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IIT JEE Practice Papers
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FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com
FULL TEST IV
Time Allotted: 3 Hours Maximum Marks: 432 P lease read the inst ruct ions care fu l ly. You are a l lo t ted 5 minutes
spec i f ica l ly for th is purpose. You are not a l lowed to leave the Examinat ion Hal l before the end of
the test .
INSTRUCTIONS
A. General Instructions 1. Attempt ALL the questions. Answers have to be marked on the OMR sheets. 2. This question paper contains Three Parts. 3. Part-I is Physics, Part-II is Chemistry and Part-III is Mathematics. 4. Each part has only one section: Section-A. 5. 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, 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 02 and 09 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.
Section-A (03 to 08) 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.
Name of the Candidate
Enrolment No.
ALL
IND
IA T
ES
T S
ER
IES
FIITJEE JEE (Main), 2013Fr
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Term
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12
AITS-FT-IV-PCM-JEE(Main)/13
FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com
2
Useful Data
PHYSICS
Acceleration due to gravity g = 10 m/s2
Planck constant h = 6.6 1034 J-s
Charge of electron e = 1.6 1019 C
Mass of electron me = 9.1 1031 kg
Permittivity of free space 0 = 8.85 1012 C2/N-m2
Density of water water = 103 kg/m3
Atmospheric pressure Pa = 105 N/m2
Gas constant R = 8.314 J K1 mol1
CHEMISTRY
Gas Constant R = 8.314 J K1 mol1 = 0.0821 Lit atm K1 mol1 = 1.987 2 Cal K1 mol1 Avogadro's Number Na = 6.023 1023 Plancks constant h = 6.625 1034 Js = 6.625 1027 ergs 1 Faraday = 96500 coulomb 1 calorie = 4.2 joule 1 amu = 1.66 1027 kg 1 eV = 1.6 1019 J 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, 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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PPhhyyssiiccss PART I
SECTION A
Single Correct Choice Type This section contains 30 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 1. An alternating voltage 220V and 50Hz is applied across a circuit containing pure resistance R,
inductance L and capacitance C in series. Across which of the following can the voltage be greater than 220V ?
(A) R, L & C (B) either L or C (C) L and C taken together (D) both (B) and (C) 2. A small ball strikes at one end of a stationary uniform frictionless rod of mass m and length l
which is free to rotate, in gravity-free space. The impact is elastic. Instantaneous axis of rotation of the rod will pass through
(A) its center of mass. (B) the center of mass of rod plus ball. (C) the point of impact of the ball on the rod. (D) the point which is at a distance 2l / 3 from the striking end. 3. The value of R for which 20 % of the main current passes
through galvanometer of resistance 80 is (A) 10 (B) 20 (C) 30 (D) 40
G
R
I
(main current)
4. A sphere of radius R is uniformly charged so that total charge on sphere is Q. Potential difference
between center of sphere and surface of sphere is (A) 0Q/4 R (B) 0Q/8 R (C) 0Q/16 R (D) 0Q/32 R 5. A large metallic plate is facing a charged sheet having charge
density placed parallel to plate at a distance A from the plate. Potential at point P at a distance x from the sheet is
(A) 0
( x)2 + A (B) 0
( x)2 A
(C) 0
( x) A (D) 0( x) + A
l
xP
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6. The circuit shown in the figure consists of a battery of emf E,
resistances R1 and R2, inductance L and a two way switch ABC. First A is connected to B for a long time and then A is connected to C. The total heat produced in R2 is
(A) 2
21
ELR
(B) 2
22
EL2R
(C) 2
21
EL2R
(D) 2
1 2
EL2R R
L
A
R2
R1E
C
B
7. As shown in figure, sphere of mass m is constrained to rotate about in
a vertical plane by connecting it to O by means of a rigid light rod of length A . What is the velocity of the sphere when it hits the wall of container ? It is given that specific gravity of ball is 2 and the container is filled with water.
(A) 2gA (B) 0.5gA
(C) gA (D) 3 g2A
ml
o
8. Consider three quantities x = EB
, 1/2
0 0
1y =
and RCz = A . Here A is the length of a wire, C is a capacitance and R is a resistance. All other symbols have standard meanings
(A) y, z have the same dimensions (B) z, x have the same dimensions (C) x, y have the same dimensions (D) all of the three have the same dimensions 9. A projectile is projected in the earths gravitational with initial kinetic energy E. The horizontal of
the projectile range is R. If the mass of the projectile is 1 kg then the angle of projection of the projectile will be equal to
(A) 1 gRsin2E
(B) 1 gR2sin
2E
(C) 1 gR0.5sin2E
(D) 1 gR4sin
2E
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10. Three rods of identical cross-sectional area are made from the same metal from the sides of an isosceles right angled triangle ABC, right angled at B. The points A and B are maintained at temperatures T and 2T respectively. In the steady state, the temperature of point C is TC. There
is a point D on BA, whose temperature is same as TC. Then the ratio BDDA
is
(A) 1 (B) 2
(C) 12
(D) 2
11. Neglect the effect of rotation of the earth. Suppose the earth suddenly stops attracting objects
placed near its surface. A person standing on the surface of the earth will (A) fly up. (B) slip along the surface. (C) fly along a tangent to the earths surface. (D) remain standing. 12. Variation of internal energy with density of one mole of monatomic gas is
depicted in the adjoining figure, corresponding variation of pressure with volume can be depicted as (Assuming the curve is rectangular hyperbola)
U
(A) P
v
(B) P
v (C) P
v
(D) P
v 13. The acceleration time graph of a particle is shown in the figure.
What is the velocity of particle at t = 8s, if initial velocity of particle is 3 m/s?
(A) 4 m/s (B) 5 m/s (C) 6 m/s (D) 7 m/s
t841O
4
a
Rough work
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14. Let B and denote induction to magnetic field and energy density at mid point of a long solenoid carrying a current i. The graph between and B will be (A)
B X O
Y
(B)
B X O
Y
(C)
B
Y
X
(D)
B X O
Y
15. In the given figure, the angle of inclination of the inclined plane is 30.
Find the horizontal velocity V0 so that the particle hits the inclined plane perpendicularly.
(A) 02gHV
5= (B) 0 2gHV 7=
(C) 0gHV5
= (D) 0 gHV 7=
90
H
3090
V0
16. A chain of mass 10 kg and length 10 m is resting on a rough horizontal surface ( = 0.2). A
constant force of 20 newton is applied at one end. The tension in the mid point of the chain is (A) 20 N (B) 15 N (C) 10 N (D) 5 N 17. Find the speed of the block when it covers a horizontal
distance A. It is given that the block never loses contact with the smooth horizontal surface, and the force always acts at an angle with the horizontal.
F0
u = 0M
smooth
(A) 0F cos
mA
(B) 02 F cos
mA
(C) 02 F cos
mA
(D) 0F cos
mA
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18. Two fixed long conducting wires cross each other perpendicularly so
that they do not actually touch but are close to each other as shown in the figure. Equal current i exist in each wire in the directions indicated. In what region(s) will there be some points with zero net magnetic field?
(A) IV, III (B) III, II (C) II, IV (D) I, IV
I II
III IV
19. A sample of hydrogen gas is excited by means of a monochromatic radiation. In the subsequent
emission spectrum, 10 different wavelengths are obtained, all of which have energies greater than or equal to the energy of the absorbed radiation. It follows that the initial quantum number of the state (before absorbing radiation) was
(A) 5 (B) 4 (C) 3 (D) 2.
20. The ratio of the min in a Coolidge tube to deBroglie of the electrons striking the target depends on
accelerating potential V as
(A) mindeBroglie
V (B)
min
deBroglieV
(C) mindeBroglie
1V
(D) min
deBroglie
1V
. 21. The minimum distance between an object and its real image formed by a thin convex lens of focal
length 30 cm is (A) 45 cm (B) 50 cm (C) 75 cm (D) 120 cm 22. A thin concave and a convex lens have the same focal length of 40 cm, and are put into contact
to form a lens combination. The combination is used to view an object of 5 mm length kept at 20 cm form the lens combination. As compared to the object, the image will be
(A) Magnified and inverted (B) Reduced and erect (C) Of the same size as the object and erect (D) Of the same size as the object but inverted 23. A uniform plank of Youngs modulus Y is moved over a smooth horizontal surface by a constant
horizontal force F0. The area of cross-section of the plank is A. The compressive strain on the plank in the direction of the force is
(A) 0FAY
(B) 02FAY
(C) 0F
2AY (D) 0
3FAY
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24. A solid ball of radius 0.2m and mass 1kg lying at rest on a smooth
horizontal surface is given an instantaneous impulse of 50 N-s at point P as shown. The number of rotations made by the ball about its diameter before hitting the ground is
(A) 625 32 (B)
2500 32
50N-sP
30
(C) 3125 32 (D)
1250 32
25. A thin uniform spherical shell and a uniform solid cylinder of
the same mass and radius are allowed to roll down a fixed incline, without slipping, starting from rest. The times taken by them to roll down the same distance are in the ratio, tsph : tcyl
(A) 1415
(B) 1514
(C) 310
(D) 103
.
26. A particle A is attached to a spring and the time period for small oscillations is observed to be T;
when an additional mass dm is added on, the time period becomes T + dT. The mass of particle A is
(A) 12dT dTdm 2
T T
+ (B)
12dT dTdm 2 2T T
+ +
(C) 1dTdm 2
T
(D) none of these
Rough work
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27. The equation of a vibrating string, fixed at both ends, is given by
xy (3 mm)sin sin(400 t)15 =
where x is the distance (in cm) measured from one end of the string, t is the time (in seconds), and y gives the displacement. The string vibrates in 4 loops.
The speed of transverse waves along the string equals, for the fundamental mode, (A) 60 m/s (B) 120 m/s (C) 30 m/s (D) 15 m/s. 28. The vibrations produced in a sitar wire are (A) progressive transverse (B) progressive longitudinal (C) stationary transverse (D) stationary longitudinal. 29. Given 2 2 F (xy )i (x y) j= +G newton. Find the work done by FG when a particle is taken along the
semicircular path OAB where the co-ordinates of B are (4, 0).
(A) 65 J3
(B) 75 J2
(C) 73 J4
(D) 0 J
30. A small cube of mass m slides down a circular path of radius R cut into a
larger block of mass M, as shown in the figure. M rests on a table, and both blocks move without friction. The blocks are initially at rest, and m starts from the top of the path. The velocity v of the cube as it leaves the block is
mR
M
(A) 2mgRM
(B) 2gR
(C) 2mgRm M+ (D)
2MgRm M+
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CChheemmiissttrryy PART II
SECTION A Single Correct Choice Type
This section contains 30 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 1. One of the reaction coordinate energy diagrams of two nucleophilic substitution reactions are
given below:
Ene
rgy
R - XNuc
R - Nuc X-
Reaction coordinates(A)
R - XNuc
R - Nuc X-
Reaction coordinates(B) Which of the following statement(s) is/are true? (A) (A) corresponds to SN1 and (B) corresponds to SN2. (B) (A) corresponds to SN2 and (B) corresponds to SN1. (C) Both (A) and (B) correspond to SN2. (D) Both (A) and (B) correspond to SN1. 2. A liquid (A) is treated with Na2CO3 solution. A mixture of two salts (B) and (C) are produced in the
solution. The mixture on acidification with sulphuric acid produces the liquid A again (A) is (A) H2O2 (B) Br2 (C) CS2 (D) CCl4 3. The 22NO Br 2NOBr+ follows the mechanism below (i) 2 2NO Br NOBr+ fast (ii) 2NOBr NO 2NOBr+ slow If the concentration of both NO and Br2 are increased two times the rate of reaction would
become (A) 4 times (B) 6 times (C) 16 times (D) 8 times
Rough work
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4. Which of the following statement is/are correct? (i) The ligand thiosulphato 22 3S O
can give rise to linkage isomers. (ii) In metal carbonyls the ligand CO molecule acts both as donor and acceptor. (iii) The complex K[Cu(CN)2] has coordination number 2. (iv) The complex ferricyanide does not follow effective atomic number (EAN) rule. (A) (i), (ii) and (iii) (B) (i), (ii) and (iv) (C) (i), (iii) and (iv) (D) (ii), (iii) and (iv) 5. In the following reaction sequence:
O
N
H
+ [ ]6 63
3
(i) p TSA, C H Heat(ii) CHCl , KOH(iii) H O
X+
What is the major product [X]?
(A) O O
OH
(B) O
Cl
Cl
(C) O O
H
(D) OCl
6. The 2s-orbital of H-atom has a radial node at 2a0 because 2s is proportional to (A) (1/2 + r/a0) (B) (2 + r/a0) (C) (2 r/2a0) (D) (2 r/a0) 7. The standard emf E0 for the reaction 2 2fumarate lactate succinate pyruvate + + is on the basis of the following information 2 2 0fumarate 2H 2e succinate E 0.031 V + + + = 0pyruvate 2H 2e lactate E 0.185 V + + + = (A) 0.154 V (B) 0.216 V (C) 0.216 V (D) 0.154 V
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8.
O
OLi
( )
( )
2R CuLi A
HydrolysisC
( )
( )
RMgXB
HydrolysisD
Identify the compounds (C) and (D) respectively. (A) O
R
OH
O
OH
R
and
(B) O
OH
R
O
R
OH
and
(C) O
OH
R
O
OH
R
and
(D) OOHR
O
OH
R
and
9. Consider the following statement(s): (i) Hypophosphorous acid shows reducing properties. (ii) Pyrophophoric acid can show basicities from one to four because it has 4-OH groups
attached to the phosphorous. (iii) Orthophosphoric acid is obtained during disproportionation of hypophosphorous acid. (iv) Metaphosphoric acid on heating gives P4O10. Which of the following is/are correct: (A) (i), (ii) and (iii) are correct (B) (i), (ii) and (iv) are correct (C) (i), (iii) and (iv) are correct (D) All are correct 10. What is the correct order of bond angle in the following? (A) CH4 > NF3 > NH3 > H2O (B) CH4 > NH3 > NF3 > H2O (C) CH4 > NH3 > H2O > NF3 (D) NF3 > CH4 > NH3 > H2O
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11. How many chiral centre(s) (excluding N centre) is/are there in morphine?
O
OH
OH
N CH3H
H
(A) 4 (B) 5 (C) 6 (D) more than 6 12. The products formed when a mixture of the following two amides (I and II) are treated with
bromine and aqueous KOH are
CONH2 CONH2
NO2
15
(A) NH2 NH2
NO2
and
(B) NH2 NH2
NO2
and
15
(C)
NH2 NH2
NO2
and
15
(D) A mixture of all of there.
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13. Which of the following cannot be taken as unit cell of any lattice?
P
S
Q
R
(A) P (B) Q (C) R (D) S 14. Which of the following graphs is/are inconsistent with ideal gas behaviours?
(A)
V
P
T2
T1T2 >
T1
(B)
P
PV
T2
T1
(C)
T (K)
V
P1P2 > P1
P2
(D)
N
P
T1T2 >
T1
T2
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15. Which statement is true if the following reaction is spontaneous? 2 2Zn Cu Zn Cu+ ++ +ZZZXYZZZ (A) Kc is larger than 1; oG is negative, and oE is positive. (B) Kc is larger than 1; oG is positive, and oE is negative. (C) Kc is larger than 1; oG is negative, and oE is negative. (D) Kc is smaller than 1; oG is negative, and oE is positive. 16. ( ) ( )2 2SnCl HCl I A B+ + + The compounds (A) and (B) are: (A) SnI2, Cl2 (B) H2SnCl4, HI (C) SnCl4, HI (D) HSnCl3, HI 17. In an O2 molecule how many sets of degenerate molecular orbitals are present? (A) 1 (B) 2 (C) 3 (D) 4 18. In the following graph
0 5 10 15 20
0.1
0.2
Rad
ialp
roba
bilit
y di
strib
utio
n (P
n,l)
Pn3, 3A Pn2, 2A Pn1, 1A
Radius (a0) Then, what is correct set of quantum nos. for the given graphs? (A) n1 = 3 l1 = 1, n2 = 2 l2 = 2, n3 = 2, l3 = 2 (B) n1 = 3 l1 = 0, n2 = 3 l2 = 1, n3 = 3, l3 = 2 (C) n1 = 3 l1 = 2, n2 = 3 l2 = 1, n3 = 3, l3 = 0 (D) n1 = 3 l1 = 0, n2 = 3 l2 = 1, n3 = 2, l3 = 2
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19. A colourless gas X forms a brown coloured gas when mixed with air. The gas X is (A) NH3 (B) NO (C) N2O (D) NO2 20. Among the following the optically inactive compound is
(A) NCH2
CH3
CH3
(B) PCH3
(C)
CCH3
C C C
CH3
(D)
H
OHCOOH
21. Consider the ionization 3 2 3 2 2 2H H N CH COO H N CH COOH H NCH COOH H
+ + + ++ +ZZZX ZZZXYZZZ YZZZ Assuming that the following acid dissociation constants apply to the ionizations: 103 2 aNH NH H ; K 1.6 10 M
+ + + = ZZZXYZZZ 3aCOOH COO H ; K 4.0 10 M
+ + = ZZZXYZZZ Estimate a value for the equilibrium constant for the process 3 2 2 2H N CH COO H NCH COOH
+ ZZZXYZZZ is (A) 4.0 108 (B) 0.25 10+7 (C) 6.4 1013 (D) 1 108 22. The gas phase reaction between nitric oxide and oxygen is third order. The following rate
constant have been measured: T/K : 80.0 143.0 228.0 300.0 413.0 564.0 K109/cm6mol7s1: 41.8 20.2 10.1 7.1 4.0 2.8 The behaviour is interpreted in terms of a temperature dependent pre exponential factor; the rate
constant is of the form n E/RTK aT e= Where a and n are constants. Assume the activation energy to be zero. What is the value of n to the nearest half integer. (A) 1.5 (B) 1.5 (C) 0.5 (D) 0.5
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23. T
HP
for an ideal gas is equal to
(A) RTVP
(B) RT VP
(C) Zero (D) ( )V
T
24. In the freezing point depression experiment, it can be noted that: (A) Vapour pressure of the solution is less than that of pure solvent. (B) Only solute molecules solidify at freezing point. (C) Solvent molecules solidify at a temperature above the freezing point of pure solvent. (D) If weight of solute is much less than the weight of solvent and Hfus is heat of fusion per mole
of solvent than the gradient (slope) of the following plot is found to be 0f
fus
RTH , where y = mass
fraction of solute in the solution.
Tf
y
25. Calcium phosphide (Ca3P2) formed by reacting calcium orthophosphate [Ca3(PO4)2] with
magnesium was hydrolysed by water. The evolved PH3 was burnt in air to yield P2O5. How many moles of Mg(PO3)2 would be obtained, if 384 g of Mg were used for reducing calcium phosphide.
3 4 2 3 2Ca (PO ) Mg Ca P MgO+ + ( )3 2 2 32Ca P H O Ca OH PH+ + 3 2 2 5 2PH O P O H O+ + ( )2 5 3 2MgO P O Mg PO+ (A) 2 moles (B) 16 moles (C) 8 moles (D) 4 moles
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26. In the following reactions:
CH3
CH3 CH3
Cl( )NaOEt A
CH3
CH3 CH3
Cl( )NaOEt B
What are the structures of (A) and (B) respectively
(A) CH3
CH3 CH3
CH3
CH3 CH3
and
(B) CH3
CH3 CH3
CH3
CH3 CH3
and
(C) CH3
CH3 CH3
Only
(D) CH3
CH3 CH3
Only
27. Ester hydrolysis in acidic condition can follow (A) SN1 and SN2 both (B) Only SN1 (C) Only SN2 (D) SN1, SN2 and tetrahedral mechanism
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28. NO2
NaOEt2NH OH ?
+
(A) NO2
NH2
(B) NO2
NH2(C) NH2
NH2
(D) NH2NH2
29.
Cl
OMe
CN( )
2NaNHX major
What is the structure of (X)? (A) CN
OMe
(B) CN
OMe
(C)
OMe
CN
(D)
OMe
CN
30. Which of the following statements are in correct regarding Co2+ and Cr3+ ions? (i) The spin magnetic moment of both the ions is same (ii) The magnetic moment of both the ions in different (iii) The magnetic moment of both the ions is not different (iv) They both have same numbers of unpaired electrons (A) (i) only (B) (iii) only (C) (i) and (iii) both (D) All
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MMaatthheemmaattiiccss PART III
SECTION A Single Correct Choice Type
This section contains 30 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 1. The roots of 1 + z + z3 + z4 = 0 are represented by the vertices of (A) a square (B) an equilateral triangle (C) a rhombus (D) a rectangle 2. The equation of plane containing the lines r a b= + GG G and r b a= + GG G is (A) [r ab] 0=GG G (B) [r ab] a.b=G GG G G (C) [r ba] a.b=G GG G G (D) [r ab] 2a.b=G GG G G
3. If sin4 + cos4+ 2 = 4 sin cos , 0 , ,2 then sin + cos is equal to
(A) 2 (B) 0 (C) 2 (D) 1 4. Number of points with integral coordinates lying on the circle x2 + y2 = 2008 is (A) 0 (B) 4 (C) 8 (D) 16 5. If the equation ax2 bx + 12 = 0 where a and b are +ve integers not exceeding 10, has both roots
greater than 2 then the number of ordered pair (a, b) is (A) 0 (B) 1 (C) 3 (D) 5
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6. The number of different ordered triplets (a, b, c), a, b, cI such that these can represent sides of a triangle whose perimeter is 21, is
(A) 12 (B) 31 (C) 55 (D) 91 7. If the cube of a natural number ends with a prime digit then the probability of its fourth power
ending not with a prime digit, is (A) 3/10 (B) 9/10 (C) 4/9 (D) 3/4 8. The number of different permutations of all the letters of the word 'PERMUTATION' such that any
two consecutive letters in the arrangement are neither both vowels nor both identical is (A) 63 6! 5! (B) 57 5! 5! (C) 33 6! 5! (D) 7 7! 5! 9. When 3233 is divided by 34, it leaves the remainder (A) 2 (B) 4 (C) 8 (D) 32
10. The value of 1 1111 11
2 3 431 2 CC C2 2 2 ...2 3 3 4 4 5
+ + upto 12 terms is (A) 1/10 (B) 1/11 (C) 1/12 (D) 1/13
11. The maximum value of f(x) = 2| x | 2 x
| x | 1
+ , (xR), is (A) 4 (B) 3 (C) 2 (D) 1 12. Maximum number of common normals of y2 = 4ax and x2 = 4by can be equal to (A) 3 (B) 4 (C) 6 (D) 5
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13. The equation 2x3 15x2 + 36x + k = 0 has three real and distinct roots. The total number of integral values of k is
(A) 0 (B) 1 (C) 2 (D) none of these 14. A cubic f(x) vanishes at x = 2 and has relative minimum/maximum at x = 1 and 1/3 such that
( )11
14f x dx3
= . Then f(x) is equal to (A) 3 2x x x+ (B) 3 2x x x 1+ + (C) 3 2x x x 2+ + (D) 3 2x x x 2+
15. The function ( ) ( )( ) ( ) ( ){ }x 3 2 21f x 2 t 1 t 2 3 t 1 t 2 dt= + attains its maximum at x is equal to (A) 1 (B) 2 (C) 3 (D) 4
16. The value of 2
x 0
xlimsinx tanx
, where [] denotes greatest integer function, is equal to (A) 0 (B) 1 (C) limit does not exist (D) none of these 17. Let A be a square Matrix all of whose entries are integers. Then the true statement is (A) If det A = 1, then A1 exists but all its entries are not necessarily integers (B) If det A 1, then 1A exists and all its entries are non integers (C) If det A 1= , then 1A exists and all its entries are integers (D) If det A 1= , 1A need not exist
18. If A is a square Matrix such that ( )4 0 0
A. AdjA 0 4 00 0 4
= then
(A) A 5= (B) adjA 18=
(C) ( )adj adjA
16adjA
= (D) adj2A 128=
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19. 1 1sin cos x cos sinx a = has at least one solution iff a (A) { }0 (B) 0,
2
(C) 3,2 2 (D) ( )0,
20. Area bounded by the minimum of { }x 1, y 1 1 = and the maximum of { }x 1, y 1 4 = in
square units is (A) 16 (B) 36 (C) 25 (D) 45
21. The range of function f(x) defined by ( ) ( )2
2e
x 1f xlog x 1
+= + , { }x R 0 is (A) ( )0,1 (B) ( )0,2 (C) [ )e, (D) ( ), 22. The area bounded by the parabola x2 + 8x +12y + 4 = 0 and its latus rectum is 6k. Then the value
of k is (A) 1 (B) 2 (C) 3 (D) 4 23. Let ( ) 3 2f x x ax bx c= + + + and x1, x2 be the roots of ( )f x 0 = , if 1 2x x< then ( )f x 0= will have (A) No real root if ( ) ( )1 2f x 0 or f x 0< > (B) Only one real root if ( ) ( )1 2f x 0 or f x 0< > (C) Three real roots if ( ) ( )1 2f x 0 or f x 0< > (D) cannot say any thing 24. If a, b, c are unit vectors then 2 2 2| a b | | b c | | c a |+ + + + + can not be less than (A) 3 (B) 6 (C) 9 (D) 12
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25. If f(x) is a twice differentiable function such that f(0) f(2) 3;f (0) f (2) 1 = = = = then 2
0
x f (x)dx is
(A) 3 (B) 2 (C) 1 (D) 0
26. The value of x 0
100x 99sinxlimsinx x
+ where [ ]. denotes the greatest integer function is (A) 197 (B) 196 (C) 198 (D) does not exist 27. General solution of (xy sin xy + cos xy)y dx + (xy sin xy cos xy) x dy = 0 is (A) y.sin (xy) = cx (B) x.sec (x y) = cy (C) y tan (xy) = cx (D) y cot (xy) = cx 28. The general solution of the differential equation (3x + 2y2) y dx + 2x (2x + 3y2) dy = 0, is (A) x6y2 + x4y3 = c (B) x6y2 + x3y4 = c (C) x2y6 + x4y3 = c (D) x2y6 + x3y4 = c
29. Let x 1, x 0
f(x)x 1, x 0+