**Physics**

**Section-A**

1. The vernier scale used for measurement has a positive zero error of 0.2 mm. If while taking a measurement it was noted that ‘o’ on the vernier scale lies between 8.5 cm and 8.6 cm, vernier coincidence is 6, then the correct value of the measurement is ……………………. cm. (least count = 0.01 cm)

(a) 8.36 cm

(b) 8.56 cm

(c) 8.58 cm

(d) 8.54 cm

2. For what value of displacement do the kinetic energy and potential energy of a simple harmonic oscillation become equal?

(a) x = A/2

(b) x = 0

(c) x = ±A

(d) x = ± A/√2

3. An electron of mass m and a photon have the same energy E. The ratio of the wavelength of an electron to that of the photon is : (c being the velocity of light)

(a) (E/2m)^{1/2}

(b)

(c) c(2mE)^{1/2}

(d)

4. A car accelerates from rest at a constant rate for some time after which it decelerates at a constant rate to come to rest. If the total time elapsed is t seconds, the total distance travelled is :

(a)

(b)

(c)

(d)

5. A Carnot’s engine working between 400 K and 800 K has a work output of 1200 J per cycle. The amount of heat energy supplied to the engine from the source in each cycle is :

(a) 1800 J

(b) 3200 J

(c) 2400 J

(d) 1600 J

6. A mass M hangs on a massless rod of length l which rotates at a constant angular frequency. The mass M moves with the steady speed in a circular path of constant radius. Assume that the system is in a steady circular motion with constant angular velocity ω. The angular momentum of M about point A is L_{A} which lies in the positive z-direction and the angular momentum of M about point B is L_{B}. The correct statement for this system is:

(a) L_{A} and L_{B} are both constant in magnitude and direction

(b) L_{B} is constant, both in magnitude and direction

(c) L_{A} is constant, both in magnitude and direction

(d) L_{B} is constant in direction with varying magnitude

7. Two ideal polyatomic gases at temperatures T_{1} and T_{2} are mixed so that there is no loss of energy. If F_{1} and F_{2}, m_{1} and m_{2}, n_{1} and n_{2} be the degrees of freedom, masses, the number of molecules of the first and second gas respectively, the temperature of the mixture of these two gases is :

(a)

(b)

(c)

(d)

8. The output of the given combination gates represents :

(a) XOR Gate

(b) NOR Gate

(c) NAND Gate

(d) AND Gate

9. A triangular plate is as shown. A force is applied at point P. The torque at point P with respect to point ‘O’ and ‘Q’ are :

(a) 15 – 20√3, 15 + 20√3

(b) 15 + 20√3, 15 – 20√3

(c) –15 + 20√3, 15 + 20√3

(d) –15–20√3, 15 – 20√3

10. A modern grand-prix racing car of mass m is travelling on a flat track in a circular arc of radius R with a speed v. If the coefficient of static friction between the tyres and the track is μs, then the magnitude of negative lift F_{1}acting downwards on the car is : (Assume forces on the four tyres are identical and g = acceleration due to gravity)

(a)

(b)

(c)

(d)

11. The thickness at the centre of a plano-convex lens is 3 mm and the diameter is 6 cm. If the speed of light in the material of the lens is 2 × 108 mm^{−}^{1}. The focal length of the lens kept in the air is ______.

(a) 0.30 cm

(b) 1.5 cm

(c) 15 cm

(d) 30 cm

12. If an electron is moving in the nth orbit of the hydrogen atom, then its velocity (vn) for the nth orbit is given as :

(a) v_{n} ∝ n

(b)

(c) v_{n} ∝ n^{2}

(d)

13. An AC current is given by I = I_{1} sin ωt + I2 cos ωt. A hot wire ammeter will give a reading :

(a)

(b)

(c)

(d)

14. Two identical metal wires of thermal conductivities K_{1} and K_{2} respectively are connected in series. The effective thermal conductivity of the combination is :

(a)

(b)

(c)

(d)

15. A boy releases a 0.5 kg ball on the frictionless floor with the speed of 20 ms^{−}^{1}. The ball gets deflected by an obstacle on the way. After deflection it moves with 5% of its initial kinetic energy. What is the speed of the ball now ?

(a) 14.41 ms^{−}^{1}

(b) 1.00 ms^{−}^{1}

(c) 19.0 ms^{−}^{1}

(d) 4.47 ms^{−}^{1}

16. A polyatomic ideal gas has 24 vibrational modes. What is the value of γ?

(a) 1.03

(b) 1.30

(c) 10.3

(d) 1.37

17. A current of 10 A exists in a wire of cross-sectional area of 5 mm^{2} with a drift velocity of 2 × 10^{−}^{3} ms^{−}^{1}. The number of free electrons in each cubic meter of the wire is ________

(a) 1 × 10^{23}

(b) 2 × 10^{6}

(c) 2 × 10^{25}

(d) 625 × 10^{25}

18. A solenoid of 1000 turns per metre has a core with a relative permeability of 500. Insulated windings of the solenoid carry an electric current of 5 A. The magnetic flux density produced by the solenoid is: (Permeability of free space = 4 × 10^{−}^{7} H/m)

(a) 2 × 10^{−}^{3} πT

(b) π/5

(c) 10^{−}^{4}πT

(d) πT

19. When two soap bubbles of radii a and b (b > a) coalesce, the radius of curvature of common surface is –

(a)

(b)

(c)

(d)

20. Which level of the single ionized carbon has the same energy as the ground state energy of hydrogen atom?

(a) 8

(b) 1

(c) 6

(d) 4

**Section-B**

21. A parallel plate capacitor whose capacitance C is 14pF is charged by a battery to a potential difference V = 12 V between its plates. The charging battery is now disconnected and a porcelin plate with k = 7 is inserted between the plates, then the plate would oscillate back and forth between the plates, with a constant mechanical energy of ____ pJ. (Assume no friction)

22. If 2.5 × 10^{−}^{6} N average force is exerted by a light wave on a non-reflecting surface of 30 cm^{2} area during 40 minutes of time span, the energy flux of light just before it falls on the surface is _______ W/cm^{2}. (Round off to the nearest integer) (Assume complete absorption and normal incidence conditions are there)

23. The following bodies

(a) a ring

(b) a disc

(c) a solid cylinder

(d) a solid sphere

of same mass ‘m’ and radius ‘R’ are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of the inclined plane is________.

[Mark the body as per their respective numbering given in the question]

24. For VHF signal broadcasting, _______ km^{2} of the maximum service area will be covered by an antenna tower of height 30 m, if the receiving antenna is placed on the ground. Let the radius of the earth be 6400 km. (Round off to the nearest integer) (Take π as 3.14)

25. Consider two identical springs each of spring constant k and negligible mass compared to the mass M as shown. Fig.1 shows one of them and Fig. 2 shows their series combination. The ratio of the time period of oscillation of the two SHM is Tb/Ta = √x, where the value of x is ______. (Round off to the nearest integer)

26. Two blocks (m = 0.5 kg and M = 4.5 kg) are arranged on a horizontal frictionless table as shown in figure. The coefficient of static friction between the two blocks is 3/7. Then the maximum horizontal force that can be applied on the larger block so that the blocks move together is _______ N. (Round off to the nearest integer) [Take g as 9.8 ms^{−}^{2}]

27. The radius in kilometre to which the present radius of the earth (R = 6400 km) to be compressed so that the escape velocity is increased 10 times is ______.

28. The equivalent resistance of a series combination of two resistors is ‘s’. When they are connected in parallel, the equivalent resistance is ‘p’. If s = np, then the maximum value for n is ______. (Round off to the nearest integer)

29. The angular speed of the truck wheel is increased from 900 rpm to 2460 rpm in 26 seconds. The number of revolutions by the truck wheel during this time is ______.

(Assuming the acceleration to be uniform).

30. Four identical rectangular plates with length, l =2 cm and breadth, b = 3/2 cm are arranged as shown in figure. The equivalent capacitance between A and C is x∈_{0}/d. The value of x is ______. (Round off to the nearest integer)

**Chemistry**

**Section-A**

1. The INCORRECT statement(s) about heavy water is (are)

(A) Used as a moderator in a nuclear reactor

(B) Obtained as a by-product in the fertilizer industry

(C) Used for the study of the reaction mechanism

(D) Has a higher dielectric constant than water

Choose the correct answer from the option given below:

(a) (B) only

(b) (B) and (D) only

(c) (C) only

(d) (D) only

2. Given below are two statements:

Statement I: Potassium permanganate on heating at 573 K forms potassium manganate.

Statement II: Both potassium permanganate and potassium manganate are tetrahedral and paramagnetic in nature.

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

(a) Both statement I and statement II are true

(b) Both statement I and statement II are false

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

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

3. Which of the following is the correct structure of tyrosine?

4. Given below are two statements:

Statement I: Retardation factor (Rf) can be measured in meter/centimetre

Statement II: Rf value of a compound remains constant in all solvents.

Choose the most appropriate answer from the options given below:

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

(b) Both statement I and statement II are false

(c) Both statement I and statement II are true

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

5. Mesityl oxide is a common name of:

(a) 3-Methyl cyclohexane carbaldehyde

(b) 4-Methyl pent-3-en-2-one

(c) 2,4-Dimethyl pentan-3-one

(d) 2-Methyl cyclohexanone

6. What is the spin-only magnetic moment value (BM) of a divalent metal ion with atomic number 25, in its aqueous solution?

(a) 5.92

(b) 5.26

(c) Zero

(d) 5.0

7. A central atom in a molecule has two lone pairs of electrons and forms three single bonds. The shape of this molecule is:

(a) Trigonal pyramidal

(b) T-shaped

(c) See-saw

(d) Planar triangular

8. Product “A” in the below chemical reaction is:

9. The point of intersection and sudden increase in the slope, in the diagram given below respectively, indicates:

(a) ΔG = 0 and melting or boiling point of the metal oxide.

(b) ΔG < 0 and decomposition of the metal oxide.

(c) ΔG = 0 and reduction of the metal oxide.

(d) ΔG > 0 and decomposition of the metal oxide.

10. The reaction given below requires which of the following reaction conditions:

(a) 623 K, 300 atm

(b) 573 K, 300 atm

(c) 573 K, Cu, 300 atm

(d) 623 K, Cu 300 atm

11. The correct order of conductivity of ions in water is:

(a) Cs^{+} > Rb^{+} > K^{+} > Na^{+}

(b) K^{+} > Na^{+} > Cs^{+} > Rb^{+}

(c) Rb^{+} > Na^{+} > K^{+} > Li^{+}

(d) Na^{+} > K^{+} > Rb_{+} > Cs^{+}

12. A colloidal system consisting of a gas dispersed in a solid is called a/an:

(a) Aerosol

(b) Solid Sol

(c) Foam

(d) Gel

13. The absolute value of the electron gain enthalpy of halogen satisfies:

(a) I > Br >Cl> F

(b) F >Cl> Br > I

(c) Cl> F > Br > I

(d) Cl> Br > F > I

14. Which of the following reaction is an example of ammonolysis?

(a) C_{6}H_{5}CH_{2}CN → C_{6}H_{5}CH_{2}CH_{2}NH_{2}

(b) C_{6}H_{5}COCl + C_{6}H_{5}NH_{2} → C_{6}H_{5}CONHC_{6}H_{5}

(c) C_{6}H_{5}CH_{2}Cl +NH_{3} → C_{6}H_{5}CH_{2}NH_{2}

(d) C_{6}H_{5}NH_{2} → C_{6}H_{5}NH_{3}^{+}Cl^{−}

15. Reducing smog is a mixture of:

(a) Smoke, fog and N_{2}O_{3}

(b) Smoke, fog and O_{3}

(c) Smoke, fog and SO_{2}

(d) Smoke, fog and CH_{2}=CH–CHO

16. Which of the following is an aromatic compound?

17. With respect to drug-enzyme interaction, identify the wrong statement.

(a) Allosteric inhibitor competes with the enzyme’s active site

(b) Competitive inhibitor binds to the enzyme’s active site

(c) Non-competitive inhibitor binds to the allosteric site

(d) Allosteric inhibitor changes the enzyme’s active site

18. Hoffmann bromamide degradation of benzamide gives product A, which upon heating with CHCl_{3} and NaOH gives product B. The structures of A and B are:

19. The product “A” in the above reaction is:

20. Which of the following compound CANNOT act as a Lewis base?

(a) ClF_{3}

(b) PCl_{5}

(c) NF_{3}

(d) SF_{4}

**Section-B**

21. A certain orbital has n = 4 and m_{l} = –3. The number of radial nodes in this orbital is ____.(Round off to the Nearest Integer).

22. 15 mL of an aqueous solution of Fe2+ in the acidic medium completely reacted with 20 mL of 0.03 M aqueous Cr ^{2}O_{7}^{2}^{−}. The molarity of the Fe^{2+} solution is _____× 10^{–2}(Round off to the Nearest Integer).

23. The reaction of white phosphorus on boiling with alkali in an inert atmosphere resulted in the formation of product ‘A’. The reaction of 1 mol of ‘A’ with an excess of AgNO_{3} in an aqueous medium gives _______ mol(s) of Ag. (Round off to the Nearest Integer).

24. The oxygen dissolved in water exerts a partial pressure of 20 kPa in the vapour above water. The molar solubility of oxygen in water is _____ × 10^{–5} mol dm–3.

(Round off to the Nearest Integer).

[Given : Henry’s law constant = KH = 8.0 × 10^{4} kPa for O_{2}. Density of water with dissolved oxygen = 1.0 kg dm^{–3} ]

25. The standard enthalpies of formation of Al_{2}O_{3} and CaO are –1675 kJ mol^{–1} and –635 kJ mol^{–1} For the reaction 3CaO + 2Al → 3Ca + Al_{2}O_{3} the standard reaction enthalpy Δ_{r}H_{0}=______ kJ. (Round off to the Nearest Integer)

26. For a certain first-order reaction 32% of the reactant is left after 570s. The rate constant of this reaction is _______ × 10^{–3} s^{–1}. (Round off to the Nearest Integer).

[Given: log_{10}2 = 0.301, ln10 = 2.303]

27. The pressure exerted by a non-reactive gaseous mixture of 6.4 g of methane and 8.8 g of carbon dioxide in a 10 L vessel at 27°C is _____ kPa. (Round off to the Nearest Integer). [Assume gases are ideal, R = 8.314 J mol^{–1} K^{–1} Atomic masses: C : 12.0u, H : 1.0u, O : 16.0 u]

28. The mole fraction of a solute in a 100 molal aqueous solution is _____ × 10^{–2}. (Round off to the Nearest Integer). [Given : Atomic masses : H : 1.0 u, O : 16.0 u]

29. In the above reaction, 3.9 g of benzene on nitration gives 4.92 g of nitrobenzene. The percentage yield of nitrobenzene in the above reaction is _____%. (Round off to the Nearest Integer). (Given atomic mass : C : 12.0 u, H : 1.0 u, O : 16.0 u, N : 14.0 u)

30. 0.01 moles of a weak acid HA (Ka = 2.0 × 10^{–6} ) is dissolved in 1.0 L of 0.1 M HCl solution. The degree of dissociation of HA is_______ × 10^{–5} (Round off to the Nearest Integer). Assume degree of dissociation << 1

**Mathematics**

**Section-A**

1. Which of the following is true for y(x) that satisfies the differential equation

(a) y(1) = 1

(b) y(1) = e^{1/2} – 1

(c) y(1) = e^{1/2} – e^{1/2}

(d) y(1) = e^{1/2} – 1

2. The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k2 has no solution if k is equal to

(a) −2

(b) −1

(c) 1

(d) 0

3. The value of is:

(a)

(b)

(c)

(d)

4. If the Boolean expression (p ⇒ q) ⇔ (q * (~ P)) is a tautology, then the Boolean expression p * (~q) is equivalent to:

(a) p ⇒ ~ q

(b) p ⇒ q

(c) q ⇒ p

(d) ~q ⇒ p

5. Choose the incorrect statement about the two circles whose equations are given below:

x^{2} + y^{2} − 10x − 10y + 41 = 0 and x^{2} + y^{2} − 16x − 10y + 80 = 0

(a) Distance between two centres is the average radii of both the circles

(b) Circles have two intersection points

(c) Both circles’ centres lie inside the region of one another

(d) Both circles pass through the centre of each other

6. The sum of possible values of x for is:

(a) −33/4

(b) −32/4

(c) −31/4

(d) −30/4

7. Let If then is equal to:

(a) 10

(b) 13

(c) 12

(d) 8

8. The equation of the plane which contains the y-axis and passes through the point (1, 2, 3) is:

(a) 3x + z = 6

(b) 3x − z = 0

(c) x + 3z = 10

(d) x + 3z = 0

9. If and then a possible value of α is:

(a) π/6

(b) π/2

(c) π/3

(d) π/4

10. The line 2x − y + 1 = 0 is a tangent to the circle at the point (2, 5) and the centre of the circle lies on x − 2y = 4. Then, the radius of the circle is:

(a) 4√5

(b) 3√5

(c) 5√3

(d) 5√4

11. Team ‘A’ consists of 7 boys and n girls and Team ‘B’ has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n is equal to

(a) 5

(b) 6

(c) 2

(d) 4

12. In a triangle PQR, the coordinates of the points P and Q are (−2, 4) and (4, −2) respectively. If the equation of the perpendicular bisector of PR is 2x – y + 2 = 0, then the centre of the circumcircle of the △PQR is:

(a) (−2, −2)

(b) (0, 2)

(c) (−1, 0)

(d) (1, 4)

13. If cot^{−1} (ɑ) = cot^{−1} 2 + cot^{−1} 8 + cot^{−1} 18 + cot^{−1} 32 + …….. upto 100 terms, then ɑ is:

(a) 1.03

(b) 1.00

(c) 1.01

(d) 1.02

14. Which of the following statements is incorrect for the function g (ɑ) for ɑ ∈ R such that

(a) g(α) is a strictly decreasing function

(b) g (ɑ) has an inflexion point at ɑ = −1/2

(c) g(ɑ) is an even function

(d) g(ɑ) is a strictly increasing function

15. If the fourth term in the expansion of is 4480, then the value of x when x ∈ N is equal to:

(a) 4

(b) 3

(c) 2

(d) 1

16. Two dice are rolled. If both dices have six faces numbered 1, 2, 3, 5, 7 and 11, then the probability that the sum of the numbers on the top faces is less than or equal to 8 is:

(a) 17/36

(b) 4/9

(c) 5/12

(d) 1/2

17. The inverse of y = 5^{logx} is:

(a) x = 5^{logy}

(b) x = y^{log5}

(c) x = y^{1/log5}

(d) x = 5^{1/logy}

18. In a school, there are three types of games to be played. Some of the students play two types of games, but none play all three games. Which Venn diagrams can justify the above statements.

(a) P and R

(b) P and Q

(c) None of these

(d) Q and R

19. The area of the triangle with vertices A (z), B (iz) and C (z + iz) is:

(a)

(b) 1

(c) 1/2

(d)

20. The value of where [x] denotes the greatest integer ≤ x is :

(a) 0

(b) π/4

(c) π/2

(d) π

**Section-B**

__Integer Type:__

21. Let there be three independent events E_{1}, E_{2} and E_{3}. The probability that only E_{1} occurs is ɑ, only E_{2} occurs is β and only E_{3} occurs is 𝛾. Let ‘p’ denote the probability of none of the events that occur that satisfies the equations (ɑ − 2β) p = ɑβ and (β − 3𝛾) p = 2β𝛾. All the given probabilities are assumed to lie in the interval (0, 1) Then, is equal to _______

22. If the equation of the plane passing through the line of intersection of the planes 2x − 7y + 4z − 3 = 0, 3x − 5y + 4z + 11 = 0 and the point (−2, 1, 3) is ax + by + cz − 7 = 0, then the value of 2a + b + c − 7 is __________

23. If then the value of det (A^{4}) + det (A^{10} − Adj (2A))^{10}) is equal to _______

24. The minimum distance between any two points P_{1} and P_{2} while considering point P_{1} on one circle and point P_{2} one the other circle for the given circles’ equations x^{2} + y^{2} − 10x − 10y + 41 = 0 and x^{2} + y^{2} − 24x − 10y + 160 = 0 ________

25. If (2021)^{3762} is divided by 17, then the remainder is______

26. If [ . ] represents the greatest integer function, then the value of is_______

27. If and its first derivative with respect to x is when x = 1, where a and b are integers, then the minimum value of |a^{2} – b^{2}| is_______

28. If the function is continuous at each point in its domain and f(0) = 1/k, then k is_______

29. The maximum value of z in the following equation z = 6xy + y^{2}, where 3x + 4y ≤ 100 and 4x + 3y ≤ 75 for x ≥ 0 and y ≥ 0 is _________

30. If and such that then is equal to_________

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