UPSC Indian Forest Service (IFoS) Main Previous Year Question Paper 2024 MATHEMATICS PAPER-I

UPSC Indian Forest Service (IFoS) Main Previous Year Question Paper 2024 MATHEMATICS PAPER-I

1. (a) Let V = ℝ⁴. Find a basis and dimension of the subspace

W = {(a,b,c,d) ∈ V : a = b + c, c = b + d}   (8)

(b) Describe explicitly a linear transformation from ℝ³ to ℝ³, which has its range spanned by (1,0,−1) and (1,2,2).

(c) Find the relation between the radii of a right circular cylinder and a cone if the former with maximum possible curved surface area is inscribed in the latter.

(d) Find the limit of (cot x − tan x)^(logₑ x), when x → 0.

(e) Show that if ax² + 2hxy + by² + 2gx + 1 = 0 represents two straight lines, then b < 0 and bg² + h² = ab.

UPSC Indian Forest Service (IFoS) MAIN Previous Year Question Paper 2024 – MATHEMATICS-I

2. (a) Let W₁ = { [x y; z 0] ; x,y,z ∈ ℂ } and W₂ = { [x 0; 0 y] ; x,y ∈ ℂ } be two subspaces of the vector space of all 2 × 2 matrices over the complex field ℂ. Show that

dim((W₁ + W₂)/W₂) = dim(W₁/(W₁ ∩ W₂))   (10)

(b) Evaluate the volume of the solid formed by rotating the curve r = a(1 + cos θ) about the initial line.

(c) (i) Reduce the equation (c² + d²)(x² + y²) = (cx + dy + 2f)² to its canonical form and show that it represents a parabola. Find the latus rectum of the parabola.

(ii) A variable sphere passes through the points (0,0,±c) and cuts the lines

y − x tan θ = 0 = z − c
y + x tan θ = 0 = z + c

in the points P and Q. If |PQ| = 2a (where a is a +ve number), then show that the centre of all such spheres lies on the circle x² + y² = (a² − c²) cosec² 2θ, z = 0.

3. (a) If u = exp{ sin⁻¹((x + y)/(√x − √y)) }, then show that x ∂u/∂x + y ∂u/∂y = (1/2) u tan(logₑ u). 10

(b) Reduce the matrix [IMAGE: Matrix missing/not provided in text] to echelon form and then to row canonical form. 15

(c) (i) Show that the equation of the plane containing the line y/b + z/c = 1, x = 0 and parallel to the line x/a − z/c = 1, y = 0 is x/a − y/b − z/c + 1 = 0. Further show that if 2d is the shortest distance between the given lines, then

1/a² + 1/b² + 1/c² = 1/d²

(ii) A variable plane is parallel to the plane x/a + y/b + z/c = 0 and meets the coordinate axes in A, B, C respectively. Prove that the circle ABC lies on the cone

yz(b/c + c/b) + zx(c/a + a/c) + xy(a/b + b/a) = 0

4. (a) Find the equations of the generating lines of the hyperboloid

x²/4 + y²/9 − z²/16 = 1

passing through the point (2,3,−4).

(b) Let T : ℝ³ → ℝ³ be defined by

T(x, y, z) = (5x − y + 3z, −6x + 4y − 6z, −6x + 2y − 4z)

Find all the eigenvalues and corresponding eigenvectors. 15

(c) (i) How many loops are generated of the curve r = a sin 3θ? Find the sum of the areas of all the loops. 8

(ii) Deduce the asymptote of the curve r logₑ θ = a. 7

5. (a) Solve the differential equation

p² + (x + y − 2y/x)p + xy + y²/x² − y − y²/x = 0, where p = dy/dx   (8)

(b) Solve the differential equation d²y/dx² − 3 dy/dx + 2y = cosh x. 8

(c) A particle moves from rest at a distance a from a centre of force where repulsion at distance x is μ x⁻². Show that its velocity at distance x is

√(2μ(x − a)/(a x))

and that the time it has taken is

√(a/(2μ)) [ √(x² − a x) + a logₑ( √(x/a) + √(x/a − 1) ) ]   (8)

(d) Two uniform steel rods of equal size l hang from their junction and rest on a symmetrically placed smooth vertical circular base of radius a. If each of the rods subtends an angle φ with the vertical line passing through the centre of the circular base, show that, applying the principle of virtual work, the relation obtained is l = 2a cot φ cosec² φ. 8

(e) If F⃗ is a solenoidal vector, then show that

curl curl curl curl F⃗ = ∇² ∇² F⃗ = ∇⁴ F⃗   (8)

6. (a) Solve the differential equation

d²y/dx² + 10 dy/dx + 29y = x e⁵ˣ + sin 2x   (10)

(b) A particle moves in a path so that its acceleration is always directed to a fixed point and is equal to μ/(distance)². Show that its path is a conic section and distinguish between the three cases that arise. Further show that the square of the periodic time varies as the cube of the major axis. 15

(c) (i) If r⃗ = x î + y ĵ + z k̂, then find curl(r⃗/|r⃗|). 6

(ii) Find the curvature and torsion of the curve x = a cos t, y = a sin t, z = b t. 9

7. (a) If near the surface of a celestial body having atmosphere, the gravity is almost constant and the absolute temperature in its atmosphere is given by

T = T₀ √(1 − z²/(n²H²))

H being the height of the homogeneous atmosphere and n a constant quantity, show that the pressure in the atmosphere will be given by

p = p₀ exp( sin⁻¹(z/(nH)) )

where p₀ is the pressure at z = 0.

(b) Solve the differential equation

d²y/dx² − cot x dy/dx − (1 − cot x)y = eˣ sin x   (15)

(c) (i) If Φ satisfies ∇²Φ = 0, then show that ∇Φ is both solenoidal and irrotational.

(ii) Verify the divergence theorem for the function

F⃗ = (x² − yz) î + (y² − zx) ĵ + (z² − xy) k̂

taken over the parallelepiped 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c.

8. (a) Verify Green's theorem for

∫_C [(xy + y²) dx + x² dy]

where C is bounded by the curves y = x and y = x².

(b) Using the method of variation of parameters, solve the differential equation

x² d²y/dx² + x dy/dx − y = x² log x, x > 0

(c) Establish a stability criterion if a rigid body is lying on another rigid body at a point of contact, and also both have rough surfaces to prevent sliding and a small area around the point of contact of both of them is circular.

A solid frustum of a paraboloid of revolution of height h and latus rectum 2a rests with its vertex on that of a paraboloid of revolution of latus rectum 2b. Find the stability condition.

Indian Forest Service (Main) Examination, 2024-MATHEMATICS PAPER I

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