SECTION-A 
PHYSICS

Question 1:

1. 2√2 < k ≤ 3
2. 2√3 < k ≤ 3√2
3. 2√3 < k < 3√3
4. 2√2 < k < 2√3

Question 2:

The distance of the point (7,−2,11) from the line x−6/1 = y−4/0 = z−8/3 is:

1. 12
2. 14
3. 18
4. 21

Question 3:

Let x=x(t) and y=y(t) be solutions of dx/dt +ax=0 and dy/dt +by=0 respectively. Given x(0)=2, y(0)=1, and 3y(1)=2x(1), find t for which x(t)=y(t).

1. log2 3/2
2. log4 3
3. log3 4
4. log4 2/3

Question 4:

If (a,b) is the orthocenter of a triangle with vertices (1,2), (2,3), (3,1), then 36I1/I2 is equal to:

1. 72
2. 88
3. 80
4. 66

Question 5:

If A denotes the sum of all the coefficients in the expansion of (1 − 3x + 10x²)ⁿ and B denotes the sum of all the coefficients in the expansion of (1 + x²)ⁿ, then:

1. A = B³
2. 3A = B
3. B = A³
4. A = 3B

Question 6:

Number of common terms in sequences 4, 9, 14... up to 25th term and 3, 6, 9... up to 37th term:

1. 9
2. 5
3. 7
4. 8

Question 7:

The shortest distance of parabola y² = 4x from circle x² + y² - 4x - 16y + 64 = 0 is d. Find d²:

1. 16
2. 24
3. 20
4. 36

Question 8:

The shortest distance between the lines (x - 4)/1 = (y + 1)/2 = (z - 3)/1 and (x - λ)/2 = (y + 1)/4 = (z - 2)/(-5) is 6√5. The sum of all possible values of λ is:

1. 5
2. 8
3. 7
4. 10

Question 9:

Evaluate the integral ∫₁⁰ (1 / [√(3 + x) + √(1 + x)]) dx in the form a + b√2 + c√3; find 2a + 3b - 4c.

1. 4
2. 10
3. 7
4. 8

Question 10:

Let S = {1,2,3...10}. M is the set of all subsets, and relation R = {(A,B): A∩B=∅}. R is:

1. symmetric and reflexive only
2. reflexive only
3. symmetric and transitive only
4. symmetric only

Question 11:

If S = {z ∈ C : |z−i| = |z+i| = |z−1|}, then n(S) is:

1. 1
2. 0
3. 3
4. 2

Question 12:

Four points (2k,3k), (1,0), (0,0), and (0,1) lie on a circle for k:

1. 2/13
2. 3/13
3. 5/13
4. 1/13

Question 13:

Consider f(x)= a(7x−12−x²), b |x²−7x+12| for continuity. Find n(S).

1. 2
2. Infinitely many
3. 4
4. 1

Question 14:

Let a₁, a₂, ..., a₁₀ be 10 observations with ∑aₖ=50 and ∑(aₖ · aⱼ)=1100. Find the standard deviation.

1. 5
2. √5
3. 10
4. √115

Question 15:

The length of the chord of the ellipse x²/25 + y²/16 = 1, with midpoint (1, 2/5), is:

1. √169/5
2. √200/9
3. √174/5
4. √154/5

Question 16:

The portion of the line 4x + 5y = 20 in the first quadrant is trisected by lines L₁ and L₂ passing through the origin. The tangent of the angle between L₁ and L₂ is:

1. 8/5
2. 25/41
3. 2/5
4. 30/41

Question 17:

Let a = i + 2j + k, b = 3(i - j + k). Let c satisfy a × c = b and a · c = 3. Then a · (c × b) - b · c is:

1. 32
2. 24
3. 20
4. 36

Question 18:

If a = lim(x→0) [ (√(1 + √(1 + x⁴))) - √2) / x⁴ ] and b = lim(x→0) [ (sin²(x)) / (√2 - √(1 + cos x)) ], then ab³ is:

1. 36
2. 32
3. 25
4. 30

Question 19:

Given f(x)=
cos(x) -sin(x) 0
sin(x) cos(x) 0
0 0 1
Evaluate: Statement I: f(-x) is the inverse of f(x).
Statement II: f(x)·f(y)=f(x+y).

1. I is false, II is true
2. Both I and II are false
3. I is true, II is false
4. Both I and II are true

Question 20:

The function f: N-{1}→N defined by f(n)=highest prime factor of n, is:

1. one-one and onto
2. one-one only
3. onto only
4. neither one-one nor onto

Question 21:

Least positive integral value of α, for which the angle between vectors αi−2j+2k and αi+2αj−2k is acute, is:

1. 5
2. 6
3. 7
4. 8

Question 22:

For differentiable function f: (0, ∞) → R, if f(x) - f(y) ≥ log_e (x/y) + x - y, find ∑f'(1/n) from n=1 to 20.

1. 2890
2. 2900
3. 3000
4. 3200

Question 23:

If the solution for Differential equation (2x + 3y − 2)dx + (4x + 6y − 7)dy = 0, y(0) = 3, has solution αx + βy + 3log|2x + 3y − γ| = 6. Find α+2β+3γ.

1. 29
2. 30
3. 28
4. 27

Question 24:

Let the area of region {(x,y): x − 2y + 4 ≥ 0, x + 2y² ≥ 0, x+4y² ≤ 8, y ≥ 0} be m/n with m and n coprime. Find m+n.

1. 119
2. 120
3. 121
4. 122

Question 25:

If 8 = 3 + 1/4 (3 + p) + 1/4² (3 + 2p) + 1/4³ (3 + 3p) + ..., then the value of p is:

1. 9
2. 8
3. 7
4. 6

Question 26:

A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required, and let a = P(X = 3), b = P(X ≥ 3), and c = P(X ≥ 6 | X > 3). Then (b + c)/a is equal to:

1. 12
2. 11
3. 13
4. 14

Question 27:

Let the set of all a in ℝ such that the equation cos(2x) + a sin(x) = 2a - 7 has a solution be [p, q], and r = tan(9°) - tan(27°) - 1/(cot(63°) + tan(81°)). Then pqr is equal to:

1. 48
2. 47
3. 49
4. 50

Question 28:

Given f(x) = x³ + x² f'(1) + x f''(2) + f'''(3), x in ℝ. Then f'(10) is equal to:

1. 202
2. 200
3. 204
4. 206

Question 29:

Let A denote the matrix A = [ [2, 0, 1], [1, 1, 0], [1, 0, 1] ], and B = [B₁, B₂, B₃] where B₁, B₂, and B₃ are column matrices such that:
A B₁ = [ [1], [0], [0] ], A B₂ = [ [2], [3], [0] ], A B₃ = [ [3], [2], [1] ]. If α = |B| and β is the sum of all diagonal elements of B, then α³ + β³ is equal to:

1. 28
2. 29
3. 30
4. 31

Question 30:

If α satisfies x² + x + 1 = 0 and (1 + α)⁷ = A + Bα + Cα², where A, B, C ≥ 0, then 5(3A - 2B - C) is equal to:

1. 5
2. 6
3. 7
4. 8

Question 31:

Position of an ant (S in metres) moving in the Y-Z plane is given by S = 2t² ˆj + 5t ˆk (where t is in seconds). The magnitude and direction of velocity of the ant at t = 1 s will be:

1. 16 m/s in y-direction
2. 4 m/s in x-direction
3. 9 m/s in z-direction
4. 4 m/s in y-direction

Question 32:

Given below are two statements:

Statement (I): Viscosity of gases is greater than that of liquids.

Statement (II): Surface tension of a liquid decreases due to the presence of insoluble impurities. In the light of the above statements, choose the most appropriate answer from the options given below:

1. Statement I is correct but Statement II is incorrect
2. Statement I is incorrect but Statement II is correct
3. Both Statement I and Statement II are incorrect
4. Both Statement I and Statement II are correct

Question 33:

If the refractive index of the material of a prism is cot(A/2), where A is the angle of the prism, then the angle of minimum deviation will be:

1. π - 2A
2. π/2 - 2A
3. π - A
4. π/2 - A

Question 34:

A proton moving with a constant velocity passes through a region of space without any change in its velocity. If E⃗ and B⃗ represent the electric and magnetic fields respectively, then the region of space may have:

1. E = 0, B = 0
2. E = 0, B ≠ 0
3. E ≠ 0, B = 0
4. E ≠ 0, B ≠ 0

Question 35:

The least positive integral value of α for which the angle between the vectors αi - 2j + 2k and αi + 2αj - 2k is acute is:

1. 3
2. 4
3. 5
4. 6

Question 36:

A rectangular loop of length 2.5 m and width 2 m is placed at 60° to a magnetic field of 4 T. The loop is removed from the field in 10 s. The average emf induced in the loop during this time is:

1. -2V
2. +2V
3. +1V
4. -1V

Question 37:

The refractive index of the material of a prism is cot(A/2), where A is the angle of the prism. The angle of minimum deviation δm is:

1. A
2. 2A
3. A/2
4. 3A

Question 38:

A spherometer has a circular base of radius 3.5 cm. The central screw moves 2 mm for every complete rotation. How many rotations should be given to the central screw to raise it from the base by 4.2 mm?

1. 3
2. 5
3. 6
4. 7

Question 39:

A charged particle of mass m and charge q is projected perpendicular to a magnetic field B with speed v. The pitch of the helical path followed by the particle is:

1. 2πmv/qB
2. 2mv/qB
3. 2πqB/mv
4. 2qB/πmv

Question 40:

For a reaction, if the equilibrium constant at 500 K is 4, then the value of the standard Gibbs free energy ΔG° at this temperature is:

1. -1155 J
2. 1386 J
3. -1386 J
4. 1155 J

Question 41:

The element with the highest first ionization enthalpy among the following is:

1. B
2. Al
3. Ga
4. In

Question 42:

For a given reaction, the rate of appearance of B is four times the rate of disappearance of A. The balanced reaction is:

1. A → 4B
2. 4A → B
3. 2A → 2B
4. 4A → 4B

Question 43:

Among the following, the most acidic compound is:

1. Benzene
2. Phenol
3. Ethanol
4. Acetylene

Question 44:

The correct order of bond angle for NH3, PH3, and AsH3 is:

1. NH3 > PH3 > AsH3
2. PH3 > NH3 > AsH3
3. AsH3 > PH3 > NH3
4. All have the same bond angle

Question 45:

Which one of the following complex ions is diamagnetic?

1. [Fe(CN)6]3-
2. [Co(NH3)6]3+
3. [NiCl4]2-
4. [CuCl4]2-

Question 46:

In a hypothetical reaction A → B, the rate of formation of B is 0.04 mol L^-1 s^-1. The rate of disappearance of A is:

1. 0.02 mol L^-1 s^-1
2. 0.04 mol L^-1 s^-1
3. 0.08 mol L^-1 s^-1
4. 0.01 mol L^-1 s^-1

Question 47:

In which of the following molecules/ions does the central atom obey the octet rule?

1. BeCl2
2. BF3
3. SO2
4. NO2

Question 48:

The reaction Zn + H2SO4 → ZnSO4 + H2 is an example of:

1. Combination reaction
2. Decomposition reaction
3. Displacement reaction
4. Redox reaction

Question 49:

If the boiling point of a solution containing 1 mole of glucose in 1000 g of water is 100.52°C, the ebullioscopic constant (Kb) of water is:

1. 0.52 K kg/mol
2. 1.52 K kg/mol
3. 2.52 K kg/mol
4. 3.52 K kg/mol

Question 50:

The IUPAC name of the compound CH3CH2CH(OH)CH3 is:

1. 1-Butanol
2. 2-Butanol
3. tert-Butanol
4. Isobutanol

Question 51:

A particle starts from origin at t = 0 with a velocity 5ˆi m/s and moves in x-y plane under action of a force which produces a constant acceleration of (3ˆi + 2ˆj) m/s². If the x-coordinate of the particle at that instant is 84 m, then the speed of the particle at this time is √α m/s. The value of α is:

Question 52:

A thin metallic wire having cross-sectional area of 10^-4 m² is used to make a ring of radius 30 cm. A positive charge of 2π C is uniformly distributed over the ring, while another positive charge of 30 pC is kept at the center of the ring. The tension in the ring is:

Question 53:

Two coils have mutual inductance 0.002 H. The current changes in the first coil according to the relation i = i₀ sin(ωt), where i₀ = 5 A and ω = 50π rad/s. The maximum value of emf in the second coil is π.

Question 54:

Two immiscible liquids of refractive indices 8/5 and 3/2 respectively are put in a beaker as shown in the figure. The height of each column is 6 cm. A coin is placed at the bottom of the beaker. For near normal vision, the apparent depth of the coin is α 4 cm. The value of α is:

Question 55:

In a nuclear fission process, a high mass nuclide (A ≈ 236) with binding energy 7.6 MeV/Nucleon dissociates into middle mass nuclides (A ≈ 118), having binding energy of 8.6 MeV/Nucleon. The energy released in the process would be MeV.

Question 56:

Four particles each of mass 1 kg are placed at four corners of a square of side 2 m. Moment of inertia of the system about an axis perpendicular to its plane and passing through one of its vertex is kg m².

Question 57:

A particle executes simple harmonic motion with an amplitude of 4 cm. At the mean position, the velocity of the particle is 10 cm/s. The distance of the particle from the mean position when its speed becomes 5 cm/s is √α cm, where α = ?

Question 58:

Two long, straight wires carry equal currents in opposite directions as shown in the figure. The separation between the wires is 5.0 cm. The magnitude of the magnetic field at a point P midway between the wires is µT.

Question 59:

The charge accumulated on the capacitor connected in the following circuit is µC _____

Question 60:

If the average depth of an ocean is 4000 m and the bulk modulus of water is 2×10⁹ Nm⁻², then the fractional compression ∆V/V of water at the bottom of the ocean is α × 10⁻². The value of α is:

Question 61:

Two nucleotides are joined together by a linkage known as:

1. Phosphodiester linkage
2. Glycosidic linkage
3. Disulphide linkage
4. Peptide linkage

Question 62:

Highest enol content will be shown by:

1. Structure 1
2. Structure 2
3. Structure 3
4. Structure 4

Question 63:

Element not showing variable oxidation state is:

1. Bromine
2. Iodine
3. Chlorine
4. Fluorine

Question 64:

Which of the following is strongest Bronsted base?

1. Structure 1
2. Structure 2
3. Structure 3
4. Structure 4

Question 65:

Which of the following electronic configuration would be associated with the highest magnetic moment?

1. [Ar] 3d⁷
2. [Ar] 3d⁸
3. [Ar] 3d³
4. [Ar] 3d⁶

Question 66:

Which of the following has highly acidic hydrogen?

1. Structure 1
2. Structure 2
3. Structure 3
4. Structure 4

Question 67:

A solution of two miscible liquids showing negative deviation from Raoult’s law will have:

1. Increased vapour pressure, increased boiling point
2. Increased vapour pressure, decreased boiling point
3. Decreased vapour pressure, decreased boiling point
4. Decreased vapour pressure, increased boiling point

Question 68:

Consider the following complex ions:
P = [FeF₆]³⁻
Q = [V(H₂O)₆]²⁺
R = [Fe(H₂O)₆]²⁺
The correct order of the complex ions, according to their spin-only magnetic moment values (in B.M.), is:

1. R < Q < P
2. R < P < Q
3. Q < R < P
4. Q < P < R

Question 69:

Choose the polar molecule from the following:

1. CCl₄
2. CO₂
3. CH₂=CH₂
4. CHCl₃

Question 70:

Given below are two statements:
Statement I: The 4f and 5f series of elements are placed separately in the Periodic table to preserve the principle of classification.
Statement II: s-block elements can be found in pure form in nature.
In the light of the above statements, choose the most appropriate answer from the options given below:

1. Statement I is false but Statement II is true
2. Both Statement I and Statement II are true
3. Statement I is true but Statement II is false
4. Both Statement I and Statement II are false

Question 71:

Given below are two statements: Assertion (A): Melting point of Boron (2453 K) is unusually high in group 13 elements. Reason (R): Solid Boron has very strong crystalline lattice. In the light of the above statements, choose the most appropriate answer from the options given below:

1. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
2. Both (A) and (R) are correct and (R) is the correct explanation of (A)
3. (A) is true but (R) is false
4. (A) is false but (R) is true

Question 72:

The ascending order of acidity of –OH group in the following compounds is:

1. (A) < (D) < (C) < (B) < (E)
2. (C) < (A) < (D) < (B) < (E)
3. (C) < (D) < (B) < (A) < (E)
4. (A) < (C) < (D) < (B) < (E)

Question 73:

Given below are two statements: Assertion (A): Melting point of Boron (2453 K) is unusually high in group 13 elements. Reason (R): Solid Boron has very strong crystalline lattice. In the light of the above statements, choose the most appropriate answer from the options given below:

1. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
2. Both (A) and (R) are correct and (R) is the correct explanation of (A)
3. (A) is true but (R) is false
4. (A) is false but (R) is true

Question 74:

Cyclohexene is a type of an organic compound:

1. Benzenoid aromatic
2. Benzenoid non-aromatic
3. Acyclic
4. Alicyclic

Question 75:

Yellow compound of lead chromate gets dissolved on treatment with hot NaOH solution. The product of lead formed is a:

1. Tetraanionic complex with coordination number six
2. Neutral complex with coordination number four
3. Dianionic complex with coordination number six
4. Dianionic complex with coordination number four

Question 76:

Given below are two statements: Statement I: Aqueous solution of ammonium carbonate is basic. Statement II: Acidic/basic nature of salt solution of a salt of weak acid and weak base depends on Ka and Kb values of acid and the base forming it. In the light of the above statements, choose the most appropriate answer from the options given below:

1. Both Statement I and Statement II are correct
2. Statement I is correct but Statement II is incorrect
3. Both Statement I and Statement II are incorrect
4. Statement I is incorrect but Statement II is correct

Question 77:

IUPAC name of the following compound (P) is:

1. 1-Ethyl-5, 5-dimethylcyclohexane
2. 3-Ethyl-1,1-dimethylcyclohexane
3. 1-Ethyl-3, 3-dimethylcyclohexane
4. 1,1-Dimethyl-3-ethylcyclohexane

Question 78:

NaCl reacts with conc. H2SO4 and K2Cr2O7 to give reddish fumes (B), which react with NaOH to give yellow solution (C). (B) and (C) respectively are:

1. CrO2Cl2, Na2CrO4
2. Na2CrO4, CrO2Cl2
3. CrO2Cl2, KHSO4
4. CrO2Cl2, Na2Cr2O7

Question 79:

The electronic configuration for Neodymium is: (Atomic Number for Neodymium 60)

1. [Xe] 4f4 6s2
2. [Xe] 5f4 7s2
3. [Xe] 4f6 6s2
4. [Xe] 4f5 5d1 6s2

Question 80:

The electronic configuration for Neodymium is: (Atomic Number for Neodymium 60)

1. [Xe] 4f⁴ 6s²
2. [Xe] 5f⁴ 7s²
3. [Xe] 4f⁴ 6s²
4. [Xe] 4f⁴ 5d¹ 6s²

Question 81:

The mass of silver (Molar mass of Ag: 108 g/mol) displaced by a quantity of electricity which displaces 5600 mL of O₂ at S.T.P. will be g.

Question 83:

Mass of methane required to produce 22 g of CO₂ after complete combustion is g.

Question 84:

If three moles of an ideal gas at 300 K expand isothermally from 30 dm³ to 45 dm³ against a constant opposing pressure of 80 kPa, then the amount of heat transferred is J.

Question 85:

3-Methylhex-2-ene on reaction with HBr in presence of peroxide forms an addition product (A). The number of possible stereoisomers for 'A' is:

Question 87:

Among the following, total number of meta-directing functional groups is (Integer based):
– OCH₃, –NO₂, –CN, –CH₃–NHCOCH₃, – COR, –OH, – COOH, –Cl

Question 88:

The number of electrons present in all the completely filled subshells having n = 4 and s = ±1/2 is:

Question 89:

Sum of bond order of CO and NO⁺ is:

Question 90:

From the given list, the number of compounds with +4 oxidation state of Sulphur is:
SO₃, H₂SO₃, SOCl₂, SF₄, BaSO₄, H₂S₂O₇