**Physics**

1. A ring is hung on a nail. It can oscillate, without slipping or sliding

(i) in its plane with a time period T_{1} and,

(ii) back and forth in a direction perpendicular to its plane, with a period T_{2}.

The ratio T_{1}/T_{2} will be:

(1) 3/√2

(2) √2/3

(3) 2/√3

(4) 2/3

2. The correct match between the entries in column I and column II are:

I |
II |

Radiation |
Wavelength |

(a) Microwave | (i) 100 m |

(b) Gamma rays | (ii) 10^{–15} m |

(c) A.M. radio waves | (iii) 10^{–10} m |

(d) X-rays | (iv) 10^{–3} m |

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

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

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

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

3. In an experiment to verify Stokes law, a small spherical ball of radius r and density falls under gravity through a distance h in the air before entering a tank of water. If the terminal velocity of the ball inside water is the same as its velocity just before entering the water surface, then the value of h is proportional to (ignore viscosity of air)

(1) r^{4}

(2) r

(3) r^{3}

(4) r^{2}

4. Ten charges are placed on the circumference of a circle of radius R with constant angular separation between successive charges. Alternate charges 1, 3, 5, 7, 9 have charge (+q) each, while 2, 4, 6, 8, 10 have charge (–q) each. The potential V and the electric field E at the centre of the circle are respectively: (Take V= 0 at infinity)

(1) V = 0; E = 0

(2)

(3)

(4)

5. A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate where v(t) is its instantaneous velocity. The instantaneous acceleration of the satellite is:

(1) −bv^{3}(t)

(2)

(3)

(4)

6. Two different wires having lengths L_{1} and L_{2}, and respective temperature coefficient of linear expansion α_{1} and α_{2,} are joined end-to-end. Then the effective temperature coefficient of linear expansion is:

(1)

(2)

(3)

(4)

7. In the circuit, given in the figure currents in different branches and the value of one resistor are shown. Then potential at point B with respect to the point A is:

(1) +2 V

(2) −2 V

(3) +1 V

(4) −1 V

8. The velocity (v) and time (t) graph of a body in a straight line motion is shown in the figure. The point S is at 4.333 seconds. The total distance covered by the body in 6 s is:

(1) 37/3 m

(2) 49/4 m

(3) 12 m

(4) 11 m

9. An infinitely long straight wire carrying current I, one side opened rectangular loop and a conductor C with a sliding connector are located in the same plane, as shown in the figure. The connector has length l and resistance R. It slides to the right with a velocity v. The resistance of the conductor and the self-inductance of the loop are negligible. The induced current in the loop, as a function of separation r, between the connector and the straight wire is:

(1)

(2)

(3)

(4)

10. Two Zener diodes (A and B) having breakdown voltages of 6 V and 4 V respectively, are connected as shown in the circuit below. The output voltage VO variation with input voltage linearly increasing with time is given by (V_{input} = 0 V at t = 0) (figures are qualitative)

11. In an adiabatic process, the density of a diatomic gas becomes 32 times its initial value. The final pressure of the gas is found to be n times the initial pressure. The value of n is:

(1) 32

(2) 1/32

(3) 326

(4) 128

12. A galvanometer is used in the laboratory for detecting the null point in electrical experiments. If on passing a current of 6 mA it produces a deflection of 2°, its figure of merit is close to:

(1) 6 × 10^{–3} A/div.

(2) 3 × 10^{–3} A/div.

(3) 666° A/div.

(4) 333° A/div.

13. In the circuit shown, charge on the 5μF capacitor is:

(1) 5.45 μc

(2) 18.00 μc

(3) 10.90 μc

(4) 16.36 μc

14. A parallel plate capacitor has a plate of length ‘l’, width ‘w’ and separation of plates is ‘d’. It is connected to a battery of emf V. A dielectric slab of the same thickness ‘d’ and of dielectric constant k = 4 is being inserted between the plates of the capacitor. At what length of the slab inside plates, will the energy stored in the capacitor be two times the initial energy stored?

(1) 2I/3

(2) I/2

(3) I/4

(4) I/3

15. A radioactive nucleus decays by two different processes. The half-life for the first process is 10 s and that for the second is 100 s. The effective half-life of the nucleus is close to:

(1) 55 sec.

(2) 6 sec.

(3) 12 sec.

(4) 9 sec.

16. A driver in a car, approaching a vertical wall, notices that the frequency of his car horn has changed from 440 Hz to 480 Hz when it gets reflected from the wall. If the speed of sound in air is 345 m/s, then the speed of the car is:

(1) 24 km/hr

(2) 36 km/hr

(3) 54 km/hr

(4) 18 km/hr

17. An iron rod of volume 10^{–3}m^{3} and relative permeability 1000 is placed as core in a solenoid with 10 turns/cm. If a current of 0.5 A is passed through the solenoid, then the magnetic moment of the rod will be:

(1) 0.5 × 10^{2} Am^{2}

(2) 50 × 10^{2} Am^{2}

(3) 5 × 10^{2} Am^{2}

(4) 500 × 10^{2} Am^{2}

18. Two coherent sources of sound, S_{1} and S_{2}, produce sound waves of the same wavelength, λ = 1 m, in phase. S_{1} and S_{2} are placed 1.5 m apart (see fig). A listener, located at L, directly in front of S_{2} finds that the intensity is at a minimum when he is 2 m away from S_{2}. The listener moves away from S_{1}, keeping his distance from S_{2} The adjacent maximum of intensity is observed when the listener is at a distance d from S_{1}. Then, d is :

(1) 12 m

(2) 2 m

(3) 3 m

(4) 5 m

19. The quantities are defined where C – capacitance, R – Resistance, L – length, E – Electric field, B – magnetic field and ε_{0}, μ_{0} – free space permittivity and permeability respectively. Then:

(1) Only y and z have the same dimension

(2) x, y and z have the same dimension

(3) Only x and y have the same dimension

(4) Only x and z have the same dimension

20. The acceleration due to gravity on the earth’s surface at the poles is g and angular velocity of the earth about the axis passing through the pole is ω. An object is weighed at the equator and at a height h above the poles by using a spring balance. If the weights are found to be same, then h is : (h < < R, where R is the radius of the earth)

(1) R^{2}ω^{2}/g

(2) R^{2}ω^{2}/8g

(3) R^{2}ω^{2}/4g

(4) R^{2}ω^{2}/2g

21. Nitrogen gas is at 300° C temperature. The temperature (in K) at which the rms speed of a H_{2} molecule would be equal to the rms speed of a nitrogen molecule, is _________. (Molar mass of N_{2} gas 28 g).

22. The surface of a metal is illuminated alternately with photons of energies E_{1} = 4 eV and E_{2} = 2.5 eV respectively. The ratio of maximum speeds of the photoelectrons emitted in the two cases is 2. The work function of the metal in (eV) is _________.

23. A prism of angle A = 1° has a refractive index μ = 1.5. A good estimate for the minimum angle of deviation (in degrees) is close to N/10. Value of N is

24. A body of mass 2 kg is driven by an engine delivering a constant power of 1 J/s. The body starts from rest and moves in a straight line. After 9 seconds, the body has moved a distance (in m) _____________.

25. A thin rod of mass 0.9 kg and length 1 m is suspended, at rest, from one end so that it can freely oscillate in the vertical plane. A particle of mass 0.1 kg moving in a straight line with velocity 80 m/s hits the rod at its bottom most point and sticks to it (see figure). The angular speed (in rad/s) of the rod immediately after the collision will be __________.

**Chemistry**

1. The major product formed in the following reaction is:

(1) CH_{3}CH(Br)CH_{2}CH(CH_{3})_{2}

(2) CH_{3}CH_{2}CH_{2}C(Br)(CH_{3})_{2}

(3) CH_{3}CH_{2}CH(Br)CH(CH_{3})_{2}

(4) Br(CH_{2})_{3}CH(CH_{3})_{2}

2. Hydrogen peroxide, in the pure state, is:

(1) Linear and blue in color

(2) Linear and almost colorless

(3) Non-planar and almost colorless

(4) Planar and blue in color

3. Boron and silicon of very high purity can be obtained through:

(1) Liquation

(2) Electrolytic refining

(3) Zone refining

(4) Vapour phase refining

4. The following molecule acts as an:

(1) Anti-histamine

(2) Antiseptic

(3) Anti-depressant

(4) Anti-bacterial

5. Among the following compounds, geometrical isomerism is exhibited by:

6. Adsorption of a gas follows Freundlich adsorption isotherm. If x is the mass of the gas adsorbed on mass m of the adsorbent, the correct plot of x/m versus p is:

7. The compound that has the largest H–M–H bond angle (M=N, O, S, C) is:

(1) CH_{4}

(2) H_{2}S

(3) NH_{3}

(4) H_{2}O

8. The correct statement about probability density (except at infinite distance from nucleus) is :

(1) It can be zero for 3p orbital

(2) It can be zero for 1s orbital

(3) It can never be zero for 2s orbital

(4) It can negative for 2p orbital

9. The rate constant (k) of a reaction is measured at different temperatures (T), and the data are plotted in the given figure. The activation energy of the reaction in kJ mol^{–1} is: (R is gas constant)

(1) R

(2) 2/R

(3) 1/R

(4) 2R

10. The variation of molar conductivity with concentration of an electrolyte (X) in aqueous solution is shown in the given figure.

The electrolyte X is:

(1) HCl

(2) CH + COOH

(3) NaCl

(4) KNO_{3}

11. The final major product of the following reaction is:

12. The major product of the following reaction is :

13. Lattice enthalpy and enthalpy of solution of NaCl are 788 kJ mol^{–1}, and 4 kJ mol^{–1}, respectively. The hydration enthalpy of NaCl is:

(1) −780kJ mol^{−}^{1}

(2) 784 kJ mol^{−}^{1}

(3) −784kJ mol^{−}^{1}

(4) 780 kJ mol^{−}^{1}

14. Reaction of ammonia with excess Cl_{2} gives:

(1) NH_{4}Cl and N_{2}

(2) NH_{4}Cl and HCl

(3) NCl_{3} and HCl

(4) NCl_{3} and NH_{4}Cl

15. Which one of the following polymers is not obtained by condensation polymerisation?

(1) Bakelite

(2) Nylon 6

(3) Buna-N

(4) Nylon 6, 6

16. Consider the comples ions, trans-[Co(en)_{2}Cl_{2}]^{+} (A) and cis-[Co(en)_{2}Cl_{2}]^{+} (B). The correct statement regarding them is:

(1) Both (A) and (B) can be optically active.

(2) (A) can be optically active, but (B) cannot be optically active.

(3) Both (A) and (B) cannot be optically active.

(4) (A) cannot be optically active, but (B) can be optically active.

17. An element crystallises in a face-centred cubic (fcc) unit cell with cell edge a. The distance between the centres of two nearest octahedral voids in the crystal lattice is:

(1) a

(2) a/2

(3) √2a

(4) a/√2

18. The correct order of the ionic radii of O^{2–}, N^{3–}, F^{–}, Mg^{2+}, Na^{+} and Al^{3+} is:

(1) N^{3–} < O^{2–} < F^{–} < Na^{+} < Mg^{2+} < Al^{3+}

(2) N^{3–} < F^{–} < O^{2–} < Mg^{2+} < Na^{+} < Al^{3+}

(3) Al^{3+} < Na^{+} < Mg^{2+} < O^{2–} < F^{–} < N^{3–}

(4) Al^{3+} < Mg^{2+} < Na^{+} < F^{–} < O^{2–} < N^{3–}

19. The increasing order of boiling points of the following compounds is:

(1) I < III < IV < II

(2) IV < I < II < III

(3) I < IV < III < II

(4) III < I < II < IV

20. The one that is NOT suitable for the removal of permanent hardness of water is:

(1) Ion-exchange method

(2) Calgon’s method

(3) Treatment with sodium carbonate

(4) Clark’s method

21. For a reaction X + Y ⇌ 2Z, 1.0 mol of X, 1.5 mol of Y and 0.5 mol of Z were taken in a 1 L vessel and allowed to react. At equilibrium, the concentration of Z was 1.0 mol L^{–1}. The equilibrium constant of reaction is ________ x/15. The value of x is _________.

22. The volume, in mL, of 0.02 M K_{2}Cr_{2}O_{7} solution required to react with 0.288 g of ferrous oxalate in acidic medium is ________. (Molar mass of Fe= 56 g mol^{–1})

23. Considering that ∆_{0} > P, the magnetic moment (in BM) of [Ru(H_{2}O)_{6}]^{2+} would be ________.

24. For a demerization reaction, 2A(g) → A_{2}(g) at 298 K, ∆U^{⊝} = −20 kJ mol^{−}^{1}, ∆S^{⊝} = −30 kJ mol^{−}^{1}, then the ∆G^{⊝} will be__________ J

25. The number of chiral carbons present in sucrose is ______.

**Mathematics**

1. If x = 1 is a critical point of the function f(x) = (3x^{2} + ax – 2 – a)e^{x}, then:

(1) x = 1 is a local minima and x = −2/3 is a local maxima of f.

(2) x = 1 is a local maxima and x = −2/3 is a local minima of f.

(3) x = 1 and x = −2/3 are local minima of f.

(4) x = 1 and x = −2/3 are local maxima of f.

2.

(1) is equal to √e

(2) is equal to 1

(3) is equal to 0

(4) does not exist

3. The statement (p → (q → p)) → (p → (p ˅ q)) is:

(1) equivalent to (p˅q)˄ (~ p)

(2) equivalent to (p˄q)˅(~ p)

(3) a contradiction

(4) a tautology

4. If and then:

(1)

(2)

(3)

(4)

5. If the sum of the first 20 terms of the series is 460, then x is equal to:

(1) 7^{1/2}

(2) 7^{2}

(3) e^{2}

(4) 7^{46/21}

6. There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is:

(1) 2250

(2) 2255

(3) 1500

(4) 3000

7. If the mean and the standard deviation of the data 3,5,7,a,b are 5 and 2 respectively, then a and b are the roots of the equation:

(1) x^{2} – 20x + 18 = 0

(2) x^{2} – 10x + 19 = 0

(3) 2x^{2} – 20x + 19 = 0

(4) x^{2} – 10x + 18 = 0

8. The derivative of with respect to is:

(1) 2√3/3

(2) 2√3/5

(3) √3/12

(4) √3/10

9. If where C is a constant of integration, then can be:

(1)

(2)

(3)

(4)

10. If the length of the chord of the circle, x^{2} + y^{2} = r^{2} (r > 0) along the line, y-2x = 3 is r, then r2 is equal to:

(1) 12

(2) 24/5

(3) 9/5

(4) 12/5

11. If α and β are the roots of the equation, 7x^{2} – 3x – 2 = 0, then the value of is equal to:

(1) 27/32

(2) 1/24

(3) 27/16

(4) 3/8

12. If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243, then the sum of the first 50 terms of this G.P. is:

(1)

(2)

(3)

(4)

13. If the line y = mx + c is a common tangent to the hyperbola and the circle x^{2} + y^{2} = 36, then which one of the following is true?

(1) 4c^{2} = 369

(2) c^{2} = 369

(3) 8m + 5 = 0

(4) 5m = 4

14. The area (in sq. units) of the region A = {(x,y): (x – 1) [x] ≤ y ≤ 2√x, 0 ≤ x ≤ 2} where [t] denotes the greatest integer function, is:

(1)

(2)

(3)

(4)

15. If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non-zero distinct real numbers, then is equal to:

(1) y(a – b)

(2) 0

(3) y(b – a)

(4) y(a – c)

16. If for some α ∈ R, the lines and coplanar, then the line L_{2} passes through the point:

(1) (2, −10, −2)

(2) (10, −2, −2)

(3) (10, 2, 2)

(4) (−2, 10, 2)

17. The value of is:

(1) 2^{15}i

(2) −2^{15}

(3) −2^{15}i

(4) 6^{5}

18. Let y = y(x) be the solution of the differential equation If y(π/3) = 0, then y(π/4) is equal to:

(1) 2 + √2

(2) √2 – 2

(3)

(4) 2 – √2

19. If the system of linear equations

x + y + 3z = 0

x + 3y + k^{2}z = 0

3x + y + 3z = 0

has a non-zero solution (x, y, z) for some k ∈ R, then is equal to:

(1) −9

(2) 9

(3) −3

(4) 3

20. Which of the following points lies on the tangent to the curve at the point (1, 0) ?

(1) (2, 6)

(2) (2, 2)

(3) (−2, 6)

(4) (−2, 4)

21. Let A = {a, b, c} and B = {1, 2, 3, 4}. Then the number of elements in the set C = {f : A → B 2 ∈ f(A)and f is not one-one} is_______

22. The coefficient of x^{4} in the expansion of (1 + x + x^{2} + x^{3})^{6} in powers of x, is_______

23. Let the vectors, such that If the projection of is equal to the projection of is perpendicular to is______

24. If the lines x+y = a and x-y = b touch the curve y = x^{2} − 3x + 2 at the points where the curve intersects the x-axis, then a/b is equal to______

25. In a bombing attack, there is 50% chance that a bomb will hit the target. At least two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that there is at least 99% chance of completely destroying the target, is_____

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