Careers

March

1. Find the particular solution to the differential equation 8dx/dt= −2cos(2t)−cos(4t) that passes through (π,π) , given that x= C−18sin(2t)−132sin(4t) is a general solution.
2. Let X be a square matrix. Consider the following two statements on X.
   
   I. X is invertible.
   II. Determinant of X is non-zero.
   
   Which one of the following is TRUE?
   
   a. I implies II; II does not imply I
   b. II implies I; I does not imply II
   c. I does not imply II; II does not imply I
   d. I and II are equivalent statements
3. Find the value of ∫2x cos (x² – 5).
4. If G is a connected simple graph, and is not a complete graph or a cycle graph with an odd number of vertices, then the chromatic number χ(G)≤Δ(G).
5. Find target 19 in the list: 1 2 3 5 6 7 8 10 12 13 15 16 18 19 20 22 ( using Binary search method )
6. State order completeness property of real numbers. show that the set of rational number is not order complete.
7. Let V be n-dimensional vector space over the filed F and w be m-dimensional vector space over F. Then the L(v,w) is finite dimensional and has dimension mn.
8. Differentiate the following expressions (with respect to x): (4x²−3x−7)log(x)
   Integrate the following expressions (with respect to x): sinxln(cosx).
9. The sum of two numbers is 82 and their product is 1456, find the two numbers.
10. Show that the sum 1 + 2 + · · · + n of the first n positive integers is O(n2).

1. Find the equation of a curve passing through (1, π/4) if the slope of the tangent to the curve at any point P (x, y) is y/x – cos2(y/x).
2. (Euler’s Formula for Planar Graphs). For any connected planar graph G embedded in the plane with V vertices, E edges, and F faces, then prove it must be the case that V + F = E + 2.
3. Solve the ODE combined with initial condition:dx/dt = 5x−3, x(2)=1
4. Let F be a field and T be the linear equation of 𝐹² defined by T(x,y) = ( x+y,x) then T is a non singular and T is onto and also find 𝑇^-1
5. Find the P.I of (𝐷²− 2𝐷𝐷¹+ 𝐷^︲)𝑦 = 𝑐𝑜𝑠(𝑥 − 3𝑦)
6. Let α be a linear map on V . Then the following conditions are equivalent for an element λ of K:
   (a) λ is an eigenvalue of α;
   (b) λ is a root of the characteristic polynomial of α;
   (c) λ is a root of the minimal polynomial of α
7. Under regularity conditions (like those specified above) for the model {fθ : θ ∈ *}, then (n I(θ ))1/2 (θˆ − θ) D → N(0, 1) as
   n → ∞.
8. Solve (𝐷³ − 8𝐷 + 4)𝑦 = 𝑒^2𝑥 + 𝑐𝑜𝑠𝑒𝑐2𝑥
9. If the graph is semi-Eulerian, x and y are the two vertices of odd degree, and the shortest pathbetween x and y has length P then the shortest postman route has length L + P
10. Let G be a simple graph with n > 3 vertices. Suppose that every pair of distinct non-adjacentvertices (u, v) satisfies the inequality d(u) + d(v) > n. Then G is Hamiltonian

April

1. For the following linear system has the origin as an isolated critical point
   
   (a) Find the general solution
   (b) Find the differential equation of the paths
   (c) Solve the equation found in (b):
   {dx/dt = 4y}
   {dy/dt = -x}
2. Show that R be an integral domain with unit element and suppose that a,b∈R botha and b are true. Also prove that a=ub, where u is unit in R.
   (or)
   If R is commutative ring with unit element and M is an idel of R, then show that M is maximal ideal of R if and only if R/M is a filed.
3. Derive Bernoulli's solution of the wave equation.
   (or)
   Let y(x) be a non-trivial solution of y"+q(x)y = 0 on a closed interval [a,b]; Then prove that y(x) has at most a finite number of zeros in this interval.
4. Let x be an ordinary point of the diffrential euation y"+P(x)y'+Q(x)y=0 and let a and b are arbitary constants. Then there exists a unique function y(x) that is analytic at x, is a solution of y"+P(x)y'+Q(x)y=0 in a certain neighbourhood of this point and satisfies the initial condition y(x)=a and y'(x)=b. Furthermore, if the power series expansion of P(x) and Q(x) are valid on an interval |X-Y|(R,R)0, then prove that the power series expansion of this solution is also valid on the same interval.
5. For a connected graph G, the following statements are equivalent:
   
   (a) G is Eulerian.
   (b) The degree of each vertex of G is an even positive integer.
   (c) G is an edge-disjoint union of cycles.
6. Prove that for every real x>0 and every integer n>0 there is one and only one positive real y such that y=x
7. Reduce to canonical form and find the general solution:
   u_xx + 5u_xy + 6u_yy = 0.
8. Find the Fundamental Solution of the Laplace Operator for n = 3.
9. Solve the following initial-boundary value problem for the wave equation with a potential term,
   
   { u_tt − u_xx + u =0 0 < x < π, t < 0 }
   { u(0, t) = u(π, t)=0 t > 0 }
   { u(x, 0) = f(x), u_t(x, 0) = 0 0 < x < π}
10. State and prove Taylor's theorem.
    (or)
    State and prove root test and ratio test.

1. Solve y²p-xyq=x(z-2y).
2. (a) Find the general solution of x(y²-z²)p+y(z²-x²)q=z(x²-y²)
   (b)Consider the logistic equation,
   du/dt = ru(1 − u) with u(t = 0) = u₀. Find the solution of the initial value problem.
3. Abel’s differential equation of the first kind is written in the form du/dt = a₀(t) + a₁(t)u + a₂(t)u² + a₃(t)u³ where aj (j = 0, 1, 2, 3) are known smooth functions of t.
   
   (i) Show that du/dt = a₀(t) + a₁(t)u + a₂(t)u² + a₃(t)u³ can be put into the standard form as dz/dx = z³ + p(t)
   (ii) Show that if a3 = 0, (1) reduces to the Ricatti equation.
   Show that for a⁰ = 0, a¹ 6= 0 and either a² = 0, a³ 6= 0 or a² 6= 0, a³ = 0, it becomes the nonlinear differential equation of Bernoulli type, which has an explicit general solution.
4. Solve the initial value problem of the differential equation du/dt = k(a − u)(b − u),a > b > 0 by direct integration if u(t = 0) = 0.
5. Solve the following PDE for f(x, y, t):
   f_t + xf_x + 3t² f^y = 0
   f(x, y, 0) = x² + y².
6. Solve:(3z-4y)p+(4x-2z)q=2y-3x.
7. Solve the following initial-boundary value problem in the domain x > 0, t > 0, for the unknown vector U = u(1) / u(2).
8. What is the solution with initial value y(0)=−1?
   y︲+(tanx)y=cos²x
9. Find the second order derivative of x²⁰.
10. A trigonometric curve C satisfies the differential equation dy/dx (cosx + ysinx) = cos³x .
    (a) Find a general solution of the above differential equation.
    (b) Given further that the curve passes through the Cartesian origin O , sketch the graph of C for 0≤ x ≤2 π .
    The sketch must show clearly the coordinates of the points where the graph of C meets the x axis.