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# IISER

## IISER Aptitude Test Free Model Paper 2 (English) by IISER

**Biology**

1.** **Offspring formed by sexual reproduction exhibit more variation than those formed by asexual reproduction because

A. each gamete has unique genetic composition.

B. sexual reproduction is a lengthy process.

C. genetic material comes from parents of two different species.

D. greater amount of DNA is involved in sexual reproduction.

2. In the human body which element is most abundant by weight and by number respectively?

A. carbon, hydrogen.

B. oxygen, hydrogen.

C. oxygen, carbon.

D. carbon, nitrogen.

3. Approximately how many cells of staphylococci will be able to fit in the volume of a human red blood cell?

A. 10.

B. 1000.

C. 10000.

D. 50000.

4. Although intracellular bacterial infection can be treated by giving an antibiotic that blocks protein synthesis, it doesn’t affect human cells. Why?

A. Antibiotic molecules can’t enter human cells.

B. Antibiotic gets degraded by human cell.

C. Human ribosomes are different from bacterial ribosomes.

D. Human genetic code is different from bacterial genetic code.

5. Which of the following groups of animals you would expect to evolve chemical defences against predators?

A. slow moving with hard shell.

B. sedentary without a hard shell.

C. burrowing.

D. arboreal.

6. Which of the following techniques will be useful for tracing the origin of Onge tribe in Andamans?

A. blood grouping.

B. mitochondrial DNA analysis.

C. DNA fingerprinting.

D. karyotyping.

7. Under which of the following conditions is Semelparous reproduction (where organisms produces all its offspring in a single reproductive event) is most likely to be favoured?

A. Adult survival rate is low.

B. Adult survival rate is high.

C. Breeding is perennial.

D. Breeding is seasonal.

8. Which of the following characters is seen only in prokaryotes and not in eukaryotes?

A. Antibiotic production.

B. Unicellular life.

C. Reproduction by budding.

D. Nitrogen fixation.

9. A scientist wants to express human protein Y in bacteria. For effective expression of this protein he should use

A. promoter of human gene Y.

B. promoter of bacterial gene.

C. operator of any human gene.

D. operator of any bacterial gene.

10. During growth of an individual animal some components of the body grow in size but not in number (type 1) while some others increase in number but not in size (type 2). Which of the following is correct?

A. type 1: bones and muscle cells; type 2: hair follicles, red blood cells and epithelial cells.

B. type 1: bones and red blood cells; type 2: hair follicles, muscle cells and epithelial cells.

C. type 1: hair follicles and muscle cells; type 2: bones, red blood cells and epithelial cells.

D. type 1: epithelial cells and bones; type 2: hair follicles, red blood cells and muscle cells.

11. Small mammals are not found in polar region because

A. they have small surface to volume ratio.

B. they have large surface to volume ratio.

C. they cannot make burrows because of ice cover.

D. of scarcity of food.

12. Which of the following growth curves represents growth of bacteria in a culture medium that contains both glucose and lactose?

A.

B.

C.

D.

13. Most stable nucleic acid is

A. DNA.

B. mRNA.

C. rRNA.

D. tRNA.

14. A matured mRNA has 300 bases with a single stop codon. What would be the length of the polypeptide synthesized from this mRNA?

A. always 100 amino acids.

B. always 99 amino acids.

C. maximum of 99 amino acids.

D. maximum of 100 amino acids.

15. A diploid organism is heterozygous for four unlinked loci. How may types of gametes can be produced?

A. 8.

B. 16.

C. 32.

D. 128.

**Chemistry**

16. The correct stability order of −C≡N, −C≡As, and −C≡Sb bonds would be

A. −C≡N > −C≡As > −C≡Sb > −C≡P.

B. −C≡As > −C≡N > −C≡P > −C≡Sb.

C. −C≡N > −C≡P > −C≡As > −C≡Sb.

D. −C≡Sb > −C≡As > −C≡P > −C≡N.

17. As predicted by VSEPR theory, the molecular shapes of XeF_{2} and XeF_{4} are respectively

A. Bent and square planar.

B. Linear and tetrahedral.

C. Bent and tetrahedral.

D. Linear and square planar.

18. Which of the following statements holds true for Cu(I) and Cu(II) complexes?

A. Cu(II) complexes are diamagnetic but Cu(I) complexes are paramagnetic.

B. Both Cu(I) and Cu(II) complexes are paramagnetic.

C. Both Cu(I) and Cu(II) complexes are diamagnetic.

D. Cu(II) complexes are paramagnetic but Cu(I) complexes are diamagnetic.

19. What is the relationship between the two molecules shown below?

A. Enantiomers.

B. Diastereomers.

C. Geometrical isomers.

D. Both are identical molecules.

20. Which of the following are aromatic?

A. I, II, and IV.

B. I, III, and V.

C. I, III, and IV.

D. I, IV, and V.

21. Arrange the following molecules in increasing order of acidity.

A. IV < II < I < III.

B. IV < I < II < III.

C. III < IV < I < II.

D. I < II < IV < III.

22. What will be the final outcome of the following sequence of reactions?

A.

B.

C.

D.

23. Predict the final product in the following sequence of reactions?

A.

B.

C.

D.

24. For the He^{+} ion which of the following options is true?

A. Energy of 3s is less than 3p.

B. Energy of 3p is less than 3d.

C. Energies of 3s, 3p, and 3d are all the same.

D. Energy of 3s is same as 3p, but lower than 3d.

25. For a free expansion of an ideal gas in an isolated chamber, which of the following statements is true?

A. Entropy of the system increases.

B. Temperature of the system decreases.

C. Internal energy of the system decreases.

D. Positive work is done by the system.

26. When an aqueous solution was treated with AgNO_{3}, a white precipitate was obtained which was soluble in NH_{4} The aqueous solution contained

A. Sulfate.

B. Chloride.

C. Acetate.

D. Carbonate.

27. A scientist measured the cell length of a cubic crystalline substance to be 3.0 × 10^{−8} . The substance was also found to have a density of 11 g/cc and an atomic mass of 60 u. The number of atoms per unit cell based on the data given above is:

A. 4.

B. 3.

C. 2.

D. 1.

28. The van der Waals coefficient a (expressed in atm ∙ dm^{6} ∙ mol^{−}^{2}) for four different gases are: He 0.0341; H_{2}242; Kr 5.125; O_{2} 1.364. Based on the data given above, the gas that will be expected to have the lowest critical temperature T_{c}:

A. He.

B. H_{2}.

C. Kr.

D. O_{2}.

29. 1 mL of 10^{−5} M HCl was diluted to 1 L by adding water. The pH of the resultant solution is

A. 8.

B. 6.9.

C. 5.

D. 7.1.

30. A, B, and C are in equilibrium as shown in the diagram. Which of the following relations among the rate constants is true?

A. k_{1}k_{2}k_{−3} = k_{3}k_{−1}k_{−2}.

B. k_{1}k_{2}k_{3} = k_{−3}k_{−1}k_{−2}.

C. k_{1}k_{−2}k_{3} = k_{−3}k_{−1}k_{2}.

D. k_{−1}k_{2}k_{3} = k_{−3}k_{1}k_{−2}.

**Mathematics**

31. The equations of two lines are x + 2y = 1 and 2x – y = 3.

A. These lines are parallel.

B. These lines are perpendicular.

C. These lines are at an angle of 60°.

D. These lines are at an angle of 30°.

32. Suppose the matrix with a and b integers has determinant 1. A possible value for b is

A. 1.

B. 2.

C. 3.

D. 9.

33. There are five students. Arul and Neeraj are Mathematics students. Sunny likes Biology. Mathematics students dislike Chemistry. Students who like Biology cannot dislike Chemistry. Students who want to take Physics must like all other subjects. Swati and Manas want to take Physics. The number of students who dislike Chemistry is

A. 1.

B. 2.

C. 3.

D. 4.

34. Let a_{1}, a_{2} ∙∙∙, a_{10} be 10 observations with median 3. Then

A. at least 5 observations are less than or equal to 3.

B. at most 5 observations are less than or equal to 3.

C. exactly 5 observations are less than or equal to 3.

D. all observations are less than or equal to 3.

35. The value of the integral is

A. positive.

B. negative.

C. 0.

D. ∞.

36. The sum of the infinite series

A. ∞.

B.

C. 1.

D.

37. The angle between the tangents to the circles x^{2} + y^{2} = 1 and x^{2} + (y – 1)^{2} = 2 at the point (1, 0) is

A. 90°.

B. 30°.

C. 30°.

D. 45°.

38. The length of the shortest path in space between the z-axis and the line given by the equations z = 0 and y = 1 is

A. 1.

B. 2.

C. √2.

D. 1/2.

39. Consider the inequalities x + y < 3, x − y < 5, 5x − 3y < 15 and 2x + y > 2. The region in the plane that consists of points satisfying these inequalities is

A. empty.

B. rectangular.

C. triangular.

D. square.

40. The number of solutions of tan x = x – x^{3} with −1 ≤ x ≤ 1 is

A. 1.

B. 2.

C. 3.

D. 4.

41. Let f(x) = |x|^{a} where a is a non-zero real number. For what values of a is f(x) differentiable at x = 0?

A. For all non-zero a.

B. For al a > 1.

C. For no values of a.

D. For all a different from 1 and 0.

42. The set of solutions in complex numbers of the equation z^{4} = −1 is

A. the empty set.

B. {i, −i}

C.

D.

43. Let denotes the greatest integer less than or equal to x. Then

A. f is continuous for all x.

B. f is continuous only when x is an integer.

C. f is continuous only when x is not an integer.

D. f is not continuous at any value of x.

44. An unbiased usual six-sided die is thrown three times. The sum of the numbers coming up is 10. The probability that 2 has appeared at least once is

A.

B.

C.

D.

45. The number of solutions of a + b + c = 15 with non-negative integer values for a, b and c is

A.

B.

C.

D.

**Physics**

46. Dimension of Planck’s constant is equivalent to the dimension of which of the following quantities?

A. Force.

B. Energy.

C. Linear momentum.

D. Angular momentum.

47. The figure below shows the displacement vs time plot of a particle undergoing simple harmonic motion. The acceleration of the particle at time would be

A.

B.

C.

D.

48. Two infinitely long rods are arranged on a plane making an angle θ with each other, as shown in the figure. The rod B starts to move with a uniform velocity in a directions perpendicular to rod A. Thus, the point of intersection P moves with a horizontal velocity Which of the following statements is true?

A. vp increases with increasing θ.

B. vp decreases with increasing θ.

C. vp is independent of θ.

D. vp is independent of v.

49. The figure shows a block of mass M, attached to an uncompressed spring of spring constant k, resting on a frictionless surface. A bullet of mass m, travelling with a constant horizontal velocity v hits the block and gets embedded inside. What is the maximum compression of the spring?

A.

B.

C.

D.

50. An electron in a hydrogen atom falls from an orbit with principal quantum number n = 2 to n = 1. What is the wavelength (λ) of the emitted radiation? (where R denotes the Rydberg constant).

A.

B.

C.

D.

51. The figure below shows a cylindrical tank filled with water upto a height h. A hole is punctured on the side of the tank, at a distance x below the water level. Water exits the hole and hits a point P (at a distance r from the base of the tank). Which of the following is true

A. r is maximum when

B. r is maximum when

C. r is maximum when

D. r is maximum when

52. A tank with uniform cross-sectional area A is filled with a liquid of density ρ upto a height h. What is the potential energy stored in the system? (where g denotes the acceleration due to gravity)

A.

B. Agρh^{2}.

C.

D. Agρh.

53. A sphere of radius R carries a positive charge density (ρ) that increases linearly with radial distance r from the centre (ρ ∝ r). The radial dependence of the magnitude of electric field inside the sphere is given by

A. E ∝ r.

B. E ∝ r^{2}.

C. E ∝ r^{−}^{1}.

D. E ∝ r^{−2}.

54. A magnet falling vertically under gravity, passes through a metal ring on its way. Which of the following is true about the acceleration a of the magnet? (where g denotes the acceleration due to gravity)

A. a > g everywhere.

B. a > g before crossing the ring and a < g after crossing the ring.

C. a < g before crossing the ring and a > g after crossing the ring.

D. a < g everywhere.

55. A point charge Q, is placed at the center of a cube of side a, in vacuum (permitivity ϵ_{0}). The flux of the electric field through the shaded face is given by

A.

B.

C.

D.

56. Two isolated conducting spheres of radii 10 cm and 20 cm have charges Q_{A} and Q_{B}, respectively. What is the final charge on the first sphere if the spheres are brought in contact and separated subsequently?

A.

B.

C.

D. remains unchanged.

57. Heat (Q) is supplied to a solid to raise its temperature (T) across its melting point (T_{M}) and boiling point (T_{B}). Which of the following graphs correctly represents the relation between heat supplied and temperature?

A.

B.

C.

D.

58. For the circuit given below, what is the effective resistance between points a and b.

A. 3R/4.

B. 3R/5.

C. 4R/3.

D. 5R/3.

59. Consider the ray diagrams for a concave mirror (focal length f_{M}) and a convex lens (focal length f_{L}) in air, as shown in the figure below.

What happens to their focal lengths when they are immersed in a medium of refractive index higher than that of air?

A. f_{M} changes and f_{L} stays same.

B. f_{M} stays same and f_{L} changes.

C. both f_{M} and f_{L} changes.

D. both f_{M} and f_{L} stay same.

60. Consider a ray of light passing through a rectangular slab of refractive index μ(μ > 1) and thickness h as shown below.

This leads to a parallel shift d in the path of the ray, which varies between 0 to d_{max} as θ varies. How does d_{max} change with μ.

A. d_{max} increases with μ.

B. d_{max} decreases with μ.

C. d_{max} first increases then decreases with μ.

D. does not change with μ.

## IISER Aptitude Test Free Model Paper 1 (English) by IISER

**Biology**

1. If an interphase cell is treated with cyanide (a metabolic poison), the cell does not divide by mitosis. However, if cyanide is added right after mitosis has started, the same cell completes mitosis. Which of the following explains this observation?

A. Metabolic activity ceases during mitosis

B. Cell division does not require metabolic activity

C. Energy required for mitosis is produced and stored in the cell during interphase

D. Mitotic cells make factors that make them resistant to cyanide

2. The symplast pathway is most easily disrupted when

A. Water transport channels in the plasma membrane of the root hair cells malfunction

B. Water transport channels in the plasma membrane of the root cortex malfunction

C. Water transport channels in the plasma membrane of the root endodermis malfunction

D. Water transport channels in the plasma membrane of the guard cells malfunction

3. Paleontological studies use fossil pollen because

A. Pollen retains viability for long periods of time unlike male gametes in animals

B. The intine of pollen is very hard and stable and can be used in rescuing plant populations on a decline

C. The exine of pollen retains its structure for long periods of time

D. Soil pollen banks, unlike soil seed banks, stay dormant for long periods of time

4. Which of the following is not an assumption of the Hardy-Weinberg principle?

A. Mating is random in the population

B. There are no mutations

C. All individuals have an equal opportunity to survive and reproduce

D. Immigration and emigration occurs in the population

5. The lens of many vertebrate eyes is a crystallized form of a protein that also functions in digestion as a metabolic enzyme. This shows that

A. Vision and digestion co-evolved

B. Digestion necessarily evolved prior to vision since it is a more basic function

C. Evolution in opportunistic

D. Vision and digestion evolved around the same time

6. On which segment of the human chromosome is the enzyme Reverse Transcriptase located?

A. Centromere

B. Telomere

C. Kinetochore

D. Satellite

7. Muscle X and muscle Y are of the same size, but muscle X is capable of much finer control than muscle Y. Which of the following is likely to be true of muscle X?

A. It is controlled by more neurons than muscle Y

B. It contains fewer motor units than muscle Y

C. It is controlled by fewer neurons than muscle Y

D. Each of its motor units consists of more cells than the motor units of muscle Y

8. A National Park associated with rhinoceros is

A. Kaziranga

B. Corbett

C. Ranthambore

D. Valley of Flowers

9. During HIV infection

A. Number of helper T-lymphocytes increase

B. Number of helper T-lymphocytes decrease

C. Number of red blood cells increase

D. Number of red blood cells decrease

10. If the blood groups of the father and mother are AB and B respectively, then which one of the following statements is true with respect to their children’s blood group?

A. Blood group is either A or B

B. Blood group is either B or AB

C. Blood group is AB only

D. Blood group can be A or B or AB

11. The sequence of DNA is 5’-ATGGTTCCATC-3’. What is the sequence of the complimentary RNA strand?

A. 5’-TACCAAGGTAG-3’

B. 3’-TACCAAGGTAG-5’

C. 3’-UACCAAGGUAG-5’

D. 5’-UACCAAGGUAG-3’

12. Which one of the following statements is correct with respect to Biochemical Oxygen Demand (BOD)?

A. Secondary treatment of effluent decreases the BOD

B. Secondary treatment of effluent increases the BOD

C. Secondary treatment of effluent does not change the BOD

D. Secondary treatment of effluent first increases and then decreases the BOD

13. Which of the following is a general nature of plant-pollinator interactions?

A. Tight one to one co-evolutionary partnership

B. A plant species is pollinated by a few pollinator species

C. Plants rely on deceit to achieve pollination by pollinator species

D. Most pollinators benefit the plant by providing pollinator services, but disadvantage the plant at the same time by laying eggs into the flower and thereby negatively aects fruit formation

14. Sickle cell anemia is a disease resulting from altered haemoglobin structure. This alteration is because of the replacement of a glutamic acid with valine. Indentify the protein structure level where this change has been made

A. Primary

B. Secondary

C. Tertiary

D. Quaternary

15. Which of the following life history adaptations is least likely when predation pressure, on a fish species that grows in size continuously throughout its lifespan, is concentrated on the larger individuals

A. Allocate more resources preferentially to early reproduction than to growth

B. Allocate more resources preferentially to growth than to early reproduction

C. Sexual maturity at an early age

D. Produce more ospring in very few reproductive seasons

**Chemistry**

16. What is the potential of a cell containing two hydrogen electrodes, in which the anode is in contact with 10^{–?}5M HCl and the cathode is in contact with 1000 times the concentration of HCl as that of the anode?

A. 0:36 V.

B. 0:18 V.

C. ?0:36 V.

D. ?0:18 V.

17. Phosphorus pentoxide, P_{4}O_{10}, has each phosphorus linked to:

A. 5 oxygen atoms with P ? P bonds.

B. 5 oxygen atoms.

C. 4 oxygen atoms with P ? P bonds.

D. 4 oxygen atoms.

18. The radius of an atom of He is 0:05 nm. Assuming that one mole of a gas occupies 22:4 litres at STP, the fraction of the volume occupied by the atoms in a mole of He gas at STP is:

A. 1.4 × 10^{–?4}.

B. 1.4 × 10^{–5}.

C. 7.1 × 10^{–?4}.

D. 7.1 × 10^{–?5}.

19. The number of degenerate orbitals present in an energy level of a H-atom characterized by where R is the Rydberg constant is:

A. 16. B. 9. C. 4. D. 1.

20. Formation of ammonia in Haber’s process, N_{2} + 3H_{2} → 2NH_{3} (ΔH =–?ve) can be increased by:

A. increase in temperature and pressure.

B. increase in temperature.

C. increase in the concentration of ammonia.

D. increase in pressure.

21. Choose the correct ordering for the dipole moments of the following molecules:

A. CO_{2} ≤ BF_{3} < H_{2}O < H_{2}S.

B. BF_{3} < CO_{2} < H_{2}S < H_{2}O.

C. CO_{2} = BF_{3} < H_{2}S < H_{2}O.

D. CO_{2} < BF_{3} < H_{2}S < H_{2}O.

22. Which among the following complexes of Mn given below has the spin only magnetic moment (μ_{s}) value of 5:9 BM?

A. [Mn(CN)_{6}]^{4–}

B. [Mn(Br)_{4}]^{2?–}

C. [Mn(en)_{3}]^{2+}; en = ethylenediamine

D. Mn_{2}(CO)_{10}

23. Schottky as well as Frenkel defects are observed in:

A. NaCl.

B. ZnS.

C. AgBr.

D. KCl.

24. A black mineral A on heating in air gives a gas B. The mineral A on reaction with H_{2}SO_{4} gives a gas C and a compound D. Bubbling C into an aqueous solution of B gives white turbidity. The aqueous solution of compound D, on exposure to air, with NH_{4}SCN gives a red compound E.

The compounds A and E respectively, are:

A. PbS and Pb(SCN)_{2}.

B. NiS and Ni(SCN)_{2}.

C. FeS and Fe(SCN)_{3}.

D. CoS and Co(SCN)_{2}.

25. Using the diagram given below, the relation between k_{1} and k_{2} for the reaction A → C is:

A. k_{1} = k_{2}.

B. k_{2} <<< k_{1}.

C. k_{1} ≤ k_{2}.

D. k_{1} <<< k_{2}.

26. The structure of IV in the following sequence is:

A. 1.

B. 2.

C. 3.

D. 4.

27. Arrange the following chloroarenes in increasing order of their reactivity in nucleophilic substitution to form their corresponding phenols.

A. II < V < III ~ IV < I.

B. II < V < III < I < IV.

C. I ~ III < IV < V < II.

D. I < IV < III < V < II.

28. Which of the following methods is suitable for the preparation of 1; 3; 5-tribromobenzene from benzene?

A. (i) AlBr_{3}/Br_{2}, light (ii) separation of isomers.

B. (i) HNO_{3}/H_{2}SO_{4} (ii) Sn/HCl (iii) Br_{2} (iv) NaNO_{2}/HCl (v) C_{2}H_{5}OH, Δ.

C. (i) HNO_{3}/H_{2}SO_{4} (ii) NaBH_{4} (iii) Br_{2}/CH_{3}COOH (iv) NaNO_{2}/HCl (v) H_{3}PO_{2}.

D. (i) HNO_{3}/H_{2}SO_{4} (ii) H_{2}/Pd (iii) NaNO_{2}/HCl (iv) CuBr/HBr.

29. The order of reactivity of the following ketones towards nucleophilic addition of water is:

A. III < IV < V < I < II.

B. I < V < IV < III < II.

C. I < III < IV < V < II.

D. II < I < V < IV < III.

30. Which among the solutions given below will not show a change in pH on dilution? (I). 0.1MNH_{4}COOCH_{3}, (II). 0.1 M NaCl, (III). 0.1 M NH_{4}OH, (IV). 0.01 M H_{2}SO_{4}.

A. I and II.

B. I, II and IV.

C. I and III.

D. III and IV

**Mathematics**

31. If then I_{3} + I_{5} + I_{7} + I_{9} equals.

A.

B.

C.

D.

32. Which is the largest number in the following sequence?

A.

B.

C.

D. The sequence is unbounded.

33. Assuming that the interchange of limit and integration is permissible, the value of

is

A. 0.

B. 1/2

C. 1.

D. ∞.

34. Let *f* : ℝ → ℝ be a function such that |*f*(x) – *f*(y)| ≤ 6|x – y|^{2} for all x, y ϵ ℝ. If *f*(3) = 6 then *f*(6) equals :

A. 6.

B. 9.

C. 12.

D. 18.

35. Let An be the area bounded by the curves y = x and y = nx^{2} in the first quadrant. Then the value of is :

A. 110.

B. 220.

C. 330.

D. 440.

36. In how many ways can four distinguishable pieces be placed on an 8 × 8 chessboard so that no two pieces are in the same row or column?

A.

B.

C.

D.

37. A and B are playing a game by alternately rolling a die, with A starting first. Each player’s score is the number obtained on his last roll. The game ends when the sum of scores of the two players is 7, and the last player to roll the die wins. What is the probability that A wins the game?

A.

B.

C.

D.

38. The binomial coefficients 0 ≤ r ≤ n – 2 .

A. Can be in A.P. or in G.P.

B. Can be in A.P. but never in G.P.

C. Can be in G.P. but never in A.P.

D. Can never be in A.P. or G.P.

39. The sum of the infinite series cot^{−1} 2 + cot^{−1} 8 + cot^{−1} 18 + ∙ ∙ ∙ + cot^{−1}(2n^{2}) + ∙ ∙ ∙ is :

A. π/3.

B. π/4.

C. π/6.

D. π/8.

40. The number of integer values of k for which the equation 7 cos θ + 5 sin θ = 2k + 1 has real solutions is :

A. 6.

B. 8.

C. 10.

D. 12.

41. The complex solutions of (z + i)^{2011} = z^{2011} lie on:

A. A circle.

B. An ellipse.

C. A hyperbola.

D. A straight line.

42. How many 2 × 2 matrices A satisfy both A^{3} = I_{2} and A^{2} = A^{t}, where I_{2} denotes the 2 × 2 identity matrix and A^{t} denotes the transpose of A?

A. 0.

B. 1.

C. 2.

D. 3.

43. Let C be the circle that touches the X-axis and whose centre coincides with the circumcentre of the triangle defined by 4|x| + 3y = 12; y ≥ 0. How many points with both co-ordinates integers are there in the interior of C?

A. 0.

B. 1.

C. 2.

D. 3.

44. Let P and Q be the centres of the circles that pass through (0; 2) and (0; 8) and touch the X-axis. Then the equation of the ellipse with P and Q as foci and touching the X-axis is:

A.

B.

C.

D.

45. Let *f* : ℝ → ℝ be a function such that *f*(x + y) + *f*(x – y) = *f*(xy) for all x, y ∈ ℝ. Then *f* is :

A. Strictly increasing.

B. Strictly decreasing.

C. Identically zero.

D. Constant but not necessarily zero.

**Physics**

46. A physical quantity f depends on the dimensionful quantities x and y as follows:

f = Ax + Bexp(c y).

Which of the following do not have the same dimensions:

A. f and B

B. c and y^{−}^{1}

C. x and B/A

D. x and A

47. The period of oscillation of a simple pendulum of length L suspended from the roof of a rocket accelerating upwards with a constant acceleration (g) is given by:

A. ∞

B. 0

C.

D.

48. The moment of inertia of a uniform solid disc of mass M and radius R about an axis normal to the disk and passing through its center is What is the moment of inertia of the same disc about an axis lying in its plane and tangent to it (as shown in the figure)?

A.

B.

C.

D.

49. A pendulum is made of a rigid rod (mass m, length l) and a small bob of mass M attached at one end (as shown in the figure). The rod is pivoted on the other end. What should be the minimum speed of the bob at its lowest point so that the pendulum completes a full circle?

A.

B.

C.

D.

50. A proton of mass 1 a.m.u. collides with a Carbon-12 nucleus (mass = 12 a.m.u.) at rest. Assuming that the collision is perfectly elastic and that the Newton’s laws of motion hold, what fraction of the proton’s kinetic energy is transferred to the Carbon nucleus?

A.

B.

C.

D. 1

51. The magnitude of the gravitational force experienced by a small spaceship of mass M inside an inter-galactic dust cloud (assumed to be spherically symmetric but not necessarily uniform) when it at a distance of r from the center of the cloud is found to be

The density of the dust cloud is (G is Newton’s gravitational constant)

A.

B.

C.

D.

52. A body cools from 67℃ to 37℃. If this takes time t when the surrounding temperature is 27℃, what will be the time taken if the surrounding temperature is 7℃?

A. 2t

B. t/3

C. t/2

D. t/4

53. Three rods (lengths 2l, l, l) made of the same material and having the same area of cross-section are joined as shown in figure. The end points A, B and C are maintained at constant temperatures 100℃, 50℃ and 0℃, respectively. Assuming that there is no loss of heat from the surface of the rods, find the temperature that the junction P ultimately reaches.

A. 50℃

B. 40℃

C. 30℃

D. 20℃

54. An AC voltage source of frequency 50 Hz and amplitude v_{0} is turned on at time t = 0 . A second voltage source of the same frequency and amplitude is turned on at a later time t = 5 ms. For both sources, the voltage is found to increase immediately after being turned on. The instantaneous voltage of the two sources can be represented respectively by v_{1}(t) and v_{2}(t), where

A. v_{1}(t) = −v_{0} sin (100πt) and v_{2}(t) = +v_{0} cos (100πt)

B. v_{1}(t) = +v_{0} sin (100πt) and v_{2}(t) = +v_{0} cos (100πt)

C. v_{1}(t) = −v_{0} sin (100πt) and v_{2}(t) = −v_{0} cos (100πt)

D. v_{1}(t) = +v_{0} sin (100πt) and v_{2}(t) = −v_{0} cos (100πt)

55. A charged particle (mass m, charge +q) is moving in a region of uniform magnetic field . If at time t = 0 the particle is at the origin and has a velocity what is the position vector of the particle at a later time ?

A.

B.

C.

D.

56. A bar magnet of mass m is suspended from the ceiling with a massless string and is set into oscillations. A gold metal plate is brought close to the oscillating pendulum. The oscillations will damp due to induction of eddy currents in the metal. Which one of the following statements is true if the gold plate is replaced by a steel plate having the same physical dimensions. (Of the two, note that gold is a better conductor of electricity.)

A. the amplitude of oscillations will decrease faster

B. the amplitude of oscillations will decrease slower

C. the amplitude of oscillations will increase

D. the amplitude of oscillations will not be affected

57. A charged particle is moving away from a uniformly charged infinite wire along a direction perpendicular to it. Initially, the particle is at a distance L from the wire moving with a velocity u. When it is at a distance 2L, its velocity is found to be 2u. What will be the velocity of the particle when it is at a distance 4L from the wire ?

A. √6u

B. √7u

C. √8u

D. √9u

58. A steel wire of length 1 meter is under a tension of 10 newtons. The speed of the transverse wave excited in this wire is v. The wire is replaced by another steel wire of the same length but half the diameter. What should be the tension in the replaced wire, so that, the speed of the wave stays the same?

A. 40 N

B. 20 N

C. 5 N

D. 2.5 N

59. Consider the circuit shown in the figure. In which resistor the amount of power dissipated is the largest?

A. R1

B. R2

C. R3

D. R4

60. A biconvex lens with focal length f in air and refractive index of 1.5 is floating on the surface of a deep pond of water (refractive index 1.33). If an object is placed at a height of 2 f vertically above the lens, then the distance between the lens and the image is

A. f

B. 2f

C. less than 2f

D. greater than 2f