NEST Exam 2012 Question Paper

NEST – 2012

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A
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NATIONAL ENTRANCE SCREENING TEST (NEST – 2012)
Total Marks: 200 Time: 3 hours

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General instructions

1. This question booklet contains five sections. Each section carries 50 marks.

2. First section is a general section and it is compulsory.

3. Section 2 to 5 are subject sections (biology, chemistry, maths and physics). Choose any
three, that is, omit any one of the four subject sections.

4. Pocket calculators, cell phones, log tables etc. are NOT permitted in the examination
hall.

5. Answers to the questions are to be marked in the supplied OMR sheet.

6. Please make sure that question booklet code (A or B) matches with OMR
sheet code (A or B) respectively. In case of discrepancy please inform the invigilator
immediately.
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7. Rough work may be done on separate blank sheets provided.

8. Return the OMR sheet to the invigilator at the end of examination.

9. Read the instructions given at the beginning of each section carefully.

Instructions for writing on OMR sheet

1. Read and follow the instructions given on OMR sheet.

2. Write your name, roll number and other required information with ball point pen in
appropriate boxes provided. Sign your name with ball point pen in the box provided.

3. In the remaining part of OMR sheet, use HB pencil only (as instructed). Make
sure the bubbles are filled properly (as indicated in OMR sheet).

4. As far as possible, fill in the answers only after you are sure that you do not need to
change them. In case you have to change the answer after filling, erase the mark properly
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so that no black spot is left inside the bubble.

5. Ensure that you are filling the bubbles corresponding to correct sections and
answers.

6. Your roll number (as given in the admit card) MUST BE ENTERED COR-
RECTLY. If entered wrongly or not entered, the OMR sheet will be invalid
and will not be graded.

Section 1: GENERAL

Marks for Section 1: 50

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This section contains 19 questions.
For each question, only one of the four options is a correct answer. For questions, 1.1 to 1.12, a
correct answer will earn 3 marks. For questions, 1.13 to 1.19, a correct answer will earn 2 marks.
For this GENERAL section, a wrong answer or an unattempted question will earn 0 marks.

Read the following passage carefully and answer questions 1.1 to 1.3.

In mathematics and computer science, graph theory is the study of graphs, mathematical
structures used to model pairwise relations between objects from a certain collection. A
graph in this context is a collection of vertices or nodes and a collection of edges that
connect pairs of vertices. An edge may be undirected, meaning that there is no distinction
between the two vertices associated with it, or directed, in which it is directed from one
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vertex to another. The graph, in addition to vertices, may contain only undirected edges
or only directed edges or a combination of both. The graphs studied in graph theory
should not be confused with the graphs of functions or other kinds of graphs.

Graphs can be used in modelling and treatment of many problems of practical inter-
est in physical, biological and social systems. In computer science, graphs are used to
represent networks of communication, data organization, computational devices, flow of
computation, etc. In chemistry a graph makes a natural model for a molecule, where
vertices represent atoms and edges represent bonds. This approach is especially used in
computer processing of molecular structures. Likewise, graph theory is useful in conser-
vation biology where a vertex can represent regions where certain species exist and the
edges represent migration paths, or movement between the regions. This information is
important when looking at breeding patterns or tracking the spread of diseases or how
changes in the movement of one species can affect the other species.

1.1 Some situations from daily life are described below. Which one of them cannot be
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represented as a problem in graph theory?
(A) A salesman wants to represent annual sale of computers over last 10 years.
(B) An electrician wants to connect 10 computers in the school to the Internet.
(C) A Web-master wants to design a website with 10 web-pages linked to each
other.
(D) A postman has to deliver letters to 10 households taking the shortest path.

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1.2 Three points P , Q and R lie in the plane. Which of these is not a “graph”?
(A) The triangle formed by P , Q and R.
(B) A line segment between Q and R.

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(C) A vector from P to Q.
(D) A circle centred at P .

1.3 Among the following statements, the false statement is
(A) In a study of interacting particles, mutual forces can be taken as edges.
(B) In a study of a bus network, zonal boundaries can be taken as edges.
(C) In the nervous system, sensory receptors can be taken as vertices.
(D) In a study of government aid programme, aid recipients can be taken as ver-
tices.

Read the following passage carefully and answer questions 1.4 to 1.6.
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Silicon is a tetravalent metalloid, comprising about 28% of the earth’s crust and is
the eighth most common element in the universe by mass. However, it rarely occurs
in the free form in nature. It is combined with oxygen and exists in various forms.
Silicate minerals are the most common minerals on the earth, and include quartz,
feldspar, mica, amphibole, etc. The silicates contain SiO4−
4 units.

Purification of silicon is of paramount importance and is achieved by multi step
processes. In one such process, metallurgical grade silicon is produced by treating
silica with coke at very high temperature (Equation 1). The product consists of about
98% silicon and about 1-2% of impurities in the form of metals and metal oxides.
Further purification is done by converting silicon to trichlorosilane (Equation 2). The
trichlorosilane is purified by distillation. Finally, pure silicon is obtained by reducing
the trichlorosilane by hydrogen (Equation 3).

SiO2 + 2C → Si + 2CO (1)
Si + 3HCl → SiCl3 H + H2 (2)
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SiCl3 H + H2 → Si + 3HCl (3)

1.4 In the first step (Equation 1), which of the following oxide impurity would also reduce
to the corresponding metal?
(A) Iron oxide. (B) Aluminium oxide.
(C) Calcium oxide. (D) Sodium oxide.

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1.5 The most likely geometry for a silicate unit is
(A) square planar. (B) square pyramidal.
(C) tetrahedral. (D) octahedral.

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1.6 In step 2 (Equation 2), silicon is converted to trichlorosilane as the trichlorosilane
(A) can be oxidised easily.
(B) is further reduced by the hydrogen formed in the process.
(C) can be separated by dissolving in water.
(D) has a lower boiling point than that of the solid impurities.

1.7 In the graph below, the scale on x-axis shows consumption of electric energy per person
per year and y-axis shows human development index (HDI) indicating average living
conditions for the citizens. The data corresponds to the year 2003-2004. Choose the
statement which is not consistent with the information presented in the graph.
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(A) USA’s position is significantly away from the overall trend curve.
(B) India’s position on the graph indicates both low electric consumption as well
as low HDI.
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(C) A cluster of countries that includes Germany roughly indicates optimal electric
consumption for high HDI.
(D) As per the overall trend, Saudi Arabia shows excess electric consumption in
relation to its HDI.
1.8 Consider the polynomial, (x−a)m2 +(a+x)m−2x. One of the factors of this polynomial
is
(A) m + 1. (B) m − 1. (C) x − a. (D) x + a.

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1.9 Mr. A and Mrs. A have two daughters. Mr. A’s paternal grandmother has a genetic
disorder related to the X-chromosome, whereas Mrs A’s family lineage has no history
for the same disorder. The probability that at least one of the daughters of the couple
will show the same disorder is

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(A) 0.33. (B) 0.125. (C) 0.0625. (D) 0.

1.10 Following is the dxy orbital diagram. The X and Y axes are shown, while Z-axis is
directed perpendicular to the plane of the paper. Under which of the following operations
would the orbital remain identical?

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(A) Rotation of 180◦ about Y axis.
(B) Rotation of 180◦ about X axis.
(C) Reflection in YZ plane.
(D) Reflection in XY plane.

1.11 Which one of the following numbers is a perfect square?
(A) 436928. (B) 452929. (C) 469232. (D) 478637.

1.12 Marks of 20 students in a class were written in descending order in a mark sheet.
However, the marks of first 2 and last 2 students got erased due to a mistake. Which of
the following will definitely remain unaffected?
(A) Mean. (B) Mode. (C) Median. (D) none of these.
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For rest of the questions in this section, each correct answer will earn 2 marks.

1.13 Which of the following group of diseases is water-borne?
(A) Jaundice, dysentery, cholera.
(B) Cholera, malaria, filaria.
(C) Jaundice, cholera, tetanus.
(D) Malaria, diarrhea, tetanus.

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1.14 The number of moles of solute present in 1 kg of a solvent is called its
(A) molality. (B) molarity. (C) normality. (D) formality.

1.15 The year 2011 was celebrated as the International Year of Chemistry and it marked

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the centenary of Nobel prize in Chemistry for
(A) Otto Hahn’s discovery of fission of heavy elements.
(B) Marie Curie’s discovery of Radium & Polonium.
(C) Dmitri Mendeleev’s discovery of the periodic table.
(D) Svante August Arrhenius’s electrolytic theory of dissociation.

1.16 Recent studies using DNA sequencing reveal that Homo sapiens originated in
(A) Middle East. (B) East Asia.
(C) Central Africa. (D) Indian subcontinent.

1.17 Recently, India has been declared free of Polio by the World Health Organisation. The
causative organism for Polio is a
(A) bacterium. (B) virus. (C) protozoan. (D) parasitic worm.
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1.18 In the context of computer networks, the acronym LAN stands for
(A) Large Area Network. (B) Linked Area Network.
(C) Local Area Network. (D) Limited Area Network.

1.19 India has a major Antarctic programme under which 30 scientific expeditions have been
carried out. The research centre set up in Antarctica by India in 1989 and operating till
today is named
(A) Dakshin Gangotri. (B) Rohini. (C) Maitri. (D) Megha.
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Section 2: BIOLOGY

Marks for Section 2: 50

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This section contains 14 questions.
For questions 2.1 to 2.10 only one of the 4 options is correct. A correct answer will
earn 3 marks, a wrong answer will earn (−1) mark, and an unattempted question will earn
0 marks.
2.1 The results obtained after fol-
lowing the experimental de-
sign of Meselson and Stahl
to study DNA replication are
shown in the adjacent figure.
These results suggest that the
mode of DNA replication is
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(A) conservative. (B) semi-conservative
(C) dispersive (D) none of the these.

2.2 Desert animals need to conserve water. Kidneys regulate the concentration of salt and
water in blood through the formation and excretion of urine. A kidney is composed of
approximately one million units called nephrons. The kidneys of desert animals have
modified nephrons which help them survive long periods without water. Which one of
these options would be the expected modification?
(A) A very short collecting duct. (B) A very long loop of Henle.
(C) A very short distal tubule. (D) A very large Bowman’s capsule.

2.3 In an in vitro transcription experiment, the transcript and both the strands of DNA
were separated and analyzed for their base composition. The result obtained is shown in
the following table. Determine which DNA strand is serving as the template for mRNA
synthesis.
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G A T C U
DNA Strand 1 23.8 30 26 20 0
DNA Strand 2 20 26.2 30 24.1 0
mRNA 24 29.6 0 20 25.8

(A) Strand 1. (B) Strand 2.
(C) Both strands 1 and 2. (D) Neither strand 1 nor 2.

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2.4 Telomeres are nucleotide sequences at the ends of DNA molecules of eukaryotic cells.
Since linear chromosomal DNA gets shortened naturally after each round of replication,
telomeres protect the genes near the ends of DNA from being eroded. An enzyme,
telomerase, catalyses the lengthening of telomeres. In which of the following cells do you

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expect the activity of telomerase to be the highest?
(A) Differentiated cells. (B) Stem cells.
(C) Actively respiring cells. (D) Aging cells.

2.5 The graph below depicts serum antibody levels in a child after vaccination. Which
antibody types do the curves A and B represent, respectively?

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(A) IgA, IgM. (B) IgM, IgG. (C) IgA, IgE. (D) IgG, IgM.

2.6 Insectivorous plants have evolved an ability to capture and digest insects in addition to
their ability to photosynthesize. Which of the following nutritional requirements is most
likely being partially met by this “carnivorous” ability of plants?
(A) Carbon. (B) Oxygen. (C) Nitrogen. (D) Hydrogen.

2.7 The phenomenon of “linked genes” contradicts which of the following fundamental prin-
ciples of genetics?
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(A) Law of segregation.
(B) Principle of dominance and recessivity.
(C) Principle of independent assortment.
(D) Incomplete dominance.

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2.8 The endosymbiotic theory of evolution is based on a phenomenon in which one organism
engulfs another but does not digest it. Both organisms continue to live together in an
arrangement that is mutually beneficial. It is considered to be one of the key evolutionary
events that led to the development of higher eukaryotic organisms including plants and

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animals. The organelles of a eukaryotic cell that are believed to have originated through
(such) early endosymbiotic events are
(A) Golgi bodies and chloroplasts.
(B) chloroplasts and endoplasmic reticulum.
(C) mitochondria and chloroplasts.
(D) mitochondria and lysosomes.

2.9 Below is a generalized representation of the process of anaerobic cellular respiration.
Which molecules represent X, Y, Z, respectively?
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(A) ATP, ethanol, lactate. (B) Ethanol, lactate, ATP.
(C) Ethanol, ATP lactate. (D) ATP, lactate, ethanol.

2.10 Flowering plants can reproduce either sexually (through fusion of haploid gametes
that are produced following meiotic divisions in reproductive organs) or asexually (from
diploid vegetative organs). The following statements represent a mix of correct and
incorrect general features of both these modes of reproduction.

1. In sexual reproduction, progeny are genetically different from each other.
2. In asexual mode of reproduction, progeny are genetically identical to each other
but different from the parent.
3. Development of greater adaptive ability is possible in case of progeny derived by
sexual reproduction.
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4. A minor change in the habitat may adversely affect all offspring derived by asexual
reproduction.
5. A bisexual plant grown in isolation is incapable of sexual reproduction.

Which of the following combinations of the above statements correctly represents general
features of both sexual and asexual reproduction of plants?
(A) 1, 3 and 4. (B) 1, 2 and 5. (C) 1, 4 and 5. (D) 1, 2 and 3.

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For questions 2.11 to 2.14 one or more than one of the 4 options may be correct. Your
answer is regarded correct only if you choose all the correct option(s) and no incor-

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rect option(s). A correct answer will earn 5 marks, a wrong answer or an unattempted
question will earn 0 marks.

2.11 An operon comprises of regulatory DNA sequences such as activator and repressor
binding sites near the promoter and controls the transcription of structural genes as
a single mRNA. Lac operon is a classical example of gene regulation related to the
primary utilization of carbon source for energy in the gut bacterium, Escherichia coli.
The transcription of structural genes is governed by availability of different concentration
of glucose and/or lactose in the medium. Which of the following statements related to
the Lac operon is/are true.
(A) Repressor binds to the RNA polymerase.
(B) cAMP level will be low in the absence of glucose.
(C) Lac operon works on the principle of negative regulation.
(D) If glucose is absent and lactose is present, high levels of lac mRNA will be
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synthesized.

2.12 E. coli is a bacterium that is routinely used in cloning experiments and can be grown
on artificially synthesized media in the laboratory. One of the strains of this bacterium,
XL-1 Blue, is naturally resistant to the antibiotic, tetracycline. A genetic transformation
experiment was performed to introduce a plasmid that encodes an ampicillin resistance
gene into this strain. Following transformation, an equal number of cells were grown
independently on two types of growth media - (A) growth medium with tetracycline
alone and (B) growth medium with tetracycline and ampicillin. The number of colonies
that grew on medium A was much higher than that obtained on medium B. Which of
the following statement(s) explain the above observation?
(A) The transformation efficiency of cells is lower than their viability.
(B) Untransformed cells also grew on growth medium “A”.
(C) Tetracycline was inhibitory to the growth of bacterial cells.
(D) Ampicillin did not allow transformed cells to grow on growth medium “B”.
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2.13 Birth records of 4 babies were Blood Type
mixed up in a hospital. The ABO father AB
blood groups of the four babies were Parents I
mother O
known to be A, B, AB and O. To father A

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determine their respective parents, Parents II
mother O
blood groups of all parents, except father Unknown
the father of baby III were tested. Parents III
mother A
Choose the correct statements: father O
Parents IV
mother O

(A) The blood group of the Father III could be A and baby III is A.
(B) The blood group of Father III could be B and baby III is AB.
(C) The blood group of Father III could be either AB or B.
(D) The blood group of Father II could be O and baby III A.

2.14 Presynaptic neurons generate nerve impulses (ac-
tion potentials) which are transmitted to the post-
synaptic neuron via synapses (junctions). Presy-
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naptic neurons release neurotransmitters in the
synapse which bind to the specific receptors on the
postsynaptic neuron. A diagrammatic figure show-
ing connectivity of neurons in a neural circuit is
given below. The dendrites and cell body of a post-
synaptic neuron (A) receive synaptic contacts from
5 different presynaptic neurons (1, 2, 3, 4, and 5).
The presynaptic neurons 1 and 5 release an excita-
tory neurotransmitter, acetylcholine (Ach) to act
on the postsynaptic neuron A, whereas presynap-
tic neurons 2, 3 and 4 release an inhibitory neuro-
transmitter, γ-amino butyric acid (GABA) to act
on ‘A’. The ability of the postsynaptic neuron to
generate the action potential depends upon the relative contribution from the excitatory
and inhibitory neurotransmitters. Choose the correct answer(s) from the options given
below.
(A) Neuron A will generate an action potential if only neurons 1 and 5 release Ach
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on it.
(B) Neuron A will generate an action potential if neuron 1 releases Ach and at the
same time neurons 2, 3, and 4 release GABA on it.
(C) Neuron A will not generate an action potential but instead gets inhibited if
neuron 1 releases Ach and at the same time neurons 2, 3, and 4 release GABA
on it.
(D) All of the above.

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Section 3: CHEMISTRY

Marks for Section 3: 50

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This section contains 14 questions.
For questions 3.1 to 3.10 only one of the 4 options is correct. A correct answer will
earn 3 marks, a wrong answer will earn (−1) mark, and an unattempted question will earn
0 marks.

3.1 The correct order of acidity of the following compounds is
(A) HOCl < HClO2 < HClO4 ≈ HIO4
(B) HOCl < HClO2 < HIO4 < HClO4
(C) HIO4 ≈ HClO4 < HClO2 < HOCl
(D) HOCl < HClO2 < HClO4 < HIO4

3.2 The probability distribution for the speed (v) of molecules for an ideal gas at temperature
T , is given by
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m
3/2 
exp −
mv 2


2πkB T 2kB T
where m is the mass of a molecule and kB is the Boltzmann constant. Among the
following, the correct statement is
(A) The most probable velocity is proportional to T 3/2 .
(B) For two different gases A and B, the average velocity of the molecules of A is
same as that of B, at a given temperature.
(C) For two different gases A and B, the average kinetic energy of the molecules
of A is same as that of B, at a given temperature.
(D) The most probable velocity is higher than the average velocity.

3.3 For the complete electrolysis of 1 mole of water using a voltage of 100V across two
electrodes, the minimum amount of electrical energy (in Joule) to be spent is (F is the
Faraday constant)
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(A) 200F (B) 100F (C) F/50 (D) F/100

3.4 The relative stabilities of N2 , N+ −
2 , and N2 are

(A) N2 > N+ −
2 ≈ N2

(B) N2 ≈ N+ −
2 > N2

(C) N2 < N+ −
2 ≈ N2

(D) N2 ≈ N+ −
2 < N2

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3.5 Which of the following pentachlorides does not exist?
(A) PCl5 (B) SbCl5 (C) AsCl5 (D) BiCl5

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3.6 The number and type of interstices present in a cubic closed-pack structure of N atoms
are
(A) N, tetrahedral. (B) 2N, tetrahedral.
(C) 2N, octahedral. (D) 4N, tetrahedral.

3.7 A mixture of two isomeric pentoses on reaction with hydroxylamine gives 5 millimoles
of a mixture of oximes. The same mixture of pentoses on treatment with bromine water
produces a mixture with one of the pentoses unchanged (0.45g). The other product
reacts with NaHCO3 . The mixture of the pentoses consists of
(A) 0.45g aldose and 0.45g ketose.
(B) 0.3g aldose and 0.45g ketose.
(C) 0.45g ketose and 0.40g aldose.
(D) 0.45g aldose and 0.25g ketose.
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3.8 The IUPAC name of the following compound is

(A) 4-vinyl-2-pentyne. (B) 4-methylhex-2-yn-5-ene.
(C) 3-methylhex-4-yn-1-ene. (D) 3-methylhex-1-en-4-yne.

3.9 Consider the four compounds (a) - (d)

Br

(a) (b) (c) (d)
Br Br
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Br

Each of them is subjected to the treatment of the following sequence of reagents: (i) Mg,
ether (ii) CO2 (iii) H+ , H2 O (iv) hot KMnO4 . The two compounds that give the same
product are
(A) a and b. (B) b and c.
(C) c and d. (D) a and d.

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3.10 The structure of the most predominant species present in the aqueous solution of serine
(a natural amino acid) at pH 1 is
COO− + COOH

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(A) HO (B) H2 O
+
NH3 NH2
+ COO− COOH
(C) H2 O (D) HO
+ +
NH3 NH3

For questions 3.11 to 3.14 one or more than one of the 4 options may be correct. Your
answer is regarded correct only if you choose all the correct option(s) and no incor-
rect option(s). A correct answer will earn 5 marks, a wrong answer or an unattempted
question will earn 0 marks.

3.11 Naphthalene (X) is an aromatic compound and is the largest single component of coal
tar.
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8 1
2
COOH

X: Y:
6 3
COOH
5 4

To prove the structure of naphthalene, Graebe (1869) carried out oxidation studies on
it. X on oxidation gives phthalic acid (Y). X can be shown as a resonance hybrid of
three principal structures (assumed to be equally contributing). Given the bond lengths
(Å): Csp3 −Csp3 = 1.54; Csp3 −Csp2 = 1.50; Csp2 −Csp2 = 1.48; Csp2 =Csp2 = 1.34, choose
the correct statement(s) from the following.
(A) There are ten isomers of dichloronaphthalene.
(B) 2-Nitronaphthalene on oxidation gives nitrophthalic acid, while 2-aminonaphthalene
gives phthalic acid.
(C) The 1,2-bond in naphthalene has a bond length of 1.39 Å.
(D) Complete reduction of naphthalene gives decalin (C10 H18 ). There are two
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isomeric decalins.

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3.12 A reaction A + B → C, follows first order kinetics with respect to each reactant, with
an overall rate constant, k = 2.0 × 10−3 M−1 s−1 . The initial concentrations of A and B
are 0.10M and 6.93M, respectively. Given ln2 = 0.693, choose the correct statement(s).

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(A) The concentration of A remaining after 100 seconds is approximately 2.5 ×
10−2 M.
(B) The concentration of B remaining after 100 seconds is approximately 1.73M.
(C) The half-life for A is about 50 seconds.
(D) The half-life for B is about 50 seconds.

3.13 Boron is a fairly rare element. However, it has many important applications. It is a
nonmetal and has a very high melting point of 2180 ◦ C. It is extracted from borax and it
forms a large number of electron deficient compounds. Choose the correct statement(s).
(A) Boron can form BF3−
6 ion.

(B) B2 O3 is a basic oxide.
(C) Amorphous boron may be obtained by the reduction of B2 O3 at high temper-
ature using sodium or magnesium.
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(D) Crystalline boron may be obtained by thermal decomposition of diborane
(B2 H6 ).

3.14 The energy profiles for the two possible mechanisms of the following reaction,

R1 R1

R2 C Br + CN− −→ R2 C CN + Br−
R3 R3

are given below
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Figure X Figure Y

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Consider the following substrates (I to IV) for the above reaction
I: R1 = R2 = R3 = H
II: R1 = C2 H5 ; R2 = R3 = H
III: R1 = C2 H5 ; R2 = CH3 ; R3 = H

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IV: R1 = C2 H5 ; R2 = R3 = CH3

Choose the correct statement(s).
(A) The reaction mechanism changes from the one shown in Figure Y to that in
Figure X, as one changes the substrate from I to IV.
(B) Considering the mechanism corresponding to Figure X for substrates I to IV,
the correct order of ∆E is: ∆EI < ∆EII < ∆EIII < ∆EIV
(C) The reaction for substrate I follows 2nd order kinetics and that for IV follows
1st order kinetics.
(D) If CN− is taken in large excess, it is difficult to ascertain if the reaction follows
mechanism corresponding to energy profile in Figure X or that in Figure Y.
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Section 4: MATHEMATICS
Marks for Section 4: 50

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This section contains 14 questions.
For questions 4.1 to 4.10 only one of the 4 options is correct. A correct answer will
earn 3 marks, a wrong answer will earn (−1) mark, and an unattempted question will earn
0 marks.
  −1  
cot θ −1 cot θ 1 α −β
4.1 If = then
1 cot θ −1 cot θ β α
(A) α = 1 and β = 1. (B) α = 1 and β = 0.
(C) α = cos 2θ and β = sin 2θ. (D) α = cos θ and β = sin θ.

4.2 The domain and range of the function log(|log x|) are respectively,
(A) R and R. (B) (0, ∞) and R.
(C) (0, 1) ∪ (1, ∞) and R. (D) R and (0, ∞).
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4.3 If f is a polynomial function which is odd, then
(A) f + 1 is an odd function.
(B) the derivative f ′ is an even function.
(C) the number of real roots is always more than 1.
(D) there is no real root.
Z x
4.4 If f (x) = te−t dt then
0
(A) it has a minimum but not a maximum.
(B) it has a maximum but not a minimum.
(C) it has neither a maximum nor a minimum.
(D) the function f is not defined.

∞
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X
4.5 For x ∈ R with |x| < 1, the value of (1 + n)xn is
n=0
2
x +x+1 1 x2 − x + 1 1
(A) . (B) . (C) . (D) .
(1 − x)2 1−x (1 − x)2 (1 − x)2

4.6 Let S denote the set {1, {1, 2}, {1, 2, 3}}. Then,
(A) {1, 2} ∪ {1} ⊆ S. (B) S \ {1, 2, 3} = {1, {1, 2}}.
(C) {1, 2, 3} ∪ {1, 2} ∈ S. (D) {1, 2, 3} \ {2, 3} ∈ S.

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4.7 The examination co-ordinator of NEST-2011 said “it is not true that some candidates of
NEST-2011 neither attempted Mathematics nor attempted Physics”. Then we deduce
that

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(A) all candidates attempted Mathematics or Physics.
(B) all candidates attempted both Mathematics and Physics.
(C) some candidates attempted both Mathematics and Physics.
(D) some candidates attempted Mathematics or Physics.

4.8 The number of ways of choosing 2 boys and 2 girls in a class for a game of mixed doubles
is 1620. Then, the number of ways of choosing 2 students in the class is
(A) 153. (B) 171. (C) 182. (D) 212.

4.9 In the figure, PQRS is a square of side length 12 and XPQ is a traingle constructed on
QR outwardly such that XQ = XR. If area of the figure PQXRS is 192, then area of the
figure (PRX) lies between
ST2 X

R Q

12

S P

(A) 50 and 60. (B) 60 and 70.
(C) 70 and 80. (D) 80 and 90.

4.10 A fair coin is repeatedly tossed. Let X be the event that we get at least 2 heads in 4
tosses and Y be the event that we get at least 3 heads in 6 tosses. Then
(A) X is a more likely event than Y .
(B) Y is a more likely event than X.
(C) both events are equally likely.
NE

(D) given data is insufficient to draw any such conclusion.

For questions 4.11 to 4.14 one or more than one of the 4 options may be correct. Your
answer is regarded correct only if you choose all the correct option(s) and no incor-
rect option(s). A correct answer will earn 5 marks, a wrong answer or an unattempted
question will earn 0 marks.

Page 19

4.11 For any real number t belonging to the open interval (0, 12 ) consider the quadratic
equation
pt (x) = (t − t2 )x2 − (1 − t)x + t2 = 0

012
in x. Let αt , βt denote the roots of pt ., Then
(A) αt and βt are always imaginary.
(B) For infinitely many values of t, the corresponding root αt is rational and for
infinitely many values of t, αt is irrational.
(C) αt and βt are real and positive; one of αt and βt is greater than 1 and other is
less than 1.
(D) If t → ( 12 )−, then |αt − βt | → 0.

4.12 The composite function g ◦ f is a constant function if
(A) f (x) = (cos−1 x, sin−1 x) and g(x, y) = x + y.
(B) f (x) = [x] and g(x) = x − [x], where [x] denotes the largest integer less than
or equal to x.
x
(C) f (x) =
ST2
|x|
and g(x) = cos( π2 x).

(D) f (x) = |sin x| and g(x) = x − [x].

4.13 For any real number x, let [x] denote the largest integer less than or equal to x and
hxi = x − [x]. If x, y, z are real numbers, then which of the following statements are
correct ?
(A) [x + y + z] = [x + y] + [z + hx + yi].
(B) hx + y + zi = hhx + yi + zi.
(C) [xyz] = [xy][zhxyi].
(D) If 0 < z < 1, then hxyzi < hxyi.

4.14 Let X, Y be points on the same side of a straight line ℓ in a plane. Suppose P is a point
on ℓ such that XP + PY is the minimum. Then
(A) The line joining P and the midpoint of XY is perpendicular to ℓ.
NE

(B) XP and YP are equally inclined to ℓ.
(C) The ellipse with X and Y as foci passing through P has ℓ as its tangent at P.
(D) The circumcircle of triangle XYP has ℓ as its tangent at P.

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Section 5: PHYSICS

Marks for Section 5: 50

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This section contains 14 questions.
For questions 5.1 to 5.10 only one of the 4 options is correct. A correct answer will
earn 3 marks, a wrong answer will earn (−1) mark, and an unattempted question will earn
0 marks.

5.1 A tape is wound tightly around a cylinder of radius 1 cm. After winding, the radius of
the cylinder is found to be 2 cm. If the length of the tape is 10 m then the thickness of
the tape should be about
(A) 0.001 mm. (B) 0.01 mm. (C) 0.1 mm. (D) 1.0 mm.

5.2 A thin ring, a disk and an annular cylinder, of same mass M, are released from a point
3.6 m from the ground up an inclined plane of 30o inclination. The ring and the disk
have the same radius R. For the annular cylinder, the outer radius is R and the inner
ST2
radius is R/2. Times taken by the ring, disk and annular cylinder, respectively, to reach
the ground are in the ratio,
√ √ √ √ √ √
(A) √ 2 : √1.5 : √1.6. (B) √1.4 : √1.5 : √2.
(C) 2 : 1.5 : 1.5. (D) 1.4 : 1.5 : 1.6.

5.3 A hot-air balloon is rising vertically with a constant velocity of 10 m/sec. An object is
dropped from the balloon when it is 60 m above the ground. The angle of elevation of
the Sun is 45o . When the falling object is 33.75 m from the ground, the rate at which
the shadow of the object is traveling along the ground is (take g = 10 m/sec2 )
(A) 15 m/sec. (B) 20 m/sec. (C) 25 m/sec. (D) 30 m/sec.

5.4 A massless rod of length L is hanging from a horizontal ceiling with help of two wires,
of equal length and diameter, attached to the ends of the rod. The Young’s modulus
of the first wire is Y and the second wire is 1.2Y . A mass M is attached to the rod
at a distance X from the first wire. If the stretching of the two wires is equal then X/L is
NE

///////////////

Y 1.2Y
L
X
b
M
5 6 7 9
(A) . (B) . (C) . (D) .
11 11 11 11

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5.5 Water in a dam is 15 m deep. A horizontal pipe of 4 cm diameter passes through the
wall of the dam 6 m below the surface of the water. A plug secures the pipe opening.
The frictional force between the plug and the pipe wall is about (Take density of water
to be 103 Kg/m3 and g is 10 m/s2 .)

012
(A) 65 N. (B) 70 N. (C) 75 N. (D) 80 N.

5.6 A uniformly charged sphere, of total charge Q, is expanding uniformly about its center.
The magnitude of the electric field E at a point r from the center of the sphere as a
function of radius of the sphere varies as,
(A) (B)
E E

r r
(C) (D)
E
ST2 E

r r

5.7 A disk of mass M and radius R is attached to a spring of constant k as shown in the
figure below. The disk rolls back and forth without slipping. The angular frequency of
the motion of the disk is,

k R
b

p p p p
(A) k/2M. (B) 2k/3M. (C) 3k/4M. (D) k/M .
NE

5.8 Two coherent point radio sources S1 and S2 , separated by a distance d = 2.3 cm, are
radiating radio waves of wavelength λ = 1 cm. A detector is moved in the plane of the
sources along a circular path of radius R with center at midway between the sources
and R >> d. The number of interference maxima that will be detected as the detector
completes one circle is
(A) 8. (B) 10. (C) 12. (D) 14.

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5.9 Consider a 3-level system with energies E1 , E2 and E3 in ascending order. λ1 , λ2 and
λ3 are the wavelengths of radiation corresponding to the transitions E2 → E1 , E3 → E2
and E3 → E1 respectively. The wavelengths are related by,

012
(A) λ1 = λ2 λ3 /(λ3 − λ2 ). (B) λ2 = λ3 − λ1 .
(C) λ2 = λ1 λ3 /(λ1 + λ3 ). (D) λ3 = λ1 λ2 /(λ1 + λ2 ).

5.10 Mean free path of cosmic ray proton in water at 4o C is λ. If water is cooled to 0o C,
then the change in mean free path in water
(A) will be positive. (B) will be negative.
(C) will be zero. (D) cannot be determined.

For questions 5.11 to 5.14 one or more than one of the 4 options may be correct. Your
answer is regarded correct only if you choose all the correct option(s) and no incor-
rect option(s). A correct answer will earn 5 marks, a wrong answer or an unattempted
question will earn 0 marks.
ST2
5.11 In two experiments A and B, the container I contains ice at −30o C and is kept in
thermal contact with another container II containing water at 40o C. The contents,
however, are not allowed to mix. The variation of temperature T with time in the two
experiments is shown below. Choose the correct statement(s) from the following.
Temperature
Temperature

I
I

(0,0) (0,0)
Time Time
II
II
NE

(A) (B)
(A) In experiment A, ice melts completely.
(B) In experiment A, water freezes after some time.
(C) In experiment B, water freezes partially.
(D) In experiment B, the total mass of ice at P is more than the initial amount.

Page 23

5.12 A and B are two radioactive substances. The half-life of A is same as the average life
of B. The decay constant of B is 3.33 per day. (assume ln 2 = 0.70)
(A) If initially A and B have the same number of nuclei, then the ratio of the

012
number of nuclei of A to that of B will be 1/e after 1 day.
(B) If the initial amount of A and B are the same, then the ratio of the number
of nuclei of A to that of B will grow exponentially.
(C) If the initial amount of B is twice that of A, then the number of nuclei of A
and B will be the same after 0.7 days.
(D) 85.7% of A will disintegrate in about 3 days.

5.13 Two concentric, coplanar rigid circular rings of radii r and 2r carry currents of magni-
tude 4i and i respectively, in opposite directions. Choose the correct statement(s) from
the following.
(A) The net torque on outer coil is zero.
(B) The net magnetic dipole moment of the coils is zero.
ST2
(C) The net force exerted by outer coil on inner coil is zero.
(D) The net magnetic field at center is zero.

5.14 Light of intensity I0 is incident on air-water interface with an incident angle θ. The
angle between the refracted and the reflected beams is 90o . A polarization analyser is
inserted in water at a distance of 5cm from the interface. The axis of the analyser makes
an angle 45o with respect to the normal to the plane of incidence. Choose the correct
statement(s) from the following.
(A) Intensity of the unpolarized light transmitted through the analyser is less than
I0 /2.
(B) Intensity of the light polarized along the normal to the plane of incidence
transmitted through the analyser is less than I0 /2.
(C) Intensity of the light polarized along the plane of incidence transmitted through
the analyser is less than I0 /2.
(D) Intensity of the light polarized along the plane of incidence reflected from the
NE

interface is less than I0 /2.

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