UPHESC Assistant Professor Agricultural Statistics PYQs

Concept: Each ‘-culture’ term names a kind of cultivation.

1. Viticulture (Latin vitis = vine) – grapes → (4).
2. Pisciculture (piscis = fish) – fish farming → (2).
3. Olericulture (olus = pot-herb) – vegetable farming → (1).
4. That leaves Vegeculture – cultivation of vegetatively propagated trees/plants → (3). Only code (B) fits.

Short trick: Viti = vine, Pisci = fish, Oleri = vegetables.

Concept: The Indian Institute of Ecology and Environment (IIEE), founded in 1980, is in New Delhi.

1. It runs courses and publishes on ecology, environment and sustainable development.
2. Do not confuse with the Wildlife Institute of India (Dehradun) or the Centre for Environment Education (Ahmedabad).

Short trick: IIEE – national capital, New Delhi.

Concept: The Tropic of Cancer (23½°N) crosses 8 states: Gujarat, Rajasthan, MP, Chhattisgarh, Jharkhand, West Bengal, Tripura, Mizoram.

1. Jharkhand, Tripura and Mizoram are cut by the line, so they lie on both sides.
2. Manipur (about 23.8°N–25.7°N) lies wholly north of it.

Short trick: “Gu-Ra-M-Ch-J-WB-T-M” = 8 states on the Tropic; Manipur is not one of them.

Concept: Census 2011: Haryana had the lowest sex ratio among states (879 females per 1000 males) though its per-capita income is among the highest.

1. National average was 943; Kerala highest (1084).
2. MP (931), Maharashtra (929), West Bengal (950) are all well above Haryana.

Short trick: Rich Haryana, poor sex ratio – 879 (lowest state, Census 2011).

Concept: Troposphere is the lowest layer (≈8 km at poles, ≈18 km at equator) where all weather occurs.

1. Order upward: Troposphere → Stratosphere (ozone) → Mesosphere → Thermosphere/Ionosphere → Exosphere.
2. Exosphere is the outermost layer.

Short trick: “T-S-M-T-E” – Troposphere touches the earth.

Concept: NSO (Ministry of Statistics & PI) estimated Q1 (April–June) 2020-21 GDP growth at –23.9%.

1. It was the sharpest contraction since quarterly data began (1996).
2. –7.5% was Q2 2020-21; +0.4% was Q3 2020-21 (return to growth).

Short trick: Q1 lockdown quarter = –23.9% (later revised to –24.4%).

Concept: RBI (2014–15) licensed ‘differentiated’ (niche) banks that may do only a limited set of banking activities.

1. Payments Banks – accept deposits up to a limit, no lending.
2. Small Finance Banks – serve small borrowers/MSMEs.
3. Commercial (universal) banks provide the full range, so they are not differentiated banks.

Short trick: Differentiated = Payments + Small Finance (Nachiket Mor committee idea).

Concept: Financial emergency is provided under Article 360 of the Constitution.

1. It has never been proclaimed till date.
2. National emergency (Art. 352) was proclaimed thrice – 1962, 1971, 1975.
3. President’s rule (Art. 356) has been used many times.

Short trick: 352 – 3 times; 356 – 100+ times; 360 – never.

Concept: Wildlife Institute of India (WII), est. 1982, is at Chandrabani, Dehradun (Uttarakhand), under MoEFCC.

1. Dehradun also hosts the Forest Research Institute and Indira Gandhi National Forest Academy.
2. Bhopal has the Indian Institute of Forest Management; Almora has the G.B. Pant Institute of Himalayan Environment.

Short trick: Dehradun = forests & wildlife capital (FRI, WII).

Concept: Pradhan Mantri Van Dhan Yojana (2018, TRIFED, Ministry of Tribal Affairs) sets up Van Dhan Kendras for value addition of minor forest produce.

1. In 2020 TRIFED hailed Manipur as the model/champion state – it set up the most Van Dhan Kendras and trained thousands of tribal gatherers.
2. Assam, Meghalaya and UP were not given this title.

Short trick: Van Dhan model state = Manipur (TRIFED, 2020).

Concept: For a function with continuous second partials, mixed partial derivatives are equal (Schwarz/Clairaut theorem).

1. $\frac{\partial z}{\partial y}=2x^{2}y$, then $\frac{{\partial }^{2}z}{\partial x\partial y}=4xy$ — a function of x and y, not the constant 4. Statement 1 is false.
2. $\frac{\partial z}{\partial x}=2x{y}^2$, then $\frac{{\partial }^{2}z}{\partial y\partial x}=4xy$ — same as above, so statement 2 is true.

Short trick: Polynomials are smooth, so the order of mixed differentiation never matters.

Concept: In a repeated limit the inner limit is taken first, with the outer variable held fixed.

1. For fixed $x\ne 0$, $\sin \frac{1}{x}$ is a bounded constant, so $\underset{y\to 0}{lim}y\sin \frac{1}{x}=0$.
2. Then $\underset{x\to 0}{lim}0=0$.

Short trick: Inner limit first: y → 0 kills everything because |sin| ≤ 1.

Concept: If one row (or column) of a determinant becomes all zeros after an elementary operation, the determinant is 0.

1. Apply $C_1\to C_1+C_2+C_3$.
2. Each entry of the new first column is $(a-b)+(b-c)+(c-a)=0$.
3. A column of zeros gives determinant 0.

Short trick: Entries in every row add to zero ⇒ determinant = 0.

Concept: Multiply by the conjugate: $\frac{1}{1+\sin x}=\frac{1-\sin x}{{\cos }^{2}x}={\sec }^{2}x-\sec x\tan x$.

1. An antiderivative is $\tan x-\sec x=\frac{\sin x-1}{\cos x}$.
2. As $x\to \pi /2$: $\frac{\sin x-1}{\cos x}\to 0$; at $x=0$ it equals $-1$.
3. Value $=0-(-1)=1$.

Short trick: Use $x\to \pi /2-x$: the integral equals ${\int }_0^{\pi /2}\frac{dx}{1+\cos x}=[\tan \frac{x}{2}{]}_0^{\pi /2}=1$.

Concept: Lagrange's MVT: if f is continuous on [a, b] and differentiable on (a, b), some c in (a, b) has $f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$.

1. Geometrically, the tangent at c is parallel to the chord (secant) joining the end points — statement 1 is exactly this.
2. MVT underlies monotonicity tests, Taylor's theorem and error bounds — a central result of real analysis, so statement 2 is also true.

Short trick: Tangent ∥ chord = Mean Value Theorem (Rolle's theorem is the special case f(a) = f(b)).

Concept: Interpolation assumes no sudden jumps in the data; methods are graphic or algebraic (binomial expansion, Newton, Lagrange, parabolic curve).

1. Pair 1: the basic assumption of interpolation/extrapolation — correct.
2. Pair 2: Lagrange's formula works for unequal intervals of the independent variable — correct.
3. Pair 3: direct binomial expansion needs equal intervals (equally spaced x) — correct.
4. Pair 4: the parabolic curve method is an algebraic method (fitting a parabola), not the graphic method — wrongly matched.

Short trick: Graphic = plot and read off; parabolic curve = algebraic fitting.

Concept: Rank–nullity: rank(A) + nullity(A) = n; rank(A²) ≤ rank(A).

1. Nullity ≥ 2 ⇒ rank(A) ≤ 3 ⇒ rank$(A^2)$ ≤ 3: I is true.
2. A is singular ⇒ det$(A^2)=(detA{)}^2=0$: IV is true.
3. III (rank$(A^2)$ ≤ 5) is also true — it holds for every 5×5 matrix.
4. II is false: A = diag(1, 1, 1, 0, 0) has nullity 2 but rank$(A^2)=3$.
5. So the true statements are I, III and IV. Option (B) 'III, IV' and option (C) 'I, IV' both list only true statements, and no option lists all three — there is no single correct option.

Short trick: Nullity k in an n×n matrix ⇒ rank ≤ n − k and determinant = 0.

Concept: $E[g(X)]=\sum g(x_i)p_i$.

1. $E\left(\frac{1}{X^2}\right)=1\cdot \frac{1}{4}+\frac{1}{4}\cdot \frac{1}{2}+\frac{1}{16}\cdot \frac{1}{4}$
2. $=\frac{16}{64}+\frac{8}{64}+\frac{1}{64}=\frac{25}{64}$.

Short trick: Write every term over 64 before adding.

Concept: Any m.g.f. must satisfy $M_x(0)=E(e^0)=1$.

1. As printed, $M_x(0)=\frac{(3+1{)}^3}{6^4}=\frac{64}{1296}=\frac{4}{81}\ne 1$, so the given function is not the m.g.f. of any random variable.
2. Taking it literally, $E(x)=M^{\prime }(0)=\frac{48}{1296}=\frac{1}{27}$ and $E(x^2)=M^{″}(0)=\frac{72}{1296}=\frac{1}{18}$, giving $v(x)=\frac{1}{18}-\frac{1}{729}\approx 0.054$ — none of the options.
3. Only with the constant changed to $\frac{1}{4^3}$ does it become ${(\frac{3}{4}+\frac{1}{4}{e}^t)}^3$, i.e. B(3, 1/4) with variance $npq=\frac{9}{16}$ (option A). That needs a change in the printed data, so no option is correct as printed.

Short trick: First test M(0) = 1; then $(q+p{e}^t{)}^n$ ⇒ Binomial, V = npq.

Concept: The two-sample (Student's) t-test pools the two sample variances, which is valid only when the population variances are equal.

1. Assumptions: normal populations, independent random samples, and ${\sigma }_1^2={\sigma }_2^2$ (homoscedasticity).
2. Pooled $S^2=\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}$, $t=\frac{{\overline{x}}_1-{\overline{x}}_2}{S\sqrt{1/n_1+1/n_2}}$ with $n_1+n_2-2$ d.f.
3. Equality of variances is checked first with the F-test; if unequal, the Behrens–Fisher problem arises (Welch/Cochran test).

Short trick: F-test first (equal variances?), then pooled t-test.