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IITJEE Home page 2007 IITJEE Paper - 2

 

1) A speherical portion has been removed from a solid sphere having a charge distributed uniformly in its volume as shown in the figure. The electric field inside the emptied space is

A )   zero everywhere
B )   non-zero and uniform
C )   non-uniform
D )   zero only at its center

 

2) A small object with uniform density rolls up a curved surface with an initial velocity v. It reaches up to a maximum height of
3 v2
4 g


    with respect to the initial position. The object is

A )   ring
B )   solid sphere
C )   hollow sphere
D )   disc

 

3) A magnetic field B = B0j exists in the region a < x < 2a and B = -B0j, in the region 2a < x < 3a, where B0 is a positive constant. A positive point charge moving with a velocity v = v0i where v0 is a positive constant, enters the magnetic field at x = a. The trajectory of the charge in this region can be like,

A )  
B )  
C )  
D )  

 

4) Electrons with de-Broglie wavelength λ fall on the target in an X-ray tube. The cut-off wavelength of the emitted Xrays is
A )  
λ0 =   2mcλ2
h
B )  
λ0 =   2h
mc
C )  
λ0 =   2m2c2λ3
h2
D )   λ0 = λ

 

5) A student performs an experiment to determine the Young's modulus of a wire, exactly 2 m long, by Searle's method. In a particular reading, the student measures the extension in the length of the wire to be 0.8 mm with an uncertainty of ±0.05 mm at a load of exactly 1.0 kg. The student also measures the diameter of the wire to be 0.4 mm with an uncertainty of ±0.01 mm. Take g = 9.8 m/s22 (exact). The Young's modulus obtained from the reading is
A )   (2.0 ± 0.3) x 1011 N/m2
B )   (2.0 ± 0.2) x 1011 N/m2
C )   (2.0 ± 0.1) x 1011 N/m2
D )   (2.0 ± 0.05) x 1011 N/m2

 

6) Positive and negative point charges of equal magnitude are kept at (0, 0, a/2) and (0, 0, -a/2) , respectively. The work done by the electric field when another positive point charge is moved from (−a, 0, 0) to (0, a, 0) is
A )   positive
B )   negative
C )   zero
D )   depends on the path connecting the initial and final positions

 

7) In the experiment to determine the speed of sound using a resonance column,
A )   prongs of the tuning fork are kept in a vertical plane
B )   prongs of the tuning fork are kept in a horizontal plane
C )   in one of the two resonances observed, the length of the resonating air column is close to the wavelength of sound in air
D )   in one of the two resonances observed, the length of the resonating air column is close to half of the wavelength of sound in air

 

8) Water is filled up to a height h in a beaker of radius R as shown in the figure. The density of water is ρ , the surface tension of water is T and the atmospheric pressure is P0. Consider a vertical section ABCD of the water column through a diameter of the beaker. The force on water on one side of this section by water on the other side of this section has magnitude
A )   | 2P0Rh + π R2 ρgh - 2RT |
B )   | 2P0Rh + R ρgh2 - 2RT |
C )   | P0πR2 + R ρgh2 - 2RT |
D )   | P0πR2 + R ρgh2 + 2RT |

 

9) A particle moves in the X-Y plane under the influence of a force such that its linear momentum is
p(t) = A [ i cos(kt) - jsin(kt) ], where A and k are constants. The angle between the force and the momentum is
A )   0°
B )   30°
C )   45°
D )   90°

 

10) STATEMENT-1:

A vertical iron rod has a coil of wire wound over it at the bottom end. An alternating current flows in the coil. The rod goes through a conducting ring as shown in the figure. The ring can float at a certain height above the coil.

Because
STATEMENT-2

In the above situation, a current is induced in the ring which interacts with the horizontal component of the magnetic field to produce an average force in the upward direction.

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 there is no external torque on a body about its center of mass, then the velocity of the center of mass remains constant.

because

STATEMENT-2

The linear momentum of an isolated system remains constant.

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:
The total translational kinetic energy of all the molecules of a given mass of an ideal gas is 1.5 times the product of its pressure and its volume.

because

STATEMENT-2

The molecules of a gas collide with each other and the velocities of the molecules change due to the collision.

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:
A cloth covers a table. Some dishes are kept on it. The cloth can be pulled out without dislodging the dishes from the table.

because

STATEMENT-2

For every action there is an equal and opposite reaction.

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) The figure shows a surface XY separating two transparent media, medium -1 and medium -2. The lines ab and cd represent wavefronts of a light wave travelling in medium-1 and incident on XY. The lines ef and gh represent wavefronts of the light wave in medium-2 after refraction.

Light travels as a

A )   parallel beam in each medium
B )   convergent beam in each medium
C )   divergent beam in each medium
D )   divergent beam in one medium and convergent beam in the other medium.

 

15) The figure shows a surface XY separating two transparent media, medium -1 and medium -2. The lines ab and cd represent wavefronts of a light wave travelling in medium-1 and incident on XY. The lines ef and gh represent wavefronts of the light wave in medium-2 after refraction.

The phases of the light wave at c, d, e and f are Φc, Φd, Φe and Φf respectively. It is given that Φc ≠ Φf

A )   Φc cannot be equal to Φd
B )   Φd can be equal to Φe
C )   (Φd - Φf) is equal to (Φc - Φe)
D )   (Φd - Φc) is not equal to (Φf - Φe)

 

16) The figure shows a surface XY separating two transparent media, medium -1 and medium -2. The lines ab and cd represent wavefronts of a light wave travelling in medium-1 and incident on XY. The lines ef and gh represent wavefronts of the light wave in medium-2 after refraction.

Speed of the light is

A )   the same in medium-1 and medium-2
B )   larger in medium-1 than in medium-2
C )   larger in medium-2 than in medium-1
D )   different at b and d

 

17) Two trains A and B are moving with speeds 20 m/s and 30 m/s respectively in the same direction on the same straight track, with B ahead of A. The engines are at the front ends. The engine of train A blows a long whistle.

Assume that the sound of the whistle is composed of components varying in frequency from f1 = 800 Hz to f2 = 1120 Hz, as shown in the figure. The spread in the frequency (highest frequency - lowest frequency) is thus 320 Hz. The speed of sound in still air is 340 m/s.

The speed of sound of the whistle is

A )   340 m/s for passengers in A and 310 m/s for passengers in B
B )   360 m/s for passengers in A and 310 m/s for passengers in B
C )   310 m/s for passengers in A and 360 m/s for passengers in B
D )   340 m/s for passengers in both the trains

 

18) Two trains A and B are moving with speeds 20 m/s and 30 m/s respectively in the same direction on the same straight track, with B ahead of A. The engines are at the front ends. The engine of train A blows a long whistle.

Assume that the sound of the whistle is composed of components varying in frequency from f1 = 800 Hz to f2 = 1120 Hz, as shown in the figure. The spread in the frequency (highest frequency - lowest frequency) is thus 320 Hz. The speed of sound in still air is 340 m/s.

The distribution of the sound intensity of the whistle as observed by the passengers in train A is best represented by

A )  
B )  
C )  
D )  

 

19) Two trains A and B are moving with speeds 20 m/s and 30 m/s respectively in the same direction on the same straight track, with B ahead of A. The engines are at the front ends. The engine of train A blows a long whistle.

Assume that the sound of the whistle is composed of components varying in frequency from f1 = 800 Hz to f2 = 1120 Hz, as shown in the figure. The spread in the frequency (highest frequency - lowest frequency) is thus 320 Hz. The speed of sound in still air is 340 m/s.

The spread of frequency as observed by the passengers in train B is

A )   310 Hz
B )   330 Hz
C )   350 Hz
D )   290 Hz

 

20) Column I describe some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situation in Column I with the characteristics 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)
The object moves on the x-axis under a conservative force in such a way that its "speed" and "position" satisfy

where c1 and c2 are positive constants. 		(p)
The object executes a simple harmonic motion.
(B)
The object moves on the x-axis in such a way that its velocity and its displacement from the origin satisfy v = -kx, where k is a positive constant. 		(q)
The object does not change its direction.
(C)
The object is attached to one end of a massless spring of a given spring constant. The other end of the spring is attached to the ceiling of an elevator. Initially everything is at rest. The elevator starts going upwards with a constant acceleration a. The motion of the object is observed from the elevator during the period it maintains this acceleration. 		(r)
The kinetic energy of the object keeps on decreasing.
(D)
The object is projected from the earth’s surface vertically upwards with a speed

where, Me is the mass of the earth and Re is the radius of the earth. Neglect forces from objects other than the earth. 		(s)
The object can change its direction only once.
Answer:
A → (p) , B → (q) & (r), C → (p), D → (r) & (q)
A )  
B )  
C )  
D )  

 

21) Two wires each carrying a steady current I are shown in four configurations in Column I. Some of the resulting effects are described in Column II. Match the statements in Column I with the statements 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)
Point P is situated midway between the wires.

(p)
The magnetic fields (B) at P due to the currents in the wires are in the same direction.
(B)
Point P is situated at the midpoint of the line joining the centers of the circular wires, which have same radii.

(q)
The magnetic fields (B) at P due to the currents in the wires are in opposite directions.
(C)
Point P is situated at the midpoint of the line joining the centers of the circular wires, which have same radii.

(r)
There is no magnetic field at P.
(D)
Point P is situated at the common center of the wires.

(s)
The wires repel each other.
Answer:
A → (q, r) , B → (p), C → (q, r), D → (q)
A )  
B )  
C )  
D )  

 

22) Column I gives some devices and Column II gives some process on which the functioning of these devices depend. Match the devices in Column I with the processes 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)
Bimetallic strip 		(p)
Radiation from a hot body
(B)
Steam engine 		(q)
Energy conversion
(C)
Incandescent lamp 		(r)
Melting
(D)
Electric fuse 		(s)
Thermal expansion of solids
Answer:
A → (s) , B → (q), C → (p), D → (r)
A )  
B )  
C )  
D )  

 

23) Among the following, the least stable resonance structure is
A )  
B )  
C )  
D )  

 

24) For the process H2O(l) (1 bar, 373 K) → H2O(g) (1 bar, 373 K), the correct set of thermodynamic parameters is
A )   ΔG = 0 , ΔS = +ve
B )   ΔG = 0 , ΔS = -ve
C )   ΔG = +ve , ΔS = 0
D )   ΔG = -ve , ΔS = +ve

 

25) Cyclohexene on ozonolysis followed by reaction with zinc dust and water gives compound E. Compound E on further treatment with aqueous KOH yields compound F. Compound F is
A )  
B )  
C )  
D )  

 

26) Consider a reaction aG + bH → Products. When concentration of both the reactants G and H is doubled, the rate increases by eight times. However, when concentration of G is doubled keeping the concentration of H fixed, the rate is doubled. The overall order of the reaction is
A )   0  
B )   1
C )   2
D )   3

 

27) Among the following metal carbonyls, the C - O bond order is lowest in
A )   [Mn(CO)6]+
B )   [Fe(CO)5]
C )   [Cr(CO)6]
D )   [V(CO)6]-

 

28) A positron is emitted from 23Na11 . The ratio of the atomic mass and atomic number of the resulting nuclide is
A )   22/10
B )   22/11
C )   23/10
D )   23/12

 

29) Consider a titration of potassium dichromate solution with acidified Mohr's salt solution using diphenylamine as indicator. The number of moles of Mohr's salt required per mole of dichromate is
A )   3
B )   4
C )   5
D )   6

 

30) The number of stereoisomers obtained by bromination of trans-2-butene is
A )   1
B )   2
C )   3
D )   4

 

31) A solution of metal ion when treated with KI gives a red precipitate which dissolves in excess KI to give a colourless solution. Moreover, the solution of metal ion on treatment with a solution of cobalt(II) thiocyanate gives rise to a deep blue crystalline precipitate. The metal ion is
A )   Pb2+
B )   Hg2+
C )   Cu2+
D )   Co2+

 

32) STATEMENT-1:
Molecules that are not superimposable on their mirror images are chiral

because

STATEMENT-2

All chiral molecules have chiral centres.

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:
Alkali metals dissolves in liquid ammonia to give blue solution

because

STATEMENT-2

Alkali metals in liquid ammonia give solvated species of the type [M(NH3)n]+ (M = alkali metals).

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:
Band gap in germanium is small.

because

STATEMENT-2

The energy spread of each germanium atomic energy level is infinitesimally small.

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:
Glucose gives a reddish-brown precipitate with Fehling's solution.

because

STATEMENT-2

Reaction of glucose with Fehling's solution gives CuO and gluconic acid.

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) Reimer-Tiemann reaction introduces an aldehyde group, on to the aromatic ring of phenol, ortho to the hydroxyl group. This reaction involves electrophilic aromatic substitution. This is a general method for the synthesis of substituted salicyladehydes as depicted below.

Which one of the following reagents is used in the above reaction?

A )   aq. NaOH + CH3Cl
B )   aq. NaOH + CH2Cl2
C )   aq. NaOH + CHCl3
D )   aq. NaOH + CCl4

 

37) Reimer-Tiemann reaction introduces an aldehyde group, on to the aromatic ring of phenol, ortho to the hydroxyl group. This reaction involves electrophilic aromatic substitution. This is a general method for the synthesis of substituted salicyladehydes as depicted below.

The electrophile in this reaction is

A )   : CHCl
B )   +CHCl2
C )   : CCl2
D )   . CCl3

 

38) Reimer-Tiemann reaction introduces an aldehyde group, on to the aromatic ring of phenol, ortho to the hydroxyl group. This reaction involves electrophilic aromatic substitution. This is a general method for the synthesis of substituted salicyladehydes as depicted below.

The structure of the intermediate I is

A )  
B )  
C )  
D )  

 

39) Redox reactions play a pivotal role in chemistry and biology. The values of standard redox potential (E°) of two half-cell reactions decide which way the reaction is expected to proceed. A simple example is a Daniel cell in which zinc goes into solution and copper gets deposited. Given below are a set of half-cell reactions (acidic medium) along with their E° (V with respect to normal hydrogen electrode) values. Using this data obtain the correct explanations to Questions 39-41.

I2 + 2e- → 2I-       E° = 0.54
Cl2 + 2e- → 2Cl-       E° = 1.36
Mn3+ + e- → Mn2+       E° = 1.50
Fe3+ + e- → Fe2+       E° = 0.77
O2 + 4H+ + 4e- → 2H2O       E° = 1.23

Among the following, identify the correct statement.

A )   Chloride ion is oxidized by O2
B )   Fe2+ is oxidized by iodine
C )   Iodide ion is oxidized by chlorine
D )   Mn2+ is oxidized by chlorine

 

40) Redox reactions play a pivotal role in chemistry and biology. The values of standard redox potential (E°) of two half-cell reactions decide which way the reaction is expected to proceed. A simple example is a Daniel cell in which zinc goes into solution and copper gets deposited. Given below are a set of half-cell reactions (acidic medium) along with their E° (V with respect to normal hydrogen electrode) values. Using this data obtain the correct explanations to Questions 39-41.

I2 + 2e- → 2I-       E° = 0.54
Cl2 + 2e- → 2Cl-       E° = 1.36
Mn3+ + e- → Mn2+       E° = 1.50
Fe3+ + e- → Fe2+       E° = 0.77
O2 + 4H+ + 4e- → 2H2O       E° = 1.23

While Fe3+ is stable, Mn3+ is not stable in acid solution because

A )   O2 oxidises Mn2+ to Mn3+
B )   O2 oxidises both Mn2+ and Fe2+ to Fe3+
C )   Fe3+ oxidizes H2O to O2
D )   Mn3+ oxidises H2O to O2

 

41) Redox reactions play a pivotal role in chemistry and biology. The values of standard redox potential (E°) of two half-cell reactions decide which way the reaction is expected to proceed. A simple example is a Daniel cell in which zinc goes into solution and copper gets deposited. Given below are a set of half-cell reactions (acidic medium) along with their E° (V with respect to normal hydrogen electrode) values. Using this data obtain the correct explanations to Questions 39-41.

I2 + 2e- → 2I-       E° = 0.54
Cl2 + 2e- → 2Cl-       E° = 1.36
Mn3+ + e- → Mn2+       E° = 1.50
Fe3+ + e- → Fe2+       E° = 0.77
O2 + 4H+ + 4e- → 2H2O       E° = 1.23

Sodium fusion extract, obtained from aniline, on treatment with iron (II) sulphate and H2SO4 in presence of air gives a Prussian blue precipitate. The blue colour is due to the formation of

A )   Fe4[Fe(CN)6]3
B )   Fe3[Fe(CN)6]2
C )   Fe4[Fe(CN)6]2
D )   Fe3[Fe(CN)6]3

 

42) Match the reactions in Column I with nature of the reactions/type of the products 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)
O2- → O2 + O22- 		(p)
redox reaction
(B)
CrO42- + H+ → 		(q)
one of the products has trigonal planar structure
(C)
MnO4- + NO2- + H+ → 		(r)
dimeric bridged tetrahedral metal ion
(D)
NO3- + H2SO4 + Fe2+ 		(s)
disproportionation
Answer:
A → (p, s) , B → (r), C → (p, q), D → (p)
A )  
B )  
C )  
D )  

 

43) Match the compounds/ions in Column I with their properties/reactions 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)
C6H5CHO 		(p)
gives precipitate with 2, 4-dinitrophenylhydrazine
(B)
CH3C ≡ CH 		(q)
gives precipitate with AgNO3
(C)
CN- 		(r)
is a nucleophile
(D)
I- 		(s)
is involved in cyanohydrin formation
Answer:
A → (p, q, s) , B → (q), C → (q,r,s), D → (q, r)
A )  
B )  
C )  
D )  

 

44) Match the crystal system/unit cells mentioned in Column I with their characteristic features mentioned 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)
Simple cubic and face-centred cubic 		(p)
have these cell parameters a = b = c and α = β = γ
(B)
cubic and rhombohedral 		(q)
are two crystal systems
(C)
cubic and tetragonal 		(r)
have only two crystallography angles of 90°
(D)
hexagonal and monoclinic 		(s)
belong to same crystal system
Answer:
A → (p, s) , B → (p, q), C → (q), D → (q, r)
A )  
B )  
C )  
D )  

 

45) Let a, b, c be unit vectors such that a + b + c = 0 . Which one of the following is correct?
A )   a x a = b x c = c x a = 0
B )   a x a = b x c = c x a ≠ 0
C )   a x a = b x c = a x c = 0
D )   a x a , b x c , c x a are mutually perpendicular

 

46)
A )  
B )  
C )  
D )  

 

47)
d2x   equals  
dy2
A )  
(   d2y   )-1  
dx2
B )  
- (   d2y   )-1   ( dy   )-3
dx2 dx
C )  
(   d2y  )   ( dy  )-2
dx2 dx
D )  
- (   d2y  )   ( dy  )-3
dx2 dx

 

48) The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is
A )   360
B )   192
C )   96
D )   48

 

49) If |z| = 1 and z ≠ ± 1, then all the values of
z
1 - z2
lie on
A )   a line not passing through the origin
B )   |z| =
C )   the x-axis
D )   the y-axis

 

50) Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is
A )   3
B )   2
C )   3/2
D )   1

 

51) Let O(0, 0), P(3, 4), Q(6, 0) be the vertices of the triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are
A )  
(   4   ,   3 )
3
B )  
( 3 ,   2   )
3
C )  
( 3 ,   4   )
3
D )  
(   4   ,   2   )
3 3

 

52) The differential equation
dy   =  
dx y
determines a family of circles with
A )   variable radii and a fixed centre at (0, 1)
B )   variable radii and a fixed centre at (0, -1)
C )   fixed radius 1 and variable centres along the x-axis
D )   fixed radius 1 and variable centres along the y-axis

 

53) Let Ec denote the complement of an event E. Let E, F, G be pairwise independent events with P(G) > 0 and
P(E ∩ F ∩ G) = 0. Then P(Ec ∩ Fc | G) equals
A )   P(Ec) + P(Fc)
B )   P(Ec) - P(Fc)
C )   P(Ec) - P(F)
D )   P(E) - P(Fc)

 

54) Let f(x) = 2 + cosx for all real x.

STATEMENT-1 : For each real t, there exists a point c in [t, t + π ] such that f'(c) = 0.

because

STATEMENT-2 : f(t) = f(t + 2π) for each real t.

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) Consider the planes 3x - 6y - 2z = 15 and 2x + y - 2z = 5.

STATEMENT-1 : The parametric equations of the line of intersection of the given planes are x = 3 + 14t, y = 1 + 2t, z = 15t

because

STATEMENT-2 : The vectors 14i + 2j + 15k is parallel to the line of intersection of the given planes.

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) Lines L1 : y - x = 0 and L2 : 2x + y = 0 intersect the line L3 : y + 2 = 0 at P and Q, respectively. The bisector of the acute angle between L1 and L2 intersects L3 at R.

STATEMENT-1 : The ratio PR : RQ equals 2 :

because

STATEMENT-2 : In any triangle, bisector of an angle divides the triangle into two similar triangles.

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) STATEMENT-1 : The curve
y =   -x2   + x + 1
2
is symmetric with respect to the line x = 1.

because

STATEMENT-2 : A parabola is symmetric about its axis.

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) If a continuous f defined on the real line R, assumes positive and negative values in R then the equation f(x) = 0 has a root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum values is negative then the equation f(x) = 0 has a root in R. Consider f(x) = kex - x for all real x where k is a real constant.

The line y = x meets y = kex for k ≤ 0 at

A )   no point
B )   one point
C )   two points
D )   more than two points

 

59) If a continuous f defined on the real line R, assumes positive and negative values in R then the equation f(x) = 0 has a root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum values is negative then the equation f(x) = 0 has a root in R. Consider f(x) = kex - x for all real x where k is a real constant.

The positive value of k for which kex - x = 0 has only one root is

A )   1/e
B )   1
C )   e
D )   loge2

 

60) If a continuous f defined on the real line R, assumes positive and negative values in R then the equation f(x) = 0 has a root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum values is negative then the equation f(x) = 0 has a root in R. Consider f(x) = kex - x for all real x where k is a real constant.

For k > 0, the set of all values of k for which kex - x = 0 has two distinct roots is

A )  
( 0 , 1   )
e
B )  
(   1   , 1 )
e
C )  
(   1   , ∞ )
e
D )   ( 0, 1 )

 

61) Let A1, G1, H1 denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n ≥ 2, let An-1 and Hn-1 has arithmetic, geometric and harmonic means as An, Gn, Hn respectively.

Which one of the following statements is correct?

A )   G1 > G2 > G3 > ...
B )   G1 < G2 < G3 < ...
C )   G1 = G2 = G3 = ...
D )   G1 < G3 < G5 < ... and G2 > G4 > G6 > ...

 

62) Let A1, G1, H1 denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n ≥ 2, let An-1 and Hn-1 has arithmetic, geometric and harmonic means as An, Gn, Hn respectively.

Which one of the following statements is correct?

A )   A1 > A2 > A3 > ...
B )   A1 < A2 < A3 < ...
C )   A1 > A3 > A5 > ... and A2 < A4 < A6 < ...
D )   A1 < A3 < A5 < ... and A2 > A4 > A6 > ...

 

63) Let A1, G1, H1 denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n ≥ 2, let An-1 and Hn-1 has arithmetic, geometric and harmonic means as An, Gn, Hn respectively.

Which one of the following statements is correct?

A )   H1 > H2 > H3 > ...
B )   H1 < H2 < H3 < ...
C )   H1 > H3 > H5 > ... and H2 < H4 < H6 < ...
D )   H1 < H3 < H5 < ... and H2 > H4 > H6 > ...

 

64) Let
f(x) = x2 - 6x + 5
x2 - 5x + 6

Match the conditions / expressions in Column I with statements in Column II and indicate y our answers by darkening the appropriate bubbles in 4 X 4 matrix given in the ORS.
Column I Column II
(A)
If -1 < x < 1, then f(x) satisfies 		(p)
0 < f(x) < 1
(B)
If 1 < x < 2, then f(x) satisfies 		(q)
f(x) < 0
(C)
If 3 < x < 5, then f(x) satisfies 		(r)
f(x) > 0
(D)
If x > 5, then f(x) satisfies 		(s)
f(x) < 1
Answer:
A → (p, r, s) , B → (q, s), C → (q, s), D → (p, r, s)
A )  
B )  
C )  
D )  

 

65) Let (x, y) be such that
sin-1(ax) + cos-1(y) + cos-1(bxy) =     π
  2

Match the statements in Column I with the statements 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)
If a = 1 and b = 0, then (x, y) 		(p)
lies on the circle x2 + y2 = 1
(B)
If a = 1 and b = 1, then (x, y) 		(q)
lies on (x2 - 1) (y2 - 1) = 0
(C)
If a = 1 and b = 2, then (x, y) 		(r)
lies on y = x
(D)
If a = 2 and b = 2, then (x, y) 		(s)
lies on (4x2 - 1) (y2 - 1) = 0
Answer:
A → (p) , B → (q), C → (p), D → (s)
A )  
B )  
C )  
D )  

 

66) Match the statements 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)
Two intersecting circles 		(p)
have a common tangent
(B)
Two mutually external circles 		(q)
have a common normal
(C)
two circles, one strictly inside the other 		(r)
do not have a common tangent
(D)
two branches of a hyperbola 		(s)
do not have a common normal
Answer:
A → (p, q) , B → (p, q), C → (q, r), D → (q, r)
A )  
B )  
C )  
D )  

ANSWERS

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