**Physics**

1. A block of mass m = 1 kg slides with velocity v = 6 m/s on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about O and swings as a result of the collision making angle θ before momentarily coming to rest. If the rod has mass M = 2 kg, and length l = 1m, the value θ of is approximately: (take g = 10 m/s^{2})

(1) 49°

(2) 63°

(3) 69°

(4) 55°

2. A uniform thin rope of length 12 m and mass 6 kg hangs vertically from a rigid support and a block of mass 2 kg is attached to its free end. A transverse short wave train of wavelength 6 cm is produced at the lower end of the rope. What is the wavelength of the wave train (in cm) when it reaches the top of the rope?

(1) 12

(2) 3

(3) 9

(4) 6

3. When a diode is forward biased, it has a voltage drop of 0.5 V. The safe limit of current through the diode is 10 mA. If a battery of emf 1.5 V is used in the circuit, the value of minimum resistance to be connected in series with the diode so that the current does not exceed the safe limit is:

(1) 300 Ω

(2) 200 Ω

(3) 50 Ω

(4) 100 Ω

4. Using screw gauge of pitch 0.1 cm and 50 divisions on its circular scale, the thickness of an object is measured. It should correctly be recorded as:

(1) 2.125 cm

(2) 2.124 cm

(3) 2.123 cm

(4) 2.121 cm

5. Model a torch battery of length l to be made up of a thin cylindrical bar of radius ‘a’ and a concentric thin cylindrical shell of radius ‘b’ filled in between with an electrolyte of resistivity ρ (see figure). If the battery is connected to a resistance of value R, the maximum joule heating in R will take place for:

(1)

(2)

(3)

(4)

6. Consider a gas of triatomic molecules. The molecules are assumed to be triangular and made of massless rigid rods whose vertices are occupied by atoms. The internal energy of a mole of the gas at temperature T is:

(1)

(2) 3RT

(3)

(4)

7. An elliptical loop having resistance R, of semi major axis a, and semi minor axis b is placed in magnetic field as shown in the figure. If the loop is rotated about the x-axis with angular frequency ω, the average power loss in the loop due to Joule heating is:

(1)

(2)

(3)

(4) Zero

8. A balloon filled with helium (32° C and 1.7 atm.) bursts. Immediately afterwards the expansion of helium can be considered as:

(1) reversible isothermal

(2) irreversible isothermal

(3) reversible adiabatic

(4) irreversible adiabatic

9. When the wavelength of radiation falling on a metal is changed from 500 nm to 200 nm, the maximum kinetic energy of the photoelectrons becomes three times larger. The work function of the metal is close to:

(1) 1.02 eV

(2) 0.61 eV

(3) 0.52 eV

(4) 0.81 eV

10. Two isolated conducting spheres S_{1} and S_{2} of radius (2/3) R and (1/3) R have 12 μC and –3μC charges, respectively, and are at a large distance from each other. They are now connected by a conducting wire. A long time after this is done the charges on S_{1} and S_{2} are respectively:

(1) 6 μC and 3 μC

(2) 4.5 μC on both

(3) + 4.5 μC and –4.5 μC

(4) 3 μC and 6 μC

11. In a radioactive material, fraction of active material remaining after time t is 9/16. The fraction that was remaining after t/2 is:

(1) 3/4

(2) 7/8

(3) 4/5

(4) 3/5

12. Moment of inertia of a cylinder of mass M, length L and radius R about an axis passing through its centre and perpendicular to the axis of the cylinder is If such a cylinder is to be made for a given mass of a material, the ratio L /R for it to have minimum possible I is:

(1) 2/3

(2) 3/2

(3) √2/3

(4) √3/2

13. A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to that of the earth’s radius R_{e}. By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion so that it become √3/2 times larger. Due to this the farthest distance from the centre of the earth that the satellite reaches is R. Value of R is:

(1) 2R_{e}

(2) 3R_{e}

(3) 4R_{e}

(4) 2.5 R_{e}

14. Pressure inside two soap bubbles are 1.01 and 1.02 atmosphere, respectively. The ratio of their volumes is:

(1) 4 : 1

(2) 2 : 1

(3) 0.8 : 1

(4) 8 : 1

15. In a Young’s double slit experiment, light of 500 nm is used to produce an interference pattern. When the distance between the slits is 0.05 mm, the angular width (in degree) of the fringes formed on the distance screen is close to:

(1) 0.17°

(2) 0.07°

(3) 0.57°

(4) 1.7°

16. A 750 Hz, 20 V (rms) source is connected to a resistance of 100 Ω, an inductance of 0.1803 H and a capacitance of 10 μ F all in series. The time in which the resistance (heat capacity 2 J/°C) will get heated by 10°C. (assume no loss of heat to the surroundings) is close to:

(1) 245 s

(2) 365 s

(3) 418 s

(4) 348 s

17. Magnitude of magnetic field (in SI units) at the centre of a hexagonal shape coil of side 10 cm, 50 turns and carrying current I (Ampere) in units of μ_{0}I/π is:

(1) 250√3

(2) 50√3

(3) 500√3

(4) 5√3

18. The magnetic field of a plane electromagnetic wave is where c = 3 × 10^{8} ms^{−}^{1} is the speed of light.

The corresponding electric field is:

(1)

(2)

(3)

(4)

19. A charged particle carrying charge 1 μC is moving with velocity If an external magnetic field of exists in the region where the particle is moving then the force on the particle is The vector is:

(1)

(2)

(3)

(4)

20. In the circuit shown in the figure, the total charge is 750 μC and the voltage across capacitor C_{2} is 20 V. Then the charge on capacitor C_{2} is:

(1) 650 μC

(2) 450 μC

(3) 590 μC

(4) 160 μC

21. A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating about its axis at 5 revolutions per minute (rpm). The person now starts moving towards the centre of the platform. What will be the rotational speed (in rpm) of the platform when the person reaches its centre _____?

22. An observer can see through a small hole on the side of a jar (radius 15 cm) at a point at height of 15 cm from the bottom (see figure). The hole is at a height of 45 cm. When the jar is filled with a liquid up to a height of 30 cm the same observer can see the edge at the bottom of the jar. If the refractive index of the liquid is N/100, where N is an integer, the value of N is _____.

23. A cricket ball of mass 0.15 kg is thrown vertically up by a bowling machine so that it rises to a maximum height of 20 m after leaving the machine. If the part pushing the ball applies a constant force F on the ball and moves horizontally a distance of 0.2 m while launching the ball, the value of F(in N) is (g = 10 ms^{–2})_____.

24. When a long glass capillary tube of radius 0.015 cm is dipped in a liquid, the liquid rises to a height of 15 cm within it. If the contact angle between the liquid and glass to close to 0°, the surface tension of the liquid, in milli Newton m^{–1} is [ρ_{(liquid)} =900 kgm^{–3} , g = 10 ms^{–2}] (Give answer in closes integer)____.

25. A bakelite beaker has volume capacity of 500 cc at 30°C. When it is partially filled with V_{m} volume (at 30°C) of mercury, it is found that the unfilled volume of the beaker remains constant as temperature is varied. If γ_{(beaker) }= 6 × 10^{–6} °C^{–1} and γ_{(mercury)}= 1.5 × 10^{–4} °C^{–1}, where γ is the coefficient of volume expansion, then V_{m} (in cc) is close to ____.

**Chemistry**

1. It is true that:

(a) A second order reaction is always a multistep reaction

(b) A first order reaction is always a single step reaction

(c) A zero order reaction is a multistep reaction

(d) A zero order reaction is a single step reaction

2. An acidic buffer is obtained on mixing:

(a) 100 mL of 0.1 M HCl and 200 mL of 0.1 M CH_{3}COONa

(b) 100 mL of 0.1 M HCl and 200 mL of 0.1 M NaCl

(c) 100 mL of 0.1 M CH_{3} COOH and 100 mL of 0.1 M NaOH

(d) 100 mL of 0.1 M CH_{3}COOH and 200 mL of 0.1 M NaOH

3. The Kjeldahl method of Nitrogen estimation fails for which of the following reaction products?

(a) (a), (c) and (d)

(b) (b) and (c)

(c) (c) and (d)

(d) (a) and (d)

4. If the boiling point of H_{2}O is 373 K, the boiling point of H_{2}S will be :

(a) greater than 300 K but less than 373 K

(b) equal to 373 K

(c) more than 373 K

(d) less than 300 K

5. The complex that can show optical activity is :

(a) cis– [CrCl_{2} (ox)_{2}]^{3−} (ox – oxalate)

(b) trans – [Fe (NH_{3})_{2} (CN)_{4}]^{−}

(c) trans – [Cr (Cl_{2}) (ox)_{2}]^{3−}

(d) cis– [Fe (NH_{3})_{2} (CN)_{4}]−

6. Which one of the following compounds possesses the most acidic hydrogen?

7. Aqua regia is used for dissolving noble metals (Au, Pt, etc.). The gas evolved in this process is :

(a) N_{2}O_{3}

(b) N_{2}

(c) N_{2}O_{5}

(d) NO

8. The antifertility drug “Novestrol” can react with:

(a) Br_{2} / water; ZnCl_{2} / HCl; FeCl_{3}

(b) Br_{2} / water; ZnCl_{2} / HCl; NaOCl

(c) Alcoholic HCN; NaOCl; ZnCl_{2} / HCl

(d) ZnCl_{2} / HCl; FeCl_{3}; Alcoholic HCN

9. Which of the following compounds produces an optically inactive compound on hydrogenation?

10. Of the species, NO, NO^{+}, NO^{2+} and NO^{−}, the one with minimum bond strength is:

(a) NO^{−}

(b) NO^{+}

(c) NO^{2+}

(d) NO

11. Glycerol is separated in soap industries by:

(a) Fractional distillation

(b) Distillation under reduced pressure

(c) Differential extraction

(d) Steam distillation

12. Thermal power plants can lead to:

(a) Ozone layer depletion

(b) Blue baby syndrome

(c) Eutrophication

(d) Acid rain

13. Henry’s constant (in kbar) for four gases ɑ, β, 𝛾, δ and δ in water at 298 K is given below:

α | β | γ | δ | |

K_{H} |
50 | 2 | 2 × 10^{−5} |
0.5 |

(density of water = 10^{3} kg m^{−}^{3} at 298 K)

This stable implies that :

(a) solubility of γ at 308 K is lower than at 298 K

(b) The pressure of a 55.5 molal solution of δ is 250 bar

(c) α has the highest solubility in water at a given pressure

(d) The pressure of a 55.5 molal solution of γ is 1 bar

14. The electronic spectrum of [Ti(H_{2}O)_{6}]^{3+} shows a single broad peak with a maximum at 20,300 cm-1. The crystal field stabilization energy (CFSE) of the complex ion, in kJ mol^{−1}, is:

(1 kJ mol^{−1} = 83.7 cm^{−1})

(a) 83.7

(b) 242.5

(c) 145.5

(d) 97

15. The atomic number of the element unnilennium is:

(a) 109

(b) 102

(c) 119

(d) 108

16. An organic compound [A], molecular formula C_{10}H_{20}O_{2} was hydrolyzed with dilute sulphuric acid to give a carboxylic acid [B] and an alcohol [C]. Oxidation of [C] with CrO_{3}−H_{2}SO_{4} produced [B]. Which of the following structures are not possible for [A]?

17. The mechanism of S_{N}^{1} reaction is given as:

A student writes general characteristics based on the given mechanism as:

(1) The reaction is favoured by weak nucleophiles.

(2) R^{Θ} would be easily formed if the substituents are bulky.

(3) The reaction is accompanied by racemization.

(4) The reaction is favoured by non-polar solvents. Which observations are correct?

(a) (1) and (2)

(b) (1), (2) and (3)

(c) (1) and (3)

(d) (2) and (4)

18. Tyndall effect is observed when:

(a) The diameter of dispersed particles is much smaller than the wavelength of light used.

(b) The diameter of dispersed particles is much larger than the wavelength of light used.

(c) The refractive index of the dispersed phase is greater than that of the dispersion medium.

(d) The diameter of dispersed particles is similar to the wavelength of light used.

19. Let C_{NaCl} and C_{BaSO4} be the conductances (in S) measured for saturated aqueous solutions of NaCl and BaSO4, respectively, at a temperature T. Which of the following is false?

(a) C_{NaCl} (T_{2}) > C_{BaSo4} (T_{1}) for T_{2} > T_{1}

(b) C_{BaSo4} (T_{2}) > C_{NaCl} (T_{1}) for T_{2} > T_{1}

(c) Ionic mobilities of ions from both salts increase with T.

(d) C_{NaCl} >> C_{BaSo4} at a given T

20. In a molecule of pyrophosphoric acid, the number of P−OH, P = O and P − O − P bonds / moiety(ies) respectively are:

(a) 3, 3 and 3

(b) 4, 2 and 1

(c) 2, 4 and 1

(d) 4, 2 and 0

21. The mole fraction of glucose (C_{6}H_{12}O_{6}) in an aqueous binary solution is 0.1. The mass percentage of water in it, to the nearest integer, is _______.

22. The volume strength of 8.9 M H_{2}O_{2} solution calculated at 273 K and 1 atm is ______. (R = 0.0821 L atm K^{−}^{1} mol^{−}^{1}) (rounded off to the nearest integer)

23. An element with molar mass 2.7 10^{−2} kg mol^{−1} forms a cubic unit cell with edge length 405 pm. If its density is 2.7 10^{3} kg m^{−3}, the radius of the element is approximately ______ 10^{−12} m (to the nearest integer).

24. The total number of monohalogenated organic products in the following (including stereoisomers) reaction is ______.

25. The photoelectric current from Na (Work function, w_{0} = 2.3 eV) is stopped by the output voltage of the cell Pt(s) H_{2} (g, 1 Bar) HCl (aq. pH =1) AgCl s Ag(s). The pH of aq. HCl required to stop the photoelectric current form K(w_{0} = 2.25 eV), all other conditions remaining the same, is _______ 10^{−}^{2} (to the nearest integer). Given,

**Mathematics**

1. The value of (2.1P_{0}−^{2}P_{1}+4.^{3}P_{2}_{−}….. up to 51th term) +(1!-2!+3!-…… up to 51th term) is equal to:

(1) 1 – 51(51)!

(2) 1 + (52)!

(3) 1

(4) 1 5 (51)!

2. Let P be a point on the parabola, y^{2} = 12x and N be the foot of the perpendicular drawn from P on the axis of the parabola. A line is now drawn through the mid-point M of PN, parallel to its axis which meets the parabola at Q. If the y-intercept of the line NQ is 4/3 , then:

(1) PN = 4

(2) MQ = 1/3

(3) PN = 3

(4) MQ = 1/4

3. If = AX^{3} + BX^{2} + Cx + D, then B + C is equal to:

(1) 1

(2) −1

(3) −3

(4) 9

4. The foot of the perpendicular drawn from the point (4,2,3) to the line joining the points (1,2,3) and (1,1,0) lies on the plane:

(1) x – y – 2z = 1

(2) x – 2y + z = 1

(3) 2x + y – z = 1

(4) x + 2y – z = 1

5. If y^{2} + log_{e}(cos^{2}x) = y, x∈(−π/2, π/2), then

(1) |y´(0)|+|y´´(0)| = 1

(2) |y´´(0)| = 0

(3) |y´(0)|+|y´´(0)| = 3

(4) |y´´(0)| = 2

6. is equal to:

(1) 5π/4

(2) 3π/2

(3) 7π/4

(4) π/2

7. A hyperbola having the transverse axis of length √2 has the same foci as that of the ellipse 3x^{2 }+ 4y^{2} = 12, then this hyperbola does not pass through which of the following points ?

(1)

(2)

(3)

(4)

8. For the frequency distribution

Variant (X): x_{1} x_{2} x_{3}…x_{15}

Frequency (f): f_{1} f_{2} f_{3}…f_{15}

where 0 < x_{1} < x_{2} < x_{3} <… x_{15} ≤ 10 and the standard deviation cannot be:

(1) 1

(2) 4

(3) 6

(4) 2

9. A die is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of 4. Then the conditional probability that the score 4 has appeared atleast once is:

(1) 1/3

(2) 1/4

(3) 1/8

(4) 1/9

10. If the number of integral terms in the expansion of (3^{1/2} +5^{1/8})^{n} is exactly 33, then the least value of n is:

(1) 128

(2) 248

(3) 256

(4) 264

11.

(1) π^{2}

(2) π^{2}/2

(3) √2π^{2}

(4) 2π^{2}

12. Consider the two sets:

A = {m ∈ R : both the roots of x^{2} − (m + 1) x + m + 4 = 0 are real} and B = [−3, 5).

Which of the following is not true ?

(1) A-B = (−∞,−3) ∪ (5, ∞)

(2) A ∩ B = {−3}

(3) B-A = (−3, 5)

(4) A U B = R

13. The proposition p − > ∼ (p ˄ q) is equivalent to:

(1) (∼p) ˅(∼ q)

(2) (∼ p) ˄q

(3) q

(4) (∼ p) ˅q

14. The function, f(x) = (3x − 7)x^{2/3}, x ∈ R is increasing for all x lying in:

(1)

(2)

(3)

(4)

15. If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is:

(1) 1/6

(2) 1/5

(3) 1/4

(4) 1/7

16. The solution curve of the differential equation, which passes through the point (0, 1), is:

(1)

(2)

(3)

(4)

17. The area (in sq. units) of the region is

(1) 23/16

(2) 79/16

(3) 23/6

(4) 79/24

18. If α and β are the roots of the equation x^{2} + px + 2 = 0 and 1/α and 1/β are the roots of the equation 2x^{2} + 2qx + 1 = 0, then is equal to:

(1)

(2)

(3)

(4)

19. The lines and

(1) do not intersect for any values of l and m

(2) intersect when l = 1 and m = 2

(3) intersect when l = 2 and m = 1/2

(4) intersect for all values of l and m

20. Let [t] denote the greatest integer ≤t. If for some λ ∈ R −{0, 1}, then L is equal to:

(1) 0

(2) 2

(3) 1/2

(4) 1

21. If then the value of k is …….

22. The diameter of the circle, whose centre lies on the line x + y = 2 in the first quadrant and which touches both the lines x = 3 and y = 2, is ……

23. The value of is equal to………..

24. Let and A^{4}[a_{ij}]. If a_{11} = 109, then a_{22} is equal to…………….

25. If (m, n ∈ N) then the greatest common divisor of the least values of m and n is……………..