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, 0 0 1 p o T n i 3 4 , 0 2 2 p 1 o T 0 2 n i 0 E 1 E , J 0 1 T p I o I T n n i i 4 s n m o a r i g t o c r e P l e m s o l o r a s t s o a t l C 2 t r 4 o 5 h 3 S & / s k m n u i a d R e 0 M 0 d 5 n p a o s T n m i a r 9 g 5 o 1 r , P 0 0 m 2 o p o o r s T s n a i l C 5 7 m r e T g n o L m o r F
FIITJEE
CONCEPT RECAPITULATION TEST (Set – V)
Time Allotted: 3 Hours
S E I R E S T S E T A I D N I L L A
FIITJEE
JEE (Main), 2013
Maximum Marks: 432
Please read the instructions carefully. You are allotted 5 minutes specifically for this purpose. You are not allowed to leave the Examination Hall before the end of the test.
INSTRUCTIONS
A. General Instructions
1. 2. 3. 4. 5.
Attempt ALL the questions. Answers Answers have to be marked on the OMR sheets. This question paper contains Three Parts. Part-I is Physics, Part-II is Chemistry and Part-III is Mathematics. Each part has only one section: Section-A. Section-A. Rough spaces are provided for rough work inside the question paper. No additional sheets will be provided for rough work. 6. Blank Papers, clip boards, log tables, slide rule, calculator, cellular phones, phones, pagers and electronic devices, in any form, are not allowed.
B. Filling of OMR Sheet
1. Ensure matching of OMR sheet with the Question paper before you start marking your answers on OMR sheet. 2. On the OMR sheet, darken the appropriate bubble with black pen for each character of your Enrolment No. and write your Name, Test Centre and other details at the designated places. 3. OMR sheet contains alphabets, numerals & special characters for marking answers.
C. Marking Scheme For All Three Parts.
(i)
Section-A (01 to 03 and 10 to 12) contains 6 multiple choice questions which have only one correct answer. Each question carries +8 marks for correct answer and – 2 mark for wrong answer. Section-A (04 to 09 and 13 to 30) contains 24 multiple choice questions which have only one correct answer. Each question carries +4 marks for correct answer and – 1 mark for wrong answer.
Name of the Candidate
Enrolment No.
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Useful Data
PHYSICS Acceleration due to gravity
2
g = 10 m/s
Planck constant
h = 6.6 ×10−
Charge of electron
e = 1.6 × 10− C
Mass of electron
me = 9.1 × 10−
Permittivity Permittivity of free space
ε0 = 8.85 × 10−12 C2/N-m2
Density of water
ρwater = 103 kg/m3
Atmospheric pressure
Pa = 10 N/m
Gas constant
R = 8.314 J K− mol−
34
J-s
19
31
5
kg
2
1
1
CHEMISTRY Gas Constant
R
Avogadro's Number Na Planck’s constant h 1 Faraday 1 calorie 1 amu 1 eV
= = = = = = = = = =
8.314 J K− mol− 1 1 0.0821 Lit atm K− mol− 1 1 1.987 ≈ 2 Cal K− mol− 23 6.023 × 10 34 6.625 × 10− J⋅s –27 6.625 × 10 erg⋅s 96500 coulomb 4.2 joule –27 1.66 × 10 kg –19 1.6 × 10 J 1
1
Atomic No:
H=1, He = 2, Li=3, Be=4, B=5, C=6, N=7, O=8, N=9, Na=11, Mg=12, Si=14, Al=13, P=15, S=16, Cl=17, Ar=18, K =19, Ca=20, Cr=24, Mn=25, Fe=26, Co=27, Ni=28, Cu = 29, Zn=30, As=33, Br=35, Ag=47, Sn=50, I=53, Xe=54, Ba=56, Pb=82, U=92. Atomic masses: H=1, H=1, He=4, Li=7, Be=9, B=11, C=12, N=14, O=16, F=19, Na=23, Mg=24, Al = 27, Si=28, P=31, S=32, Cl=35.5, K=39, Ca=40, Cr=52, Mn=55, Fe=56, Co=59, Ni=58.7, Cu=63.5, Zn=65.4, As=75, Br=80, Ag=108, Sn=118.7, I=127, Xe=131, Ba=137, Pb=207, U=238.
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Ph ysics
AITS-CRT(Set-V)-PCM-JEE(Main)/13
PART – I
SECTION – A Single Correct Choice Type This section contains 30 multiple choice questions. questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. correct . 1.
A block of metal weighing 2 kg is resting on a frictionless plane. It is struck by a jet releasing water at a rate of 1 kg/s and at a speed of 5 m/s. The initial acceleration of the block is (A) (C)
2.
3.
5
2
m/s
3 25 8
(B) 2
m/s
(D)
25 4 5 2
2
m/s 2
m/s
A body is fired from point P and strikes at Q inside a smooth circular wall as shown in the figure. It rebounds to point S (diametrically opposite to P ). ). The coefficient of restitution will be (A) cot α (B) 1 2 (C) tan α (D) tan α
A wedge of mass M resting M resting on a horizontal frictionless surface is given a force F in F in the horizontal direction. The net horizontal force acting on the shaded portion of the wedge is (A) F (C)
F 6
(B)
Block 2kg
Q
S
P
h 3
F
F 3
(D) zero
h h 3
h
Rough work
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4.
Conductor ABC which consist of two rigid quarter circular wires of radius R, lies in X-Y plane plane and carries current I as shown. A uniform magnetic G G ˆ on field B is switched on in the region that exert force F = 2 IRB k
Y R R I
0
G
A
conductor ABC. B can be
B0
(A)
( −ˆ i )
(B)
2 B0 ˆ ˆ (i + j) (C) 2 5.
In the fusion reaction
B0
jˆ
2
O
B R I R
X
C
(D) both (B) and (C)
2 1
H +12 H →32 He +10 n , the masses of deuteron, helium and neutron
expressed in amu are 2.015, 3.017 and 1.009 respectively. If 1 kg of deuterium undergoes 2 complete fusion, find the amount of total energy release (D) 1 amu = 931.5 MeV/c (A) (C) 6.
≈ 6.02 ×1013 J ≈ 9.0 ×1013 J
The compound unstable nucleus 236 92
(B) (D) 236 92
≈ 5.6 ×1013 J ≈ 0.9 ×1013 J
U often decays in accordance with the following reaction
particles U →15440 Xe +9348 Sr + other
In the nuclear reaction presented above, the other particles might be (A) an alpha particle, which consists consists of two protons and two neutrons (B) two protons (C) one proton and one neutron (D) two neutrons 7.
A point object is placed at a distance of 1000 mm from a concave mirror of focal length 400 mm. If the object is moved towards the mirror by 200 mm, the image moves by a distance of (A) 133.3 mm towards mirror (B) 133.3 mm away from the mirror (C) 30 mm towards mirror (D) 30 mm away from the mirror Rough work
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AITS-CRT(Set-V)-PCM-JEE(Main)/13
y
If an object is placed 20 cm in front of a half thin convex lens of focal length 10 cm, as shown in figure, then co-ordinate of image taking object position as origin
20cm 2mm
(A) [20 cm, 0.2 cm] (C) [40 cm, − 0.2 cm] 9.
10.
(B) [40 cm, 0.4 cm] (D) [20 cm, 0.4 cm]
When an elastic material with Young’s modulus Y subjected to a stretching stress X, the elastic energy stored per unit volume of the material is (A)
2X Y
(B)
Y2 2X
(C)
X2 Y 2
(D)
X2 2Y
From a disc of mass 2 kg and radius 4m a small disc of radius 1 m with center O ' is extracted. The new moment of inertia. about an axis passing through O perpendicular to plane of disc is 2 2 (A) 16 kg m (B) 12 kg m (C)
11.
x
0
255 16
kg m 2
(D)
247 16
14 + 3π 3π + 6 3 Mr 2 π 4 2
2
(B) Mr
(C)
(D)
1 2
’
kg m 2
A wire frame AOPQB, lying in the horizontal plane, is free to rotate about a vertical axis passing through center C of the same circle and ⊥ to plane of AOPQB. The mass M of the frame is uniformly distributed over its whole length. The moment of inertia of the frame about this axis, is (OA = QB = r and CP = r the radius of semicircular part) (A) Mr
2m O O 4m
π + r π + 2r
A
r O C Q r
r
P
B
Mr 2
Rough work
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12.
13.
A block of mass m is suspended from a spring. Its frequency of oscillation is f . The spring is cut into two identical halves and the same block is suspended from one of the two pieces of the spring, such that it just touches the other spring below in its equilibrium position. The frequency of small oscillations of the mass will be (B)
(C) 2f
(D) 2(2f –
2f )
The angle between the directions of the particle velocity and the wave velocity, in a transverse wave, is
(C)
15.
m
2f
(A) f
(A)
14.
6
π
(B) zero
4
π
(D)
2
π 2
.
A block of 1 kg is kept on a rough surface of an elevator moving up with constant velocity of 5 m/s. In 10 second work done by normal reaction (no sliding on incline surface) (i) from ground frame is 320 J (ii) is equal to work work done by friction force force in elevator frame frame (iii) is equal to work done by friction in ground frame correct answer is (A) (i) (B) (ii), (iii) (C) (i), (ii) (D) only (iii).
1kg
5m/s
37º
Average power delivered by an AC source when a resistor of resistance R , an inductor of inductance L and a capacitor of capacitance C are connected in series is (choose the most appropriate option) (A) minimum when the frequency of the source is
(B) maximum when the frequency of the source is
1
1
2π
LC
1
1
2π
LC
.
.
(C) zero when there is no resistor in the circuit. (D) both (B) & (C) Rough work
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16.
17.
AITS-CRT(Set-V)-PCM-JEE(Main)/13
B
A uniform rod AB of length 1 m is placed at edge of a smooth table as shown in figure. It is hit horizontally at point B from left. If the magnitude of the displacement of the centre of mass in 1 s is 5 2 m, the angular speed of rod is (A) 30 rad/s (B) 20 rad/s (C) 10 rad/s (D) 5 rad/s Four resistors are connected in a square formation. Two batteries of emf, emf, E and 2E are connected along the diagonals as shown in the figure. The current flowing in the resistor (R 1) is (R1 = R) (A) (C)
4 E
(B)
3 R 3 E
(D)
2 R
E
Table
A
R
R E
R 1
R
R 3 E R
2E
18.
The rate at which heat is dissipated in the 1.5 Ω resistance in the previous question is given by -t/2 (A) 37.5 W (B) (37.5 W) . e , where t is the time in µS -t -t/2 (C) (37.5 W) e , where t is time in µS (D) (50 W) e , where t is the time in µS
19.
An electric dipole with dipole moment p
G
= (3iˆ + 4ˆj)
G C-m, is kept in electric field E
Cˆi . = 0.4kN / Ci
What is the torque acting on it & the potential energy of the dipole ?
ˆ , −1200J (A) 1600( N × m ) k (C) 20.
−1600( N × m)kˆ , − 1200J
(B) (D)
ˆ −1600(N × m)k,12 m)k,120 00J ˆ 1200J 1600(N × m)k, m)k,
In the arrangement of a pair of parallel plates having separation 1 cm as shown. What is electric field in the region between the plates ? (A) 15 kN/C towards right (B) 15 kN/C towards left (C) 25 kN/C towards right (D) 25 kN/C towards left
1cm
-50V
+200V
Rough work
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21.
22.
P In the figure ab and cd are two long conducting wires kept parallel to each other at a separation A in a uniform time b a varying magnetic field. The ends a and c are connected together by a resistor of resistance R. The magnetic F R induction B perpendicular to the plane of the wires varies with time according to the relation B = B 0t, where B0 is a positive constant with proper unit and t is the time in c d second. A conducting wire PQ placed on the wires ab and Q cd and dragged on the wire with constant speed. v along the length of the wires by applying a constant force F. Find the value of F in terms of the other given parameters. At time t = 0, PQ is very close to ac. Ignore any resistance (electrical as well as mechanical) other than R.
(A)
B02 A2 vt 2
(C)
4B02 A2 vt 2
R R
(B)
2B02 A2 vt 2
(D)
8B02 A2 vt 2
R R
.
A small air bubble of radius r is r is found to form at a depth of H from the open surface of the liquid contained in a beaker. If S is the surface tension and ρ , the density of the liquid and po, the atmospheric pressure the pressure inside the bubble is (A) (C)
23.
4S r 4S r
+
+ po ρ gH + p
(B)
–
+ po ρ gH + p
(D)
2S r 2 S r
–
+ po ρ gH + p
+
+ po ρ gH + p
Three identical spheres having mass M and radius R each are kept in contact at rest as shown in figure. On a frictionless horizontal plane. The net force acting on any one sphere is (A) (C)
GM2
(B)
R2 GM2
2R2 (where G is universal gravitational constant)
3
GM2 4R2
(D) zero.
Rough work
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24.
Two adiabatic vessels, each containing the same mass m of water but at different temperatures, are connected by a rod of length L, cross–section A, and thermal conductivity K. the ends of the rod are inserted into the vessels, while the rest of the rod is insulated so that there is negligible loss of heat into the atmosphere. The specific heat capacity of water is s, while that of the rod is negligible. 1 The temperature difference between the two vessels reduces to of its original value after a e time, ∆t. The thermal conductivity (K) of the rod may be expressed by msL emsL (A) (B) A∆t A∆t msL msL (C) (D) . 2eA∆t 2A∆t
25.
The heat (Q) supplied to a solid, which is otherwise thermally isolated from its surroundings, is plotted as a function of its absolute temperature, θ. It is found that they are related by the equation, 2 4 Q = aθ + bθ . (a, b are constants). The heat capacity of the solid is given by (A) a (C) a
26.
θ3 3
θ 3
+b
+b
θ5
(B) aθ + bθ3
5 3
θ
3
(D) 2aθ + 4bθ .
5
The K.E. (K) of a particle moving along a circle of radius R depends on the distance covered 2 s as K = as . The force acting on particle is (A)
2as 2
(B)
R
2as 1/ 2
s 2 1 + R
1/ 2
(C)
s 2 2as 1 + 2 R
(D) none of these.
Rough work
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27.
10
Two particles are simultaneously projected with the same speed in the same vertical plane, but perpendicular to each other, in a uniform gravitational field. Their times of flight are T 1 , T 2 while the horizontal ranges are R1 , R2 . Then (choose one option) (A) R1 (C)
28.
= R2
R1 T1
=
(B) R1 + R2
R 2
(D) both (A) and (B) (B) are true
T2
A block of mass
= gT1T 2
3 10
kg is placed on a rough horizontal surface as
shown in the figure. A force of 1 N is applied at on end of the block and the block remains stationary. The normal force exerted by the 2 surface on the block acts (g ( g = = 10 m/s ) (A) through the centre of mass of the block. (B) through point A point A.. (C) through point B. (D) through the point at a distance 5 cm. from A from A.. 29.
30.
F = 1N 30º
A uniform conducting rectangular loop of sides A , b and mass m carrying current i is hanging horizontally with the help of two vertical strings. There exists a uniform horizontal magnetic field B which is parallel to the longer side of loop. The value of b tension which is least is m g − Bb m g + Bb (A) (B) 2 2 mg − 2iBb mg + 2Bb (C) (D) . 2 2
c 0 2
B
A 20 cm
B A
The mean lives of a radio-active substances are 1620 years and 405 years for α-emission and βemission respectively. Find out the time during which three fourth of a sample will decay if it is decaying both by α-emission and β-emission simultaneously (A) 324 years (B) 449 years (C) 480 years (D) 425 years Rough work
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Chemistr y
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PART – II
Straight Objective Type This section contains 30 multiple choice questions numbered 1 to 30. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct. 1.
In an atomic BCC, what fraction of edge is not covered by atoms: (A) 0.144 (B) 0.130 (C) 0.279 (D) 0.134
2.
A complex [CoA6] red coloured while [CoB6] is green coloured, which of the following is correct statement? (A) Ligand A is producing larger crystal crystal field splitting 3+ (B) [CoB6] is easily oxidized (C) Both are equally oxidized (D) Ligand and B is producing larger crystal field splitting
3.
A solution of H2O2 labelled as ’20 V’ was left open. Due to this, some H 2O2 decomposed and volume strength of the solution decreased. To determined the new volume strength of the H 2O2 solution, 10 ml of the solution was taken and it was diluted to 100 ml. 10 ml of this diluted solution required 25 ml of 0.0245 M KMnO4 solution for titration under acidic condition. Calculate the new volume strength of H2O2 solution? (A) 17.15 V (B) 18.4 V (C) 19.4 V (D) 16.5 V
4.
A white crystalline substance dissolves in water. On passing H2S in this solution, a black precipitate is obtained. The black ppt. dissolves completely in hot HNO 3. On adding a few drops of concentrated H2SO4, a white precipitate is obtained which is soluble in ammonium acetate. The white precipitate is that of: (A) BaSO4 (B) SrSO 4 (C) PbSO4 (D) Ag2SO4
5.
When 1 mole of A (g) is introduced in a closed 1 L vessel maintained at constant temperature. The following equilibria are established: ZZ X A ( g ) Y K c1 = ? ZZ Z Z B ( g ) + 2C ( g ) ;
3+
3+
ZZ X C ( g) Y ZZ Z Z 2D ( g ) + 3B ( g ) ;
The pressure at equilibrium is Calculate K c & K c if 1
2
(A) 0.14 and 0.14 (C) 0.19 and 0.16
Kc2
=?
13 6 times the initial pressure.
[C]eq. 4 = . [ A ]eq. 9 (B) 0.28 and 0.18 (D) 0.11 and 0.14 Rough Work
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6.
ZZ X For the reaction, N2O 4 ( g ) Y ZZ Z Z 2NO2 ( g ) , if percentage dissociation of N2O4 are 20%, 45%, 65%
and 80% then the sequence of observed densities at given extent of dissociation are: (A) (d20 = d45) > (d65 = d80) (B) d80 > d65 > d45 > d20 (C) d20 > d45 > d65 > d80 (D) d20 = d45 = d65 = d80 7.
One mole of naphthalene was burnt in oxygen gas at constant volume to give carbondioxide gas o and liquid water at 25 C. The heat evolved was found to be 5138.8 kJ. Calculate the enthalpy of reaction at constant pressure: (A) - 5143.8 kJ (B) +6538.3 kJ (C) - 4148 kJ (D) - 3398 kJ
8.
For the set of reactions: K
1 ZZ Z Z (i) A + B Y Z ZX Z ZC K
2 →D (ii) C + B
K
−1
K1 [ A ][B] − K −1 [ C] − K2 [ C][ B] is equal to:
(A) (C)
−
d[ A]
−
(B)
dt
d [C]
(D)
dt
d[B] dt
d [D] dt
OH
9.
1. H S O + CH2 ( OEt )2 → Product. 2. conc.HCl 2
4
NO2 OH
OH CH2Cl
Cl
(A)
(B) NO 2
NO2 CH2Cl
(C)
OH
(D) CH2Cl
NO2
Rough Work
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O H C
10. KCN,EtOH Productt is: → →Produc H O 2
OMe
O
(A)
OH
O
(B)
OH
C HC
C HC OMe MeO
MeO O
(C)
OH
(D)
O
OH
C HC
C HC
OMe
OMe OMe
O
11.
( ) → X →Y H+ CH3CO CO 2 O
Li
O
X and Y are: (A) O
;
O
OCOCH3
O [ Y]
[ X] OH
(B)
;
O
OCOCH3
O [Y]
[ X] (C) O
OCH3
;
O
O [Y]
[ X]
O
(D) ; O
[ X]
OH
OCOCH3 O [ Y]
Rough Work
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12.
For the cell: Pt | H2(g) | sol X || KCl (saturated)| Hg 2Cl2(s) | Hg|Pt o The observed emf at 25 C was 612 mV. When solution X was replaced by a standard phosphate buffer whose assigned pH is 6.86, the emf was 741 mV. Find pH of the solution X. (A) 4.68 (B) 5.13 (C) 6.12 (D) 4.24
13.
The pKa of acetylsalicylic acid (aspirin) is 3.5. The pH of gastric juice in human stomach is about 2 – 3 and the pH in the small instestine is about 8. Aspirin will be: (A) Unionised in the small intestine intestine and the stomach (B) Completely ionized in the small small intestine and in the stomach (C) Ionised in the stomach and almost unionized in the small intestine (D) Ionised in the small intestine intestine and almost unionized in the stomach
14.
Calculate wavelength of He atom whose speed is equal to the r.m.s. at 20 C? -12 -11 (A) 7.64 × 10 m (B) 5.28 × 10 m -11 -10 (C) 7.38 × 10 m (D) 2.28 × 10 m
15.
Which one of the following ylides will be most stable and least reactive for nucleophilic addition reaction with aldehyde or ketones?
o
(+ )
(−)
(A) ( C6H5 )3 − P − CH CH3 ( +)
( −)
(C) Ph3 P − CH− CH2 16.
(B)
(+)
(+)
− CH3
(− )
( C6H5 )3 P − CH− CH3 ( −)
(D) Ph3 P − CH− COOC2H 5
In the Perkin reaction which one of the following intermediates gives compound (I): O H 5C 6 C H
CH
O
C
O
C
CH3
(I)
(A)
O
O H5C6 CH
CH2
C
O
C
O
(B)
O CH3
O
C CH3
H5C6 HC CH2 O
(C) O
C
O
(D) H5C6 HC
CH2
CH2
COCH 3 O
CH2
H5C6 HC
COO
C
O O
C
CH3
COOH
Rough Work
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17.
AITS-CRT(Set-V)-PCM-JEE(Main)/13
When the deuterium labeled compound (I) is subjected to dehydrohalogenation: H H3C
Br D
−
+
C H O Na →X 2 5
H
H
The only product X is: (A) Methylcyclohexene (C) Cycloheptene
(B) 3-Methylcyclohexene (D) 2-Methylcyclohexene
18.
Which of the following is/are non-reducing sugars? (A) Fructose (B) Glucose (C) Sucrose (D) Lactose
19.
Which of the following statement is not true? (A) O-F bond length in OF2 is less than O-F bond length in O 2F2. (B) In HCO3− , all C – O bond length are not identical. (C) In diborane, two different B – H bond lengths are observed although the hybridization hybridization of both boron atoms are same. (D) In hydrazine, the N – N bond length is larger than normal N – N bond length.
20.
How many stereoisomers are possible for the following molecule H CH
CH
CHCOOH
H3C
(A) 2 (C) 4 21.
(B) 3 (D) 6 o
The vapour pressure of water at 20 C is 17.54 mm Hg. What will be the vapour pressure of the water, in the apparatus shown after the piston is lowered, decreasing the volume of the gas above the liquid to one half of its initial volume (assume T constant): (A) (B) (C) (D)
8.77 mm Hg 17.54 mm Hg 35.08 mm Hg Between 8.77 and 17.54 mm Hg.
Water vapour Liquid water
Rough Work
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22.
Which one of the following is most stable? H
(A)
H
(B)
Br
Br
H
H
H
H
H
H
H
H
NH2
H
(C)
(D)
Br
H
NH2 Br
H
H
H H
H
H
H
H
NH2
23.
NH2
The acid strength order is: COOH I
I
(a)
COOH
COOH
COOH
I
I
I
(b)
(c)
(A) (a) > (b) > (c) > (d) (C) (a) > (b) > (d) > (c)
I
I (d)
I
(B) (a) > (c) > (b) > (d) (D) (b) > (a) > (d) > (c)
24.
In a space shuttle, the CO2 output per astronaut has been estimated as 44 g per hour. An experimental catalytic converter reduces CO2 at a rate of 600 mL (STP) per min into H 2O. What fraction of the time would such a converter have to operate in order to keep up the CO 2 output of one astronaut? (A) 0.622 (B) 0.782 (C) 0.382 (D) 0.411
25.
The correct statements are: The planar shape of N(SiH3)3 is explained by the 1. Type of hybrid orbitals of nitrogen. 2. Additional pπ − dπ overlap along the N-Si bond. 3. Higher electronegativity of nitrogen. (A) 1, 2, and 3 (B) 1 and 2 (C) 2 and 3 (D) 1 and 3
26.
The compound (X) obtained in the following reaction is most likely to be: Li/NH ( ) →( X) 3 A
(A)
(B)
(C)
(D)
CH
Rough Work
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OH
27.
In the acid catalysed dehydration of CH2
CH3
(I)
(II)
the product(s) formed is/are: CH3
(IV)
(III)
(A) I and IV (C) III and IV O C
(B) II and IV (D) I, II and III CH3 Ph O
28.
IO ( a ) Na IO Productt is: is: → Produc (b ) H+ ,∆ 4
(A)
HOOC
(B)
Ph O
O
COOH
(C)
COOH
(D)
OH
O
Ph
29.
Consider the following statement: S1 : Fluorine does not form any polyhalide as it has low F – F bond energy. S2 : The chlorine has the most negative electron gain enthalpy. S3 : The first ionization potential of N is greater than O. Which of the above statements are correct? (A) S1, S2 and S3 (B) S1 and S2 (C) S1 and S3 (D) S2 and S3
30.
MnO24− (1 mo mole ) in neutral aqueous medium, disproportionates to:
(A) (C)
2 3 1 3
mole of MnO 4− and mole of Mn2O7 and
1 3 1 3
mole of MnO2
(B)
mole of MnO2
(D)
1 3 2 3
mole of MnO 4− and mole of Mn2O7 and
2 3 1 3
mole of MnO2 mole of MnO2
Rough Work
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M athematics
18
PART – III
SECTION – A Straight Objective Type This section contains 30 multiple choice questions numbered 1 to 30. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct. 1.
2.
The curve y = ax2 (A) 3, –12 (C) –3, 12
The point on the curve y
1 5 2, 4 5 (C) 2, − 3 (A)
3.
(B) –3, –12 (D) 3, 12
= x2 + 4x + 3 which is nearest from the line y = 3x + 2 is 1 5 (B) − , 2 4 5 (D) 2, 3
The sides of the greatest rectangle that can be inscribed in the ellipse (A) a 2 , b 2 (C) a, b
4.
+ bx has minimum at (2, –12) then the values of a, b are
x2 a2
+
y2 b2
= 1 are
(B)
a, b a b (D) , 2 2
The maximum area of the triangle with vertices at (a, 0), (a cos θ, b sin θ) and (a cos θ, –b sin θ) is (A)
3 3 ab 4
(C)
3ab 4
(B) 3 ab (D) 3 3ab Rough work
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5.
If the perimeter of a sector of a circle is 16 cms then it’s maximum area is (A) 16 sq. cms (B) 32 sq. cms (C) 48 sq. cms (D) 64 sq. cms
6.
The shortest distance of the point (0, c) from the curve y
7.
c−4
(B)
c−
(C)
4c − 1
(D)
c −1
The function f ( x ) =
9.
x 2 1 + sin x (A) cosx x + si n x (C) 1 + cos x
= x tan , then
4
sinx is x
π 2 (C) stationary at x = π/2 If y
1 is 2
1
(A)
(A) increasing in 0,
8.
= x2 , where c ≥
dy dx
π 2
(B) decreasing in 0,
(D) both increasing and decreasing at x = π/2
is equal to (B) (D)
If f is a differentiable function with f (1) = 8 , f ′ (1) =
1 8
x + cos x sinx 1 + cos x x + sin x
. If f is invertible and g = f −1 , then
(A) g′ (1) = 8
(B) g′ (1) =
(C) g′ ( 8 ) = 8
(D) g′ ( 8 ) =
1 8 1 8
Rough work
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10.
Let f ( x ) =
x 1 + x tan x
, then f ( x ) has
0, π 2 π (C) no critical point in 0, 2
0, π 2 −π , π (D) exactly one critical point in 2 2
(A) one point of minima in
11.
(B) one point of maxima in
2/3
The critical points of f ( x ) = ( x − 2 )
(2x + 1)
is(are)
(A) 1 only (C) 2 only 12.
The height of the right circular cylinder of maximum volume that can be inscribed in a sphere of radius ‘a’ is a 2a (A) (B) 3 3 (C)
13.
(B) 1, 2 (D) 0, 1, 2
2
(D) a 2
a
The stationary point of y
= x2 +
250 is x
(A) (1, 51) (C) (5, 25) 14.
The function f ( x ) = 2x 2
(B) (5, 1) (D) (5, 75)
9 − 4 x + 1 in the interval 3, has minimum value 2
(A) –1 (C) 1 15.
The turning points of f ( x ) = sin x + (A) x = π/4 only (C) x = π/4, 2π/3 only
(B) 7 (D) 17 1 2
sin 2x +
1 3
sin 3x in (0, π) are at
(B) x = π/4, π/3 (D) x = π/4, 2π/3, 3π/4 Rough work
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16.
If a > b, then the maximum and minimum values of a sin2 x + b cos2 x is (A) a, b (B) b, a a2
(C) 17.
AITS-CRT(Set-V)-PCM-JEE(Main)/13
+ b2 , – a 2 + b2
The greatest value of f ( x ) = 2x 2 (A)
17 2
(C) 0
(D)
+
2 x2
ab , – ab
, for x ∈ [ −2, 0 ) ∪ (0, 2] and f ( 0 ) = 1 is (B) 1 (D)
11 2
18.
If x and y are strictly positive such that that x + y = 1, then the the minimum minimum value of x log log x + y log y is (A) log 2 (B) –log 2 (C) 2 log 2 (D) 0
19.
The minimum value of E(x, y) = cos x + cos y + 2 cos (x + y) where x, y ∈ R is equal to 9 9 (A) − (B) − 2 4 3 3 (C) − (D) − 2 4
20.
If A, B, C are interior angles of triangle ABC such that 2 2 (cos A + cos B + cos C) + (sin A + sin B + sin C) = 9 then number of possible triangle is (A) 0 (B) 1 (C) 3 (D) infinite
21.
Let a, b, c ∈ R and 1 be a root of the equation ax + bx + c = 0, then the equation 2 4ax + 3bx + 2c = 0 has (A) imaginary roots (B) real and equal roots (C) real and unequal roots (D) rational roots
2
Rough work
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22.
Sandra is writing her Christmas list. If she writes writes n items items in her list, the probability of getting each 1 1 item is for example if she writes 3 items, she will get each item with a probability of . If 6 2n Sandra write infinitely many gifts on her list then the probability of her getting no gifts, is 1 –1 (A) e (B) 2e 1 (C) e−1/ 2 (D) 2
23.
If the roots of cubic equation x – 6x + β1x – β2 = 0 are in A.P. with positive integral common difference, them the maximum value of (β1 + β2), is (A) 17 (B) 18 (C) 19 (D) 20
24.
Let α1 and α2 he two real values of α for which the numbers 2α , α , 24 taken in that order from 2 2 an arithmetic progression if β1 and β2 are two real values of β for which the numbers 1, β , 6 – β
3
2
2
4
taken in that order form a geometric progression, then the value of ( α12 (A) 10 (C) 12
+ α 22 + β12 + β22 ) is equal to
(B) 11 (D) 13 2
25.
Let f(x) = x + ax + b cos x, a being an integer and b is a real number. Find the number of ordered pairs (a, b) for which the equations f(x) = 0 and f(f(x)) ≡ 0 have the same (non–empty) set of real roots (A) 1 (B) 2 (C) 3 (D) 4
26.
Number of integral values of ‘a’ for which every solution of the inequality x + 1 > 0 is also the 2 solution of the inequality (a – 1)x – (a + |a – 1| + 2)x + 1 ≥ 0, is (A) 0 (B) 1 (C) 2 (D) 3
2
Rough work
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27.
The expression
2
2
2
2
2
2
( 10 C0 ) − ( 10 C1 ) + ( 10 C2 ) − .......... + ( 10 C8 ) − ( 10 C9 ) + ( 10 C1 0 )
(A) 0 (C)
AITS-CRT(Set-V)-PCM-JEE(Main)/13
(B)
−10 C5
is equal to
2
( 10 C5 )
(D) 2 ⋅ 9C5 2012
28.
If C0, C 1, ….., C 2012 are binomial coefficients in the expansion of (1 + x) and a0, a 1, ….., a 2012 are real numbers in arithmetic progression then value of a 0C0 – a1C1 + a2C2 – a3C3 + ….. + a2012C2012, is (A) 1 (B) 2012 (C) –1 (D) 0
29.
If tan9θ =
3 4
(where 0 <
θ<
π 18
) then the value of (3 cosec 3θ – 4 sec 3θ) is equal to
(A) 3 (C) 5 30.
(B) 4 (D) 10
Let n be the number of ordered quadruples (x1, x2, x3, x4) of positive odd integers that satisfy 4
n is equal to ∑ x = 98 , then the value of 100 i
i=1
(A) 144 (C) 196
(B) 169 (D) 225 Rough work
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