UPSC Mathematics 2022

(a) Prove that any set of n linearly independent vectors in a vector space V of dimension n constitutes a basis for V. 10 (b) Let T : ℝ² → ℝ³ be a linear transformation such that…

(a) Find all solutions to the following system of equations by row-reduced method : x₁ + 2x₂ − x₃ = 2, 2x₁ + 3x₂ + 5x₃ = 5, − x₁ − 3x₂ + 8x₃ = − 1. 15 (b) A wire of length l is cu…

(a) Let the set P = x y z middle| l x - y - z = 0 and 2x - y + z = 0 be the collection of vectors of a vector space R³(R). Then (i) prove that P is a subspace of R³. (ii) find a b…

(a) Find a linear map T : R² → R² which rotates each vector of R² by an angle θ. Also, prove that for θ = (π)/2, T has no eigenvalue in R. 15 (b) Trace the curve y²x² = x² – a², w…

(a) Show that the general solution of the differential equation dy/dx + Py = Q can be written in the form y = Q/P - e⁻∫ P dxC + ∫ e^∫ P dx d(Q/P), where P, Q are non-zero function…

(a) A cable of weight w per unit length and length 2lhangs from two points P and Q in the same horizontal line. Show that the span of the cable is2l(1 - (2h²)/(3l²)), whereh is th…

(a) Verify Stokes' theorem for F⃗ = xî + z^2ĵ + y^2k̂ over the plane surface : x + y + z = 1 lying in the first octant. 20 (b) Solve the following initial value problem by using…

(a) (i) Find the general and singular solutions of the differential equation : (x² - a²)p² - 2xyp + y² + a² = 0, where p = dy/dx. Also give the geometric relation between the gene…

(a) Show that the multiplicative group G = {1, -1, i, -i}, where i = √(-1), is isomorphic to the group G' = ({0, 1, 2, 3}, +₄). 10 marks (b) If f(z) = u + iv is an analytic functi…

(a) Let f(x) = x² on [0, k], k > 0. Show that f is Riemann integrable on the closed interval [0, k] and ∫₀ᵏ f dx = k³/3. 15 marks (b) Prove that every homomorphic image of a group…

(a) Evaluate ∫_C (z+4)/(z² + 2z + 5) dz, where C is |z + 1 - i| = 2. (15 marks) (b) Find the maximum and minimum values of x²/a⁴ + y²/b⁴ + z²/c⁴, when lx + my + nz = 0 and x²/a² +…

(a) Let R be a field of real numbers and S, the field of all those polynomials f(x) ∈ R[x] such that f(0) = 0 = f(1). Prove that S is an ideal of R[x]. Is the residue class ring R…

(a) It is given that the equation of any cone with vertex at (a, b, c) is f((x-a)/(z-c), (y-b)/(z-c)) = 0. Find the differential equation of the cone. (10 marks) (b) Solve, by Gau…

(a) Solve the heat equation ∂u/∂t = ∂²u/∂x², 0 < x < l, t > 0 subject to the conditions u(0, t) = u(l, t) = 0, u(x, 0) = x(l-x), 0 ≤ x ≤ l. (20 marks) (b) Find a combinatorial cir…

(a) Find the general solution of the partial differential equation (D² + DD' - 6D'²)z = x² sin(x+y) where D ≡ (∂)/(∂ x) and D' ≡ (∂)/(∂ y). (15 marks) (b) The velocity of a train…

(a) Reduce the following partial differential equation to a canonical form and hence solve it: yuₓₓ + (x+y)u_xy + xu_yy = 0 (15 marks) (b) Using Runge-Kutta method of fourth order…