12th t t : u t # 04

JEE(Advanced) TEST # 04
Test Type : Unit Test # 04 12th Undergoing/Pass Students
READ THE INSTRUCTIONS CAREFULLY
GENERAL / lkekU; %
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.
5. After breaking the seal of the booklet, verify that the booklet contains 25 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. / g = 10 m/s2
QUESTION PAPER FORMAT
16. The question paper has three parts : Physics, Chemistry and Mathematics.
D O
N OT
B RE
A K
TH E
SE A
LS W
IT HO
U T
BE IN
G IN
ST RU
CT ED
T O
D O
SO B
Y TH
E IN
VI GI
LA TO
PART-1 : PHYSICS
SECTION-I (i) : (Maximum Marks: 18)
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. Force acting on a particle is N. Work done by this force is zero, when a particle is moved on
the line 3y + kx = 5. Here value of k is
(A) 2 (B) 4 (C) 6 (D) 8
2. A circular conducting loop of radius R carries a current I. Another straight infinite conductor carrying current I passes through the diameter of this loop as shown in the figure. The magnitude of force exerted by the straight conductor on the loop is :-
(A) πµ I (B) µ I (C) (D)
BEWARE OF NEGATIVE MARKING
This section contains SIX (06) questions. Each question has FOUR options. 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 :
(2 + 3 )i j
2π μ0 I 2
ALLEN Target : JEE(Main + Advanced) 2022/14-11-2021
3. The potential energy of a particle varies according to the graph shown. Force acting on it varies according to which of the following graphs :-
(A) (B)
(C) (D)
4. Two bodies are connected by an elastic spring. A net external force F acts on the system of bodies. The
(A) work done by F is equal to increase in kinetic energy of the bodies.
(B) work done by F and spring force is equal to increase in kinetic energy plus spring potential energy of bodies and spring system
(C) work done by F and spring force is equal to increase in kinetic energy of the two bodies.
(D) work done by spring force is always zero.
5. A force has components F = axy – by , F = –axy + bx where a = 2 N/m and b = 2 N/m (figure).
Calculate the work done by the force on an object of mass 4kg if it is moved on a closed path from (x, y) values of (0, 1) to (4, 1) to (4, 3) to (0, 3) and back to (0, 1) (all coordinates in meters). The path between the points is always the shortest straight one and all the distances are given in meters :-
(A) 64 J (B) 0 J (C) 72 J (D) 18 J
6. An electron is revolving around a proton in a circular orbit of diameter 1Å. If it produces a magnetic field of 14 wb/m at the proton, then its angular velocity will be about :-
(A) 4.375 × 10 rad/s (B) 2.25 × 10 rad/s
(C) 4 × 10 rad/s (D) 8.75 × 10 rad/s
F x
2 y
SECTION-I (ii) : (Maximum Marks: 16)
Full Marks : +4 If only (all) the correct option(s) is (are) chosen. Zero Marks : 0 In all other cases.
7. A body of mass m is thrown at an angle θ above the horizontal ground with the initial velocity v . Which of the following statements is/are correct?
(A) Work done by the gravity up to the instant when speed is minimum is
(B) Instantaneous power of gravity when speed is minimum is zero.
(C) Instantaneous power of gravity as a function of time t is mg (gt – v sinθ)
(D) Average power of gravity over the whole time of motion is zero.
8. A block of mass 1.0 kg is pulled up on a rough slope of inclination 37° with the horizontal. The coefficient of kinetic friction between the block and the slope is 0.25. The block moves with acceleration 2.0 m/s . After 5.0 s from the instant the block starts, which of the following statements is/are true.
(A) Power delivered by the pulling force is 100 W.
(B) Power delivered by the weight of the block is –60 W.
(C) Power delivered by the friction between the slope and the block is –20W.
(D) Power delivered by the contact force between the slope and the block is –20W.
This section contains FOUR (04) questions. Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of these four option(s) is (are) correct option(s). For each question, choose the correct option(s) to answer the question. Answer to each question will be evaluated according to the following marking scheme:
0
9. A body starts from rest on a smooth horizontal surface under the action of a constant horizontal force F and a resistance force F = kv where k is a positive constant and v is instantaneous speed. Choose the correct option as a function of time t :
(A) Velocity achieved by body at time t is
(B) Curve between velocity and time t and that between power of force F and time t are both straight line
(C) Power of force F increase continuously and becomes constant finally after long time
(D) Rate of change of power of force F decreases continuously and becomes zero finally after long time
10. In the given diagram, there are two semicircular parts one having radius ‘a’ and another having radius ‘3a’ as shown. If resistance of ACB part is R and resistance of ADB part is 3R. Select the correct option(s) :-
(A) Current through ACB is I/4
(B) Current through ADB is I/4
(C) Magnitude of at ‘O’ due to ACB part is
(D) Magnitude of at ‘O’ due to ADB part is
R
SECTION-I (iii) : (Maximum Marks: 12)
Full Marks : +3 If ONLY the correct option is chosen. Zero Marks : 0 In all other cases.
Paragraph for Questions 11 and 12
A person wants to roll a solid non-conducting spherical ball of mass m and radius r on a surface whose coefficient of static friction is µ. He placed the ball on the surface wrapped with n turns of closely packed conducting coils of negligible mass at the diameter. By some arrangement he is able to pass a current i through the coils in the anticlockwise direction as seen from +y-axis. A constant horizontal
magnetic field is present throughout the space as shown in the figure. (Assume µ is large enough to
help rolling motion)
11. If current i is passed through the coils the maximum torque in the coil is :-
(A) –πnir B (B) πnir B
(C) (D)
12. The angular velocity of the ball when it has rotated through an angle θ is (θ < 180°) is :-
(A) (B)
(C) (D)
This section contains FOUR (04) questions. This section contains TWO paragraphs. Based on each paragraph, there are TWO questions. Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is correct. For each question, choose the option corresponding to the correct answer. Answer to each question will be evaluated according to the following marking scheme:
B
sin θ 10 7
Two labour Raju & Anil constructed two parallel walls each consisting of 96 bricks [Weight 500 gm & size 8" × 4" × 2"]. The length of the wall made is 4' but the height is different as Raju made 4" thick wall but Anil made 2" thick wall. If Raju completed the work in 2hr & Anil in 3hr then [take g = 10m/s , 1'' = 2.5 cm & 1' = 30 cm]
13. The difference in work performed by Raju & Anil is–
(A) 384 J (B) 192 J
(C) 200 J (D) 0
14. The work get stored in the form of potential energy in walls made by Raju & Anil are respectively–
(A) 192 J, 384 J (B) 192 J, 292 J
(C) 192 J, 392 J (D) 192 J, 192 J
2
SECTION-I (iv) : (Maximum Marks: 6)
15. There exists uniform magnetic field in space. A particle of mass m = 10 kg and charge q = πC is
projected from origin with velocity m/s. The position of particle is observed at t = 0.5 sec,
1 sec and 2 sec respectively. Following table coordinates of the particles at given moments
x-coordinate y-coordinate z-coordinate

(P) x (1) 0
(Q) x (2) 1
(R) y (3) 2
(S) z (4) 4
(A) P → 3;Q → 2;R → 1;S → 1 (B) P → 2;Q → 3;R → 1;S → 1
(C) P → 1;Q → 3;R → 2;S → 4 (D) P → 1;Q → 2;R → 3;S → 4
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 :
Space for Rough Work
1 1 1
2 2 2
3 3 3
16. A lift of mass m kg has to be taken through a height h from state of rest to state of rest. This process is done in shortest possible time with the help of cables which can bear a tension 0 ≤ T ≤ 3 mg. In list-I are given statements regarding variation of a given quantity with respect to time and in list II are given some graphs showing variation of these quantities with time. Match the List-I with List-II.
List–I List–II
(P) Variation of velocity of lift with time (1)
(Q) Variation of power delivered by tension in cable with time.
(2)
(3)
(S) Variation of acceleration of lift with time (4)
(A) P → 3;Q → 2;R → 1;S → 4 (B) P → 3;Q → 2;R → 4;S → 1
(C) P → 2;Q → 3;R → 4;S → 1 (D) P → 2;Q → 3;R → 1;S → 4
SECTION-III : (Maximum Marks: 24)
Full Marks : +4 If only the correct answer is given. Zero Marks : 0 In all other cases
1. A small object can move in a vertical plane along a semi–circle of radius 0.1 m without friction. The speed at which the ball must be launched from point A so that its acceleration is 3g at point B is given by 'v' m/s. Find the value of v (in m/s).
2. In a region, magnetic field along X-axis changes with time according to the given graph. A charged
particle of mass 1 kg & charge 1C is located at origin starts moving with initial velocity
m/s. It completes one revolution in T time. Find the displacement (in m) of particle at time .
This section contains SIX (06) 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 :
( + 2 )2 π i 3–√ j
0 5T0
3. The chain of length L is released from rest on a smooth fixed incline with x = 0 as shown in the figure.
Determine velocity v of the chain when a half of the length has fallen (Take L = ) in m/s. (Neglect
edge effect of inclined)
4. Find the magnetic field (in µT) at point A. (Given : Amp, R = 1m)
5. Figure shows a square loop 10 cm on each side in the x-y plane with its centre at the origin. An infinite wire is at z = 12 cm above y-axis. What is torque on loop due to magnetic force? If torque is expressed as 12x ×10 N-m, fill value of x.
6. Suppose a hypothetical magnetic field exists in space above the earth surface where is a
unit vector directed radially outward from origin. Origin is on surface of earth. A light charged particle has to perform uniform circular motion in the combined uniform (vertical) gravitational field of earth and magnetic field with speed v and radius r. Height of the plane of motion from earth surface will be
. Find n.
h = nv2
SECTION-IV : (Maximum Marks: 8)
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

(A) At point O (0, 0, 0) (P)
(B) At point P (a, 0, 0) (here a > > L) (Q)
(C) At point P (0, a, 0) (here (a > > L) (R)
(D) At point P (a, a, 0) (here a > > L) (S)
(T)
This section contains ONE (01) question. 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 question, marks will be awarded in one of the following categories :
PART-2 : CHEMISTRY
1. In which of the following species presence of double bond does not affect idealised bond angle :- (I) POCl (II) XeO F (III) XeO F (IV) XeO
(A) II, III, IV (B) I, II only (C) III, IV only (D) I, II, III
2. In which of the following reactions is Incorrect according to its products formed.
(A) PCl + 3H O → H PO + 3HCl (B) NCl + 3H O → NH + 3HOCl
(C) ICl + H O → HClO + HI (D) N O + H O → HNO + HNO
3. The CORRECT order of hybridisation of the N, P and B in the following species is respectively N(SiH ) , PCl , H BO
(A) sp , sp , sp (B) sp , sp , sp (C) sp , sp , sp (D) sp , sp , sp
4. Radius of cation and anion are 2.5 Å & 2.6 Å respectively. If a cubic crystal system is prepared by combination of above cation and anion then edge length of unit cell is (Take )
(A) 2.7 Å (B) 2 Å (C) 3 Å (D) 6 Å
5. A cubic unit cell contains manganese ions at the corners and fluoride ions at the centres of each edge. The coordination number of maganese ion is-
(A) 8 (B) 2 (C) 4 (D) 6
6. The decomposition of a gaseous substance (A) to yield gaseous products (B), (C) follows first order kinetics. If initially only (A) is present and l0 minutes after the start of the reaction the pressure of (A) is 200 mm Hg and that of total pressure is 300 mm Hg, then the rate constant for A → B + C is
(A) (1/600) n 1.25 sec (B) (2.303/10) log 1.5 min
(C) (1/10) n 1.25 sec (D) sec
3 3 2 2 4 4
3 2 3 3 3 2 3
2 2 4 2 3 2
3 3
3 3 3
3 3 2 2 3 3 2 3 2 3 2 2
= 1.73–√
–1
7. Which among the following molecule(s) have peroxy linkage :-
(A) CrO (B) H P O (C) H S O (D) HClO
8. Which of the following statement(s) is/are CORRECT -
(A) [B O ] ion is non-planar but B N H is planar
(B) (H Si) N is planar but (SiH ) P is pyramidal
(C) H C – NCS is bent but SiH - NCS is linear
(D) (CH ) N is pyramidal but (GeH ) N is planar
9. Potassium crystallizes in a body centred cubic lattice with edge length a = 5.0 Å. Then choose the correct statements in given option -
(A) The number of nearest neighbour to any K atom is 8
(B) The number of next nearest neighbour to any K atom is 6
(C) Distance between nearest neighbours is 4.33 Å
(D) Distance between next nearest neighbours is 5.0 Å
10. For the reaction 2A(g) B(s) + 2C(g) + 3D(g)
Volume of C collected at t = 70 sec is 37.5 ml and volume of D collected after sufficient time is 112.5 ml, then which of the following are correct (ln2 = 0.7) [A follows first order kinetics]
(A) Rate constant of reaction is 5 × 10 sec (B) Half life time of A is 70 sec
(C) Volume of C at t = 140 sec is 56.25 ml (D) Volume of D at t = 140 sec is 75 ml
This section contains FOUR (04) 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:
5 4 2 8 2 2 8 4
3 6 –3
3 3 6
Qualitatively, the formation of molecular orbitals can be understood in terms of the constructive or destructive interference of the electron waves of the combining atoms. In the formation of bonding molecular orbital, the two electron waves of the bonding atoms reinforce each other due to constructive interference while in the formation of antibonding molecular orbital, the electron waves cancel each other due to destructive interference.
11. Which of the following combinations give(s) antibonding sigma molecular orbital if z-axis is the internuclear axis ?
(A) (B)
(C) (D)
(A) HeH (B) Be (C) C (D) NO
+ 2 2
ALLEN Leader Test Series/Joint Package Course/Phase-II/14-11-2021
Consider the reaction represented by the equation : CH Cl(g) + H O(g) —→ CH OH(g) + HCl(g) These kinetic data were obtained for the given reaction concentrations :
Initial conc. (M) Initial rate of disappearance of
CH Cl M s [CH Cl] [H O]
0.2 0.2 1 0.4 0.2 2 0.4 0.4 8
Answer the following question based on these data :
13. The rate law for the reaction will be
(A) r = k[CH Cl] [H O] (B) r = k[CH Cl] [H O]
(C) r = k[CH Cl] [H O] (D) r = k[CH Cl] [H O]
14. Unit of rate constant will be :
(A) sec (B) litre mol sec
(C) litre mol sec (D) mol litre sec
3 2 3
15.
(Q) XeO F (2) orbital participated in hybridisation
(R) (3) Maximum lone pair of electron present on central atom out of given species
(S) (4) Only orbital participated in
(A) P → 2;Q → 4;R → 3;S → 1 (B) P → 3;Q → 4;R → 2;S → 1
(C) P → 2;Q → 4;R → 1;S → 3 (D) P → 4;Q → 3;R → 2;S → 1
16.
List-II (Nearest distance between two cations in terms of edge
length (a) of unit cell)
(P) NaCl (1)
(S) Na O (4) a
(A) P → 1;Q → 1;R → 4;S → 2 (B) P → 2;Q → 3;R → 1;S → 4
(C) P → 1;Q → 2;R → 3;S → 2 (D) P → 4;Q → 3;R → 2;S → 1
5
NH⊕4
1. Find the number of chemical species, in which 2pπ–3dπ back bond is present B(OH) , N(SiH ) , P(CH ) , N(GeH ) , BeCl , O(SiH ) , OCl , BCl
2. Which of the following pairs have dipole-dipole interaction (BF + BF ), (CCl + Na ) (HCl + HCl), (CHCl + CHCl ), (K + HCl), (Na + Cl )
3. Calculate the number of σ bond in given compound C N (N ) . Fill your answer as sum of digits (excluding decimal places) till you get the single digit answer.
4. An element A (molecular weight 80 gm/mole) crystallize in BCC and FCC lattice. If density of FCC lattice is 4 gm/cm and distance between next nearest neighbour in BCC is d than what is (in
cm).
5. In first order reaction (A → P) calculate if fraction of reactant is decayed in 't' time where t is
half life.
6. The energy of activation for a reaction is 10 kJ/mol. The presence of catalyst lowers the energy of activation by 75%. Find the factor by which rate of reaction increases at 27ºC due to catalyst. (Take R = 25/3 J/mol-k) If your answer is e then fill x in OMR sheet
3 3 3 3 3 3 3 2 3 2 2 3
3 3 4 +
1. Column-I Column-II
(B) Frenkel defect (Q) Conductivity increases
(C) Metal deficiency defect (R) F-centre appears
(D) Metal excess defect (S) Stoichiometry changes
(T) Density remains same
PART-3 : MATHEMATICS
1. The line joining A(3,1) and B(5,3) is rotated about the mid point C of AB in the anticlockwise direction through 135º. The coordinates of B in the new position are-
(A) (B) (C) (D)
2. is equal to
(A) (B)
(C) (D)
4. If the variable line (a + 2b)x – (a – b)y + (3a + 6b) = 0 (a, b ∈ R) passes through a fixed point, then coordinates of fixed point is :-
(A) (–3, 0) (B) (3, 0) (C) (1, 0) (D) (0, 3)
5. If , (where c is constant of integration), where , then
is equal to

(A) (B) x sec(logx). tan(logx)
(C) (D) x sec(log x)
(4, 2 + )2–√ (4 − , 2)2–√ (3 − 2 , 1)2–√ (4, 2 − )2–√
∫ (1 − x). cosec2xdxtan 2
2
x + 4
+ + C+ 8x + 15x2− −−−−−−−−−√ tan −1 + 8x + 15x2− −−−−−−−−−√ − + C+ 8x + 15x2− −−−−−−−−−√ tan −1 + 8x + 15x2− −−−−−−−−−√
+ + Cx + 4− −−−√ tan −1 + 8x + 15x2− −−−−−−−−−√ − + Cx + 4− −−−√ tan −1 + 8x + 15x2− −−−−−−−−−√
∫ = g(x) + c xdxsec2
n2 100
1 + sec(log x)
2 tan(log x)
1 + sec(log x)
x tan(log x)
7. If area of the triangle formed by the points A(3, 0), B(0, 3) & P(a, b) is sq. units, then identify the
correct statement(s)
(A) Locus of P will be a union of two parallel lines
(B) Locus of P will be a union of two perpendicular lines
(C) Locus of P is given by x + y + 2xy – 6(x + y) + 8 = 0
(D) Locus of P is given by x – y + 2y – 6x + 8 = 0
8. If = = , where C is constant of integration and p,q N, then (p – q – 8)
is equal to
(C) (D)
9. A variable line segment PQ of length 2 moves in xy plane such that the end P always lies on x-axis and Q lies
on the line y = 2x. If locus of mid point of PQ is S(x,y) = 0 where S(x,y) = x + by + 2hxy + 2gx + 2ƒy + c, then-
(A) S(1,1) > 0 (B) S(1,1) = 0 (C) b + c = 1 (D) g = ƒ = 0
10. Coordinates of point A is (1,2) and P(α,β) lies on the line L : 2x + 3y + 5 = 0. Point B(γ,δ) is point
of intersection of L : 3x – 2y = 0 & L : λx + 2y = 10, λ ∈ R such that γ & δ are positive
integers. Let s = α + β + γ + δ + λ , then which of following is/are correct-
(A) If |PA – PB| is minimum, then s = 11 (B) If |PA – PB| is maximum, then 5s = 24
(C) If |PA + PB| is minimum, then 31s = 156 (D) If |PA + PB| is minimum, then 5s = 24
This section contains FOUR (04) 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:
3 2
2 2
2 2
2
Let ƒ,g,h be 3 functions such that ƒ'(x) > 0 & g(x) > 0 ∀ x ∈ R where &
, x ≠ 0.
On the basis of above information, answer the following questions :
11. is equal to -
12. ƒ(x), g(x), h(x) are in -
(A) A.P. (B) G.P. (C) H.P. (D) None of these
If the lines ax + by + c = 0, bx + cy + a = 0 and cx + by + a = 0 are concurrent (a, b, c are distinct and non-zero real numbers). On the basis of above information answer the following.
13. If the point of concurrency is (α, β), then (α + β ) is :-
(A) 0 (B) 1 (C) 2 (D) 3
14. For quadratic equation ax + bx + c = 0, which of the following is always true :-
(A) Both roots are imaginary (B) Both roots are negative
(C) At least one root is positive (D) Both roots are equal
∫ ƒ(x). g(x)dx = + Cx4
g(x)
g(x)
h(x)
15. Match List-I with List-II and select the correct answer using the code given below the list.
List-I List-II
(P)
An equilateral triangle has one vertex at (0,0) and another at
the x-coordinate of third
vertex is
(1) 1
(Q) (1,–2), (3,6), (5,10), (3,y) in order are the vertices of a parallelogram, then y is
(2) 2
(R) x-coordinate of circumcentre of triangle with vertices (–2,–3), (–1,0), (7,–6) is
(3) 3
(S)
y-coordinate of a point which divides the line segment joining (5,– 2) and (9,6) internally in the ratio 3 : 1, is
(4) 4
(A) P → 1;Q → 4;R → 3;S → 2
(B) P → 4;Q → 3;R → 1;S → 2
(C) P → 3;Q → 2;R → 3;S → 4
(D) P → 3;Q → 1;R → 4;S → 2
16. List I List II
(P) (1)
(Q) (2)
(R) (3)
(S) (4)
Where C is the constant of integration.
(A) P → 4;Q → 1;R → 2;S → 3 (B) P → 4;Q → 1;R → 3;S → 2
(C) P → 1;Q → 2;R → 3;S → 4 (D) P → 1;Q → 3;R → 4;S → 2
∫ dx 1 + x2− −−−−√
1 2
2 2–√
1. Let , (,m,n ∈ R), then is equal to
2. Let m and m are slopes of lines represented by the line pair qx – 2pxy + y – 2x + 3y + λ = 0, then value
of is
3. If = ,
where C is constant of integration, then is equal to
4. Sum of slopes of all possible lines passing through origin (O) and intersecting the lines x + y = 1 & x + 2y = 1 at A & B respectively such that , is
5. The point (x,y) lies on line 3x + 4y = 6, such that the smallest value of 25|x – 2x + y +1| is m, then the
value of is
6. If (where c is integration constant & , m, n ∈ N), then ( – m – n)
is equal to
∫ = + C ( − 4) dxx6
.( + 2)x6 1/4 x4
( + 2)x6 m
xn n m
2 2 2
1− 9x2− −−−−−√ ( + )+C1
(α + 3β)
1. Column-I Column-II
is equal to (P) 1
(B) If then km is
equal to (Q) 3
(D) Let , then [g (1)] is
equal to (where [.] denotes greatest integer function) (where C is constant of integration)
(S) Rational number
(T) Irrational number

tan m
1
x m

12th t t : u t # 04

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