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IITJEE Home page 2007 IITJEE Paper - 1 2007 IITJEE Paper - 1 (online test)

 

1) A circuit is connected as shown in the figure with the switch S open. When the switch is closed the total amount of charge that flows from Y to X is
A )   0  
B )   54 μ C
C )   27 μ C
D )   81 μ C

 

2) A long hollow conducting cylinder is kept coaxially inside another long, hollow conducting cylinder of larger radius. Both cylinders are initially electrically neutral.
A )   A potential difference appears between the two cylinders when a charge density is given to the inner cylinder.
B )   A potential difference appears between the two cylinders when a charge density is given to the outer cylinder.
C )   No potential difference appears between the two cylinders when a uniform line charge is kept along the axis of the cylinders
D )   No potential difference appears between the two cylinders when same charge density is given to both the cylinders

 

3) In the options given below, let E denote the resr mass energy of a nucleus and "n" a neutron. The correct option is :
A )   E (23692 U) > E (13753 I) + E (9739 Y) + 2 E (n)
B )   E (23692 U) < E (13753 I) + E (9739 Y) + 2 E (n)
C )   E (23692 U) < E (14056 Ba) + E (9436 Kr) + 2 E (n)
D )   E (23692 U) = E (14056 Ba) + E (9436 Kr) + 2 E (n)

 

4) In an experiment to determine the focal length (f) of a concave mirror by the u-v method, a student places the object pin A on the principal axis at the distance "x" from the pole P. The student looks at the pin and its inverted image from a distance keeping his/her eye in line with PA. When the student shifts his/her eye towards left, the image appears to the right of the object pin. Then ,
A )   x < f
B )   f < x < 2f
C )   x = 2f
D )   x > 2f

 

5) The largest wavelength in the ultraviolet region of the hydrogen spectrum is 122 nm. The smallest wavelength in the infrared region of the hydrogen spectrum (to the nearest integer) is
A )   802 nm
B )   823 nm
C )   1882 nm
D )   1648 nm

 

6) A resistance of 2 Ω is connected across one gap of a meter-bridge (the length of the wire is 100 cm) and an unknown resistance, greater than 2 Ω is connected across the other gap. When these resistance are interchanged, the balance point shifts by 20 cm. Neglecting any corrections, the unknown resistance is
A )   3 Ω
B )   4 Ω
C )   5 Ω
D )   6 Ω

 

7) A ray of light travelling In water is incident on its surface open to air. The angle of incidence is θ, which is less than the critical angle. Then there will be
A )   only a reflected ray and no refracted ray
B )   only a refracted ray and no reflected ray
C )   a reflected ray and a refracted ray and the angle between them would be less than 180° - 2 θ
D )   a reflected ray and a refracted ray and the angle between them would be greater than 180° - 2 θ

 

8) Two particles of mass 'm' each are tied at the ends of a light string of length 2a. The whole system is kept on a frictionless horizontal surface with the string held tight so that each ease is at a distance a from the centre P (as shown in the figure). Now, the mid-point d the string is pulled vertically upwards with a small but constant force F. As a result, the parrt1cles move towards each other on the surface. The magnitude of acceleration, when the separation between them becomes 2x, is
A )  
B )  
C )  
D )  

 

9) Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. The net charge on the sphere is then,
A )   negative and distributed uniformly over the surface of the sphere
B )   negative and appears only at the point on the sphere closest to the point charge
C )   negative and distributed non-uniformly over the entire surface of the sphere
D )   zero

 

10) STATEMENT-1
The formula connecting u, v and f for a spherical mirror is valid only for mirrors whose sizes are very small compared to their radii of curvature. because
STATEMENT-2
Laws of reflection are strictly valid for plane surfaces, but not for large spherical surfaces.
A )   Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
B )   Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
C )   Statement-1 is True, Statement-2 is False.
D )   Statement-1 is False, Statement-2 is True.

 

11) STATEMENT-1
If the accelerating potential in an X-ray tube is increased, the wavelengths of the characteristic X-rays do not change. because
STATEMENT-2
When an electron beam strikes the target in an X-ray tube, part of the kinetic energy is converted into X-ray energy.
A )   Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
B )   Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
C )   Statement-1 is True, Statement-2 is False.
D )   Statement-1 is False, Statement-2 is True.

 

12) STATEMENT-1
A block of mass m starts moving on a rough horizontal surface with a velocity v. It stops due to friction between the block and the surface after moving through a certain distance. The surface is now tilted to an angle of 300 with the horizontal and the same block is made to go up on the surface with the same initial velocity v. The decrease in the mechanical energy in the second situation is smaller than that in the first situation.
because
STATEMENT-2
The coefficient of friction between the block and the surface decreases with the increase in the angle of inclination.
A )   Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
B )   Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
C )   Statement-1 is True, Statement-2 is False.
D )   Statement-1 is False, Statement-2 is True.

 

13) STATEMENT-1
In an elastic collision between two bodies, the relative speed of the bodies after collision is equal to the relative speed before the collision.
because
STATEMENT-2
In an elastic collision, the linear momentum of the system is conserved.
A )   Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
B )   Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
C )   Statement-1 is True, Statement-2 is False.
D )   Statement-1 is False, Statement-2 is True.

 

14) A fixed thermally conducting cylinder has a radius R and height L0. The cylinder is open at its bottom and has a small hole at its top. A piston of mass M is held at a distance L from the top surface, as shown in the figure. The atmospheric pressure is P0.

The piston is now pulled out slowly and held at a distance 2L from the top. The pressure in the cylinder between its top and the piston will then be
A )   P0
B )  
P0
2
C )  
P0   +   Mg
2 π R2
D )  
P0 - Mg
2 π R2

 

15) A fixed thermally conducting cylinder has a radius R and height L0. The cylinder is open at its bottom and has a small hole at its top. A piston of mass M is held at a distance L from the top surface, as shown in the figure. The atmospheric pressure is P0.

While the piston is at a distance 2L from the top, the hole at the top is sealed. The piston is then released, to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top is
A )  
B )  
C )  
D )  

 

16) A fixed thermally conducting cylinder has a radius R and height L0. The cylinder is open at its bottom and has a small hole at its top. A piston of mass M is held at a distance L from the top surface, as shown in the figure. The atmospheric pressure is P0.

The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank is brought below the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the cylinder as shown in the figure. The density of the water is ρ. In equilibrium, the height H of the water column in the cylinder satisfies
A )   ρg(L0 - H)2 + P0(L0 - H) + L0P0 = 0
B )   ρg(L0 - H)2 - P0(L0 - H) - L0P0 = 0
C )   ρg(L0 - H)2 + P0(L0 - H) - L0P0 = 0
D )   ρg(L0 - H)2 - P0(L0 - H) + L0P0 = 0

 

17) Two discs A and B are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I respectively about the common axis. Disc A is imparted an initial angular velocity 2 ω using the entire potential energy of a spring compressed by a distance x1. Disc B is imparted an angular velocity ω by a spring having the same spring constant and compressed by a distance x1. Both the discs rotate in the clockwise direction. The ratio of x1/x2 is
A )   2
B )   1/2
C )  
D )   1/

 

18) Two discs A and B are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I respectively about the common axis. Disc A is imparted an initial angular velocity 2 ω using the entire potential energy of a spring compressed by a distance x1. Disc B is imparted an angular velocity ω by a spring having the same spring constant and compressed by a distance x1. Both the discs rotate in the clockwise direction.

When disc B is brought in contact with disc A, they acquire a common angular velocity in time t. The average frictional torque on one disc by the other during this period is

A )  
2 I ω
3 t
B )  
9 I ω
2 t
C )  
9 I ω
4 t
D )  
3 I ω
2 t

 

19) Two discs A and B are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I respectively about the common axis. Disc A is imparted an initial angular velocity 2 ω using the entire potential energy of a spring compressed by a distance x1. Disc B is imparted an angular velocity ω by a spring having the same spring constant and compressed by a distance x1. Both the discs rotate in the clockwise direction.

The loss of kinetic energy during the above process is

A )  
I ω2
2
B )  
I ω2
3
C )  
I ω2
4
D )  
I ω2
6

 

20) Some physical quantities are given in Column I and some possible SI units in which these quantities may be expressed are given in Column II. Match the physical quantities in Column I with the units in Column II and indicate your answer by darkening appropriate bubbles in the 4 X 4 matrix given in the ORS.
Column I Column II
(A)
GMeMs
G → universal gravitational constant, Me → mass of the earth,
Ms → mass of the sun, 		(p)
(volt) (coulomb) (metre)
(B)
3 R T
M
R → universal gas constant, T → absolute temperature, M → molar mass 		(q)
(kilogram) (metre)3 (second)-2
(C)
F2
q2B2
F → force, q → charge, B → magnetic field 		(r)
(meter)2 (second)-2
(D)
G Me
Re
G → universal gravitational constant, Me → mass of the earth, Re → radius of the earth 		(s)
(farad) (volt)2 (kg)-1
Answer:
A → (p)&(q) , B → (r) & (s), C → (r) & (s), D → (r) & (s)
A )  
B )  
C )  
D )  

 

21) Some laws/processes are given in Column I. Match these with the physical phenomena given in Column II and indicate your answer by darkening appropriate bubbles in the 4 X 4 matrix given in the ORS.
Column I Column II
(A)
Transition between two atomic energy levels 		(p)
Characteristic X - rays
(B)
Electron emission from a material 		(q)
Photoelectric effect
(C)
Mosley's law 		(r)
Hydrogen spectrum
(D)
Change of photon energy into kinetic energy of electrons 		(s)
β - decay

Answer:
A → (p) & (r), B → (q) & (s), C → (p), D → (q)

A )  
B )  
C )  
D )  

 

22) Column I gives certain situations in which a straight metallic wire of resistance R is used and Column II gives some resulting effects. Match the statements in Column I with the statements in Column II and indicate your answer by darkening appropriate bubbles in the 4 × 4 matrix given in the ORS.
Column I Column II
(A)
A charged capacitor is connected to the ends of the wire 		(p)
A constant current flows through the wire
(B)
The wire is moved perpendicular to its length with a constant velocity in a uniform magnetic field perpendicular to the plane of motion 		(q)
Thermal energy is generated in the wire
(C)
The wire is placed in a constant electric field that has a direction along the length of the wire. 		(r)
A constant potential difference develops between the ends of the wire
(D)
A battery of constant emf is connected to the ends of the wire 		(s)
Charges of constant magnitude appear at the ends of the wire

Answer:
A → (q), B → (r) & (s), C → (r) & (s), D → (p), (q) & (r)

A )  
B )  
C )  
D )  

 

23) The number of structural isomers for C6H14 is
A )   3
B )   4
C )   5
D )   6

 

24) In the following reaction,

the structure of the major product 'X' is
A )  
B )  
C )  
D )  

 

25) When 20 g of naphthoic acid (C11H8O2) is dissolved in 50 g of benzene (Kf = 1.72 K kg mol-1), a freezing point depression of 2 K is observed. The van’t Hoff factor (i) is
A )   0.5
B )   1
C )   2
D )   3

 

26) Among the following, the paramagnetic compound is
A )   Na2O
B )   O3
C )   Na2O
D )   KO2

 

27) The value of log10K for a reaction A B is
(Given : ΔrH°298K = -54.07kJ mol-1 , ΔrS°298K = 10JK-1 mol-1; and R = 8.314 JK-1 mol-1 ; 2.303 X 8.314 X 298 = 5705)
A )   5
B )   10
C )   95
D )   100

 

28) The species having bond order different from that in CO is
A )   NO-
B )   NO+
C )   CN-
D )   N2

 

29) The percentage of p-character in the orbitals forming P-P bonds in P4 is
A )   25
B )   33
C )   50
D )   75

 

30) Extraction of zinc from zinc blende is achieved by
A )   electrolytic reduction
B )   roasting followed by reduction with carbon
C )   roasting followed by reduction with another metal
D )   roasting followed by self-reduction

 

31) The reagent(s) for the following conversion,

is/are
A )   alcoholic KOH
B )   alcoholic KOH followed by NaNH2
C )   aqueous KOH followed by NaNH2
D )   Zn/CH3OH

 

32) STATEMENT-1 :
p-Hydroxybenzoic acid has a lower boiling point than o-hydroxybenzoic acid.
because
STATEMENT-2
o-Hydroxybenzoic acid has intramolecular hydrogen bonding.
A )   Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
B )   Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
C )   Statement-1 is True, Statement-2 is False.
D )   Statement-1 is False, Statement-2 is True.

 

33) STATEMENT-1 :
Micelles are formed by surfactant molecules above the critical micellar concentration (CMC).
because
STATEMENT-2
The conductivity of a solution having surfactant molecules decreases sharply at the CMC.
A )   Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
B )   Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
C )   Statement-1 is True, Statement-2 is False.
D )   Statement-1 is False, Statement-2 is True.

 

34) STATEMENT-1 :
Boron always forms covalent bond.
because
STATEMENT-2
The small size of B3+ favours formation of covalent bond.
A )   Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
B )   Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
C )   Statement-1 is True, Statement-2 is False.
D )   Statement-1 is False, Statement-2 is True.

 

35) STATEMENT-1 :
In water, orthoboric acid behaves as a weak monobasic acid.
because
STATEMENT-2
In water, orthoboric acid acts as a proton donor.
A )   Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
B )   Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
C )   Statement-1 is True, Statement-2 is False.
D )   Statement-1 is False, Statement-2 is True.

 

36) Chemical reactions involve interaction of atoms and molecules. A large number of atoms/molecules (approximately 6.023 X 1023) are present in a few grams of any chemical compound varying with their atomic/molecular masses. To handle such large numbers conveniently, the mole concept was introduced. This concept has implications in diverse areas such as analytical chemistry, biochemistry, electrochemistry and radiochemistry. The following example illustrates a typical case, involving chemical/electrochemical reaction, which requires a clear understanding of the mole concept.

A 4.0 molar aqueous solution of NaCl is prepared and 500 mL of this solution is electrolysed. This leads to the evolution of chlorine gas at one of the electrodes (atomic mass: Na = 23, Hg = 200; 1 Faraday=96500 coulombs)


The total number of moles of chlorine gas evolved is

A )   0.5
B )   1.0
C )   2.0
D )   3.0

 

37) Chemical reactions involve interaction of atoms and molecules. A large number of atoms/molecules (approximately 6.023 X 1023) are present in a few grams of any chemical compound varying with their atomic/molecular masses. To handle such large numbers conveniently, the mole concept was introduced. This concept has implications in diverse areas such as analytical chemistry, biochemistry, electrochemistry and radiochemistry. The following example illustrates a typical case, involving chemical/electrochemical reaction, which requires a clear understanding of the mole concept.

A 4.0 molar aqueous solution of NaCl is prepared and 500 mL of this solution is electrolysed. This leads to the evolution of chlorine gas at one of the electrodes (atomic mass: Na = 23, Hg = 200; 1 Faraday=96500 coulombs)


If the cathode is a Hg electrode, the maximum weight (g) of amalgam formed from this solution is

A )   200
B )   225
C )   400
D )   446

 

38) Chemical reactions involve interaction of atoms and molecules. A large number of atoms/molecules (approximately 6.023 X 1023) are present in a few grams of any chemical compound varying with their atomic/molecular masses. To handle such large numbers conveniently, the mole concept was introduced. This concept has implications in diverse areas such as analytical chemistry, biochemistry, electrochemistry and radiochemistry. The following example illustrates a typical case, involving chemical/electrochemical reaction, which requires a clear understanding of the mole concept.

A 4.0 molar aqueous solution of NaCl is prepared and 500 mL of this solution is electrolysed. This leads to the evolution of chlorine gas at one of the electrodes (atomic mass: Na = 23, Hg = 200; 1 Faraday=96500 coulombs)


The total charge (coulombs) required for complete electrolysis is

A )   24125
B )   48250
C )   96500
D )   193000

 

39) The noble gases have closed-shell electronic configuration and are monoatomic gases under normal conditions. The low boiling points of the lighter noble gases are due to weak dispersion forces between the atoms and the absence of other interatomic interactions.

The direct reaction of xenon with fluorine leads to a series of compounds with oxidation numbers +2, +4 and +6. XeF4 reacts violently with water to give XeO3. The compounds of xenon exhibit rich stereochemistry and their geometries can be deduced considering the total number of electron pairs in the valence shell.


Argon is used in arc welding because of its

A )   low reactivity with metal
B )   ability to lower the melting point of metal
C )   flammability
D )   high calorific value

 

40) The noble gases have closed-shell electronic configuration and are monoatomic gases under normal conditions. The low boiling points of the lighter noble gases are due to weak dispersion forces between the atoms and the absence of other interatomic interactions.

The direct reaction of xenon with fluorine leads to a series of compounds with oxidation numbers +2, +4 and +6. XeF4 reacts violently with water to give XeO3. The compounds of xenon exhibit rich stereochemistry and their geometries can be deduced considering the total number of electron pairs in the valence shell.


The structure of XeO3 is

A )   linear
B )   planar
C )   pyramidal
D )   T-shaped

 

41) The noble gases have closed-shell electronic configuration and are monoatomic gases under normal conditions. The low boiling points of the lighter noble gases are due to weak dispersion forces between the atoms and the absence of other interatomic interactions.

The direct reaction of xenon with fluorine leads to a series of compounds with oxidation numbers +2, +4 and +6. XeF4 reacts violently with water to give XeO3. The compounds of xenon exhibit rich stereochemistry and their geometries can be deduced considering the total number of electron pairs in the valence shell.


XeF4 and XeF6 are expected to be

A )   oxidizing
B )   reducing
C )   unreactive
D )   strongly basic

 

42) Match the complexes in Column-I with their properties listed in Column-II. Indicate your answer by darkening the appropriate bubbles of the 4 X 4 matrix given in the ORS.
Column I Column II
(A)
[Co(NH3)4(H2O)2]Cl2 		(p)
geometrical isomers
(B)
[Pt(NH3)2Cl2] 		(q)
paramagnetic
(C)
[Co(H2O)5Cl]Cl 		(r)
diamagnetic
(D)
[Ni(H2O)6]Cl2 		(s)
metal ion with +2 oxidation state

Answer:
A → (p, q, s), B → (p, r, s), C → (q, s), D → (q, s)

A )  
B )  
C )  
D )  

 

43) Match gases under specified conditions listed in Column-I with their properties/laws in Column-II. Indicate your answer by darkening the appropriate bubsbles of the 4 X 4 matrix given in the ORS.
Column I Column II
(A)
hydrogen gas (P = 200 atm, T = 273K) 		(p)
compressibility factor ≠ 1
(B)
hydrogen gas (P ~ 0, T = 273K) 		(q)
attractive forces are dominant
(C)
CO2 (P = 1 atm, T = 273K) 		(r)
PV = nRT
(D)
real gas with very large molar volume 		(s)
P(V - nb) = nRT

Answer:
A → (p, s), B → (r), C → (p, q), D → (p, s)

A )  
B )  
C )  
D )  

 

44) Match the chemical substances in Column-I with type of polymers/type of bonds in Column-II. Indicate your answer by darkening the appropriate bubbles of the 4 X 4 matrix given in the ORS.
Column I Column II
(A)
cellulose 		(p)
natural polymer
(B)
nylon-6, 6 		(q)
synthetic polymer
(C)
protein 		(r)
amide linkage
(D)
sucrose 		(s)
glycoside linkage

Answer:
A → (p, s), B → (q, r), C → (p, r), D → (s)

A )  
B )  
C )  
D )  

 

45) A hyperbola, having the transverse axis of length 2 sinθ , is confocal with the ellipse 3x2 + 4y2 = 12. Then its equation is
A )   x2cosec2θ - y2sec2θ = 1
B )   x2sec2θ - y2cosec2θ = 1
C )   x2sin2θ - y2cos2θ = 1
D )   x2cos2θ - y2sin2θ = 1

 

46) The tangent to the curve y = ex drawn at the point (c, ec) intersects the line joining the points (c − 1, ec-1) and (c + 1, ec+1)
A )   on the left of x = c
B )   on the right of x = c
C )   at no point
D )   at all points

 

47) A man walks a distance of 3 units from the origin towards the north-east (N 45° E) direction. From there, he walks a distance of 4 units towards the north-west (N 45° W) direction to reach a point P. Then the position of P in the Argand plane is
A )   3 eiπ/4 + 4 i
B )   (3 - 4 i)eiπ/4
C )   (4 + 3 i)eiπ/4
D )   (3 + 4 i)eiπ/4

 

48) Let f(x) be differentiable on the interval (0, ∞) such that f(1) = 1, and

for each x > 0. Then f(x) is
A )  
1   +   2x2
3x 3
B )  
  -   1   +   4x2
3x 3
C )  
  -   1   +   2
x x2
D )  
1
x

 

49) The number of solutions of the pair of equations
2 sin2θ - cos 2θ = 0
2 cos2θ - 3 sin xθ = 0
in the interval [0, 2π] is
A )   zero
B )   one
C )   two
D )   four

 

50) Let α , β be the roots of the equation x2 - px + r = 0 and α/2 2 β be the roots of the equation x2 - qx + r = 0. Then the value of r is
A )  
2  (p - q) (2q - p)
9
B )  
2  (q - p) (2p - q)
9
C )  
2  (q - 2p) (2q - p)
9
D )  
2  (2p - q) (2q - p)
9

 

51) The number of distinct real values of λ , for which the vectors
- λ2i + j + k ,
i - λ2j + k and
i + j - λ2k
are coplanar, is
A )   zero
B )   one
C )   two
D )   three

 

52) One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is
A )   1/2
B )   1/3
C )   2/5
D )   1/5

 

53)

equals
A )  
8   f(2)
π
B )  
2   f(2)
π
C )  
2   f (   1   )  
π 2
D )   4 f(2)

 

54)
Let the vectors represent the sides of a regular hexagon.

STATEMENT-1 :

because
STATEMENT-2 :

A )   Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
B )   Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
C )   Statement-1 is True, Statement-2 is False.
D )   Statement-1 is False, Statement-2 is True.

 

55) Let F(x) be an indefinite integral of sin2x.

STATEMENT-1 :
The function F(x) satisfies F(x + π) = F(x) for all real x.
because
STATEMENT-2 :
sin2(x + π) = sin2x for all real x.

A )   Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
B )   Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
C )   Statement-1 is True, Statement-2 is False.
D )   Statement-1 is False, Statement-2 is True.

 

56) Let H1, H2, ... Hn be mutually exclusive and exhaustive events with P(Hi) > 0, i = 1, 2,...,n. Let E be any other event with 0 < P(E) < 1.

STATEMENT-1 :
P(Hi | E) > P(E | Hi) . P(Hi) for i = 1, 2,....,n
because
STATEMENT-2 :

A )   Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
B )   Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
C )   Statement-1 is True, Statement-2 is False.
D )   Statement-1 is False, Statement-2 is True.

 

57) Tangents are drawn from the point (17, 7) to the circle x2 + y2 = 169.

STATEMENT-1 :
The tangents are mutually perpendicular.
because
STATEMENT-2 :
The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is x2 + y2 = 338.

A )   Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
B )   Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
C )   Statement-1 is True, Statement-2 is False.
D )   Statement-1 is False, Statement-2 is True.

 

58) Consider the circle x2 + y2 = 9 and the parabola y2 = 8x. They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S.

The ratio of the areas of the triangles PQS and PQR is

A )   1 :
B )   1 : 2
C )   1 : 4
D )   1 : 8

 

59) Consider the circle x2 + y2 = 9 and the parabola y2 = 8x. They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S.

The radius of the circumcircle of the triangle PRS is

A )   5
B )   3
C )   3
D )   2

 

60) Consider the circle x2 + y2 = 9 and the parabola y2 = 8x. They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S.

The radius of the incircle of the triangle PQR is

A )   4
B )   3
C )   8/3
D )   2

 

61) Let Vr denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r - 1). Let

Tr = Vr+1 - Vr - 2 and Qr = Tr+1 - Tr for r = 1, 2, ...

The sum V1 + V2 + ...+ Vn is

A )  
1   n(n + 1) (3n2 - n + 1)
12
B )  
1   n(n + 1) (3n2 + n + 1)
12
C )  
1   n (2n2 - n + 1)
2
D )  
1   n (2n3 - 2n + 3)
3

 

62) Let Vr denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r - 1). Let

Tr = Vr+1 - Vr - 2 and Qr = Tr+1 - Tr for r = 1, 2, ...

Tr is always

A )   an odd number
B )   an even number
C )   a prime number
D )   a composite number

 

63) Let Vr denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r - 1). Let

Tr = Vr+1 - Vr - 2 and Qr = Tr+1 - Tr for r = 1, 2, ...

Which one of the following is a correct statement?

A )   Q1, Q2, Q3, ...are in A.P. with common difference 5
B )   Q1, Q2, Q3, ...are in A.P. with common difference 6
C )   Q1, Q2, Q3, ...are in A.P. with common difference 11
D )   Q1 = Q2 = Q3 = ...

 

64) Consider the following linear equations

ax + by + cz = 0
bx + cy + az = 0
cx + ay + bz = 0
Match the conditions / expressions in Column I with statements in Column II and indicate your answers by darkening the appropriate bubbles in 4 X 4 matrix given in the ORS.
Column I Column II
(A)
a + b + c ≠ 0 and a2 + b2 + c2 = ab + bc + ca 		(p)
the equations represent planes meeting only at a single point.
(B)
a + b + c = 0 and a2 + b2 + c2 ≠ ab + bc + ca 		(q)
the equations represent the line x = y = z.
(C)
a + b + c ≠ 0 and a2 + b2 + c2 ≠ ab + bc + ca 		(r)
the equations represent identical planes.
(B)
a + b + c = 0 and a2 + b2 + c2 = ab + bc + ca 		(s)
the equations represent the whole of the three dimensional space.
Answer:
A → (r) , B → (q), C → (p), D → (s)

A )  
B )  
C )  
D )  

 

65) Match the integrals in Column I with the values in Column II and indicate your answer by darkening the appropriate bubbles in the 4 X 4 matrix given in the ORS.
Column I Column II
(A)
(p)
1   log (   2   )
2 3
(B)
(q)
2  log (   2   )
3
(C)
(r)
π
3
(D)
(s)
π
2
Answer:
A → (s) , B → (s), C → (p), D → (r)
A )  
B )  
C )  
D )  

 

66) In the following [X] denotes the greatest integer less than or equal to x. Match the functions in Column I with the properties Column II and indicate your answer by darkening the appropriate bubbles in the 4 X 4 matrix given in the ORS.
Column I Column II
(A)
x |x| 		(p)
continuous in (-1, 1)
(B)
(q)
differentiable in (-1, 1)
(C)
x + [x] 		(r)
strictly increasing in (-1, 1)
(D)
|x - 1| + |x + 1| 		(s)
not differentiable at least at one point in (-1, 1)
Answer:
A → (p,q,r) , B → (p,s), C → (r,s), D → (p,q)
A )  
B )  
C )  
D )  

ANSWERS

1) C 2) A 3) A 4) B
5) B 6) A 7) C 8) B
9) D 10) C 11) C 12) C
13) B 14) A 15) D 16) C
17) C 18) A 19) B 20)
21) 22) 23) C 24) B
25) A 26) D 27) B 28) A
29) D 30) B 31) B 32) D
33) B 34) A 35) C 36) B
37) D 38) D 39) A 40) C
41) A 42) 43) 44)
45) A 46) A 47) D 48) C
49) C 50) D 51) C 52) C
53) C 54) C 55) D 56) D
57) A 58) C 59) B 60) D
61) D 62) D 63) B 64)
65) 66)
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