1.  Given is a thin convex lens of glass (refractive index μ) and each side having radius of curvature R. One side is polished for complete reflection. At what distance from the lens, an object be placed on the optic axis so that the image gets formed on the object itself?  (b) R 2μ-1 (c) R 2μ-3 (d) R μ

2. A bob of mass m is suspended at a point O by a light string of length l and left to perform vertical motion (circular) as shown in figure. Initially, by applying horizontal velocity vo  at the point 'A', the string becomes slack when, the bob reaches at the point 'D. The ratio of the kinetic energy of the bob at the points B and C is

3. Given below are two statements: Statement-I: The equivalent emf of two non-ideal batteries connected in parallel is smaller than either of the two emfs. Statement-II: The equivalent internal resistance of two nonideal batteries connected in parallel is smaller than the internal resistance of either of the two batteries. In the light of the above statements, choose the correct answer from the options given below.

4. An electron is made to enter symmetrically between two parallel and equally but oppositely charged metal plates, each of 10 cm length. The electron emerges out of the electric field region with a horizontal component of velocity 10^6 m/s. If the magnitude of the electric field between the plates is 9.1 V/cm, then the vertical component of velocity of electron is (mass of electron = 9.1 x 10 ^-31 kg and charge of electron = 1.6 x 10^-19 C)

5. The work functions of cesium (Cs) and lithium (Li) metals are 1.9 eV and 2.5 eV, respectively. If we incident a light of wavelength 550 nm on these two metal surfaces, then photo-electric effect is possible for the case of

6. An electron in the ground state of the hydrogen atom has the orbital radius of 5.3 x 10^-11 m while that for the electron in third excited state is 8.48 x 10^-10 m. The ratio of the de Broglie wavelengths of electron in the ground state to that in the excited state is

7. A parallel-plate capacitor of capacitance 40 µF is connected to a 100 V power supply. Now the intermediate space between the plates is filled with a dielectric material of dielectric constant K = 2. Due to the introduction of dielectric material, the extra charge and the change in the electrostatic energy in the capacitor, respectively, are

8. A uniform circular disc of radius 'R' and mass 'M' is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of radius R/2 is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above.

9. Which of the following circuits represents a forward biased diode?

10. Two spherical bodies of same materials having radii 0.2 m and 0.8 m are placed in same atmosphere. The temperature of the smaller body is 800 K and temperature of the bigger body is 400 Κ. If the energy radiated from the smaller body is E, the energy radiated from the bigger body is (assume, effect of the surrounding temperature to be negligible)

11. An amount of ice of mass 10^-3 kg and temperature -10°C is transformed to vapour of temperature 110°C by applying heat. The total amount of work required for this conversion is,(Take, specific heat of ice = 2100 J kg^-1 K^-1, specific heat of steam = 1920 J kg^-1 K^-1, Latent heat of ice = 3.35 ×10^5 J kg ^-1 and Latent heat of steam = 2.25 × 10^6 J kg ^-1)

SECTION -B (NUMERICAL ABILITY) Q.NO- 12 to15

12. Two soap bubbles of radius 2 cm and 4 cm. respectively. are in contact with each other. The radius of curvature of the common surface, in cm, is _____________.

13. Three conductors of same length having thermal conductivity k1, k₂ and k3 are connected as shown in figure. Area of cross sections of 1st and 2nd conductor are same and for 3rd conductor it is double of the 1st conductor The temperature are given the figure. In steady state condition, the value of  ∅ is _________ ºC. 100°C

(Given k₁= 60 J s^-1 m^-1  K^-1, k2 = 120 Js^-1 m^-1 K^-1, k3=135 Js^-1 m^-1 K^-1)

14. The driver sitting inside a parked car is watching vehicles approaching from behind with the help of his side view mirror, which is a convex mirror with radius of curvature R = 2 m. Another car approaches him from behind with a uniform speed of 90 km/hr. When the car is at a distance of 24 m from him, the magnitude of the acceleration of the image of the car in the side view mirror is 'a'. The value of 100 a is _________________ m/s².

15. A particle is projected at an angle of 30° from horizontal at a speed of 60 m/s. The height traversed by the particle in the first second is ho, and height traversed in the last second, before it reaches the maximum height, is h1. The ratio h1:h2 is_________________ (Take, g = 10 m/s²)

Chemistry

Section-A(Multiple choice questions)

1. Match List -I with List II

Choose the correct answer from the options given below:

2. Which of the following electrolyte can be used to obtain H2S2O3 by the process of electrolysis?

3. Given below are two statements:

Statement I: One mole of propyne reacts with excess of sodium to liberate half a mole of H₂ gas. Statement II: Four g of propyne reacts with NaNH, to liberate NH3 gas which occupies 224 mL at STP. In the light of the above statements, choose the most appropriate answer from the options given below :

4. A vessel at 1000 K contains CO₂ with a pressure of 0.5 atm. Some of CO2 is converted into CO on addition of graphite. If total pressure at equilibrium is 0.8 atm, then K, is

5. The products formed in the following reaction sequance are

6. A liquid when kept inside a thermally insulated closed vessel at 25 °C was mechanically stirred from outside. What will be the correct option for the following thermodynamic parameters?

7. Which of the following electronegativity order is incorrect?

8. Arrange the following increasing boiling points. solutions in order of their

(i) 10^-4 M NaCl             (ii) 106-4 M Urea   (iii) 10^-3 M NaCl (iv) 10^-2 M NaCl

9. The IUPAC name of the following compound is

10. From the magnetic behaviour of [NiCl4]^2- (paramagnetic) and [Ni(CO)4] (diamagnetic), choose the correct geometry and oxidation state.

11. Which of the following statement is not true for radioactive decay?

12. Lanthanoid ions with 4f^7 configuration are:

(A) Eu^2+    (B) Gd^³+         (C) Eu^3+     (D) Tb^³+   (E) Sm^²+

Choose the correct answer from the options given below:

13. . How many different stereoisomers are possible for the given molecule?

14. In which of the following complexes the CFSE Δ0 will be equal to zero?

15. Which of the following acids is a vitamin?

MATHEMATICS - SECTION - A (MUTIPLE CHOICE QUESTIONS)

1. Let x = x(y) be the solution of the differential equation y^2dx+(x-1/y)dy=0. If x(1) = 1, then x (1/2) is:

2. Let A = {1, 2, 3,...., 10} and B ={m/n: m, n∈ A, m

3. The area of the region, inside the circle (x-2√3)² + y²=12 and outside the parabola y² = 2√3x is:

4. Let a1, a2, a3,... be a G.P. of increasing positive terms. If a1 a5 = 28 and a₂ + a4 = 29, then a6 is equal to:

5. Let the foci of a hyperbola be (1, 14) and (1, -12). If it passes through the point (1, 6) then the length of its latus-rectum is:

6. From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is :

7. The number of non-empty equivalence relations on the set {1, 2, 3} is:

8. Let f(x) be a real differentiable function such that f(0) = 1 and f(x + y) = f(x)f'(y)+f'(x)f(y) for all x, y ∈R. 100 Then Eloge f(n) is equal to : n=1

9. Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of 16((sec^-1 x)² + (cosec^¯¹ x)²) is:

10. A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point (2,5) and intersects the circle C at exactly two points. If the set of all possible values of r is the interval (α,β)  then 3β-2α is equal to :

11. Let the triangle PQR be the image of the triangle with vertices (1, 3), (3, 1) and (2, 4) in the line x + 2y = 2. If the centroid of ΔPQR is the point (a, ẞ), then 15(α - β) is equal to:

12. Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected balls is black, given that the second selected ball is also black is m/n, where gcd(m, n) = 1, then m + n is equal to:

13. A coin is tossed three times. Let X denote the number of times a tail follows a head. If µ and σ² denote the mean and variance of X, then the value of 64 (μ + σ²) is :

14. Let the parabola y = x² + px - 3, meet eet the coordinate axes at the points P, Q and R. If the circle C with centre at (-1,-1) passes through the points P, Q and R, then the area of ΔPQR is:

15. Let f: R→R be a twice differentiable function such that f(x + y) = f(x)f(y) for all x, y∈ R If f' (0) = 4a and f satisfies f" (x) = 3a f'(x)-f(x) = 0, a > 0, then the area of the region R = {(x, y) | 0≤ y ≤ f(ax), 0 ≤ x ≤2} is: