**Physics**

1. For a plane electromagnetic wave, the magnetic field at a point x and time t is The instantaneous electric field E corresponding to B is:

(speed of light c = 3 × 10^{8} ms^{−}^{1)}

(1)

(2)

(3)

(4)

2. Particle A of mass m_{1} moving with velocity collides with another particle B of mass m_{2} which is at rest initially. Let be the velocities of particles A and B after collision respectively. If m_{1} = 2m_{2} and after collision the angle between is:

(1) 105°

(2) 15°

(3) −45°

(4) 60°

3. When a car is at rest, its driver sees raindrops falling on it vertically. When driving the car with speed v, he sees that raindrops are coming at an angle 60° from the horizontal. On further increasing the speed of the car to (1 + β) v, this angle changes to 45°. The value β is close to:

(1) 0.50

(2) 0.73

(3) 0.37

(4) 0.41

4. A charged particle going around in a circle can be considered to be a current loop. A particle of mass m carrying charge q is moving in a plane with speed v under the influence of magnetic field The magnetic moment of this moving particle:

(1)

(2)

(3)

(4)

5. A double convex lens has power P and same radii of curvature R of both the surfaces. The radius of curvature of a surface of a plano-convex lens made of the same material with power 1.5 P is:

(1) R/3

(2) 3R/2

(3) R/2

(4) 2R

6. A circuit to verify Ohm’s law uses ammeter and voltmeter in series or parallel connected correctly to the resistor. In the circuit:

(1) Ammeter is always connected in series and voltmeter in parallel.

(2) Both, ammeter and voltmeter must be connected in series.

(3) Both ammeter and voltmeter must be connected in parallel.

(4) The ammeter is always used in parallel and voltmeter in series.

7. A square loop of side 2a and carrying current I is kept in the xz plane with its centre at the origin. A long wire carrying the same current I is placed parallel to the z-axis and passing through the point (0, b, 0), (b >> a). The magnitude of the torque on the loop about z-axis will be:

(1)

(2)

(3)

(4)

8. In a dilute gas at pressure P and temperature T, the mean time between successive collisions of a molecule varies with T as:

(1) √T

(2) 1/T

(3) T

(4) 1/√T

9. When a particle of mass m is attached to a vertical spring of spring constant k and released, its motion is described by y (t) = y_{0}sin^{2}ωt, where ‘y’ is measured from the lower end of unstretched spring. Then ω is:

(1)

(2)

(3)

(4)

10. The linear mass density of a thin rod AB of length L varies from A to B as where x is the distance from A. If M is the mass of the rod then its moment of inertia about an axis passing through A and perpendicular to the rod is:

(1)

(2)

(3)

(4)

11. A fluid is flowing through a horizontal pipe of varying cross-section, with speed v ms^{–1} at a point where the pressure is P pascal. At another point where pressure is P/2 Pascal its speed is V ms^{–1}. If the density of the fluid is kg m^{–3} and the flow is streamlined, then V is equal to:

(1)

(2)

(3)

(4)

12. Three rods of identical cross-section and lengths are made of three different materials of thermal conductivity K_{1}, K_{2} and K_{3}, respectively. They are joined together at their ends to make a long rod (see figure). One end of the long rod is maintained at 100°C and the other at 0°C (see figure). If the joints of the rod are at 70°C and 20°C in steady-state and there is no loss of energy from the surface of the rod, the correct relationship between K_{1}, K_{2} and K_{3} are:

(1) K_{1} : K_{2} = 5 : 2, K_{1} : K_{3} = 3 : 5

(2) K_{1} < K_{2} < K_{3}

(3) K_{1} : K_{3} = 2 : 3, K_{2} : K_{3} = 2 : 5

(4) K_{1} > K_{2} > K_{3}

13. Assuming the nitrogen molecule is moving with r.m.s. velocity at 400 K, the de-Broglie wavelength of nitrogen molecule is close to (Given: nitrogen molecule weight: 4.64 × 10^{–26} kg, Boltzman constant: 1.38 × 10^{–23} J/K, Planck constant: 6.63 × 10^{–34}s)

(1) 0.44 Å

(2) 0.34 Å

(3) 0.20 Å

(4) 0.24 Å

14. Consider the force F on a charge ‘q’ due to a uniformly charged spherical shell of radius R carrying charge Q distributed uniformly over it. Which one of the following statements is true for F, if ‘q’ is placed at distance r from the centre of the shell?

(1)

(2)

(3)

(4)

15. Two identical electric point dipoles have dipole moments are held on the x-axis at distance ‘a’ from each other. When released, they move along the x-axis with the direction of their dipole moments remaining unchanged. If the mass of each dipole is ‘m’, their speed when they are infinitely far apart is:

(1)

(2)

(3)

(4)

16. Two planets have masses M and 16 M and their radii are a and 2a, respectively. The separation between the centres of the planets is 10a. A body of mass m is fired from the surface of the larger planet towards the smaller planet along the line joining their centres. For the body to be able to reach at the surface of the smaller planet, the minimum firing speed needed is:

(1)

(2)

(3)

(4)

17. In the figure shown, the current in the 10 V battery is close to:

(1) 0.21 A from positive to the negative terminal

(2) 0.36 A from negative to the positive terminal

(3) 0.42 A from positive to the negative terminal

(4) 0.71 A from positive to the negative terminal

18. A student measuring the diameter of a pencil of circular cross-section with the help of a vernier scale records the following four readings 5.50 mm, 5.55 mm, 5.45 mm, 5.65 mm. The average of these four readings is 5.5375 mm and the standard deviation of the data is 0.07395 mm. The average diameter of the pencil should therefore be recorded as:

(1) (5.54 ± 0.07) mm

(2) (5.5375 ± 0.0740) mm

(3) (5.5375 ± 0.0739) mm

(4) (5.538 ± 0.074) mm

19. Given the masses of various atomic particles m_{p} = 1.0072 u, m_{n} = 1.0087 u, me = 0.000548 u, m_{v}[bar] = 0, m_{d} = 2.0141 u, where p ≡ proton, n ≡ neutron, e ≡ electron, v [bar] ≡ antineutrino and d ≡ deuteron. Which of the following processes is allowed by momentum and energy conservation?

(1) n + n deuterium atom (electron bound to the nucleus)

(2) e^{+} + e^{–} → γ

(3) p → n + e^{+} + v [bar]

(4) n + p → d + γ

20. A particle moving in the xy plane experiences a velocity-dependent force, where v_{x} and v_{y} are the x and y components of its velocity v. If a is the acceleration of the particle, then which of the following statements is true for the particle?

(1) The kinetic energy of the particle is constant in time.

(2) quantity v × a is constant in time

(3) quantity v . a is constant in time

(4) F arises due to a magnetic field

21. A Young’s double-slit experiment is performed using monochromatic light of wavelength λ. The intensity of light at a point on the screen, where the path difference is λ, is K units. The intensity of light at a point where the path difference is λ / 6 is given by nK/12, where n is an integer. The value of n is __________.

22. The centre of mass of the solid hemisphere of radius 8 cm is x from the centre of the flat surface. Then the value of x is __________.

23. The output characteristics of a transistor is shown in the figure. When V_{CE} is 10V and I_{C} = 4.0 mA, then the value of β_{ac} is __________.

24. An engine operates by taking a monatomic ideal gas through the cycle shown in the figure. The percentage efficiency of the engine is close to __________.

25. In a series LR circuit, power of 400W is dissipated from a source of 250 V, 50 Hz. The power factor of the circuit is 0.8. In order to bring the power factor to unity, a capacitor of value C is added in series to the L and R. Taking the value of C as (n/3π) μF, then the value of n is __________.

**Chemistry**

1. Match the following :

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

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

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

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

2. The IUPAC name of the following compound is:

(1) 2-nitro-4-hydroxymethyl-5-amino benzaldehyde

(2) 3-amino-4-hydroxymethyl-5-nitro benzaldehyde

(3) 4-amino-2-formyl-5-hydroxymehtyl nitrobenzene

(4) 5-amino-4-hydroxymethyl-2-nitro benzaldehyde

3. For the given cell;

Cu(s)|Cu^{2+}(C_{1}M)||Cu^{2+}(C_{2}M)|Cu(s)

Change in Gibbs energy (ΔG) is negative, if :

(1) C_{2} = √2 C_{1}

(2) C_{2} = C_{1}/√2

(3) C_{1} = 2C_{2}

(4) C_{2} = C_{1}

4. Reaction of an inorganic sulphite X with dilute H_{2}SO_{4} generates compound Y. Reaction of Y with NaOH gives X. Further, the reaction of X with Y and water affords compound Z. Y and Z, respectively, are :

(1) SO_{2} and NaHSO_{3}

(2) S and Na_{2}SO_{3}

(3) SO_{2} and Na_{2}SO_{3}

(4) SO_{3} and NaHSO_{3}

5. The value of KC is 64 at 800 K for the reaction N_{2}(g) + 3H_{2}(g) ⇌ 2NH_{3}(g)

The value of K_{C} for the following reaction is:

(1) 1/4

(2) 8

(3) 1/8

(4) 1/64

6. The correct match between Item – I (Starting material) and Item – II (reagent) for the preparation of benzaldehyde is :

7. For a d_{4} metal ion in an octahedral field, the correct electronic configuration is :

(1) e^{2}_{g} t^{2}_{2g} when ∆_{0} < P

(2) t^{4}_{2g} e^{0}_{g} when ∆_{0} < P

(3) t^{3}_{2g} e^{1}_{g} when ∆_{0} > P

(4) e^{1}_{g} t^{3}_{2g} when ∆_{0} < P

8. The correct match between Item – I and Item – II is :

Item-I Item-II

(a) Natural rubber (I) 1, 3-butadiene + styrene

(b) Neoprene (II) 1, 3-butadiene + acrylonitrile

(c) Buna-N (III) Chloroprene

(d) Buna-S (IV) Isoprene

(1) (a) – (III), (b) – (IV), (c) – (I), (d) – (II)

(2) (a) – (IV), (b) – (III), (c) – (II), (d) – (I)

(3) (a) – (IV), (b) – (III), (c) – (I), (d) – (II)

(4) (a) – (III), (b) – (IV), (c) – (II), (d) – (I)

9. Which one of the following statement is not true?

(1) Lactose contains α-glycosidic linkage between C_{1} of galactose and C_{4} of glucose.

(2) Lactose is a reducing sugar and it gives Fehling’s test.

(3) On acid hydrolysis, lactose gives one molecule of D(+)-glucose and one molecule of D(+)-galactose.

(4) Lactose(C_{11}H_{22}O_{11}) is a disaccharide and it contains 8 hydroxyl groups.

10. The element that can be refined by distillation is :

(1) tin

(2) gallium

(3) zinc

(4) nickel

11. Match the following compounds (Column-I) with their uses (Column-II) :

Column-I Column-II

(I) Ca(OH)_{2} (A) Casts of statues

(II) NaCl (B) White wash

(III) (C) Antacid

(IV) CaCO_{3} (D) Washing soda preparation

(1) (I)-(B),(II)-(C),(III)-(D),(IV)-(A)

(2) (I)-(C),(II)-(D),(III)-(B),(IV)-(A)

(3) (I)-(B),(II)-(D),(III)-(A),(IV)-(C)

(4) (I)-(D),(II)-(A),(III)-(C),(IV)-(B)

12. Mischmetal is an alloy consisting mainly of :

(1) lanthanoid and actinoid metals

(2) lanthanoid metals

(3) actinoid metals

(4) actinoid and transition metals

13. For a reaction, 4M(s) + n O_{2}(g) → 2 M_{2} On (s) the free energy change is plotted as a function of temperature. The temperature below which the oxide is stable could be inferred from the plot as the point at which :

(1) the free energy change shows a change from negative to positive value.

(2) the slope changes from positive to zero

(3) the slope changes from positive to negative.

(4) the slope changes from negative to positive.

14. The increasing order of the boiling points of the major products A,B and C of the following reaction will be :

(1) A < B < C

(2) C < A < B

(3) A < C < B

(4) B < C < A

15. The average molar mass of chlorine is 35.5g mol^{−}^{1}. The ratio of ^{35}Cl to ^{37}Cl in naturally occurring chlorine is close to :

(1) 1 : 1

(2) 3 : 1

(3) 2 : 1

(4) 4 : 1

16. Which of the following compound can be prepared in good yield by Gabriel phthalimide synthesis?

17. The reaction of NO with N_{2}O_{4} at 250 K gives:

(1) N_{2}O

(2) NO_{2}

(3) N_{2}O_{5}

(4) N_{2}O_{3}

18. A set of solution is prepared using 180 g of water as a solvent and 10g of different nonvolatile solutes A, B and C. The relative lowering of vapour pressure in the presence of these solutes are in the order [Given, molar mass of A = 100 g mol^{−}^{1} ; B = 200g mol^{−}^{1} ; C = 10,000g mol^{−}^{1}]

(1) A > C > B

(2) B > C > A

(3) C > B > A

(4) A > B > C

19. Dihydrogen of high purity (> 99.95%) is obtained through :

(1) the electrolysis of acidified water using Pt electrodes.

(2) the reaction of Zn with dilute HCl

(3) the electrolysis of brine solution

(4) the electrolysis of warm Ba(OH)_{2}

20. A crystal is made up of metal iron ‘M_{1}‘ and ‘M_{2}‘ and oxide ions. Oxide ions form a ccp lattice structure. The cation ‘M_{1}‘ occupies 50% of octahedral voids and the cation ‘M_{2}‘ occupies 12.5% of tetrahedral voids of oxide lattice. The oxidation number of ‘M_{1}‘ and ‘M_{2}‘ are, respectively:

(1) +2, +4

(2) +3, +1

(3) +4, +2

(4) +1, +3

21. For Freundlich adsorption isotherm, a plot of log (x/m) (y-axis) and log p (x-axis) gives a straight line. The intercept and slope for the line is 0.4771 and 2, respectively. The mass of gas, adsorbed per gram of adsorbent if the initial pressure is 0.04 atm, is…………….×10^{−}^{4} (log 3=0.4771)

22. The atomic number of Unnilunium is …………….

23. A solution of phenol in chloroform when treated with aqueous NaOH gives compound P as a major product. The mass percentage of carbon in P is ……………. (to the nearest integer)

(Atomic mass : C = 12; H = 1; O = 16)

24. The rate of a reaction decreased by 3.555 times when the temperature was changed from 40°C to 30oC. The activation energy (in KJ mol^{−}^{1}) of the reaction is……… [Take; R = 8.314 J mol^{−}^{1} K^{−}^{1} In 3.555 = 1.268]

25. If the solubility product of AB^{2} is 3.20 × 10^{−}^{11} M^{3}, then the solubility of AB^{2} in pure water is ……….× 10^{−}^{4} mol L^{−}^{1} [Assuming that neither kind of ion reacts with water].

**Mathematics**

1. If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfies:

(1) e^{4} + 2e^{2} – 1 = 0

(2) e^{2} + 2e – 1 = 0

(3) e^{4} + e^{2} – 1 = 0

(4) e^{2} + e – 1 =0

2. The set of all real values of λ for which the function f(x) = (1− cos^{2} x) (λ + sin x), has exactly one maxima and exactly one minima, is:

(1) (−3/2, 3/2)

(2) (−1/2, 1/2)

(3) (−3/2, 3/2)

(4) (−1/2, 1/2)

3. The probabilities of three events A, B and C are given by P(A) = 0.6, P(B) = 0.4 and P(C) = 0.5. If P (A ∪ B) = 0.8, P (A ∩ C) = 0.3, P (A ∩ B ∩ C) = 0.2, P (B ∩ C) = β and P(A ∪ B ∪C) = α , where 0.85 ≤ α ≤ 0.95 , then β lies in the interval:

(1) [0.36,0.40]

(2) [0.25,0.35]

(3) [0.35,0.36]

(4) [0.20,0.25]

4. The common difference of the A.P. b_{1}, b_{2},….. bm is 2 more than the common difference of A.P. a_{1}, a_{2}, …an. If a_{40} = −159, a100 = −399 and b_{100} = a_{70}, then b_{1} is equal to:

(1) −17

(2) 81

(3) 127

(4) −81

5. The integral equals:

(1) e(4e – 1)

(2) e(4e + 1)

(3) 4e^{2} – 1

(4) e(2e – 1)

6. If the tangent to the curve, y = f(x) = xlog_{e}x, (x > 0) at a point (c, f(c)) is parallel to the line-segment joining the points (1,0) and (e, e), then c is equal to:

(1)

(2)

(3)

(4)

7. If is the solution of the differential equation, then the function p(x) is equal to:

(1) cosec x

(2) cot x

(3) tan x

(4) sec x

8. If α and β are the roots of the equation 2x(2x + 1) = 1, then β is equal to:

(1) 2α(α – 1)

(2) −2α(α + 1)

(3) 2α^{2}

(4) 2α(α + 1)

9. For all twice differentiable functions f: R→ R, with f(0) = f(1) = f´(0) = 0,

(1) f”(x) = 0, at every point x∈(0,1)

(2) f”(x) ≠ 0, at every point x∈(0,1)

(3) f”(x) = 0, for some x∈(0,1)

(4) f”(0) = 0

10. The area (in sq. units) of the region enclosed by the curves y = x^{2}−1 and y = 1−x^{2} is equal to:

(1) 4/3

(2) 7/2

(3) 16/3

(4) 8/3

11. For a suitably chosen real constant a, let a function, f:R−{−a}→R be defined by Further suppose that for any real number x ≠ −a and f(x) ≠ −a, (fof) (x) = x. Then f(−1/2) is equal to:

(1) −3

(2) 3

(3) 1/3

(4) −1/3

12. Let If B = A + A^{4}, then det (B):

(1) is one

(2) lies in (1, 2)

(3) lies in (2, 3)

(4) is zero

13. The centre of the circle passing through the point (0,1) and touching the parabola y = x2 at the point (2, 4) is :

(1) (3/10, 16/5)

(2) (6/5, 53/10)

(3) (−16/5, 53/10)

(4) (−53/10, 16/5)

14. A plane P meets the coordinate axes at A, B and C respectively. The centroid of a triangle ABC is given to be (1, 1, 2). Then the equation of the line through this centroid and perpendicular to the plane P is:

(1)

(2)

(3)

(4)

15. Let f : R→ R be a function defined by f(x) = max {x, x^{2}}. Let S denote the set of all points in R, where f is not differentiable. Then

(1) {0, 1}

(2) an empty set

(3) {1}

(4) {0}

16. The angle of elevation of the summit of a mountain from a point on the ground is 450. After climbing up one km towards the summit at an inclination of 30° from the ground, the angle of elevation of the summit is found to be 60°. Then the height (in km) of the summit from the ground is:

(1)

(2)

(3)

(4)

17. If the constant term in the binomial expansion of is 405, then |k| equals:

(1) 1

(2) 9

(3) 2

(4) 3

18. Let z = x + iy be a non-zero complex number such that z^{2} = i|z|^{2}, where i = √−1 , then z lies on the

(1) line, y = x

(2) real axis

(3) imaginary axis

(4) line, y = −x

19. Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point (−1, −4) in this line is:

(1) (11/5, 28/5)

(2) (8/5, 29/5)

(3) (29/5, 11/5)

(4) (29/5, 8/5)

20. Consider the statement : “For an integer n, if n^{3} − 1 is even, then n is odd.” The contrapositive statement of this statement is:

(1) For an integer n, if n is even, then n^{3}−1 is even

(2) For an integer n, if n is odd, then n^{3}−1 is even

(3) For an integer n, if n^{3}−1 is not even, then n is not odd.

(4) For an integer n, if n is even, then n^{3}−1 is odd

21. The number of words (with or without meaning) that can be formed from all the letters of the word “LETTER” in which vowels never come together is:

22. If be two non-zero vectors such that is perpendicular to then the value of λ is________

23. Consider the data on x taking the values 0, 2, 4, 8, …..,2^{n} with frequencies ^{n}C_{0}, ^{n}C_{1}, ^{n}C_{2}… ^{n}C_{n}, respectively. If the mean of this data is 728/2^{n}, then n is equal to:________

24. Suppose that function f : R→R satisfies f(x+y) = f(x) f(y) for all x, y ∈ R and f(1) = 3. If then n is equal to ………..

25. The sum of distinct values of λ for which the system of equations

(λ − 1)x + (3λ + 1)y + 2λz = 0

(λ − 1) x + (4λ − 2)y + (λ + 3)z = 0

2x + (3λ + 1)y + 3(λ − 1)z = 0 has non-zero solutions, is: