IISER IAT 2023 Question Paper with Answers & Solutions

Find in this paper

Try each question before opening its answer. Answer letters refer to the option order displayed here. A marked qualification identifies a source defect or a modelling assumption; see the explanation before scoring that item.

Open the displayed answer key

Subject	Question: answer
Biology	1: D · 2: B · 3: D · 4: C† · 5: D · 6: A · 7: B · 8: C · 9: B · 10: C† · 11: D · 12: A · 13: D · 14: B · 15: D
Chemistry	1: D · 2: A · 3: B · 4: C · 5: B · 6: B · 7: C · 8: A · 9: C · 10: A · 11: D · 12: B · 13: A · 14: C · 15: D
Mathematics	1: A · 2: A · 3: D · 4: A · 5: C · 6: C · 7: B · 8: A · 9: B · 10: C · 11: C · 12: D · 13: A · 14: B · 15: D
Physics	1: C · 2: A · 3: B · 4: A · 5: D · 6: B · 7: C · 8: C · 9: A · 10: D · 11: C · 12: B · 13: A · 14: B · 15: B

— = no valid listed option. † = read the qualification beside the solution.

Biology

Question 1

IAT 2023 · Biology

$Match the entries in Column I with their functions described in Column II.$

	Column I		Column II
P.	Squamous epithelium	(i)	The nucleus is at the basal side of the cell; also helps in movement of particles and mucous.
Q.	Cuboidal epithelium	(ii)	The nucleus is at the basal side of the cell; also helps in secretion and absorption.
R.	Columnar epithelium	(iii)	The nucleus is at the center of the cell; also helps in secretion and absorption.
S.	Ciliated epithelium	(iv)	It serves as a diffusion barrier.

P-(i); Q-(ii); R-(iii); S-(iv)
P-(ii); Q-(iv); R-(i); S-(iii)
P-(iii); Q-(i); R-(iv); S-(ii)
P-(iv); Q-(iii); R-(ii); S-(i)

Answer & worked solution

Answer D

Squamous epithelium provides a thin barrier for diffusion, P–iv. Cuboidal cells typically have central nuclei and perform secretion and absorption, Q–iii. Columnar cells have basally placed nuclei and the stated absorptive/secretory roles, R–ii. Cilia move material over the surface, S–i. The resulting combination is D.

Question 2

IAT 2023 · Biology

Which one of the following best describes peptones?

Zymogen form of pepsin
Partially digested proteins
Activated form of pepsin
An intestinal mixture of proteins, mucous and ${HCO}_{3}^{-}$

Answer & worked solution

Answer B

Peptones are mixtures of shorter, soluble peptide products formed by partial protein hydrolysis. They are digestion products rather than an inactive or activated form of the enzyme pepsin. The inactive pepsin precursor is pepsinogen, so B is the correct description.

Question 3

IAT 2023 · Biology

Match the biomolecules given in Column I with their corresponding chemical nature given in Column II.

	Column I		Column II
P.	Insulin	(i)	Secondary metabolite
Q.	Inulin	(ii)	Homopolymer
R.	Lectin	(iii)	Quaternary ammonium derivative
S.	Lecithin	(iv)	Hetropolymer

P-(i); Q-(iii); R-(ii); S-(iv)
P-(ii); Q-(iv); R-(iii); S-(i)
P-(iii); Q-(i); R-(iv); S-(ii)
P-(iv); Q-(ii); R-(i); S-(iii)

Answer & worked solution

Answer D

Insulin is a protein made of different amino-acid residues, so it is a heteropolymer. Inulin is principally a fructose polymer and is classified as a homopolymer in this context. Lectins are included among the listed secondary metabolites, while lecithin contains the quaternary-ammonium choline group. Thus P–iv, Q–ii, R–i and S–iii gives D.

Question 4

IAT 2023 · Biology

A mitotic drug inhibits microtubule formation. Which one of the following stages of karyokinesis will be the first to get affected by the drug?

Prophase
Anaphase
Metaphase
Telophase

Answer & worked solution

Answer C · qualification below

Metaphase is the expected exam answer; “first affected” can include prophase.

The expected exam answer is metaphase, C: a spindle-disrupting drug prevents correct chromosome attachment/alignment and blocks progression to anaphase. However, the phrase “first to get affected” is imprecise. Mitotic microtubule organisation begins during prophase, so literal inhibition of their formation can already affect prophase, option A. Preserve that distinction rather than claiming that microtubules first form in metaphase.

Question 5

IAT 2023 · Biology

Which one of the following statements regarding seed structure is INCORRECT?

In dicots, the outer layer of the seed coat is called testa.
The endosperm is not present in some of the mature dicot seeds.
Coleoptile and coleorhiza are found in monocotyledonous seeds.
In monocot seeds, the membranous seed coat that is fused with the fruit wall is called the aleurone layer.

Answer & worked solution

Answer D

The aleurone layer is the protein-rich outer layer of the endosperm in cereal grains. It is not the seed coat fused to the fruit wall. The fused seed-coat/pericarp arrangement characterises the caryopsis, whereas testa, loss of endosperm in some dicots, coleoptile and coleorhiza are correctly described in the other options.

Question 6

IAT 2023 · Biology

Which one of the following anatomical features of wood can be used to estimate the age of a tree growing in a temperate climate?

Spring wood and late wood.
Heart wood and sap wood.
Spring wood and heart wood.
Autumn wood and sap wood.

Answer & worked solution

Answer A

In a temperate climate, the cambium typically produces lighter early or spring wood followed by denser late or autumn wood each year. One pair forms an annual growth ring. Counting these successive rings estimates the tree's age; heartwood and sapwood alone do not provide annual boundaries.

Question 7

IAT 2023 · Biology

Which one of these statements is CORRECT about biological nitrogen fixation in plants?

Atmospheric nitrogen is fixed by Nitrogenase by converting $N_{2}$ to ${NO}_{3}^{-}$ .
The catalytic redox center of Nitrogenase contains Mo and Fe as cofactors.
Nitrogenase can function optimally only in the presence of molecular oxygen.
The transport of important amides, like asparagine and glutamine, produced by transamination, to different parts of the plant body is mediated by phloem.

Answer & worked solution

Answer B

The conventional nitrogenase enzyme contains an iron–molybdenum cofactor at its catalytic centre. It reduces nitrogen to ammonia, not directly to nitrate, and is oxygen-sensitive. Thus B is correct. The alternative about amides also confuses their formation and the transport description used in this context.

Question 8

IAT 2023 · Biology

Which one of the following is an example of genetic diversity?

Greater variation of ecosystems found in India than in Scandinavia.
Higher diversity of amphibians in the Western Ghats than in the Eastern Ghats.
Variation in potency and concentration of reserpine produced by Rauwolfia vomitoria.
The greater diversity of plant species found in India compared to Central Asia.

Answer & worked solution

Answer C

Genetic diversity is variation within a species. Differences among Rauwolfia populations in the potency and concentration of the same alkaloid illustrate this level. Differences in amphibian or plant species counts concern species diversity, while ecosystem variety concerns ecological diversity.

Question 9

IAT 2023 · Biology

When the ribosome encounters a stop codon in the mRNA, during translation, which one of the following binds to the stop codon?

Rho factor
Release factor
Sigma factor
Termination factor

Answer & worked solution

Answer B

A release factor recognises the stop codon in the ribosomal A site and promotes hydrolysis of the bond linking the polypeptide to its tRNA. No aminoacyl-tRNA normally decodes a stop codon. Rho and sigma are associated with bacterial transcription, so the translation-termination factor requested is the release factor.

Question 10

IAT 2023 · Biology

$Match the terms in column I with their physiological roles given in Column II.$

	Column I		Column II
P.	Sertoli cells	(i)	Secretion of chorionic gonadotropin
Q.	Follicle stimulating hormone	(ii)	Carries urine away from bladder
R.	Placenta	(iii)	Carries urine away from kidney
S.	Urethra	(iv)	Provides nutrition to developing spermatozoa
		(v)	Triggers ovulation

Which one of the following combinations is correct?

P-(i); Q-(iv); R-(iii); S-(v)
P-(iii); Q-(ii); R-(v); S-(iv)
P-(iv); Q-(v); R-(i); S-(ii)
P-(v); Q-(i); R-(iv); S-(iii)

Answer & worked solution

Answer C · qualification below

C is the intended matching key, but the FSH–ovulation pair is biologically imprecise.

Three matches are unambiguous: Sertoli cells nourish developing sperm, P–iv; placenta secretes chorionic gonadotropin, R–i; urethra carries urine away from the bladder, S–ii. These select the intended key C. The remaining Q–v pair is a source defect: the direct ovulatory trigger is the LH surge, whereas FSH primarily supports follicular growth. Therefore C is the intended combination, but no listed option is fully correct if “triggers ovulation” is interpreted precisely.

Question 11

IAT 2023 · Biology

For which one of the following diseases does the causative agent require the splicing of their hnRNA to generate mature mRNA?

Tuberculosis
Pertussis
Typhoid
Malaria

Answer & worked solution

Answer D

The malaria parasite Plasmodium is a eukaryote, so its gene-expression machinery includes pre-mRNA processing and splicing. The causative agents of tuberculosis, pertussis and typhoid are bacteria, which do not use the typical eukaryotic hnRNA-to-mRNA splicing pathway. Thus malaria is the appropriate choice.

Question 12

IAT 2023 · Biology

$The diagram represents an enzyme, its substrate and potential inhibitors (𝑃, 𝑄, 𝑅, 𝑆, 𝑇) .$

Which one of the following combinations is the best pair of competitive inhibitors for the enzyme?

P, S
$𝑄, 𝑅$
S, T
R, T

Answer & worked solution

Answer A

A competitive inhibitor must occupy the substrate-binding active site, not merely either allosteric site. The active site and substrate in the diagram have a complementary triangular feature. P and S include the matching projecting shape and can compete for that site; Q, R and T better match other shapes. Hence P and S form the best pair.

Question 13

IAT 2023 · Biology

In an island with 10,000 individuals, four have sickle cell anemia, a recessive autosomal disease. Assuming that the locus is in Hardy-Weinberg equilibrium, how many individuals in that island are expected to be heterozygous for the disease allele?

4
9608
9996
392

Answer & worked solution

Answer D

For an autosomal recessive disorder, affected frequency is

q^2=4/10000=0.0004

, so q = 0.02 and p = 0.98. Heterozygote frequency is

2pq=2(0.98)(0.02)=0.0392

. Multiplying by 10000 gives 392 expected carriers.

Question 14

IAT 2023 · Biology

A 1000 base-pair DNA fragment was cloned between Hind III and EcoRI sites of the plasmid vector (plAT) of size 3500 base-pair. The cloned fragment had a Not I site as shown in the figure. In order to confirm the presence of the insert, the recombinant plasmid was digested completely with (a) Not I and EcoR I, and (b) Not I and Hind III.

IAT 2023, Biology, question 14 — question diagram

In lane 1 the products of the digestion by Not I and EcoRI was loaded. In lane 2 the products of the digestion by Not I and Hind III was loaded. Which one of the following correctly represents the agarose gel electrophoresis profile of the digested recombinant plasmid for (a) and (b), respectively?

Answer & worked solution

Answer B

The recombinant circle is 3500 + 1000 = 4500 bp. The insert's NotI site divides it into 400 and 600 bp segments. From the indicated vector intervals, NotI + EcoRI produces 400, 500, 1700 and 1900 bp fragments; NotI + HindIII produces 600, 900, 1100 and 1900 bp fragments. Each set totals 4500 bp. B shows these bands in the specified lane order.

Question 15

IAT 2023 · Biology

The following pedigree chart shows the inheritance of a genetic disorder. I-2 and III-2 are the only affected individuals. IAT 2023, Biology, question 15 — question diagram

Which one of the following is the correct pattern of inheritance of the disorder, and the genotype of the II- 3 individual?

Autosomal recessive, homozygous for the normal allele
Autosomal dominant, homozygous for the normal allele
X-linked recessive, heterozygous
Autosomal recessive, heterozygous

Answer & worked solution

Answer D

Unaffected parents II-2 and II-3 have an affected daughter, supporting recessive inheritance. X-linked recessive inheritance would require that daughter's father to be affected, contrary to the diagram. The unaffected mother II-2 inherits the recessive allele from her affected father, and II-3 must also contribute it to III-2. Thus the disorder is autosomal recessive and II-3 is heterozygous.

Chemistry

Question 1

IAT 2023 · Chemistry

$How many radial nodes does {Ca}^{+} have in its 4 s orbital?$

Answer & worked solution

Answer D

For an orbital with principal quantum number n and angular quantum number l, the radial-node count is

n-l-1

. For 4s, n = 4 and l = 0, so there are

4-0-1=3

radial nodes. The ion's charge does not change this count for the named orbital.

Question 2

IAT 2023 · Chemistry

Amongst

O_{2}, N_{2}, F_{2}

, and

B_{2}

, which molecules will be attracted to an external magnetic field?

$O_{2}$ and $𝐵_{2}$
$O_{2}$ and $𝐹_{2}$
$𝑂_{2}, 𝐵_{2}$ , and $𝑁_{2}$
$𝐹_{2}, 𝑁_{2}$ , and $𝐵_{2}$

Answer & worked solution

Answer A

O2 has two unpaired electrons in its π* orbitals, and B2 has two unpaired electrons in its degenerate π bonding orbitals. Both are paramagnetic and attracted by an applied field. N2 and F2 have all electrons paired in their ground-state molecular-orbital configurations.

Question 3

IAT 2023 · Chemistry

What is the smallest P-P-P bond angle in the highly reactive allotrope of phosphorus?

$45^{\circ}$
$60^{\circ}$
$109^{\circ}$
$120^{\circ}$

Answer & worked solution

Answer B

The highly reactive white-phosphorus allotrope consists of P4 tetrahedra. Each triangular face gives a P–P–P angle of 60°, much smaller than a normal unstrained tetrahedral bond angle. This angle strain contributes to its reactivity.

Question 4

IAT 2023 · Chemistry

Which of the following is an ore of iron?

Malachite
Bauxite
Siderite
Quartz

Answer & worked solution

Answer C

Siderite is iron(II) carbonate, FeCO3, and is an iron ore. Malachite is a copper mineral, bauxite is an aluminium ore and quartz is SiO2. Therefore the iron-bearing option is C.

Question 5

IAT 2023 · Chemistry

Which parameters are plotted in the Ellingham diagram?

$Δ_{𝑟} 𝐻^{\circ}$ vs $𝑇$
$Δ_{𝑟} 𝐺^{\circ}$ vs $𝑇$
$Δ_{𝑟} 𝑆^{\circ}$ vs $𝑇$
$Δ_{𝑟} 𝑆^{\circ}$ vs $𝑃$

Answer & worked solution

Answer B

An Ellingham diagram plots the standard Gibbs free-energy change for oxidation against temperature. It helps compare oxide stability and the thermodynamic feasibility of reduction. The plotted pair is therefore

\Delta_rG^\circ

versus T.

Question 6

IAT 2023 · Chemistry

Which of the following compounds will NOT undergo the Finkelstein reaction with NaI via

𝑆_{𝑁} 2

pathway?

IAT 2023, Chemistry, question 6 — question diagram

I and III
II and III
II and IV
I and IV

Answer & worked solution

Answer B

II is an aryl bromide and III is a vinylic bromide; their C–Br bonds are on sp2 carbons and do not undergo ordinary backside SN2 displacement in the Finkelstein reaction. I is benzylic and IV is a primary alkyl bromide, both compatible with SN2 substitution by iodide. Thus II and III are excluded.

Question 7

IAT 2023 · Chemistry

Which one amongst the following is the most efficient way of synthesizing

𝑛

-propyl benzene?

Answer & worked solution

Answer C

Direct Friedel–Crafts alkylation with a propyl electrophile can rearrange to an isopropyl group. Friedel–Crafts acylation instead installs the unrearranged propanoyl group, and Wolff–Kishner reduction then converts its carbonyl to CH2. The acylation–reduction sequence in C gives n-propylbenzene efficiently without carbocation rearrangement of the chain.

Question 8

IAT 2023 · Chemistry

$Which amongst the following are chiral compounds?$

II and IV
I and IV
II and III
I and II

Answer & worked solution

Answer A

In II, the carbon bearing NH2 is attached to four different groups, so it is a stereogenic centre. IV also has an alcohol-bearing carbon with inequivalent paths around the substituted ring. I and III have internal symmetry that makes the apparently relevant ring paths equivalent. The chiral compounds are therefore II and IV.

Question 9

IAT 2023 · Chemistry

$Which one amongst the following bases is NOT present in RNA?$

Answer & worked solution

Answer C

Structure III is thymine, identified by the methyl-substituted pyrimidinedione ring. RNA normally uses uracil in its place. The other structures are cytosine, guanine and uracil, all standard RNA bases, so III is the exception.

Question 10

IAT 2023 · Chemistry

$Which amongst the following are aromatic?$

I and II
I and III
II and IV
III and IV

Transcription note: Restored the missing distractor from the 2023 paper; the secondary source repeated II and IV.

Answer & worked solution

Answer A

I is a substituted cyclopropenyl cation with two delocalised π electrons, satisfying

4n+2

with n = 0. In II, two double bonds and an oxygen lone pair form a continuous six-π-electron ring, also aromatic. III has an eight-π-electron count if fully conjugated and IV has four; neither meets the aromatic count. Thus I and II are aromatic. II is an oxygen-containing six-membered ring, not furan.

Question 11

IAT 2023 · Chemistry

Why is it harder to compress liquids and solids relative to gases?

Solids and liquids have definite volume.
Due to the presence of electron-nuclear attraction in solids and liquids.
Due to the absence of electron-nuclear attraction in solids and liquids
Molecules are closer to each other in solids and liquids.

Answer & worked solution

Answer D

Particles in liquids and solids are already packed much closer together than those in a gas. Compressing them further encounters strong short-range repulsion, while a gas initially has much more intermolecular space to reduce. D explains the lower compressibility rather than merely restating that condensed phases have a nearly fixed volume.

Question 12

IAT 2023 · Chemistry

Related to the Freundlich adsorption isotherm, which one of the following statements is NOT correct?

The value of $\frac{1}{𝑛}$ is between 0 and 1 .
It holds good over a wide range of pressures.
The Freundlich adsorption isotherm equation is an empirical equation.
It is used for the adsorption of both gases and solutions.

Answer & worked solution

Answer B

The Freundlich equation is empirical and works over a limited pressure or concentration range. It does not reproduce saturation correctly at sufficiently high pressure. Thus the claim that it remains valid over a wide pressure range without qualification is the incorrect one, B; its empirical nature and use for gases and solutions are valid.

Question 13

IAT 2023 · Chemistry

Consider the following reaction:

{CH}_{4} (g) \to C (g) + 4 H (g); Δ_{𝑎} 𝐻^{0} = 1665 {kJmol}^{-} 1

Which of the statements is FALSE?

$Δ_{𝑎} 𝐻^{0}$ is the mean bond enthalpy of a C-H bond.
All four $C - H$ bonds in ${CH}_{4}$ are identical in bond length and energy.
The energy required to break individual $C - H$ bonds in successive steps is different.
Mean C-H bond enthalpies differ slightly from compound to compound

Answer & worked solution

Answer A

The stated reaction separates all four C–H bonds into gaseous atoms. Its 1665 kJ/mol enthalpy is the total atomisation enthalpy, not the value per C–H bond. The mean bond enthalpy for methane is

1665/4=416.25

kJ/mol. Sequential bond removals have different energies because the remaining species changes after each step.

Question 14

IAT 2023 · Chemistry

In two solutions

𝑋

(hexane and benzene) and

𝑌

(water and HCl ), what types of deviations from Raoult's law are observed?

Positive deviation and no deviation (ideal behaviour), respectively.
Negative deviation and positive deviation, respectively.
Positive deviation and negative deviation, respectively
No deviation (ideal behaviour) and negative deviation, respectively.

Answer & worked solution

Answer C

Mixing hexane and benzene weakens the favourable like-molecule interactions enough to give positive deviation from Raoult's law. Water and HCl interact strongly through ionisation and hydration, reducing escaping tendency and giving negative deviation in the stated comparison. Thus X is positive and Y negative, option C.

Question 15

IAT 2023 · Chemistry

In aqueous solution, the hydronium ion gets further hydrated to give which of the following species?

$H_{3} O_{2}^{+}$
$H_{5} O_{3}^{+}$
$H_{7} O_{4}^{+}$
$H_{9} O_{4}^{+}$

Answer & worked solution

Answer D

Adding three water molecules to hydronium gives

\mathrm{H_3O^+\cdot3H_2O}

, whose combined formula is

\mathrm{H_9O_4^+}

. This is the further-hydrated species among the choices. Other hydration structures can occur in water, but only D has the appropriate formula here.

Mathematics

Question 1

IAT 2023 · Mathematics

Let

𝑓 : 𝐑 \to (0, \infty)

be a continuous decreasing function. Suppose

𝑓 (0), f (1), \dots, 𝑓 (10)

are in a geometric progression with common ratio

\frac{1}{5}

. In which of the following intervals does the value of

\int_{0}^{10} 𝑓 (𝑥) 𝑑 𝑥

lie?

$(0, 2 𝑓 (0))$
$(4 𝑓 (0), 6 𝑓 (0))$
$(8 𝑓 (0), 10 𝑓 (0))$
$(12 𝑓 (0), 14 𝑓 (0))$

Transcription note: The given value is f(1), not a dotted derivative; checked against the 2023 paper.

Answer & worked solution

Answer A

Because f is positive, the integral is positive. Since it is decreasing, on each interval [k,k+1] its integral is at most f(k). Therefore

\int_0^{10}f(x)\,dx\leq\sum_{k=0}^9f(k)=f(0)\sum_{k=0}^95^{-k}<\frac54f(0)<2f(0)

. This puts it in interval A.

Question 2

IAT 2023 · Mathematics

Let $𝑀$ be a $3 \times 3$ matrix with real entries such that

${[\begin{matrix} 𝑥_{1} \\ 𝑥_{2} \\ 𝑥_{3} \end{matrix}] : 𝑀 [\begin{matrix} 𝑥_{1} \\ 𝑥_{2} \\ 𝑥_{3} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]} = {[\begin{matrix} 𝑥_{1} \\ 𝑥_{2} \\ 𝑥_{3} \end{matrix}] : 𝑥_{1} + 𝑥_{2} = 0 = 𝑥_{2} + 𝑥_{3}}$

What is the value of the determinant of M ?

Answer & worked solution

Answer A

The given nullspace contains the nonzero vector (1,−1,1), so M has a nontrivial kernel. It is therefore singular and its determinant is zero. More specifically, that nullspace is one-dimensional, but nontriviality alone is sufficient.

Question 3

IAT 2023 · Mathematics

What is the locus of the center of circles passing through the origin

(0, 0)

with fixed radius?

Line
Hyperbola
Parabola
Circle

Answer & worked solution

Answer D

If a circle has fixed radius r and passes through the origin, its centre must lie at distance r from the origin. Conversely, every such centre defines a radius-r circle through the origin. The locus is therefore the circle

x^2+y^2=r^2

Question 4

IAT 2023 · Mathematics

Let

𝛼

be a real number. What is the total number of distinct point(s) of intersection between the parabola

𝑦 = 𝑥^{2} + 4 𝑥 \sin 𝛼 + 6

and the pair of lines

𝑦^{2} = 1

Zero
One
Two
Four

Answer & worked solution

Answer A

Complete the square:

y=(x+2\sin\alpha)^2+6-4\sin^2\alpha

. Its minimum is at least 2 because

\sin^2\alpha\leq1

. The pair of lines is y = ±1, both below that minimum. Hence there are no intersection points for any real α.

Question 5

IAT 2023 · Mathematics

Let

𝐿

be a straight line passing through the origin, and it makes angles

𝛼, 𝛽

and

𝛾

with the positive

𝑋, 𝑌

and

𝑍

-axes, respectively. What is the value of

\cos 2 𝛼 + \cos 2 𝛽 + \cos 2 𝛾

-3
3
-1
1

Answer & worked solution

Answer C

Direction cosines satisfy

\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1

. Using

\cos2\theta=2\cos^2\theta-1

, the required sum is

2(1)-3=-1

Question 6

IAT 2023 · Mathematics

$What is the total number of distinct divisors of 2^{9} \times 3^{19} ?$

Answer & worked solution

Answer C

Every positive divisor has the form

2^a3^b

with 0 ≤ a ≤ 9 and 0 ≤ b ≤ 19. There are 10 independent choices for a and 20 for b, giving

10\times20=200

distinct positive divisors.

Question 7

IAT 2023 · Mathematics

Suppose the mean and the standard deviation of the data

{𝑥_{1}, 𝑥_{2}, \dots, 𝑥_{9}}

are

𝜇

and

𝜎 (\neq 0)

, respectively. After including one more data value

𝑥_{10}

, the mean of the data

{𝑥_{1}, 𝑥_{2}, \dots, 𝑥_{9}, 𝑥_{10}}

remains

𝜇

. What is the standard deviation of the data

{𝑥_{1}, 𝑥_{2}, \dots, 𝑥_{9}, 𝑥_{10}}

$\frac{\sqrt{10}}{3} 𝜎$
$\frac{3}{\sqrt{10}} 𝜎$
$\frac{10}{9} 𝜎$
$\frac{9}{10} 𝜎$

Answer & worked solution

Answer B

Keeping the mean at μ requires the added observation to equal μ. It contributes zero squared deviation, while the original sum of squared deviations remains

9\sigma^2

. The new variance is therefore

9\sigma^2/10

and the new standard deviation is

3\sigma/\sqrt{10}

Question 8

IAT 2023 · Mathematics

Consider three biased coins. Let the probability of getting head be

\frac{1}{3}

and the probability of getting tail be

\frac{2}{3}

in each of the coins. Consider the experiment of tossing the threee coins one by one, and the following events:

𝐸

:"At least two heads show up" F:"First coin shows tail". What is the conditional probability of

𝐸

given that

𝐹

has already occurred?

$1 / 9$
$2 / 9$
$1 / 4$
$4 / 9$

Answer & worked solution

Answer A

Given that the first toss is a tail, at least two heads among all three tosses can occur only if both remaining tosses are heads. Independence makes that conditional probability

(1/3)(1/3)=1/9

Question 9

IAT 2023 · Mathematics

Which of the following differential equations has

𝑦 = 𝑒^{𝑥}

as one of its particular solutions?

$𝑦 \frac{𝑑^{2} 𝑦}{𝑑 𝑥^{2}} + 𝑒^{𝑥} \frac{𝑑 𝑦}{𝑑 𝑥} + 𝑦^{2} = 𝑒^{2 𝑥}$
$𝑦 \frac{𝑑^{2} 𝑦}{𝑑 𝑥^{2}} - 𝑒^{𝑥} \frac{𝑑 𝑦}{𝑑 𝑥} + 𝑦^{2} = 𝑒^{2 𝑥}$
$𝑦 \frac{𝑑^{2} 𝑦}{𝑑 𝑥^{2}} - 𝑒^{𝑥} \frac{𝑑 𝑦}{𝑑 𝑥} + 𝑦^{2} = 𝑒^{𝑥}$
$𝑦 \frac{𝑑^{2} 𝑦}{𝑑 𝑥^{2}} + 𝑒^{𝑥} \frac{𝑑 𝑦}{𝑑 𝑥} + 𝑦^{2} = 𝑒^{𝑥}$

Answer & worked solution

Answer B

For

y=e^x

, both derivatives y′ and y″ equal

e^x

. In B, substitution gives

e^{2x}-e^{2x}+e^{2x}=e^{2x}

, identical to the right-hand side for every x. The plus-sign alternatives give three such terms, and C has the wrong right-hand side.

Question 10

IAT 2023 · Mathematics

What is the area of the region

{(𝑥, 𝑦) : 0 \leq 𝑦 \leq 𝑥 𝑒^{𝑥^{2}}, 0 \leq 𝑥 \leq 1}

$\frac{1}{2} 𝑒$
$𝑒 - 1$
$\frac{1}{2} (𝑒 - 1)$
$𝑒 - 2$

Answer & worked solution

Answer C

The area is

\int_0^1xe^{x^2}\,dx

. Substitute

u=x^2

so that du = 2x dx; the integral becomes

\frac12\int_0^1e^u\,du=\frac12(e-1)

. The integrand is non-negative on the interval, so this integral is the area directly.

Question 11

IAT 2023 · Mathematics

Consider the objective function $𝑍 = 𝑥 - 𝑦$ subject to the constraints

$\begin{matrix} 𝑥 + 2 𝑦 \leq 10 \\ 𝑥 + 𝑦 \geq 2 \\ 𝑥 \geq 0, 𝑦 \geq 0 \end{matrix}$

What is the minimum value of $𝑍$ subject to the above constraints?

2
-2
-5
-10

Answer & worked solution

Answer C

From

x+2y\leq10

, we have

y\leq(10-x)/2

. Hence

Z=x-y\geq3x/2-5\geq-5

because x ≥ 0. The feasible point (0,5) satisfies the remaining constraint x+y ≥ 2 and attains Z = −5, proving the minimum.

Question 12

IAT 2023 · Mathematics

Let

𝑝 (𝑥) = 𝑥^{2} + 𝑏 𝑥 + 𝑐

be quadratic polynomial with real coefficients

𝑏

and

𝑐

. Suppose

𝑝 (1) = 5

and

𝑝 (- 1) = 3

. What is the product of the roots of

𝑝 (𝑥) = 0

1
-1
2
3

Answer & worked solution

Answer D

The two conditions give

1+b+c=5

and

1-b+c=3

. Adding yields

2+2c=8

, so c = 3. For the monic quadratic, the product of its roots is c, hence 3.

Question 13

IAT 2023 · Mathematics

Let

𝑓 : (- 1, 2) \to 𝐑

be a differentiable function such that

𝑓^{'} (𝑥) = \frac{2}{𝑥^{2} - 5}

and

𝑓 (0) = 0

. Then in which of the following intervals does

𝑓 (1)

lie?

$(- \infty, 0)$
$(0, 2)$
$(2, 4)$
$(4, \infty)$

Answer & worked solution

Answer A

On [0,1],

x^2-5<0

, so

f'(x)=2/(x^2-5)

is negative. Therefore

f(1)<f(0)=0

, placing f(1) in

(-\infty,0)

. Integration or the mean-value theorem gives the same sign conclusion.

Question 14

IAT 2023 · Mathematics

Let

𝑓 (𝑥) = \sin (3 𝑥), 𝑥 \in [0, \frac{𝜋}{2}]

. Which of the following statements is true

$𝑓$ is increasing on $(\frac{𝜋}{4}, \frac{𝜋}{2})$ .
$𝑓$ is decreasing on $(\frac{𝜋}{4}, \frac{𝜋}{2})$
$𝑓$ is increasing on $(\frac{𝜋}{4}, \frac{𝜋}{3})$ and decreasing on $(\frac{𝜋}{3}, \frac{𝜋}{2})$ .
$𝑓$ is decreasing on $(\frac{𝜋}{4}, \frac{𝜋}{3})$ and increasing on $(\frac{𝜋}{3}, \frac{𝜋}{2})$ .

Answer & worked solution

Answer B

The derivative is

f'(x)=3\cos3x

. For

\pi/4<x<\pi/2

, the angle 3x lies between 3π/4 and 3π/2, where cosine is negative. Thus f is strictly decreasing throughout that interval, option B.

Question 15

IAT 2023 · Mathematics

Which one of the following functions is differentiable at

𝑥 = 0

$| 𝑥 |$
$| 𝑥 |^{\frac{1}{2}}$
$\sin | 𝑥 |$
$\cos | 𝑥 |$

Answer & worked solution

Answer D

Cosine is even, so

\cos|x|=\cos x

for all real x and is differentiable at zero with derivative zero. The function |x| has a corner, its square root has an unbounded slope, and sin|x| has unequal one-sided derivatives. Therefore D is the only differentiable option.

Physics

Question 1

IAT 2023 · Physics

A ball is thrown vertically upwards with an initial speed

𝑢

from a height

ℎ

above the ground. The ball eventually hits the ground with a speed

𝑣

. The acceleration due to gravity is

𝑔

and air resistance is negligible. What is the average speed of the ball over its entire trajectory?

$\frac{𝑔 ℎ}{2 (𝑢 + 𝑣)}$
$\frac{𝑢 + 𝑣}{2}$
$\frac{𝑢^{2} + 𝑣^{2}}{2 (𝑢 + 𝑣)}$
$\frac{𝑢^{2} + 𝑔 ℎ}{2 (𝑢 + 𝑣)}$

Answer & worked solution

Answer C

The upward part has length

u^2/(2g)

and the downward part has length

v^2/(2g)

. Their times are u/g and v/g. Total distance divided by total time is therefore

(u^2+v^2)/[2(u+v)]

, not the average of the endpoint speeds.

Question 2

IAT 2023 · Physics

Two cubes

𝐴

and

𝐵

of same dimensions are made of different materials with densities

𝜌_{𝐴}

and

𝜌_{B}

, respectively. Cube A floats in water with a fraction

𝜂

of its volume immersed. When cube

𝐵

is placed on top of cube

𝐴

, it is found that cube

𝐴

is just entirely immersed, while cube

𝐵

is entirely above the surface of the water. What is the ratio

𝜌_{B} / 𝜌_{A}

$(1 - 𝜂) / 𝜂$
$𝜂 / (1 - 𝜂)$
$𝜂$
$1 / 𝜂$

Answer & worked solution

Answer A

For cube A alone, buoyancy gives

\rho_A=\eta\rho_w

. With A fully submerged and B above water, the supported mass requires

\rho_A+\rho_B=\rho_w

. Thus

\rho_B/\rho_A=(\rho_w-\rho_A)/\rho_A=(1-\eta)/\eta

Question 3

IAT 2023 · Physics

An object is placed in front of a convex lens. A real inverted image of double its size is formed. When the object is moved closer to the lens by a distance

𝑑

, the image shifts away from the lens by a distance

8 𝑑

from its previous position. What is the magnitude of the magnification in the final setup?

$1 / 2$
4
8
1

Answer & worked solution

Answer B

Let the initial object distance be u, so the real image distance is 2u. The lens equation gives f = 2u/3. After the shift, distances are u−d and 2u+8d. Equating their lens formula to the original gives d = u/6. The final magnification magnitude is

(2u+8d)/(u-d)=(10u/3)/(5u/6)=4

Question 4

IAT 2023 · Physics

The frequency of the whistle of a train moving with a constant speed is observed by a stationary detector on the platform to be

𝑣_{1}

. The frequency of the same whistle is detected to be

𝑣_{2}

inside another train moving on a parallel track, at a speed

𝑣

towards the frst train. If the speed of sound in air is taken to be

𝑣_{𝑠}

ound, what is the ratio

𝑣 / 𝑣_{𝑠}

ound?

$\frac{𝑣_{2} - 𝑣_{1}}{𝑣_{1}}$
$\frac{𝑣_{2} - 𝑣_{1}}{𝑣_{2}}$
$\frac{𝑣_{2}}{𝑣_{1}}$
$\frac{𝑣_{1}}{𝑣_{2}}$

Answer & worked solution

Answer A

The stationary detector measures the incoming wave frequency ν1. Moving towards the same arriving wavefronts at speed v changes it to

\nu_2=\nu_1(1+v/v_{\rm sound})

. The source-motion factor is already included in ν1 and cancels in this ratio. Thus

v/v_{\rm sound}=(\nu_2-\nu_1)/\nu_1

Question 5

IAT 2023 · Physics

A current

𝐼

flows through a cylindrical cable of length

𝐿

and uniform cross-sectional area

𝐴

. The power dissipated due to the current is

𝑃_{1}

. The cable is cut into two equal halves. If the cross-sectional area and the current flowing through the two halves remain unchanged and the power dissipated in each half is

𝑃_{2}

, which of these options is correct?

$𝑃_{2} = 𝑃_{1} / 4$
$𝑃_{2} = 2 𝑃_{1}$
$𝑃_{2} = 𝑃_{1}$
$𝑃_{2} = 𝑃_{1} / 2$

Answer & worked solution

Answer D

Resistance is

R=\rho L/A

. Halving length while retaining material and cross-section halves the resistance. Each half still carries the same current, so its dissipation is

P_2=I^2(R/2)=P_1/2

Question 6

IAT 2023 · Physics

Particle A with charge

𝑄

and particle B with charge

2 𝑄

are fixed at positions

{\vec{𝑟}}_{𝐴}

and

{\vec{𝑟}}_{𝐵}

, respectively. The force on

𝐴

due to

𝐵

{\vec{𝐹}}_{𝐵 𝐴}

, and that on

𝐵

due to

𝐴

{\vec{𝐹}}_{𝐴 𝐵}

. Which of the following options is correct?

${\vec{𝐹}}_{AB} = {\vec{𝐹}}_{BA}$
${\vec{𝐹}}_{AB} = - {\vec{𝐹}}_{BA}$
${\vec{𝐹}}_{AB} = 2 {\vec{𝐹}}_{BA}$
${\vec{𝐹}}_{AB} = - 2 {\vec{𝐹}}_{BA}$

Answer & worked solution

Answer B

The two Coulomb forces have the same product of charges and the same separation, so their magnitudes are equal. Their directions are opposite along the joining line, as required by Newton's third law. Consequently

\vec F_{AB}=-\vec F_{BA}

even though the two charges differ.

Question 7

IAT 2023 · Physics

The magnetic fux

𝜙_{𝐵} (𝑡)

through a coil at a time

𝑡

is given by

𝜙_{𝐵} (𝑡) = 𝜙_{0} \cos 𝜔 𝑡

, where

0 \leq

𝜔 𝑡 \leq 𝜋

and

𝜙_{0}

is a non-zero constant. At what time is the magnitude of the induced emf a maximum?

0
$\frac{𝜋}{𝜔}$
$\frac{𝜋}{2 𝜔}$
$\frac{𝜋}{4 𝜔}$

Answer & worked solution

Answer C

Faraday's law gives

\mathcal E=-d\phi_B/dt=\phi_0\omega\sin\omega t

. In the interval 0 ≤ ωt ≤ π, its magnitude is greatest at ωt = π/2. Thus the maximum occurs at

t=\pi/(2\omega)

Question 8

IAT 2023 · Physics

The intrinsic electron and hole concentrations of a Si-based intrinsic semiconductor are

𝑛_{𝑒}^{(0)}

and

𝑛_{ℎ}^{(0)}

, respectively. Upon doping with trivalent impurities the respective carrier concentrations become

𝑛_{𝑒}

and

𝑛_{ℎ}

. Which of the following options is true?

$𝑛_{𝑒} = 𝑛_{ℎ}$
$𝑛_{𝑒} > 𝑛_{𝑒}^{(0)}$
$𝑛_{ℎ} > 𝑛_{ℎ}^{(0)}$
$𝑛_{𝑒}^{(0)}$ and $𝑛_{ℎ}^{(0)}$ are independent of temperature

Answer & worked solution

Answer C

Trivalent impurities act as acceptors in silicon and produce p-type material. Holes become the majority carriers, so their concentration increases above the intrinsic value. The electron concentration decreases under the mass-action relation at a fixed temperature. Therefore C is correct.

Question 9

IAT 2023 · Physics

The velocity $𝑣 (𝑡)$ of a particle moving in one dimension is given by:

$𝑣 (𝑡) = {\begin{matrix} 𝛼 𝑡, & 0 \leq 𝑡 \leq 𝑇 / 3 \\ 𝛼 𝑇 / 3, & 𝑇 / 3 \leq 𝑡 \leq 2 𝑇 / 3 \\ 𝛼 (𝑇 - 𝑡), & 2 𝑇 / 3 \leq 𝑡 \leq 𝑇 \end{matrix}$

where $𝛼 (\neq 0)$ is a constant. What is the displacement of the particle from time $𝑡 = 0$ to $𝑇$ ?

$\frac{2 𝛼 𝑇^{2}}{9}$
$\frac{4 𝛼 𝑇^{2}}{9}$
$\frac{8 𝛼 𝑇^{2}}{9}$
$\frac{7 𝛼 𝑇^{2}}{9}$

Answer & worked solution

Answer A

Displacement equals the signed area under the velocity–time graph. The two triangular sections each contribute

\frac12(T/3)(\alpha T/3)=\alpha T^2/18

, and the middle rectangle contributes

\alpha T^2/9

. Their sum is

2\alpha T^2/9

, valid for either sign of α.

Question 10

IAT 2023 · Physics

Two fixed point particles, each of charge $+ 𝑞$ , are separated by a distance $2 𝐿$ . Another point charge $+ 𝑞$ of mass $𝑚$ is oscillating about its equilibrium position as indicated in the figure below. The time period of oscillation is given by $𝑇 =$ $2 𝜋^{3 / 2} 𝛼 \sqrt{𝑚} / 𝑞$ . Given that $𝜖_{0}$ is the

permittivity of free space, which of the following options is the dimensionally correct expression for $𝛼$ in SI units?

IAT 2023, Physics, question 10 — question diagram

$𝜖_{0}^{3 / 2} 𝐿$
$𝜖_{0}^{1 / 2} 𝐿$
$𝜖_{0}^{3 / 2} 𝐿^{1 / 2}$
$𝜖_{0}^{1 / 2} 𝐿^{3 / 2}$

Transcription note: Restored the diagram from the 2023 paper: both fixed charges are +q. The secondary diagram incorrectly labelled one as +2q.

Answer & worked solution

Answer D

Coulomb's law implies

[\varepsilon_0]=Q^2T^2/(ML^3)

. Therefore

[\varepsilon_0^{1/2}L^{3/2}]=QT/M^{1/2}

. Multiplying this by

\sqrt m/q

, as in the given period, leaves time units. Only D has the required dimensions.

Question 11

IAT 2023 · Physics

$What is the effective resistance between 𝐴 and 𝐵 in the circuit shown below?$

$𝑅 / 3$
$2 R / 3$
$4 R / 3$
$𝑅 / 6$

Answer & worked solution

Answer C

The top, central and bottom A-to-B paths each have resistance R + 3R = 4R. Their division ratios are identical, so the three intermediate nodes are at equal potential and no current flows through the two vertical 2R links. The remaining three 4R branches are in parallel, giving

R_{\rm eq}=4R/3

Question 12

IAT 2023 · Physics

Consider a Carnot heat engine operating between a heat reservoir at temperature 600 K , and an external atmosphere at temperature 300 K . In one cycle, 1000 kJ of heat is extracted from the heat reservoir, and the associated work is input to a reversible refrigerator that operates between 200 K and the same external atmosphere at 300 K . The refrigerator completes one cycle and releases heat into the atmosphere. How much heat is released into the atmosphere at the end of one cycle of the combined system?

2500 Kj
2000 kJ
1000 kJ
500 kJ

Answer & worked solution

Answer B

The engine's efficiency is

1-300/600=1/2

, so it supplies 500 kJ of work and rejects 500 kJ to the atmosphere. The refrigerator has COP

200/(300-200)=2

, extracting 1000 kJ and rejecting

1000+500=1500

kJ. The combined atmospheric heat release is

500+1500=2000

kJ.

Question 13

IAT 2023 · Physics

Two spherical bodies A and B each of mass

𝑀

and radius

𝑅

are located such that their centers are apart by a distance of

4 𝑅

as shown in the figure. An object

𝐶

of mass

𝑚

is thrown from the surface of A directly towards the center of B with a speed

𝑣_{0} = 2 𝑣_{min}

, where

𝑣_{𝑚}

in is the minimum speed needed for

𝐶

to reach the surface of

𝐵

. Given that

𝐺

is the universal gravitational constant, how does the speed

𝑣 (𝑥)

of C change as a function of its distance from the center of A? IAT 2023, Physics, question 13 — question diagram

$𝑣 (𝑥) = \frac{2 \sqrt{2 𝐺 𝑀 𝑅}}{𝑥^{\frac{1}{2}} (4 𝑅 - 𝑥)^{\frac{1}{2}}}$
$𝑣 (𝑥) = \frac{2 𝑅 \sqrt{2 𝐺 𝑀 𝑅}}{𝑥 (4 𝑅 - 𝑥)}$
$𝑣 (𝑥) = \frac{\sqrt{2 𝐺 𝑀 𝑅}}{𝑥^{\frac{1}{2}} (4 𝑅 - 𝑥)^{\frac{1}{2}}}$
$𝑣 (𝑥) = \frac{6 𝑅^{2} \sqrt{2 𝐺 𝑀 𝑅}}{𝑥^{\frac{3}{2}} (4 𝑅 - 𝑥)^{\frac{3}{2}}}$

Answer & worked solution

Answer A

Along the joining line, potential per unit mass is

-GM[1/x+1/(4R-x)]

. Its maximum is at the midpoint x = 2R. From the launch point x = R, the barrier requires

v_{\mathrm{min}}^2=2GM/(3R)

. Since

v_0=2v_{\mathrm{min}}

, the total mechanical energy is zero. Thus

v^2=2GM[1/x+1/(4R-x)]=8GMR/[x(4R-x)]

, giving A.

Question 14

IAT 2023 · Physics

Consider the Bohr model of an atom where an electron of charge

- 𝑒

revolves around a nucleus of charge

+ 𝑒

in an orbit of radius

𝑟

. The electron has an orbital angular momentum

2 ℏ

. If the nucleus had charge

+ 2 𝑒

, what would have been the radius of the orbit of the electron with the same principal quantum number?

$2 𝑟$
$𝑟 / 2$
$𝑟$
$\sqrt{2} 𝑟$

Transcription note: Restored the reduced Planck constant in the stated angular momentum 2ℏ using the 2023 paper.

Answer & worked solution

Answer B

For a hydrogen-like ion in the same principal quantum state, the Bohr radius scales as

r_n\propto n^2/Z

. Doubling nuclear charge from +e to +2e halves the orbit radius while n remains fixed. Hence the new radius is r/2.

Question 15

IAT 2023 · Physics

A quantity has been measured to have a value of 0.00230 in some units. How many significant figures does the measured value have?

Answer & worked solution

Answer B

Leading zeros only locate the decimal point and are not significant. In 0.00230, the digits 2 and 3 are significant, and the final zero after the decimal records stated precision. There are therefore three significant figures.

Read the review method and correction register →

Continue exploringIAT 2022 solved paper

Back to home

IISER IAT 2023 question paper with solutions

Biology

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Chemistry

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Mathematics

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Physics

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15