1. science
2. physics

eODEm
P2-15-1
PAPER-2
552701
Maximum Marks: 240
Time: 3 Hours
READ THE INSTRUCTIONS CAREFULLY
GENERAL:
1. This sealed booklet is your Question Paper. Do not break the seal till you are told to do so.
2.
The question paper CODE is printed on the left hand top corner of this sheet and the right
hand top corner of the back cover of this booklet.
Use the Optical Response Sheet (ORS) provided separately for answering the questions.
3.
4.
The ORS CODE is printed on its left part as well as the right part. Ensure that both these
codes are identical and same as that on the question paper booklet. If not, contact the
invigilator.
5. Blank spaces are provided within this booklet for rough work.
6. Write your name and roll number in the space provided on the back cover of this booklet.
7.
After breaking the seal of the booklet, verify that the booklet contains 32 pages and that all
the 60 questions along with the options are legible.
QUESTION PAPER FORMAT AND MARKING SCHEME:
The question paper has three parts: Physics, Chemistry and Mathematics. Each part has
three sections.
9. Carefully read the instructions given at the beginning of each section.
10. Section 1 contains 8 questions. The answer to each question is a single digit integer ranging
from 0 to 9 (both inclusive).
Marking scheme: +4 for correct answer and 0 in all other cases.
11. Section 2 contains 8 multiple choice questions with one or more than one correct option.
Marking scheme: +4 for correct answer, 0 if not attempted and -2 in all other cases.
12. Section 3 contains 2 "paragraph" type questions. Each paragraph describes an experiment,
a situation or a problem. Two multiple choice questions will be asked based on this
paragraph. One or more than one option can be correct.
Marking scheme: +4 for correct answer, 0 if not attempted and -2 in all other cases.
OPTICAL RESPONSE SHEET:
13. The ORS consists of an original (top sheet) and its carbon-less copy (bottom sheet).
14. Darken the appropriate bubbles on the original by applying sufficient pressure. This will leave
an impression at the corresponding place on the carbon-less copy.
15. The original is machine-gradable and will be collected by the invigilator at the end of the
examination.
16. You will be allowed to take away the carbon-less copy at the end of the examination.
17. Do not tamper with or mutilate the ORS.
18. Write your name, roll number and the name of the examination center and sign with pen in
the space provided for this purpose on the original. Do not write any of these details
anywhere else. Darken the appropriate bubble under each digit of your roll number.
~
'Please see the last page of this booklet for rest of the instructio"ns ...
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PART I :.PHYSICS
SECTION 1 (Maximum Marks: 32)
•
•
•
o
This section contains EIGHT questions
The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both
inclusive
For each question, darken the bubble corresponding to the correct integer in the ORS
Marking scheme:
+4 If the bubble corresponding to the answer is darkened
o In all other cases
,Q:('" A large spherical mass M is fIxed at one position and two identical point masses m are kept
on a line passing through the centre of M (see figure). The point masses are connected by a
rigid massless rod of length l and this assembly is free to move along the line connecting
them. All three masses interact only through their mutual gravitational interaction. When
the point mass nearer to M is at a distance r = 3l from M, the tension in the rod is zero for
1.
m = k ( M ). The value of k is
288
The energy of a system as a function of time t is given as E(t)=A2exp(-at),
a = 0.2
S-I.
where
The measurement of A has an error of 1.25%. If the error in the measurement of
time is 1.50%, the percentage error in the value of E(t) at t = 5 s is
~
The densities of two solid spheres A and B of the same radii R vary with radial distance r as
PA(r)=k(~)
and
PB(r)=k(~J,
respectively, where
k
is a constant. The moments of
inertia of the individual spheres about axes passing through their centres are IA and IB,
respectively. If IB = ~,
IA
the value of n is
10
,
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PHYSICS
Four harmonic waves of equal frequencies and equal intensities 10 have phase angles)/, 11/3,
211/3 and/When
they are superposed, the intensity of the resulting wave is n1o. The value of
nis
~
For a radioactive material, its activity A and rate of change of its activity R are defined as
dN and R = - dA, where N(t) is the number of nuclei at time t. Two radioactive
dt
dt
sources P (mean life ,) and Q (mean life 2,) have the same activity at t = O. Their rates of
A
=-
p
change of activities at t = 2, are Rp and RQ, respectively. If R
RQ
= n,
e
then the value of n is
A monochromatic beam of light is incident at 60° on one face of an equilateral prism of
refractive index n and emerges from the opposite face making an angle B(n) with the normal
(see the figure). For n =.J3 the value of B is 60° and dB
dn
= m. The value of m is
2-
",(),(I
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(,0
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Q;(,\J
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(;
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PHYSICS
In the following circuit, the current through the resistor R (= 2Q) is I Amperes, The value
of! is
20
ii
An electron in an excited state of Li2+ ion has angular momentum 3h/2Jr, The de Broglie
wavelength of the electron in this state is pJrao (where Go is the Bohr radius), The value
ofp is
2-
r'V' .
('
I ,
,v \
'i
(1
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-----------------------PHYSICS
SECTION 2 (Maximum Marks:
•
•
•
•
32)
This section contains EIGHT questions
Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of
these four option(s) is(are) correct
For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS
Marking scheme:
+4 If only the bubble(s) corresponding to all the correct option(s) is(are) darkened
o If none of the bubbles is darkened
-2 In all other cases
Two spheres P and Q of equal radii have densities PI and P2' respectively. The spheres are
connected by a massless string and placed in liquids L1 and L2 of densities 0"1 and 0"2 and
viscosities 171 and 172 , respectively. They float in equilibrium with the sphere P in L1 and
sphere Q in L2 and the string being taut (see figure). If sphere Palone in(LJhas terminal
~
_~__~-~-_.
~
---,-.~~.
velocity Vp and Q alone in L1 has terminal velocity VQ ' then
.-
..
..
--
...~.,
-
,_.,,,.,'----'-
-
.."",
\
If
~
(B)
I Vp I
--=-
172
(C)
~
IVQ
I
171
In terms of potential difference V, electric current I, permittivity
speed of light c, the dimensionally correct equation(s) is(are)
¥{
",1' = So V'
(B) sol = '" Vi'.
(fl('
t
= So' V
&0'
permeability
J.10
and
(D)
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PHYSICS
Consider a uniform spherical charge distribution of radius ~ centred at the origin O.
In this distribution, a spherical cavity of radius R2, centred at P with distance
OP = a
=~ -
r
R2 (see figure) is made. If the electric field inside the cavity at position
is
--'> --'>
E (1') , then the correct statement(s) is(are)
'
I '\
.\
.f
~"~"I,
I,
f,_.. ~\
'"
V
....
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-
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(A)
E
is uniform, its magnitude is independent of R2 but its direction depends on
(B)
E
is uniform, its magnitude depends on R2 and its direction depends on
(C)
E
is uniform, its magnitude is independent of a but its direction depends on
£p('
E
is uniform and both its magnitude and direction depend on
r
r
)<;,
a
)<7
a
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V:: ML!-T-31).
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(
PHYSICS
1
G Instrainplottingon thestressy-axisversusandstrain
curves for two materials
and
a student by mistake puts
stress on the x-axis as shown in the figure. Then the correct
P
Q,
statement(s) is(are)
- ---_._--_._--~
Stress
y~ > ~
tf,)
P has more tensile strength than Q
\.E:E( P is more ductile than Q
(C) P is more brittle than Q /:>..
(D) The Young's modulus of P is more than that of Q ~
--1
1
ytS5=U9hWO~
J-K- '
b \c ~
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Q.13
Q.14
f
PHYS::S ]
. t f' fl 'd f
d'
dl.
A sp h enca 1 b0 dy 0 f ra d'lUS R conSlS
S 0 a Ul 0 constant enslty an IS m eqUl 1 rlUm
under its own gravity. If per) is the pressure at r(r < R), then the correct option(s) is(are)
(A)
P(r=O)=O
(C)
P(r
P(r
= 3R/5)
= 2R/5)
_~
21
(B)
P(r
P(r
= 3R/4) 63
= 2R/3) = 80
(D)
P(r
P(r
= R/2) 20
= R/3) = 27
li-
A parallel plate capacitor having plates of area S and plate separation d, has capacitance Cl
in air. When two dielectrics of different relative permittivities (£1 = 2 and £2 = 4) are
introduced between the two plates as shown in the figure, the capacitance becomes C2• The
C2
1) "
ratio C is
1
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(B)
5/3
(C)
7/5
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PHYSICS
Q.15 An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston
(as shown in the figure). Initially the gas is at temperature TI, pressure PI and volume VI
and the spring is in its relaxed state. The gas is then heated very slowly to temperature Tz,
pressure Pz and volume Vz. During this process the piston moves out by a distance x.
Ignoring the friction between the piston and the cylinder, the correct statement(s) is(are)
.................
. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . ..
. .. ... .. . .. .. . .. .
. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . ..
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
. . . .. . . .. . .... .. .
. . . .. . .. . . . .. . . . .
. . .. . ... ... .. .. ..
.. .. . .. . .. .. . . . ..
. . . ... .. . .. .~
........................ '.
..
. . . . . . . . . . . . . . . . . . . . . . ... . . . . ... .
,
~\
(A) If Vz= 2VI and Tz = 3Tv then the energy stored in the spring is ~ PI VI
~
If Vz= 2VI and Tz = 3TI, then the change in internal energy is 3PI VI
(C)
If Vz= 3VI and Tz = 4TI, then the work done by the gas is 'i PI VI
3
)0
".
(D) If Vz= 3 VI and Tz = 4TI, then the heat supplied to the gas is .!2. PI VI
6
k'lt -1((~
_.
.- PtA
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4
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10/32
"
PHYSICS
A fission reaction is given by 2~~U~ l:~Xe + ~:Sr + x + y, where x and yare
1
two particles.
Considering 2~~U to be at rest, the kinetic energies of the products are denoted by Kxe, Ksr,
Kx (2 MeV) and Ky (2 MeV), respectively. Let the binding energies per nucleon of 2~~U, l:~Xe
and ~~Sr be 7.5 MeV, 8.5 MeV and 8.5 MeV, respectively. Considering different conservation
laws, the correct option(s) is(are)
y1'
x = n, y
= n, KSr= 129 MeV, KXe = 86 MeV
..w)
x = p, y
= e-, KSr= 129 MeV, KXe = 86 MeV
J"
.JP5
x
= p, y = n, KSr= 129 MeV, KXe = 86 MeV
(D) x
= n, y = n, KSr= 86 MeV, KXe = 129 MeV
Space for rough work
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2. t{- .., Q
...
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11/32
PHYSICS
SECTION 3 (Maximum Marks:
•
•
•
•
•
16)
This section contains TWO paragraphs
Based on each paragraph, there will be TWO questions
Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of
these four option(s) is(are) correct
For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS
Marking scheme:
+4 If only the bubble(s) corresponding to all the correct option(s) is(are) darkened
o If none ofthe bubbles is darkened
-2 In all other cases
Space for rough work
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p ..
ct-£:
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BIQ.
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12/32
PHYSICS
PARAGRAPH
1
In a thin rectangular metallic strip a constant current I flows along the positive x-direction, as
shown in the figure. The length, width and thickness of the strip are l, wand d, respectively.
->
A uniform magnetic field B is applied on the strip along the positive y-direction. Due to this, the
charge carriers experience a net deflection along the z-direction. This results in accumulation of
charge carriers on the surface PQRS and appearance of equal and opposite charges on the face
opposite to PQRS. A potential difference along the z-direction is thus developed. Charge
accumulation continues until the magnetic force is balanced by the electric force. The current is
assumed to be uniformly distributed on the cross section of the strip and carried by electrons .
--1.
••••
1--
I
L_
Q.17
Q
P
z
Consider two different metallic strips (1 and 2) of the same material. Their lengths are the
same, widths are WI and W2 and thicknesses are dI and d2, respectively. Two points K and M
are symmetrically located on the opposite faces parallel to the x-y plane (see figure). VI and
V2 are the potential differences between K and M in strips 1 and 2, respectively. Then, for a
given current I flowing through them in a given magnetic field strength B, the correct
statement(s) is(are)
(A) If WI = W2 and dI = 2d2, then V2 = 2VI
~..B(If WI = W2 and
dI = 2d2, then V2 = VI
(C) If WI = 2w2 and dI = d2, then V2 = 2VI
(D) If WI = 2W2 and dI = d2, then V2 = VI
Q.18 Consider two different metallic strips (1 and 2) of same dimensions (length l, width wand
thickness d) with carrier densities nI and n2, respectively. Strip 1 is placed in magnetic field
BI and strip 2 is placed in magnetic field B2, both along positive y-directions. Then VI and V2
are the potential differences developed between K and M in strips 1 and 2, respectively.
Assuming that the current I is the same for both the strips, the correct option(s) is(are)
(A) If BI = B2 and nI = 2n2, then V2 = 2VI
(B) If BI = B2 and nI = 2n2' then V2 = VI
(C) If BI = 2B2 and nI = n2, then V2 = O.5VI
(D) If BI = 2B2 and nI = n2, then V2 = VI
Space for rough work
** 1
13/32
\
.
PHYSICS
PARAGRAPH 2
Light guidance in an optical fiber can be understood by considering a structure comprising of thin
solid glass cylinder of refractive index n] surrounded by a medium of lower refractive index n2.
The light guidance in the structure takes place due to successive total internal reflections at the
interface of the media n] and n2 as shown in the figure. All rays with the angle of incidence i less
than a particular value im are confined in the medium of refractive index ~. The numerical
aperture (NA) of the structure is defined as sin i",.
d
_
"
l
Air
Cladding
\
Q.19
For two structures namely 8] with ~ = fi5/4 and n2 = 3/2, and 82 with I~ = 8/5 and
n2 = 7/5 and taking the refractive index of water to be 4/3 and that of air to be 1, the correct
option(s) is(are)
/
JA) NA of 8] immersed in water is the same as that of 82 immersed in a liquid of refractive
. d
16
~
m ex
~
lIP'
3",15
~~',
..t);
(B)
immersed in liquid of refractive index
immersed in water
(ci
/
',.
6
.J15
]s the same as that of 82
'r
NA of 8] placed in air is the same as that of 82 immersed in liquid of refractive index
4
\
.J15 .
.',', \' '
A
(D)
",
NA of 8] placed in air is the same as that of 82 placed in water "I
"
Q.20 If two structures of same cross-sectional area, but different numerical apertures NA1 and
NA2 (NA2 < NA]) are joined longitudinally, the numerical aperture of the combined
structure is
't I' '
"
".
NA1NA2
(A)
-.\~
** 1
(B)
NA] +NA2
8
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~fI • "
,
~,J ....
v
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j:
,
"('''1
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-
(C)
NA1 +NA2
JlJ{
NA]
NA2
,t \'
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h
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f
y(
END OF PART I : PHYSICS
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SECTION 1 (Maximum Marks: 32) .
vi
•
•
This section contains EIGHT questions
The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both
inclusive
For each question, darken the bubble corresponding to the correct integer in the ORB
•
•
Marking scheme:
+4 If the bubble corresponding to the answer is darkened
o In all other cases
"
Q.21 The number of hydroxyl group(s) in Q is
H+
aqueous dilute KMn04 (excess)
----.
p
heat
Q.22
• Q
OOC
Among the following, the number ofreaction(s) that produce(s) benzaldehyde is
o
CMO
j
CO, HCI
! J
V
Anhydrous AICI3/CuCI
!
\lJ/
COCI
I
()
H
_2__
//
(YC02Me
V
n
~
(."
Pd-BaS04
DIBAL-H
,
-ro-lu-e-n-e,---7S-0-C--'
H20
,(3l
,~,
"\
, "\ If
'o\"\
W .• /
.-
\\ l:;
..)',
'
In the complex acetylbromidodicarbonylbis( triethy lphosphine )iron(II), the
\
Fe-C bond(s) is
"')
.••..
~
/
'
(.J
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15/32
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( CHEMISTRY ]
eYt
.:J.
/
V
-le)
/
3
tt? complex ions, [Co(NH2.CH2-CH2.NH2)2Cl~+,
[CrC12(C2.0r.,4)21
[Fe(H20MO~92r,~
-,
3
[Fe(NH3MCN)4r, [Co(NH2-CH2.CH2-NH2MNH~9l12+and [CO(NH~20)Cl12+, the number of
complex ion(s) that show(s) c~isomerism
is -6
-+'3
Thre_~_moles of B2H6 are completely reacted with methanol. The number of moles of boron
containing produCt formed is ~ ~
J26
The molar conductivity of a solution of a weak acid HX (0.01 M) is 10 times smaller than the
molar conductivity of a solution of a weak acid HY (0.10 M). If A~_ '" A~_, the difference in
II
"
their pKa values, pKa (HX) - pKa (HY), is (consider degree of ionization of both acids to
be «1)
fc{Y\
9.-27 A closed vessel with rigid walls contains
-
\
/
1 mol of 2~~Uand 1 mol of air at 298 K. Considering
complete decay of 2~~Uto 2~~Pb,the ratio of the final pressure to the initial pressure of the
system at 298 K is
~
.~.
~
In dilute aqueous H2S04, the complex diaquodioxalatoferrate(II) is oxidized by MnO~. For
this reaction, the ratio of the rate of change of [H+]to the rate of change of [MnO~] is
3
i,/
-- L1
-j r)(
-.
cJ.
Space for rough work
-.J
iIT] 1Ll1 I Ij
()
(!c ~
QV\---------
C\
I ---~J
L.
r
r' ~
l./I,
C. vi,
f.'O'
t,\
Co'
i
-
-I ,-'
(F e (~t~):.((! r.\ .:;
\
Yi=$5
--.---:.,
I. ,?;>f
l
)
0f
-Li ,~'
,
(Q."
4'
,
MIn 11
8
'20C
...--;:;
' . ',i
{•. f'
I
I
.,
v;/
~.. I),
16/32
'
[ CHEMISTRY]
SECTION 2 (Maximum Marks: 32)
•
•
•
•
\
I
\
This section contains EIGHT questions
Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of
these four option(s) is(are)correct
For each question, darken the bubble(s) corresponding to all the correct option(s) in the
ORS
Marking scheme:
+4 If only the bubble(s) corresponding to all the correct option(s) is (are) darkened
o If none of the bubbles is darkened
-2 In all other cases
In the following reactions, the product S is
i. 03
----
Q.30
.•• R ----.
S
The major product ~Jg!1Q!\:iQg react0ns i!3
o
CH2=CH-CH3, H+
--------...
T
-----~.
U
(A)
\.
~CH3
.
l
1
0
<£
H3C CH3
era,a'H
1
0
~
0-0
(C)
(-(--c
high pressure, heat
~
0-0
(-
([)
radical initiator, O2
~CH2
I~
CH2
>a
(D)
~a'a,H
I
0
17/32
[ CHEMISTRY]
In the following reactions,
(Y NH
V
2
NaN02, HCI
(A)
\..jJ?,{
(JJ(
(D)
Q.33
~OH
~,NaOH
w
• V
aOc
The cor~tstatement(s)
Q.32
the major product W is
regarding,
(i) HCIO, (ii) HCIOz, (iii) HCI03 and (iv) HCI04, is (are)
The number of CI=O bonds in (ii) and (iii) together is two
The number oflone pairs of electrons on CI in (ii) and(iii)
The hybridization
of CI in (iv) is
Amongst (i) to (iv), the strongest
...
,'J
B'3+ F
1 ,e
together is three
Sp3
acid is (i)~
The pair(s) of ions where BOTH the ions are precipitated
of dilute HCI, ~s(are)
(A) Baz+, ZnzX,
(B)
~
3+
}17
upon passing, HzS gas in presence
(D'''' H g,Z+ B.13+
'VI
Space for rough work
\
1
()
II
f.
,
.. 'C' -
** 1
18/32
I
J
[ CHEMISTRY]
Q.34 Under hydrolytic conditions, the compounds used for preparation of linear polymer and for
chain termination,
respectively, are './
..
(A) CHaSiCla and Si(CHa)4 YJ
\.{B) (CHa)2SiC12and (CHa)aSiCl
--_._._
(C)
'
(D) SiC14 and (CHahSiCl f<
(CHa)2SiC12and CHaSiCla
~'
When O2 is adsorbed on a metallic surface, e~
t~n;;(~r occurs from the metal to O2,The
• k TRUE stateme~t(s) regarding this adsorpti"is(are) .
At&
::..\~</----.t.~:@) O2 IS physlsorbed;<
ckv,V\,<.
{d3) heat IS released
. \~()
v<.<S)
occupancy of
of O2 is increased
ff;p
Jd
bond length of O2 is increased
One mole of a monoatomic real gas satisfies the equation p(V - b) = RT where b is a
constant. The relationship of interatomic potential V(r) and interatomic distance r for the
gas is given by
VCr)
(A)
VCr)
(B)
o
o~---r
r
!...I',
VCr)
VCr)
(C)
(D)
of---L----
o
r
r
Space for rough work
I
..,
_6'-)'(-'0
"-
\
,,
\,
t
I!)'
i'
.,
.1
r. , I:'
!
v.
"
.
It-I
I '!
C' {"'I' ".'
I
(I
** 1
19/32
[CHEMIST!YJ
SECTION 3 (Maximum Marks: 16)
•
•
•
o
•
This section contains TWO paragraphs
Based on each paragraph, there will be TWO questions
Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of
these four option(s) is (are) correct
For each question, darken the bubble(s) corresponding to all the correct option(s) in the
ORS
Marking scheme:
+4 If only the bubble(s) corresponding to all the correct option(s) is(are) darkened
o
-2
If none of the bubbles is darkened
In all other cases
PARAGRAPH 1
In the following reactions
Pd-BaS04
---~.
CsHs
-------.
x
I H20
~ HgS04, H2S04
i. EtMgBr, H20
CsHsO
• Y
ii. H+, heat
Q.37
~-c::-
Compound X is
OH
°
(A)
~CH'
(B)
d'CH,
cd
(J0H
1
0
(D)
U
~
(B)
(C)
~CH'
0
~)
\
C
CHO
(j'/I
@J-
~CH3
.
dCH'
gI) 1: 1CH,
\
C
(::
a
( •••.c
0
oH
C2:r.c \
Space for rough work
o
** 1
CY\
\\
CH2
1
I
c-
Q.38. The major compound Y is
(A) ~CH3
C
0-~b~C
I
C
\ C
Et
20/32
[ CHEMISTRY]
PARAGRAPH 2
When 100 mL of 1.0 M HCI was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at
constant pressure, a temperature increase of 5.7 °C was measured for the beaker and its
contents (Expt. 1). Because the enthalpy of neutralization of a strong acid with a strong base is a
constant (-57.0 kJ morl), this experiment could be used to measure the calorimeter constant.
In a second experiment (Expt. 2), 100 mL of 2.0 M acetic acid (Ka = 2.0 x 10-5) was mixed with
100 mL of 1.0 M NaOH (under identical conditions to Expt. 1) where a temperature rise of 5.6 °C
was measured.
(Consider heat capacity of all solutions as 4.2 J g-l K-l and density of all solutions as 1.0 g mL-I)
Q.39 Enthalpy of dissociation (in kJ mol-I) of acetic acid obtained from the Expt. 2 is
ci
1.0
(B)
10.0
.
(C)
24.5
(D) 51.4
(C)
5.0
(D)
The pH of the solution after Expt. 2 is
QAO
(A)
rI
2.8
4.7
7.0
END OF PART II: CHEMISTRY
~XL{'2
A
~;4
--- ~
"J
:5"'1
~ (f'I)
>(
j
2ifVCj
(;
j
-
-
5-~jL
b,
!"
.r
"
I~
I~J
r~l;
J ~ el-
tJ.3
~
'.
f tl"" ';
-I
I L','" . •'.'
1
It,
).
t
-*,,'
2_
. --'
,-
I U -I
I ( .'- 2.
y
!r
, 1"
.
V!. 1 l•
/ j
** 1
21/32
PART III : MATHE~ATICS
SECTION 1 (Maximum Marks: 32)
•
•
This section contains EIGHT questions
The answer to each question is a SINGLE DIGIT INTEGER ranging from to 9, both
inclusive
For each question, darken the bubble corresponding to the correct integer in the ORS
Marking scheme:
+4 If the bubble corresponding to the answer is darkened
o In all other cases
°
•
•
, I I,
e.,
Q.41
For any integer
A. •.•",_.
1-
k, let a = cos(k; ) + sin (k; lwhere i = ~
. The value of the expression
i
k
V
u
~
paM' -a,1
I/ a
4k-1
-
"j.
,/
/
Q.:43
~
-.j..
.-tJ
a4k-Z!
k=l
Q.42
~
~~
Suppose that
ratio of the
6 : 11 and the
A.P. is
all the terms of an arithmetic progression (A.P.) are natural numbers. If the
sum of the first seven terms to the sum of the first eleven terms is
seventh term lies in between 130 and 140, then the common difference of this
~
The coefficient of x9 in the expansion of (1+ x)(l + xZ)(l + x3)
(1+ x100) is
..•
Space for rough work
~
._----
.~if
-
GI,O
l:f
'7
i
.
(2-a.
+
:. . ~
f
(
~
j'
llt,
\
~
!:)
('H
6J
l2a.
,-
,1\
oj i
dJ
Jtfd
'f'
,.
J
6 ')
(I-Ix)
C lb) (t.i'X/
:;'\/
(I~';(~)
.~
~,)
I..'':( ) '.1:( /
i.iCJ
( H
•.;':)I If./
f)
"/(0; ) (
.. ~1:.."
X) ( ,I.r I .
,
'f
\-11( /
-',
H XI )
Cit
"
)-\ ( ,s '\
(~~+~r
..
** 1
1tbtflJ
.'
( '4?:
c'.
C'
\
! (
I~
"I)
22/32
I
..• -I
i
(MATHEMATICS) .
Q.44
i
x2
Suppose that the foci of the ellipse - + - = 1 are (f1' 0) and (f2' 0) where ft > 0 and (2 < O.
9
5
Let ~ and P2 be two parabolas with a common vertex at (0, 0) and with foci at (ft, 0) and
(2(2,0), respectively. Let T" be a tangent to ~ which passes through (2(2,0) and T2 be a
tangent to P2 which passes through (ft, 0). If m1 is the slope of T" and m2 is the slope of
T2, then the value of (~~ + m~J is
Q.45
2-
Let m and n be two positive integers greater than 1. If
lim (_eCOs(a_n) -eJ= _(~)
a.->O
am
2
then the value of m is
n
12_\
Q.46
If
a
(12+ 9x J dx
2
= J1(e9X+3tan-l
X)
1+x2
o
where tan -1 x takes only principal values, then the value of (lOgell + a
. )- i-
t
(
,.•..
rr
(,
(0,'
~l'
\
/-j
/
j
\,...
-
~.•.-
']
..•.
~v.
I
,
/"
Q!\
n ),
Y'2
/
"
-le
,./
,
I
~
__
,.....
\
K .- j
r
1- 3:) is
-1-I,¥,
_., --.-r' - -.
c,.C.(
** 1
C~'J
c<
(
..,'
f\~)
'I .
r
;(r\
J
23/32
(MATHEMA TI CS )
Let [: IR ~ IR be a continuous o!<!Junction, which vanishes exactly at one point and
Qd7
1
x
[(1)=2'
Suppose that
F(x)
x
= f[(t)dt
for all xE[-1,2]
and G(x)= ftl [(f(t»Idt
-1
xE[-I,Z].
Q.48
If limF(X)=~,thenthevalUeOf[(!)iS
x ..••1 G(x)
14
Suppose that
vector
...•
S
p ", q :rd Tit are. three
s along
(-t-t-t
-t-t--t
are x, y and z, respectively, then the
Space for rough work
e'G,
-;
-
tl ..~";
~t'Fj •
"';1f
-t--t-t
(-p-q+r)
0
--,'
•.
3
non-coplanar vectors in 1R . Let the components of a
and
(p-q+r)
value of 2x + y + z is
•~
1-
2
~, ~ and' T be 4, 3 and 5, respectively. If the components of this vector
along -p+q+r),
..,.,......,
for all
-1
..
t;;.
:.;$1
(
,
-
F -i L
,
.-1 . t )"
p
I.~
Il
n., ".
.-
,,
\.;
8.jb-2
-2
._
"
5 .-
4~ 13j
A
f'\
-\ O~
f
r
I
I
.
,.( 7:
(I.
1
/
, \.
.I,
I,..
•.
-,2-1--/0
q-
- (P
** 1
'.
2
-I Lf~'
24/32
(MATHEMATICS 1
SECTION 2 (Maximum Marks: 32)
•
This section contains EIGHT questions
•
Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of
these four option(s) is(are) correct
•
For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS
•
Marking scheme:
+4
If only the bubble(s) corresponding
If none of the bubbles is darkened
o
-2
Q.49
In all other cases
Let
S
2
ax
-
be the set of all non-zero
real
x + a = 0 has two distinct real roots
Which of the following intervals
(B)
Q.50
to all the correct option(s) is(are) darkened
and
Xl
satisfying
X2
the quadratic
equation
I Xl
the inequality
x21 < 1.
-
S?
a subset(s) of
I
(1J5' 2"1)
where the in~erse trigonometric
functions
35,0)
(0')
Yl
C
If a=3sin-I
(-
is(~
a such that
numbers
6
1)
and ,8=3COS-I(~).
take only
the principal values, then the correct option(s) is(are)
(A)
cos,8 > 0
(B)
sin,8 < 0
cos(a + ,8) > 0
(C)
cosa < 0
(D)
Space for rough work
.
,
I.
:. i
I
"'. ').
--I
't{!~~2
i.
;:
~.
j
';I
/...7
''-
'
J
f,.
1
1:!-
-I
,,':'.
3
<'.
C(
.f','
1
S-
<-.
'\.
I
I ,
-J',
,
.
'- ~
-l
**
L
()\-
! ,lc!!J
\6 .
(,1 i..
.1 , '
I
A
,)
.-
/
,/.- .
,I
~-
..
~,
25/32'
[MATHEMATICS]
Q.51
Let El and E2 be two ellipses whose centers are at the origin. The major axes of El and E2
2
lie along the x -axis and the y -axis, respectively. Let S be the circle x + (y _1)2 = 2. The
straight line x + y = 3 touches the curves S, E1 and E2 at P, Q and R, respectively.
Suppose that PQ = PR =
2.J2 . If el
3
then the correct expression(s) is(are)
(A)
Q.52
2
2
e1 +e2
43
=-
and e2 are the eccentricities of El and E2, respectively,
(D)
(C)
(B)
40
Consider the hyperbola H: x2 -l = 1 and a circle S with center N(xz,O). Suppose that
Hand S touch each other at a point P(Xl'Yl) with Xl > 1 and Yl > O. The common tangent to
Hand S at P intersects the x -axis at point M. If (1, m) is the centroid of the triangle !1PMN,
then the correct expression(s) is(are)
(A)
. dl
1
= 1 - -2 for
dXl
3xl
Xl
>1
(B)
(C)
dl
1
-=1+-2
for
dX1
3xl
Xl
>1
(D)
dm
1
dYl
3
-=-
for Yl >0
Space for rough work
• I
•
~ \
n
",'
,
,'\
•
.\
'(vi /(
-I
I
J
..
~)
,
'-
.•..i,
1
oJ-
')'1(.\
',)(1.,\,1'
'-'
'l
o
I
I..'.
~.I
.~
'I
_i.
-,
** 1
.
26/32
[MATHEMATICS)
Q.53 The option(s) with the values of a and L that satisfy the following equation is (are)
4"
f e (sin
t
6
at+ cos4 at)dt
= L?
o
"
f e (sin
t
6
at+ cos4 at)dt
o
e4" -1
a=2 L=-,
e" -1
(B)
e4" +1
a=2, L=-e" +1
e4" -1
(C) a=4, L=-e" -1
(D)
e4" + 1
a=4, L=-e" +1
(A)
Q.54 Let I, g: [-1, 2] --+ lR be continuous functions which are twice differentiable on the interval
(-1, 2). Let the values of I and g at the points -1, 0 and 2 be as given in the following table:
x=-1
x=O
x=2
I(x)
3
6
0
g(x)
0
1
-1
In each of the intervals (-1, 0) and (0, 2) the function (f - 3g)" never vanishes. Then the
correct statement(s) is(are)
(A)
{'(x) - 3g' (x) = 0 has exactly three solutions in (-1, 0) u (0, 2)
(B)
f'(x) -3g'(x)
(C)
('(x) - 3g' (x) = 0 has exactly one solution in (0,2)
(D)
('(x) - 3g' (x)
=0
has exactly one solution in (-1,0)
= 0 has exactly two solutions in (-1,0) and exactly two solutions in (0, 2)
Space for rough work
** 1
27/32
[MATHEMATICS)
Q.55
Let
f(x)=7tan8x+7tan6x-3tan4x-3tan2x
expression(s)
1
dx =-'
12
fo xf(x)dx=-
K/4
(C)
Q.56
Let f'(x)
=
all
XE(_Jr,
Jr).
\ 2 2
Then
the
correct
is (are)
fo xf(x)
K/4
(A)
for
K/4
(B)
f f(x)dx
=0
o
1
K/4
(D)
6
f f(x)dx
=1
o
1
19~x43 for all x E JR with f(-2 ) = o. If
2+ sm 7lX
II f(x)dx
m =0;
=0; M,
then the possible
1/2
values of m and Mare
(A)
(C)
m=13,
M =24
m = -11, M
=0
(B)
(D)
m=m
1
4'
= 1,
M=-
M
1
2
= 12
Space for rough work
\
')
\ I
./ (I)t) ,
** 1
28/32
v\
[MATHEMATICS]
SECTION 3 (Maximum Marks:
•
•
•
16)
This section contains TWO paragraphs
Based on each paragraph, there will be TWO questions
Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of
these four option(s) is(are) correct
For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS
•
•
Marking scheme:
+4 If only the bubble(s) corresponding to all the correct option(s) is(are) darkened
o If none of the bubbles is darkened
-2 In all other cases
\
PARAGRAPH 1
\
~
~~
'
Let n and nz be the number of red and black balls, respectively, in box I. Let n3 and n4 be the
number of red and black balls, respectively, in box II.
j
Q.57
One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly
out of this box. The ball was found to be red. If the probability that this red ball was drawn
from box II is .!, then the correct option(s) with the possible values of ~,nz,na
. 3
is(are)
(A)
nj = 3, nz = 3, n3 = 5, n4 = 15 {<>
= 3, nz = 6, n3 = 10, n4 = 50
and n4
$f ~
(C)
nj = 8, nz = 6, n3 = 5, n4 = 20(0
(D)
~ = 6, nz = 12, n3 = 5, n4 = 20
A ball is drawn at random from box I and transferred to box II. If the probability of drawing
Q.58
.!, then
a red ball from box I, after this tr::sfer,
values of nj and nz is(are)
is
(A)
(B)
nj = 2 and nz = 3
I5.P?'
n = 3 and nz = 6
~ = 4 and nz = 6 >c
~ = 10 and nz = 20
,yc)
the correct option(s) with the possible
3
i
l'
..
--
.~
+
.
.7
'r,
I
~
--'
'i<
G-W
!
"7/
JE.,
-8
~
j;.:'.,
8
:Ie:
-"
~ i~
5
---! -~
;;0
tr
'-;;l,z1
J-
\-,/
_.:Ef. __ .
/'
-.
..
]'1
":E.? \.
')
3C
-;,,\
2.(
i
~
$;-
01,:;
** 1
/i
J;-,"
.'
~~
2
"t-{)
10'
6
- ;'"
":r
')
~:.
,-.,
Y
, ~r'-"
.'
2
j.
9
~S '\
j
' __
.t
\' ~
2:--,:t. ~
iT'
r
'f..
f)
10...
Space for rough work
<
("<J
v
29/32
/J..(l"J
9(:)
,. :-.'
Ii)
<,
/
a
I
ft
Ii'.
t'
J
[MATHEMATICS 1
PARAGRAPH
r- (/)
2
F: lR --* lR be a thrice differentiable function. Suppose
F'(x) < 0 for all x E (112 ,3). Let f(x) = xF(x) for all x E lR.
Let
that
F(I)
J:.{
= 0, F(3) = -4
and
";( F' (-£i
'/
Q.59
The correct statement(s)
is (are)
A
\J...A1'
('(1) < 0
\j!65
~
('(x)"* 0 for any x
E (1,3)
3
Q.60
r
(I)
{(2)<0
(D)
('(x) = 0 for some x
E (1,3)
3
If f x2F'(x)dx=-I2
and fx3 F"(x)dx=40,
1
then the correct expression(s)
is(are)
1
3
9('(3) + ('(1) - 32 = 0
(A)
(B)
f f(x)dx
= 12
1
\.(9,(
9('(3) - ('(1) + 32
at
=0
1
ff(X)dX=-I2
1
.
t
I C.
;
,
,j
.I
(
0,
,
I i I
'.
-
it
.J
-" !
I
\ ti
/
'J
'-'
J
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END OF THE QUESTION PAPER
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DARKENING THE BUBBLES ON THE ORS :
19. Use a BLACK BALL POINT PEN to darken the bubbles in the upper sheet.
20. Darken the bubble COMPLETELY.
21. Darken the bubbles ONLY if you are sure of the answer.
22. The correct way of darkening a bubble is as shown here: _
23. There is NO way to erase or "un-darken" a darkened bubble.
24. The marking scheme given at the beginning of each section gives details of how darkened
and not darkened bubbles are evaluated.
NAME OF THE CANDIDATE
tS;ARA.N
5.1.M.'I
ROLL NO .. Q.OjD.1~.i
.
.
I HAVE READ ALL THE INSTRUCTIONS
AND SHALL ABIDE BY THEM
I have verified the identity, name and roll
number of the candidate, and that question
paper and ORS codes are the same
Signature of the Invigilator
Signature of the Candidate
Space for rough work
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