Kerala PSC Range Forest Officer – Kerala Forest and Wildlife Department Previous Question Paper – 020/2026

The Commission publishes this paper without an answer key, so no answers are shown.

1. Let R₁ be the relation on ℤ defined by aR₁b iff ab ≥ 0, and let R₂ be the relation on ℤ \ {0} defined by aR₂b iff ab > 0. (a) Test whether R₁ is an equivalence relation on ℤ. (b) Test whether R₂ is an equivalence relation on ℤ \ {0}. Justify your answers. (5)

2. Is it true that there exists a bijective function from ℕ to ℤ? If yes, define such a function explicitly and prove that it is bijective. Justify your reasoning with clear steps, (where ℕ is the set of natural numbers). (5)

3. Using the Euclidean algorithm, find the greatest common divisor (gcd) of 252 and 198. Then Express this gcd as a linear combination of 252 and 198. (5)

4. Using Fermat’s Little Theorem, find the remainder when 5101 is divided by 17. (5)

5. Solve the reciprocal equation x⁴ − 5x³ + 6x² − 5x + 1 = 0 using the reciprocal transformation. (5)

6. Let f(x) = x⁴ − 5x² + 6. Let the roots be α, β, γ, δ. Compute Σα², and also find Σα²β². (5)

7. Let ℝ be the group of real numbers under addition. Determine to which of the following groups it is isomorphic. Justify your answer with proof.
(a) The group ℝ+ of positive real numbers under multiplication.
(b) The group ℂ* of non-zero complex numbers under multiplication. (5)

8. Find all ideals N of the ring ℤ₆. For each ideal N, compute ℤ₆/N and identify a known ring (up to isomorphism) to which it is isomorphic. Justify your answers. (5)

9. Find a basis for the vector space M 2×2 , the space of all 2 × 2 matrices with real entries. (5)

10. Let T: ℝ³ → ℝ³ be a linear transformation defined by T(1, 0, 0) = (1, −3/2, 2), T(0, 1, 0) = (−3, 9/2, −6), T(0, 0, 1) = (2, −3, 4). Find nullity of T. (5)

11. Find the rank of the following matrix. 1 1 1 1 1   1 2 2 2 2 1 2 3 3 3   1 2 3 4 4  (5)

12. Let V be the vector space of all complex polynomials p with deg(p) ≤ n. Let T : V → V be defined by (Tp)(x) = p′(1). Prove that dim(ker T) = n. (5)

13. For any nonempty bounded set A ⊂ ℝ, define −A = {−a : a ∈ A}. Prove that inf(−A) = −sup(A). (5)

14. Let 0 < b < 1. Is the sequence (nbⁿ) convergent? Justify. (5)

15. Discuss the convergence of the series Σ(n=1 to ∞) n!e^(−n²). (5)

16. Prove that if S and T are countable, then S ∪ T is countable. (5)

17. Let f: ℝ → ℝ be differentiable and lim(x→∞) f′(x) = 1. Show that f is unbounded. (5)

18. State TRUE or FALSE by giving a proper justification : There exists a continuous function f : R → R such that f ( x ) ∈ Q for all x ∈ R − Q and f ( x ) ∈ R − Q for all x ∈ Q . (5)

19. Let f : ℝ → ℝ satisfy f(x + y) = f(x) + f(y) for all x, y ∈ ℝ. If f is continuous at 0, show that f(x) = f(1)x for all x ∈ ℝ. (5)

20. Let p be an odd-degree polynomial with real coefficients and let g : ℝ → ℝ be bounded and continuous. Show that there exists x₀ ∈ ℝ such that p(x₀) = g(x₀). Apply this to prove that x⁹ − 4x⁶ + x⁵ + 1/(1 + x²) = sin 3x + 17 has at least one real root. (5)

21. Show that the Dirichlet function
1, if x is rational f (x ) =  0, if x is irrational is not integrable on [0, 1]. (5)

22. Find the area of the region enclosed by y = x² and y = x + 6. (5)

23. Find the length of the loop of the curve 3ay² = x(x − a)². (5)

24. Use the method of slicing, find the volume a right circular cone whose height is h and base is a circle of radius r. (5)

25. Show that the locus of foot of the perpendicular from the origin to plane passing through a fixed point passes through the origin. (5)

26. Determine the point at which the plane 2x + 2y + z − 8 = 0 is tangent to the sphere x² + y² + z² − 4x + 6y − 2z + 5 = 0. (5)

27. Find the line of intersection of the planes 2x + 3y + 4z − 5 = 0 and x + 2y + 3z − 4 = 0. (5)

28. Using the geometrical interpretation of scalar triple product of vectors, verify whether the points (1, − 1, − 5), (4, 2, 1), (5, 3, − 3) and ( −2, − 2, − 9) are coplanar. (5)

29. Find the volume of the solid bounded by the cylinder x² + y² = 9 and the planes y + z = 6 and z = 0. (5)

30. Use a polar double integral to find the area enclosed by the cardioid r = 1 + cos θ. (5)

31. Use Green’s Theorem to find the work done by the force field F⃗(x, y) = (x² + y)i⃗ + (y² − x)j⃗ on a particle that travels once counter clockwise around the triangle bounded by x = 0, y = 0, x + y = 2. (5)

32. Use the Divergence Theorem to find the outward flux of the vector field F⃗(x, y, z) = 4xi⃗ + yj⃗ + z²k⃗ across the unit cube given by the inequalities 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1. (5)

33. Find the orthogonal trajectories of the family of curves y = x + ce⁻ˣ. (5)

34. Find the integrating factor and solve y dx + (2x − yeʸ) dy = 0. (5)

35. (a) Define (i) slack variable and (ii) unrestricted variable. (b) Convert to standard form: maximize Z = 2x₁ + 2x₂ + 4x₃ + x₄, subject to 3x₁ + 2x₂ + x₃ + 9x₄ ≤ 16; 7x₁ + 2x₂ + 4x₃ + x₄ ≤ 24; x₁, x₂, x₃ ≥ 0; and x₄ unrestricted. (5)

36. Obtain an initial basic feasible solution using north west corner rule.
D1 D2 D3 D4 Supply O1 6 4 1 5 14 O2 8 9 2 7 16 O3 4 3 6 2 5 Demand 6 10 15 4 (5)

37. Prove that U(x, y) = 2y³ − 6x²y + 4x² − 7xy − 4y² + 3x + 4y − 4 is harmonic. Find an analytic function f(z) whose real part is U(x, y). (5)

38. Find the maximum value of f(z) = eᶻ/z on the rectangle −1 ≤ Re(z) ≤ 1, 1 ≤ Im(z) ≤ 2. (5)

39. Expand f(z) = 1/((z − 1)(z − 2)) as a power series in the regions: (a) |z| < 1, (b) 1 < |z| < 2, (c) |z| > 2. (5)

40. Using Cauchy’s Residue theorem, evaluate ∮_C [ze^(πz)/(z⁴ − 16) + zⁿe^(π/z)] dz, where n is an integer with n ≥ 1 and C is the ellipse 9x² + y² = 9. (20)