61.71% C, 4.03% H, and 16.00% N by mass. What is its empirical formula?

A) C5H4NO B) C9H7N2O2 C) C10H8N2O C) C11H8NO2

42. Consider the following sequence of reactions:

The functional groups undergoing change in the conversion of A to E respectively are

A) COOH,  NC,  CONH 2 ,  NH 2 ,  CH 2 OH

B) COOH,  CN,  CONH 2 ,  NH 2 ,  CH 2 OH
C) COOH,  CONH 2 ,  CN,  CH 2 NH 2 ,  CH 2 OH
D) CONH 2 ,  COOR,  NC,  NHR,  CHOH

Page. No. 12
43.

The compounds which can be reduced with formaldehyde and conc. AqKOH, are

A) only II and V B) only I and V

C) only II and III D) only I, II and IV

44. What is the X for given reaction?

A) B) C) D) All

45. The correct sequence of reactions to get ‘Q’ as the only product from ‘P’ is

A) (i) H2&Pt catalyst (ii) C2H5Cl & AlCl3

B) (i) Mg in ether (ii) aqueous alcohol (iii) C2H5Cl & AlCl3

C) (i) Mg in ether (ii) C2H5Cl & AlCl3

D) (i) C2H5Cl & AlCl3 (ii) Mg in ether (iii) aqueous alcohol

Page. No. 13
46. When acid ‘X’ is heated to 230°C, along with CO 2 and , a compound ‘Y’ is formed. If
‘X” is HOOC  CH 2 2 CH  COOH 2 , the structure of ‘Y’ is

(A) HOOC  CH 2 3 COOH (B)

(C) CH3CH 2 CH  COOH 2 (D)

47. The best reagents and conditions to accomplish the following conversion is?

A) (i) LiAlH4 in ether, (ii) 3 moles of CH 3 I followed by heating with AgOH

B) (i) LiAlH4 in ether; (ii) P2O5 and heat

C) (i) 20 % H2SO4& heat, (ii) P2O5 and heat

D) H2 and Lindlar catalyst

48. Two students did a set of experiments on ketones ‘X’ and ‘Y’ independently and
obtained the following results.

The ketones ‘X’ and ‘Y’ are respectively

Page. No. 14
 (A) 2-ethylcyclobutanone and 3-ethylcyclobutanone

(B) 2-methylcyclopentanone and 3-methylcyclopentanone

(C) 3-methylcyclopentanone and 2-methylcyclopentanone

(D) 3-methyl-4-penten-2-one and 4-methyl-1-penten-3-one

49. RNA and DNA are chiral molecules, their chiralityis due to:

A) D-sugar compound B) L-sugar component

C) chiral bases D) Chiral phosphate ester groups

50. The electronic configurations of bivalent europium and trivalent cerium are
(atomic number : Xe = 54, Ce = 58, Eu = 63)

(A)  Xe 4f 2 and  Xe  4f 7 (B)  Xe 4f 7 and  Xe  4f 1

(C)  Xe 4f 7 6s 2 and  Xe  4f 2 6s 2 (D)  Xe 4f 4 and  Xe 4f 9

SECTION-II
(NUMERICAL VALUE ANSWER TYPE)
This section contains 10 questions. The answer to each question is a Numerical value. If the Answer in the
decimals , Mark nearest Integer only. 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, -1 in all other cases.
51. Geraniol is an acyclic unsaturated alcohol C10H18O, a terpene found in rose oil, adds two
moles of bromine to form a tetrabromide, C10H18OBr4 . It can be oxidized to a ten-carbon
aldehyde or to a ten-carbon carboxylic acid. Upon vigorous oxidation, geraniol yields:

Number of double bonds in Geraniol is?

Page. No. 15
52. The solubility product of AgC 2 O 4 at 25o C is 2.3  1011 M 2 . A solution of K 2 C 2 O 4
containing 0.15 moles in 500 ml water is shaken at 25o C with Ag 2 CO3 till the following
equilibrium is reached.


Ag 2 CO3  K 2C 2O 4  Ag 2 C2O 4  K 2 CO3

At equilibrium, the solution contains 0.035 mole of K 2 CO3 . Assuming the degree of
dissociation of K 2 C 2 O 4 and K 2 CO3 to be equal, The solubility product of Ag 2 CO3 is
x 1012 , then x is ______

53. The volume, in mL, of 0.02 M K 2 Cr2 O 7 solution required to react with 0.288 g of ferrous
oxalate in acidic medium is ______. (Molar mass of Fe = 56 g mol1 )

54. Equivalence conductance at infinite dilution of NH 4 Cl, NaOH and NaCl are
129.8, 217.4 and 108.9  1 cm 2 mol 1 , respectively. If the equivalent conductance of 0.01 N

solution of NH 4 OH is 9.532  1cm 2 mol1 , then the percentage (%) dissociation of NH 4 OH

at this temperature is

55. Glycine (C2H5O2N) is the simplest of amino acids. Molecular formula of the linear
oligomer synthesized by linking ten glycine molecules together via a condensation
reaction is CwHxOyNz

What is the value of w+x+y+z?

56. For silver, C p  JK 1 mol1  = 23 + 0.01T. If the temperature (T) of 3 moles of silver is
raised from 300 K to 1000 K at 1 atm pressure, the value of H will be close to

57.

If X is the moles of KOH consumed per mole of reactant in reaction-1 and Y is the
moles of KOH consumed per mole of reactant in reaction 2, what is (X+Y)

Page. No. 16
58. Lovastatin, a drug used to reduce the risk of cardio vascular diseases has the following
structure

The number of stereogeniccenters present in lovastatin is

59. How many of the following are an example of aromatic electrophilic substitution
reactions

60. Total number of paramagnetic species in which unpaired electron(s) is/are present in
 * M.O.

O 2 ;O 2 ;O 2 ; N 2 ; N 2 ; N 22  ; B2

Page. No. 17
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. The number of points in the complex plane that satisfy the conditions z  2  2,

z 1  i   z 1  i   4 (where i  1 ) is

A) 0 B) 1 C) 2 D) more than two

62. Let A and B are square matrices of same order satisfying AB  A and BA  B , then

A 
2020
2019
 B 2019 is equal to:

A) A  B B) 2020  A  B  C) 22019  A  B  D) 22020  A  B 

x 2  3x x 1 x  3
4 3 2
63. If px + qx + rx + sx + t = 2
x 1 2 x x 3 then p is equal to
x2  3 x  4 3x

A) –5 B) –4 C) –3 D) –2.

64. If  is the root of the equation x  x  2  0 then the value of

2 
6 3  22    is
  3  3  2
5 4 3

equal to:
A) 3 B) 6 C) 9 D) 12
65. Number of 4 digit numbers of the form N = abcd which satisfy following three
conditions:
i) 4000  N  6000 ii) N is a multiple of 5

iii) 3  b  c  6 is equal to:

A) 12 B) 18 C) 24 D) 48

66. If a and b are chosen randomly by throwing a pair of fair cubical dice, then the
2
 a x  bx x
probability that lim    6 equals:
x 0
 2 

4 2 3 1
A) B) C) D)
9 9 9 9

Page. No. 18
67. A random variable X takes values x = 0, 1, 2, 3, ... with probability proportional to
x
1
  . Then ____ (E(X) denotes the mean of the random variable X)
5

A) P(X = 0) = 16/25 B) P(X =1) = 4/25

C) P(X  1) = 7/25 D) E(X) = 25/32

 x 2  2 x  a , x  1
68. Let f  x    , then number of positive integral value(s) of ‘a’ for
 6  x, x 1

which f  x  has local minima at x = 1, is ______

A) 6 B) 7 C) 8 D) 9

69.
1
  12  
Let g  x   f 2x 2  1  f 1  x 2 x  R, where f " x   0x  R , g(x) is necessarily
4
increasing in the interval

 2 2
A)   ,  B)   2   2 
,0    , 
 3 3  3   3 

 2
C)  1,1 D)  ;  
 3 

  
70. Let a, b, c are three vectors having magnitudes 1,2,3 respectively satisfy the
    
relation a  b  .c  6 . If d is a unit vector coplanar with b and c such that b.d  1 then
   2    2
 
the value of a  c .d  a  c  d is  
9 9
A) 9 B) 3 C)  D)
2 2

71. Three straight lines mutually perpendicular to each other meet in a point P and one of
them intersects the x-axis and another intersects the y-axis, while the third line passes
through a fixed point (0, 0, c) on the z-axis. Then the locus of P is

A) x 2  y 2  z 2  2cx  0 B) x 2  y 2  z 2  2cy  0

C) x 2  y 2  z 2  2cz  0 D) x 2  y 2  z 2  2c  x  y  z   0

Page. No. 19
72. Area of the trapezium whose vertices lies on the parabola y 2  4 x and its diagonals pass
25
through 1, 0  and having length unit each, is
4

75 625 25 25
A) sq.units B) sq.units C) sq.units D) sq.units
4 16 4 8

73. Let f be a differentiable function defined in 0,1 such that f  f  x    x and f  0   1. If

1
p
  x  f  x 
2018
the value dx  where p and q are relatively prime then the value of
0
q

 p  q  , is ___________

A) 2023 B) 2024 C) 2020 D) 2022

74. Let f(x) = Maximum {x2, (1  x)2, 2x(1  x)}, where 0  x  1. Determine the area of the
A
region bounded by the curves y = f(x), x-axis, x = 0 and x = 1 is then find the value
54

of A.

A) 30 B) 36 C) 32 D) 34

75. Let y  y  x  satisfy the differential equation

 y3 
 2xy  x 2
y 
3 
2 2
 
 dx  x  y dy  0 . If y 1  1 and the value of  y  0    ke k  N 
3


Then k is _________
A) 3 B) 4 C) 1 D) 2

76. If a1,a2 ,......, a4001 are in arithmetic progression and

1 1 1
  .....   10 and a2  a4000  50 . Find the value of a1  a 4001 .
a1a 2 a 2a 3 a 4000a 4001

A) 20 B) 32 C) 60 D) 30
n n
1 r
77. If an   n C n  n C , then the number of ordered pairs  p, q  such that
, b 
r 0 r r 0 r

ap
c p  cq  1, where c p  , is:
bp

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

Page. No. 20
 If  sin 250  3 cos 850  sin 850   a  b cos 500 , a, b, are rational numbers in their
2
78.

simplest form, then a  b  

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

 3  x; x 1
 x  1; x  0  2
79. . Let f  x    and g  x   x  2x  2; 1  x  2 then:_______
2  x ; x  0  x  5; x 2


Note: [k] denotes greatest integer function less than or equal to k.

A) lim g  f  x    2 B) lim g  f  x    3
x 0 x 0

C) lim f  f  x     0 D) lim g  g  x     1

x 0 x 0

x
dt
80. Let f be real-valued function such that e 2 x
f x   x  3   for all x   1,1 and
0 t6 1

let y  g  x  be a function whose graph is reflection of the graph of y  f  x  w. r. t. line

y = x, then g '  3  is equal to:

1 1 1
A) 1 B) C) D)
2 4 8

SECTION-II
(NUMERICAL VALUE ANSWER TYPE)
This section contains 10 questions. The answer to each question is a Numerical value. If the Answer in the
decimals , Mark nearest Integer only. 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, -1 in all other cases.
x 3  1 x 5 3x  2
 
81.  
Let A   y  1 6x 2  2 z  1  and A  3 . If f  x   tr . B 1 and B = adj(A), then
 2 3 9x  6
 

global maximum value of f  x  in x  0,6 is:

82. Let f : A  A where A  1,2,3,4,5,6,7 , then number of function f such that

 
f f  f  x    x x  A , is:

Page. No. 21
83. The point on the line r   2i  6j  34k   t  2i  3j  10k  , t  R which is nearest to the line

   
r  6i  7 j  7k   4i  3j  2k ,   R is a , b, c  , then the value of a  b  c is equal to

x 2 dx
1 1
ex I
84. Let I1   dx and I 2   3 . Then 1 is
0
1 x 0 e
x

2 x 3

e.I 2

85. A, B, C are the vertices of triangle with right angled at A and P  4,0  ;Q  0,6  are two
given points. If the ratio of the distances from each vertex of triangle to P, to that of Q is
 r 
2:3, the centroid of triangle ABC lies on a circle with radius ‘ r ‘ then   is equal to
 13 
_____ [ . ] represent GIF
86. Let A,B,C,D are four points, in I, II, III, IV quadrants respectively, lying on the circle

 x  12   y  22  25 so that AC and BD are perpendicular diameters. Circles, touching

the given circle, are drawn at A, B, C, D with radii 1, 2, 3, 4 respectively. If A is  4,6  ,
then the sum of the coordinates of centers of the circles corresponding to B and D is

87. Mr. A either walks to school or take bus to school everyday. The probability that he
takes a bus to school is 1/4. If he takes a bus, the probability that he will be late is 2/3. If
he walks to school, the probability that he will be late is 1/3. The probability that Mr. A
p
will be on time for at least one out of two consecutive days is , where p and q are co-
q
prime, find the value of  q  p  .

cos ec 2x  2019  f x  

88. If   cos x 2019 dx   C , then find the value of f    g  0  . (where C
 g  x 
2019
4

is constant of integration)

89. Let y1, y2 , y3.....yn be n observations. Let wi  lyi  k , i  1,2,3....n where l , k are
constants. If the mean of yi ' s is 48 and their standard deviation is 12, the mean of wi ' s
is 55 and standard deviation of wi ' s is 15, then values of l + k + 0.75 should be ______

90. Let f be an invertible function from R  R satisfying the equation

     
f 3  x   x 3  2 f 2  x   2x 3  1 f  x   x 3  0 . Then the value of f '  8  f 1 '  8  , is:

Page. No. 22