Kerala PSC Range Forest Officer – Forest and Wildlife – (Mathematics) Previous Question Paper – 045/2025

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

1. Determine whether R is an equivalence relation on ℤ\{0}, where n R m iff nm > 0 for all n, m ∈ ℤ\{0}.

2. Prove using basic Boolean operations on sets: A′B + B′C′ + AB + B′C = 1.

3. Compute the remainder of 8¹⁰³ when divided by 13. [Hint: use Fermat’s theorem.]

4. Find φ(18), where φ is Euler’s phi function.

5. Form the quadratic equation whose roots are −2 and −3.

6. Check whether x² − 5x + 5 = 0 is a reciprocal equation.

7. Find all left and right cosets of the subgroup 3ℤ of ℤ under usual addition. (5)

8. Consider the group ℤ24 under addition modulo 24. Is ℤ24 is a cyclic group? If so what are the generators of ℤ24? (5)

9. Find the value of the determinant using row reduction
x 2x 3x x+y 3x + 2 y 6x + 3 y x + y + z 4 x + 3 y + 2z 10x + 6 y + 3z
1 1 2 (5)

10. Find the inverse matrix of A = 9 2 0 by using Cayley-Hamilton 5 0 3 theorem. (5)

11. Show that {( x1 , x 2 , x 3 , x 4 ) ∈ ℝ4 |x1 + x 3 = 0, x 2 − x 4 = 0 } is a subspace of ℝ4 . Find the dimension of this subspace. (5)

12. Let V be the vector space of all 2 by 2 real matrices and let T : V → ℝ2 be  a b   the linear transformation given by T   c d   = ( a + b + c, d ) , If is the   standard basis for V and /={(1, 1), (–1, 0)} is a basis for ℝ2, find the matrix of T relative to these bases. (5)

13. Show that every infinite set A contains an infinite sequence of distinct terms. Deduce that there is a function f : A → A that is one to one but not onto. (5)

14. Let (xₙ) be a sequence of integers such that xₙ ≠ xₙ₊₁ for all n. Prove or disprove: (a) (xₙ) is a Cauchy sequence; (b) (xₙ) cannot have a convergent subsequence. (5)

15. Determine whether the series Σ(n=0 to ∞) [2ⁿ sin²(4n) / (3ⁿ + cos²(2n))] converges or diverges. (5)

16. Show that the series Σ(k=0 to ∞) [(-1)^(k+1) k² / (2 + k³)] is conditionally convergent. (5)

17. Evaluate lim . x →0 x2 (5)

18. For what values of a, m and b does the function f(x) = {3 for x = 0; −x² + 3x + a for 0 < x < 1; mx + b for 1 ≤ x ≤ 2} satisfy the hypotheses of the mean value theorem on the interval [0, 2] ? (5)

19. Find the absolute maximum and absolute minimum of f ( x ) = 2x 3 − 9x 2 − 24 x + 2 on the interval 0 ≤ x ≤ 5 . (5)

20. Let S_A = 15t² + 10t + 20 and S_B = 5t² + 40t, t ≥ 0, be the position functions of cars A and B that are moving along parallel straight lanes of a highway. At what instant of time do they have the same velocity? Which car is ahead at this instant? (5)

21. Find the length of the curve x = y³/3 + 1/(4y) from y = 1 to y = 3. (5)

22. Using definite integrals, find the lateral surface area of the cone generated by revolving y = x/2, 0 ≤ x ≤ 4, about the x-axis. (5)

23. Find the area of the region enclosed by the parabola y = 2 − x 2 and the line y = −x . (5)

24. Consider a triangular plate with vertices (0, 0), (1, 0) and (1, 2) having constant density of 3 g/cm2. Find the moment of the plate about y-axis. (5)

25. Find the equation of the tangent plane to the sphere x 2 + y 2 + z 2 + 2ux + 2vy + 2wz + d = 0 at the point ( x1 , y1 , z1 ) on it. (5)

26. Find a unit vector perpendicular to the vectors 3 i + j + 2 k and 2 i − 2 j + 4 k . Also find the angle between the given vectors.
x −3 y −5 z −7 (5)

27. Find the shortest distance between the lines = = and 1 −2 1 x +1 y +1 z +1 = = . 7 −6 1 (5)

28. Find the equation of the plane passing through the point (2, –3, 1) and perpendicular to the line joining the point (3, 4, –1) and (3, –1, 5). (5)

29. Suppose that a geyser, centered at the origin of a polar co-ordinate system, sprays water in a circular pattern in such a way that the depth D of water that reaches a point at a distance of r feet from the origin in 1 hour is D = kre −r . Find the total volume of water that the geyser sprays inside a circle of radius R centered at the origin. (5)

30. Find the mass of the solid between x² + y² + z² = 2 and x² + y² + z² = 8 if ρ(x, y, z) = 1/√(x² + y² + z²). (5)

31. Evaluate  ( yz − 1) dx + ( z + xz + z ) dy + ( y + xy + 2 yz ) dz . (1, 2, 2 )
3 2 (5)

32. Determine ∮_C (y³ − xy) dx + (xy + 3xy²) dy, where C is the boundary of the region in the first quadrant enclosed by x = 0, y = 1 − x² and y = x². (5)

33. Solve the differential equation : = e 3x − 2 y + x 2e − 2 y . dx
 e −2 x y  dx (5)

34. Solve the differential equation :  − = 1.  x x  dy   (5)

35. Find the general solution of the equation : y′′ − 5 y′ + 4 y = 65 sin 2x . (5)

36. A manufacturing unit makes doors and windows and intends to produce no more than 30 pieces per week. No more than 15 doors per week can be sold, and at least 10 windows per week must be supplied. Net profit is Rs. 100 per window and Rs. 150 per door. Formulate a linear programming problem that maximizes total profit; do not solve it. (5)

37. Discuss the convergence of Σ(k=0 to ∞) [zᵏ / ((1 − z^(k+1))(1 − z^(k+2)))], |z| ≠ 1. (5)

38. Find all the poles and singularities of f ( z ) = . z 2 − (4 + 3 i )z + (1 + 5 i ) (5)

39. Verify whether u( x , y ) = x 2 − y 2 is Harmonic. If so find its harmonic conjugate. (5)

40. Evaluate  x dz where Ω is the boundary of the square [0, 1] × [0, 1] with Ω ℂ considered as ℝ2. (5)