IISER IAT 2020 Question Paper with Answers & Solutions

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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.

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

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

Biology

Question 1

IAT 2020 · Biology

Which of the following root modifications is observed in maize?

Fibrous roots
Lateral root
Prop roots
Stilt

Answer & worked solution

Answer D

Maize develops supporting adventitious roots from its lower stem nodes. These grow obliquely into the soil and are called stilt roots. Fibrous roots describe its general root-system type, but the supporting modification asked for here is the stilt root; prop roots, as in banyan, descend from branches.

Question 2

IAT 2020 · Biology

Which of the following chemical intermediates are products of light in photosynthesis?

ATP and NADPH
$H_{2} O$ and NADPH
${CO}_{2}$ and NADPH
${CO}_{2}$ and ATP

Answer & worked solution

Answer A

The light reactions convert light energy into chemical energy by producing ATP and reducing NADP⁺ to NADPH. These products supply energy and reducing power for carbon fixation. Water is consumed as an electron source, while carbon dioxide is fixed in the subsequent carbon-assimilation reactions.

Question 3

IAT 2020 · Biology

Which of the following organisms produce Cry proteins?

Bacillus thuringiensis
Saccharomyces cerevisiae
Escherichia coli
Pseudomonas syringe

Answer & worked solution

Answer A

Cry proteins are insecticidal crystal proteins naturally produced by Bacillus thuringiensis. Genes encoding selected Cry proteins are used in Bt crops. Although laboratory bacteria can be engineered to express these genes, the naturally associated organism among the options is B. thuringiensis.

Question 4

IAT 2020 · Biology

A hypothetical plant species shows two flower colour phenotypes- red and white. Red is dominant over white. In a population, this flower colour locus was found to be in HardyWeinberg equilibrium. If 490 out of a total population size of 1000 plants were found to be white-flowered, how many of the red-flowered plants will be heterozygous?

Answer & worked solution

Answer D

White flowers are recessive, so

q^2=490/1000=0.49

. Thus

q=0.7

and

p=0.3

. Heterozygotes have frequency

2pq=2(0.3)(0.7)=0.42

. There are therefore 420 heterozygous red-flowered plants. The other 90 red plants are homozygous dominant.

Question 5

IAT 2020 · Biology

Troponin controls binding of actin to myosin motor, thereby regulating muscle contraction. Binding of which metal ions to troponin is crucial for this purpose?

$𝑁 𝑎$
$𝐾$
$𝑀 𝑔$
Ca

Answer & worked solution

Answer D

Calcium ions bind to troponin C and change the position of the troponin–tropomyosin complex. This exposes the myosin-binding sites on actin, allowing cross-bridge formation. Sodium and potassium help generate electrical signals, but calcium is the direct regulatory ion in this step of contraction.

Question 6

IAT 2020 · Biology

The following word diagram depicts a food chain in a forest.

$Trees (Aspen, willow) \to Herbivore (Elk) \to Predator (Wolf)$

Which is the most likely outcome of depletion in predator population?

Increased browsing by the herbivores stimulates the plant growth leading to increased primary productivity
Overgrowth of herbivore population causing reduction in tree populations and therefore, if the process continues the entire ecosystem will breakdown
No change in the ecosystem as the herbivore and the tree populations will continue to maintain equilibrium
Overgrowth of herbivore population without any change in the remaining part of the ecosystem

Answer & worked solution

Answer B · qualification below

The predicted direction is sound; complete ecosystem collapse is not inevitable.

Reducing the wolf population weakens predation on elk. More elk can then browse the aspen and willow, reducing tree recruitment and abundance. This is a trophic cascade. B identifies the expected direction of the changes; its prediction of complete ecosystem breakdown is a possible extreme outcome, rather than an inevitable consequence of every predator decline.

Question 7

IAT 2020 · Biology

Which of the following is the correct description of interferon?

A synthetic neurotransmitter used to treat neuronal diseases
A class of proteins secreted by B-lymphocytes that categorically attack and destroy virusinfected cells
Signalling molecules that interfere with its own synthesis inside a secretory cell
A class of proteins secreted by virus infected cells and are involved in protecting uninfected cells

Answer & worked solution

Answer D

Virus-infected cells release interferons that signal neighbouring cells to activate antiviral defences. These proteins help limit viral replication and spread. They are not antibodies secreted by B cells, nor are they synthetic neurotransmitters. Their name reflects interference with viral infection, not inhibition of their own synthesis.

Question 8

IAT 2020 · Biology

Which of the following products are expected when lipase acts on diglycerides?

Fatty acids and glucose
Glucose and galactose
Fatty acids and glycerol
Monoglycerides and glycerol

Answer & worked solution

Answer C

A diglyceride contains two fatty-acid ester bonds to glycerol. Complete hydrolysis of both bonds by lipase releases two fatty acids and glycerol. Partial hydrolysis can produce a monoglyceride plus a fatty acid, but the complete-hydrolysis products represented among these options are fatty acids and glycerol.

Question 9

IAT 2020 · Biology

Radial symmetry is NOT found in which of the following taxa?

Echinodermata
Cnidaria
Ctenophora
Cyclostomata

Answer & worked solution

Answer D

Cyclostomes are vertebrates with a bilateral body plan. Radial organisation occurs in cnidarians and adult echinoderms; ctenophores are commonly described as biradial. Echinoderm larvae are bilateral, but the adult body plan is the relevant comparison in this question.

Question 10

IAT 2020 · Biology

Reabsorption of which of these molecules in kidney occurs via passive transport?

Urea
Glucose
Sodium ions
Amino acids

Answer & worked solution

Answer A

Urea can move passively down its concentration gradient, including through facilitated urea transporters in appropriate nephron segments. Glucose and amino-acid reabsorption depend primarily on sodium-linked transport, and sodium transport is driven by active mechanisms. Therefore urea is the intended passive-reabsorption example.

Question 11

IAT 2020 · Biology

$Match the human cell types (column I) to the number of chromatids (column II)$

Column I		Column II
(a)	Col $𝐺_{2} phase cells$	(i)	46
(b)	$𝐺_{1} phase cells$	(ii)	92
(c)	Spermatogonia (pre-replication)	(iii)	23
(d)	Ovum

a-ii, b-i, c-iii, d-iii
$𝑎 - 𝑖 𝑖, 𝑏 - 𝑖, 𝑐 - 𝑖, 𝑑 - 𝑖 𝑖 𝑖$
a-ii, b-iii, c-i, d-iii
a-i, b-ii, c-iii, d-iii

Transcription note: Removed stray table text introduced by transcription; cell types and counts are unchanged.

Answer & worked solution

Answer B

DNA has already replicated in a diploid G₂ cell, giving 46 chromosomes but 92 chromatids. A G₁ cell and a spermatogonium before replication each have 46 chromatids. A mature ovum is haploid with 23 single-chromatid chromosomes. Hence a–ii, b–i, c–i and d–iii is the correct match.

Question 12

IAT 2020 · Biology

Which of the following processes is NOT involved in functionalisation of mRNA (from hnRNA) in eukaryotes?

Amino acylation
$5^{'}$ -end capping
Polyadenylation
Splicing

Answer & worked solution

Answer A

Eukaryotic pre-mRNA processing includes addition of a 5′ cap, a poly-A tail and removal of introns by splicing. Aminoacylation attaches an amino acid to its corresponding tRNA. It is essential to translation but is not a step that converts hnRNA into mature mRNA.

Question 13

IAT 2020 · Biology

Which of the following processes increases the uptake of water and minerals by roots in legumes?

Active uptake by xylem
Pumping through plasmodesmata
Pumping through cortical cells
Formation of mycorrhizae

Answer & worked solution

Answer D

Mycorrhizal fungi extend the effective absorbing network beyond the root surface. Their hyphae improve access to water and mineral nutrients, particularly phosphate, in exchange for plant-derived carbon. This differs from nitrogen fixation in legume nodules, which addresses a different nutrient-acquisition process.

Question 14

IAT 2020 · Biology

Which of the following graphs correctly represents the relationship between population growth rate

(dN / dt)

and population size

(N)

, if the population shows logistic growth?

Answer & worked solution

Answer C

For logistic growth,

dN/dt=rN(1-N/K)

. As a function of N this is a downward-opening quadratic: it is zero at

N=0

and

N=K

and reaches its maximum at

N=K/2

. C is the qualitative rise-and-fall sketch. The ideal mathematical curve is a parabola, even though the reproduced sketch is not drawn precisely to that shape.

Question 15

IAT 2020 · Biology

Which of the following processes gives rise to new organisms from unfertilized ova in some species of plants and animals?

Autogamy and parthenogenesis
Parthenogenesis and apomixis
Syngamy and apomixis
Xenogamy and autogamy

Answer & worked solution

Answer B

Parthenogenesis is development of an individual from an unfertilised egg. In plants, apomixis includes forms of seed formation without fertilisation, including parthenogenetic development of an egg. Autogamy and xenogamy are pollination processes, and syngamy is fusion of gametes, so they do not describe the requested unfertilised development.

Chemistry

Question 1

IAT 2020 · Chemistry

$How many atoms are present in a hexagonal close-packed unit cell?$

Answer & worked solution

Answer D · qualification below

Uses the conventional hexagonal-prism unit cell.

In the conventional hexagonal-prism HCP cell, 12 corner atoms contribute

12/6=2

, the two basal-face centres contribute

2/2=1

, and three atoms lie wholly inside. The total is

2+1+3=6

. This answer uses the conventional cell; the smaller primitive HCP cell has a two-atom basis.

Question 2

IAT 2020 · Chemistry

The molar conductivity of 0.00241 M acetic acid is

32.87 {Scm}^{2} {mol}^{- 1}

. If the limiting molar conductivity at 298 K is

390.5 {Scm}^{2} {mol}^{- 1}

, what is the dissociation constant (

mol L^{- 1}

$5.36 \times 10^{- 5}$
$1.86 \times 10^{- 5}$
$5.36 \times 10^{- 4}$
$1.86 \times 10^{- 4}$

Answer & worked solution

Answer B

The degree of dissociation is

\alpha=\Lambda_m/\Lambda_m^\circ=32.87/390.5\approx0.08417

. For a weak monoprotic acid,

K_a=c\alpha^2/(1-\alpha)

. Substituting

c=0.00241

gives

K_a\approx1.86\times10^{-5}\ \mathrm{mol\,L^{-1}}

, matching B.

Question 3

IAT 2020 · Chemistry

$What is the most appropriate Lewis structure of {NO}_{2} F ?$

Answer & worked solution

Answer D

The total valence-electron count is

5+2(6)+7=24

. Nitrogen cannot exceed an octet, so it should have one N=O bond, one N–O bond and one N–F bond. This assigns formal charges

+1

to N and

-1

to the singly bonded oxygen, with resonance between the two oxygens. D represents this arrangement; two N=O bonds plus N–F would put ten electrons around nitrogen.

Question 4

IAT 2020 · Chemistry

Common disaccharide sucrose is held together by a glycosidic linkage between

C1 of $𝛽$ -D-glucose and C2 of $𝛼$ -D-fructose.
C2 of $𝛼$ -D-glucose and C1 of $𝛽$ -D-fructose.
C1 of $𝛼$ -D-glucose and C2 of $𝛽$ -D-fructose.
C1 of $𝛼$ -D-glucose and C2 of $𝛼$ -D-fructose.

Answer & worked solution

Answer C

Sucrose joins the anomeric carbon C1 of α-D-glucose to the anomeric carbon C2 of β-D-fructose. The linkage is therefore α(1→2)β. Because both anomeric centres participate, sucrose has no free reducing hemiacetal group.

Question 5

IAT 2020 · Chemistry

Select the correct option based on the increasing order of boiling point of the molecules given below. IAT 2020, Chemistry, question 5 — question diagram

$𝐼 < 𝐼 𝐼 𝐼 < 𝐼 𝑉 < 𝐼 𝐼$
$𝐼 𝐼 < 𝐼 < 𝐼 𝑉 < 𝐼 𝐼 𝐼$
$𝐼 < 𝐼 𝐼 𝐼 < 𝐼 𝐼 < 𝐼 𝑉$
III $<$ I $<$ II $<$ IV

Answer & worked solution

Answer C

The alkane I has only dispersion forces and boils first. Aldehyde III has dipole–dipole attraction but cannot hydrogen-bond strongly to itself as a donor. Alcohol II forms intermolecular hydrogen bonds. Carboxylic acid IV forms strongly associated hydrogen-bonded dimers. For these specific compounds the order is

I<III<II<IV

Question 6

IAT 2020 · Chemistry

Identify the major product that forms easily and faster in the following reaction. (Tetrahydrofuran= solvent) IAT 2020, Chemistry, question 6 — question diagram

Answer & worked solution

Answer B

Borane generated from diborane in THF reduces carboxylic acids rapidly and selectively. The free –COOH group becomes –CH₂OH, while the ester and nitro substituents remain in the product selected here. This gives B. Reducing the ester or nitro group instead would miss the reagent's preferential reaction with the acid under these conditions.

Question 7

IAT 2020 · Chemistry

A first-order reaction is found to have a half-life of one second (sec). What will be the time required for

99.9 %

completion of the reaction?

0.693 sec
2 sec
10 sec
69.3 sec

Answer & worked solution

Answer C

A first-order reaction leaves fraction

(1/2)^{t/t_{1/2}}

unreacted. For 99.9% conversion the remaining fraction is

0.001

. With a one-second half-life,

t=\log_2(1000)=9.966\ldots

s, approximately 10 s. This is about ten half-lives, not one thousand half-lives.

Question 8

IAT 2020 · Chemistry

The total number of nodes for a given principal quantum number

𝑛

, and azimuthal quantum number

ℓ

, are

sum of $ℓ$ angular nodes and $𝑛 + ℓ - 1$ radial nodes
sum of $ℓ$ angular nodes and $𝑛 - ℓ - 1$ radial nodes
sum of $ℓ + 1$ angular nodes and $𝑛 - ℓ - 1$ radial nodes
sum of $ℓ + 1$ angular nodes and $𝑛 + ℓ - 1$ radial nodes

Answer & worked solution

Answer B

A hydrogen-like orbital has

\ell

angular nodes and

n-\ell-1

radial nodes. Adding them gives a total of

n-1

. The radial count excludes the origin and infinity, where the wavefunction may vanish without defining an additional radial nodal surface.

Question 9

IAT 2020 · Chemistry

Which among the following is NOT correct?

$H_{3} {PO}_{4}$ is a tribasic acid
$H_{3} {PO}_{3}$ is a dibasic acid.
$H_{3} {PO}_{2}$ is a monobasic acid.
$H_{3} {PO}_{2}$ is a tribasic acid

Answer & worked solution

Answer D

For these phosphorus oxyacids, acidic hydrogens are those attached to oxygen.

\mathrm{H_3PO_4}

has three O–H groups,

\mathrm{H_3PO_3}

has two and

\mathrm{H_3PO_2}

has one. P–H hydrogens are not counted in their ordinary acid basicity. Thus calling hypophosphorous acid tribasic is incorrect.

Question 10

IAT 2020 · Chemistry

Which among the following is NOT correct regarding the first ionization potential

({IP}_{1})

of period

𝐼 𝐼 𝐼

elements

𝑀 𝑔, 𝐴 𝑙, 𝑆 𝑖, 𝑃

and

𝑆

$𝑃 < 𝑆$
$𝐴 𝑙 < 𝑀 𝑔$
$𝐴 𝑙 < 𝑆 𝑖$
$𝑆 𝑖 < 𝑃$

Answer & worked solution

Answer A

Phosphorus has a half-filled

3p^3

subshell. Sulfur has

3p^4

, with one paired set of electrons; removing a paired electron is relatively easier. Consequently the first ionisation energy of P is greater than that of S, so

P<S

is the incorrect comparison. The other stated inequalities agree with the period's trend and the Mg–Al subshell exception.

Question 11

IAT 2020 · Chemistry

Which one of the following aqueous solutions gives the most intense colour?

$0.1 M [Mn {(H_{2} O)}_{6}] {Cl}_{2}$
$0.1 M [Mn {({NH}_{3})}_{6}] {Cl}_{3}$
$0.1 M [Mn {({NH}_{3})}_{5} Cl] Cl$
$0.1 M {KMnO}_{4}$

Answer & worked solution

Answer D

Permanganate has intense ligand-to-metal charge-transfer absorption, producing a strong purple colour even though Mn(VII) is

d^0

. Such allowed charge-transfer transitions are much stronger than ordinary, often forbidden,

d

–

d

transitions in the listed manganese complexes. The number of unpaired electrons alone does not determine colour intensity.

Question 12

IAT 2020 · Chemistry

What is the correct relation between the heat capacity at constant volume

(𝐶_{𝑣})

and heat capacity at constant pressure

(𝐶_{𝑝})

for an ideal gas? (

𝑅 =

Universal gas constant)

$𝐶_{𝑣} + 𝐶_{𝑝} = 2 𝑅$
$𝐶_{𝑃} - 𝐶_{𝑉} = 𝑅$
$𝐶_{𝑣} - 𝐶_{𝑝} = 2 𝑅$
$𝐶_{𝑣} + 𝐶_{𝑝} = 𝑅$

Answer & worked solution

Answer B

For one mole of ideal gas,

H=U+RT

. Differentiating with respect to temperature gives the molar relation

C_p=C_v+R

. Hence

C_p-C_v=R

. If the symbols instead meant total heat capacities of n moles, the difference would be

nR

; the options use molar heat capacities.

Question 13

IAT 2020 · Chemistry

$The number of unpaired electron(s) in the square planar complex [Ni (CN)_{4}]^{2 -} is/are$ :

Answer & worked solution

Answer D

[\mathrm{Ni(CN)_4}]^{2-}

, nickel is

+2

and has a

d^8

configuration. The strong-field square-planar splitting leaves the highest

d_{x^2-y^2}

orbital empty while the lower four d orbitals are doubly occupied. All electrons are paired, so the number of unpaired electrons is zero.

Question 14

IAT 2020 · Chemistry

$Select the most appropriate compound I and the reagents for the synthesis of compound II$

Answer & worked solution

Answer C

The cyclic ketal first hydrolyses in aqueous acid to acetophenone I. A methyl ketone undergoes the haloform reaction with iodine and base, producing benzoate and iodoform. Acidification then gives benzoic acid II. Therefore I must be acetophenone and the reagent set must include

\mathrm{I_2/NaOH}

, as in C.

Question 15

IAT 2020 · Chemistry

$Identify the final major product III in the following reaction sequence:$

Answer & worked solution

Answer C · qualification below

Intended formal product; the unprotected nitro group makes routine Grignard preparation problematic.

The formal exam sequence first nitrates bromobenzene mainly para to Br. Replacing that Br by a magnesium-halide group, then carboxylating and acidifying, would give para-nitrobenzoic acid, C. There is a chemical limitation: an unprotected nitro group is incompatible with routine Grignard preparation using Mg in ether. Thus C is the intended formal product, but the stated sequence should not be taught as a reliable practical synthesis without special conditions or a redesigned route.

Mathematics

Question 1

IAT 2020 · Mathematics

If S is the set of all points

(𝑥, 𝑦)

satisfying the equation

\sin 𝑥 = \cos 𝑦

, then which of the following statements is correct?

$𝑆$ contains all points of a parabola.
$𝑆$ contains all points of a circle.
$𝑆$ contains all points of a hyperbola.
$𝑆$ contains all points of a straight line.

Answer & worked solution

Answer D

Use

\cos y=\sin(\pi/2-y)

. One family of solutions is

x=\pi/2-y+2k\pi

, so every point on each line

x+y=\pi/2+2k\pi

belongs to S. Another family is also linear. Thus S contains an entire straight line, making D correct.

Question 2

IAT 2020 · Mathematics

𝐴 = [\begin{matrix} 1 & 𝑎 & 0 \\ 0 & 1 & 𝑏 \\ 0 & 0 & 1 \end{matrix}]

, then the determinant of

𝐼 - 𝐴 + 𝐴^{2} - 𝐴^{3} + 𝐴^{4} - \dots + 𝐴^{2020}

2020
$𝑎^{2020} - 𝑏 𝑎^{2019} + \dots - 𝑏^{2019} 𝑎 + 𝑏^{2020}$
$2020^{3}$
1

Answer & worked solution

Answer D

A and every power of A are upper triangular with diagonal entries 1. The alternating sum has 2021 terms, beginning and ending with a positive sign. Each diagonal entry is therefore

1-1+1-\cdots+1=1

. Its determinant, the product of the three diagonal entries, is 1, regardless of a and b.

Question 3

IAT 2020 · Mathematics

Let

𝑆 = {(𝑎, 𝑏) : 𝑎, 𝑏 \in 𝑄}

and

ℓ

be the line

𝑦 = 𝑚 𝑥 + 𝑐

. Which of the following statements is not correct?

If $ℓ$ passes through two points of $𝑆$ , then $𝑚$ must be rational.
If $𝑚$ is rational and $𝑐$ is irrational, then $ℓ$ passes through at least one point of $𝑆$ .
If $ℓ$ passes through exactly one point of $𝑆$ , then $𝑚$ must be irrational.
If $𝑚$ is irrational and $𝑐$ is rational, then $ℓ$ passes through exactly one point of $𝑆$ .

Answer & worked solution

Answer B

If m and x are rational while c is irrational, then

mx+c

is irrational. No point on that line can have both coordinates rational, so B is false. Two distinct rational points give a rational slope; with irrational slope and rational intercept, the only rational point is

(0,c)

. These observations verify the other statements.

Question 4

IAT 2020 · Mathematics

If $𝑝 (𝑡) = \frac{𝑡 (𝑡 - 1) \dots (𝑡 - 2019)}{2019!}$ , then the value of

$\int_{0}^{1} (\frac{1}{𝑡 + 1} + \frac{1}{𝑡 + 2} + \dots + \frac{1}{𝑡 + 2020}) 𝑝 (- 𝑡 - 1) 𝑑 𝑡$

is:

$2019^{2}$
2019
$2020^{2}$
2020

Answer & worked solution

Answer C

Define

F(t)=p(-t-1)=\prod_{j=1}^{2020}(t+j)/2019!

, since the number of minus signs is even. Its logarithmic derivative is

F'/F=\sum_{j=1}^{2020}1/(t+j)

. The integral is therefore

F(1)-F(0)=2021!/2019!-2020!/2019!=2020^2

Question 5

IAT 2020 · Mathematics

Let

𝑎, 𝑏

be nonzero real numbers. If

𝑝 (𝑥) = 𝑥^{𝑛} + 𝑐_{𝑛 - 1} 𝑥^{𝑛 - 1} + \dots + 𝑐_{0}

, where

𝑐_{𝑛 - 1}, \dots, 𝑐_{0}

are integers and

𝑛 \geq 3

, then which of the following statements is correct?

If $𝑎 + 𝑖 𝑏$ is a root of $𝑝 (𝑥)$ , then $𝑛$ must be even.
If $𝑎$ is a root of $𝑝 (𝑥)$ , then $𝑛$ must be odd.
If $𝑎 + 𝑖 𝑏$ is a root of $𝑝 (𝑥)$ , then $(𝑎 + 𝑖 𝑏)^{2}$ can never be a root of $𝑝 (𝑥)$ .
If $𝑎$ is a root of $𝑝 (𝑥)$ , then $𝑎$ must be an integer.

Answer & worked solution

No valid listed option

No valid listed option for the stated real-number condition.

None of the four statements is always true for the real numbers stated. A fails for the cubic

(x-1)(x^2-2x+2)

, which has root

1+i

. B and D both fail for

(x^2-2)(x^2+1)

, an even-degree monic integer polynomial with real root

\sqrt2

. C fails for

(x^2-2x+2)(x^2+4)

: both

1+i

and

(1+i)^2=2i

are roots. D would follow from the rational-root theorem if a were required to be rational, but the reproduced question says real. The source's D key is therefore not valid for the wording shown.

Question 6

IAT 2020 · Mathematics

The number of skew-symmetric matrices

𝐴 = {[𝑎_{𝑖} 𝑗]}_{3 \times 3}

, where

𝑎_{𝑖} 𝑗 \in {- 3, - 2, - 1, 0, 1, 2, 3}

is:

$7^{3}$
$3^{7}$
$21^{3}$
$7^{6}$

Answer & worked solution

Answer A

Skew-symmetry requires

A^T=-A

. Every diagonal entry is zero, while the three entries above the diagonal can each be chosen freely from the seven allowed values. The corresponding entries below are then fixed as their negatives. Thus there are

7^3

matrices.

Question 7

IAT 2020 · Mathematics

$Define binary operation * on 𝑄 by 𝑎 * 𝑏 = 𝑎 + 𝑏 - 3 . The inverse of 2 with respect to * is$

Answer & worked solution

Answer A

The identity e must satisfy

a*e=a

, so

a+e-3=a

and

e=3

. The inverse b of 2 must then satisfy

2*b=3

, giving

2+b-3=3

and

b=4

. Checking

4*2

also gives the identity 3.

Question 8

IAT 2020 · Mathematics

Let $[𝑥]$ denote the greatest integer less than or equal to $𝑥$ . The number of positive integer solutions of the equation

$[\frac{𝑥}{19}] = [\frac{𝑥}{20}]$

are:

Transcription note: Removed a stray solution fragment appended to the option; the scanned paper gives only 187.

Answer & worked solution

Answer C

Let the common floor be k. Then

20k\le x\le19k+18

. For

k=0

, the positive solutions are 1 through 18. For

k=1,\ldots,18

, there are

19-k

solutions; larger k give none. The total is

18+(18+17+\cdots+1)=18+171=189

Question 9

IAT 2020 · Mathematics

Let $𝑓 : ℝ \to ℝ$ and $𝑔 : ℝ \to ℝ$ be two functions such that

$𝑔 \circ 𝑓 (𝑥) = {\begin{matrix} 𝑥^{2}, & if 𝑥 \geq 0 \\ 𝑒^{𝑥} - 1, & if 𝑥 < 0 \end{matrix}$

Then which of the following statements is correct?

$𝑔$ is onto.
$𝑓$ is onto.
$𝑔$ is one-one.
$𝑓$ is one-one.

Answer & worked solution

Answer D

The composite is strictly increasing on each side of zero. Its negative-side outputs lie in

(-1,0)

and its nonnegative-side outputs in

[0,\infty)

, so it is also injective across the join. If

f(x_1)=f(x_2)

, applying g gives equal composite values and therefore

x_1=x_2

. Thus f must be one-to-one. The properties listed for g or surjectivity of f are not forced.

Question 10

IAT 2020 · Mathematics

$𝐹 (𝑥) = \int_{0}^{𝑒^{𝑥}} (𝑡^{3} + 2 𝑡^{2} - 𝑡 - 2) 𝑑 𝑡$ , then for how many real numbers $𝑥$ does $𝐹^{'} (𝑥) = 0$ ?

Answer & worked solution

Answer D

By the fundamental theorem and chain rule,

F'(x)=e^x[(e^x)^3+2(e^x)^2-e^x-2]

. The cubic factors as

(t+2)(t+1)(t-1)

. Since

t=e^x

is positive, only

t=1

is possible, giving exactly one real solution,

x=0

Question 11

IAT 2020 · Mathematics

Let $𝐶$ be a circle which passes through $(- 2, 1)$ and $(0, 3)$ and whose centre lies on the line $𝑦 - 𝑥 = 1$ . Then which of the following points lies on $𝐶$ ?

$(\sqrt{2} + 1, 1)$
$(1, \sqrt{2} + 1)$
$(1, \sqrt{3} + 1)$
$(1, \sqrt{3} - 1)$

Answer & worked solution

Answer C

Write the centre as

(h,h+1)

. Equating its squared distances from

(-2,1)

and

(0,3)

gives

(h+2)^2+h^2=h^2+(h-2)^2

, so

h=0

. The circle is

x^2+(y-1)^2=4

. At

x=1

y=1\pm\sqrt3

, and C is the listed point satisfying this equation.

Question 12

IAT 2020 · Mathematics

For what value of

𝑡

does

𝑓 (𝑥) = 𝑥^{3} + 𝑡 𝑥^{2} + (2 - 𝑡) 𝑥 - 3

have only real roots?

$𝑡 = 0$ .
$𝑡 = 4$ .
$𝑡 = 2$ .
$𝑡 = - 1$

Answer & worked solution

Answer B

The polynomial always has root 1 and factors as

(x-1)[x^2+(t+1)x+3]

. The remaining two roots are real when

(t+1)^2-12\ge0

. Among

t=0,4,2,-1

, only

t=4

satisfies this condition. Repeated roots, if present at a boundary value, would also count as real.

Question 13

IAT 2020 · Mathematics

𝑓 (𝑥) = 𝑎 𝑥^{2} + 𝑏 𝑥 + 𝑐

and

𝑓 (\frac{1}{𝑛}) = \frac{𝑛 + 1}{𝑛^{2}}

for all

𝑛 \in 𝑁

, then what is the value of

lim_{𝑥 \to 0} 𝑓^{'} (𝑥)

2
0
1
-1

Answer & worked solution

Answer C

Multiplying the given identity by

n^2

gives

a+bn+cn^2=n+1

for infinitely many positive integers n. The two polynomials must be identical, so

a=1

b=1

and

c=0

. Therefore

f'(x)=2x+1

and its limit at zero is 1.

Question 14

IAT 2020 · Mathematics

A bag contains 8 balls, of which 5 are green and 3 are red. The probability that two balls chosen at random are of the same colour is :

$12 / 28$
$14 / 28$
$15 / 28$
$13 / 28$

Answer & worked solution

Answer D

Without replacement, there are

\binom82=28

unordered pairs. Same-colour pairs number

\binom52+\binom32=10+3=13

. The required probability is

13/28

. Green–red pairs are the complementary 15 outcomes.

Question 15

IAT 2020 · Mathematics

Consider the differential equation

𝑓^{''} + 𝑝 𝑓^{'} + 𝑞 𝑓 = 0

where

𝑝

and

𝑞

are constants,

𝑝^{2} = 4 𝑞

and

𝑞 \neq 0

. If y is a nonzero solution of the equation, then which of the following statements is correct?

$(1 + 𝑥) 𝑦$ can never be a solution
$𝑦^{2}$ can never be a solution
$𝑦^{'}$ can never be a solution
$𝑥 𝑦$ can never be a solution

Answer & worked solution

Answer B

The repeated characteristic root is

r=-p/2\ne0

, so every solution is

(A+Bx)e^{rx}

. Squaring a nonzero solution produces

(A+Bx)^2e^{2rx}

, which cannot have that same exponential-linear form when

r\ne0

. In contrast, derivatives of solutions are solutions, and for

y=e^{rx}

both

xy

and

(1+x)y

are solutions. Hence only B is always true.

Physics

Question 1

IAT 2020 · Physics

The potential energy of a point particle of mass $𝑚$ undergoing rectilinear motion along the $𝑥$ -axis is given by

$𝑉 (𝑥) = 𝐴 𝑥 + 𝐵 𝑥^{2}$

What is the maximum speed attained by the particle if it starts from rest at $𝑥 = \frac{𝐴}{𝐵}$ ?

$\frac{3 𝐴}{\sqrt{2 𝐵 𝑚}}$
$\frac{𝐴}{\sqrt{2 𝐵 𝑚}}$
$\frac{2 𝐴}{\sqrt{𝐵 𝑚}}$
$𝐴 \sqrt{\frac{2}{𝐵 𝑚}}$

Answer & worked solution

Answer A · qualification below

The finite maximum assumes B > 0; the listed sign assumes A > 0.

For

B>0

, complete the square:

V=B(x+A/(2B))^2-A^2/(4B)

. Its minimum is

-A^2/(4B)

, while the initial energy is

V(A/B)=2A^2/B

. Thus

mv_{\mathrm{max}}^2/2=9A^2/(4B)

and

v_{\mathrm{max}}=3|A|/\sqrt{2Bm}

. Option A assumes positive A, as the options implicitly do. If B is not positive, a bounded oscillation with this finite maximum is not guaranteed.

Question 2

IAT 2020 · Physics

Two identical containers kept at the same temperature carry the same gas with different molecular mean free paths $𝑙_{1}$ and $𝑙_{2}$ . The gas from the first container is completely transferred into the second at the same temperature. What is the molecular mean free path of the resultant mixture in the second container?

$\frac{𝑙_{1} 𝑙_{2}}{𝑙_{1} + 𝑙_{2}}$
$\sqrt{𝑙_{1} 𝑙_{2}}$
$𝑙_{1} + 𝑙_{2}$
$\sqrt{𝑙_{1}^{2} + 𝑙_{2}^{2}}$

Answer & worked solution

Answer A

For the same gas, mean free path is inversely proportional to number density:

\ell=1/(\sqrt2\,\sigma n)

. Equal container volumes imply that transferring all gas adds the number densities,

n=n_1+n_2

. Therefore

1/\ell=1/\ell_1+1/\ell_2

, giving

\ell=\ell_1\ell_2/(\ell_1+\ell_2)

Question 3

IAT 2020 · Physics

Consider the circuit with a Zener diode whose breakdown voltage $𝑉_{𝑧} = 4 V$ . What is the voltage $𝑉_{0}$ if $𝑉_{𝑖} = 20 V$ ?

Answer & worked solution

Answer B

In reverse breakdown the ideal Zener maintains 4 V across itself. The remaining

20-4=16

V is shared by the two equal series resistors, so each drops 8 V. The output includes the Zener plus the lower resistor:

V_0=4+8=12

V. It is not merely the Zener voltage because the output terminals span both components.

Question 4

IAT 2020 · Physics

Consider a mass-pulley system as shown in the figure. There is a wedge of mass $𝑀$ and equal wedge angles $𝜃$ lying on a rigid horizontal table. The coefficient of friction between the wedge and the table is $𝜇$ . There are two blocks of mass $𝑚_{1}$ and $𝑚_{2}$ lying on the incline of the wedge. The coefficients of friction between the blocks and wedge are $𝜇_{1}$ and $𝜇_{2}$ as shown in the figure. Consider $𝑚_{1} > 𝑚_{2}$ and the coefficients of friction ( $𝜇, 𝜇_{1}$ and $𝜇_{2}$ ) to be less than $\tan 𝜃$ . Gravity is acting downwards with acceleration due to gravity $𝑔$ . What should be the value of $\frac{𝑚_{1}}{𝑚_{2}}$ so that the system is in equilibrium?

$\frac{𝜇_{1} 𝜇_{2} \cos 𝜃 + \sin 𝜃}{\sin 𝜃 - 𝜇 \cos 𝜃}$
$\frac{𝜇_{1} \cos 𝜃 + \sin 𝜃}{\sin 𝜃 - 𝜇_{2} \cos 𝜃}$
$\frac{𝜇_{2} \cos 𝜃 + \sin 𝜃}{\sin 𝜃 - 𝜇_{1} \cos 𝜃}$
$\frac{𝜇 \cos 𝜃 + \sin 𝜃}{\sin 𝜃 - 𝜇_{1} \cos 𝜃}$

Answer & worked solution

Answer C · qualification below

The listed result is the limiting upper mass ratio, not every possible equilibrium ratio.

At the limiting condition where the heavier block is about to descend, friction on

m_1

acts up its slope and friction on

m_2

acts down its slope. Equal tension gives

m_1g(\sin\theta-\mu_1\cos\theta)=m_2g(\sin\theta+\mu_2\cos\theta)

. Their ratio is the expression in C. More generally, static equilibrium permits a range of ratios; C is its upper limiting value for the specified tendency. The table friction does not enter this balance for an entirely stationary system.

Question 5

IAT 2020 · Physics

What is the maximum angle of incidence $𝜃_{𝑖}$ in the figure for which the ray would satisfy the condition for total internal reflection at the boundary of media of refractive indices $𝑛_{1}$ and $𝑛_{2}$ , where $𝑛_{1} > 𝑛_{2}$ ?

$𝜃_{𝑖} = \sin^{- 1} (𝑛_{1}^{2} - 𝑛_{2}^{2})$
$𝜃_{𝑖} = \sin^{- 1} {(𝑛_{1} - 𝑛_{2})}^{2}$
$𝜃_{𝑖} = \sin^{- 1} (\sqrt{\frac{𝑛_{2}}{𝑛_{1}}})$
$𝜃_{𝑖} = \sin^{- 1} (\sqrt{𝑛_{1}^{2} - 𝑛_{2}^{2}})$

Answer & worked solution

Answer D · qualification below

The inverse-sine expression requires its argument to be at most one.

Let the refracted ray make angle r with the horizontal normal at entry. Snell's law gives

\sin\theta_i=n_1\sin r

. Its incidence angle on the horizontal interface is

90^\circ-r

. At the limiting critical condition,

\cos r=n_2/n_1

, so

\sin\theta_i=\sqrt{n_1^2-n_2^2}

. This gives D when the square root is at most one. Otherwise all accessible entry angles satisfy the condition; strict total reflection occurs on the appropriate side of the critical boundary.

Question 6

IAT 2020 · Physics

What is the equivalent resistance between the points A and B in the figure? IAT 2020, Physics, question 6 — question diagram

$39 𝑅 / 10$
$39 𝑅 / 29$
$68 𝑅 / 87$
$87 𝑅 / 68$

Transcription note: Restored the circuit from the scanned 2020 paper; the uppermost right resistor is R, not 2R as in the secondary diagram.

Answer & worked solution

Answer B

The dangling middle resistor on the left carries no current, leaving

2R\parallel2R=R

. In the upper right group,

R\parallel2R\parallel R=2R/5

. The lower bridge is balanced, so its middle resistor carries no current and the two

5R

paths give

5R/2

. These two right-hand groups in parallel give

10R/29

. Adding the left R yields

39R/29

. The uppermost resistor label has been restored to R from the scanned paper; a secondary diagram incorrectly labelled it 2R.

Question 7

IAT 2020 · Physics

Consider a solid rod of mass

𝑚

and uniform density resting against a vertical wall and horizontal floor as shown in the figure. The coefficients of friction of the rod with the wall and with the floor are given to be

𝜇_{1}

and

𝜇_{2}

respectively. Gravity is acting downwards with acceleration due to gravity

𝑔

. What should be the value of the inclination angle

𝛼

so that the rod stays in equilibrium? IAT 2020, Physics, question 7 — question diagram

$\tan^{- 1} (\frac{𝜇_{1}}{𝜇_{2}})$
$\tan^{- 1} (\frac{1 - 𝜇_{2}}{2 𝜇_{1} 𝜇_{2}})$
$\tan^{- 1} (\frac{1 - 𝜇_{1} 𝜇_{2}}{2 𝜇_{2}})$
$\tan^{-1}(\mu_2/\mu_1)$

Transcription note: Removed a stray force-balance equation appended to option D; it is not part of that option in the scanned paper.

Answer & worked solution

Answer C · qualification below

The result is a limiting minimum angle, not a unique static-equilibrium angle.

At limiting equilibrium, horizontal balance gives

N_w=\mu_2N_f

, and vertical balance gives

N_f+\mu_1N_w=mg

. Taking moments about the lower end gives

N_wL\sin\alpha+\mu_1N_wL\cos\alpha=(mgL/2)\cos\alpha

. Eliminate the reactions to obtain

\tan\alpha=(1-\mu_1\mu_2)/(2\mu_2)

. C is the limiting minimum angle when this is positive; larger angles can also remain at rest. It is not a unique angle for all static equilibria.

Question 8

IAT 2020 · Physics

The charge distribution inside a sphere of radius

𝑅 = 2 𝑎 / 𝑏

is given by the volume charge density

𝜌 = 𝑎 𝑟^{2} - 𝑏 𝑟^{3}

, where

𝑎

and

𝑏

are constants and

𝑟

is the radial distance from the center of the sphere. At what distance from the center of the sphere, will the electric field go to zero?

$4 𝑎 / 3 𝑏$
$3 𝑎 / 2 𝑏$
$𝑎 / 𝑏$
$6 𝑎 / 5 𝑏$

Answer & worked solution

Answer D

The charge enclosed within radius r is

Q(r)=4\pi\int_0^r(as^2-bs^3)s^2\,ds=4\pi(ar^5/5-br^6/6)

. Gauss's law gives

E(r)=ar^3/(5\epsilon_0)-br^4/(6\epsilon_0)

. Apart from the centre, it vanishes at

r=6a/(5b)

, which lies inside

R=2a/b

for the intended positive constants.

Question 9

IAT 2020 · Physics

The acceleration due to earth's gravity on a point particle at a height $ℎ$ above the surface of the earth is denoted by $𝑔_{𝑜} (ℎ)$ and at a depth $𝑑$ below the surface of the earth is denoted by $𝑔_{𝑖} (𝑑)$ . Consider the earth to be a sphere of radius $𝑅$ with uniform mass density. Which of the following correctly represents the ratio $𝑔_{𝑖} (𝑑) / 𝑔_{𝑜} (𝑑)$ ?

Answer & worked solution

Answer A

For a uniform Earth,

g_i(d)=g_s(1-d/R)

, while

g_o(d)=g_s/(1+d/R)^2

. With

x=d/R

, their ratio is

(1-x)(1+x)^2

. It starts at 1, reaches a maximum

32/27

x=1/3

, and falls to zero at

x=1

. A has these features; D has an incorrect near-surface curvature and location of the peak.

Question 10

IAT 2020 · Physics

Consider an infinite one-dimensional wire carrying a uniform current $𝐼$ as shown in the figure. A square loop of side $𝑎$ is initially placed such that the center of the loop is at a distance $𝑅$ from the wire, where $𝑅 > \frac{𝑎}{2}$ . The square loop is then moved rightwards with a uniform speed as shown in the figure. What is the induced emf in the loop as a function of time $𝑡$ ?

IAT 2020, Physics, question 10 — question diagram

$\frac{𝜇_{0} 𝐼 𝑎^{2} 𝑣}{2 𝜋 [(𝑅 + 𝑣 𝑡)^{2} - \frac{𝑎^{2}}{4}]}$
$\frac{3 𝜇_{0} 𝐼 𝑎^{2} 𝑣}{4 𝜋 [(𝑅 + 𝑣 𝑡)^{2} - \frac{𝑎^{2}}{4}]}$
$\frac{𝜇_{0} 𝐼 𝑎^{2} 𝑣}{4 𝜋 [(𝑅 + 𝑣 𝑡)^{2} - \frac{𝑎^{2}}{4}]}$
$\frac{3 𝜇_{0} 𝐼 𝑎^{2} 𝑣}{2 𝜋 [(𝑅 + 𝑣 𝑡)^{2} - \frac{𝑎^{2}}{4}]}$

Answer & worked solution

Answer A

At time t the loop centre is

s=R+vt

from the wire. Integrating

B=\mu_0I/(2\pi r)

over the square gives

\Phi=(\mu_0Ia/2\pi)\log[(s+a/2)/(s-a/2)]

. Differentiate and take the magnitude:

|\mathcal E|=\mu_0Ia^2v/[2\pi(s^2-a^2/4)]

, matching A. Lenz's law determines the circulation direction.

Question 11

IAT 2020 · Physics

The electric field of an electromagnetic wave is given by $\vec{𝐸} (𝑥, 𝑡) = 𝐸_{0} \hat{𝑧} \cos (𝑎 𝑥 + 𝑏 𝑡)$

If $𝑐$ is the speed of light, then the value of $𝑏$ is

$𝑎 𝑐$
$2 𝜋 𝑎 𝑐$
$\sqrt{2} 𝑎 𝑐$
$\frac{𝑎 𝑐}{2 𝜋}$

Answer & worked solution

Answer A

The wave equation requires

\omega^2=c^2k^2

. In the phase

ax+bt

, a is the wave number and b the angular frequency, so

|b|=c|a|

. With the positive quantities used in the options,

b=ac

. The plus sign between position and time indicates propagation in the negative x direction.

Question 12

IAT 2020 · Physics

An electron of mass $𝑚_{𝑒}$ has a speed $𝑢_{𝑛}$ in the $𝑛^{th}$ Bohr orbit of a hydrogen atom. The normalized speed $𝑉_{𝑛}$ is given as $𝑉_{𝑛} = 𝑢_{𝑛} / 𝑐 = 1 / 685$ and the mass of the electron in the units of energy is given as $𝑀_{𝑒} = 𝑚_{𝑒} 𝑐^{2} = 0.51 \times 10^{6} eV$ , where $𝑐$ is the velocity of light in a vacuum. It is given that $\frac{𝑒^{2}}{2 𝜖_{0} ℎ 𝑐} = 1 / 137$ . The Planck's constant h is given as $4.13 \times 10^{- 15} eV - \sec$ . The electron makes a transition from $𝑛^{th}$ orbit to the ground state of the atom. What is the frequency of the emitted photon?

$3.29 \times 10^{15} Hz$
$2.47 \times 10^{15} Hz$
$2.93 \times 10^{15} Hz$
$3.16 \times 10^{15} Hz$

Answer & worked solution

Answer D

The Bohr speed is

u_n/c=\alpha/n=1/(137n)

. Equating this to

1/685

gives

n=5

. The ground-state binding energy is

M_e\alpha^2/2\approx13.586

eV. The transition releases

13.586(1-1/25)\approx13.043

eV, so

\nu=\Delta E/h\approx3.16\times10^{15}

Hz.

Question 13

IAT 2020 · Physics

In the photoelectric effect experiment, the frequency of the incident light is changed from red to blue while keeping the energy per unit area per unit time of the incident radiation fixed. What will happen to the saturation current?

Saturation current doesn't depend on frequency and therefore will be same for red and blue light
Saturation current depends on intensity and therefore will be same for red and blue light
Saturation current for blue colour light is larger than red colour light
Saturation current for blue colour light is smaller than red colour light

Answer & worked solution

Answer D · qualification below

The comparison assumes equal quantum efficiency and both colours above threshold.

At fixed incident power per unit area I, the photon arrival rate is

I/(h\nu)

. Blue photons carry more energy, so fewer arrive per second than red photons. Assuming both colours exceed threshold and have the same photoemission efficiency, the saturation current is smaller for blue light. A frequency-dependent quantum efficiency would require additional material data; the stated comparison uses the standard one-photon model.

Question 14

IAT 2020 · Physics

An ideal rigid diatomic gas undergoes expansion at constant pressure when

Δ 𝑄

heat is added to it. What is the ratio of the change in the internal energy

Δ 𝑈

to the heat supplied

Δ 𝑄

$\frac{Δ 𝑈}{Δ 𝑄} = \frac{3}{5}$
$\frac{Δ 𝑈}{Δ 𝑄} = \frac{5}{7}$
$\frac{Δ 𝑈}{Δ 𝑄} = \frac{7}{9}$
$\frac{Δ 𝑈}{Δ 𝑄} = \frac{2}{3}$

Answer & worked solution

Answer B

A rigid diatomic gas has five active degrees of freedom, so

C_V=5R/2

and

C_P=7R/2

. At constant pressure,

\Delta Q=nC_P\Delta T

, while

\Delta U=nC_V\Delta T

. Their ratio is

C_V/C_P=5/7

. Vibrational modes are excluded by the rigid-diatomic assumption.

Question 15

IAT 2020 · Physics

The fundamental constants are given to be permittivity constant

(𝜖_{0})

, Planck's constant

(ℎ)

, velocity of light

(𝑐)

, Newton's constant

(𝐺)

and Boltzmann's constant

(𝑘_{𝐵})

. Which of the following quantities constructed out of the above constant has the dimensions of an electric charge?

$\sqrt{\frac{ℎ 𝐺}{𝜖_{0}^{3}}}$
$\sqrt{ℎ 𝑐}$
$𝑘_{𝐵} \sqrt{ℎ 𝑐}$
$\sqrt{𝜖_{0} ℎ 𝑐}$

Answer & worked solution

Answer D

From Coulomb's law,

[\epsilon_0]=Q^2T^2/(ML^3)

. Also

[h]=ML^2/T

and

[c]=L/T

. Multiplying gives

[\epsilon_0hc]=Q^2

, so

\sqrt{\epsilon_0hc}

has the dimensions of charge. The other proposed combinations do not supply the required electrical dimension.

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IISER IAT 2020 question paper with solutions

Biology

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Question 11

Question 12

Question 13

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Question 15

Chemistry

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Question 15

Mathematics

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Question 15

Physics

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Question 14

Question 15