Test Pattern

(1001CJA102922060) )1001CJA102922060)
English

CLASSROOM CONTACT PROGRAMME JEE(Advanced)

PART TEST
(Academic Session : 2022 - 2023) 06-12-2022
JEE(Main+Advanced) : ENTHUSIAST COURSE [PHASE : TEAS, TOAS, TAAS, TNAS, I(A) & LIVE-I(A)] (SCORE-I)

Time : 3 Hours PAPER-2 Maximum Marks : 180

READ THE INSTRUCTIONS CAREFULLY

GENERAL :

1. This sealed booklet is your Question Paper. Do not break the seal till you are told
to do so.

2. Use the Optical Response sheet (ORS) provided separately for answering the questions.

3. Blank spaces are provided within this booklet for rough work.

DO NOT BREAK THE SEALS WITHOUT BEING INSTRUCTED TO DO SO BY THE INVIGILATOR

4. Write your name, form number and sign in the space provided on the back cover of this
booklet.

5. After breaking the seal of the booklet, verify that the booklet contains 28 pages and
that all the 18 questions in each subject and along with the options are legible. If not,
contact the invigilator for replacement of the booklet.

6. You are allowed to take away the Question Paper at the end of the examination.

OPTICAL RESPONSE SHEET :

7. The ORS will be collected by the invigilator at the end of the examination.

8. Do not tamper with or mutilate the ORS. Do not use the ORS for rough work.

9. Write your name, form number and sign with pen in the space provided for this purpose
on the ORS. Do not write any of these details anywhere else on the ORS. Darken
the appropriate bubble under each digit of your form number.

DARKENING THE BUBBLES ON THE ORS :

10. Use a BLACK BALL POINT PEN to darken the bubbles on the ORS.

11. Darken the bubble COMPLETELY.

12. The correct way of darkening a bubble is as :

13. The ORS is machine-gradable. Ensure that the bubbles are darkened in the correct
way.

14. Darken the bubbles ONLY IF you are sure of the answer. There is NO WAY to erase
or "un-darken" a darkened bubble.

15. Take g = 10 m/s2 unless otherwise stated.

QUESTION PAPER FORMAT :
16. The question paper has three parts : Physics, Chemistry and Mathematics.

Please see the last page of this booklet for rest of the instructions
 Target : JEE (Main + Advanced) 2023/06-12-2022/Paper-2
ALLEN
SOME USEFUL CONSTANTS
Atomic No. : H = 1, B = 5, C = 6, N = 7, O = 8, F = 9, Al = 13, P = 15, S = 16,
Cl = 17, Br = 35, Xe = 54, Ce = 58
Atomic masses : H = 1, Li = 7, B = 11, C = 12, N = 14, O = 16, F = 19, Na = 23, Mg = 24,
Al = 27, P = 31, S = 32, Cl = 35.5, Ca=40, Fe = 56, Br = 80, I = 127,
Xe = 131, Ba=137, Ce = 140

· Boltzmann constant k = 1.38 × 10–23 JK–1

1
· Coulomb's law constant = 9 ×10 9
4 pe0
· Universal gravitational constant G = 6.67259 × 10–11 N–m 2 kg–2
· Speed of light in vacuum c = 3 × 108 ms–1
· Stefan–Boltzmann constant s = 5.67 × 10–8 Wm–2 –K–4
· Wien's displacement law constant b = 2.89 × 10–3 m–K
· Permeability of vacuum µ0 = 4p × 10–7 NA–2
1
· Permittivity of vacuum Î0 =
m0 c2
· Planck constant h = 6.63 × 10–34 J–s

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BEWARE OF NEGATIVE MARKING

PART-1 : PHYSICS
SECTION-I (i) : (Maximum Marks: 12)
This section contains FOUR (04) questions.
Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four 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
1. In the circuit shown in the figure, the value of VB + VD :-

(A) – 3V (B) 7V
(C) – 7V (D) 3V
2. Choose correct statement(s) :
(1) The number of significant figures in 0.31 × 103 m is two.
(2) The length and breadth of a rectangular sheet are 4.234 m and 1.00 m then the perimeter of rectangle
with correct number of significant figures is 10.46 m.
(3) When 97.52 m is divided by 2.54 s, the result with correct number of significant figures is 38.393.
(4) When 0.2 J is subtracted from 7.26 J, then result with correct number of significant figure is 7.1 J.
(A) Only 1 and 2 are correct. (B) Only 1, 2 and 3 are correct.
(C) Only 2 and 3 are correct. (D) Only 1 and 4 are correct.
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3. In the circuit shown, the voltmeter reads 30 V when connected across the 400 Ω resistance. The same
voltmeter when connected across 300 Ω will measure :

(A) 20 V (B) 22.5 V (C) 25 V (D) 40 V

4. The diagram shows two galvanometers G1 and G2. When current I = 1A both G1 and G2 shows full scale
deflection. It is given that G1 shows full scale deflection for 5 mA and G2 shows full scale deflection for 2 mA.
The values of r1 and r2 are (G1 and G2 are of negligible resistances)

2R R
(A) r1 = and r 2 =
9 995
2R 7R
(B) r1 = and r 2 =
3 995
3R 7R
(C) r1 = and r 2 =
2 995
4R 4R
(D) r1 = and r 2 =
9 90

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SECTION-I (ii) : (Maximum Marks: 24)
This section contains SIX (06) questions.
Each question has FOUR options. 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 and
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 none of the options is chosen (i.e. the question is unanswered).
Negative Marks : –2 In all other cases.
For Example : If first, third and fourth are the ONLY three correct options for a question with second
option being an incorrect option; selecting only all the three correct options will result in +4 marks.
Selecting only two of the three correct options (e.g. the first and fourth options), without selecting any
incorrect option (second option in this case), will result in +2 marks. Selecting only one of the three
correct options (either first or third or fourth option), without selecting any incorrect option (second
option in this case), will result in +1 marks. Selecting any incorrect option(s) (second option in this
case), with or without selection of any correct option(s) will result in –2 marks.

5. A block of mass ‘m’ is attached by means of a spring to the bottom of a tank of water as shown in figure.
At equilibrium, the spring is under compression. If the tank is now allowed to fall freely, then choose the
correct alternative(s) :-

(A) the block comes to rest when the spring is at its natural length.
(B) the spring compresses more than its equilibrium compression.
(C) the buoyant force becomes zero.
(D) there will be some elongation is the spring.
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6. A piston is pressed with a force F on a hydraulic cylinder containing water (ρ = 103 kg/m3). The cross-sectional
area of the cylinder is A = 100 cm2. The water is forced into pipe with a cross-sectional area of 1 cm2 that rises
a height h = 50 cm (Atmospheric pressure P0 = 105 N/m2)

(A) Force F required to make water eject with a speed of 10 m/s is 550 N.
(B) Force F required to make water eject with a speed of 10 m/s at the top is 5500 N.
(C) The pressure inside the cylinder at the bottom when water ejects at 10 m/s is 1.55 × 105 Pa.
(D) none of the above
7. An isolated parallel plate capacitor have unequal charges +4Q and –2Q on its plates. If capacitance of the
capacitor is C and P is a point outside of the capacitor. Then :-

(A) a point charge at point P will not experience any electric force due to capacitor plates.
3Q
(B) the potential difference between the plates will be .
C
9Q 2
(C) the energy stored in the electric field in the region between the plates is .
2C
2Q 2
(D) the force on one plate due to the other plate is .
Cd

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8. A cylindrical vessel C1 of area of cross-section A and height 1 m is filled fully up to a height 0.4 m by
water (density 1000 kg-m−3) and remaining by liquid of density 800 kg-m−3. The cylinder has small
orifice at the bottom, which is closed initially. The cylinder C1 is then immersed in another bigger
cylindrical vessel C2 of area of crosssection 6A, containing liquid of density 800 kg-m−3 to the height of
0.5 m. If the orifice of C1 is opened, then,
(A) The speed with which the water comes out from the orifice (in ms−1) will be 4
(B) After certain interval of time, water from C1 stops flowing out, thereby establishing the steady
state of the system. The length of water column in cylinder C1 is 6.67 cm
(C) After certain interval of time, water from C1 stops flowing out, thereby establishing the steady
state of the system. The length of water column in cylinder C1 is 9.5 cm
(D) In steady state, the pressure exerted on the bottom of cylinder C2 will be about 1.09 × 105 Nm−2
[Atmospheric pressure, p0 = 1 × 105 Nm−2 ]
9. In the circuit given below, if R1 = 240 Ω and R2 = 120 Ω respectively, L is a lamp of 300 W then

(A) Power dissipated in R1 is 60 W (B) Power dissipated in R2 is 120 W

(C) Effective resistance of the circuit is 50 Ω (D) The ammeter A reads 6 A
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10. A battery of emf E and internal resistance r is connected across a resistance R. Resistance R can be
adjusted to any value greater than or equal to zero. A graph is plotted between the current passing through
the resistance (i) and potential difference across the terminals of the battery (V). Select the correct
alternative (s).

(A) Internal resistance of the battery is 4 Ω.

(B) emf of the battery is 16 V.
(C) Maximum current which can be taken from the battery is 2A.
(D) Maximum thermal power dissipated across R is 16 J.

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SECTION-III : (Maximum Marks: 24)
This section contains EIGHT (08) questions.
The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive.
For each question, enter the correct integer value of the answer in the place designated to enter the
answer.
For each question, marks will be awarded in one of the following categories :
Full Marks : +3 If only the correct answer is given.
Zero Marks : 0 If no answer is given.
Negative Marks : –1 In all other cases
1. Find the attraction force (in Newton) between two parallel glass plates separated by a distance h = 0.10 mm,
after a water drop of mass m = 70 g was introduced between them. The wetting is assumed to be complete.
1
(g = 10 m/s2, ρ = 103 kg/m3, T = × 10–3 N/m)
14
2. The reading of the ideal ammeter connected in the given circuit is 0.39 × N. The Value of N is :-
(Assume that the cells have negligible internal resistance)

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3. The figure shows two identical parallel plate capacitors connected to a battery with switch S closed. The
switch is now opened and the free space between the plates of the capacitors is filled with a dielectric of
dielectric constant (or relative permittivity) 3. The ratio of the total electrostatic energy stored in both
capacitors before and after the introduction of the dielectric is 3/N. Then find the value of N.

4. A resistor of resistance R and a capacitor of capacitance C is connected in series with the voltage source.
The EMF of the source changes periodically as shown in the figure. The maximum charge on the
capacitor is e × CE . The value of N is – (τ is time constant)
e+N

5. The space between the plates of a parallel plate capacitor is completely filled with a material of resistivity
2 × 1011 Ω–m and dielectric constant 6. Capacity of the capacitor with the given dielectric medium
between the plates is 20 nF. Find the leakage current (in µA) if a potential difference 2500 V is applied
across the capacitor. (Give your answer in nearest integer.)

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6. A tube of conical bore of length 10cm is just dipped inside the water. The diameters of upper and
lower ends are 0.04 cm and 0.06 cm respectively. Find the height (in cm) to which the liquid rises in
the tube. Given surface tension of water is 70 dyne cm–1 and angle of contact 0°. (Take : g = 9.8 m/s2)
(Give your answer in nearest integer.)
7. In the circuit shown in Figure,

current in the 3 Ω resistor is

8. A open organ pipe containing air resonates in fundamental mode due to a tuning fork. The measurement
value of length ℓ (in cm) of the pipe are ℓ = 94.0 ± 0.1, r = 5.0 ± 0.05. The velocity of the sound in air is
accurately known. The maximum percentage error in the measurement of the frequency of that tuning fork
by thus experiment, will be α, then find the value of 25α.
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PART-2 : CHEMISTRY
SECTION-I (i) : (Maximum Marks: 12)
This section contains FOUR (04) questions.
Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four 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
1. Consider following four reactions and given plot showing variation of equilibrium constant (Keq) with
temperature :
A + B ⇌ C ΔH1°
D + E ⇌ F ΔH2°
G⇌H ΔH3°
I⇌J ΔH4°

The correct graph showing the variation of % yield of product, C, F, H and J with temperature is
(ΔH° is independent of temperature, lnK is the natural log of equilibrium constant)

(A) (B)

(C) (D)

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2. Consider the four different cells having equal cell potential is given conditions :
Ag | Ag+ (Ag2CrO4, saturated sol.) || Ag+ (0.1M) | Ag, Ecell = xV and K sp Ag2 CrO4 = 'p'
,

Ag | Ag+(AgCl saturated sol.) || Ag+(0.1M) | Ag, Ecell = xV and Ksp,AgCl = 'q'

+ +
Ag | Ag (Ag3PO4 saturated sol.) || Ag (0.1M) | Ag, Ecell = xV and K sp Ag3 P O4 = 'r'
,

+ +
Pt | H2(g), 1 bar | H (1M) || (Ag2SO4 saturated sol.) Ag | Ag, Ecell = xV and K sp Ag2 SO4 = 's'
,

The correct option for solubility product for all the four cases is (by considering all cells are in
equilibrium stage)
(A) s > q > p > r (B) s < p < q < r (C) r > s > q > p (D) r > q > p > s
3. 1 g of charcoal adsorbs 100 ml of 0.5 M CH3COOH to form a monolayer, and thereby the molarity of
CH3COOH reduces to 0.49. Find the surface area of the charcoal adsorbed by each molecule of acetic
acid. Surface area of charcoal = 3.01 × 102m2/g.
(A) 5 × 10–19 m2 (B) 9 × 10–19 m2 (C) 20 × 10–19 m2 (D) 25 × 10–19 m2
4. 10 g of a weak monocarboxylic acid in 100 g of benzene freezes at 3.35°C. Assume that acid undergoes
complete association in benzene. Find the degree of dissociation of 10 g of acid in 100 g of water whose
freezing point is found to be –1.75°C. Given: Kf (H2O) = 1.86 K-Kg/mole, Kf(benzene) = 5.12 K-Kg/mole
and freezing point of benzene = 5.51°C.
(A) 0.05 (B) 0.115 (C) 0.621 (D) 0.825

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SECTION-I (ii) : (Maximum Marks: 24)
This section contains SIX (06) questions.
Each question has FOUR options. 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 and
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 none of the options is chosen (i.e. the question is unanswered).
Negative Marks : –2 In all other cases.
For Example : If first, third and fourth are the ONLY three correct options for a question with second
option being an incorrect option; selecting only all the three correct options will result in +4 marks.
Selecting only two of the three correct options (e.g. the first and fourth options), without selecting any
incorrect option (second option in this case), will result in +2 marks. Selecting only one of the three
correct options (either first or third or fourth option), without selecting any incorrect option (second
option in this case), will result in +1 marks. Selecting any incorrect option(s) (second option in this
case), with or without selection of any correct option(s) will result in –2 marks.

5. The chromate ion, CrO 22 oxidizes the stannite ion, HSnO 2 to the stannate ion
− −
HSnO 3
−
and is itself
reduced to chromite ion, CrO 2 in basic medium as per the following reaction:
−

CrO 42 −
+ HSnO 2
−
+ H 2 O → HSnO 3−
+ CrO 2
−
+ OH −

If 400 ml of M/5 chromate ion react with 500 ml of M/4 stannite ion then:
(A) Chromate ion. CrO 42 is the limiting reagent
−

(B) Stannite ion HSnO 2 is the limiting reagent

−

(C) At the end of reaction, the concentration of CrO 2 ions ≈ 0.09M

−

(D) At the end of reaction, the concentration of HSnO 3 ions ≈ 0.13M

−

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6. One mole of an ideal monoatomic gas initially at a temperature T1, pressure P1 and volume V1 in a
thermally insulated cylinder closed at one end and fitted with a movable piston at the other end is
suddenly compressed placing a weight on the piston. The gas pressure is raised to 2P1. Which of the
following is correct?
2 −3 RT1
(A) Rise in temperature = ΔT = T1 (B) Change in volume = ΔV = 10 P 1
5
7
(C) Final temperature = T1 (D) P 1 V11 2P 1 ) V21
= (
5
7. A sample of H2O2 solution labelled as 34.05 volume has density of 1.128 gm/ml. Mark the correct option(s)
representing concentration of same solution in other units. [Solution contains only H2O and H2O2]
(A) Mole fraction of H2O2 in the solution = 0.5
(B) %w/v = 10.2%
(C) M H 2 O2 = 6M

(D) m H 2 O2 = 2.924 m

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8. For phase transition involving 3 moles of benzene -

C6H6(ℓ, 77°C, 1 atm) → C6H6(g, 77°C, 0.5 atm) ; ΔHvap = 1.4 kJ/mole
If 77°C is normal boiling point of benzene then correct value(s) of change in thermodynamic parameters
for process is : [Use : R = 8 J/K-mole , ln 2 = 0.7]
(A) ΔS = 20.8 J/K (B) ΔG = –5.88 kJ (C) ΔU < 0 (D) ΔU > 0
9. Which of following statement(s) is/are correct for Zeolites :
(A) They are hydrated alumino-silicates which can be used for shape selective catalysis
(B) They have general formula Mx/n[(AlO2)x(SiO2)y].mH2O where, n is the charge on the metal
cation Mn+
(C) They have pore sizes between 260 pm to 740 pm
(D) ZSM-5 which is used in petroleum industry as a catalyst, converts hydrocarbons into alcohols
is an example of Zeolites.
10. Choose the correct statement(s) among the following -
(A) Internal energy of H2O remains constant during conversion of liquid into its vapour at constant
temperature
(B) During fusion of ice into water, enthalpy change and internal energy change are almost same at
constant temperature
(C) Molar heat capacity of diatomic gases are temperature dependent
(D) During boiling of H2O at 1 atm, 100°C, average kinetic energy of H2O molecules increases
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SECTION-III : (Maximum Marks: 24)
This section contains EIGHT (08) questions.
The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive.
For each question, enter the correct integer value of the answer in the place designated to enter the
answer.
For each question, marks will be awarded in one of the following categories :
Full Marks : +3 If only the correct answer is given.
Zero Marks : 0 If no answer is given.
Negative Marks : –1 In all other cases
1
1. The variation of vs C λm for RCOOH at 298 K has been shown in the graph. Find pKa of RCOOH.
λm

2. A 150 ml solution of I2 is divided into two unequal parts. First part reacts with hypo solution in acidic
medium. 15 ml of 0.4 M hypo was consumed. Second part was treated with 100 ml of 0.3M NaOH
solution yielding NaI and NaIO3. Residual base required 10 ml of 0.3M H2SO4 solution for complete
neutralization. Calculate the value of "10 times" the initial concentration of I2?
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3. An aromatic compound contains 69.4% carbon and 5.8% hydrogen. A sample of 0.303 g of this
compound was analyzed for nitrogen by kjeldahl's method. The ammonia evolved was absorbed in 50 ml
of 0.05M H2SO4. The excess of the acid required 25 ml of 0.1 M NaOH for neutralization. Its molecular
w+x−y−z
mass is 121. Determine the molecular formula and express the answer in the form of if the
2
molecular formula of the compound is CwHxNyOz.
4. 0.00012% MgSO4 and 0.000111% CaCl2 is present in water. What is the measured hardness of water in
ppm of CaCO3. [Atomic mass : Mg = 24, S = 32, Ca = 40, Cl = 35.5]
5. When Aluminium pieces are added to aqueous solution of Tetrathionic acid (H2S4O6) a pungent gas is
produced. Find number of moles of Tetrathionic acid consumed per mole of Aluminium in this reaction ?
6. Calculate the quantity (in gram) of CO2 in 550 mL of soda water when packed under 2.5 Bars
CO2 pressure at 298 K (KH for CO2 = 1.67 × 108 Pa at 298 K)
7. A 10 ml sample of hydroxylamine solution was diluted to one litre. 50 mL of this diluted solution was
boiled with excess of Fe (III) solution and the resulting solution required 10 mL of 0.02M
KMnO4 solution for complete oxidation of Fe(II). Calculate the mass (in g) hydroxylamine in 91 mL of
the original hydroxylamine solution.
Balanced reaction is, 2NH2OH + 4Fe3+ → N2O(g) + H2O + 4Fe2+ + 4H+
8. In the following reactions:
Δ
CaOCl2 −→ a Ca(ClO3)2 + b CaCl2
NaCl + H2SO4 → Na2SO4 + c HCl
Na2S2O3 + d HCl → e NaCl + H2O + S ↓ + SO2
What will be the value of (a + b + c) – (d + e) in the balanced chemical reactions?

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PART-3 : MATHEMATICS
SECTION-I (i) : (Maximum Marks: 12)
This section contains FOUR (04) questions.
Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four 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
1. The value(s) of the real parameter ‘a’ for which the inequality x6 – 6x5 + 12x4 + ax3 + 12x2 – 6x + 1 ≥ 0 is
satisfied for all real x
(A) [–12, 37] (B) [–12, 38] (C) [–17, 14] (D) [–17, 35]
π π
2. Let f(x) = sin3 x + λ sin2 x for − < x < and λ > 0. The interval in which λ should lie in order that
2 2
f(x) has exactly one minimum and one maximum is
3 3
(A) 0 < λ < (B) 0 < λ < 1 (C) 1 < λ < 2 (D) 1 < λ <
2 2
−−−−−− −−−−−−−
x 7
√( + ) − 3√(2x − 3)
3. The value of lim is equal to
x→2 3 −
x
−−−−
6
−
√( + ) − 2√
3 −−−
(3x − 5)
−−−−

34 34 23
(A) (B) − (C) (D) 0
23 23 34

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⎡
cos2n x if x<0
⎢
⎢ n −−−−−
n
4. The function defined as f (x) = nlim ⎢ √1 + x if 0⩽x⩽1 which of the following does hold good.
→∞⎢
⎢
1
⎣
1+x n
if x>1

(A) Continuous at x = 0, but discontinuous at x = 1

(B) Continuous at x = 1, but discontinuous at x = 0
(C) Continuous at both x = 1 and x = 0
(D) Discontinuous at both x = 1 and x = 0
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SECTION-I (ii) : (Maximum Marks: 24)
This section contains SIX (06) questions.
Each question has FOUR options. 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 and
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 none of the options is chosen (i.e. the question is unanswered).
Negative Marks : –2 In all other cases.
For Example : If first, third and fourth are the ONLY three correct options for a question with second
option being an incorrect option; selecting only all the three correct options will result in +4 marks.
Selecting only two of the three correct options (e.g. the first and fourth options), without selecting any
incorrect option (second option in this case), will result in +2 marks. Selecting only one of the three
correct options (either first or third or fourth option), without selecting any incorrect option (second
option in this case), will result in +1 marks. Selecting any incorrect option(s) (second option in this
case), with or without selection of any correct option(s) will result in –2 marks.

5. A function f(x) is defined as

Tangents are made on f(x) at (x,y) where x > 0 & y > 0. Let tangents T1 : y = m1x + b1 and T2 : y = m2x + b2
have highest and lowest y intercepts of all tangents made in first quadrant then
(A) b1 + b2 = 36
(B) b1 + b2 = 28
(C) m1 + m2 = –4
(D) Area made by y axis, T1 = 0 and T2 = 0 is 54 square units

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6. Suppose that f(x) is defined and positive for all real numbers x. If f(x) increases for x < 0 and decreases
1
for all x > 0, and g(x) = , then
f ( 1x )

(A) g increases for all real x (B) g increases for x < 0

(C) g decreases for x > 0 (D) g increases for x > 0
7. Let f(x) be double differentiable function such that |f′′(x)| ≤ 5 ∀ x ∈ [0, 4] and f takes its largest value at an
interior point of this interval. Then the value of |f′(0)| + |f′(4)| can be
(A) 18 (B) 19 (C) 20 (D) 21
8. If F(x) = f(x) g(x) and f ′ (x) g′ (x) = c , then

f g
(A) F′ = c[ ′ + ] (B)
f g′
F ′′′ f ′′′ g′′′ F ′′′ f ′′′ g′′′
(C) = + (D) = +
F f g F ′′ f ′′ g′′

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9. Let f: [0, 1] → R be a double differentiable function with f(0) = f(1) = 0, and f′′(x) + 2f′(x) + f(x) ≥ 0 ∀ x ∈ (0, 1),
then
(A) f(x) ≤ 0 ∀ x ∈ [0, 1]
1 1
(B) If ex f(x) assumes its minimum value in [0, 1] at x = – , thenf(x) + f′(x) < 0 ∀ x ∈ 0
( ,
–)
√ 2 √ 2
(C) If g(x) = ex f(x), then number of real solution of the equation g(g(g(x))) = 0 is 2
(D) f(x) > 0 ∀ x ∈ [0, 1]
10. Which of the following are true for ∀ x ∈ (0, ∞)
x2 x
(A) ln(1 + x) > x − (B) ln(1 + x) >
2 x+1
(C) 4cos x + x > 0 (D) 2tan–1 x < x + 1

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SECTION-III : (Maximum Marks: 24)
This section contains EIGHT (08) questions.
The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive.
For each question, enter the correct integer value of the answer in the place designated to enter the
answer.
For each question, marks will be awarded in one of the following categories :
Full Marks : +3 If only the correct answer is given.
Zero Marks : 0 If no answer is given.
Negative Marks : –1 In all other cases
ln([1 + 9x 2 ]) − 9n x 2n cos(9x 2 )
1. For the function f (x) = nlim number of points of discontinuity in (–3, 3)
→∞ 1 + 9n x 2n
is: (where [.] represents greatest intger function)
2. Let C be the curve f(x) = (ln x)2 + 2 ln x and A(a, f(a)), B(b, f(b)) where (a < b) are the points of tangency
of two tangents drawn from origin to the curve C. The value of the product ab is equal to
3. The smallest integral value of P for which the function f(x) = 6Px – P sin 4x – 5x – sin 3x is
monotonically increasing and has no critical point on R is:
2x − 1 x2 + 1 f x 2+1
( ( ))
4. Let f(x) be a real valued function such that f (x) = ∀ x ∈ (2, ∞) and g(x) = +
x−2 x f (x)
∀ x > 2, then minimum value of g(x) is _____ .

5. Let f : R → R be a twice differentiable function such that lim f(x) = a (a ≠ 0), then lim x2f′′(x) is equal to
x→∞ x→∞

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6. A continuous and differentiable function y = f(x) is such that it’s graph cuts line y = mx + c at 11 distinct
points. Then the minimum number of points at which f′′(x) = 0 is
7. The set of all points where f(x) is increasing in (a, b) ∪ (c, ∞) then [a + b + c] is (where [.] denotes
x2 2
the greatest integer function). Given that f (x) = 2f ( ) + f(6 – x ) ∀ x ∈ R and f′′(x) > 0 ∀ x ∈ R
2
8. The equation k + x3ex+3 = 0 has 2 distinct real roots and k is a prime natural number, then the number of
possible value of k is
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ALLEN
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ALLEN

NAME OF THE CANDIDATE ................................................................................................

FORM NO. .............................................

I have read all the instructions I have verified the identity, name and Form
number of the candidate, and that question
and shall abide by them.
paper and ORS codes are the same.

____________________________ ____________________________
Signature of the Candidate Signature of the Invigilator

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