General Instructions

1. The question paper consists of 3 Subjects (Subject I: Physics, Subject II: Chemistry, Subject III: Mathematics).
Each Subject has two sections (Section 1 & Section 2).
2. Section 1 contains 3 types of questions [Type A, Type B and Type C].
Type A contains Five (05) Multiple Choice Questions. Each question has 4 choices (A), (B), (C) and (D), out of
which ONLY ONE CHOICE is correct.
Type B contains Five (05) Multiple Correct Answers Type Questions. Each question has 4 choices (A), (B), (C)
and (D), out of which ONE OR MORE THAN ONE CHOICE is correct.

Type C contains ONE (01) paragraph. Based on the paragraph, there are TWO (02) questions. Each question has
FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct answer.
3. Section 2 contains 6 Numerical Value Type Questions. The answer to each question is a NUMERICAL VALUE. For
each question, enter the correct numerical value of the answer. If the answer is a decimal numerical value, then
round-off the value to TWO decimal places. If the answer is an Integer value, then do not add zero in the decimal
places. In the OMR, do not bubble the  sign for positive values. However, for negative values, Θ sign should be
bubbled. (Example: 6, 81, 1.50, 3.25, 0.08)
4. For answering a question, an ANSWER SHEET (OMR SHEET) is provided separately. Please fill your Test Code,
Roll No. and Group properly in the space given in the ANSWER SHEET.

Syllabus
Physics : Simple Harmonic Motion, Wave Motion, Electrostatics
Chemistry : Solid state, Theory of Solutions, Chemical Bonding, States of Matter
Mathematics : Function, Trigonometry, Quadratic Equation, Binomial Theorem
 MARKING SCHEME
SECTION-1 | Type A
 This section contains Five (05) questions. Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of
these four options is the answer. For each question, choose the option corresponding to the correct answer.
 Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +3 If ONLY the correct option is chosen;
Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered);
Negative Marks : –1 In all other cases.

SECTION-1 | Type B
 This section contains Five (05) questions. Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE
THAN ONE of these four option(s) is (are) correct answer(s).
 For each question, choose the option(s) corresponding to (all) the correct answer(s).
 Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +4 If only (all) the correct option(s) is(are) chosen;
Partial Marks : +3 If all the four options are correct but ONLY three options are chosen;
Partial Marks : +2 If three or more options are correct but ONLY two options are chosen, both of which are
correct;
Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct
option;
Zero Marks : 0 If unanswered;
Negative Marks : –2 In all other cases.
 For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers,
then
choosing ONLY (A), (B) and (D) will get +4 marks; choosing ONLY (A) and (D) will get +2 marks;
choosing ONLY (A) will get +1 mark;
choosing no option(s) (i.e. the question is unanswered) will get 0 marks and
choosing any other option(s) will get –2 marks.

SECTION-1 | Type C
 This section contains ONE paragraphs. Based on each paragraph, there are TWO questions.
 Each question has FOUR options (A), (B), (C) and (D). ONLY ONE options is the correct answer.
 For each question, choose the option corresponding to the correct answer.
 Answer to each question will be evaluated according to the following marking scheme;
Full Marks : +3 If ONLY the correct option is chosen;
Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered);
Negative Marks : −1 In all other cases.

SECTION - 2
 This section contains 6 Integer Type Questions. The answer to each question is a NUMERICAL VALUE. For
each question, enter the correct numerical value of the answer. If the answer is a decimal numerical value,
then round-off the value to TWO decimal places. If the answer is an Integer value, then do not add zero in the
decimal places. In the OMR, do not bubble the  sign for positive values. However, for negative values, Θ sign
should be bubbled. (Example: 6, 81, 1.50, 3.25, 0.08)
 Answer to each question will be evaluated according to the following marking scheme:
Full Marks: +3 If ONLY the correct Integer value is entered. There is NO negative marking.
Zero Marks: 0 In all other cases.
SUBJECT I : PHYSICS 59 MARKS

SECTION-1 | Type A
This section consists of 5 Multiple Choice Questions. Each question has 4 choices (A), (B), (C) and (D), out of which
ONLY ONE CHOICE is correct.
1. A metal ball of radius R is placed concentrically inside a hollow metal sphere of inner radius 2R and outer
radius 3R. The ball is given a charge +Q and the hollow sphere a total charge –2Q. The electrostatic
potential energy of this system is:

Q2 5Q 2 5Q 2 7Q 2
(A) (B) (C) (D)
60 R 480 R 80 R 240 R
2. Two short dipoles of dipole moment ‘p’ are kept antiparallel as shown at the origin. Length of each dipole
is ‘a’. Electric field at point P is:

6ap 3ap 3ap 4ap

(A) (B) (C) (D)
40r 4 40r 4  0r 4 30 r 4

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3. In the shown figure an infinitely long V shaped, uniformly charged wire
is placed. Linear charge density on it is . The ratio of the electric field
intensity (magnitude) at A and B is: (AB bisects the angle and points A
and B are at equal distances from the point O).
(A) 2
(B) 1

(C) cos  
2

(D) tan  
2
 a
4. A point charge q is kept at a point  a, 2a,  in the space.
 2
Consider a triangular plane surface whose co-ordinates of
 3a   3a 5a 
vertices are (a,2a,0),  , 2a,0  and  , ,0  . The
 2   2 2 
electric flux passing through the considered triangular surface is:
q q
(A) (B)
12 0 240
q q
(C) (D)
480 960
5. A system of two identical rods (L-shaped) of mass m and length l are resting on a peg P as shown in the
figure. If the system is displaced in its plane by a small angle , find the period of oscillations:

4 2l 2 2l
(A) 2 (B) 2
3g 3g

2l l
(C) 2 (D) 3
3g 3g

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 SECTION-1 | Type B
This section consists of 5 Multiple Correct Answers Type Questions. Each question has 4 choices (A), (B), (C) and
(D), out of which ONE OR MORE THAN ONE CHOICE is correct.
6. Block A of mass m is placed over a plank B of mass 5m. Plank is connected with a spring of force constant
K. The system is oscillating. The block A just loses contact with the plank when plank is momentarily at
rest. Then choose the correct option(s).

6m
(A) Time period of oscillation is T  2
K
6mg
(B) Amplitude of oscillation is A 
K
12mg
(C) Maximum compression of spring is A 
K
16mg 8mg
(D) Velocity of spring is when its compression is
3K K
7. Two conducting spheres A and B of radius R and 2R are connected by a thin conducting wire.
 21 
The conducting sphere ‘B’ is surrounded by a grounded thin concentric shell ‘C’ of radius  R  . The
 10 
connecting wire between the sphere A and B is not touching the thin shell ‘C’. If conducting sphere ‘B’ is
given a charge ‘Q’ and the separation between the spheres A and B is much larger than their radii.
Then choose the correct option(s).

Q
(A) The charge appearing on the conducting sphere ‘A’ is
43
Q
(B) The charge appearing on the conducting sphere ‘A’ is
101
100Q
(C) The charge appearing on the conducting sphere ‘B’ is
101
42Q
(D) The charge appearing on the conducting sphere ‘B’ is
43
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 
8. A uniform electric field of strength E exists in a region. An electron enters a point A with velocity v as
shown. It moves through the electric field and reaches at point B. Velocity of particle at B is 2v and it is
moving at an angle 30° with x-axis shown. Then which of the below option(s) are correct.

 mv 2  mv 2 
(A) Electric field E   i j
2ea 8ea
 mv 2 
(B) Electric field E  (i )
2ea
(8 3  1)mv 3
(C) Rate of doing work at B is
16a
3mv3
(D) Rate of doing work at B is
2a
9. Consider a system of 3 charges q/2, q/4 and –3q/4 placed at point A, B and C respectively as shown in the
figure. Take O to be the centre of the circle of radius R and CAB  30. Which of the following
option(s) are correct?

13q
(A) The electric field at O is
160 R 2
(B) Potential energy of the system is zero

3q 2
(C) Magnitude of force between charges at B and C is
640 R 2
(D) Potential at ‘O’ is zero

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10. The air column in a pipe closed at one end is made to vibrate in its first overtone by a tuning fork of
frequency 550 Hz. The speed of sound in air is 330 m/s. End correction may be neglected. Let P0 denotes
the mean pressure at any point in the pipe and P0 the maximum amplitude of pressure variation in the
pipe. Then choose the correct option(s).
(A) The length of the pipe is 45 cm
(B) The length of the pipe is 75 cm

 P0 
(C) The maximum pressure at the middle of the pipe is  P0  
 2
(D) The maximum pressure at the closed end of the pipe is ( P0  P0 )

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 SECTION-1 | Type C

This section consists of ONE (01) paragraph. Based on each paragraph, there are TWO (02) questions. Each question
has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct answer.
PARAGRAPH FOR Q-11 & 12

A solid sphere having charge density  and radius R is shown in figure. A spherical cavity of radius R/2 is made in
it. The distance between centre of sphere and spherical cavity is R/4.

11. The potential at O is:

11R 2 35R 2 13R 2 23R 2

(A) (B) (C) (D)
240 960 2480 480
12. The magnitude of electric field at O is:
R 2R 5R R
(A) (B) (C) (D)
12 0 150 180 210

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 SECTION-2
This section consists of 6 Numerical Value Type Questions. The answer to each question is a NUMERICAL VALUE. For
each question, enter the correct numerical value of the answer. If the answer is a decimal numerical value, then round-
off the value to TWO decimal places. If the answer is an Integer value, then do not add zero in the decimal places. In the
OMR, do not bubble the  sign for positive values. However, for negative values, Θ sign should be bubbled. (Example:
6, 81, 1.50, 3.25, 0.08)

3A
1. Three waves 1  A sin[ kx  t ],  2  sin(kx  t  ) and  3  2 A sin(kx  t   /2) are
2
superimposed. If resultant wave is given by   R.sin(kx  t   /4). Where R is some constant.
Find  (in degrees). [Take sin 48‫ۥ‬° = 1 / 3 ]

2. A Charge 2q0 is placed at a distance of d = 12 cm from the centre of annular disc of inner radius 9 cm and
q0
outer radius 16 cm. If flux of electric field through the disc area is . Find n?
n 0

3. A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a
speed of 2 ms 1 in front of the open end of the pipe and parallel to it, the length of the pipe should be
changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is 320 ms 1 ,
the smallest value of the percentage change required in the length of the pipe is n/2. Find the value of n.

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4. A non-conducting uniformly charged sphere is kept on a rough inclined plane. A uniform electric field is
present in the horizontal direction as shown in the figure. The angle of inclination is 45° and coefficient of
friction between the sphere and the inclined plane is 1/7. As the sphere is released, it is in the state of
impending motion and starts pure rolling. For this the ratio of gravitational force to electrostatic force on
the sphere is:

5. Three identical springs of force constant k have been joined to the three identical balls (each of mass m), as
shown in the figure which are in at the three vertices of an equilateral triangle. In the shown arrangement,
each of the spring is in its natural length. When all three balls are simultaneously given small displacements
of equal magnitude along the directions (in the plane of triangle) as shown in the figure, the oscillation
1 k
frequency for the balls will be then   _________.
2 2m

6. The speed of sound in hydrogen is 1270 m/s. The speed of sound (in m/s) in a mixture of oxygen and
1270
hydrogen with no. of moles in ratio 1 : 14 at same temperature is m / s Find n .
n
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SUBJECT II : CHEMISTRY 59 MARKS

SECTION-1 | Type A
This section consists of 5 Multiple Choice Questions. Each question has 4 choices (A), (B), (C) and (D), out of which
ONLY ONE CHOICE is correct.

1. What is the mass of N 2 in a 15 L gaseous mixture at 20°C and 740 mm pressure. If the composition of the
mixture by volume is: H 2  10%, O 2  20% and N 2  70%.
(A) 11.90 g (B) 16.2 g (C) 21.91 g (D) 28.00 g
2. C60 (buckyball) is cubic closest packed (face-centred cubic) in its crystalline form. If you insert sodium
atoms into all the tetrahedral and octahedral holes of the C60 structure, the formula would become
Na x C60 . What is the value of x?
(A) 1 (B) 2 (C) 3 (D) 4
3. Lithium Borohydride crystalises in an orthorhombic system with four molecules per unit cell. The unit cell
dimensions are a = 6.8 Å, b = 4.4 Å and C = 7.2 Å. If the molar mass is 21.76 g/mol, then the density of
crystals is:
(A) 0.67 g/cc (B) 1.67 g/cc (C) 2.67 g/cc (D) 14.5 g/cc
4. Dry air is passed through a solution containing 10 g of the solute in 90 g of water and then through pure
water. The loss in weight of the solution is 2.5 g and that of pure solvent is 0.05 g. Calculate the molecular
weight of the solute.
(A) 50 (B) 180 (C) 100 (D) 25
5. When a liquid that is immiscible with water was steam distilled at 95.2°C at a total pressure of 748 torr, the
distillate contained 1.25 g of the liquid per gram of water. The vapour pressure of water is 648 torr at
95.2°C, what is the molar mass of liquid?
(A) 7.975 g/mol (B) 166 g/mol (C) 145.8 g/mol (D) 14.58 g/mol

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 SECTION-1 | Type B
This section consists of 5 Multiple Correct Answers Type Questions. Each question has 4 choices (A), (B), (C) and
(D), out of which ONE OR MORE THAN ONE CHOICE is correct.
6. In the depression in freezing point experiment which of the following is/are correct for urea solution in
aqueous medium.
(A) The vapour pressure of solution is less than that of pure solvent
(B) Only solvent molecules solidify at the freezing point
(C) The freezing temperature of solution depends upon concentration of solution
(D) The freezing temperature of solution also depends on external pressure

7. A cubic unit cell contains manganese ions at the corners and fluoride ions at the centre of each edge. Which
of the following is/are correct?
(A) The empirical formula of the compound is MnF3
(B) The coordination number of the manganese ion is 6
(C) The coordination number of the fluoride ion is 2
(D) The edge length of the unit cell is 4.02 Å, if the radius of manganese ion is 0.65Å and fluoride ion
is 1.36Å
8. For a sample containing O 2 and SO 2 mark out the correct statement(s):

(A) At same temperature average speed of O 2 is greater than that of SO 2

(B) If absolute temperature of O 2 is half to that of SO 2 gas than the speed distribution curves will be
different
(C) At same temperature and at lower speed, fraction of molecules of SO 2 will be greater than that of
O2
(D) At same temperature and at higher speed, fraction of molecules of O 2 will be greater than that of
SO 2

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9. Benzene and naphthalene form an ideal solution at room temperature. For this process, the true statement(s)
is(are).
(A) G is negative (B) Stotal is positive
(C) H   ve (D) Ssurrounding  0
10. Equal weights of diborane and ethylene are kept in two vessels of equal volume maintained at
T K and 2 T K temperature respectively. Identify the correct statements.
(A) Density of diborane is equal to that of ethylene
(B) Kinetic energy of diborane is greater than that of ethylene
(C) Rate of effusion of diborane is less than that of ethylene
(D) Root mean square velocity of diborane is less than that of ethylene

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 SECTION-1 | Type C

This section consists of ONE (01) paragraph. Based on each paragraph, there are TWO (02) questions. Each question
has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct answer.
PARAGRAPH FOR Q-11 & 12
According to V.B.T. atoms of element form bond only to pair up their unpaired electrons present in ground state or
excited state. This pairing of unpaired electron will take place by overlapping of orbitals each one having one
unpaired electron with opposite spin.
11. Which of the following orbital combinations does not form - bond?
(A) p x  p x sideways overlapping (B) d  p y sideways overlapping
z2

(C) d xy  d xy sideways overlapping (D) d yz  p y sideways overlapping

12. Which of the following can form -bond ? (z-axis is internuclear axis).

(A) p x orbital (B) p y orbital

(C) d orbital (D) d zx orbital

z2

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 SECTION-2
This section consists of 6 Numerical Value Type Questions. The answer to each question is a NUMERICAL VALUE. For
each question, enter the correct numerical value of the answer. If the answer is a decimal numerical value, then round-
off the value to TWO decimal places. If the answer is an Integer value, then do not add zero in the decimal places. In the
OMR, do not bubble the  sign for positive values. However, for negative values, Θ sign should be bubbled. (Example:
6, 81, 1.50, 3.25, 0.08)
1. Find the number of correct statements about azeotropic mixture.
(i) When a sizeable deviation from ideality occurs, the boiling point curve for a binary mixture may
exhibit a maximum or minimum.
(ii) Mixtures having maximum or minimum boiling point are known as azeotropic mixture.
(iii) These mixtures distil without change in composition.
(iv) It follows that, at a given pressure, the maximum separation of a real solution that can be achieved
by fractional distillation is to azeotropic mixture and not for any one of the pure component.
(v) The azeotropic composition may vary significantly with applied pressure.

2. The freezing point of an aqueous solution of KCN containing 0.2 mole/kg water was –0.80°C. On adding
0.1 mole of Hg(CN) 2 in the solution containing 1 kg of water, the freezing point of the solution
was –0.6°C. Assuming that the complex is formed according to the following equation
Hg(CN) 2  mCN   Hg(CN) m
m
 2 and Hg(CN) 2 is the limiting reactant, the value of m is_________.

3. The formula of a spinel is MgAl2O4 . How many effective number of anionic vacancy for unit cell will be
8/3
created if the oxide ions of the spinel are replaced by X ions?

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4. Consider that AgX crystallizes in rock salt structure. The density of AgX is 6477 kg/m3 and length of unit
cell is 577 pm. Atomic weight of Ag is 108 g/mol. The atomic weight of X (in g/mol, rounded off two
23
decimal places) is: [Take N A  6  10 ]

5. How many of the following species are linear with zero dipole-moment.
XeF2 , CO 2 , I3 , N 2O, NO 2 , SCN  , OCN  , N3

6. Silver has a cubic unit cell with edge length of 408 pm. If density is 10.6 g/cm 3 , how many atoms of
silver are there in one unit cell?

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SUBJECT - III : MATHEMATICS 59 MARKS

SECTION-1 | Type A
This section consists of 5 Multiple Choice Questions. Each question has 4 choices (A), (B), (C) and (D), out of which
ONLY ONE CHOICE is correct.
1. Let X be the set of all positive integers greater than or equal to 8 and let f : X  X be a function such
that f ( x  y )  f ( xy ) for all x  4, y  4, if f (8)  9. Then f (9) is equal to:
(A) 19 (B) 9 (C) 13 (D) 8

3
2. The solution set of log|sin x| ( x 2  8 x  23)  is:
log 2 | sin x |
 3   3 
(A) x  (3, )   ,    , 5  (B) x  (3, )  (, 5)
 2   2 
 5   3   3   5 
(C) x   3,  (D) x  (3, )   ,    , 5    , 
 2   2   2   2 
3. A function is defined for all real numbers and satisfies f (2  x)  f (2  x) and f (7  x)  f (7  x) for
all real x. If f (0)  0, what is the least number of roots f ( x )  0 must have in the interval
1000  x  1000.
(A) 402 (B) 401 (C) 200 (D) 201

4. The sum of all integral values of a values of a where a  [10, 10] such that the graph of the function
f ( x )  x  2  a  3 has exactly three solutions, is:
(A) 5 (B) 13 (C) 3 (D) 0

5. If f ( x )  2 x3  3x 2  1 then number of distinct real solution(s) of the equation f ( f ( x ))  0 is:

(A) 0 (B) 2 (C) 6 (D) 3

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 SECTION-1 | Type B
This section consists of 5 Multiple Correct Answers Type Questions. Each question has 4 choices (A), (B), (C) and
(D), out of which ONE OR MORE THAN ONE CHOICE is correct.
6. Which of the following statement(s) is/are correct?
100
(A) The coefficient of x 2 in the expansion of  100 Cr ( x  3)100r  4r is equal to 5050
r 0
100
(B) The coefficient of x 2 in the expansion of  100Cr ( x  3)100r  4r is equal to 4950
r 0
n n 1
Ck
(C) If Ak  n
Ck  nCk 1
and 3  Ak  4, then n is equal to 64
k 0

n n 1
Ck
(D) If Ak  n
Ck  nCk 1
and 3  Ak  4, then n is equal to 128
k 0

1  
7. Let f ( x)  sgn(cot x)  tan  [ x]  , where [x] is the greatest integer function less than or equal to x.
2 
Then which of the following alternatives is/are true?
(A) f ( x ) is many one but not even function
(B) f ( x ) is periodic function
(C) f ( x ) is bounded function
(D) Graph of f ( x ) remains above x-axis

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8. The equation sin( a 2 x )  cos( ax) ; x  [0, 2] has:
(A) 18 solutions if a = 3 (B) 8 solutions if a = 2
(C) 2 solutions if a = 1 (D) 4 solutions if a = –1

9. Let a, b and c are positive real numbers forming an A.P. The equation ax 2  2bx  5c  0 will have real
roots for:
a c a c
(A)  2 5 (B)  2 5
c a c a

a c a c
(C)  4 (D)  4
c a c a

10. The set S of all real x for which ( x 2  x  1) x 1  1 contains.

(A) (–5, –1) (B) (–1, 1) (C) (–1, 0) (D) (–3, 1)

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 SECTION-1 | Type C

This section consists of ONE (01) paragraph. Based on each paragraph, there are TWO (02) questions. Each question
has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct answer.
PARAGRAPH FOR Q-11 & 12
If x, y, z  R , x  y  z  4, x 2  y 2  z 2  6 . If the range of z is [p, q], then answer the following questions.

q 2 x ( x 2  1)
11. If f ( x )  , x  0 , then range of f ( x ) is:
x 2  ( x 2  1)2
 1  8  5  12 
(A)  0, 5  (B)  0, 5  (C)  0, 8  (D)  0, 5 

12. Number of integral values of a for which exactly one root of the equation x2 + (a + 1) x – 4a = 0 lies in (p,
q) is:
(A) 0 (B) 2 (C) 3 (D) 4

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 SECTION-2
This section consists of 6 Numerical Value Type Questions. The answer to each question is a NUMERICAL VALUE. For
each question, enter the correct numerical value of the answer. If the answer is a decimal numerical value, then round-
off the value to TWO decimal places. If the answer is an Integer value, then do not add zero in the decimal places. In the
OMR, do not bubble the  sign for positive values. However, for negative values, Θ sign should be bubbled. (Example:
6, 81, 1.50, 3.25, 0.08)

1. If ( x – y ) f ( x  y ) – ( x  y ) f ( x  y )  4 xy ( x 2 – y 2 ), for all x, y  R and f (1)  –2, then

f (5)  f (4)  f (0)  f (2)  f (3)  f (4)  f (5) –11 is equal to _____________.

 1 
2. The number of common points of the curves y  sin 1 (2 x )  tan 1   2  and
 [ 2 x] 
y  cos 1 (2 x  5)  1, where [  ] denotes the greatest integer function, is ____________.

x 3  1 
3. If the function f ( x)  sin( x  1)     x  4 x and g ( x)  f –1 ( x ), then the value of   equals
2  g (5) 
(where [  ] represents the greatest integer function).

2 4 8
4. The sum of sin 2  sin 2  sin 2 is S, then the value of 4S is ____________.
7 7 7
10
5. Let 1   (3r  10Cr  r  10Cr )  210 (  45  ) where ,   N and f ( x)  x 2  2 x  k 2  1.
r 1
If ,  lie between the roots of f ( x )  0, then find the smallest positive integral value of k.

 e x  e x 
6. If f ( x )  ax17  b sin x  sin 2 x  sin 3 x  cx 2 sgn(sin x )  d log( x  1  x 2 )  x(| x  1|  | x  1|)  x  x  be
e e
 
defined on the set of real numbers, (a  0, b, c, d  R ) if f (7)  7, f (5)  5, f (2)  3, then the
minimum number of zeroes of the equation f ( x )  0 is equal to____________.

SPACE FOR ROUGH WORK