**Physics**

**Section-A**

1. A 25 m long antenna is mounted on an antenna tower. The height of the antenna tower is 75 m. The wavelength (in meter) of the signal transmitted by this antenna would be:

(a) 200

(b) 400

(c) 100

(d) 300

2. A block of mass m slides along a floor while a force of magnitude F is applied to it at an angle θ as shown in the figure. The coefficient of kinetic friction is μK. Then, the block’s acceleration ‘a’ is given by (g is the acceleration due to gravity)

(a)

(b)

(c)

(d)

3. Four equal masses, m each is placed at the corners of a square of length (l) as shown in the figure. The moment of inertia of the system about an axis passing through A and parallel to DB would be:

(a) ml^{2}

(b) 3ml^{2}

(c) √3ml^{2}

(d) 2ml^{2}

4. The stopping potential in the context of the photoelectric effect depends on the following property of incident electromagnetic radiation:

(a) Amplitude

(b) Phase

(c) Frequency

(d) Intensity

5. One main scale division of a vernier callipers is ‘a’ cm and nth division of the vernier scale coincides with (n–1)th division of the main scale. The least count of the callipers in mm is:

(a)

(b)

(c)

(d)

6. A plane electromagnetic wave of frequency 500 MHz is travelling in vacuum along y-direction. At a particular point in space and time, The value of electric field at this point is: (speed of light = 3×10^{8} ms^{–1}) Assume x, y, z are unit vectors along are unit vectors along x, y and z directions.

(a)

(b)

(c)

(d)

7. The maximum and minimum distances of a comet from the Sun are 1.6 × 10^{12} m and 8.0 × 10^{10} m respectively. If the speed of the comet at the nearest point is 6 × 10^{4} ms^{–1}, the speed at the farthest point is :

(a) 1.5 × 10^{3} m/s

(b) 4.5 × 10^{3} m/s

(c) 3.0 × 10^{3} m/s

(d) 6.0 × 10^{3} m/s

8. A block of 200 g mass moves with a uniform speed in a horizontal circular groove, with vertical sidewalls of a radius of 20 cm. If the block takes 40 s to complete one round, the normal force by the sidewalls of the groove is:

(a) 6.28 × 10^{−}^{3} N

(b) 0.0314 N

(c) 9.859 × 10^{−}^{2} N

(d) 9.859 × 10^{−}^{4} N

9. An RC circuit as shown in the figure is driven by an AC source generating a square wave. The output wave pattern monitored by CRO would look close to:

10. In thermodynamics, heat and work are:

(a) Intensive thermodynamics state variables

(b) Extensive thermodynamics state variables

(c) Path functions

(d) Point functions

11. A conducting wire of length ‘l’, area of cross-section A and electric resistivity ρ is connected between the terminals of a battery. A potential difference V is developed between its ends, causing an electric current. If the length of the wire of the same material is doubled and the area of cross-section is halved, the resultant current would be:

(a)

(b)

(c)

(d)

12. The pressure acting on a submarine is 3 × 10^{5} Pa at a certain depth. If the depth is doubled, the percentage increase in the pressure acting on the submarine would be :(Assume that atmospheric pressure is 1 × 10^{5} Pa, the density of water is 10^{3} kg m^{–3}, acceleration due to gravity g = 10 ms^{–2})

(a)

(b)

(c)

(d)

13. A bar magnet of length 14 cm is placed in the magnetic meridian with its north pole pointing towards the geographic north pole. A neutral point is obtained at a distance of 18 cm from the centre of the magnet. If BH = 0.4 G, the magnetic moment of the magnet is (1 G = 10^{–4 }T)

(a) 28.80 J T^{−}^{1}

(b) 2.880 J T^{−}^{1}

(c) 2.880 × 10^{3 }J T^{−}^{1}

(d) 2.880 × 10^{2} J T^{−}^{1}

14. The volume V of an enclosure contains a mixture of three gases, 16 g of oxygen, 28 g of nitrogen and 44 g of carbon dioxide at absolute temperature T. Consider R as a universal gas constant. The pressure of the mixture of gases is:

(a) 4RT/V

(b) 88RT/V

(c)

(d) 3RT/V

15. A conducting bar of length L is free to slide on two parallel conducting rails as shown in the figure

Two resistors R_{1} and R_{2} are connected across the ends of the rails. There is a uniform magnetic field pointing into the page. An external agent pulls the bar to the left at a constant speed v. The correct statement about the directions of induced currents I_{1} and I_{2} flowing through R_{1} and R_{2} respectively is :

(a) I_{1} is in the clockwise direction and I_{2} is in the anticlockwise direction

(b) Both I_{1} and I_{2} are in a clockwise direction

(c) I_{1} is in the anticlockwise direction and I_{2} is in a clockwise direction

(d) Both I_{1} and I_{2} are in the anticlockwise direction

16. The velocity-displacement graph describing the motion of a bicycle is shown in the figure.

The acceleration-displacement graph of the bicycle’s motion is best described by:

17. For changing the capacitance of a given parallel plate capacitor, a dielectric material of dielectric constant K is used, which has the same area as the plates of the capacitor. The thickness of the dielectric slab is where ‘d’ is the separation between the plates of parallel plate capacitors. The new capacitance (C’) in terms of original capacitance (C_{0}) is given by the following relation:

(a)

(b)

(c)

(d)

18. For an electromagnetic wave travelling in free space, the relation between average energy densities due to electric (U_{e}) and magnetic (U_{m}) fields is :

(a) U_{e} ≠ U_{m}

(b) U_{e} = U_{m}

(c) U_{e} > U_{m}

(d) U_{e} < U_{m}

19. Time period of a simple pendulum is T inside a lift when the lift is stationary. If the lift moves upwards with an acceleration g/2, the time period of the pendulum will be:

(a)

(b)

(c)

(d)

20. The angle of deviation through a prism is minimum when

(A) Incident ray and emergent ray are symmetric to the prism

(B) The refracted ray inside the prism becomes parallel to its base

(C) Angle of incidence is equal to that of the angle of emergence

(D) When the angle of emergence is double the angle of incidence

Choose the correct answer from the options given below:

(a) Only statement (D) is true

(b) Statements (A), (B) and (C)

(c) Statements (B) and (C) are true

(d) Only statement (A) and (B) are true

**Section-B**

21. A fringe width of 6 mm was produced for two slits separated by 1 mm apart. The screen is placed 10 m away. The wavelength of light used is ‘x’ nm. The value of ‘x’ to the nearest integer is ________.

22. The value of power dissipated across the Zener diode (Vz = 15 V) connected in the circuit as shown in the figure is x × 10^{–1}

The value of x, to the nearest integer, is _________.

23. The resistance R=V/I, where V = (50 ± 2) V and I = (20 ± 0.2) A. The percentage error in R is ‘x’ %. The value of ‘x’ to the nearest integer is _________.

24. A sinusoidal voltage of peak value 250 V is applied to a series LCR circuit, in which R = 8Ω, L=24 mH and C=60 μF. The value of power dissipated at resonant conditions is ‘x’ kW. The value of x to the nearest integer is _________.

25. A ball of mass 10 kg moving with a velocity 10√3 ms–1 along the X-axis, hits another ball of mass 20 kg which is at rest. After the collision, the first ball comes to rest and the second one disintegrates into two equal pieces. One of the pieces starts moving along the Y-axis at a speed of 10 m/s. The second piece starts moving at a speed of 20 m/s at an angle θ (degree) with respect to the X-axis. The configuration of pieces after the collision is shown in the figure. The value of θ to the nearest integer is ________.

26. In the figure given, the electric current flowing through the 5 kΩ resistor is ‘x’ mA.

The value of x to the nearest integer is ________.

27. Consider a 20 kg uniform circular disk of radius 0.2 m. It is pin supported at its centre and is at rest initially. The disk is acted upon by a constant force F=20 N through a massless string wrapped around its periphery as shown in the figure.

Suppose the disk makes n number of revolutions to attain an angular speed of 50 rad s^{–1}. The value of n, to the nearest integer is ________. [Given: In one complete revolution, the disk rotates by 6.28 rad]

28. The first three spectral lines of H-atom in the Balmer series are given λ_{1}, λ_{2}, λ_{3} considering the Bohr atomic model, the wavelengths of first and third spectral lines (λ_{1}/λ_{3}) are related by a factor of approximately ‘x’ × 10^{–1}. The value of x, to the nearest integer, is ________.

29. Consider a frame that is made up of two thin massless rods AB and AC as shown in the figure. A vertical force of magnitude 100 N is applied at point A of the frame.

Suppose the force is resolved parallel to the arms AB and AC of the frame. The magnitude of the resolved component along the arm AC is x N. The value of x, to the nearest integer, is ________. [Given: sin (35°) = 0.573, cos (35°) = 0.819, sin (110°) = 0.939, cos (110°) = −0.342]

30. In the logic circuit shown in the figure, if input A and B are 0 to 1 respectively, the output at Y would be ‘x’. The value of x is ________.

**Chemistry**

**Section-A**

1. Among the following, the aromatic compounds are:

Choose the correct answer from the following options:

(a) (A) and (B) only

(b) (A), (B) and (C) only

(c) (B), (C) and (D) only

(d) (B) and (C) only

2. Given below are two statements:

Statement I: H_{2}O_{2} can act as both oxidizing and reducing agent in the basic medium.

Statement II: In the hydrogen economy, energy is transmitted in the form of dihydrogen.

In the light of the above statements, choose the correct Ans: from the options given below:

(a) Statement I is false but statement II is true

(b) Both Statement I and Statement II are true

(c) Statement I is true but statement II is false

(d) Both Statement I and Statement II are false

3. Which of the following is Lindlar catalyst?

(a) Zinc chloride and HCl

(b) Partially deactivated palladised charcoal

(c) Sodium and Liquid NH_{3}

(d) Cold dilute solution of KMnO_{4}

4. In chromatography technique, the purification of a compound is independent of:

(a) Length of the column or TLC plate

(b) Mobility or flow of solvent system

(c) Physical state of the pure compound

(d) Solubility of the compound

5. Which among the following pairs of Vitamins is stored in our body relatively for longer duration?

(a) Ascorbic acid and Vitamin D

(b) Thiamine and Ascorbic acid

(c) Vitamin A and Vitamin D

(d) Thiamine and Vitamin A

6. In the below chemical reaction, intermediate “X” and reagent/condition “A” are:

7. Which of the following reaction/reactions DOES NOT involve Hoffmann bromamide degradation?

8. A group 15 element, which is a metal and forms a hydride with strongest reducing power among group 15 hydrides. The element is:

(a) Bi

(b) As

(c) P

(d) S

9. Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R:

Assertion A: The size of the Bk^{3+} ion is less than the Np^{3+} ion.

Reason R: The above is a consequence of the lanthanoid contraction.

In the light of the above statements, choose the correct Ans: from the options given below:

(a) A is false but R is true

(b) Both A and R are true but R is not the correct explanation of A

(c) A is true but R is false

(d) Both A and R are true and R is the correct explanation of A

10. The products “A” and “B” formed in the below reactions are:

11. The type of pollution that gets increased during the day time and in the presence of O_{3} is:

(a) Global warming

(b) Reducing smog

(c) Acid rain

(d) Oxidizing smog

12. The product “P” in the below reaction is:

13. Match List-I with List-II:

List-I
Industrial process |
List-II
Application |
||

(a) | Haber’s Process | (i) | HNO_{3} synthesis |

(b) | Ostwald’s process | (ii) | Aluminium extraction |

(c) | Contact process | (iii) | NH_{3} synthesis |

(d) | Hall-Heroult process | (iv) | H_{2}SO_{4} synthesis |

Choose the correct answer from the options given below:

(a) (a)-(iv), (b)-(i), (c)-(ii), (d)-(iii)

(b) (a)-(ii), (b)-(iii), (c)-(iv), (d)-(i)

(c) (a)-(iii), (b)-(iv), (c)-(i), (d)-(ii)

(d) (a)-(iii), (b)-(i), (c)-(iv), (d)-(ii)

14. Given below are two statements:

Statement I: The Eo value for Ce^{4+}/Ce^{3+} is +1.74 V.

Statement II: Ce is more stable in Ce^{4+} state than Ce^{3+} state.

In the light of the above statements, choose the correct answer from the options given below:

(a) Both Statement I and Statement II are correct

(b) Statement I is incorrect but statement II is correct

(c) Both Statement I and Statement II are incorrect

(d) Statement I is correct but statement II is incorrect

15. Given below are two statements:

Statement I: Both CaCl_{2}.6H_{2}O and MgCl_{2}.8H_{2}O undergo dehydration on heating.

Statement II: BeO is amphoteric whereas the oxides of other elements in the same group are acidic.

In the light of the above statements, choose the correct Ans: from the options given below:

(a) Statement I is true but statement II is false

(b) Both Statement I and Statement II are false

(c) Statement I is false but statement II is true

(d) Both Statement I and Statement II are true

16. Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: Enol form acetone [CH_{3}COCH_{3}] exists in <0.1% quantity. However, the enol forms acetylacetone [CH_{3}COCH_{2}OCCH_{3}] that exists in approximately 15% quantity.

Reason R: Enol form of acetylacetone is stabilized by intramolecular hydrogen bonding, which is not possible in enol form of acetone.

In the light of the above statements, choose the correct statement:

(a) A is true but R is false

(b) Both A and R are true but R is the correct explanation of A

(c) A is false but R is true

(d) Both A and R are true but R is not the correct explanation of A

17. Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R

Assertion A: The H–O–H bond angle in a water molecule is 104.5°

Reason R: The lone pair – lone pair repulsion of electrons is higher than the bond pair-bond pair repulsion.

In the light of the above statements, choose the correct answer from the options given below:

(a) A is false but R is true

(b) A is true but R is false

(c) Both A and R are true, and R is the correct explanation of A

(d) Both A and R are true, but R is not the correct explanation of A

18. Match List – I with List – II:

List-I
Name of oxo acid |
List-II
Oxidation state of ‘P’ |
||

(a) | Hypophosphorous acid | (i) | +5 |

(b) | Orthophosphoric acid | (ii) | +4 |

(c) | Hypophosphoric acid | (iii) | +3 |

(d) | Orthophosphorous acid | (iv) | +2 |

(v) | +1 |

Choose the correct answer from the options given below:

(a) (a)-(iv), (b)-(v), (c)-(ii), (d)-(iii)

(b) (a)-(v), (b)-(iv), (c)-(ii), (d)-(iii)

(c) (a)-(v), (b)-(i), (c)-(ii), (d)-(iii)

(d) (a)-(iv), (b)-(i), (c)-(ii), (d)-(iii)

19. The process that involves the removal of sulphur from the ores is:

(a) Refining

(b) Roasting

(c) Smelting

(d) Leaching

20. The functions of antihistamine are:

(a) Antiallergic and Analgesic

(b) Antacid and antiallergic

(c) Antiallergic and antidepressant

(d) Analgesic and antacid

**Section-B**

21. If the equation below is balanced with integer coefficients, the value of c is__________. (Round off to the nearest integer)

2MnO_{4}^{−} + bC_{2}O_{4}^{2}^{−} + cH^{+} → xMn2+ yCO_{2} + zH_{2}O.

22. Complete combustion of 750 g of an organic compound provides 420 g of CO_{2} and 210 g of H_{2} The percentage composition of carbon and hydrogen in organic compounds is 15.3 and _______ respectively. (Round off to the Nearest Integer).

23. AB2 is 10% dissociated in water to A^{2+} and B^{–}. The boiling point of a 10.0 molal aqueous solution of AB_{2} is ____________ ° (Round off to the Nearest Integer).

24. A certain element crystallizes in a bcc lattice of unit cell edge length 27Å. If the same element under the same conditions crystallises in the fcc lattice, the edge length of the unit cell in Å will be _________. (Round off to the Nearest Integer).

[Assume each lattice point has a single atom]

[Assume √3 = 1.73, √2 = 1.41]

25. The equivalents of ethylene diamine required to replace the neutral ligands from the coordination sphere of the trans-complex of CoCl_{3}.4NH_{3} is __________. (Round off to the nearest Integer).

26. For the reaction A(g) ⇌ B(g) at 495 K, Δ_{r}G° = –9.478 kJ mol^{–(1)}

If we start the reaction in a closed container at 495 K with 22 millimoles of A, the amount of B in the equilibrium mixture is ________millimoles. (Round off to the nearest Integer).

[R = 8.314] mol^{–1} K^{–1}; ln 10 = 2.303]

27. When light of wavelength 248 nm falls on a metal of threshold energy 3.0 eV, the de-Broglie wavelength of emitted electrons is ____________Å. (Round off to the Nearest Integer).

[Use : √3 = 1.73, h = 6.63×10^{–34} Js

me = 9.1×10^{–31} kg; c = 3.0 × 10^{8}ms^{–1}; 1eV = 1.6×10^{–19}J]

28. A 6.50 molal solution of KOH (aq.) has a density of 1.89 g cm^{–3}The molarity of the solution is __________ moldm^{–3} (Round off to the Nearest Integer).

[Atomic masses : K : 39.0 u; O: 16.0 u; H: 1.0 u]

29. Two salts A_{2}X and MX have the same value of solubility product of 4.0 × 10^{–12}. The ratio of their molar solubilities i.e. ________. (Round off to the Nearest Integer).

30. The decomposition of formic acid on gold surface follows first order kinetics. If the rate constant at 300 K is 1.0 × 10^{–3} s^{–1} and the activation energy Ea = 11.488 kJ mol^{–1}, the rate constant at 200 K is __________ × 10^{–5} s^{–1}. (Round off to the Nearest Integer).

**Mathematics**

1. Consider three observations a, b and c such that b = a+c. If the standard deviation of a + 2, b + 2, c + 2 is d, then which of the following is true?

(a) b^{2} = a^{2} + c^{2} + 3d^{2}

(b) b^{2} = 3 (a^{2} + c^{2}) − 9d^{2}

(c) b^{2} = 3 (a^{2} + c^{2}) + 9d^{2}

(d) b^{2} = 3 (a^{2} + c^{2} + d^{2})

2. Let vector be obtained by rotating the vector by an angle 45° about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices (ɑ, β), (0, β) and (0, 0) is equal to:

(a) 1

(b) 1/2

(c) 1/√2

(d) 2√2

3. If for a > 0, the feet of perpendiculars from the points A (a, –2a, 3) and B (0, 4, 5) on the plane lx + my + nz = 0 are points C (0, –a, –1) and D respectively, then the length of line segment CD is equal to :

(a) √41

(b) √31

(c) √55

(d) √66

4. The range of a ∈ R for which the function x ≠ 2nπ, n ∈ N has critical points, is:

(a) [–4/3, 2]

(b) [1, ∞)

(c) (–∞,–1]

(d) (–3, 1)

5. Let the functions f: R→R and g: R →R be defined as:

and

Then, the number of points in R where (fog)(x) is NOT differentiable is equal to:

(a) 1

(b) 2

(c) 3

(d) 0

6. Let a complex number z, |z| ≠ 1, satisfy Then, the largest value of |z| is equal to ____

(a) 5

(b) 8

(c) 6

(d) 7

7. A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade is:

(a) 3/4

(b) 52/867

(c) 39/50

(d) 22/425

8. If n is the number of irrational terms in the expansion of [3^{1/4} + 5^{1/8}]^{60}, then (n − 1) is divisible by

(a) 8

(b) 26

(c) 7

(d) 30

9. Let the position vectors of two points P and Q be, respectively. Let R and S be two points such that the direction ratios of lines PR and QS are (4, –1, 2) and (–2,1,–2) respectively. Let lines PR and QS intersect at T. If the vector TA is perpendicular to both the length of vector is √5 units, then the modulus of a position vector of A is:

(a) √5

(b) √227

(c) √171

(d) √482

10. If the three normals drawn to the parabola, y^{2} = 2x pass through the point (a, 0) a ≠ 0, then ‘a’ must be greater than

(a) 1

(b) −1/2

(c) 1/2

(d) −1

11. Let Then is equal to :

(a) tan^{−}^{1} (3/2)

(b) π/2

(c) cot^{−}^{1}(3/2)

(d) tan^{−}^{1}(3)

12. The number of roots of the equation, in the interval [0, π] is equal to :

(a) 3

(b) 4

(c) 2

(d) 8

13. If y = y(x) is the solution of the differential equation, then the maximum value of the function y(x) over R is equal to :

(a) 8

(b) −15/4

(c) 1/2

(d) 1/8

14. Which of the following Boolean expression is a tautology?

(a) (p ∧ q) ∧ (p → q)

(b) (p ∧ q) ∨ (p ∨ q)

(c) (p ∧ q) ∨ (p → q)

(d) (p ∧ q) → (p → q)

15. Let Then, the system of linear equations has:

(a) No solution

(b) A unique solution

(c) Exactly two solutions

(d) Infinitely many solutions

16. If for log_{10}sin x + log_{10} cos x =−1 and then the value of n is equal to :

(a) 16

(b) 12

(c) 20

(d) 9

17. The locus of the midpoints of the chord of the circle, x^{2} + y^{2} = 25 which is tangent to the hyperbola, is:

(a) (x^{2} + y^{2})^{2} − 16x^{2} + 9y^{2} = 0

(b) (x^{2} + y^{2})^{2} − 9x^{2} + 144y^{2} = 0

(c) (x^{2} + y^{2})^{2} − 9x^{2} − 16y^{2} = 0

(d) (x^{2} + y^{2})^{2} − 9x^{2} + 16y^{2} = 0

18. Let [x] denote the greatest integer less than or equal to x. If for n ∈ N, then is equal to:

(a) 1

(b) 2^{n}^{−}^{1}

(c) n

(d) 2

19. Let P be a plane lx + my + nz = 0 containing the line, If plane P divides the line segment AB joining points A (–3, –6, 1) and B (2, 4, –3) in ratio k : 1 then the value of k is equal to :

(a) 1.5

(b) 2

(c) 4

(d) 3

20. The number of elements in the set {x ∈ R : (|x| − 3) |x + 4| = 6} is equal to

(a) 2

(b) 3

(c) 1

(d) 4

**Section-B**

21. Let f: (0, 2) → R be defined as Then, is equal to_______

22. The total number of 3 × 3 matrices A having entries from the set {0, 1, 2, 3} such that the sum of all the diagonal entries of AA^{T} is 9, is equal to ____

23. Let f:R→R be a continuous function such that f (x) + f (x + 1) = 2, for all x ∈ If then the value of I_{1} + 2I_{2} is equal to______

24. Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four-digit numbers, then the number of common terms in these two series is equal to ____

25. If the normal to the curve at a point (a, b) is parallel to the line x + 3y = – 5, a > 1, then the value of |a + 6b| is equal to ______

26. If then a + b + c is equal to_______

27. Let ABCD be a square of the side of unit length. Let a circle C_{1} centred at A with a unit radius is drawn. Another circle C_{2} which touches C_{1} and the lines AD and AB is tangent to it, is also drawn. Let a tangent line from point C to the circle C_{2} meet the side AB at E. If the length of EB is ɑ + √3β, where ɑ, β are integers, then ɑ + β is equal to ______

28. Let z and w be two complex numbers such that and Re (w) has a minimum value. Then, the minimum value of n ∈ N, for which w^{n} is real, is equal to ____

29. Let and and I3 be the identity matrix of order 3. If the determinant of the matrix (P^{−}^{1}AP − I_{3})^{2} is ɑω^{2}, then the value of ɑ is equal to _____

30. Let the curve y = y(x) be the solution of the differential equation, If the numerical value of area bounded by the curve y = y(x) and the x-axis is then the value of y(1) is equal to_______

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