**Physics**

**Section-A**

1. A mosquito is moving with a velocity and accelerating in uniform conditions. What will be the direction of the mosquito after 2s?

(a)

(b)

(c)

(d)

2. Statement I: A cyclist is moving on an unbanked road with a speed of 7 kmh^{−}^{1} and takes a sharp circular turn along a path of radius of 2m without reducing the speed. The static friction coefficient is 0.2. The cyclist will not slip and pass the curve.

(g = 9.8 m/s^{2})

Statement-II: If the road is banked at an angle of 45°, cyclists can cross the curve of 2m radius with the speed of 18.5 kmh^{−}^{1} without slipping.

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

(a) Both statement I and statement II are false

(b) Both statement I and statement II are true

(c) Statement I is correct and statement II is incorrect

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

3. Calculate the time interval between 33% decay and 67%decay if half-life of a substance is 20 minutes:

(a) 40 minutes

(b) 20 minutes

(c) 60 minutes

(d) 13 minutes

4. A large block of wood of mass M = 5.99 kg is hanging from two long massless cords. A bullet of mass m = 10 g is fired into the block and gets embedded in it. The (block + bullet) then swing upwards, their centre of mass rising a vertical distance h = 9.8 cm before the (block + bullet) pendulum comes momentarily to rest at the end of its arc. The speed of the bullet just before the collision is: (take g = 9.8 ms^{−}^{2})

(a) 831.4 m/s

(b) 841.4 m/s

(c) 811.4 m/s

(d) 821.4 m/s

5. What will be the nature of the flow of water from a circular tap, when its flow rate increased from 0.18 L/min to 0.48 L/min? The radius of the tap and viscosity of water is 0.5 cm and 10^{−}^{3} Pa s, respectively. (Density of water: 10^{3} kg/m^{3})

(a) Remains steady flow

(b) Unsteady to steady flow

(c) Steady flow to unsteady flow

(d) Remains turbulent flow

6. The refractive index of a converging lens is 1.4. What will be the focal length of this lens if it is placed in a medium of the same refractive index? Assume the radii of curvature of the faces of the lens are R_{1} and R_{2}

(a) Zero

(b)

(c) Infinite

(d) 1

7. For the given circuit, comment on the type of transformer used.

(a) Step down transformer

(b) Auxiliary transformer

(c) Step-up transformer

(d) Autotransformer

8. Red light differs from blue light as they have:

(a) Same frequencies and same wavelengths

(b) Different frequencies and different wavelengths

(c) Different frequencies and same wavelengths

(d) Same frequencies and different wavelengths

9. The de-Broglie wavelength associated with an electron and a proton were calculated by accelerating them through the same potential of 100 V. What should nearly be the ratio of their wavelengths? (mp = 1.00727u; me = 0.00055u)

(a) (1860)2 : 1

(b) 43 : 1

(c) 1860 : 1

(d) 41.1 : 1

10. A charge Q is moving distance in the magnetic field Find the value of work done by

(a) Infinite

(b) 1

(c) −1

(d) Zero

11. Amplitude of a mass spring system, which is executing simple harmonic motion decreases with time. If mass = 500g, Decay constant = 20 g/s then how much time is required for the amplitude of the system to drop to half of its initial value?

(ln 2 = 0.693)

(a) 15.01 s

(b) 17.32 s

(c) 0.034 s

(d) 34.65 s

12. A resistor develops 500 J of thermal energy in 20s when a current of 1.5A is passed through it. If the current is increased from 1.5A to 3A, what will be the energy developed in the 20s.

(a) 500 J

(b) 1000 J

(c) 2000 J

(d) 1500 J

13. Calculate the value of mean free path (λ) for oxygen molecules at temperature 27°C and pressure 1.01 × 10^{5} Assume the molecular diameter 0.3nm and the gas is ideal. (k = 1.38 × 10^{−}^{23} jK^{−}^{1})

(a) 102 nm

(b) 32 nm

(c) 58 nm

(d) 86 nm

14. A bimetallic strip consists of metals A and B. It is mounted rigidly as shown. The metal A has higher coefficient of expansion compared to that of metal B. When the bimetallic strip is placed in a cold bath, it will:

(a) Not bend but shrink

(b) Neither bend nor shrink

(c) Bend towards the right

(d) Bend towards the left

15. The following logic gate is equivalent to:

(a) NOR Gate

(b) AND Gate

(c) OR Gate

(d) NAND Gate

16. Two identical antennas mounted on identical towers are separated from each other by a distance of 45 km. What should nearly be the minimum height of the receiving antenna to receive the signals in line of sight? (Assume radius of earth is 6400 km)

(a) 19.77 m

(b) 79.1 m

(c) 158.2 m

(d) 39.55 m

17. The magnetic field in a region is given by A square loop of side d is placed with its edges along the x and y axes. The loop is moved with a constant velocity The emf induced in the loop is:

(a)

(b)

(c)

(d)

18. The half-life of Au198 is 2.7 days. The activity of 1.50 mg of Au198 if its atomic weight is 198 g mol^{−}^{1} is (NA = 6 × 10^{23}/mol).

(a) 252 Ci

(b) 357 Ci

(c) 240 Ci

(d) 535 Ci

19. In order to determine the Young’s Modulus of a wire of radius 0.2 cm (measured using a scale of least count = 0.001 cm) and length 1m (measured using a scale of least count = 1 mm), a weight of mass 1 kg (measured using a scale of least count = 1 g) was hanged to get the elongation of 0.5 cm (measured using a scale of least count 0.001 cm). What will be the fractional error in the value of Young’s Modulus determined by this experiment.

(a) 9%

(b) 1.4%

(c) 0.9%

(d) 0.14%

20. Find out the surface charge density at the intersection of point x = 3 m plane and x-axis in the region of uniform line charge of 8 nC/m lying along the z-axis in free space.

(a) 47.88 C/m

(b) 0.07 nC m^{−}^{2}

(c) 0.424 nC m^{−}^{2}

(d) 4.0 nC m^{−}^{2}

**Section-B**

21. A closed organ pipe of length L and an open organ pipe contains gases of densities ρ_{1} and ρ_{2} The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with the same frequency. The length of the open pipe is where x is ______(Round off to the Nearest Integer)

22. A deviation of 2° is produced in the yellow ray when the prism of crown and flint glass are chromatically combined. Taking dispersive powers of crown and flint glass as 0.02 and 0.03 respectively and refractive index for yellow light for these glasses are 1.5 and 1.6 respectively. The refracting angles for crown glass prism will be _______ ° (in degree). (Round off to the Nearest Integer)

23. If one wants to remove all the mass of the earth to infinity in order to break it up completely. The amount of energy that needs to be supplied will be where x is __________. (Round off to the Nearest Integer). (M is the mass of earth, R is the radius of earth, G is the gravitational constant)

24. In a parallel plate capacitor set up, the plate area of the capacitor is 2 m^{2} and the plates are separated by 1 m. If the space between the plates is filled with a dielectric material of thickness 0.5 m and area 2 m^{2} (see fig) the capacitance of the set-up will be ________∈_{0}. (Dielectric constant of the material = 3.2) (Round off to the Nearest Integer)

25. For an ideal heat engine, the temperature of the source is 127ºC. In order to have 60% efficiency the temperature of the sink should be _________ºC. (Round off to the Nearest Integer)

26. A force is applied on an intersection point of x = 2 plane and x-axis. The magnitude of torque of this force about a point (2, 3, 4) is _______. (Round off to the Nearest Integer)

27. The energy dissipated by a resistor is 10 ml in 1 s when an electric current of 2mA flows through it. The resistance is ________. (Round off to the Nearest Integer)

28. A swimmer can swim with a velocity of 12 km/h in still water. Water flowing in a river has a velocity of 6 km/h. The direction with respect to the direction of flow of river water he should swim in order to reach the point on the other bank just opposite to his starting point is ______°. (Round off to the Nearest Integer) (Find the angles in degrees)

29. A solid disc of radius ‘a’ and mass ‘m’ rolls down without slipping on an inclined plane making an angle θ with the horizontal. The acceleration of the disc will be where b is _______ . (Round off to the Nearest Integer). (g = acceleration due to gravity; θ = angles as shown in figure)

30. A body of mass 2 kg moves under a force of It starts from rest and was at the origin initially. After 4s, its new coordinates are (8, b, 20). The value of b is _______. (Round off to the Nearest Integer)

**Chemistry**

**Section-A**

1. Identify the reagent(s) ‘A’ and condition(s) for the reaction:

(a) A = HCl; Anhydrous AlCl_{3}

(b) A = HCl, ZnCl_{2}

(c) A = Cl_{2}, dark, Anhydrous AlCl_{3}

(d) A = Cl_{2}; UV Light

2. The INCORRECT statement regarding the structure of C_{60} is:

(a) It contains 12 six-membered rings and 24 five-membered rings.

(b) Each carbon atom forms three sigma bonds.

(c) The five-membered rings are fused only to six-membered rings.

(d) The six-membered rings are fused to both six and five-membered rings.

3. Match List-I with List-II:

List-I
Test/Reagents/Observation(s) |
List-II
Species detected |

(a) Lassaigne’s Test | (i) Carbon |

(b) Cu(II) oxide | (ii) Sulphur |

(c) Silver nitrate | (iii) N, S, P and halogen |

(d) The sodium fusion extract gives black precipitate with acetic acid & lead acetate | (iv) Halogen Specifically |

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

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

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

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

4. The structure of X is:

5. Ammonolysis of alkyl halides followed by the treatment with NaOH solution can be used to prepare primary, secondary and tertiary amines. The purpose of NaOH in the reaction is:

(a) to remove basic impurities

(b) to activate NH_{3} used in the reaction

(c) to increase the reactivity of alkyl halide

(d) to remove acidic impurities

6. Arrange the following metal complex/compounds in the increasing order of spin only magnetic moment. Presume all three, high spin systems.

(Atomic numbers Ce = 58, Gd = 64 and Eu = 63)

(1) (NH_{4})_{2}[Ce(NO_{3})_{6}]

(2) Gd(NO_{3})_{3}

(3) Eu(NO_{3})_{3}

(a) (1) < (3) < (2)

(b) (1) < (2) < (3)

(c) (3) < (1) < (2)

(d) (2) < (1) < (3)

7. Identify the elements X and Y using the ionisation energy values given below:

1^{st} 2^{nd}

X 195 4563

Y 731 1450

(a) X = F; Y = Mg

(b) X = Mg; Y = F

(c) X = Na; Y = Mg

(d) X = Mg; Y = Na

8. The INCORRECT statement below regarding colloidal solutions is:

(a) A colloidal solution shows colligative properties

(b) An ordinary filter paper can stop the flow of colloidal particles

(c) A colloidal solution shows the Brownian motion of colloidal particles

(d) The flocculating power of Al^{3+} is more than that of Na^{+}

9. The characteristics of elements X, Y and Z with atomic numbers, respectively, 33, 53 and 83 are:

(a) X and Z are non-metals and Y is a metalloid.

(b) X and Y are metalloids and Z is a metal.

(c) X, Y and Z are metals.

(d) X is a metalloid, Y is a non-metal and Z is a metal.

10. The exact volumes of 1 M NaOH solution required to neutralise 50 mL of 1 M H_{3}PO_{3} solution and 100 mL of 2 M H_{3}PO_{2} solution, respectively, are:

(a) 100 mL and 50 mL

(b) 50 mL and 50 mL

(c) 100 mL and 100 mL

(d) 100 mL and 200 mL

11. Which of the following reduction reaction CANNOT be carried out with coke?

(a) Fe_{2}O_{3} → Fe

(b) ZnO → Zn

(c) Al_{2}O_{3} → Al

(d) Cu_{2}O → Cu

12. An unsaturated hydrocarbon X on ozonolysis gives A. Compound A when warmed with ammoniacal silver nitrate forms a bright silver mirror along the sides of the test tube. The unsaturated hydrocarbon X is:

13. Two statements are given below:

Statement-I: Sodium hydride can be used as an oxidising agent.

Statement-II: The lone pair of electrons on nitrogen in pyridine makes it basic.

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

(b) Both statement I and statement II are false

(c) Both statement I and statement II are true

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

14. Which of the following polymer is used in the manufacture of wood laminates?

(a) Melamine formaldehyde resin

(b) Cis-poly isoprene

(c) Phenol and formaldehyde resin

(d) Urea-formaldehyde resin

15. The correct statements about H_{2}O_{2} are:

(1) used in the treatment of effluents.

(2) used as both oxidizing and reducing agents.

(3) the two hydroxyl groups lie in the same plane.

(4) miscible with water.

Choose the correct answer from the options given below:

(a) (1), (3) and (4) only

(b) (1), (2) and (4) only

(c) (1), (2), (3) and (4)

(d) (2), (3) and (4) only

16. The greenhouse gas/es is (are)

(1) Carbon dioxide

(2) Oxygen

(3) Water vapour

(4) Methane

Choose the most appropriate answer from the options given below:

(a) (1) and (2) only

(b) (1), (3) and (4) only

(c) (1) and (3) only

(d) (1) only

17. In the below reaction, the reagent “A” is:

(a) NaBH_{4}, H_{3}O^{+}

(b) HCl, Zn-Hg

(c) Alkaline KMnO_{4}, H^{+}

(d) LiAlH_{4}

18. Which of the following is least basic?

(a)

(b)

(c)

(d)

19. Fex_{2} and Fey_{3} are known when x and y are:

(a) x=Cl, Br, I and y=F, Cl, Br, I

(b) x=F, Cl, Br, I and y=F, Cl, Br

(c) x=F, Cl, Br, I and y=F, Cl, Br, I

(d) x=F, Cl, Br and y =F, Cl, Br, I

20. The secondary structure of protein is stabilized by:

(a) van der Waals forces

(b) Peptide bond

(c) Hydrogen bonding

(d) Glycosidic bond

**Section-B**

21. At 25°C, 50 g of iron reacts with HCl to form FeCl_{2}. The evolved hydrogen gas expands against a constant pressure of 1 bar. The work done by the gas during this expansion is ________________ J. (Round off to the nearest integer).

[Given: R = 8.14 J mol^{–1} K^{–1}. Assume, hydrogen is an ideal gas]

[Atomic mass of Fe is 55.85 u]

22. A 5.0 m moldm^{–3} aqueous solution of KCl has a conductance of 0.55 mS when measured in a cell of cell constant 1.3 cm^{–1}. The molar conductivity of this solution is ___________ mSm^{2}mol^{1}. (Round off to the nearest integer).

23. The number of orbitals with n = 5, m_{1} = +2 is ___________. (Round off to the nearest integer).

24. A and B decompose via first-order kinetics with half-lives 54.0 min and 18.0 min respectively. Starting from an equimolar non-reactive mixture of A and B, the time taken for the concentration of A to become 16 times that of B is __________ min. (Round off to the nearest integer).

25. [Ti(H_{2}O)_{6}]^{3+} absorbs light of wavelength 498 nm during a d–d transition. The octahedral splitting energy for the above complex is ___________ × 10^{–19} (Round off to the nearest integer). h = 6.626 × 10^{–34} Js; c = 3 × 10^{8} ms^{–1}.

26. Sulphurous acid (H_{2}SO_{3}) has K_{a1} = 1.7 × 10^{–2} and K_{a2} = 6.4 × 10^{–8}. The pH of 0.588 M H_{2}SO_{3} is _________. (Round off to the nearest integer)

27. In Duma’s method of estimation of nitrogen, 0.1840 g of an organic compound gave 30 mL of nitrogen collected at 287 K and 758 mm of Hg pressure. The percentage composition of nitrogen in the compound is __________. (Round off to the nearest integer). [Given: Aqueous tension at 287 K = 14 mm of Hg]

28. Ga (atomic mass 70 u) crystallizes in a hexagonal close packed structure. The total number of voids in 0.581 g of Ga is ________ × 10^{21}. (Round off to the nearest integer). [Given: NA = 6.023 × 10^{23}]

29. When 35 mL of 0.15 M lead nitrate solution is mixed with 20 mL of 0.12 M chromic sulphate solution, _________ × 10^{–5} moles of lead sulphate precipitate out. (Round off to the nearest integer).

30. At 363 K, the vapour pressure of A is 21 kPa and that of B is 18 kPa. One mole of A and 2 moles of B are mixed. Assuming that this solution is ideal, the vapour pressure of the mixture is _______ kPa. (Round off to the nearest integer).

**Mathematics**

**Section-A**

1. The least value of |z| where z is a complex number which satisfies the inequality is equal to:

(a) 2

(b) 3

(c) 8

(d) √5

2. Let f : S → S where S = (0, ∞) be a twice differentiable function such that f (x + 1) = x f (x). If g : S → R be defined as g (x) = log_{e} f(x), then the value of |g”(5) – g”(1)| is equal to:

(a) 197/144

(b) 187/144

(c) 205/144

(d) 1

3. If y = y (x) is the solution of the differential equation with y(0) = 0, then equal to:

(a) log_{e} 2

(b)

(c)

(d)

4. If the foot of the perpendicular from point (4, 3, 8) on the line ℓ ≠ 0 is (3, 5, 7) then the shortest distance between the line L_{1} and line is equal to:

(a)

(b) 1/√3

(c) 1/2

(d) 1/√6

5. If (x, y, z) be an arbitrary point lying on a plane P which passes through the points (42, 0, 0), (0, 42, 0) and (0, 0, 42), then the value of the expression is equal to:

(a) 3

(b) 0

(c) 39

(d) −45

6. Consider the integral where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to:

(a) 45(e – 1)

(b) 45(e + 1)

(c) 9(e – 1)

(d) 9(e + 1)

7. Let A (–1, 1), B (3, 4) and C(2, 0) be given three points. A line y = mx, m > 0, intersects lines AC and BC at point P and Q respectively. Let A_{1} and A_{2} be the areas of ΔABC and ΔPQC respectively, such that A_{1} = 3A_{2}, then the value of m is equal to:

(a) 4/15

(b) 1

(c) 2

(d) 3

8. Let f be a real-valued function, defined on R – {–1, 1} and given by Then in which of the following intervals, function f (x) is increasing?

(a) (–∞,–1) ⋃ ([1/2, ∞) – {1})

(b) (–1, 1/2]

(c) (–∞, ∞) – {–1, 1}

(d) (–∞, 1/2] – {–1}

9. Let the lengths of intercepts on x-axis and y-axis made by the circle x^{2} + y^{2} + ax + 2ay + c = 0, (a < 0) be 2 √2 and 2 √5, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line x + 2y = 0, is equal to:

(a) √10

(b) √6

(c) √11

(d) √7

10. Let A denote the event that a 6-digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions, be divisible by 3. Then the probability of event A is equal to:

(a) 4/9

(b) 9/56

(c) 3/7

(d) 11/27

11. Let α ∈ R be such that the function is continuous at x = 0, where {x} = x – [x], [x] is the greatest integer less than or equal to x. Then:

(a) α = π/4

(b) No such α exists

(c) α = 0

(d) α = π/√2

12. The maximum value of , x ∈ R is:

(a) √7

(b) √5

(c) 5

(d) 3/4

13. Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let α be the number of triangles having these points from different sides as vertices and β be the number of quadrilaterals having these points from different sides as vertices. Then (β – α) is equal to:

(a) 1890

(b) 795

(c) 717

(d) 1173

14. Let C be the locus of the mirror image of a point on the parabola y^{2} = 4x with respect to the line y = x. Then the equation of tangent to C at P(2, 1) is:

(a) 2x + y = 5

(b) x + 2y = 4

(c) x + 3y = 5

(d) x – y = 1

15. Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy is:

(a) 1

(b) 2

(c) 3

(d) 0

16. Let C_{1} be the curve obtained by the solution of the differential equation Let the curve C_{2} be the solution of If both the curves pass through (1, 1), then the area enclosed by the curves C_{1} and C_{2} is equal to :

(a)

(b)

(c) π − 1

(d) π + 1

17. Let If and then the value of is equal to:

(a) 11

(b) 15

(c) 9

(d) 13

18. Let P(x) = x^{2} + bx + c be a quadratic polynomial with real coefficients such that and P(x) leaves remainder 5 when it is divided by (x – 2). Then the value of 9(b + c) is equal to :

(a) 7

(b) 11

(c) 15

(d) 9

19. If the points of intersections of the ellipse and the circle x^{2} + y^{2} = 4b, b > 4 lie on the curve y^{2} = 3x^{2}, then b is equal to :

(a) 5

(b) 6

(c) 12

(d) 10

20. Let A = {2, 3, 4, 5, ……, 30} and ‘≅’ be an equivalence relation on A×A, defined by (a, b) ⩭ (c, d), if and only if ad = bc. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4, 3) is equal to :

(a) 7

(b) 5

(c) 6

(d) 8

**Section-B**

21. Let be a vector perpendicular to the vectors and If then the value of is equal to_____

22. In ΔABC, the lengths of sides AC and AB are 12 cm and 5 cm, respectively. If the area of ΔABC is 30 cm^{2} and R and r are respectively the radii of circumcircle and incircle of ΔABC, then the value of 2R + r (in cm) is equal to ______

23. Consider the statistics of two sets of observations as follows:

Size | Mean | |

Observation I | 10 | 2 |

Observation II | N | 3 |

If the variance of the combined set of these two observations is 17/9, then the value of n is equal to ______

24. Let up to n-terms, where a >1. If S_{24}(x) = 1093 and S_{12}(2x) = 265, then value of a is equal to______

25. Let n be a positive integer. Let If then n is equal to ________

26. Let f : R → R and g : R → R be defined as where a, b are non-negative real numbers. If (gof) (x) is continuous for all x ∈ R, then a + b is equal to ______

27. If the distance of the point (1, – 2, 3) from the plane x + 2y – 3z + 10 = 0 measured parallel to the line, then the value of |m| is equal to________

28. Let 1/16, a and b be in G.P. and 1/a, 1/b, 6 be in A.P., where a, b, > 0. Then 72 (a + b) is equal to ______

29. Let be two 2 × 1 matrices with real entries such that A = XB, where and k ∈ If and then the value of k is ________

30. For real numbers α, β, γ and δ, if

where C is an arbitrary constant, then the value of 10 (α + βγ + δ) is equal to______.

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