UPSC Mathematics 2023 — Paper I

(a) Let V₁ = (2, -1, 3, 2), V₂ = (-1, 1, 1, -3) and V₃ = (1, 1, 9, -5) be three vectors of the space ℝ⁴. Does (3, -1, 0, -1) ∈ span {V₁, V₂, V₃} ? Justify your answer. (10 marks) (b) Find the rank and nullity of the linear transformation : T : ℝ³ → ℝ³ given by T(x, y, z) = (x + z, x + y + 2z, 2x + y + 3z) (10 marks) (c) Find the values of p and q for which limₓ→₀ [x(1 + p cos x) - q sin x]/x³ exists and equals 1. (10 marks) (d) Examine the convergence of the integral ∫₀¹ (log x)/(1+x) dx (10 marks) (e) A variable plane which is at a constant distance 3p from the origin O cuts the axes in the points A, B, C respectively. Show that the locus of the centroid of the tetrahedron OABC is 9(1/x² + 1/y² + 1/z²) = 16/p². (10 marks)

(a) If the matrix of a linear transformation T : IR³→IR³ relative to the basis (1, 0, 0), (0, 1, 0), (0, 0, 1) is 1 & 1 & 2 -1 & 2 & 1 0 & 1 & 3 , then find the matrix of T relative to the basis (1, 1, 1), (0, 1, 1), (0, 0, 1). (15 marks) (b) Evaluate the triple integral which gives the volume of the solid enclosed between the two paraboloids Z = 5(x² + y²) and Z = 6 – 7x² – y². (15 marks) (c)(i) Show that the equation 2x² + 3y² – 8x + 6y – 12z + 11 = 0 represents an elliptic paraboloid. Also find its principal axis and principal planes. (10 marks) (c)(ii) The plane x/a+y/b+z/c=1 meets the coordinate axes in A, B, C respectively. Prove that the equation of the cone generated by the lines drawn from the origin O to meet the circle ABC is yz(b/c+c/b)+zx(c/a+a/c)+xy(b/a+a/b)=0. (10 marks)

Let A = 1 & 0 & 0 1 & 0 & 1 0 & 1 & 0 (i) Verify the Cayley-Hamilton theorem for the matrix A. (ii) Show that Aⁿ = Aⁿ⁻² + A² - I for n ≥ 3, where I is the identity matrix of order 3. Hence, find A⁴⁰. 10+10 (b) Justify whether (0, 0) is an extreme point for the function f(x, y) = 2x⁴ - 3x^2y + y². 15 (c) Find the equation of the sphere through the circle x² + y² + z² - 4x - 6y + 2z - 16 = 0; 3x + y + 3z - 4 = 0 in the following two cases. (i) the point (1, 0, -3) lies on the sphere. (ii) the given circle is a great circle of the sphere. 15

Find the rank of the matrix A = 1 & 2 & -1 & 0 -1 & 3 & 0 & -4 2 & 1 & 3 & -2 1 & 1 & 1 & -1 by reducing it to row-reduced echelon form. 15 (b) Trace the curve y²(x² - 1) = 2x - 1. 20 (c) Prove that the locus of a line which meets the lines y = mx, z = c; y = -mx, z = -c and the circle x² + y² = a², z = 0 is c²m²(cy - mzx)² + c²(yz - cmx)² = a²m²(z² - c²)². 15

(a) Obtain the solution of the initial-value problem dy/dx - 2xy = 2, y(0) = 1 in the form y = eˣ²[1 + √π erf(x)]. (10 marks) (b) Given that L{f(t); p} = F(p). Show that ∫₀^∞ f(t)/t dt = ∫₀^∞ F(x)dx. Hence evaluate the integral ∫₀^∞ (e⁻ᵗ - e⁻³ᵗ)/t dt. (10 marks) (c) A cylinder of radius 'a' touches a vertical wall along a generating line. Axis of the cylinder is fixed horizontally. A uniform flat beam of length 'l' and weight 'W' rests with its extremities in contact with the wall and the cylinder, making an angle of 45° with the vertical. If frictional forces are neglected, then show that a/l = (√5 + 5)/(4√2). Also, find the reactions of the cylinder and wall. (10 marks) (d) A particle is moving under Simple Harmonic Motion of period T about a centre O. It passes through the point P with velocity v along the direction OP and OP = p. Find the time that elapses before the particle returns to the point P. What will be the value of p when the elapsed time is T/2 ? (10 marks) (e) If ā = sinθî + cosθĵ + θk̂, b̄ = cosθî - sinθĵ - 3k̂, c̄ = 2î + 3ĵ - 3k̂, then find the values of the derivative of the vector function ā × (b̄ × c̄) w.r.t. θ at θ = π/2 and θ = π. (10 marks)

(a) Solve the differential equation: d³y/dx³ - 3d²y/dx² + 4dy/dx - 2y = eˣ + cos x. (15 marks) (b) When a particle is projected from a point O₁ on the sea level with a velocity v and angle of projection θ with the horizon in a vertical plane, its horizontal range is R₁. If it is further projected from a point O₂, which is vertically above O₁ at a height h in the same vertical plane, with the same velocity v and same angle θ with the horizon, its horizontal range is R₂. Prove that R₂ > R₁ and (R₂-R₁):R₁ is equal to (1/2){√(1 + 2gh/v²sin²θ) - 1}:1. (15 marks) (c) Evaluate the integral ∬ₛ (3y²z²î + 4z²x²ĵ + z²y²k̂)·n̂ dS, where S is the upper part of the surface 4x² + 4y² + 4z² = 1 above the plane z = 0 and bounded by the xy-plane. Hence, verify Gauss-Divergence theorem. (20 marks)

(a)(i) Find the solution of the differential equation : dy/dx=-(2xy³+2)/(3x^2y²+8e^4y) 10 (a)(ii) Reduce the equation x^2p²+y(2x+y)p+y²=0 to Clairaut's form by the substitution y=u and xy=v. Hence solve the equation and show that y+4x=0 is a singular solution of the differential equation. 10 (b) A solid hemisphere is supported by a string fixed to a point on its rim and to a point on a smooth vertical wall with which the curved surface is in contact. If θ is the angle of inclination of the string with vertical and φ is the angle of inclination of the plane base of the hemisphere to the vertical, then find the value of (tanφ-tanθ). 15 (c) If the tangent to a curve makes a constant angle θ with a fixed line, then prove that the ratio of radius of torsion to radius of curvature is proportional to tanθ. Further prove that if this ratio is constant, then the tangent makes a constant angle with a fixed direction. 15

(a) Solve the following initial value problem by using Laplace transform technique : (d^2y)/(dt²) - 4dy/dt + 3y(t) = f(t), y(0) = 1, y'(0) = 0 and f(t) is a given function of t. 15 (b) A particle is projected from an apse at a distance √c from the centre of force with a velocity √((2λ)/3c³) and is moving with central acceleration λ(r⁵ - c^2r). Find the path of motion of this particle. Will that be the curve x⁴ + y⁴ = c² ? 20 (c) For a scalar point function φ and vector point function f⃗, prove the identity ∇ · (phif⃗) = ∇φ · f⃗ + φ(∇ · f⃗). Also find the value of ∇ · ((f(r))/rr⃗) and then verify stated identity. 15