PAPER CODE C J A 1 0 1 2 1 2 30 0 0 4

CLASSROOM CONTACT PROGRAMME

(ACADEMIC SESSION 2023-2024)

TIME : 3 HOURS Maximum Marks : 198

PAPER -2
Please read the instructions carefully. You are allotted 5 minutes specifically for this purpose.
INSTRUCTIONS
A. General :
1. The booklet is your Question Paper. Do not break the seal of this booklet before being instructed to do
so by the invigilator.

DO NOT BREAK THE SEALS WITHOUT BEING INSTRUCTED TO DO SO BY THE INVIGILATOR

2. The question paper CODE is printed on the bottom page corner of this sheet.

3. Blank spaces and blank pages are provided in the question paper for your rough work. No additional sheets
will be provided for rough work.

4. Blank papers, clipboards, log tables, slide rules, calculators, cameras, cellular phones, pagers and
electronic gadgets are NOT allowed inside the examination hall.

5. Write your name and Form number in the space provided on the back cover of this booklet.

6. The answer sheet, a machine-readable Optical Response Sheet (ORS), is provided separately.

7. DO NOT TAMPER WITH/MUTILATE THE ORS OR THE BOOKLET.

8. On breaking the seal of the booklet check that it contains 24 pages and all the 18 questions in each subject
and corresponding answer choices are legible. Read carefully the instructions printed at the beginning of
each section.

B. Filling the ORS :

9. A candidate has to write his / her answers in the ORS sheet by darkening the appropriate bubble with the
help of Black ball point pen as the correct answer(s) of the question attempted.

C. Question Paper Formate :

The question paper consists of 3 parts (Physics, Chemistry and Mathematics).
10. SECTION – I Contains 6 multiple choice questions. Each question has FOUR choices (A), (B), (C) and
(D) out of which ONE or MORE are correct.
11. SECTION – II (i) contains 6 questions. The answer to each question is a SINGLE DIGIT INTEGER
ranging from 0 TO 9, BOTH INCLUSIVE. For each question, enter the correct integer corresponding
to the answer using the mouse and the on-screen vi numeric keypad in the place designated to enter
the answer.
12. SECTION – II (ii) contains 6 questions. The answer to which is a Numerical Value. For each
question, enter the correct numerical value to answer two decimal place, If the numerical value has
more than two decimal places, truncate/round-off the value to TWO decimal places.

Please read the last page of this booklet for rest of the instructions
 SOME USEFUL CONSTANTS
Atomic No. H = 1, B = 5, C = 6, N = 7, O = 8, F = 9, Al = 13, P = 15, S = 16, Cl = 17,
Br = 35, Xe = 54, Ce = 58,
Atomic masses : H = 1, Li = 7, B = 11, C = 12, N = 14, O = 16, F = 19, Na = 23, Mg = 24,
Al = 27, P = 31, S = 32, Cl = 35.5, Ca=40, Fe = 56, Br = 80, I = 127,
Xe = 131, Ba=137, Ce = 140,

 Boltzmann constant k = 1.38 × 10 –23 JK –1

1
 Coulomb's law constant = 9 ×10 9
4 0
 Universal gravitational constant G = 6.67259 × 10 –11 N–m 2 kg –2
 Speed of light in vacuum c = 3 × 10 8 ms –1
 Stefan–Boltzmann constant  = 5.67 × 10 –8 Wm –2 –K –4
 Wien's displacement law constant b = 2.89 × 10 –3 m–K
 Permeability of vacuum µ 0 = 4 × 10 –7 NA –2

1
 Permittivity of vacuum 0 =
0 c 2
 Planck constant h = 6.63 × 10 –34 J–s
HAVE CONTROL  HAVE PATIENCE  HAVE CONFIDENCE  100% SUCCESS
BEWARE OF NEGATIVE MARKING
PART A - PHYSICS
SECTION – I : (Maximum Marks : 24)
 This section contains SIX (06) questions.
 Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of these
fouroption(s) is (are) correct option(s).
 For each question, choose the correct option(s) to answer the question.
 Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +4 If only (all) the correct option(s) is (are) chosen.
Partial Marks : +3 If all the four options are correct but ONLY three options are chosen.
Partial Marks : +2 If three or more options are correct but ONLY two options are chosen,
both of which are correct options.
Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and
it is a correct option.
Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered).
Negative Marks : –2 In all other cases
1. Figure shows a conducting rectangular loop of electrical resistance R. There exists a

uniform magnetic field given by B  B0 10t 2  5t  kˆ in the region. The current in the loop
at

10B0
(A) t = 0 is zero (B) t = 0 is from A to B
R
1
(C) t  s is zero (D) t = 1s is 30 B 0 from B to A
4 R
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2. The figure shows two infinite parallel sheets of current, with current per unit length 1 and
2. If BP, BQ and BR represent the magnetic field at the points P, Q and R respectively, then
0
(A) BP  (1   2 ) (along x–axis)
2 P.
(1   2 ) 1
(B) BQ   0 (along x–axis)
2 y-axis
 1   2  Q. x-axis
(C) B R     0 (along x–axis)
 2 

  
2
(D) B Q   1 2 2   0 (along x–axis)
 
R.
3. A nonconducting disc having uniform positive charge Q, is rotating about its axis with
uniform angular velocity . The magnetic field at the centre of the disc is-

 0 Q
(A) directed outward (B) having magnitude
4R
 0 Q
(C) directed inwards (D) having magnitude
2R
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4. H+ He+ and O2+ all having the same kinetie energy pass through a region in which there
is a uniform magnetic field perpendicular to their velocity. The masses of H+, He+ and O2+
are 1 amu, 4 amu and 16 amu respectively. Then :
(A) H+ will be deflected most (B) O2+ will be deflected most
(C) He+ and O2+ will deflected equally (D) All will be deflected equally
5. A charge is released from rest in a region of steady and uniform electric and magnetic fields
which are parallel to each other. The charge will moves along which path:-
(A) Circular (B) Helical (C) Parabola (D) Straight line
6. The figure shows three long straight current carrying conductors. The straight parts are
th
3
long and the circular part in each case in   of a complete circle. Let Ba, Bb and Bc
4
represents the strength of field at the centre O in the three cases, then

 0i  3 1  0i  3 1 
(A) Ba     (B) Bb    
4R  2   2R  4  

 0i  3 1  3 0i
(C) Ba     (D) Bc 
4R  2   8R
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 SECTION II (i) (Maximum Marks: 18)
 This section contains SIX (06) questions.
 The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 TO 9,
BOTH INCLUSIVE.
 For each question, enter the correct integer corresponding to the answer using the mouse
and the on-screen virtual numeric keypad in the place designated to enter the answer.
 Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +3 If ONLY the correct integer is entered;
Zero Marks : 0 If the question is unanswered:
Neqative Marks : –1 In all other cases.
1. A large circular loop of radius 1 metre has total resistance 4. A variable current is flowing through
the loop. The current at any instant is given by equation I = tA, where t is time in second. A very
small circular loop of radius 1 cm having resistance 3.14 is placed coaxially with the larger loop
0 104
at a distance of 3 metre from the centre of large loop. If induced current in smaller loop is x 
16
ampere then, find x.
2. A conducting wire bent in the form of a parabola y2 = 2x carries a current i=2A as shown in figure.

This wire is placed in a uniform magnetic field B  1kˆ Tesla. The magnetic force on the wire is
(in newton)

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3. A current I flows along a thin wire coil shaped as shown in figure. The radius of the curved part of
0 I
the wire is r. If magnetic field at the centre O of the coil is (3  N) , then value of N is
8r

O
90°
A B
4. Current l is flowing in a closed equilateral triangular loop ABC of side L. If magnetic field at the centriod
k 0 l
O is given by expression , then find k.
2L

l O l

C l B

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 x2
5. A rod of length l starts sliding down the smooth parabolic curve y  from y = a as shown in figure.
a
Uniform magnetic field B0 ˆj is present in region. EMF developed across the ends of rod when it reaches

a ga
the point x  is B0l . Value of n, is
2 n


6. A uniform magnetic field B  B0ˆj exists in a space. A particle of mass and m and change q is projected
towards negative x–axis with speed v from the a point (d, 0, 0). The maximum value v for which
nBqd
the particle does not hit y–z plane is . Find n.
m
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 SECTION–II (ii) : (Maximum Marks : 24)
 This section contains SIX (06) questions. The answer to each question is a NUMERICAL
VALUE .
 For each question, enter the correct numerical value of the answer using the mouse and the on-screen
virtual numeric keypad in the place designated to enter the answer. If the numerical value has more
than two decimal places, truncate/round-off the value to TWO decimal places.
 Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +4 If ONLY the correct numerical value is entered;
Zero Marks : 0 In all other cases.

7. When magnetic flux through a coil is changed, the variation of induced current in the
coil with time is as shown in graph. If resistance of coil is 10 , then the total change
in flux of coil will be-

t(s)

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8. A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a as shown in the
figure. Both the cylinder and the cavity are infinitely long. A uniform current density J
flows along the length. If the magnitude of the magnetic field at the point P Is given by
N
 0 aJ , then the value of N is
12

9. A long insulated copper wire is closely wound as a spiral of 'N' turns. The spiral has inner
radius 'a' and outer radius 'b'. The spiral lies in the X–Y plane and a steady current 'I' flows
through the wire. The Z–component of the magnetic field at the center of the spiral is
4 0 NI  b 
n   . Find k.
k(b  a)  a 

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 
10. A particle of mass M and positive charge Q, moving with a constant velocity u1  4imsˆ 1 ,
enters a region of uniform static magnetic field normal to the x–y plane. the region of the
magentic field extends from x = 0 to x = L . After passing through this region, the particle

emerges on the other side after 10 milliseconds with a velocity u 2  2  
3iˆ  ˆj ms 1 . The

M
magnitude of the magnetic field is N 3Q . What is N?

11. A rectangular coil PQ has 2n turns, an area 2a and carries a current 2I, (refer figure). The
plan of the coil is at 60° to a horizontal uniform magnetic field of flux density B. The torque
on the coil due to magnetic force is xBnal. Find x

12. A conducting rod rotates with a constant angular velocity '' about the axis which passes
through point 'O' and perpendicular to its length. A uniform magentic field 'B' exists parallel
to the axis of the rotation. Then potential difference between the two ends of the rod is
nB 2 . Find n.

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 BEWARE OF NEGATIVE MARKING
PART B - CHEMISTRY
SECTION – I : (Maximum Marks : 24)
 This section contains SIX (06) questions.
 Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of these
fouroption(s) is (are) correct option(s).
 For each question, choose the correct option(s) to answer the question.
 Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +4 If only (all) the correct option(s) is (are) chosen.
Partial Marks : +3 If all the four options are correct but ONLY three options are chosen.
Partial Marks : +2 If three or more options are correct but ONLY two options are chosen,
both of which are correct options.
Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and
it is a correct option.
Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered).
Negative Marks : –2 In all other cases
1. The correct order of leaving group ability is/are:

(A) (B) CF3SO3Θ  CCl3SO3Θ

Θ
Θ Θ Θ
(C) C N  I (D) N H 2  O H

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2. Which will show geometrical isomerism:

(A) NO2 N = N – OR (B) N = N – SO3R

CH 3

(C) CH 3 CH CH C CH 2D (D)
CH 3 CH 2D CH 3

3. Which of the following statement are true ?

(A) Bridgehead halides are inert for both SN1 and SN2 reaction.
(B) The first step in both SN1 and El reaction is same.
(C) SN2 reaction proceed with total retention of configuration.
(D) E2 elimination are favoured by weak base
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4. ‘Spin only’ magnetic moment of Ni in [Ni(dmg)2] is same as that found in:
(A) Ni in [NiCl2(PPh3)2] (B) Mn in [MnO4]–
(C) Co in [CoBr4]2– (D) Pt in [Pt(H2O)2Br2]
5. Which of the following is/are correct order of crystal field splitting energy(0) for the following
complexes:-
(A) [Co(CN)6]3– > [Co(H2O)6]3+ (B) [Co(H2O)6]2+ < [Co(H2O)6]3+
(C) [Co(H2O)6]3+ >[Rh(H2O)6]3+ (D) [Co(NH3)6]3+ < [CoF6]3–
6. The correct statement(s) is/are:
(A) The PF6– ion can exist
(B) The NF6– ion does not exist
(C) N can form p-p bonds with itself and with other elements having small size and high E.N.
(D) The catenation tendency is weaker in N than P
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 SECTION II (i) (Maximum Marks: 18)
 This section contains SIX (06) questions.
 The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 TO 9,
BOTH INCLUSIVE.
 For each question, enter the correct integer corresponding to the answer using the mouse
and the on-screen virtual numeric keypad in the place designated to enter the answer.
 Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +3 If ONLY the correct integer is entered;
Zero Marks : 0 If the question is unanswered:
Neqative Marks : –1 In all other cases.
1. How many monochlorinated products of methylcyclohexane are optically active.
2. How many of the following compounds will give white precipitate with aqueous AgNO3.

(I) (II) (III) (IV)

Ph X[Number of substitution products

C2 H5OH including stereoisomer(all possible)]
3. CH 3 CH CH CH 3
CH 3 Br Y[Number of elimination products
including stereoisomer(all possible)]
Report your answer as (X + Y)

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4. Out of the following. How many have correct IUPAC naming:
(1) [Ni(CN)4]2– -Tetracyanonickel(II) ion
(2) [Pt(Py)4] [PtCl4] - Tetrapyridineplatinum(II) tetrachlorideplatinate(II)
(3) [Ni(dmg)2] - bis(dimethylglyoximato)nickel(II)
(4) [Fe(CO)5] - Pentacyanocarbonylferrate(O)
(5) K2[HgI4] - Potassium tetraiodidomercurate(II)
(6) [Pt(NH3)4]2+ -Tetraammineplatinum(IV) ion
(7) [Cu(gly)2] - Diglycinatecopper(II)
(8) K4[Fe(CN)6] - Potassium hexacyanidoferrate(II)
(9) [Pt(NH3)6] Cl4 - Hexaammineplatinum(IV) chloride
5. How many of the given complexes follow E.A.N, rule ?
(a) Fe(CO)5 (b) Co2(CO)8 (c) Fe(C5H5)2 (d) [K3Fe(CN)6]
(e) Fe(NO)2(CO)2 (f) [CoF6]4–

6. NH3, N2H4, HN3, PH3, AsH3, SbH3, BiH3

Number of molecules in which lone pair of electrons on the central atom is present in pure s-orbital.

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 SECTION–II (ii) : (Maximum Marks : 24)
 This section contains SIX (06) questions. The answer to each question is a NUMERICAL
VALUE .
 For each question, enter the correct numerical value of the answer using the mouse and the on-screen
virtual numeric keypad in the place designated to enter the answer. If the numerical value has more
than two decimal places, truncate/round-off the value to TWO decimal places.
 Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +4 If ONLY the correct numerical value is entered;
Zero Marks : 0 In all other cases.

7. Among the followings:

If X is the number of electrophiles and Y is the number of nucleophiles. Report your answer as
X Y
Θ
(i) CH 3 (ii) IΘ (iii) NO2 (iv) C H 3
 Θ
(v) N H3 (vi) Br  (vii) C l (viii) H


(ix) AlCl3 (x) CH3OH (xi) CH 3  C  O (xii) BH3

Θ
8. When the concentration of alkyl halide is tripled and the concentration of O H ion is reduced to half,
the rate of SN2 reaction increases by X times. Report your answer as 10 X.

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9. In the given reaction: Alkenes

Total number of alkenes(Including stereo isomers) formed will be

10. If excess of AgNO 3 solution is added to 100 mL of a 0.024 M solution of dichlorobis
(ethylenediamine)cobalt(III) chloride. How many moles of AgCl will be precipitated ?
11. What will be the theoretical value of 'spin only' magnetic moment when Fe(SCN)3 reacts with a
solution containing F– ions to yield a colourless complex ?
12. Which of the following on heating will produce an oxide of nitrogen.
(NH4)2SO4 , (NH4)2Cr2O7 , NH4NO3 , KNO3 , Pb(NO3)2, (NH4)2HPO4, NH4Cl, NH4NO2

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 BEWARE OF NEGATIVE MARKING
PART C - MATHEMATICS
SECTION – I : (Maximum Marks : 24)
 This section contains SIX (06) questions.
 Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of these
fouroption(s) is (are) correct option(s).
 For each question, choose the correct option(s) to answer the question.
 Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +4 If only (all) the correct option(s) is (are) chosen.
Partial Marks : +3 If all the four options are correct but ONLY three options are chosen.
Partial Marks : +2 If three or more options are correct but ONLY two options are chosen,
both of which are correct options.
Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and
it is a correct option.
Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered).
Negative Marks : –2 In all other cases

f (x)
1. If f(x+y) = f(x)f(y) for all x,y and f (0)  0 , and F(x)  then :
1   f  x 
2

2011 2011 2011 2010 2011

(A)  F  x  dx   F  x  dx
2010 0
(B)  F  x  dx   F  x  dx   F  x  dx
2010 0 0

2011 2010 2010

(C)  F  x  dx  0
2010
(D)   2F  x   F  x   dx  2  F  x  dx
2010 0

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 7 a b 1 
 d  . All the unknown numbers are distinct integers from the
2. Let matrix A  c  
3 e f 10 
set {2, 4, 5, 6, 8, 9} such that sum of entries of 1st row, 3rd row, 1st column and 4th column are equal
to k, then
(A) a + b + c = k + 1 (B) k = 18 (C) ef = d (D) c + d = k – 2
3. A is a square matrix of order 3. A, A , A all has the same value of determinant. Also (adj A) = AT
–1 T

then
(A) det(A–1) = 1
(B) (ABAT)2 = A2B2(AT)2, where B is a 33 matrix
(C) det {(ABAT)1000} = (detB2) det(B) = 0, 1 or –1
(D) (ABAT)–1 = AB–1AT, if B is an invertible 33 matrix
4. If A, B are two 4 × 4 matrices with real entries such that |A – B|  0 and they satisfy the equations:
A2 – 2B + I4 = 04 ……(1)
B2 – 2A + I4 = 04 ……(2)
Then :
(A) |A + B| = 2 (B) |A + B| = 16 (C) A  I4 (D) B  I4
1
5. Let  x sin x.sec x dx 
3

2
 x.f  x   g  x    k , then:
(A) f (x)   1,1 (B) g(x) = sin x has 6 solution for x  [, 2 ]

(C) g '  x   f  x  , x  R (D) f(x) = g(x) has no solution

2

6. Let f(2 – x) = f(2 + x) and f(4 – x) = f(4 + x). Function f(x) satisfies f  x  dx  5 . If

0

50

 f  x  dx  I . Then
0
I can be greater than

(A) 5 (B) 8 (C) 11 (D) 14

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 SECTION II (i) (Maximum Marks: 18)
 This section contains SIX (06) questions.
 The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 TO 9,
BOTH INCLUSIVE.
 For each question, enter the correct integer corresponding to the answer using the mouse
and the on-screen virtual numeric keypad in the place designated to enter the answer.
 Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +3 If ONLY the correct integer is entered;
Zero Marks : 0 If the question is unanswered:
Neqative Marks : –1 In all other cases.

 x 
3/2
6
  x 4  x 2
1. x  x4  x2  2x 4  3x 2  6 dx   C where C is constant, then find the value
6

18

of        .

1  1 1 1 
2. Find the value of nlim 1    ........  .
 n 2 3 n


2
dx
3.   sin x  cos x  2
 sin x cos x  sin x cos x equals
4

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4. Let f : RR be a continuous function and f(x) = f(2x) is true x  R . If f(1) = 3, then the value of
1

 f  f  x  dx is equal to
1

2i  j
5. Let An = [aij] be a square matrix of order 3, where a ij  for all i, j, 1  i, j  3 . Then
32n
lim Tr(3A1  32 A 2  33 A3  ..........  3n A n ) is equal to
n 

6. If A and B are square matrices of the same order such that |A| = |B| = 1 and A(adj A + adj B) = B.
Then the value of |A + B| is equal to:
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 SECTION–II (ii) : (Maximum Marks : 24)
 This section contains SIX (06) questions. The answer to each question is a NUMERICAL
VALUE .
 For each question, enter the correct numerical value of the answer using the mouse and the on-screen
virtual numeric keypad in the place designated to enter the answer. If the numerical value has more
than two decimal places, truncate/round-off the value to TWO decimal places.
 Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +4 If ONLY the correct numerical value is entered;
Zero Marks : 0 In all other cases.

 /4
3x 9  3x 5  7x 3  7x  2
7. 
 /4
cos 2
x
dx is

3
1 1  2x  k2
8. If  0 1 x2
sin
  2 
1 x 
dx 
9 then k is equal to

1
 x2 x3 x 2n 
9. Let I n   
1
x 1  x 
2

3
 ....... 
2n
dx where n  N . If nlim

I can be expressed as a rational
 n

p p
number in the lowest form, then find the value of is .
q q
Space For Rough Work

23/26
 1


10. A continuous real function 'f' satisfies f(2x) = 3 f(x)  R. If f (x)dx  1 , then compute the value
0


of definite integral f (x)dx
1

4n
n
lim 
11. The value of
  is equal to
n  2
r 1 r 3 r 4 n

n 1
n2
12. f(x) is a continuous function for all real values of x and satisfies n
f (x)dx 
2
 n  I , then

 f  x  dx is equal to
3

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Space for Rough Work
D. Marking scheme :
12. For each question in Section-I , you will be awarded 4 marks if you darken the bubble corresponding to the correct
answer and zero mark if no bubbles are darkened.In all other cases minus two (–2) mark will be awarded.
Partial Marks : +1 For darkening a bubble corresponding to each correct option, Provided NO incorrect option
is darkened.
 For example, if (A), (C) and (D) are all the correct options for a question, darkening all these three will result in
+4 marks; darkening only (A) and (D) will result in +2 marks; and darkening (A) and (B) will result in –2 marks,
as a wrong option is also darkened.
13. For each question in Section-I(i) , you will be awarded 3 marks if you darken all the bubble(s) corresponding to
only the correct answer(s) and zero mark if no bubbles are darkened. In all other cases minus two (–1) mark will
be awarded.
14. For each question in Section-I(ii) , you will be awarded 4 marks if you darken all the bubble(s) corresponding to
only the correct answer(s) and zero mark if no bubbles are darkened. No negative marks will be awarded for incorrect
answers in this section.
15. Take g = 10 m/s2 unless otherwise stated.

Appropriate way of darkening the bubble for your answer

to be evaluated 1 4 2 0 0 0 2 2
0 0 0 0 0
a The one and the only acceptable
1 1 1 1 1 1 1
a Part darkening
2 2 2 2 2
3 3 3 3 3 3 3 3
a a Darkening the rim 4 4 4 4 4 4 4
Answer will not be evaluated - 5 5 5 5 5 5 5 5
a Cancelling after darkening no marks, no negative marks 6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7
a Erasing after darkening 8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9
Figure-1 : Correct way of bubbling for valid answer and Figure-2 : Correct Way of Bubbling your Form Number
a few examplex of invalid answers on the ORS. (Example Form Numebr : 14200022)
Any other form of partial marking such as ticking or
crossing the bubble will be invalid

Name of the Candidate Form Number

I have read all the instructions and shall abide by I have verified all the information filled in by the Candi-
th em . d a te .

Signature of the Candidate Signature of the Invigilator

Your Hard Work Leads to Strong Foundation