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NATIONAL ENTRANCE SCREENING TEST
I. Read the following passage carefully and answer questions 1.1 to 1.3.
1. Black holes are astronomical objects that exert such a strong gravitational force that no particle – not even light – can come out of its surface; hence they are invisible to external observers.
If a black hole absorbs an in-falling material of total energy E, its mass increases by E/c2, in accordance with Einstein’s well-known mass-energy equivalence relation. The total energy of the black hole and the in-falling material is thus conserved. This is like the first law of thermodynamics. It is also known that in any physical process, the surface area A of the black hole never decreases. Now, according to the second law of thermodynamics, the entropy of the universe never decreases. Thus, the surface area of a black hole is taken to be proportional to its entropy.
A black hole can be either a rotating one or a non-rotating one. The surface area of a rotating black hole is given by where M is the mass of the black hole, c the speed of light, G the universal gravitational
constant and L the angular momentum of the black hole about the axis of rotation. According to the second law of thermodynamics, energy extraction from a black hole is possible with
maximum efficiency when the entropy of the black hole remains constant. The mechanism of this process is difficult but the process satisfies both the laws of thermodynamics.
1.1 Consider a non-rotating black hole B1 and a rotating black hole B2, both having the same mass M. Choose the correct statement.
(A) The entropy of B2 is less than the entropy of B1.
(B) The entropy of B2 is more than the entropy of B1.
(C) The entropy of B2 is the same as that of B1.
(D) The sum of the entropies of B1 and B2 will always remain constant.
1.2 Assume that the process of energy extraction from a black hole takes place with maximum efficiency. Choose the correct statement.
(A) Both the mass and angular momentum of the black hole will remain constant in the process.
(B) Both the mass and angular momentum of the black hole will decrease in the process.
(C) Only the mass of the black hole will decrease by E/c2 (where E is total energy extracted), while angular momentum of the black hole will remain constant.
(D) Both the mass and surface area of the black hole will decrease in the process.

2. The phenomenon of twins is rare among human species. Statistically, there are about 40 twins
in 1000 live births. Generally, twins are born when two separate eggs of a woman are fertilised by two sperms at the same time. These are called fraternal twins. In rarer cases, a single
fertilised egg divides into two and the two parts develop into independent embryos. Such twins are called identical twins. The frequency of identical twins is about 3 per 1000 live births.
Statistically, the genetic match between two fraternal twins is no different from that of any two siblings. The third type of twin is half-identical twin. In this case, a single egg divides into two before it is fertilised by two different sperms. In some very rare cases, identical twins are joined together and they are called Siamese twins. In any case, each fertilised egg gets 50% genetic material from father and 50% from mother. Studies on identical twins are useful because they can throw light on whether a trait in a person can be attributed to genetics or environment. For example, some diseases are supposed
to have genetic origin. In these cases, there would be strong correlation in observations made on identical twins whereas the correlation would be less in observations made on fraternal twins or siblings. However, the studies based on identical twins are only indicative of trends and not conclusive because of small sample sizes.

1.4 In the country of Raritania, 10,000 live births took place on April 1, which is its independence day. The government decided that a pair of randomly chosen children born on this day will receive full free education. The probability that the chosen pair will be a twin is about
(A) 4 × 10-2 (B) 1.6 × 10-3 (C) 4 × 10-5 (D) 4 × 10-6
1.5 In a study, three samples X, Y and Z of equal sizes are taken from a large eye hospital.X consists of pairs of identical twins, Y consists of pairs of fraternal twins and Z of sibling pairs who are not twins. P, Q, R and S are four diseases of the eye. Table below gives the number of pairs in each sample where both members test positive for a disease as indicated in the first column.
Based on the table, which of the diseases are most likely to have a genetic origin?
(A) P & Q (B) R & S (C) Q & S (D) P & R
1.6 A pole fixed on the ground is leaning away from the vertical. When the Sun is directly overhead, the length of its shadow is 7.5m. An observer standing 30m away from the base of the pole in the direction of the shadow, measures the angle of elevation of the top of the pole to be 30o. Find the length of the pole. Ignore the height of the observer.
(A) 7.5m (B) 15m (C) 22.5m (D) 30m
1.7 A number written as 213 in quadral system (number system with base 4) will be represented in hexal system (number system with base 6) as
(A) 23 (B) 39 (C) 103 (D) 303
1.8 Which of the following is thought NOT to be associated with the Homo Erectus?
(A) Use of fire (B) Hunting
(C) Use of tools (D) Language with script
1.9 The priority-dispute regarding the discovery of calculus in mathematics involved:
(A) Newton and Leibnitz (B) Fermat and Gauss
(C) Newton and Gauss (D) Leibnitz and Fermat
1.10 The existence of dark energy is inferred by the observation of
(A) rotation curves of galaxies.
(B) accelerated expansion of the universe.
(C) existence of microwave background radiation.
(D) Hawking radiation from black holes.

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