UPHESC Assistant Professor Mathematics 2025 PYQs

Concept: A floating body displaces its own weight of liquid (law of flotation).

1. Sea water (≈1.025 g/cm³, due to dissolved salts) is denser than river water (≈1.000 g/cm³).
2. The ship's weight is unchanged, so it needs a smaller displaced volume in sea water – it floats higher (rises a little).
3. Thus R is true and directly explains A.

Short trick: Denser liquid → less sinking (ships carry 'Plimsoll' load lines for fresh vs salt water).

Concept: The AICC met in Delhi on 14–15 June 1947 to ratify the Mountbatten (3 June) Plan.

1. 3 June 1947: Mountbatten announced the partition plan.
2. 14 June 1947: at the AICC meeting Gandhiji, though opposed to partition, advised Congressmen to accept it as the only way out.
3. The AICC then passed the resolution accepting the plan (157 for, 29 against).

Short trick: 3 June = Plan announced; 14 June = AICC (Gandhi) accepts.

Concept: Public health and sanitation is Entry 6 of the State List.

1. The 42nd Amendment (1976) moved five subjects from State to Concurrent List: education (Entry 25), forests, weights and measures (33A), protection of wild animals & birds, and administration of justice (11A).
2. Public health stayed in the State List.

Short trick: 42nd Amendment's five: Education, Forests, Wildlife, Weights & measures, Justice – health is NOT among them.

Concept: Megasthenes was sent by Seleucus Nicator to the court of Chandragupta Maurya at Pataliputra (c. 302 BCE).

1. He wrote 'Indica', describing Mauryan society and administration.
2. Deimachus came to Bindusara's court; Dionysius was sent by Ptolemy II (to Bindusara/Ashoka).
3. Brihadratha was the last Mauryan, killed by Pushyamitra Shunga.

Short trick: Seleucus → Megasthenes → Chandragupta; Antiochus → Deimachus → Bindusara.

Concept: Dyarchy in the provinces was introduced by the Government of India Act, 1919 (Montagu–Chelmsford Reforms), in force from 1921.

1. Provincial subjects were split into 'transferred' (ministers responsible to legislature) and 'reserved' (Governor and executive council).
2. 1909 = Morley–Minto Reforms (separate electorates); 1935 Act abolished provincial dyarchy and introduced it at the Centre; 1937 = provincial elections/ministries.

Short trick: 1919 = dyarchy in provinces; 1935 = provincial autonomy + dyarchy at Centre.

Concept: Article 21A (inserted by the 86th Amendment, 2002) guarantees free and compulsory education to children aged 6–14 years.

1. Implemented through the RTE Act, 2009 (in force 1 April 2010).
2. Article 45 (DPSP) now covers early childhood care and education below 6 years.
3. Article 19 = six freedoms; Article 32 = right to constitutional remedies.

Short trick: 21A = 'A' for Akshar (education); 86th Amendment, 6–14 years.

Concept: The 97th Constitutional Amendment Act, 2011 gave constitutional status to co-operative societies.

1. It added 'co-operative societies' to Art. 19(1)(c), inserted Art. 43B (DPSP) and Part IXB (Arts. 243ZH–243ZT).
2. 96th (2011) = 'Oriya' → 'Odia'; 99th (2014) = NJAC; 101st (2016) = GST.

Short trick: 97 → Co-operatives (Part IXB); 101 → GST.

Concept: In 2020-21 India's real GVA contracted; the contact-intensive services sector was hit hardest and agriculture was the only sector to grow.

1. Provisional estimates (May 2021): services ≈ −8.4%, industry ≈ −7.0%, agriculture ≈ +3.6%.
2. Trade, hotels, transport and communication alone shrank by about 18%.
3. Worst to least affected: Service, Industry, Agriculture.

Short trick: Lockdown hit 'contact' jobs most; the farm sector was the 'silver lining'.

Concept: Disguised (hidden) unemployment: more people are engaged than needed, so the marginal productivity of the extra workers is zero.

1. Typical case: five family members working on a farm that needs only three – removing two does not reduce output.
2. Underemployment = working below capacity/hours; seasonal = jobless in the off-season; 'formal unemployment' is not a standard category.

Short trick: Disguised = 'looks employed, adds nothing' (MP = 0).

Concept: Budget 2025-26 announced a social security scheme for the welfare of online platform (gig) workers.

1. Platform workers will get identity cards, registration on the e-Shram portal and health care under PM Jan Arogya Yojana.
2. About 1 crore gig workers were expected to benefit.
3. Farm, factory and women workers already had schemes; the 'first time' scheme was for platform workers.

Short trick: Gig workers (delivery/cab apps) → e-Shram + PM-JAY, Budget 2025-26.

Concept: A group is simple if its only normal subgroups are $\{e\}$ and $G$.

1. (R) is Lagrange's theorem – true.
2. If $|G|=p$ (prime), by Lagrange every subgroup has order 1 or $p$, so the only subgroups (hence normal subgroups) are $\{e\}$ and $G$.
3. So (A) is true and follows directly from (R).

Short trick: Prime order ⇒ only trivial subgroups (Lagrange) ⇒ simple.

Concept: Second (diamond) isomorphism theorem: $HN/N\cong H/(H\cap N)$.

1. Define $\varphi :H\to HN/N$, $\varphi (h)=hN$; it is an onto homomorphism.
2. $\ker \varphi =\{h\in H:h\in N\}=H\cap N$.
3. First isomorphism theorem gives $H/(H\cap N)\cong HN/N$.

Short trick: Diamond: 'top over N' ≅ 'H over the bottom ($H\cap N$)'.

Concept: A cyclic group of order $n$ has $\phi (n)$ generators.

1. In ${\mathbb{Z}}_8$, $k$ is a generator iff $gcd(k,8)=1$.
2. $k=1,3,5,7$ → $\phi (8)=4$.

Short trick: Generators = Euler phi of the order; $\phi (2^k)=2^{k-1}$.

Concept: $A_n$ = set of all even permutations of $S_n$; $|A_n|=n!/2$.

1. $|A_3|=3!/2=3$, so options (A)–(C) (2 elements) are out.
2. Transpositions like (1 2) are odd; 3-cycles are even (product of 2 transpositions).
3. $A_3=\{I,(123),(132)\}$.

Short trick: Even permutations only – a transposition is never in $A_n$.

Concept: Order of a permutation = LCM of the lengths of its disjoint cycles.

1. $1\to 1$, $2\to 3\to 4\to 2$.
2. So the permutation is the 3-cycle $(234)$.
3. Order = 3.

Short trick: Write disjoint cycles, take LCM of lengths.

Concept: Factor first; the roots here are roots of unity.

1. $x^4+x^2+1=(x^2+x+1)(x^2-x+1)$.
2. Roots: $\omega ,{\omega }^2$ (cube roots of unity) and $-\omega ,-{\omega }^2$ (primitive 6th roots).
3. All lie in $\mathbb{Q}(\omega )=\mathbb{Q}(\sqrt{-3})$, so degree $=2$.

Short trick: $x^4+x^2+1=\frac{x^6-1}{x^2-1}$ → only 3rd and 6th roots of unity → $\mathbb{Q}(\sqrt{-3})$.

Concept: In a commutative ring with 1: $M$ maximal ⇔ $R/M$ is a field; $P$ prime ⇔ $R/P$ is an integral domain.

1. Integral domain characterises prime ideals, not maximal ones.
2. Euclidean domain / UFD are not the criterion.
3. Hence $R/M$ must be a field.

Short trick: Maximal ↔ Field, Prime ↔ Integral domain.

Concept: On a complex Hilbert space, $T=T^{\ast }$ ⇔ $⟨Tx,x⟩\in \mathbb{R}$ for all $x$.

1. If $T=T^{\ast }$: $⟨Tx,x⟩=⟨x,Tx⟩=\overline{⟨Tx,x⟩}$, so it is real.
2. Conversely, real values give $⟨(T-T^{\ast })x,x⟩=0$ for all $x$, hence $T=T^{\ast }$ (polarisation).
3. Purely imaginary ⇔ skew-adjoint; zero ⇔ $T=0$.

Short trick: Self-adjoint = 'real' operator: $⟨Tx,x⟩$ real.

Concept: Every Hilbert space is Banach, every Banach space is a normed linear space, and every normed space is (being metric) a topological space.

1. $l_2$ with inner product $\sum x_i{\overline{y}}_i$ is complete, so it is Hilbert (II) ⇒ Banach (III) ⇒ normed linear (IV) ⇒ topological (I).
2. Read from the most special to the most general, the sequence is II, III, IV, I – option (C).
3. Read as the steps of building up structure (topological → normed → complete → inner-product), the sequence is I, IV, III, II – option (B). This is equally valid: it is the same chain in the reverse direction.
4. The question does not say which direction is wanted, so both (B) and (C) are correct and the question is treated as a bonus. Options (A) and (D) break the chain and are wrong.

Short trick: Hilbert ⊂ Banach ⊂ Normed ⊂ Metric ⊂ Topological.

Concept: Differentiable at 0 ⇔ $\frac{f(h,k)-f(0,0)-h{f}_x-k{f}_y}{\sqrt{h^2+k^2}}\to 0$.

1. $|f|\le \frac{r^2/2}{r}=\frac{r}{2}\to 0$ → continuous.
2. $f(x,0)=0,f(0,y)=0$ → $f_x(0,0)=f_y(0,0)=0$ (exist).
3. Differentiability needs $\frac{hk}{h^2+k^2}\to 0$; on $k=h$ it is $\frac{1}{2}$ → not differentiable.

Short trick: Polar form: $f=\frac{r}{2}\sin 2\theta$ – continuous, but $f/r$ depends on θ ⇒ not differentiable.