**MATHEMATICS**

PAPER - II

Section -I

Straight Objective Type

PAPER - II

Section -I

Straight Objective Type

This section contains 6 multiple choice questions. Each question has 4 choices (1), (2), (3) and (4), out of which ONLY ONE is correct.

1. Let the function g : (-, ) be given by g(u) = 2 tan

^{-1}(e

^{u}) -. Then, g is

(1) even and is strictly increasing in (0, )

(2) odd and is strictly decreasing in (-, )

(3) odd and is strictly increasing in (-, )

(4) neither even nor odd, but is strictly increasing in (-, )

Ans : (3)

2. A particle P starts from the point z

_{0}= 1 + 2i, where i = . It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z

_{1}. From z

_{1}the particle moves units in the direction of the vector and then it moves through an angle /2 in anticlockwise direction on a circle with centre at origin, to reach a point z

_{2}. The point z

_{2}is given by

(1) 6 + 7i

(2) -7 + 6i

(3) 7 + 6i

(4) -6 + 7i

Ans : (4)

3. Consider a branch of the hyperbola x

^{2}- 2y

^{2}- 2 x - 4 y - 6 = 0 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

(1) 1 -

(2) -1

(3)

(4) + 1

Ans : (2)

4.

(1)

(2)

(3)

(4)

Ans : (3)

5. Let Two non-collinear unit vector and 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 . Where P is farthest from origin O, let M be the length of and be the unit vector along . Then,

(1)

(2)

(3)

(4)

Ans : (1)

6. 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 N = 1, 2, 3, ... ,

(1)

(2)

(3)

(4)

Ans : (1)

7. The area of the region between the curves y = and y = bounded by the lines x = 0 and x = is

(1)

(2)

(3)

(4)

Ans : (2)

8. 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

(1) 2, 4 or 8

(2) 3, 6 or 9

(3) 4 or 8

(4) 5 or 10

Ans : (4)

9. Consider three points where

(1) P lies on the line segment RQ

(2) Q lies on the line segment PR

(3) R lies on the line segment QP

(4) P, Q, R are non-collinear

Ans : (4)

R1 º =

= (cos + sin) P, Q, R are non-collinear as 1 < cos + sin <

**Section - II**

Assertion - Reason Type

Assertion - Reason 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.

10. Consider

L

_{1}: 2x + 3y + p - 3 = 0

L

_{2}: 2x + 3y + p + 3 = 0,

where p is a real number, and C : x

^{2}+ y

^{2}+ 6x - 10y + 30 = 0

STATEMENT-1

If line L

_{1}is a chord of circle C, then line L

_{2}is not always a diameter of circle C.

and

STATEMENT-2

If Line L

_{1}is a diameter of circle C, then line L2 is not a chord of circle C.

(1) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

(2) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

(3) STATEMENT-1 is True, STATEMENT-2 is False

(4) STATEMENT-1 is False, STATEMENT-2 is True

Ans : (3)

distance between parallel lines is less then the radius of the circle

If one is chord then other may or may not be a diameter and if one is diameter other must be a chord.

11

STATEMENT-1

y(x) = sec

and

STATEMENT-2

y(x) is given by

(1) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

(2) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

(3) STATEMENT-1 is True, STATEMENT-2 is False

(4) STATEMENT-1 is False, STATEMENT-2 is True

Ans : (3)

Statement 2 is false.

12 Suppose four distinct positive numbers a

_{1}, a

_{2}, a

_{3}, a

_{4}are in G.P. Let b

_{1}= a

_{1}, b

_{2 }= b

_{1}+ a

_{2}, b

_{3 }= b

_{2 }+ a

_{3}and b

_{4}= b

_{3}+ a

_{4}.

STATEMENT-1

The numbers b

_{1}, b

_{2}, b

_{3}, b

_{4}are neither in A.P. nor in G.P.

and

STATEMENT-2

The numbers b

_{1}, b

_{2}, b

_{3}, b

_{4}are in H.P.

(1) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

(2) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

(3) STATEMENT-1 is True, STATEMENT-2 is False

(4) STATEMENT-1 is False, STATEMENT-2 is True

Ans : (3)

Let a

_{1}= a, a

_{2}= a

_{5}, a

_{3}= ar

^{2}, a

_{4}= ar

^{3}b

_{1}= a, b

_{2}= a + ar, b

_{3}= a + ar + ar

^{2}, b

_{4}= a + ar + ar

^{2}+ ar

^{3}. So b

_{1}, b

_{2}, b

_{3}, b

_{4 }are neither in A.P. nor in G.P. nor in H.P. So, statement-1 is true and statement 2 is false.

13 Let a, b, c, p, q be real numbers. Suppose a, b are the roots of the equation the roots of the equation ax

^{2}+ 2bx + c = 0, where

STATEMENT-1:

and

STATEMENT-2:

(1) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

(2) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

(3) STATEMENT-1 is True, STATEMENT-2 is False

(4) STATEMENT-1 is False, STATEMENT-2 is True

Ans : (1)

both roots are not common

exactly one root is common or No root common

(but as per paragraph a is a common root)

exactly one root is common.

common root must be real

both equation has real roots.

**Section -III**

Linked Comprehension Type

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 Questions Nos. 14 to 16**

14. Which of the following is true?

(1) (2 + a)

^{2}f'' (1) + (2 - a)

^{2 }f'' (-1) = 0

(2) (2 - a)

^{2}f'' (1) - (2 + a)

^{2}f'' (-1) = 0

(3) f' (1) f' (-1) = (2 - a)

^{2}

(4) f' (1) f' (-1) = -(2 + a)

^{2}

Ans : (1)

15. Which of the following is true?

(1) f(x) is decreasing on (-1, 1) and has a local minimum at x = 1

(2) f(x) is increasing on (-1, 1) and has a local maximum at x = 1

(3) f(x) is increasing on (-1, 1) but has neither a local maximm nor a local minimum at x = 1

(4) f(x) is decreasing on (-1, 1) but has neither a local maximum nor a local minimum at x = 1

Ans : (1)

16. Let

Which of the following is true?

(1) g'(x) is positive on (-, 0) and negative on (0, )

(2) g'(x) is negative no (-, 0) and positive on (0, )

(3) g'(x) changes sign on both (-, 0) and (0, )

(4) g'(x) does not change sign on (-, )

Ans : (2)

**Paragraph for Questions Nos. 17 to 19**

17. The unit vector perpendicular to both L

_{1}and L

_{2}is

(1)

(2)

(3)

(4)

Ans : (2)

18. The shortest distance between L

_{1}and L

_{2}is

(1) 0

(2)

(3)

(4)

Ans : (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 L

_{1}and L

_{2}is

(1)

(2)

(3)

(4)

Ans : (3)

**Section - IV**

Matrix Match Type

Matrix Match Type

*This section contains 3 questions. Each question contains statements given in two column 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. The answers to these questions have to be appropriately bubbles as illustrated in the following example.*

*If the correct matches are A-p, A-s, B-q, B-r, C-p, C-q and D-s, then the correctly bubbled 4 × 4 matrix should be as follows :*

20. Consider the lines given by

L

_{1}: x + 3y - 5 = 0

L

_{2}: 3x - ky - 1= 0

L

_{3}: 5x + 2y - 12 = 0

Match the statements/Expressions in Column I with the statements/Expressions in Column II. Indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

Column I | Column II | ||

(A) | L_{1}, L_{2}, L_{3} are concurrent, if | (p) | k = -9 |

(B) | One of L_{1}, L_{2}, L_{3} is parallel to at least one of the other two, if | (q) | k = |

(C) | L_{1}, L_{2}, L_{3} form a triangle, if | (r) | k = |

(D) | L_{1}, L_{2}, L_{3} do not form a triangle, if | (s) | k = 5 |

(2) (A) - (p); (B) - (s), (q); (C) - (q); (D) - (p), (r), (s)

(3) (A) - (q); (B) - (p), (s); (C) - (s); (D) - (r), (q), (s)

(4) (A) - (r); (B) - (s), (q); (C) - (p); (D) - (p), (q), (r)

Ans : (1)

x + 3y = 5

3x - ky = 1

5x + 2y = 12

(A) 13k - 65 = 0 k = 5

(B) L

_{1}|| L

_{2}- k = -9

or L

_{2}|| L

_{3 }= - 5/2 k = -6/5

(C) no two should be parallel

and no three should be concurrent k = 5/6

(D) k = -9, -6/5 , 5

21. Consider all possible permutations of the letters of the word ENDEANOEL.

Match the entries in Column I with the correctly related quantum number(s) in Column II. Indicate your answer by darkening the appropriate bubbles of 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! |

(2) (A) - (r); (B) - (s); (C) - (p); (D) - (q)

(3) (A) - (s); (B) - (q); (C) - (r); (D) - (p)

(4) (A) - (p); (B) - (s); (C) - (q); (D) - (q)

Ans : (4)

22. Match the statements/Expressions in Column I with the statements/Expressions in Column II. Indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

Column I | Column II | ||

(A) | (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} is the transpose of the matrix AB, then the possible values of k are | (q) | 1 |

(C) | Let a = log_{3} log_{3} 2. An integer k satisfying 1< 2^{(- k + 3 -a)} < 2, must be less than | (r) | 2 |

(D) | (s) | 3 |

(2) (A) - (q); (B) - (p), (r); (C) - (p); (D) - (p), (s)

(3) (A) - (p); (B) - (r), (s); (C) - (s); (D) - (p), (r)

(4) (A) - (s); (B) - (s), (q); (C) - (r); (D) - (p), (q)

Ans : (1)

**PHYSICS**

Section - I

Straight Objective Type

Section - I

Straight Objective Type

This section contains 6 multiple choice questions. Each question has 4 choices (1), (2), (3) and (4), out of which ONLY ONE is correct.

23. A radioactive sample S1 having an activity of 5Ci 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

(1) 20 years and 5 years, respectively

(2) 20 years and 10 years respectively

(3) 10 years each

(4) 5 years each.

Ans : (1)

24. 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 = 60º.

(1)

(2) the potential energy of the system is zero

(3)

(4)

Ans : (3)

The electric field due to charges at A and B are equal and opposite. Therefore, at O, the electric field is only due to C, which has a magnitude

(1) is wrong.

The potential energy of the system

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

Figure :

(1)

(2)

(3)

(4) .

Ans : (1)

is positive and can only be along y-axis.

26. 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

Figure :

(1)

(2)

(3)

(4) .

Ans : (1)

(1) is correct.

27. 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

(1) 344

(2) 336

(3) 117.3

(4) 109.3

Ans : (1)

(1) is correct.

28. A light beam is travelling from Region I to Region IV (refer figure). The refractive index in Region I, II, III and IV are respectively. The angle of incidence q for which the beam just misses entering Region IV is

Figure :

(1)

(2)

(3)

(4) .

Ans : (2)

29. A block (B) is attached to two unstretched springs S1 and S2 with spring constants k and 4k, 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 measure with respect to the equilibrium position of the block B. The ratio is

Figure :

(1) 4

(2) 2

(3)

(4)

Ans : (3)

When the block undergoes a maximum displacement y towards wall 2, the spring S1 is at its natural length but S2 is compressed.

30. A glass tube of uniform internal radius (r) has 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,

Figure :

(1) air from end 1 flows towards end 2. No change in the volume of the soap bubbles

(2) air from end 1 flows towards end 2. Volume of the soap bubble at end 1 decreases

(3) no change occurs

(4) air from end 2 flows towards end 1. Volume of the soap bubble at end 1 increases.

Ans : (2)

The excess pressure inside a soap bubble

Where R is the radius of curvature

Now radius of curvature at end 1 is less than the radius of curvature at end 2.

31. 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

Figure

(1)

(2)

(3)

(4) .

Ans : (4)

(4) is correct.

**Section - II**

Assertion - Reason Type

Assertion - Reason 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.

32. STATEMENT - 1

For practical purposes, the earth is used as a reference at zero potential in electrical circuits.

because

STATEMENT - 2

The electrical potential of a sphere of radius R with charge Q uniformly distributed on the surface is given by .

(1) Statement - 1 is True, Statement - 2 is True; Statement - 2 is a correct explanation for Statement - 1.

(2) Statement - 1 is True, Statement - 2 is True; Statement - 2 is NOT a correct explanation for Statement- 1.

(3) Statement - 1 is True, Statement - 2 is False.

(4) Statement - 1 is False, Statement - 2 is True.

Ans : (1)

Both the statements are true and statement-2 explains statement-1.

33. STATEMENT - 1

The sensitivity of a moving coil galvanometer is increased by placing a suitable magnetic material as a core inside the coil.

because

STATEMENT - 2

Soft iron has a high magnetic permeability and cannot be easily magnetized or demagnetized.

(1) Statement - 1 is True, Statement - 2 is True; Statement - 2 is a correct explanation for Statement - 1.

(2) Statement - 1 is True, Statement - 2 is True; Statement - 2 is NOT a correct explanation for Statement- 1.

(3) Statement - 1 is True, Statement - 2 is False.

(4) Statement - 1 is False, Statement - 2 is True.

Ans : (3)

Since Statement-2 is false.

34. 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.

because

STATEMENT - 2

If the observer and the object are moving at velocities respectively with reference to a laboratory frame, the velocity of the object with respect to the observer is

(1) Statement - 1 is True, Statement - 2 is True; Statement - 2 is a correct explanation for Statement - 1.

(2) Statement - 1 is True, Statement - 2 is True; Statement - 2 is NOT a correct explanation for Statement- 1.

(3) Statement - 1 is True, Statement - 2 is False.

(4) Statement - 1 is False, Statement - 2 is True.

Ans : (2)

Both the statements are true but statement-2 does not explain statement-1.

35. STATEMENT - 1

It is easier to pull a heavy object than to push it on a level ground.

because

STATEMENT - 2

The magnitude of frictional force depends on the nature of the two surfaces in contact.

(1) Statement - 1 is True, Statement - 2 is True; Statement - 2 is a correct explanation for Statement - 1.

(2) Statement - 1 is True, Statement - 2 is True; Statement - 2 is NOT a correct explanation for Statement- 1.

(3) Statement - 1 is True, Statement - 2 is False.

(4) Statement - 1 is False, Statement - 2 is True.

Ans : (2)

Both the statements are true but Statement-2 does not explain statement-1.

**Section - III**

Linked Comprehension Type

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.

Figure :

36. The electric field at r = R is

(1) independent of a

(2) directly proportional to a

(3) directly proportional to a

^{2}

(4) inversely proportional to a.

Ans : (1)

For a spherically symmetric charge distribution, the electric field at outside points is independent of how the charge is distributed in the volume.

37. For a = 0, the value of d (maximum value of as shown in the figure) is

(1)

(2)

(3)

(4) .

Ans : (2)

38. The electric field within the nucleus is generally observed to be linearly dependent on r. This implies

(1) a = 0

(2)

(3) a = R

(4) .

Ans : (3)

If is constant then the E r.

**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 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 m.

Figure :

39. The net external force acting on the disc when its centre of mass is at displacement x with respect to its equilibrium position is

(1) -kx

(2) -2kx

(3)

(4)

Ans : (4)

40. The centre of mass of the disc undergoes simple harmonic motion with angular frequency w equal to

(1)

(2)

(3)

(4) .

Ans : (4)

41. The maximum value of V

_{0}for which the disk will roll without slipping is

(1)

(2)

(3)

(4) .

Ans : (3)

(3) is correct.

**Section - IV**

Matrix Match Type

Matrix Match Type

*This section contains 3 questions. Each question contains statements given in two column 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. The answers to these questions have to be appropriately bubbles as illustrated in the following example.*

*If the correct matches are A-p, A-s, B-q, B-r, C-p, C-q and D-s, then the correctly bubbled 4 × 4 matrix should be as follows :*

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. Indicate your answer by darkening the appropriate bubbles of the 4 × 4 matrix given in the ORS.

Column I | Column II | ||

(A) | Potential energy of a simple pendulum (y axis) as a function of displacement (x axis) | (p) | |

(B) | Displacement (y axis) as a function of time (x axis) for a one dimensional motion at zero or constant acceleration when the body is moving along the positive x-direction. | (q) | |

(C) | Range of a projectile (y axis) as a function of its velocity (x axis) when projected at a fixed angle | (r) | |

(D) | The square of the time period (y axis) of a simple pendulum as a function of its length (x axis) | (s) |

(2) (A) - (s); (B) - (r), (s); (C) - (q); (D) - (p)

(3) (A) - (q); (B) - (q), (r); (C) - (p); (D) - (r)

(4) (A) - (p); (B) - (q), (s); (C) - (s); (D) - (q)

Ans : (4)

43. Column I contains a list of processes involving expansion of an ideal gas. Match this with Column II describing the thermodynamic change during this process. Indicate your answer by darkening the appropriate bubbles of the 4 × 4 given in the ORS.

Column I | Column II | ||

(A) | An insulated container has two chambers separated by a valve. Chamber I contains an ideal gas and the Chamber II has vacuum. The valve is opened. | (p) | The temperature of the gas decreases |

(B) | An ideal monoatomic gas expands to twice its original volume such that its pressure where V is the volume of the gas | (q) | The temperature of the gas increases or remains constant |

(C) | An ideal monoatomic gas expands to twice its original volume such that its pressure where V is the volume of the gas | (r) | The gas loses heat |

(D) | An ideal monoatomic gas expands such that its pressure P and volume V follows the behaviour shown in the graph | (s) | The gas gains heat |

(2) (A) - (p); (B) - (p), (s); (C) - (p), (r); (D) - (p), (s)

(3) (A) - (r); (B) - (q), (r); (C) - (p), (s); (D) - (q), (r)

(4) (A) - (s); (B) - (q), (s); (C) - (r), (q); (D) - (p), (q)

Ans : (1)

44. 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. Match all the properties of images from Column II with the appropriate components given in Column I. Indicate your answer by darkening the appropriate bubbles of the 4 × 4 matrix given in the ORS.

Column I | Column II | ||

(A) | (p) | Real image | |

(B) | (q) | Virtual image | |

(C) | (r) | Magnified image | |

(D) | (s) | Image at infinity |

(2) (A) - (p), (q), (r), (s); (B) - (p) ; (C) - (p), (q), (r), (s); (D) - (p), (q), (r), (s)

(3) (A) - (p), (q), (r), (s); (B) - (s) ; (C) - (p), (q), (r), (s); (D) - (p), (q), (r), (s)

(4) (A) - (p), (q), (r), (s); (B) - (r) ; (C) - (p), (q), (r), (s); (D) - (p), (q), (r), (s)

Ans : (1)

**CHEMISTRY**

Section - I

Straight Objective Type

Section - I

Straight Objective Type

This section contains 6 multiple choice questions. Each question has 4 choices (1), (2), (3) and (4), out of which ONLY ONE is correct.

45. Cellulose upon acetylation with excess acetic anhydride/H

_{2}SO

_{4}(catalytic) gives cellulose triacetate whose structure is

(1)

(2)

(3)

(4)

Ans : (2)

Cellulose in the straight chain polysaccharide composed of -D-glucose units which are joined by glycosidic linkage between c-1 of one glucose unit and c-4 of the next glucose unit. Since it cellulosetriacelate so, its structure is as given in B.

46. The correct stability order for the following species is

(1) (II) > (IV) > (I) > (III)

(2) (I) > (II) > (III) > (IV)

(3) (II) > (I) > (IV) > (III)

(4) (I) > (III) > (II) > (IV)

Ans : (4)

The cabocation (I) is stabilized by six -H atom as well as by +R effect of alkoxy group attached. III is stabilized by +R effect and by three -H atoms. II is stabilished only by five - H atoms while IV being 1° carbocation is stabilized by two -H atoms.

47. In the following reaction sequence, the correct structures of E, F and G are

(* implies

^{13}C labelled carbon)

(1)

(2)

(3)

(4)

Ans : (3)

48. Among the following, the surfactant that will form micelles in aqueous solution at the lowest molar concentration at ambient conditions is

(1) CH

_{3}(CH

_{2})

_{15}N

^{+}(CH

_{3})

_{3}Br

^{-}

(2) CH

_{3}(CH

_{2})

_{11 }OSO

_{3}

^{-}Na

^{+}

(3) CH

_{3}(CH

_{2})

_{6}COO

^{-}Na

^{+}

(4) CH

_{3}(CH

_{2})

_{11}N

^{+}(CH

_{3})

_{3}Br

^{-}

Ans : (2)

49. Electrolysis of dilute aqueous NaCl solution was carried out by passing 10 milli ampere current. The time required to liberate 0.01 mol of H

_{2}gas at the cathode is (1 Faraday = 96500 C mol

^{-1})

(1) 9.65 × 10

^{4}sec

(2) 19.3 × 10

^{4}sec

(3) 28.95 × 10

^{4}sec

(4) 38.6 × 10

^{4}sec

Ans : (2)

50. Solubility product constants (K

_{sp}) of salts of types MX, MX

_{2}and M

_{3}X 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

(1) MX > MX

_{2}> M

_{3}X

(2) M

_{3}X > MX

_{2}> MX

(3) MX

_{2}> M

_{3}X > MX

(4) MX > M

_{3}X > MX

_{2}

Ans : (4)

51. Among the following, the coloured compound is

(1) CuCl

(2) K

_{3}[Cu(CN)

_{4}]

(3) CuF

_{2}

(4) [Cu(CH

_{3}CN)

_{4}]BF

_{4}

Ans : (3)

Cu

^{2+}as in CuF

_{2}with incomplete d-orbital (d

^{9}- configurtation) will exhibit colour due to d-d transition. In all the other choices Cu exists as Cu

^{+}with 3d

^{10}configuration and hence colourless.

52. The IUPAC name of [Ni(NH

_{3})

_{4}] [NiCl

_{4}] is

(1) Tetrachloronickel (II) - tetraamminenickel (II)

(2) Tetraamminenickel (II) - tetrachloronickel (II)

(3) Tetraamminenickel (II) - tetrachloronickelate (II)

(4) Tetrachloronickel (II) - Tetraamminenickelate (0)

Ans : (3)

The cationic complex is named first followed by the name of anionic complex. The name of the metal in anionic complex ends with the suffix ate (See IUPAC nomenclature rules in detail).

53. Both [Ni (CO)

_{4}] and [Ni (CN)

_{4}]

^{2-}are diamagnetic. The hybridizations of nickel in these complexes, respectively, are

(1) sp

^{3}, sp

^{3 }

(2) sp

^{3}, dsp

^{2}

(3) dsp

^{2}, sp

^{3 }

(4) dsp

^{2}, dsp

^{2}

Ans : (2)

Note that both : CO and :CN

^{-}are strong field ligands and are able to induce spin pairing.

**Section - II**

Assertion - Reason Type

Assertion - Reason 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

The geometrical isomers of the complex [M(NH

_{3})

_{4}Cl

_{2 }are optically inactive.

and

STATEMENT-2

Both geometrical isomers of the complex [M (NH

_{3})

_{4}Cl

_{2}] possess axis of symmetry.

(1) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

(2) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

(3) STATEMENT-1 is True, STATEMENT-2 is False

(4) STATEMENT-1 is False, STATEMENT-2 is True

Ans : (1)

55. 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.

(1) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

(2) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

(3) STATEMENT-1 is True, STATEMENT-2 is False

(4) STATEMENT-1 is False, STATEMENT-2 is True

Ans : (3)

It is only in the cyclic process that the absorption of heat from a reservoir and its complete conversion into work is not possible and not in any process as stated.

56. STATEMENT-1

Aniline on reaction with NaNO

_{2}/HCl at 0°C followed by coupling with b - naphthol gives a dark blue coloured precipitate.

and

STATEMENT-2

The colour of the compound formed in the reaction of aniline with NaNO

_{2}/HCl at 0°C followed by coupling with b-naphthol is due to the extended conjugation.

(1) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

(2) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

(3) STATEMENT-1 is True, STATEMENT-2 is False

(4) STATEMENT-1 is False, STATEMENT-2 is True

Ans : (4)

57. STATEMENT-1

[Fe(H

_{2}O)

_{5}NO] SO

_{4 }is paramagnetic.

and

STATEMENT-2

The Fe in [Fe(H

_{2}O)

_{5 }NO]SO

_{4}has three unpaired electrons.

(1) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

(2) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

(3) STATEMENT-1 is True, STATEMENT-2 is False

(4) STATEMENT-1 is False, STATEMENT-2 is True

Ans : (1)

In the complex [Fe(H

_{2}O)

_{5}NO]SO

_{4}, the odd e

^{-}on N-atom of NO molecule is transferred to the incomplete d-orbital of Fe

^{2+}which changes Fe

^{2+}into Fe

^{+}and NO into NO

^{+}. H

_{2}O and both being weak field ligand fail to cause spin-pairing into Fe

^{+}as shown below:

As evident there are three unpaired electrons and hence paramagnetic

**Section - III**

Linked Comprehension Type

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 Questions Nos. 58 to 60**

A tertiaryalcohol 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

(1)

(2)

(3)

(4)

Ans : (2)

59. The structure of compound I is

(1)

(2)

(3)

(4)

Ans : (1)

60. The structures of compounds J, K and L, respectively, are

(1) PhCOCH

_{3}, PhCH

_{2}COCH

_{3 }and PhCH

_{2}COO

^{-}K

^{+}

(2) PhCHO, PhCH

_{2}CHO and PhCOO

^{-}K

^{+}

(3) PhCOCH

_{3}, PhCH

_{2}CHO and CH

_{3}COO

^{-}K

^{+}

(4) PhCHO, PhCOCH

_{3}and PhCOO

^{-}K

^{+}

Ans : (4)

**Paragraph for Questions 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 thee 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

(1) 4

(2) 6

(3) 12

(4) 17

Ans : (2)

Effective number of atoms present in unit cell of HCP

62. The volume of this HCP unit cell is

(1)

(2)

(3)

(4)

Ans : (10

63. The empty space in this HCP unit cell is

(1) 74%

(2) 47.6%

(3) 32%

(4) 26%

Ans : (4)

Packing fraction in HCP = 0.74

void fraction = 1 - 0.74 = 0.26

empty space in HCP unit cell = 26%

**Section - IV**

Matrix Match Type

Matrix Match Type

This section contains 3 questions. Each question contains statements given in two column 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. The answers to these questions have to be appropriately bubbles as illustrated in the following example.

This section contains 3 questions. Each question contains statements given in two column 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. The answers to these questions have to be appropriately bubbles as illustrated in the following example.

*If the correct matches are A-p, A-s, B-q, B-r, C-p, C-q and D-s, then the correctly bubbled 4 × 4 matrix should be as follows :*

64. Match the entries in Column I with the correctly related quantum number(s) in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 × 4 matrix given in the ORS.

Column I | Column II | ||

(A) | (p) | sodium fusin extract of the compound gives Prussian blue colour with FeSO_{4} | |

(B) | (q) | gives positive FeCl_{3 }test | |

(C) | (r) | gives white precipitate with AgNO_{3} | |

(D) | (s) | reacts with aldehyes to form the corresponding hydrazone derivative |

(2) (A) - (r), (q); (B) - (s), (q); (C) - (s), (q), (r); (D) - (r)

(3) (A) - (r), (p); (B) - (r), (q); (C) - (r), (q), (r); (D) - (s)

(4) (A) - (p), (s); (B) - (p), (r); (C) - (q), (q), (r); (D) - (q)

Ans : (1)

(A) Though it contains N will nto give lassaigne test as there is no carbon present in it.

Although FeCl

_{3 }has not been mentioned in p yet some Fe

^{2+}ion getting oxidized by aerial oxides into Fe

^{3+}is sufficient to form prussion blue.

Positive FeCl

_{3}test is given by phenolic OH present with molecule. While ppt with AgNO

_{3 }solution will be given by Cl

^{-}ion.

65. Match the entries in Column I with the correctly related quantum number(s) in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 × 4 matrix given in the ORS.

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 |

(2) (A) - (r); (B) - (p), (q), (r), (s); (C) - (p), (q), (s); (D) - (q)

(3) (A) - (p); (B) - (p), (q), (r), (s); (C) - (p), (s), (r); (D) - (r)

(4) (A) - (q); (B) - (p), (q), (r), (s); (C) - (p), (q), (r); (D) - (p)

Ans : (4)

Orbital angular momentum (L) =

= azimuthal Q.N.

n determines size, shape and m - orientation of orbital.

The more function of H-like atom is spherical and has radial part only. It is independent of and , the angular parameters. Pauli principle concerns with the spin of electron it states that an orbital can contain only a maximum of two electrons and that two when their direction of spin are opposite.

66. Match the entries in Column I with the correctly related quantum number(s) in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 × 4 matrix given in the ORS.

Column II | Column II | ||

(A) | PbS PbO | (p) | roasting |

(B) | CaCO_{3} CaO | (q) | calcinations |

(C) | ZnS Zn | (r) | carbon reduction |

(D) | Cu_{2}S Cu | (s) | self reduction |

(2) (A) - (q), (B) - (p), (C) - (p), (s), (D) - (p), (r), (q)

(3) (A) - (s), (B) - (r), (C) - (p), (q), (D) - (p), (r), (s)

(4) (A) - (r), (B) - (s), (C) - (p), (r), (D) - (p), (s), (q)

Ans : (1)