Question Report 117

(0999DJA161115230005) Test Pattern
)0999DJA161115230005)
DISTANCE LEARNING PROGRAMME JEE(Advanced)
TEST # 04
(Academic Session : 2023 - 2024) 10-12-2023
JEE(Main + Advanced) : LEADER TEST SERIES / JOINT PACKAGE COURSE (PHASE-II)
12th Undergoing/Pass Students
Test Type : Unit Test # 03

Time : 3 Hours Maximum Marks : 246
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.
4. Write your name, form number and sign in the space provided on the back cover of this booklet.
DO NOT BREAK THE SEALS WITHOUT BEING INSTRUCTED TO DO SO BY THE INVIGILATOR

5. After breaking the seal of the booklet, verify that the booklet contains 40 pages and that all the
20 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.
Note : No questions have been asked in the section II of this paper. Please leave section II
blank on the OMR answer sheet.
Target : JEE (Main + Advanced) 2024/10-12-2023
ALLEN®
SYLLABUS :
PHYSICS : Center of Mass, Momentum & Collison, Magnetic Effect Of Current, Rotational
Dynamics, Electromagnetic Induction ( EMI ), Elasticity, Thermal Expansion,
Calorimetry & Heat Transfer, Alternating Current.
CHEMISTRY : Electronic Displacement Effects, Acidic Strength & Basic Strength, Carbonyl
Compound, Oxidation, Reduction, Aldol & Similar Name Reactions, Ideal Gas, Real
Gas, Ionic Equilibrium, Electrochemistry, Atomic Structure, Surface Chemistry, Salt
Analysis, Heating Effect, Reactions of Salt Analysis
MATHEMATICS : Circle, Area Under The Curve, Differential Equation, Parabola, Ellipse, Hyperbola,
Matrices, Permutation & Combination, Binomial Theorem, Vectors
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
Space for Rough Work
Note : In case of any correction in the test paper, please mail to dlpcorrections@allen.in within 2 days along with Paper Code
& Your Form No.
LTS-2/40 0999DJA161115230005
Leader Test Series/Joint Package Course/Phase-II/10-12-2023
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. A block of mass m sliding on a horizontal frictionless surface has kinetic energy K. It collides head on
with another stationary identical block. The collision is perfectly inelastic. How much energy is lost in
this collision?
(A) K/4 (B) K/2

(C) K/3 (D) 3K/4
2. The diagram shows stress vs strain curve for materials A and B. From the curves we infer that
(A) A is brittle but B is ductile (B) A is ductile but B is brittle

(C) Both A and B are ductile (D) Both A and B are brittle
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3. A helium nucleus is moving in a circular path of radius 0.8m. If it takes 2 sec to complete one revolution.
Find out magnetic field produced at the centre of the circle.
(A) μ 0 × 10 –19 T (B) 10−19
T
μ0
2 × 10−19
(C) 2 × 10 –19 T (D) T
μ0
4. A conducting wire frame is placed in a magnetic field which is directed into the paper. The magnetic
field is increasing at a constant rate. The directions of induced currents in wires AB and CD are
(A) B to A and D to C (B) A to B and C to D

(C) A to B and D to C (D) B to A and C to D
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SECTION-I (ii) : (Maximum Marks: 16)
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 : –1 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 –1 marks.
5. Consider the physics of pushups: The drawing shows a simplified model of the force needed to do a pushup.
Assume that a person’s body is a bar of mass M with a moment of inertia about the pivot (the toes) of IT.
Assume that the center of mass is a distance L from the pivot and that the force shown (due to the arms) is
purely vertical and applied at a distance of 4L/3 from the pivot.
(A) The value of the applied force F needed for the object to be in equilibrium is 3
Mg
4
(B) The value of the applied force F needed for the object to be in equilibrium is Mg
(C) The value of applied force to keep the man in equilibrium is independent of the angle q and
thus independent of the height h.
(D) The value of the force exerted on the person by the ground at the toes, (not the force applied
Mg
on the arms) is so that man remain equilibrium.
4
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6. A thin copper rod rotates about an axis passing through its end and perpendicular to its length with a
speed ω 0. The temperature of the copper rod is increased by 100° C. If the coefficient of linear
expansion of copper is 2 × 10 – 5 / °C, the percentage change in the angular speed of the rod is
(A) – 2% (B) – 4% (C) – 0.2% (D) – 0.4%
7. A cube and a sphere (both solid) of same mass, same material are heated upto same temperature and then
kept in same surrounding :-
(A) Sphere will cool down at faster rate.
(B) Cube will cool down at faster rate.
1/3
3
(C) Ratio of initial rate of cooling of cube to sphere = 2[ ]
4π
1/3
1 3
(D) Ratio of initial rate of cooling of cube to sphere = [ ]
2 4π
8. A proton moving with a constant velocity passes through a region of space without any change in its
velocity. If E and B represent the electric and magnetic fields respectively, this region of space may have :
(A) E = 0, B = 0 (B) E = 0, B ≠ 0
(C) E ≠ 0, B = 0 (D) E ≠ 0, B ≠ 0
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SECTION-I (iii) : (Maximum Marks: 16)
This section contains TWO paragraphs.
Based on each paragraph, there are TWO questions
Each question has Four options (A), (B), (C) and (D) ONE OR MORE THAN ONE of these four
option(s) is(are) correct.
Partial Marks : +2 If three or more options are correct but ONLY two options are chosen, both
of which are correct options.
correct option.
Paragraph for Questions 9 and 10

A pendulum is formed by attaching a small body of mass, M = 0.2 kg to the end of a string 0.5 m long.
The other end of the string is attached to a nail so that the pendulum can swing freely in a circle in the
vertical plane. The pendulum is set in motion so that its speed at the bottom of the swing is 2 m/s.
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9. The maximum height of the mass M above its lowest position is CLOSEST to :
(A) 0.1 m
(B) 0.2 m
(C) 0.4 m
(D) 0.5 m
10. If the mass M of the body and the length of the string are each doubled, but the speed of the mass at the
bottom is still 2 m/s, which of the following quantities would change ?
(Assuming it is completing full circle)
(A) The maximum height of the mass above its lowest position
(B) Average torque about nail during one complete oscillation.
(C) The work done by the gravitational force on the mass during one complete oscillation.
(D) The tension in the string at the bottom of its swing
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A ring of mass M and radius R sliding with a velocity v0 suddenly enters into rough surface where the
coefficient of friction is μ , as shown in figure.
11. Choose the correct alternative(s) :-

3v20
(A) The linear distance moved by the centre of mass before the ring starts rolling is .
8μg
(B) The net work done by friction force is − 3 mv20 .

8
mv20
(C) The loss is kinetic energy of the ring is .
4
mv20
(D) The gain in rotational kinetic energy is + .
8
12. Choose the correct statement(s) :-
(A) As the ring enters on the rough surface, the limiting friction force acts on it.
(B) The direction of friction is opposite to the direction of motion.
(C) The friction force accelerates the ring in the clockwise sense about its centre of mass.
(D) As the ring enters on the rough surface it starts rolling.
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SECTION-I (iv) : (Maximum Marks: 6)
This section contains TWO (02) questions.
Each question has matching lists. The codes for the lists have choices (A), (B), (C) and (D) out of which
ONLY ONE is correct
For each question, marks will be awarded in one of the following categories :
13. Four rods of material X and three rods of material Y are connected as shown in figure. All the rods are of
identical lengths and cross – sectional area. Given thermal resistance of rod of material X is, Rx = R and
thermal conductivities of materials are related by relation KY = 2KX.
List – I List – II
(P) Thermal resistance between B and E (1) 500
13
(Q) Thermal resistance between A and F (2) 700
13
(R) Temperature of junction B (in °C) (3) 2R
3
(S) Temperature of junction D (in °C) (4) 13R
6
(A) P → 3;Q → 4;R → 1;S → 2 (B) P → 4;Q → 1;R → 2;S → 3
(C) P → 3;Q → 4;R → 2;S → 1 (D) P → 4;Q → 1;R → 3;S → 2
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14. A sample ‘A’ of liquid water and a sample B of ice of equal mass are kept in 2 nearby containers so that
they can exchange heat with each other but are thermally insulated from the surroundings. The graphs in
list-II show the sketch of temperature T of samples versus time t. Match with appropriate description in
list-I.
List-I List-II
Equilibrium temperature is
(P) (1)
above melting point of ice.
(Q) At least some of water freezes. (2)
(R) At least some of ice melts. (3)
Equilibrium temperature is
(S) (4)
below freezing point of water.
(A) P → 2;Q → 1,3;R → 3,4;S → 3 (B) P → 1;Q → 2;R → 3;S → 4

(C) P → 3;Q → 2;R → 4;S → 1 (D) P → 4;Q → 3;R → 2;S → 1
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SECTION-III : (Maximum Marks: 16)
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.
Full Marks : +4 If ONLY the correct answer is chosen.
Zero Marks : 0 In all other cases.
1. Figure shows a conducting loop in shape of a trapezium carrying a current i = 10A. Find the magnetic
field B (in µT) at a point P. It is given that a = 10 cm.
2. An electron is shot into one end of a solenoid. As it enters the uniform magnetic field within the
solenoid, its speed is 800 m/s and its velocity vector makes an angle of 30° with the central axis of the
solenoid. The solenoid carries 4.0 A current and has 8000 turns along its length. Find number of
revolutions made by the electron within the solenoid by the time it emerges from the solenoid’s opposite
end. (Use charge to mass ratio e for electron = 3 × 1011 C/kg). Fill your answer of y if total number
√
m
y
of revolution is × 106.
5
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3. In an AC circuit capacitive reactance of capacitor is XC = 50 2 Ω and inductive reactance of inductor is
√
XL = 50 2 Ω . If the source voltage is 100 volts. Then find the current passing through the source.
√
4. A uniform magnetic field of intensity B = 6 sin 2t directed into the plane of the paper exists in the
π
cylindrical region of radius 1 m. A loop of resistance 10 Ω folded in the form of equilateral triangle of
side length 2m is placed as shown in figure. Find the maximum potential drop in wire AB :
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SECTION-IV : (Maximum Marks: 16)
Based question contains two columns, Column-I and Column-II.
Column-I has four entries (A), (B), (C) and (D)
Column-II has five entries (P), (Q), (R), (S) and (T)
Match the entries in Column-I with the entries in Column-II.
One or more entries in Column-I may match with one or more entries in Column-II.
For each entry in Column-I, darken the options of all the matching entries. For example, if entry (A) in
Column-I matches with entries (Q), (R) and (T), then chosen these three options. Similarly, for entries
(B), (C) and (D).
For each entry in Column-I
Full Marks : +2 If only the option(s) corresponding to all the correct match(es) is (are) chosen
Zero Marks : 0 In all other cases
1. Assume that 2 bodies collide head on. The graph of their velocities with time are shown in column-I
match them with appropriate situation in column-II
Column-I Column-II
(A) (P)
(B) (Q)
(C) (R)
(D) (S)
(T)
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2. A square loop of conducting wire is placed near a long straight current carrying wire as shown. Match
the statements in column-I with the corresponding results in column-II.
Column-I Column-II
If the magnitude of current I is
(A) (P) Induced current in the loop will clockwise
increased be
If the magnitude of current I is
(B) (Q) Induced current in the loop will be anticlockwise
decreased
If the loop is moved away from the
(C) (R) wire will attract the loop
wire
If the loop is moved towards the
(D) (S) wire will repel the loop
wire
Torque about centre of mass of loop is zero due to
(T)
magnetic force
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PART-2 : CHEMISTRY
Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct
answer.
1. Correct order of basic strength for given compounds is :
(A) 3 > 2 > 1 (B) 3 > 1 > 2 (C) 2 > 1 > 3 (D) 1 > 3 > 2
2. Consider the following statements :
(i) Spectral lines of He+ ion belonging to Balmer series are not in visible range.
(ii) In the Paschen series of H-atom maximum lines are in the UV region.
(iii) 2nd line of Lyman series of He+ ion has energy 48.4 eV approx.
The above statements (i), (ii) and (iii) respectively are (T = True, F = False)
(A) T F F (B) F T T
(C) T F T (D) T T T
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3. Determine the pOH of a solution after 0.1 mole of NaOH is added to 1 litre of solution containing
0.15 M CH3COOH and 0.2 M CH3COONa. Assume no change in volume.
[Given: log 3 = 0.48, log 2 = 0.30, pKa of acetic acid = 4.74]
(A) 8.48 (B) 5.52
(C) 9.66 (D) None
4. A current of 9.65 amp. passing for 16 minute 40 sec through a molten tin salt deposits 5.95 g of tin. The
oxidation state of the tin in the salt is : (at. wt of Sn = 119)
(A) +4 (B) +3
(C) +2 (D) +1
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answer(s).
correct option.
5. Which of the following reaction(s) will produce aromatic product ?

Conc. H2 SO4
(A) (B) −−−−−−−−→
NH2 −NH2
(C) + AlBr3 → (D) −−−−−−−→
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6. In which of the following reaction, final major product gives intramolecular aldol condensation reaction ?
(A) (B)
(i)O3 (1eq) (i)O3 (1eq)

(C) −−−−−−−→ (D) −−−−−−−→
(ii)Zn/H2 O (ii)Zn/H2 O
7. Which of the following reactions produce(s) propane as a major product ?
(A)
(B)
(C)
(D)
8. If a gas is allowed to expand at constant temperature then which of the following does not hold true :
(A) the kinetic energy of the gas molecules decreases
(B) the kinetic energy of the gas molecules increases
(C) the kinetic energy of the gas molecules remains the same
(D) Cannot be predicted
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correct option.
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Compound (W) on reductive ozonolysis gives compound (P) and compound (Q). Compound (P) and
compound (Q) when heated together in the presence of NaOH gives Cinnamaldehyde as a product.
compound (P) form cinnamic acid with acetic anhydride in the presence of sodium acetate.
( 1 ) O3 NaOH
(W) −−−−−− −→ (P) + (Q) −−−−→ Cinnamaldehyde
(2)Zn−H2 O Δ
Ac O
(P) −−−−−2−−
+
→ Cinnamic acid
AcO Na
9.
Select the correct statement ?

(A) Compound (T) is Phenol.
(B) Compound (P) is primary compnent of oil of bitter almonds
(C) Compound (R) on clemmenson reduction gives toluene.
(D) all of these.
10. The compound which will form YELLOW precipitate on reaction with NaOH − I2
(A) Compound (W) (B) Compound (P)
(C) Compound (Q) (D) Compound (T)
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Resistance of a conductivity cell filled with 0.01 M HA (aq.) solution is 200 Ω at 300K. (HA is weak
monoprotic acid). If cell constant is 4 cm –1 & Λ 0m (HA) is 2 Ω –1m2mol –1. (R = 0.08 bar/mol-k)
11. Molar conductivity of 0.01M HA ( Ω –1cm2mol –1)
(A) 2 × 103 (B) 2 × 102
(C) 4.0 × 103 (D) 4.0 × 102
12. Osmotic pressure of 0.01M HA(aq.) at 300K is -
(A) 2.64 bar (B) 0.264 bar
(C) 0.24 bar (D) 0.48 bar
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ONLY ONE is correct
13. List-I List-II
(P) pH of 0.1 M-aq. BOH solution at 25°C (Kb of BOH = 10 –9) (1) 2
for the reaction : A(aq.) → products, [A] = 0.8 M at t = 10 min,
(Q) (2) 6
[A] = 0.4 M at t = 30 min, [A] = 0.2M at t = 70 min. order of reaction is
(R) Co-ordination number in hexagonal packing in two dimension (3) 3
4.00 gm silver salt of an organic acid (mol. wt. = 79) gives 3.24 gm silver
(S) (4) 9
residue on strong heating. The basicity of acid is
(A) P → 1;Q → 2;R → 3;S → 4 (B) P → 4;Q → 1;R → 2;S → 3
(C) P → 4;Q → 1;R → 3;S → 2 (D) P → 2;Q → 3;R → 4;S → 1
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14. Match the list
List-I List-II
(P) S2⊝ (1) Produces white ppt. with excess AgNO3
Evolves gas with dil. HCl which turns lime water
(Q) HSO3Θ (2)
milky
Evolves gas with dil.H2SO4 which does not turn
(R) SO23 Θ (3)
Baryta water milky
(S) S2 O23 Θ (4) Produces ppt. with Pb(OAc)2 solution.
(A) P → 3;Q → 1;R → 1,2,4,3;S → 1,4
(B) P → 3,2;Q → 2;R → 1,2,4,3;S → 1,4
(C) P → 2,4;Q → 2;R → 1,2,4;S → 1,2,4,3
(D) P → 3,4;Q → 2;R → 1,2,4;S → 1,2,4
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answer.
1.
Number of possible products (including stereoisomers)
2. Among the following cations(s) :
Hg2+, Mg2+, Cu2+, Ag+, Pb2+
(i) Number of cation(s) gives(s) yellow ppt with Na2SO4 is = (X)
(ii) Number of cation(s) gives(s) yellow ppt with KI is = (Y)
(iii) Number of cation(s) gives(s) white ppt with HCl is = (Z)
Find the value of (X + Y + Z)
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3. Find the number of radicals which produce green colour on flame
Be2+ , Mg2+ , Al3+ , Cu+2 , Ba2+ , K+
4.
If molecular mass of gas C, gas D and gas G are x, y & z respectively. Find x – y – 2z.
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(B), (C) and (D).
1. Column-I Column-II
(Reagent) (Observable change with given compounds)
(A) Na2O2 solution (P) FeCl3 solution
(B) Pb(CH3COO)2 solution (Q) (NH4)2S solution
(C) Na2CO3 solution (R) CuSO4 solution
(D) IO3 – (aq) + H+(aq) (S) KI solution
(T) Cr2(SO4)3 solution
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2. Match the column
Column-II
Column-I
A and B is differentiated
Reactions
by
(A) (P) Tollen’s test
(B) (Q) Fehling test
(C) (R) NaHCO3
(D) (S) Iodoform test
(T) FeCl3
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PART-3 : MATHEMATICS
Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is the correct
answer.
1. The sum of the squares of the lengths of the chords intercepted on the circle, x2 + y2 = 25 by the
lines, x + y = n, n ∈ N, where N is the set of all natural numbers, is :
(A) 420 (B) 160 (C) 105 (D) 210
2. The smaller area (in sq. units) included between the curves √ x + √|y| = 1 and | x | + | y | = 1 is
(A) 1 4 2 5
(B) (C) (D)
3 3 3
x2
3. If the curve y(x) satisfying xdx = ( − y 3 ) dy passes through (0, 2), then value of (y(4))2(4 – (y(4))2) is
y
(A) 8 (B) 16 (C) 24 (D) 12
4. AB, AC are tangents to a parabola y2 = 4ax. p1, p2 & p3 are the lengths of the perpendiculars from
A, B & C respectively on any tangent to the curve, then p2, p1, p3 are in:
(A) A.P. (B) G.P. (C) H.P. (D) none of these
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answer(s).
correct option.
−−
→ −→
5. Let a and b be positive real numbers. Suppose P Q = ai^ + bj^ and P S = ai^ − bj^ are adjacent sides of a
−−
→ −→
parallelogram PQRS. Let →u and →v be the projection vectors of w → = ^i + ^j along P Q and P S , respectively.
If |→u| + |→v| = |w
→ | and if the area of the parallelogram PQRS is 8, then which of the following statements
is/are TRUE?
(A) a + b = 4
(B) a – b = 2
(C) The length of the diagonal PR of the parallelogram PQRS is 4
−−
→ −→
(D) → is an angle bisector of the vectors P Q
w and P S
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6. The co-ordinates (2, 3) and (1, 5) are the foci of an ellipse which passes through the origin, then the
equation of
(A) Tangent at the origin is ( 3 √2 − 5 ) x + ( 1 − 2 √2 ) y = 0
(B) Tangent at the origin is ( 3 √2 + 5 ) x − ( 1 + 2 √2 ) y = 0
(C) Normal at the origin is ( 3 √2 + 5 ) x − ( 2 √2 + 1 ) y = 0
(D) Normal at the origin is ( 3 √2 − 5 ) x + ( 1 − 2 √2 ) y = 0
7. Let , where k is a constant and x, y, z are variables. Then which
of the following statement is/are True ?

(A) Regardless of the value of k, the matrix A is not invertible, i.e., there is no 3 × 3 matrix B such
that BA = the 3 × 3 identity matrix.
(B) There is a unique k such that determinant of A is 0.
(C) The set of solutions (x, y, z) of the matrix equation is either a line or a plane
containing the origin.
(D) If the equation has a solution, then it must be true that q = 10p.
dy
8. A solution curve of the differential equation (x2 + xy + 4x + 2y + 4) − y 2 = 0, x > 0, passes through
dx
the point (1, 3). Then the solution curve
(A) intersects y = x + 2 exactly at one point
(B) intersects y = x + 2 exactly at two points
(C) intersects y = (x + 2)2
(D) does NOT intersect y = (x + 3)2
0999DJA161115230005 LTS-31/40
correct option.

Consider ƒ(x) = sin4 π x & g(x) = ℓ nx on the coordinate plane.
On the basis of above information, answer the following questions :
9. Number of points of intersection of y = ƒ(x) & y = g(x) for x > 3/2 is equal to -
(A) 1 (B) 2 (C) 3 (D) 4
10. Area bounded by y = ƒ(x), y = g(x) & x = 0 is greater than
(A) 1 (B) 1 (C) 2 (D) 5
2 2
LTS-32/40 0999DJA161115230005

x2 y2 1
Let slope of normal at point P in first quadrant on the hyperbola − =1 is – . Tangent and
a2 b2 2
SN
normal at point P meet x-axis at points T & N respectively such that =2
ST
(S is focus nearest to point P)

11. SP
( ) is less than -
a
(A) 1 (B) 2 (C) √ 7 (D) √ 5
12. Eccentricity of hyperbola, is -
(A) √ 2 (B) √ 3 (C) √
7 (D) √
5
2 2
0999DJA161115230005 LTS-33/40
ONLY ONE is correct
13. Consider an ellipse with major & minor axis having length 4 & respectively. Let P be any variable point
on it. S & S' be the foci and C be the centre of ellipse
Match List-I with List-II and select the correct answer using the code given below the list.
List-I List-II
(P) Maximum value of PS.PS' is (1) 8
(Q) Minimum value of (PS)2 + (PS')2 is (2) 2 + √3
Minimum length of portion of tangent at P intercepted between

(R) (3) 4
principal axes is
If foot of normal from P on major axis is P' and normal at P cuts
(S) (4) 1
CN
transverse axis at N, then is 4
CP ′
(A) P → 3;Q → 1;R → 2;S → 4 (B) P → 3;Q → 1;R → 4;S → 2
(C) P → 3;Q → 4;R → 1;S → 2 (D) P → 3;Q → 2;R → 4;S → 1
LTS-34/40 0999DJA161115230005
14. List-I List-II
Number of ways in which 3 squares of unit length can be chosen on a 8 × 8
(P) (1) 441
chessboard, so that all squares are in same diagonal line is
Number of words that can be formed using the letters of the word
(Q) (2) 361
'LIVELIHOOD' if each word contains three letters is
Number of five element subset, that can be chosen from the set of the first 11
(R) (3) 392
natural numbers so that at least two of the five elements are consecutive is
Number of ways in which the letters of the word 'ABBCABBC' can be arranged
(S) (4) 264
such that the word ABBC does not appear is any word is
(A) P → 3;Q → 1;R → 2;S → 4 (B) P → 1;Q → 3;R → 4;S → 2
(C) P → 2;Q → 3;R → 4;S → 1 (D) P → 3;Q → 4;R → 1;S → 2
0999DJA161115230005 LTS-35/40
answer.
1. If volume of a regular tetrahedron of edge length k is V and shortest distance between any pair of
d3
opposite edges of same regular tetrahedron is d, then find the value of .
V
2. If the locus of the point of intersection of two tangents to the parabola y2 = 8x which with the tangent at
the vertex form a triangle of constant area unity, is the curve x2(y2 – 8x) = λ , then λ is equal to
3. A square with side 5 × 5 is divided into squares of unit length by drawing parallel lines, then number of
possible rectangles with one side odd and another side of even length is 3(k)2, then value of 'k' is
4. The value of parameter 'm' such that area bounded by y = x2 – 5 and the line y = mx + 3 attains its
minimum value
LTS-36/40 0999DJA161115230005
(B), (C) and (D).
1. Column – I Column – II
1 1 100
(A) In the expansion of ( 56 + 28 ) , if the number of irrational terms is λ then λ (P) 29

is greater than
8 1
(B) If numerically greatest term in the expansion of (1 – 3x) for x = 2
is Tr, then (Q) 58
value of (9r + 4)
K
The coefficient of x10 in (1 + x2 – x3)8 is K, then ( − 21) is less than or
(C) 4 (R) 82
equal to
If the sum of coefficients in the expansion of
2013
1
(D) ( − 15x + 15x3 ) (17x – 17x5 + 3)2017 is n, then the number of dissimilar (S) 97
3
terms in the expansion of (1 + x)n is less than
(T) 98
0999DJA161115230005 LTS-37/40
2. Let a, b, c be vectors then -
Column-I Column-II
(A) [→a + →b →b + →c →c + →a ] (P) |→b|2 [→a →c →b]
(B) [(→a × →b) × (→a × →c )].→b (Q) (→a . →b) [→a →b →c ]
(C) [→a × →b →b × →c →c × →a ] (R) 2[→a →b →c ]
(D) →b . {(→a × →b) × (→c × →b)} (S) [→a →b →c ]2
(T) 3[→a →b →c ]2
LTS-38/40 0999DJA161115230005
ALLEN®
0999DJA161115230005 LTS-39/40
ALLEN®
NAME OF THE CANDIDATE .............................................................................
FORM NO. .............................................
I have read all the instructions I have verified the identity, name and Form
and shall abide by them. number of the candidate, and that question
paper and ORS codes are the same.
____________________________ ____________________________
Signature of the Candidate Signature of the Invigilator
ALLEN CAREER INSTITUTE Pvt. Ltd.

Registered & Corporate Office : ‘SANKALP’, CP-6, Indra Vihar, Kota (Rajasthan) INDIA-324005
Ph. : +91-744-3556677, +91-744-2757575|E-mail : dlp@allen.in |Website : www.dlp.allen.ac.in, dsat.allen.ac.in
LTS-40/40 Your Target is to secure Good Rank in JEE 2024 0999DJA161115230005

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(0999DJA161115230005) Test Pattern

Test Type : Unit Test # 03

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

OPTICAL RESPONSE SHEET :

DARKENING THE BUBBLES ON THE ORS

11. Darken the bubble COMPLETELY.

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

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

QUESTION PAPER FORMAT

SOME USEFUL CONSTANTS

Space for Rough Work

BEWARE OF NEGATIVE MARKING

(A) K/4 (B) K/2

(A) A is brittle but B is ductile (B) A is ductile but B is brittle

(A) B to A and D to C (B) A to B and C to D

Space for Rough Work

Space for Rough Work

Paragraph for Questions 9 and 10

Space for Rough Work

Space for Rough Work

Paragraph for Questions 11 and 12

11. Choose the correct alternative(s) :-

(B) The net work done by friction force is − 3 mv20 .

Space for Rough Work

Space for Rough Work

(Q) At least some of water freezes. (2)

(R) At least some of ice melts. (3)

(A) P → 2;Q → 1,3;R → 3,4;S → 3 (B) P → 1;Q → 2;R → 3;S → 4

Space for Rough Work

Space for Rough Work

Space for Rough Work

Space for Rough Work

Space for Rough Work

5. Which of the following reaction(s) will produce aromatic product ?

Space for Rough Work

(i)O3 (1eq) (i)O3 (1eq)

7. Which of the following reactions produce(s) propane as a major product ?

Space for Rough Work

Paragraph for Questions 9 and 10

Select the correct statement ?

Space for Rough Work

Paragraph for Questions 11 and 12

Space for Rough Work

Space for Rough Work

Space for Rough Work

Space for Rough Work

Space for Rough Work

(A) (P) Tollen’s test

(B) (Q) Fehling test

(C) (R) NaHCO3

(D) (S) Iodoform test

Space for Rough Work

Space for Rough Work

Space for Rough Work

(B) Tangent at the origin is ( 3 √2 + 5 ) x − ( 1 + 2 √2 ) y = 0

(C) Normal at the origin is ( 3 √2 + 5 ) x − ( 2 √2 + 1 ) y = 0

(D) Normal at the origin is ( 3 √2 − 5 ) x + ( 1 − 2 √2 ) y = 0

7. Let , where k is a constant and x, y, z are variables. Then which

of the following statement is/are True ?