Physics Section-A (Multiple Choice Questions)

1.Two shorts dipoles (A, B), A having charges ± 2µC and length 1 cm and having charges B ± 4µC and length 1 cm are placed with their centres 80 cm apart as shown in the figure. The electric field at a point P, equidistant from the centres of both dipoles is ____________ N/C.

2. To compare EMF of two cells using potentiometer the balancing lengths obtained are 200 cm and 150 cm. The least count of scale is 1 cm. The percentage error in the ratio of EMFs is

3. Which of the following pair of nuclei are isobars of the element?

4. Suppose a long solenoid of 100 cm length, radius 2 cm having 500 turns per unit length, carries a current I = 10sin(ωt) A, where ω = 1000 rad/s. A circular conducting loop (B) of radius 1 cm coaxially slided through the solenoid at a speed v = 1 cm/s. The r.m.s. current through the loop when the coil B is inserted 10 cm inside the solenoid is α√2 µA. The value of α is ___________ (Resistance of the loop = 10 Ω)

5. For the given logic gate circuit, which of the following is the correct truth table?

6. The ratio of speeds of electromagnetic waves in vacuum and a medium, having dielectric constant k = 3 and permeability of μ = 2μ_o, is (μ_ο = Permeability of Vacuum)

7. A small metallic sphere of diameter 2 mm and density 10.5 g/cm³ is dropped in glycerine having viscosity 10 Poise and density 1.5 g/cm³ respectively. The terminal velocity attained by the sphere is______ cm/s.

(π= 22/7  and g = 10m/s²)

8. A prism of angle 75° and refractive index √3 is coated with thin film of refractive index 1.5 only at the back exit surface. To have total internal reflection at the back exit surface the incident angle must be (sin15° = 0.25 and sin25° = 0.43).

9. The internal energy of a monoatomic gas is 3nRT. One mole of helium is kept in a cylinder having internal cross section area of 17 cm² and fitted with a light movable frictionless piston. The gas is heated slowly by suppling 126 J heat. If the temperature rises by 4°C, then the piston will move ______ cm. (atmospheric pressure = 10^5 Pa)

10. A body of mass 14 kg initially at rest explodes and breaks into three fragments of masses in the ratio 2:2:3. The two pieces of equal masses fly off perpendicular to each other with a speed of 18 m/s each. The velocity of the heavier fragment is __________ m/s.

11. A bead P sliding on a frictionless semi-circular string (ACB) and it is at point S at t = 0 and at this instant the horizontal component of its velocity is v. Another bead Q of the same mass as P is ejected from point A at t = 0 along the horizontal string AB, with the speed v, friction between the beads and the respective strings may be neglected in both cases. Let t_p and t_Q be the respective times taken by beads P and Q to reach the point B, then the relation between t_p and t_Q is

12. The current passing through a conducting loop in the form of equilateral triangle of side 4√3 cm is 2 A. The magnetic field at its centroid is α × 10^-5 T. The value of α is _______. (Given: μ_ο = 4π × 10^-7 SI units)

13. Two charges 7µC and -2µC are placed at (-9,0,0) cm and (9, 0, 0) cm respectively in an external field E= A /r^2 r^, where A = 9 × 10^5 N/Cm². Considering the potential at infinity is 0, the electrostatic energy of the configuration is ____________J.

14. When an unpolarized light falls at a particular angle on a glass plate (placed in air), it is observed that the reflected beam is linearly polarized. The angle of refracted beam with respect to the normal is_____. (tan^-1 (1.52) = 57.7°, refractive indices of air and glass are 1.00 and 1.52, respectively.)

15. One mole of an ideal diatomic gas expands from volume V to 2 V isothermally at a temperature 27°C and does W joule of work. If the gas undergoes same magnitude of expansion adiabatically from 27°C doing the same amount of work W, then its final temperature will be (close to) ___°C. (log_e2 = 0.693)

16. A circular loop of radius 7 cm is placed in uniform magnetic field of 0.2 T directed perpendicular to plane of loop. The loop is converted into a square loop in 0.5 s. The EMF induced in the loop is _____ mV.

17. An air bubble of volume 2.9 cm³ rises from the bottom of a swimming pool of 5 m deep. At the bottom of the pool water temperature is 17°C. The volume of the bubble when it reaches the surface, where the water temperature 27°C, is _____cm³. (g = 10 m/s², density of water = 10^3 kg/m³ and 1 atm pressure is 10^5 Pa)

18. A paratrooper jumps from an aeroplane and opens a parachute after 2 s of free fall and starts deaccelerating with 3 m/s². At 10 m height from ground, while descending with the help of parachute, the speed of paratrooper is 5 m/s. The initial height of the airplane is _____ m. (g = 10 m/s²)

19. A parallel plate capacitor with plate separation 5 mm is charged by a battery. On introducing a mica sheet of 2 mm and maintaining the connections of the plates with the terminals of the battery, it is found that it draws 25% more charge from the battery. The dielectric constant of mica is ______.

20. A block is sliding down on an inclined plane of slope θ and at an instant t = 0 this block is given an upward momentum so that it starts moving up on the inclined surface with velocity u. The distance (S) travelled by the block before its velocity become zero, is _________. (g = gravitational acceleration)

SECTION-B (NUMERICAL VALUE TYPE QUESTIONS)

21. The velocity of sound in air is doubled when the temperature is raised from 0°C to α°C. The value of α is _______.

22. The average energy released per fission for the nucleus of 235^_92U is 190 MeV. When all the atoms of 47 g pure 235^_92U undergo fission process, the energy released is α × 10^23 MeV. The value of α is________. (Avogadro Number = 6 × 10^23 per mole)

23. The size of the images of an object, formed by a thin lens are equal when the object is placed at two different positions 8 cm and 24 cm from the lens. The focal length of the lens is ______cm.

24. Suppose there is a uniform circular disc of mass Mkg and radius r m shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis A of the disc is given by x/256 Mr². The value of x is ______ .

25. A ball of radius r and density ρ dropped through a viscous liquid of density σ and viscosity η attains its terminal velocity at time t, given by t = Αρ^ar^bη^cσ^d where A is a constant and a, b, c and d are integers. The  value of b+c/a+d _________.

CHEMISTRY (SECTION-A MULTIPLE CHOICE QUESTIONS)

1. It is noticed that Pb^2+ is more stable than Pb^4+, but Sn^2+ is less stable than Sn^4+. Observe the following reactions.

PbO_2 + Pb → 2PbO; Δ_r G° (1)

SnO_2 + Sn→ 2SnO; Δ_r G° (2)

Identify the correct set from the following

2. Given below are two statements:

Statement I: The second ionisation enthalpy of Na is larger than the corresponding ionisation enthalpy of Mg. Statement II: The ionic radius of O^2- is larger than that of F^-.

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

3.

Consider the above electrochemical cell where a metal electrode (M) is undergoing redox reaction by forming M^+ (M → M^+ +e^-). The cation M^+ is present in two different concentrations c_₁ and c_₂ as shown above. Which of the following statement is correct for generating a positive cell potential?

5.

Given above is the concentration vs time plot for a dissociation reaction: A→nB.

Based on the data of the initial phase of the reaction (initial 10 min), the value of n is_______.

6. Iodoform test can differentiate between

A. Methanol and Ethanol

B. CH_3COOH and CH_3CH_2COOH

C. Cyclohexene and cyclohexanone

D. Diethyl ether and Pentan-3-one E. Anisole and acetone

Choose the correct answer from the options given below:

7. Given below are two statements:

in the order (i) Acidic KMnO_4, (ii) Ammonia, (iii) Bromine and alkali.

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

8. The work functions of two metals (M_A and M_B) are in the 1: 2 ratio. When these metals are exposed to photons of energy 6 eV, the kinetic energy of liberated electrons of M_A: M_B is in the ratio of 2.642: 1. The work functions (in eV) of M_A and M_B are respectively.

9. Which statements are not true about XeO_2F_2?

A. It has a see-saw shape.

B. Xe has 5 electron pairs in its valence shell in XeO_2F_2.

C. The O-Xe - O bond angle is close to 180°.

D. The F-Xe - F bond angle is close to 180°.

E. Xe has 16 valence electrons in XeO_2F_2.

Choose the correct answer from the options given below:

10. Both human DNA and RNA are chiral molecules. The chirality in DNA and RNA arises due to the presence of

11. The oxidation state of chromium in the final product formed in the reaction between KI and acidified K_2Cr_2O_7 solution is

12. A student has been given a compound 'x' of molecular formula-C_6H_7N. 'x' is sparingly soluble in water. However, on addition of dilute mineral acid, 'x'becomes soluble in water. 'x' when treated with CHCl_3 and KOH (alc.), 'y' is produced. 'y' has a specific unpleasant smell. On treatment with benzenesulphonyl chloride, 'x' gives a compound 'z' which is soluble in alkali. The number of different H atoms present in 'z' is

13. Given below are two statements:

Statement I : (CH_3)_3^+C is more stable than ^+CH_3 as nine hyperconjugation interactions are possible in (CH_3)_3^+C.

Statement II: ^+CH_3 is less stable than (CH_3)_3^+C as only three hyperconjugation interactions are possible in ^+CH_3.

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

14. Which of the following statements are true about haloform reaction?

A. Sodium hypochlorite reacts with KI to give KOI.

B. KOI is a reducing agent.

D. Isopropyl alcohol will not give iodoform test.

E. Methanoic acid will give positive iodoform test.

Choose the correct answer from the options given below:

15. Identify the incorrect statements from the following:

A. Notation ^24_12Mg represents 24 protons and 12 neutrons.

B. Wavelength of radiation of frequency a 4.5 × 10^15 s^-1 is 6.7 × 10^-18 m.

C. One radiation has wavelength = λ_1(900 nm) and energy = E_₁. Other radiation has wavelength = λ_2(300 nm) and energy = E_2. E_₁: E_2 = 3:1.

D. Number of photons of light of wavelength 2000 pm that provides 1 J of energy is 1.006 × 10^16.

Choose the correct answer from the options given below:

16. Elements X and Y belong to Group 15. The difference between the electronegativity values of X and phosphorus is higher than that of the difference between phosphorus and Y. X and Y are respectively

17. Identify the correct set of details from the following:

A. [Co(NH_3)_6]^³+ : Inner orbital complex; d^2sp³ hybridized

B. [MnCl_6]^3-: Outer orbital complex; sp³d² hybridized

C. [CoP_6]^3-: Outer orbital complex; d²sp³ hybridized

D. [FeF_6]^3-: Outer orbital complex; sp³d² hybridized

E. [Ni(CN)_4]^²-: Inner orbital complex; sp³ hybridized

Choose the correct answer from the options given below:

18. In Carius method 0.2425 g of an organic compound gave 0.5253 g silver chloride. The percentage of chlorine in the organic compound is

19. A mixed ether (P), when heated with excess of hot concentrated hydrogen iodide produces two different alkyl iodides which when treated with aq. NaOH give compounds (Q) and (R). Both (Q) and (R) give yellow precipitate with NaOI. Identify the mixed ether (P).

20. Observe the following reactions at T (K).

I. A → products

II. 5Br^-_(aq) + BrO^-_3(aq) + 6H^+_(aq) → 3Br_2(aq) + 3H_2O_(l) Both the reactions are started at 10.00 am. The rates of these reactions at 10.10 am are same. The value of -Δ[Br^-]/Δt at 10.10 am is 2 × 10^-4 mol L^-1 min^-¹. The concentration of A at 10.10 am is 10^-2 mol L^-1.

What is the first order rate constant (in min^-1) of reaction I?

SECTION-B (NUMERICAL BASED TYPE QUESTIONS)

21. Total number of unpaired electrons present in the central metal atoms/ions of [Ni(CO)_4], [NiCl_4]^2-, [PtCl_₂(NH_3)_2], [Ni(CN)_4]^²- and [Pt(CN)_4]^2- is,  _______ .

22. 200 cc of x × 10^-3 M potassium dichromate is required to oxidise 750 cc of 0.6 M Mohr's salt solution in acidic medium. Here x = __________.

23. Two liquids A and B form an ideal solution. At 320 K, the vapour pressure of the solution, containing 3 mol of A and 1 mol of B is 500 mm Hg. At the same temperature, if 1 mol of A is further added to this solution, vapour pressure of the solution increases by 20 mm Hg. Vapour pressure (in mm Hg) of B in pure state is _______ (Nearest integer)

24. X_2(g) + Y_2(g) → 2Z_(g)

X_2(g) and Y_2(g) are added to a 1 L. flask and it is found that the system attains the above equilibrium at T (K) with the number of moles of X_2(g), Y_2(g) and Z_(g) being 3, 3 and 9 mol respectively (equilibrium moles). Under this condition of equilibrium, 10 mol of Z_(g) is added to the flask and the temperature is maintained at T(K). Then the number of moles of Z_(g) in the flask when the new equilibrium is established is _________. (Nearest integer)

25. Consider the following reaction of benzene.

In compound (Q), the percentage of oxygen is _______ %. (Nearest Integer)

Mathematics (Section-A Multiple Choice Questions)

1. Let π/2 < θ < π and cot θ  = - 1/2√2. Then the value of sin (15θ/2) (cos8θ+sin8θ)+ cos(15θ/2)(cos8θ-sin8θ) is equal to

2. Let a^→ = i^-2j^+3k^,b^→=2i^+j^-k^,c^→=λi^ +j^+k^ and v^→ =a^→× b^→. If v^→.c^→ = 11 and the length of the projection of b^→on c^→ is p, then 9p² is equal to

3. Let A(1, 2) and C(-3, -6) be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line 7x - y = 14. If B(α, β) and D(γ, δ) are the other two vertices, then |α + β + γ + δ| is equal to

4. The least value of (cos²θ - 6sinθ cosθ + 3sin^2θ + 2) is

5. Let PQ be a chord of the hyperbola x²/4 - y^2/b^2 =1, perpendicular to the x-axis such that OPQ is an equilateral triangle, O being the centre of the hyperbola. If the eccentricity of the hyperbola is √3, then the area of the triangle OPQ is

6. Let A = {0, 1, 2, ..., 9}. Let R be a relation on A defined by (x, y) ∈ R if and only if |x - y| is a multiple of 3.

Given below are two statements:

Statement I: n(R) =36

Statement II: R is an equivalence relation. In the light of the above statements,

choose the correct orrect answer from the options given below

7. Let a^→,b^→,c^→ be three vectors such that a^→ × b^→ = 2(a^→× b^→). If |a^→|=1, |b^→|=4, |c^→|= 2 and the angle between b^→ and c^→ is 60°, then |a^→.c^→ | is equal to

8. Consider two sets A = {x ∈ z:}|(|x-3|-3)| ≤ 1} and B= {x ∈ R-{1,2}: (x-2)(x-4)/x-1} log_e (|x-2|) = 0}. Then the number of onto function f: A →B is equal to

9. If the points of intersection of the ellipses x² + 2y² - 6x  - 12y + 23 = 0 and 4x² + 2y² - 20x - 12y + 35 = 0 lie on a circle of radius r and centre (a, b), then the value of ab + 18r² is

10. The system of linear equations

x+y+z=6

x + 2y+3z = b

2x + 5y + az = 36 has

11. Let

If (1/2) = a √2/b+c, where a,b,c,∈ N, gcd (a,b) =1 then a + b + c is equal to

12. An equilateral triangle OAB is inscribed in the parabola y^2 = 4x with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is

13.

14.

Continuous at x = 0, then a + b is equal to

15. Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A. Then a ball is randomly drawn from the bag A. If the probability, that the ball drawn is white, is P/q, gcd(p,q)=1, then p + q is equal to

16. The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is

17. If the mean and the variance of the data

are  μ and 19 respectively, then the value of λ+ μ  is

18. The area of the region enclosed between the circles x²+ y² = 4 and x² + (y-2)² = 4 is

19. The sum of all the real solutions of the equation log_(x+3)(6x² + 28x + 30) = 5 - 2log_(6x+10)(x² + 6x + 9) is equal to

20. If z = √3/2 + i/2,i = √-1, then (z^201-i)^8 is equal to

SECTION-B (NUMERICAL VALUE TYPE QUESTIONS)

21. If the image of the point P(a, 2, a) in the line x/2 = y + a/1 = z/ 1 is Q and the image of Q in the line x-2b/2 = y-a/1 = z+2b/-5 is P, then a + b is equal to ________.

22. The number of elements in the set

23. Let S denote the set of 4-digit numbers abcd such that a > b > c > d and P denote the set of 5-digit numbers having product of its digits equal to 20. Then n(S) + n(P) is equal to _____.

25. If the solution curve y = f(x) of the differential equation (x²-4)y′-2xy+2x(4-x^2)² =0, x>2 , passes through the point (3, 15), then the local maximum value of f is ________