Transcript
AIEEE 2011 This booklet contains 10 printed pages. Do not open this Test Booklet until you are asked to do so.
Important Instructions: 1. Immediately fill in the particulars on this page of the Test Booklet with Blue/Black Ball Point Pen. Use of pencil is strictly prohibited. 2. The Answer Sheet is kept inside this Test Booklet. When you are directed to open the Test Booklet, take out the Answer Sheet and fill in the particulars carefully. 3. The test is of 3 hours duration. 4. The Test Booklet consists of 90 questions. The maximum marks are 360. 5. There are three part in the question paper. The distribution of marks subjectwise in each part is as under for each correct response. Part A : MATHEMATICS (120 marks) - Question No. 1 to 30 consist Four (4) marks each for each correct response. Part B : CHEMISTRY (120 marks) - Question No. 31 to 60 consist Four (4) marks each for each correct response. Part C : PHYSICS (120 marks) - Question No. 61 to 90 consist Four (4) marks each for each correct response.. 6. Candidates will be awarded marks as stated above in instructions No. 5 for correct response of each question. ¼ (one fourth) marks will be deducted for indicating incorrect response of each question. No deduction from the total score will be made if no response is indicated for an item in the answer sheet. 7. Use Blue/Black Ball Point Pen only for writing particulars/marking response the Answer Sheet. Use of pencil is strictly prohibited. 8. No candidate is allowed to carry any textual material, printed or written, bits of papers, pager, mobile phone, any electronic device, etc., inside the examination hall/room. 9. Rough work is to be done on the space provided for this purpose in the Test Booklet only. This space is given at the bottom of each page and in 1 page at the end of the booklet. 10. On completion of the test, the candidate must hand over the Answer Sheet to the Invigilator on duty in the Room/Hall. However, the candidates are allowed to take away this Test Booklet with them. Name of the Candidate (in Capitals): _____________________________________________________________ Enrollment Number: __________________________________________________________________________ Candidate's Signature:____________________________ Invigilator's Signature: ___________________________
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AIEEE 2011
[1]
Part - A Mathematics 1.
Consider 5 independent Bernulli's trials each with probability of sucess p. If the probability of at least one failure is greater than or equal to lies in the interval
FG OP H Q F 3 11 O (3) G 4 , 12 P H Q 11 (1) 12 , 1
2.
3.
6.
31 , then p 32
F d y I F dy I (1) – G dx J GH dx JK H K F d y I F dy I (3) – G H dx JK GH dx JK
F GG H
x2
2
FG 1 , 3 OP H 2 4Q F 1O (4) G 0, 2 P H Q (2)
1 cos{2( x 2)} x2
I JJ K
2
(3) equals 2 4.
7.
F d yI (2) G H dx JK F d y I F dy I (4) G dx J GH dx JK H K 1
2
2
3
2
2
2
2
The number of values of k for which the linear equations 4 x ky 2 z 0
kx 4 y z 0 2x 2 y z 0 possess a non-zero solutoin is (1) zero (2) 3 (3) 2 (4) 1 8.
Statement -1 The point A (1, 0, 7) is the mirror image of the
(2) does not exist
point B(1, 6, 3) in the line
(4) equals 2
x y 1 z 2 1 2 3
Statement -2
Let R be the set of real numbers
x y 1 z 2 bisects the line 1 2 3 segment joining A (1, 0, 7) and B (1, 6, 3)
Statement -1
The line
A {( x, y) R R : y x is an integer} is an equivalence relation on R.
(1) Statement -1 is false, Statement -2 is true (2) Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for statement-1
Statement -2
B {( x, y ) R R : x y for some rational number } is an equivalence relation on R. (1) Statement -1 is false, Statement -2 is true (2) Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for statement-1
(3) Statement-1 is true, Statement -2 is true, Statement-2 is not correct explanation for Statement-1 (4) Statement -1 is true, Statement-2 is false
z 2 z 0 has two distinct roots on the line
Consider the following statements P : Suman is brilliant Q : Suman is rich R : Suman is honest The negation of the statment "Suman is brilliant and dishonest if and only if Suman is rich" can be expressed as (1) ~ ( P ~ R) Q
Re z 1 , then it is necessary that (1) (1, ) (2) (0, 1)
(3) ~ (Q ( P ~ R))
(3) Statement-1 is true, Statement -2 is true, Statement-2 is not correct explanation for Statement-1 (4) Statement -1 is true, Statement-2 is false 5.
1
2
1 (1) equals
3
2
The coefficient of x 7 in the expansion of (1 x x 2 x 3 ) 6 is (1) 132 (2)144 (3) – 132 (4) – 144
lim
d2x equal dy 2
Let , be real and z be complex number. If
(3) ( 1, 0)
AIEEE 2011
(4) 1
9. .
(2) ~ P (Q ~ R) (4) ~ Q ~ P R
[2]
10.
The lines L1: y x 0 and L2 : 2 x y 0
value of is
intersect the line L3 : y 2 0 at P and Q respectivley. The bisector of the acute angle
(1) 5 3
(2) 2 3
(3) 3 2
(4) 2 5
between L1 and L2 intersects L3 at R. Statement -1 16.
The ratio PR : RQ equals 2 2 : 5 .
f ( x)
In any triangle, bisector an angle divides the triangle into two similar triangles. (1) Statement -1 is false, Statement -2 is true (2) Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for statement-1 (3) Statement-1 is true, Statement -2 is true, Statement-2 is not correct explanation for Statement-1 (4) Statement -1 is true, Statement-2 is false
17.
The domain of the function
1
f ( x)
x x
(1) ( , )
is
(2) ( , ) {0}
(3) (0, )
(4) ( , 0)
If the mean deviation about the median of the numbers a, 2a, . . ., 50 a is 50, then a equals
( 3, 1) and has eccentricity
(1) 5 (3) 3
2
5 is
(1) 5x 2 3 y 2 32 0
(2) 3x 2 5 y 2 32 0
(3) 5x 3 y 48 0
(4) 3x 5 y 15 0
2
2
2
19.
(1)
3 13 A 4 16
(2)
(3)
13 A1 16
(4) 1 A 2
The value of
z
(1) log 2 (3)
log 2 8
F GH
j
e j e j e
j
j
(2) 5 (4) 5
The value of p and q for which the function
2
3
2
is continuous for all x in R, are,
y 1 z 3 and 2
1
e
R| sin( p 1) x sin x ,x0 x || f ( x) S q ,x0 || x x x , x 0 |T x
log 2 2
the plane x 2 y 3z 4 is cos
AIEEE 2011
20.
(2) log 2
If the angle between the line x
1 3i k and b 1 2i 3 j 6k , then If a 10 7 the value of 2a b a b a 2 b is (1) 3 (3) 3
3 A1 4
8 log(1 x ) dx is 1 x2
(4)
(2) 2 (4) 4
e
If A sin 2 x cos4 x then for all real x.
0
15.
and local minimum at 2 and 2 and 2 and local maximum at 2
Equations of the ellipse whose axes are the axes of coordinates and which passes through the point
1
14.
t sin t dt
Then f has (1) local maximum at (2) lcoal maximum at (3) lcoal minimum at (4) local minimum at
A man saves Rs 200 in each of the first three months of his services. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after (1) 21 months (2) 18 months (3) 19 months (4) 20 months
2
13.
z 0
18. 12.
IJ K
x
Statement -2
11.
FG H
5 For x 0, 2 , define
I JK
5 14 . Then
(1) p
1 3 ,q 2 2
(2) p
1 3 ,q 2 2
(3) p
5 3 ,q 2 2
3 1 (4) p , q 2 2
[3]
21.
The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty
The two circles x 2 y 2 ax and x 2 y 2 c 2 (c 0) touch each other if
22.
(1) a 2c
(2) 2 a c
(3) a c
(4) a 2c
is 9 C3 . Statement -2 The number of ways of choosing any 3 places from 9 different places is 9 C3 (1) Statement -1 is false, Statement -2 is true (2) Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for statement-1 (3) Statement-1 is true, Statement -2 is true, Statement-2 is not correct explanation for Statement-1 (4) Statement -1 is true, Statement-2 is false
Let be the purchase value of an equipment is I and V ( t ) be the value after it has been used for t years. The value V ( t ) depreciates at a rate given by differdV (t ) k (T t ) , where k 0 is dt a constant and T is the total life in years of the equipment. Then the scrap value V(T) of the equipment is
ential euqation
2 (2) T
(1) e kT (3) I 23.
24.
kT 2 2
(4) I
1 k
curve x y 2 is 4
k (T t ) 2 2
If C and D are two events such that C D and P ( D ) 0 , then the correct statement among the following is
P ( D) (1) P(C| D) P(C )
(2) P(C| D) P(C)
(3) P(C| D) P(C)
(4) P(C| D) P(C)
(1)
(3) 28.
3 4 8
3 2 8
(4)
3 2
The area of the region enclosed by the curves
b g
(1) 5/2 square units (2) 1/2 square units (3) 1 square units (4) 3/2 square units 29.
A (BA) and (AB) A are symmetric matrices
If
bg
b g
dy y 3 0 and y 0 2, the y ln 2 is equal dx
to: (1) –2 (3) 5
(2) 7 (4) 13
(1) Statement -1 is false, Statement -2 is true (2) Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for statement-1
The vectors a and b are no perpendicular and c and d are two vectors satisfying : b c b d and a d 0 . Then the vector d is equal to:
(3) Statement-1 is true, Statement -2 is true, Statement-2 is not correct explanation for Statement-1 (4) Statement -1 is true, Statement-2 is false
a c c (1) b a b
b c b (2) c a b
Statement -2 AB is symmetric matrix if matrix multiplication of A with B commutative.
If ( 1) is a cube root of unity, and
(1 ) 7 A B . Then (A, B) equals (1) (–1, 1) (2) (0, 1) (3) (1, 1) (4) (1, 0) 26
(2)
3
y x, x e, y 1 / x and the positive x-axis is
Let A and B be two symmetric matrices of order 3 Statement -1
25.
The shortest distance between line y x 1 and
27
30.
FG IJ H K F b c I (3) c G J c H a b K
F I GH JK F b c I (4) b G J c H a b K
Statement -1
AIEEE 2011
[4]
Part -B Chemistry 31.
32.
33.
In context of the lanthanoids, which of the following statements is not correct? (1) Availability of 4f electrons results in the formation of compounds in +4 state for all the members of the series. (2) There is a gradual decrease in the radii of the members with increasing atomic number in the series. (3) All the members exhibit +3 oxidation state. (4) Because of similar properties the separation of lanthanoids is not easy.
37.
and NH 4 are respectively:
38.
(2) A 2 B
(3) AB2
(4) A 2 B3
The magnetic moment (spin only) of NiCl 4
. BM (1) 141
. BM (2) 182
(3) 5.46 BM
(4) 2.82 BM
is
Which of the following facts about the complex
b
Cr NH 3
g
6
(1) The complex gives precipitate with silver nitrate solution.
35.
(1) 64 times (3) 24 times 36.
(2) 10 times (4) 32 times
'a' and 'b' are van der Waals' constants for gases. Chlorine is more easily liquefied than ethane because
Ethylene glycol is used as an antifreeze in a cold climate. Mass of ethylene glycol which should be added to 4 kg of water to prevent it from freezing at –0.6 oC
The outer electron configuration of Gd (Atomic No : 64) is (1) 4 f 7 5d 1 6s2
(2) 4 f 3 5d 5 6s2
(3) 4 f 8 5d 0 6s2
(4) 4 f 4 5d 4 6s2
The structure of IF7 is
Ozonolysis of an organic compound gives formaldehyde as one of the products. This confirms the presence of: (1) an acetylenic triple bond (2) two ethylenic double bonds (3) a vinyl group (4) an isopropyl group
42.
The degree of dissociation () of a weak electrolyte,
A x By is related to van't Hoff factor (i) by the expression: (1)
x y 1 i 1
(2)
(3)
i 1 x y 1
(4)
(1) a for Cl 2 a for C 2 H 6 but b for Cl 2 b for C 2 H 6 (2) a and b for Cl 2 a and b for C 2 H 6
(2) 804.32 g (4) 400.00 g
41.
The rate of a chemical reaction doubles for every 10 C rise of temperature. If the temperature is raised by 50C the rate of the reaction increases by about:
(4) sp, sp 3 , sp 2
(1) pentagonal bipyramid (2) square pyramid (3) trigonal bipyramid (4) octahedral
Cl 3 is wrong?
(2) The complex involves d 2 sp 3 hybridisation and is octahedral in shape. (3) The complex is paramagnetic (4) The complex is an outer orbital complex.
(3) sp 2 , sp, sp 3
(1) 304.60 g (3) 204.30 g
40. 34.
(2) sp, sp 2 , sp 3
mass of ethylene glycol 62 g mol 1 )
39. 2
(1) sp 2 , sp 3 , sp
will be: ( K f for water 186 . K kg mol 1 and molar
In a face centred cubic lattice, atom A occupies the corner positions and atom B occupies the face centre positions. If one atom of B is missing form one of the face centred points, the formula of the compound is (1) A 2 B5
The hybridisation of orbitals of N atom in NO 3 , NO 2
i 1
bx y 1g x y 1 i 1
(3) a and b for Cl 2 a and b for C 2 H 6 (4) a for Cl 2 a for C 2 H 6 but b for Cl 2 b for C2 H 6
AIEEE 2011
[5]
43.
44.
45.
46.
47.
48.
A gas absorbs a photon of 355 nm and emists at two wavelengths. If one of the emissions is at 680 nm, the other is at: (1) 518 nm (2) 1035 nm (3) 325 nm (4) 743 nm
50.
Identify the compound that exhibits tautomerism. (1) Phenol (2) 2-Butene (3) Lactic acid (4) 2-Pentanone
51.
The entropy change involved in the isothermal reversible expansion of 2 moles of an ideal gas from a volume of 10 dm3 to a volume of 100 dm3 at 27C is (1) 42.3 J mol 1 K 1
(2) 38.3 J mol 1 K 1
(3) 358 . J mol 1 K 1
(4) 32.3 J mol 1 K 1
Silver Mirror test is given by which one of the following compounds? (1) Benzophenone (2) Acetaldehyde (3) Acetone (4) Formaldehyde Trichloroacetaldehyde was subjected to Cannizszaro's reaction by using NaOH . The mixture of the products contains sodium trichloroacetate and another compound. The other compound is: (1) Chloroform (2) 2, 2, 2-Trichloroethanol (3) Trichloromethanol (4) 2, 2, 2-Trichloropropanol
b g (2) pb H g 1 atm and H (3) pb H g 1 atm and H (4) pb H g 2 atm and H
2.0 M
10 . M
2
2
2
49.
(1) MgCl 2
(2) FeCl 2
(3) SnCl 2
(4) AlCl 3
Boron cannot form which one of the following anions? (1) BO 2
(2) BF63
(3) BH 4
(4) B OH
b g
4
52.
Sodium ethoxide has reacted with ethanoyl chloride. The compound that is produced in the above reaction is (1) Ethyl ethanoate (2) Diethyl ether (3) 2-Butanone (4) Ethyl chloride
53.
Which of the following reagents may be used to distinguish between phenol and benzoic acid
54.
(1) Neutral FeCl 3
(2) Aqueous NaOH
(3) Tollen's reagent
(4) Molisch reagent
A vessel at 1000 K contains CO 2 with a pressure of 0.5 atm. Some of the CO 2 is converted into CO on the addition of graphite. If the total pressure at equilibrium is 0.8 atm the value of K is (1) 0.18 (2) 1.8 (3) 3 (4) 0.3
The reduction potential of hydrogen half cell will be negative if: (1) p H 2 2 atm and H 2.0 M
Among the following the maximum covalent character is shown by the compound:
55.
The strongest acid amongst the following compounds is (1) ClCH 2 CH 2 CH 2 COOH (2) CH 3COOH (3) HCOOH
b g
10 . M
Phenol is heated with a solution of mixture of KBr and
KBrO 3 . The major product obtained in the above reaction is: (1) 2, 4, 6-Tribromophenol (2) 2-Bromophenol (3) 3-Bromophenol (4) 4-Bromophenol
(4) CH 3CH 2 CH Cl CO 2 H 56.
Which one of the following orders presents the correct sequence of the increasing basic nature of the given oxides? (1) K 2 O Na 2 O Al 2 O 3 MgO (2) Al 2 O 3 MgO Na 2 O K 2 O (3) MgO K 2 O Al 2 O 3 Na 2 O (4) MgO K 2 O MgO Al 2 O 3
AIEEE 2011
[6]
57.
A 5.2 molal aqueous solution of methyl alcohol,
59.
CH 3OH is supplied. What is the mole fraction of methyl alcohol in the solution? (1) 0.050 (2) 1.100 (3) 0.190 (4) 0.086 58.
Which of the following statement is wrong? (1) N 2 O 4 has two resonance structures. (2) The stability of hydrides increases form NH 3 to
BiH 3 in group 15 of the periodic table. (3) Nitrogen cannot form d p bond.
The presence or absence of hydroxy group on which carbon atom of sugar differentiates RNA and DNA? (1) 4th (2) 1st (3) 2nd (4) 3rd
(4) Single N– N bond is weaker than the single P P bond. 60.
Which of the following statements regarding sulphur is incorrect? (1) The oxidation state of sulphur is never less than +4 in its compounds. (2) S2 molecular is paramagnetic (3) The vapour at 200 C consists mostly of S8 rings. (4) At 600 C the gas mainly consists of S2
AIEEE 2011
[7]
72. A resistor 'R'and 2 F capacitor in series are connected through a switch to 200 V direct supply. Across the capacitor is a neon bulb that lights up at 120 V. Calculate the value of R to make the bulb light up 5 s after the
b
switch has been closed. log10 2.5 0.4
g
(1) 3.3 107
(2) 13 . 104
(3) 17 . 105
(4) 2.7 106
73. A current I flows in an infinitely long wire with cross section in the form of a semicircular ring of radius R. The magnitude of the magnetic induction along its axis is: (1)
0I 4R
(2)
0I 2R
(3)
0I 2 2 R
(4)
0I 2R
74. An object, moving with a speed of 6.25 m / s, is decelerated at a rate given by
dv 2.5 v dt where v is the instantaneous speed. The time taken by the object, to come to rest, would be: (1) 8 s (2) 1 s (3) 2 s (4) 4 s 75. Direction: The question has paragraph fllowed by two statements, Statement-1 and Statement-2. Of the given four alternatives after the statements, choose the one that describes the statements. A thin air film is formed by putting the convex surface of a plen-convex lens over a plane glass plate. With monochromatic light, this film gives an interference pattern due to light reflected from the top (convex) surface and the bottom (glass plate) surface of the film. Statement - 1: When light reflects from the air-glass plate interface, the reflected wave surffers a phase change of . Statement - 2: The centre of the interference pattern is dark. (1) Statement - 1 is flase, Statement - 2 is true. (2) Statement - 1 is true, Statement - 2 is flase. (3) Statement - 1 is true, Statement - 2 is true and Statement - 2 is the correct explanation of Statement - 1 (4) Statement - 1 is true, Statement - 2 is true and Statement - 2 is not the correct explantion of Statement - 1. 76. Two bodies of masses m and 4m are placed at a distance r. The gravitational potential at a point on the line joining them where the gravitational field is zero is:
9Gm (1) r (3)
4Gm r
AIEEE 2011
(2) zero (4)
6Gm r
77. This equation has Statement - 1 and Statement - 2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement -1: Sky wave signals are used for long distance radio communication. These signal are in general, less stable than ground wave signals. Statement -2: The state of ionosphere varies from hour to hour, day to day and season to season. (1) Statement - 1 is flase, Statement - 2 is true. (2) Statement - 1 is true, Statement - 2 is flase. (3) Statement - 1 is true, Statement - 2 is true and Statement - 2 is the correct explanation of Statement - 1 (4) Statement - 1 is true, Statement - 2 is true and Statement - 2 is not the correct explantion of Statement - 1. 78. A fully charged capacitor C with initial charge q0 is connected to a coil of self inductance L at t 0. The time at which the energy is stored equally between the electric and the magnetic field is: (1)
LC
(2) LC
LC (4) 2 LC 4 79. This equation has Statement - 1 and Statement - 2. Of the four choices given after the statements, choose the one that best describe the two statements. Statement - 1: A metallic surface is irradiated by a monochromatic light of frequency v v0 (the threshold frequency). The maximum kinetic energy and the stopping potential are K max and V0 respectively. If the frequency incident on (3)
the surface is doubled, both the K max and V0 are also doubled. Statement - 2: The maximum kinetic energy and the stopping potential of photoelectrons emitted from a surface are linearly dependent on the frequency of incident light. (1) Statement - 1 is flase, Statement - 2 is true. (2) Statement - 1 is true, Statement - 2 is flase. (3) Statement - 1 is true, Statement - 2 is true and Statement - 2 is the correct explanation of Statement - 1 (4) Statement - 1 is true, Statement - 2 is true and Statement - 2 is not the correct explantion of Statement - 1. 80. Water is flowing continuously from a tap having an internal diameter 8 10 3 m . The water velocity as it leaves the tap is 0.4 m s 1 . The diameter of the water stream at a distance 2 10 1 m below the tap is close to: (1) 3.6 10 3 m
(2) 5.0 10 3 m
(3) 7.5 10 3 m
(4) 9.6 10 3 m
[9]
81. A mass M, attached to a horizontal spring, executes SHM with amplitude A1 . When the mass M passes through its mean position then a smaller mass m is placed over it and both of them move together with amplitude A2 . The ratio
FG A IJ H A K is: 1
of
2
F M mIJ (1) G H M K
1 2
(2)
M M m
F M IJ (4) G H M mK
M m (3) M
1 2
82. Two particles are executing simple harmonic motion of the same amplitude A and frequency along the x-axis. Their mean position is separated by distance
b
g
86. A screw gauge gives the following reading when used to measure the diameter of a wire. Main scale reading: 0mm. Circular scale reading: 52 dividisons Given that 1 mm on main scale corresponds to 100 divisions of the circular scale. The diameter of wire from the above data is (1) 0.005 cm (2) 0.52 cm (3) 0.052 cm (4) 0.026 cm 87. A mass m hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass m and radius R. Assuming pulley to be a perfect uniform ciruclar disc, the acceleration of the mass m, if the string does not slip on the pulley, is (1)
g 3
(3) g
(2)
3 g 2
(4)
2 g 3
X 0 X 0 A . If the maximum separation between them
b
g
is X 0 A , the phase difference between their motion is
b g
88. The transverse displacement y x, t of a wave on a string is given by
b g
e
ax 2 bt 2 2 abxt
j
(1) 6
(2) 2
This represents a:
(3) 3
(4) 4
(1) standing wave of frequency
y x, t e
1
83. If a wire is stretched to make it 0.1% longer, its resistance will (1) decrease by 0.05% (2) increase by 0.05% (3) increase by 0.2% (4) decrease by 0.2% 84. A water fountain on the ground sprinkles water all around it. If the speed of water coming out of the fountain is v, the total area around the fountain that gets wet is: (1) (3)
v2 g
(2)
2
v4
(4)
g2
v2 g
v4 2 g2
85. A thermally insulated vessel contains an ideal gas of molecular mass M and ratio of specific heats . It is moving with speed v and is suddenly brought to rest. Assuming no heat is lost to the surroundings, its temperature increases by: (1)
b 1g Mv K
(3)
b 1g Mv K
2
2R
2
2R
AIEEE 2011
b g b g
1 2 (2) 2 1 R Mv K (4)
Mv 2 K 2R
b
(2) wave moving in +x direction with speed
a b
(3) wave moving in –x direction with speed
b a
(4) standing wave of frequency
b
89. A car is fitted with a convex side-view mirror of focal length 20 cm. A second car 2.8 m behind the first car is overtaking the first car at a relative speed of 15 m / s. The speed of the image of the second car as seen in the mirror of the first one is: (1) 15 m / s (3)
(2)
1 m/s 15
1 m/ s 10
(4) 10 m / s
90. Let the x - z plane be the boundary between two transparent media. Medium 1 in z 0 has a refractive index of
2 and medium 2 with z < 0 has a refractive
index of 3. A ray of light in medium 1 given by the vector A 6 3i 8 3 j 10k is incident on the plane of separation. The angle of refraction in medium 2 is: (1) 75o (2) 30o (3) 45o (4) 60o
[ 10 ]
Triumph Academy
AIEEE-2011 Main Test Paper
Solutions
AIEEE 2011MAIN TEST PAPER SOLUTIONS Mathematics (Solution) 1. [4]
[For 1 , roots are equal]
P (at least one failure) =1 - P (all success) 1 1 p 5
5
0 p 1
1 0 p5 2
31 32 31 1 32 32
FG IJ 5 H K L 1O p M 0, P N 2Q
F H
2
I6 K
e
C0 C1 x C2 x ... C6 x 6
e
2
6
je
j
6
j .e
6
e
C0 C1 x C2 x ... C6 x 6
2
j e
6
je
4
C0 C5 C2 C3 C3 C1 6
6
6
6
6
6
6
12
j
j
4
1 cos 2 ( x 2 )
x2 lim x 2
k
2
k
4 1 0
2
2 1
k 2 6k 8 0
( k 2) ( k 4) 0 k 2, 4
x2
Number of solution = 2
lim
x 2
2 sin( x 2 ) x2
8. [3] mid pt of AB (1, 3, 5) lies on the line dr of AB ( 0, 6, 4) dr of given line (1, 2 , 3)
2 sin( x 2 ) x 2
d 2 y dx dx 2 dy
2 RHL = lim
x2
x 2
FG dy IJ H dx K
2
2 sin( x 2)
2 sin( x 2 )
LHL = lim
1
I JK
7. [3] For non-zero
144
x 2
d dx d 1 dy dy dy dy / dx
FG IJ H K
36 ( 300) 120
3. [2] lim
d2y 2 d 2x dx 3 2 dy dy dx
= Coeff of x 7 in the expansion of 6
2
2. [4] Coeff of x 7 in the expansion of (1 x ) (1 x )
6
F I F GH JK GH dy d F 1 I dx G J dx H dy / dx K dy d2x
6. [1]
x2
lim
x 2
2 sin( x 2 ) 2 x2
lines are perpendicular
0 (1) 6( 2) ( 4 ) ( 3) 0
AB is perpendicular to the given line B is mirror image of A Statement (1) is true . Statmeent (2) is also true But not sufficient.
LHL RHL The limit does not exist 4. [4] Statement-1 ( x , y ) A ( x x ) integer
9. [3] Given Statement is equivalent to
x x 0 x A is reflexive
( P ~ R) Q
( y x ) integer ( x y ) integer
b
g
( y x ) ( z y ) ( z x ) integer
Hence x Az A in transitive
g b
Negation is ~ ( P ~ R ) Q ~ (Q ( P ~ R)
A is symmetric ( y x ) integer and ( z y ) integer
L1
L2
A is equivalance
Statement-2 ( x , y ) B x y , for , some rational
( 0, 1) B as 0 0.1, 0 is a rational no. but (1, 0) B as 1 . 0 there is no rational existing. Hence B is not symmetric
O
10. [4]
Q
P (–2, –2)
R
(1, –2)
L3
5. [1] The coefficients of quadratic equation
z 2 z 0 are real. Therefore complex roots exists in conjugate pair. Let
z1 1 iy , z2 1 iy
z1 z2 2
z1z2 1 y 2 (1, )
PR OP 2 2 PQ OQ 5
Statement (1) is True Statment (2) is False 11. [1] Let number of months = n
PCM
81-B/3, Lohagal Road, Ajmer 305001 Tel: 0145-2628805, 2628178
[1]
Triumph Academy
AIEEE-2011 Main Test Paper
Solutions
Total saving = 200 + 200 +
z
LM MN
/4
d200 240 280... to (n 2) termsi
2I
8 log(1 tan ) log
0
n2 2 ( 200) ( n 3) 40 2
d
11040 400
d
10640 ( n 2 ) 200 ( n 3) 20
i
i
8 log 2 d 8 log 2
0
FI GH 4 JK
I log 2
532 ( n 2) (10 n 3)
z
/4
F 2 I OP d GH 1 tan JK P Q
( n 2) (n 7)
n 2 5n 546 0
dr's of line :
( n 21) (n 26) 0
dr's of normal plane (1, 2, 3)
n 21 ( n 0)
15. [2]
so,
Let angle between line and plane is
then
(1, 2, ) 1 1 2.2 3
cos( 90 )
4 1 9 4 1 2
2 12. [2] e 1
b
2
a
2
b2
2 5
14
1 e 1 2
2 3 5 5
given cos
a 2 : b2 5 : 3
Equation of ellipse
14 5 2
(1) passes through ( 3, 1)
( 3) (1) 9 1 32 k 5 3 5 3 15 Equation of ellipse 2
5 3 sin 14 14
3 5
Hence
2 x2 y k . . . (1) 5 3
3 5
sin
a2
9( 5) ( 3 5)
30 20
2
3 14
2
2
2 3
z x
2 x2 y 32 5 3 15
16. [1]
applying newton leibnitz formula
13. [2] A sin 2 x cos 4 x
f’
cos x cos x 1 4
2
FG H
cos2 x
1 2
IJ K
F Now 0 G cos H
2
I
z
1 x
Local minima
IJ K
f ( x ) ( x sin x ) 1 0 2
1 4
z z 0
2
17. [4]
dx
8 log(1 tan ) 1 tan 2
1
Now x x 0 . . . (1) case 1 x 0 then xx0
.sec dx 2
00 x
8 log(1 tan ) d .
. . (1)
0
L F IO I z 8 log M1 tan G J P . d MN H 4 K PQ L (1 tan ) OP d I z 8 log M1 N 1 tan Q F 2 I d . . . (2) I z 8 log G H 1 tan JK
case 2 x 0 then (1) x x 0
/4
also
f ( x)
x x
/4
I
f ( x ) x sin x Hence f ( x ) is maximum at x and local minima at x 2
dx sec 2 I
+ 2
Local maxima
x tan
/4
–
0
3 4
1 x 2
8 log(1 x )
0
+
f
3 A1 4 1
14. [2]
2
t sin t dt
0
3x 2 5 y 2 32
f ( x)
2 x 0 x 0 x R
Hence x ( , 0)
0
/4
18. [4]
xi a , 2a . . . 25a, 26a, . . . 50a
0
Median =
/4
0
e
25a 26a 25.5 a 2
j
Now xi x 24.5 a , 23.5 a . . . 0.5 a , 0.5 a . . .
using equation (1) and (2)
23.5 a , 24.5 a
PCM
81-B/3, Lohagal Road, Ajmer 305001 Tel: 0145-2628805, 2628178
[2]
Triumph Academy
FH
xi x 2 0.5 a a 15 . a ...24.5 a
LM N
25 0.5 a 24.5 a 2
2
AIEEE-2011 Main Test Paper
Solutions
IK
22. [3]
OP Q
d i
z zd
V T
I
Mean deviation
625 a 50
xi x
n
F t I V dT i I G kTt k J 2K H
( 2 a b ) ( a b ) ( a 2b )
23. [3]
( 2 a b ) ( a b ) a 2( a b ) b
LM N
( 2 a b ) ( a a ) b ( a b ) a 2 ( a b ) b (b b ) a
{
( 2 a b ) (b 2 a ) 2 2 4 a b
}OPQ
24. [3]
4 1 5
di
V T I
kT 2 2
F C I Pd C D i P d C i GH D JK Pd Di Pd Di O Pd Di 1 F CI P G J P dC i H DK P
d i d i d ABAi A B
20. [4]
x 0
lim
x 3/ 2
1 x 1 lim x 0 x
(1 x ) 1 1 x 0 x 1 x 1
25. [3]
lim
sin( p 1) x sin x q lim f ( x ) lim x x 0 x 0
x 0
d ABi
T
B T A T BA AB AB is symmetric statement 2 is TRUE but NOT a correct explanationof statement 1.
b
A B 1
A B 1
A 1, B 1
26 [3]
LM sin( p 1) x ( p 1) sin x OP x Q N ( p 1) x
p q2
A T ABA
statement 1 is TRUE
g e j 7
2
7
14
Statement 1 number of way's of distributing n identical things into m person when each receive at least one
n 1Cm 1 hence total ways 101 C41 9 C3
( p 1) 1 p 2
T
A B 2
1 2
lim
T
Given AB BA
x x2 x
x 0
( C D )
A BA AB A ABA T
q lim f ( x )
0
T2 kT 2 2 2
2
T
2
kT k
a 1, b 1, a . b 0
19. [2]
i
kT kt dt
0
50
a 4
T
dV
625 a
dV kT kt dt
Statement 2 is correct but not correct explanation of staement 1.
1 3 2 2 2
27 [3]
e j on the parabola. Distance of the line 2
Let a point P t , t
y x 1
from P
t2 t 1
t t 1 2
2
2
FG t 1 IJ H 2K
21. [3]
2
t2 t 1 0
3 4
2
2 2 Diameter of x y ax = Radius of
x 2 y 2 c2
a c
3 4 2
3 4 2
PCM
(when t
1 ) 2
3 2 8
81-B/3, Lohagal Road, Ajmer 305001 Tel: 0145-2628805, 2628178
[3]
Triumph Academy
30. [1] b c b d
z z 1
e
28. [4] Area x dx 1 dx x 0 1
Fx I G J d ln x i H2K 2
1
e 1
1 3 1 2 2
0
29. [2]
dy y3 dx
d i Now y d 0i 2
PCM
ln y 3 x C
y 5e x 3
AIEEE-2011 Main Test Paper
Solutions
dy dx y3
y 3 C e
23 C
d i
b c d 0
e
j
c d b
d c b
Now
a d 0
a c a b 0
a c a b
e j F a c I d c G Jb H a b K
x
C 5
y ln 2 5e
ln 2
3 7
81-B/3, Lohagal Road, Ajmer 305001 Tel: 0145-2628805, 2628178
[4]
[1]
AIEEE 2011
Mathematics 1.
Consider 5 independent Bernulli's trails each with probability of sucess p. If the probability of at least one failure is greater than or equal to lies in the interval
F 11 O (1) G 12 , 1P H Q F 3 11 O (3) G 4 , 12 P H Q Ans:
Ans:
x2
F 1 3O (2) G 2 , 4 P H Q F 1O (4) G 0, 2 P H Q
LHL = lim
x2
RHL = lim
x2
4.
5
2.
The coefficient of x 7 in the expansion of (1 x x 2 x 3 ) 6 is (1) 132 (2)144 (3) – 132 (4) – 144
Ans:
[4] Coeff of x 7 in the expansion of (1 x ) (1 x 2 )
e
j
Coeff of x 7 in the expansion of 6
0
6
je j e C j e C j C e C j e C je C j
C1x 6C2 x 2 ... 6C6 x 6 . 6 C0 6 C1x 2 6C2 x 4 ... 6C6 x12 6
6
0
6
6
5
2
6
3
6
3
36 ( 300) 120
3.
F limG GH x2
144
1 cos{2( x 2)} x2
2
(3) equals 2
1
Let R be the set of real numbers Statement -1 A {( x , y ) R R : y x is an integer} is an equivalence relation on R.
Ans:
[4] Statement-1 ( x , y ) A ( x x ) integer
x x 0 x A is reflexive ( y x ) integer ( x y ) integer A is symmetric ( y x ) integer and ( z y ) integer
( y x ) ( z y ) ( z x ) integer
Hence x Az A in transitive A is equivalance Statement-2 ( x , y ) B x y , for , some rational ( 0, 1) B as 0 0.1, 0 is a rational no.
I JJ K
but (1, 0) B as 1 . 0 there is no rational existing. Hence B is not symmetric
1 (1) equals
2 sin( x 2) 2 sin( x 2) lim 2 x2 x2 x 2
Statement -2 B {( x , y ) R R : x y for some rational number } is an equivalence relation on R. (1) Statement -1 is false, Statement -2 is true (2) Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for statement-1 (3) Statement-1 is true, Statement -2 is true, Statement-2 is not correct explanation for Statement-1 (4) Statement -1 is true, Statement-2 is false
6
Coeff of x 7 in the expansion of (1 x ) 6 (1 x 2 ) 6
eC
2 sin( x 2) 2 sin( x 2) lim x2 x2 x2
LHL RHL The limit does not exist
31 1 32 32
FG 1 IJ H 2K L 1O p M0, P N 2Q
2 sin( x 2) x2
2
31 1 1 p 32
0 p5
1 cos 2( x 2) x2
lim
5
0 p5 1
x2
31 , then p 32
[4] P (at least one failure) =1 - P (all success)
[2] lim
(2) does not exist (4) equals 2
[2]
5.
AIEEE 2011
Let , be real and z be complex number. If
7.
z 2 z 0 has two distinct roots on the line
Re z 1 , then it is necessary that (1) (1, ) (2) (0, 1)
Ans:
kx 4 y z 0 2x 2 y z 0 possess a non-zero solutoin is (1) zero (2) 3 (3) 2 (4) 1
(4) 1
(3) ( 1, 0)
The number of values of k for which the linear equations 4 x ky 2 z 0
[1] The coefficients of quadratic equation z 2 z 0 are real. Therefore complex roots exists in conjugate pair. Let
Ans:
[3] For non-zero 4
k
2
z1 1 iy , z2 1 iy
k
4 1 0
z1 z2 2
2
2 1
z1z2 1 y 2 (1, )
k 2 6k 8 0
[For 1 , roots are equal]
( k 2) ( k 4) 0 k 2, 4
Number of solution = 2
2
6.
d x equal dy 2
8.
F d y I F dy I (1) – G dx J GH dx JK H K F d y I F dy I (3) – G H dx JK GH dx JK
F d yI (2) G H dx JK F d y I F dy I (4) G dx J GH dx JK H K
3
2
2
1
2
[1]
3
2
FG IJ H K
FG H
FG H
IJ K
d d dx dy dy / dx dy
1
FG dy IJ H dx K
2
d 2 y dx 2 dx dy
d2y 2 d 2x dx 3 2 dy dy dx
FG IJ H K
x y 1 z 2 1 2 3
Statement -2 x y 1 z 2 bisects the line 1 2 3 segment joining A (1, 0, 7) and B (1, 6, 3) (1) Statement -1 is false, Statement -2 is true (2) Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for statement-1 (3) Statement-1 is true, Statement -2 is true, Statement-2 is not correct explanation for Statement-1 (4) Statement -1 is true, Statement-2 is false
2
The line
2
d 2 x d dx d d 2 dy dy dy dy / dx dy
point B(1, 6, 3) in the line
2
2
Ans:
1
2
Statement -1 The point A (1, 0, 7) is the mirror image of the
IJ K Ans:
[3] mid pt of AB ( 2 , 3, 5) lies on the line dr of AB ( 0, 6, 4) dr of given (1, 2 , 3) 0 (1) 6( 2) ( 4 ) ( 3) 0 AB is perpendicular to the given line B is mirror image of A Statement (1) is true . Statmeent (2) is also true But not sufficient.
[3]
9. .
Consider the following statements P : Suman is brilliant Q : Suman is rich R : Suman is honest The negation of the statment "Suman is brilliant and dishonest if and only if sumna is rich" can be expressed as (1) ~ ( P ~ R) Q (2) ~ P (Q ~ R) (3) ~ (Q ( P ~ R))
Ans:
AIEEE 2011
11.
A man saves Rs 200 in each of the first three months of his services. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service wil be Rs. 11040 after (1) 21 months (2) 18 months (3) 19 months (4) 20 months
Ans:
[1] Let number of months = n
(4) ~ Q ~ P R
Total saving = 200 + 200 +
d200 240 280... to (n 2) termsi n2 11040 400 c2(200) (n 3)40h 2 10640 (n 2) c200 (n 3)20h
The lines L1: y x 0 and L2 : 2 x y 0
532 ( n 2) (10 n 3)
intersect the line L3 : y 2 0 at P and Q respectivley. The bisector of the acute angle
[3] Given Statement is equivalent to ( P ~ R) Q
Negation is ~ ( P ~ R) Q
10.
between L1 and L2 intersects L3 at R. Statement -1 The ratio PR : RQ equals 2 2 : 5 . Statement -2 In any triangle, bisector an angle divides the trianle into two similar triangles. (1) Statement -1 is false, Statement -2 is true (2) Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for statement-1 (3) Statement-1 is true, Statement -2 is true, Statement-2 is not correct explanation for Statement-1 (4) Statement -1 is true, Statement-2 is false Ans:
12.
( n 2 ) (n 7 )
n 2 5n 546 0 ( n 21) (n 26) 0
n 21 (Q n 0)
Equations of the ellipse whose axes are the axes of coordinates and which passes through the point
( 3, 1) and has eccentricity
Ans:
2
5 is
(1) 5x 2 3 y 2 32 0
(2) 3x 2 5 y 2 32 0
(3) 5x 2 3 y 2 48 0
(4) 3x 2 5 y 2 15 0
2 [2] e 1
[4]
b2 a2
b2 2 3 1 e2 1 5 5 a2
a 2 : b2 5 : 3
Equation of ellipse x2 y2 k . . . (1) 5 3
(1) passes through ( 3, 1)
( 3) 2 (1) 2 9 1 32 5 3 5 3 15 Equation of ellipse k
x 2 y 2 32 5 3 15 PR OP PQ OQ
2 2 5
Statement (1) is ture Statment (2) is False
3x 2 5 y 2 32
[4]
13.
If A sin 2 x cos4 x then for all real x. (1) (3)
Ans:
AIEEE 2011
3 13 A 4 16
(2)
13 A1 16
(4) 1 A 2
15.
3 A1 4
If the angle between the line x
y 1 z 3 and 2
1 the plane x 2 y 3z 4 is cos
[2] A sin 2 x cos4 x
(1) 5 3
(2) 2 3
(3) 3 2
(4) 2 5
F GH
5 14
I JK
cos4 x cos2 x 1
FG H
cos2 x
1 2
IJ K
2
Ans:
3 2
FG H
1 2 Now 0 cos x 2
IJ K
[2] Let angle between line and plane is then dr's of line : (1, 2, )
2
dr's of normal plane (1, 2, 3)
1 4
3 A 1 4
z 1
14.
The value of
0
log 2 8
(3)
z 1
Ans:
[2] I
0
14
(4)
log 2 2
x tan
z
/4
0
I
z
LM MN
8 log 1 tan
0
z z
/4
0
/4
0
16.
8 log(1 tan ) .sec 2 dx 1 tan 2
/4
also
z LMMN
8 log(1 tan ) log
0
z
/4
8 log 2 d 8 log 2
0
I log 2
FG IJ H 4K
3 14
9( 2 5) (3 5) 2
30 20
2 3
FG H
IJ K
5 For x 0, 2 , define
z
t sin t dt
0
Then f has (1) local maximum at (2) lcoal maximum at (3) lcoal minimum at (4) local minimum at
z
and local minimum at 2 and 2 and 2 and local maximum at 2
x
Ans:
[1] f ( x )
t sin t dt
0
using equaiton (1) and (2)
2I
x
8 log 1
/4
14 5 2
f ( x)
FG IJ OP . d H 4 K PQ
LM (1 tan ) OP d N 1 tan Q F 2 IJ d . . . (2) 8 log G H 1 tan K
5 3 sin 14 14
3 5 Hence
8 log(1 x ) dx 1 x2
dx sec 2
2 5
given cos
(2) log 2
(1) log 2
4 1 9 4 1 2
3 5
sin
8 log(1 x ) dx is 1 x2
1 1 2.2 3
cos( 90 )
so,
FG 2 IJ OP d H 1 tan K PQ
applying newton leibnitz formula
[5]
AIEEE 2011
x x 2e0.5 a a 15. a ...24.5 a j i
2
f ( x) ( x sin x ) 1 0
625 a
f ( x ) x sin x
Mean deviation
Hence f ( x ) is maximum at x and local minima at x 2
17.
1
f ( x)
x x is
(3) (0, )
Ans:
The domain of the function (1)
[4]
(4) ( , 0)
f ( x)
(2) ( , ) {0} 19.
x x
case 1 x 0
xx0
Ans:
2 x 0 x 0 x R Hence x ( , 0)
(2) 2 (4) 4 xi a , 2a . . . 25a, 26a, . . .
25a 26a 25.5 a 2
i
Now xi x 24.5 a , 23.5 a . .. 0.5 a , 0.5 a . . .
23.5 a , 24.5 a
1 If a 1 3i k and b 2i 3 j 6k , then 7 10 the value of 2a b a b a 2b is
e
e
j
je
je
j
j
(2) 5 (4) 5
[2] a 1, b 1, a . b 0
o
( 2 a b ) (b 2 a )
t
2 2 4 a 4 b
bers a, 2a, . . ., 50 a is 50, then a equals
d
a 4
(2a b ) ( a a ) b (a b ) a 2 (a b ) b (b b ) a
If the mean deviation about the median of the num-
Median =
50
( 2a b ) ( a b ) a 2 ( a b ) b
x x 0
[4] 50a
i
n
(2a b ) (a b ) (a 2b )
case 2 x 0
Ans:
50
x x
(1) 3 (3) 3
00 x
(1) 5 (3) 3
625 a
e
1
Now x x 0
18.
25 0.5 a 24.5 a 2
4 1 5
20.
The value of p and q for which the function
R| sin( p 1) x sin x ,x0 x || f ( x) S q ,x0 || x x x , x 0 |T x 2
3
2
is continuous for all x in R, are, (1) p
1 3 ,q 2 2
(2) p
1 3 ,q 2 2
(3) p
5 3 ,q 2 2
3 1 (4) p , q 2 2
[6]
Ans:
AIEEE 2011
Ans:
[4] q lim f ( x ) x 0
x 0
lim
x0
z zb bg
V T
x x2 x x 3/ 2
lim
dV kT kt dt
[3]
T
dV
I
1 x 1 x
0
F GH
bg
t2 V T I kTt k 2
(1 x ) 1 1 x x0 1 x 1
lim
kT 2 k
1 2
q lim f ( x ) lim x 0
x 0
sin( p 1) x sin x x
L sin( p 1) x ( p 1) sin x OP lim M x Q N ( p 1) x
( p 1) 1 p 2
21.
1 3 pq2 2 2 2
x y c ( c 0) touch each other if 2
2
(1) a 2c
(2) 2 a c
(3) a c
(4) a 2c
Ans:
[3]
(2) P(C| D) P(C)
(3) P(C| D) P(C)
(4) P(C| D) P(C)
[3] P
kT 2 2
(4) I
[3] A BA AB A ABA
b g b g
b ABAg
1 k
k (T t ) 2 2
( C D )
b g F CI P G J PbC g H DK
differential euqation
(3) I
FG C IJ PbC Dg PbCg H D K P b Dg P b Dg
O P D 1
The value V ( t ) depreciates at a rate given by
2 (2) T
kT 2 2
Ans:
Let be the purchase value of an equaipment and V ( t ) be the value after it has been used for t years.
(1) e kT
T2 kT 2 2 2
Let A and B be two symmetric matrices of order 3 Statement -1 A (BA) and (AB) A are symmetric matrices Statement -2 AB is symmetric matrix if matrix multiplication of A with B commutative. (1) Statement -1 is false, Statement -2 is true (2) Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for statement-1 (3) Statement-1 is true, Statement -2 is true, Statement-2 is not correct explanation for Statement-1 (4) Statement -1 is true, Statement-2 is false
x 2 y 2 c2
dV (t ) k (T t ) , where dt k 0 is a constant and T is the total life in years of the equipment. Then the scrap value V(T) of the equipment is
0
24.
Diameter of x 2 y 2 ax = Radius of
22.
T
P ( D) (1) P(C| D) P(C )
The two circles x 2 y 2 ax and 2
Ans:
bg
V T I
I JK
If C and D are two events such that C D and P ( D ) 0 , then the correct statement among the following is
23.
x0
g
kT kt dt
AT B T AT ABA
statement 1 is TRUE Given AB BA
b ABg
T
T
B T AT BA AB
AB is symmetric statement 2 is TRUE but NOT a correct explanationof statement 1.
[7]
AIEEE 2011
3 4 2
If ( 1) is a cube root of unity, and
25.
(1 ) 7 A B . Then (A, B) equals (1) (–1, 1) (2) (0, 1) (3) (1, 1) (4) (1, 0)
b
[3] A B 1
Ans:
g e j 7
2
26
3 4 2
28.
The area of the region enclosed by trhe curves
b g
(1) 5/2 square units (2) 1/2 square units (3) 1 square units (4) 3/21 square units
A 1, B 1
Statement -1 The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is C3 . Statement -2 The number of ways of choosing any 3 places
z z 1
from 9 different places is C3 (1) Statement -1 is false, Statement -2 is true (2) Statement -1 is true, Statement -2 is true; Statement -2 is a correct explanation for statement-1 (3) Statement-1 is true, Statement -2 is true, Statement-2 is not correct explanation for Statement-1 (4) Statement -1 is true, Statement-2 is false
0
29.
F x I bln xg GH 2 JK 2
1
e 1
Ans: [2]
Statement 2 is correct but not correct explanation of staement 1.
curve x y 2 is 4
(2)
3
3 4
dy y3 dx
(4)
3 2
e j
Ans: [3] Let a point P t 2 , t on the parabola. Distance of the line y x 1 from P
23 C
b g
The vectors a and b are no perpendicular and c and d are two vectors satisfying : b c b d and a d 0 . Then the vector d is equal to: b c (2) b c a b
F I GH JK F b c I (4) b G J c H a b K
FG IJ H K F b c I (3) c G J c H a b K
t t 1 2
FG t 1 IJ H 2K
2
2
t t 1
3 4
2
2
t2 t 1 0
C5
y ln 2 5eln 2 3 7
y 5e x 3
b c d 0
e
Ans: [1] b c b d 2
dy dx y3
y 3 C ex
a c (1) c b a b
8
3 2 8
30.
b g
(2) 7 (4) 13
b g Now y b0g 2
hence total ways 10 1C4 1 9C3
1 3 1 2 2
bg
ln y 3 x C
The shortest distance between line y x 1 and
dy y 3 0 and y 0 2, the y ln 2 is equal dx
If
to: (1) –2 (3) 5
n 1Cm 1
(3)
1
1 dx x
0
Ans: [3] Statement 1 number of way's of distributing identical things into m person when each receive at least one
e
Ans: [4] Area x dx
9
(1)
3 2 8
y x, x e, y 1 / x and the positive x-axis is
9
27
7
14 2 1
1 ) 2
(when t
c d b
Now a d 0
a c a b
j
d c b
a c a b 0
e j F a c I d c G Jb H a b K
[1]
AIEEE 2011
PHYSICS PART C — PHYSICS 61. A Carnot engine operating between temperatures T1 and
63. Three perfect gases at absolute temperatures T1 , T2 and
1 . When T2 is lowered by 62 K, its 6
T3 are mixed. The masses of molecules are m1 , m2 and
T2 has efficiency
1 efficiency increases to . Then T1 and T2 are, 3 respectively: (1) 310 K and 248 K (2) 372 K and 310 K (3) 372 K and 330 K (4) 330 K and 268 K Sol: [2] As on reducing T2 , efficiency is increasing means T1 T2 . So, from 1
1 and 1
T2 , T1
T2 1 T1 6 T2 62 1 T1 3
m3 and the number of molecules are n1 , n2 and n3 respectively. Assuming no loss of energy, the final temperature of the mixtures is:
n12 T12 n22 T22 n32 T32 (1) n1T1 n2 T2 n3T3 (2)
1
2
3
3
(3)
n1T1 n2 T2 n3T3 n1 n2 n3
(4)
n1T12 n2 T22 n3T32 n1T1 n2 T2 n3T3
Sol: [3] From conservation of internal energy
b
g
n1Cv T1 n2 Cv T2 n3Cv T3 n1 n2 n3 TCv
Solving we get T2 310 K and T1 372 K
62. A pulley of radius 2 m is rotated about its axis by a force
e
bT T T g
j
T
n1T1 n2T2 n3T3 n1 n2 n3
F 20t 5t 2 newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about is axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is: (1) more than 9 (2) less than 3 (3) more than 3 but less than 6 (4) more than 6 but less than 9 Sol: [3]
z ze
or
jb g
e
20t 5t 2 2 I 10 t
d
0
j
4t t 2 dt
2t 2
t3 3
t3 0 3
t 6s
d t3 2t 2 dt 3
z FGH 6
0
2t 2
I JK
t3 dt 36 rad 3
So number of rotations than 6 bat greater than 3.
the speed of the boat is 150 . ms1 , the magnitude of the induced emf in the wire of aerial is: (1) 0.15 mV (2) 1 mV (3) 0.75 mV (4) 0.50 mV Sol: [1] As B v
Bv 5 105 15 . 2 0.15 10 3 V
0
2t 2
magnetic field is 5.0 10 5 N A 1 m 1 due north and horizontal. The boat carries a vertical aerial 2 m long. If
so, motional emf is
The direction of motion reverses when becomes zero.
64. A boat is moving due east in a region where the earth's
36 18 5.73 which is lesser 2
65. A thin horizontal circular disc is rotating about a vertical axis passing through its centre. An insect is at rest at a point near the rim of the disc. The insect now moves along a dimater of the disc to reach its other end. During the journey of the insect, the angular speed of the the disc: (1) first increases and then decreases (2) remains unchanged (3) continuously decreases (4) continuously increases Sol: [1] As during the said journey or insect, moment of inertia first decreases then increases so from angular momentum conservation, angular speed first increases then decreases.
[2]
AIEEE 2011
PHYSICS 66. Two identical charged spheres suspended from a common point by two massless strings of length are initially a
b
distance d d
g
Sol: [4] As the amount present is reducing by half amount in the given time interval so the time interval must be equal to a half life which according to question is 20 min.
apart because of their mutual
repulsion. The charge begins to leak from both the spheres at a constant rate. As a result the charges approach each other with a velocity v. Then as a function of distance x between them, (1) v x
(2) v x
(3) v x 1
(4) v x
1 2
1 2
69. Energy required for the electron excitation in Li from the first to the third Bohr orbit is: (1) 122.4 eV (2) 12.1 eV (3) 36.3 eV (4) 108.8 eV Sol: [4] For hydrogen like ions, the excitation energy is given by
E 136 . Z2
or
70. The electrostatic potential inside a charged spherical ball
or
l
x
3/ 2
FG 2k IJ q H mg K
l
T
x kq 2 2 mgx 2 x3
F
T F
x d mg
2
is given by ar 2 b where r is the distance from the centre ; a, b, are constants. Then the charge density inside the ball is
mg
(1) 6 a 0
(2) 24 a 0 r
(3) 6 a 0 r
(4) 24 a 0
dV we have E 2ar dr As the electric field is directly proportional to r so the charge density must be constant. For a uniform spherical
Sol: [1] From E
2 k q mg
Differentiating w.r.t. time we have 1
3 2 x v 2
vx
charge distrubition from E
2 k dq constant mg dt
r we get 3 0
r 2ar 3 0
1 2
67. 100 g of water is heated fvrom 30 oC to 50 oC. Ignoring the slight expansion of the water, the change in its internal energy is (specific heat of water is 4184 J / kg / K): (1) 2.1 kJ (2) 4.2 kJ (3) 84 kJ (4) 8.4 kJ Sol: [4] On ignoring expansion, from first law of thermodynamics, we have
Q U or
2
2
E 108.8 eV
kq 2 / x 2 mg
x/2 For d , tan
so
2 1
For Z 3, and n1 1, n2 3 , on solving we get
Sol: [2] For the figure shown, in equilibrium.
tan
F1 1I GH n n JK
U msT 01 . 20 4184 8.4 kJ
6a 0
71. Work done in increasing the size of a soap bubble from a radius of 3 cm to 5 cm is nearly (Surface tension of soap solution 0.03Nm 1 ): (1) 0.4 mJ
(2) 4 mJ
(3) 0.2 mJ
(4) 2 mJ
Sol: [1] Work done in increasing radius of a soap bubble from r1 to
r2 is
68. The half life of a radioactive substance is 20 minutes. The
b
g
approximate time interval t 2 t1 between the time t 2
2 1 of it has decayed and time t1 when of it had 3 3 decayed is (1) 28 min (2) 7 min (3) 14 min (4) 20 min when
e
j
e
j
W 8S r2 2 r12 8 0.03 52 32 104 0.4 mJ
72. A resistor 'R'and 2 F capacitor in series is connected through a switch to 200 V direct supply. Across the capacitor is a neon bulb that lights up at 120 V. Calculate the value of R to make the bulb light up 5 s after the
b
switch has been closed. log10 2.5 0.4
g
[3]
AIEEE 2011
PHYSICS (1) 3.3 107
(2) 13 . 104
(3) 17 . 105
(4) 2.7 106
where v is the instantaneous speed. The time taken by the object, to come to rest, would be: (1) 8 s (2) 1 s (3) 2 s (4) 4 s
Sol: [4] For a RC circuit during growth
F GH
V V0 1 e
so
or
t RC
F GH
120 200 1 e
I JK
t RC
I JK
t 5 Cn 2
6.25
v
t
2.5dt 0
t 2s
The question has paragraph fllowed by two statements, Statement-1 and Statement-2. Of the given four alternatives after the statements, choose the one that describes the statements.
t
or
dv
75. Direction:
t 5 n RC 2
R
z z 0
dv 2.5 v Sol: [3] Given, dt
C 2.3 log10
5 2
A thin air film is formed by putting the convex surface of a plane-convex lens over a plane glass plate. With monochromatic light, this film gives an interference pattern due to light reflected from the top (convex) surface and the bottom (glass plate) surface of the film.
5 2.7 106 6 2 10 2.3 0.4
73. A current I flows in an infinitely long wire with cross section in the form of a semicircular ring of radius R. The magnitude of the magnetic induction along its axis is: (1)
0I 4R
(2)
0I 2R
(3)
0I 2 2 R
(4)
0I 2R
dB
d
/ 2
R
Statement - 1:
0
Sol: [2] Consider an element in the form of a is a wire of width
Rd along the length of the cylinder. The current through the element is
FG I IJ b Rd g I d . H R K The magnetic field at O due to the element is
When light reflects from the air-glass plate interface, the reflected wave surfers a phase change of . Statement - 2: The centre of the interference pattern is dark. (1) Statement - 1 is false, Statement - 2 is true. (2) Statement - 1 is true, Statement - 2 is false. (3) Statement - 1 is true, Statement - 2 is true and Statement - 2 is the correct explanation of Statement - 1 (4) Statement - 1 is true, Statement - 2 is true and Statement - 2 is not the correct explanation of Statement - 1.
FI I G d J H K
Sol: [3] Statement 1 and 2 both are true and statement 2 is a correct explanation of statement-1
2R
76. Two bodies of masses m and 4m are placed at a distance r. The gravitational potential at a point on the line joining them where the gravitational field is zero is:
0
From symmetry of figure, the net field is
z
B
0
0
FG I d IJ H K sin I 2R
0 2
R
74. An object, moving with a speed of 6.25 m / s, is decelerated at a rate given by
dv 2.5 v dt
(1)
9Gm r
(2) zero
(3)
4Gm r
(4)
Sol: [1] Refer figure shown x m
r–x P
4m
6Gm r
[4]
AIEEE 2011
PHYSICS For gravitating field at P to be zero
b g b g
Gm G 4m 2 x2 rx
77. This question has Statement - 1 and Statement - 2. Of the four choices given after the statements, choose the one that best decrribes the two statements.
q0 2
for an LC circuit, from q q0 cost for q
r x 3 So, gravitational potential at P will be
F Gm IJ FG Gx4m IJ FG Gm IJ FG G 4m IJ 9Gm V G H x K H r x K H r / 3 K H 2r / 3 K r
q
cost
q0 2
we get
1 2
or
t t
4
2C 4
79. This question has Statement - 1 and Statement - 2. Of the four choices given after the statements, choose the one that best decrribes the two statements.
Statement - 1: Sky wave signals are used for long distance radio communication. These signals are in gneeral, less table than ground wave signals. Statement - 2: The state of ionosphere varies from hour to hour, day to day and season to season. (1) Statement - 1 is false, Statement - 2 is true. (2) Statement - 1 is true, Statement - 2 is false. (3) Statement - 1 is true, Statement - 2 is true and Statement - 2 is the correct explanation of Statement - 1 (4) Statement - 1 is true, Statement - 2 is true and Statement - 2 is not the correct explanation of Statement - 1. Sol: [3] Statement - 1 is true, Statement - 2 is true and
Statement - 2 is the correct explanation of Statement - 1 78. A fully charged capacitor C with iniital charge q0 is connected to a coil of self inductance L at t 0. The time at which the enrgy is stored equally between the electric and the magnetic fields is: (1) (3)
(2) LC
LC
4
(4) 2 LC
LC
Sol: [3] As per question, at the said instent U B U e U net / 2 So it at this instant if the charge on capacitor plates is q then so
F I GH JK
q 2 1 q02 Ue 2C 2 2C
Statement - 1: A metallic surface is irradiated by a monochromatic light of frequency v v0 (the threshold frequency). The maximum kinetic energy and the stopping potential are
K max and V0 respectively. If the frequency incident on the surface is doubled, both the K max and V0 are also doubled. Statement - 2: The maximum kinetic energy and the stopping potential of photoelectrons emitted from a surface are linearly dependent on the frequency of incident light. (1) Statement - 1 is false, Statement - 2 is true. (2) Statement - 1 is true, Statement - 2 is false. (3) Statement - 1 is true, Statement - 2 is true and Statement - 2 is the correct explanation of Statement - 1 (4) Statement - 1 is true, Statement - 2 is true and Statement - 2 is not the correct explanation of Statement - 1. Sol: [1] Let on doubling v, new value of maximum KE be h 2v 2 Kmax Kmax , then Kmax
b g
As, Kmax hv Thus on doubling v, maximum KE and hence stopping potential become more than double so statement 1 is false.
80. Water is flowing continuously from a tap having an internal diameter 8 10 3 m . The water velocity as it leaves the tap is 0.4 ms1 . The dimaeter of the water stream at a distance 2 10 1 m below the tap is close to: (1) 3.6 10 3 m
(2) 5.0 10 3 m
(3) 7.5 10 3 m
(4) 9.6 10 3 m
[5]
AIEEE 2011
PHYSICS
b
Sol: [1] If v2 be the speed at depth h below the tap, then
v2
v12
2 gh
phase difference between their motion is
2
(1)
6
(2)
2
v1 d1 v2
(3)
3
(4)
4
From equation of continuty v1d12
or
d2
v2 d 2
v1
ev12 2 ghj
1/ 2
Sol: [3] The positions of two particles w.r.t. a common origion O are
d1
x1 A sin t
0.4
1/ 2 . 4g b016
x0
3.6 10 m
81. A mass M, attached to a horizontal spring, executes SHM with amplitude A1 . When the mass M passes through its mean position then a smaller mass m is placed over it and
FG A IJ H A K is: 2
M M m
F M IJ (4) G H M mK
M m (3) M
gb g A F M mIF I G JG J A H M KH K F k I F M m IJ GH M m JK G H M K F kI GH M JK
b g
1
2
1
. 3
2 2 A A V Here volume of the wire remains same so for small changes in length, fractional change in resistance is given by
1 2
g b
2
b g A 2 cosbt / 2g sin / 2
Sol: [3] Resistance of a wire R
M 1 A1 M m 2 A2
or
O’
sin / 2 must be half. For that
R 2 0.2% R
Sol: [1] From momentum conservation just before and after putting the mass m, we have
b
x0
83. If a wire is stretched to make it 0.1% longer, its resistance will (1) decrease by 0.05% (2) increase by 0.05% (3) increase by 0.2% (4) decrease by 0.2%
1
(2)
O
As the maximum separation is x0 A so the value of
both of them move together with amplitude A2 . The ratio
1 2
g
x x2 x1 x0 A sin t sin t
3
F M mIJ (1) G H M K
b
P2
The separation between them is
8 103
0.2 8 103
of
P1
x2 x0 A sin t
on substituting values of v1, h and d1, we obtain d2
g
the maximum separation between them is X 0 A , the
84. A water fountain on the ground sprinkles water all around it. If the speed of water coming out of the fountain is v, the total area around the fountain that gets wet is: (1) M m M
(3)
82. Two particles are executing simple harmonic motion of the same amplitude A and frequency along the x-axis. Their
b
g
mean position is separated by distance X 0 X 0 A . If
v2
(2)
g2 v4
(4)
g2
v2 g
v4 2 g2
Sol: [3] Maximum area over which water sprinkels Rmax 2 where Rmax is the maximum possible range, given by
Rmax
v2 g
and thus maximum area is
v 4 g2
[6]
AIEEE 2011
PHYSICS 85. A thermally insulated vessel contains an ideal gas of molecular mass M and ratio of specific heats . It is moving with speed v and is suddenly brought to rest. Assuming no heat is lost to the surroundings, its temperature increases by: (1)
b 1g Mv K
(3)
b 1g Mv K
b g b g
1 2 (2) 2 1 R Mv K
2
2R
2
(4)
2R
So
T
b 1g Rv
F mR I F mR I FG a IJ . . . (ii) GH 2 JK GH 2 JK H R K 2
2
On Solving (i) and (ii) we get, a
2g 3
b g
88. The transverse displacement y x, t of a wave on a string is given by
Mv 2 K 2R
b g
y x, t e
Sol: [1] As the vessel is suddenly brought to rest, the kinetic associated with the motion of centre of mass (ordered kinetic energy) will be abosrbed by the gas and is now energy appear as the inerease in the disordered kinetic energy of gas and results in increase in the temperature of the gas, thus 1 2 mv nCv T 2
TR
FG IJ FG R IJ T H K H 1K
e
ax 2 bt 2 2 abxt
j
This represents a: 1
(1) standing wave of frequency
b
(2) wave moving in +x direction with speed
a b
(3) wave moving in –x direction with speed
b a
1 2 m mv 2 M
2
2M (where m is the mass of gas enclosed in the container)
(4) standing wave of fcrequency
b
Sol: [3] The given expression can be written
86. A screw gauge gives the following reading when used to measure the diameter of a wire Main scale reading: 0mm. Circular scale reading: 52 divisions Given that 1 mm on main scale corresponds to 100 divisions of the circular scale. The diameter of wire from the above data is (1) 0.005 cm (2) 0.52 cm (3) 0.052 cm (4) 0.026 cm Sol: [3] As, least count of screw gauge is given by pitch 1 0.01 mm Number of disions on circular scale 100
So the given reading is 52 LC 0.52 mm 0.052 cm
87. A mass m hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass m and radius R. Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass m, if the string does not slip on the pulley, is (1)
g 3
(3) g
(2)
b g
y x, t e
Sol: [4] For motion of mass m mg T ma . . . (i) For rotational motion of pulley (Disc)
e
a x bt
j
2
which represents a wave travelling in –x direction with speed
b . a
89. A car is fitted with a convex side-view mirror of focal length 20 cm. A second car 2.8 m behind the first car is overtaking the first car at a relative speed of 15 m / s. The speed of the image of the second car as seen in the mirror of the first one is: (1) 15 m / s (3)
(2)
1 m/s 15
1 m/ s 10
(4) 10 m / s
Sol: [3] On differentiating the mirror equation w.r.t. time, we obtain.
3 g 2
2 (4) g 3
or
1 dv 1 du 0 v 2 dt u 2 dt
FG IJ du H K dt F 20 IJ 15 1 m / s G H 280 20 K 15 dv v 2 du f 2 dt u dt u f 2
2
[7]
AIEEE 2011
PHYSICS 90. Let the x - y plane be the boundary between two transparent media. Medium 1 in z 0 has a refractive index of
2 and medium 2 with z < 0 has a refractive
index of 3. A ray of light in medium 1 given by the vector A 6 3i 8 3 j 10k is incident on the plane of separation. The angle of refraction in medium 2 is: (1) 75o (2) 30o o (3) 45 (4) 60o
Sol: [3] In our opinion this question can be solved only if we consider the boundary between the two transparent media to x - y plane rather than x-z plane given wrongly in question. If 1 be the angle of incident ray with the normal to z-axis then using the direction cosine formula with z-axis we have
cos 1
10
e6 3j e8 3j b10g 2
2
1 60o From Snell's law n1 sin 1 : n2 sin 2 we have
e 2 j sin 60 e 3 j sin o
sin 2
2
4
2 3
2
3 1 2 2
2
10 1 20 2
