|
MATHEMATICS: Paper-II (Code: 0)
SECTION - I
Straight Objective Type
This section contains 9 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.
Note: Questions with (*) mark are from syllabus of class XI.
*1. A particle P starts from the point  , where  It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z1. From z1 the particle moves  units in the direction of the vector  and then it moves through an angle
in anticlockwise direction on a circle with centre at origin, to reach a point z2. The point z2 is given by
(A) 6 + 7i
(B) -7 + 6i
(C) 7 + 6i
(D) -6 + 7i
Sol.:
Correct choice: (4)
2. Let the function
be given by
 Then, g is
(A) even and is strictly increasing in (0, )
(B) odd and is strictly decreasing in (-,)
(C) odd and is strictly increasing in ( -,)
(D) neither even nor odd, but is strictly increasing in (-,)
sol:
  g(u) is increasing function
g(u) is odd function
Correct choice: (3)
*3. Consider a branch of the hyperbola
 with vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is
(a)
(b)
(c)
(d)
sol:
Correct choice: (2)
4. The area of the region between the curves bounded by the lines x = 0 and is
(a)
(b)
(c)
(d)
sol:
Correct choice: (2)
*5. Consider three points  and  where
(A) P lies on the line segment RQ
(B) Q lies on the line segment PR
(C) R lies on the line segment QP
(D) P, Q, R are non-collinear
Sol.: Consider  
 
Points are non collinear.
Correct choice: (4)
6. An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is
(A) 2, 4 or 8
(B) 3, 6 or 9
(C) 4 or 8
(D) 5 or 10
Sol.: Given that n(S)=10;n(A) = 4
Correct choice: (4)
7. Let two non-collinear unit vectors
form an acute angle. A point P moves so that at any time t the position vector (where O is the origin) is given by
 When P is farthest from origin O, let M be the length of and  be the unit vector along.Then
(a) 
(b) 
(c) 
(d) 
sol:
Maximum occurs when Hence unit vector along this is
Correct choice: (1)
8. Then, for an arbitrary constant C, the value of J-I equals
(a)
(b)
(c)
(d)
Sol.:
Correct choice: (3)
9. Let g(x)=logf(x) where f(x) is a twice differentiable positive function on (0, ) such that f(x+1) = xf(x) .Then,for
(a)
(b)
(c)
(d)
sol:
Correct choice: (1)
SECTION - II
Reasoning Type
This section contains 4 reasoning type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.
*10. Suppose four distinct positive numbers a1,a2,a3,a4 are in G.P. Let b1=a1 ,b2=b1 + a2,b3=b2+ a3 and b4=b3+ a4.
STATEMENT-1: The numbers b1,b2,b3,b4 are neither in A.P. nor in G.P
and
STATEMENT-2: The numbers b1,b2,b3,b4 are in H.P.
(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.
sol:
Let a2= a1.r,a3= a1.r2 and a4= a1.r3 , where r 0 and 1
Thus b1 = a1, b2 = a1(1+r), b3 = a1(1+r+r2) and b4 = a1(1+r + r2 + r3)
 
Correct choice: (3)
*11. Let a, b, c, p, q be real numbers. Suppose  ,  are the roots of the equation x2 + 2px + q=0 and  are the roots of the equation ax2 + 2bx + c=0, where 
STATEMENT-1:
and
STATEMENT-2: bpa or cqa
(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.
Sol.:
None of a and q can be zero and aq c, aq - c
Since is the common root, hence
From equation (i) and (ii) 2(ap-b) + (aq-c)=0
For a definite value of ,ap-b0;  apb.i.e.statement-2 is correct.
Since nature of and is not given and a, b, c, p, q, R.It means and both will be real simultaneously or non-real simultaneously. Therefore discriminants of both given equation will be of same sign.
Statement-1 is also correct, which can not be explained by statement-2. Correct choice: (2)
*12. Consider L1: 2x + 3y + p- 3; L2: 2x + 3y + p + 3 = 0 where p is a real number, and C: x2 + y2 +6x - 10y + 30 = 0
STATEMENT-1: If line L1is a chord of circle C, then line L2 is not always a diameter of circle C.
and
STATEMENT-2: If line L1is a diameter of circle C,then line L2 is not always a diameter of circle C.
(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; STATEMENT2 is NOT a correct explanation for STATEMENT-1.
(C) STATEMENT-1 is True, STATEMENT-2 is False.
(D) STATEMENT1 is False, STATEMENT-2 is True.
sol:
Centre of the circle is (-3,5)
Radius, r = 2
Distance between the given lines
(greatest limiting distance between a diameter and a chord)
Correct choice: (3)
13. Let a solution y=y(x) of the differential equation satisfy
STATEMENT- 1:
and
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; STATEMENT2 is NOT a correct explanation for STATEMENT-1.
(C) STATEMENT-1 is True, STATEMENT-2 is False.
(D) STATEMENT1 is False, STATEMENT-2 is True.
Sol.:
Correct choice: (3)
SECTION - III
Linked Comprehension Type
This section contains 2 paragraphs. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.
Paragraph for Question Nos. 14 to 16
Consider the function f :(-,) (-,) defined by
14. Which of the following is true?
(a)(2+a)2f ''(1) + (2-a)2f ''(-1) = 0
(b)
(2+a)2f ''(1) - (2+a)2f ''(-1) = 0
(c) f '(1) f '(-1) +(2-a)2
(d) f '(1) f '(-1)=-(2-a)2
sol:
 
15. Which of the following is true?
(A) f(x) is decreasing on (-1,1) and has a local minimum at x = 1
(B) f(x) is increasing on (-1,1) and has a local maximum at x = 1
(C) f(x) is increasing on (-1,1) but has neither a local maximum nor a local minimum at x = 1
(D) f(x) is decreasing on (-1,1) but has neither a local maximum nor a local minimum at x = 1
Sol.:
Correct choice: (1)
16. Let  Which of the following is true?
(A) g'(x) is positive on (- ,0) and negative on (0 ,)
(B) g'(x) is negative on (- ,0) and positive on (0 ,)
(C) g'(x) changes sign on both (- ,0) and (0 ,)
(D) g'(x) does not change sign on (- ,)
sol:
correct choice(2)
Paragraph for Question Nos. 17 to 19
Consider the lines
17. The unit vector perpendicular to both L1 and L2 is
(a) 
(b) 
(c) 
(d)
Sol.:
Vector perpendicular to both L1 and L2 is = 
Unit vector perpendicular to L1 and L2 is
Correct choice: (2)
18. The shortest distance between L1 and L2 is
(A) 0
(B)
(C) 
(D)
Sol.:
Shortest distance between L1 and L2 is  
Correct choice: (4)
19. The distance of the point(1,1,1) from the plane passing through the point (-1,-2,-1) and whose normal is perpendicular to both the lines L1 and L2 is
(A) 
(
B) 
(C)
(D) 
Correct choice: (3)
SECTION - IV
Matrix-Match Type
This section contains 3 questions. Each question contains statements given in two columns which have to be matched. Statements in Column I are labeled as A, B, C and D whereas statements in Column II
*20. Consider the lines given by L1: x + 3y - 5 = 0; L2: 3x-ky-1=0 and L3: 5x + 2y-12=0.
Match the Statements/ Expressions in Column I with the Statements/ Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.
| Column I |
Column II |
| (A) |
L1,L2,L3 are concurrent, if |
(p) |
K=-9 |
| (B) |
One of L1,L2,L3 is parallel to at least one of the other two, if |
(q) |
K =  |
| (C) |
L1,L2,L3 form a triangle, if |
(r) |
K =  |
| (D) |
L1,L2,L3 do not form a triangle, if |
(s) |
K = 5 |
sol:
A-s L1,L2,L3 are concurrent if
1(12K + 2)-3(-36+5)-5(6+5K) = 0 -13K+65=0
K =5
B-p, q L1 and L3 are non parallel therefore either L2 is parallel to L1 or L2 is parallel to L3.
C-r For (p), (q), (s) no triangle is possible. Therefore (r) is the correct match. D-p, q, s
*21. Match the Statement/ Expressions in Column I with the Statement/ Expression in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.
| Column I |
Column II |
| (A) |
The minimum value of |
(p) |
0 |
| (B) |
Let A and B be 3 × 3 matrices of real numbers, where A is symmetric, B is skew-symmetric, and (A+B) (A-B) = (A-B) (A+B) .If (AB)t =(-1)K AB, where (AB)t s the transpose of the matrix AB, then the possible values of k are |
(q) |
1 |
| (C) |
Let a=log3 log3 2. An integer k satisfying  must be less than |
(r) |
2 |
| (D) |
If sin =cos ,then the possible values of are |
(s) |
3 |
sol
Clearly minimum value does not exist.
*22. Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statement/ Expressions in Column I with the Statement/ Expression in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.
| Column I |
Column II |
| (A) |
The number of permutations containing the word ENDEA is |
(p) |
5 ! |
| (B) |
The number of permutations in which the letter E occurs in the first and the last positions is |
(q) |
2 × 5 ! |
| (C) |
The number of permutations in which none of the letters D, L, N occurs in the last five positions is |
(r) |
7 × 5 ! |
| (D) |
The number of permutations in which the letters A, E, O occur only in odd positions is |
(s) |
21 × 5 ! |
sol:
SECTION - I
Straight Objective Type
This section contains 9 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.
23. Consider a system of three charges placed at points A, B and C, respectively, as shown in the figure. Take O to be the centre of the circle of radius R and angle CAB = 60°
(A) The electric field at point O is  directed along the negative x-axis
(B) The potential energy of the system is zero
(C) The magnitude of the force between the charges at C and B is
(D) The potential at point O is 
sol:
Correct choice: (3)
24. A radioactive sample S1 having an activity of 5 Ci has twice the number of nuclei as another sample S2 which has an activity of 10 Ci. The half lives of S1 and S2 can be
(A) 20 years and 5 years, respectively
(B) 20 years and 10 years, respectively
(C) 10 years each
(D) 5 years each
sol:
Correct choice: (1)
*25. A transverse sinusoidal wave moves along a string in the positive x-direction at a speed of 10 cm/s. The wavelength of the wave is 0.5 m and its amplitude is 10 cm. At a particular time t, the snap-shot of the wave is shown in figure. The velocity of point P when its displacement is 5 cm is
(a)
(b)
(c)
(d)
Sol.:
The slope at point P is negative, and vP = -v(slope)
Correct choice: (1)
*26. A block (B) is attached to two unstretched springs S1 and S2 with spring constants k and 4 k, respectively (see figure I). The other ends are attached to identical supports M1 and M2 not attached to the walls. The springs and supports have negligible mass. There is no friction anywhere. The block B is displaced towards wall 1 by a small distance x (figure II) and released. The block returns and moves a maximum distance y towards wall 2. Displacements x and y are measured with respect to the equilibrium position of the block B. The ratio 
(A) 4
(B) 2
(C) 
(D) 
Sol.:
By conservation of mechanical energy 
Correct choice: (3)
*27. A bob of mass M is suspended by a massless string of length L. The horizontal velocity V at position A is just sufficient to make it reach the point B. The angle at which the speed of the bob is half of that at A, satisfies
(A) 
(B) 
(C) 
(D) 
sol: By conservation of mechanical energy
Correct choice:(4)
*28. A glass tube of uniform internal radius (r) has a valve separating the two identical ends. Initially, the valve is in a tightly closed position. End 1 has a hemispherical soap bubble of radius r. End 2 has sub-hemispherical soap bubble as shown in figure. Just after opening the valve,
(A) air from end 1 flows towards end 2. No change in the volume of the soap bubbles
(B) air from end 1 flows towards end 2. Volume of the soap bubble at end 1 decreases
(C) no change occurs
(D) air from end 2 flows towards end 1. Volume of the soap bubble at end 1 increases
Sol.: Since radius of curvature of sub-hemispherical surface is more than that of hemispherical surface and 
Correct choice: (2)
*29. A vibrating string of certain length  under a tension T resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length 75 cm inside a tube closed at one end. The string also generates 4 beats per second when excited along with a tuning fork of frequency n. Now when the tension of the string is slightly increased the number of beats reduces to 2 per second. Assuming the velocity of sound in air to be 340 m/s, the frequency n of the tuning fork in Hz is
(A) 344
(B) 336
(C) 117.3
(D) 109.3
Sol.:
n = 344 Hz or 336 Hz Since on increasing tension, f increases and number of beats decreases,
n = 344 Hz
Correct choice: (1)
30. A parallel plate capacitor C with plates of unit area and separation d is filled with a liquid of dielectric constant K = 2. The level of liquid is  initially. Suppose the liquid level decreases at a constant speed V, the time constant as a function of time t is
(A) 
(B) 
(C)
(D) 
Sol.: Let at any time t, the level has fallen by a distance x ( = Vt)
Time constant ( ) = RCeq
Correct choice: (1)
31. A light beam is traveling from Region I to Region IV (Refer Figure). The refractive index in Regions I, II, III and IV are respectively. The angle of incidence for which the beam just misses entering Region IV is
(A) 
(B) 
(C) 
(D)
sol:
Correct choice: (2)
SECTION - II
Reasoning Type
This section contains 4 reasoning type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.
*32. STATEMENT-1 For an observer looking out through the window of a fast moving train, the nearby objects appear to move in the opposite direction to the train, while the distant objects appear to be stationary.
and
STATEMENT-2
If the observer and the object are moving at velocities  and  respectively with reference to a laboratory frame, the velocity of the object with respect to the observer is -.
(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.
Sol.: Correct choice: (2)
*33. STATEMENT-1 It is easier to pull a heavy object than to push it on a level ground.
and
STATEMENT-2 The magnitude of frictional force depends on the nature of the two surfaces in contact.
(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.
Sol.: Correct choice: (2)
34. STATEMENT-1 For practical purposes, the earth is used as a reference at zero potential in electrical circuits.
and
STATEMENT-2 The electrical potential of a sphere of radius R with charge Q uniformly distributed on the surface is given by
(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.
Sol.: Correct choice: (2)
35.
STATEMENT-1 The sensitivity of a moving coil galvanometer is increased by placing a suitable magnetic material as a core inside the coil.
and
STATEMENT-2
Soft iron has a high magnetic permeability and cannot be easily magnetized or demagnetized.
(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.
Sol.: Correct choice: (3)
SECTION -III
Linked Comprehension Type
This section contains 2 paragraphs. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has 4 choices (1), (2), (3) and (4), out of which ONLY ONE is correct.
Paragraph for Question Nos. 36 to 38
The nuclear charge (Ze) is non-uniformly distributed within a nucleus of radius R. The charge density  (r) [charge per unit volume] is dependent only on the radial distance r from the centre of the nucleus as shown in figure. The electric field is only along the radial direction
36. The electric field at r = R is
 
(A) independent of a
(B) directly proportional to a
(C) directly proportional to a2
(D) inversely proportional to a
Sol.:
which is constant.
Correct choice: (1)
37. For a = 0, the value of d (maximum value of as shown in the figure) is
(A) 
(B) 
(C) 
(D) 
Sol.: total charge
Correct choice: (2)
38. The electric field within the nucleus is generally observed to be linearly dependent on r. This implies
(A) a = 0
(B) 
(C) a = R
(D) 
Sol.: The electric field inside a spherical charge distribution is linearly dependent on distance from centre when charge density is uniform. Hence a = R . Correct choice: (3)
Paragraph for Question Nos. 39 to 41
A uniform thin cylindrical disk of mass M and radius R is attached to two identical massless springs of spring constant k which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance d from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of the each spring is L. The disk is initially at its equilibrium position with its centre of mass (CM) at a distance L from the wall. The disk rolls without slipping with velocity  .The coefficient of friction is
*39. The net external force acting on the disk when its centre of mass is at displacement x with respect to its equilibrium position is
(A) -kx
(B) -2kx
(C) 
(D)
sol:
Correct choice: (4)
*40. The centre of mass of the disk undergoes simple harmonic motion with angular frequency 
equal to
(A)
(B) 
(C) 
(D) 
Sol.:
Correct choice: (4)
*41. The maximum value of V0 for which the disk will roll without slipping is
(A) 
(B) 
(C) 
(D) 
sol:
On solving above equations we get,
Applying conservation of mechanical energy
Correct choice: (3)
SECTION- IV
Matrix-Match Type
This section contains 3 questions. Each question contains statements given in two columns, which have to be matched. Statements in Column I are labelled as A, B, C and D whereas statements in Column II
*42. Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column I with the graphs given in Column II.
Sol.: (A) - (p); (B) - (q),(s) ; (C) - (s); (D) - (q)
43. An optical component and an object S placed along its optic axis are given in Column I. The distance between the object and the component can be varied. The properties of images are given in Column II.
Sol.: (A) - (p),(q),(r),(s); (B) - (q) ; (C) - (p), (q), (r), (s); (D) - (p),(q),(r),(s)
*44. Column I contains a list of processes involving expansion of an ideal gas. Match this with Column II
Sol.: (A) For free expansion, dW = 0, dQ = 0 and dU = 0.
Temperature remains constant.
(A) - (q); (B) - (p),(r) ; (C) - (p), (s); (D) - (q), (s)
SECTION - I
Straight Objective Type
This section contains 9 multiple choice questions. Each question has 4 choices (1), (2), (3) and (4), out of which ONLY ONE is correct.
*45. The correct stability order for the following species is
(A) (II) > (IV) > (I) > (III)
(B) (I) > (II) > (III) > (IV)
(C) (II) > (I) > (IV) > (III)
(D) (I) > (III) > (II) > (IV)
Sol.: The correct stability order of the given species is (I) > (III) > (II) > (IV). Structure (I) is stabilised by resonance and hyperconjugation due to 6 -H atoms. Structure (III) is stabilised by resonance and hyperconjugation due to 3 -H atoms. Structure(II) is stabilised by hyperconjugation due to 5 -H atoms. Structure (IV) is stabilised by hyperconjugation due to 2 -H atoms.
Correct choice: (4)
46. Cellulose upon acetylation with excess acetic anhydride/H2SO4 (catalytic) gives cellulose triacetate whose structure is
(a) 
(b)
(C)
(d)
Sol.: Cellulose is a polymer of(D) glucose. The product cellulose triacetate should have three acetate substituents in each of its repeat unit.
Correct choice: (1)
47. In the following reaction sequence, the correct structures of E, F and G are
(* implies 13C labelled carbon)
(A) 
(B)
(C) 
(D) 
sol:
Correct choice: (3)
48. Among the following, the coloured compound is
(A) CuCl
(B) K3[Cu(CN)4]
(C) CuF2
(D) [Cu(CH3CN)4]BF4
Sol.:
CuF2 is coloured because Cu2+ ion has one unpaired electron in the 3d subshell. Rest of the compounds are not coloured because Cu+ present in each compound has completely filled 3d subshell. Correct choice: (3)
49. Both [Ni(CO)4] and [Ni(CN)4]2- are diamagnetic. The hybridisations of nickel in these complexes, respectively, are
(A) sp3, sp3
(B) sp3, dsp2
(C) dsp2, sp3
(D) dsp2, dsp2
sol:
The hybridisation of Ni in [Ni(CO)4] is sp3 and in [Ni(CN)4]2- is dsp2.
Correct choice: (2)
50. The IUPAC name of [Ni(NH3)4] [NiCl4] is
(A) Tetrachloronickel(II)-tetraamminenickel(II)
(B) Tetraamminenickel(II)-tetrachloronickel(II)
(C) Tetraamminenickel(II)-tetrachloronickelate(II)
(D) Tetrachloronickel(II)-tetraamminenickelate(0) Sol.: Tetraamminenickel(II)-tetrachloronickelate(II)
Correct choice: (3)
51. Electrolysis of dilute aqueous NaCl solution was carried out by passing 10 milli ampere current. The time required to liberate 0.01 mol of H2 gas at the cathode is (1 Faraday = 96500 C mol-1)
(A) 9.65 × 104 sec
(B) 19.3 × 104 sec
(C) 28.95 × 104 sec
(D) 38.6 × 104 sec
sol:
Correct choice: (2)
52. Among the following, the surfactant that will form micelles in aqueous solution at the lowest molar concentration at ambient conditions is
(A) CH3(CH2)15N+(CH3)3Br-
(B) CH3(CH2)11NaOSO3
(C) CH3(CH2)6COO-Na+
(D) CH3(CH2)11N+(CH3)3Br-
Sol.: Those surfactant molecules will be able to form associated colloids (micelles) readily which have greater hydrophobic interactions. Thus critical micellar concentration (CMC) will be lowest for such surfactant molecules. The hydrophobic end is biggest in CH3(CH2)15N+(CH3)3Br-
Correct choice: (1)
*53. Solubility product constants (Ksp) of salts of types MX, MX2 and M3X at temperature 'T' are 4.0 × 10-8, 3.2 × 10-14 and 2.7 × 10-15, respectively. Solubilities (mol dm-3) of the salts at temperature 'T' are in the order
(A) MX > MX2 > M3X
(B) M3X > MX2 > MX
(C) MX2 > M3X > MX
(D) MX > M3X > MX2
sol:
The decreasing order of their solubilities is MX > M3X > MX2.
Correct choice: (4)
SECTION - II
Reasoning Type
This section contains 4 reasoning type questions. Each question has 4 choices (1), (2), (3) and (4) out of which ONLY ONE is correct.
54. STATEMENT-1: Aniline on reaction with NaNO2/HCl at 0°C followed by coupling with -naphthol gives a dark blue coloured precipitate. and STATEMENT-2: The colour of the compound formed in the reaction of aniline with NaNO2/HCl at 0°C followed by coupling with -naphthol is due to the extended conjugation.
(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.
Sol.: STATEMENT-1 is false because the coupling of diazonium salt with -naphthol gives orange red coloured precipitate.
STATEMENT-2 is true.
Correct choice: (4)
55. STATEMENT-1: [Fe(H2O)5NO]SO4 is paramagnetic.
and
STATEMENT-2: The Fe in [Fe(H2O)5NO]SO4has three unpaired electrons.
(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.
Sol.: [Fe(H2O)5NO]SO4 has ligand NO in +1 oxidation state and metal Fe in +1 state. Since Fe+ has 3 unpaired electrons, the complex is paramagnetic. STATEMENT-2 is true and is the correct explanation of STATEMENT-1.
Correct choice: (1)
56. STATEMENT-1: The geometrical isomers of the complex [M(NH3)4Cl2] are optically inactive. and STATEMENT-2: Both geometrical of the complex [M(NH3)4Cl2] possess axis of symmetry.
(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.
Sol.: The two geometrical isomers of [M(NH3)4Cl2]are optically inactive as both of them have plane of symmetry. The trans isomer has C2 and C4 axes of symmetry whereas the cis isomer has C2 axis of symmetry. A molecule may be optically active even if it has axis of symmetry. Both the statements are true but the STATEMENT-2 is not the correct explanation of STATEMENT-1.
Correct choice: (2)
*57. STATEMENT-1: There is a natural asymmetry between converting work to heat and converting heat to work. and STATEMENT-2: No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work.
(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.
Sol.: Both the statements are true and STATEMENT-2 is the correct explanation of STATEMENT-1.
Correct choice: (1)
SECTION - IV
Linked Comprehension Type
This section contains 2 paragraphs. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has 4 choices (1), (2), (3) and (4), out of which ONLY ONE is correct.
Paragraph for Question Nos. 58 to 60
A tertiary alcohol H upon acid catalysed dehydration gives a product I. Ozonolysis of I leads to compounds J and K. Compound J upon reaction with KOH gives benzyl alcohol and a compound L, whereas K on reaction with KOH gives only M.
58. Compound H is formed by the reaction of
(A)
(B) 
(C) 
(D)
Sol.: The compound (H)  This being a tertiary alcohol can be prepared through the reaction of ketone with a Grignard reagent followed by hydrolysis via any of three possible routes.
Correct choice: (2)
59. The structure of compound I is
(A) 
(B) 
(C) 
(D) 
Sol.:
Correct choice: (1)
60. The structures of compounds J, K and L, respectively, are
(A) PhCOCH3, PhCH2COCH3 and PhCH2COO-K+
(B) PhCHO, PhCH2CHO and PhCOO-K+
(C) PhCOCH3, PhCH2CHO and CH3COO-K+
(D) PhCHO, PhCOCH3 and PhCOO-K+
Sol.:
Correct choice: (4)
Paragraph for Question Nos. 61 to 63
In hexagonal systems of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of these three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be 'r'.
61. The number of atoms in this HCP unit cell is
(A) 4
(B) 6
(C) 12
(D) 17
sol:
The number of effective atoms in the above hcp unit cell =
Correct choice: (2)
62. The volume of this HCP unit cell is
(A) 24 r3
(B) 16r3
(C) 12r3
(D) 
Sol.: Volume of unit cell = area of the base × height of unit cell = 
Correct choice: (1)
63. The empty space in this HCP unit cell is
(A) 74%
(B) 47.6%
(C) 32%
(D) 26%
Sol.:
Correct choice: (4)
SECTION - IV
Matrix-Match Type
This section contains 3 questions. Each question contains statements given in two columns which have to be matched. Statements A, B, C, D in Column I have to be matched with statements (p, q, r, s) in Column II.
64. Match the compounds in Column I with their characteristic test(s)/reaction(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 × 4 matrix given in the ORS.
Sol.: (A)- (r), (s) ; (B)-(p), (q) ; (C)-(p), (q), (r) ; (D)-(p), (s)
65.Match the conversions in Column I with the type(s) of reaction(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 × 4 matrix given in the ORS.
Sol.: (A)- (p) ; (B)-(q) ; (C)-(p), (r) ; (D)-(p), (r), (s)
*66. Match the entries in Column I with the correctly related quantum number(s) in Column II.
Column I |
Column II |
| (A) |
Orbital angular momentum of the electron in a hydrogen-like atomic orbital |
(p) |
Principal quantum number |
| (B) |
A hydrogen-like one electron wave function obeying Pauli principle |
(q) |
Azimuthal quantum number |
| (C) |
Shape, size and orientation of hydrogen-like atomic orbitals |
(r) |
Magnetic quantum number |
| (D) |
Probability density of electron at the nucleus in hydrogen-like atom |
(s) |
Electron spin quantum number |
Sol.: (A)-(q) ; (B)-(p), (q), (r), (s) ; (C)-(p), (q), (r) ; (D)- (p), (q), (r)
|
where V is the volume of the gas
where V is its volume
