Careers at Zemuria
Hiring Policy:
We welcome people of any Race, Color, Ancestry, Religion, Sex, National
origin, Sexual orientation, Gender Identity, Age, Marital or Family status,
Disability, Veteran status, Conscientious objectors and any other status.
These differences are what enables us to work towards the future we envision
for ourselves, and the World.
All hiring decisions are unanimous decisions taken by the whole team after
careful consideration.
In rare occasions where there is a requirement of overriding a veto of a
team member due to an exceptional candidate, member(s) who have voted ‘hire’
must take personal responsibility and mentor the hired candidate until they
are approved by the full team.
Join us by pitching how you can help Zemuria in Our mission.
Positions open:
These are the positions for which we regularly hire. Please do check here to
see if there’s an opening available or for new positions.
 Frontend Engineer: NIL
 Backend Engineer: 2 Positions
Note: All qualified engineers are required to pass at least level 01 of our Math Program Test. Apart from these, Math experts are encouraged to look into our Math Program and apply.
Our Math Thesis: In this world of AI, insisting on proficiency on Math for backend engineering ensures that we have a strong foundation in problemsolving, analytical thinking and algorithmic design.
Please remember, while Math qualifications are important, we know that it's also essential to consider practical programming skills, experience with relevant technologies, and the ability to work in a team when evaluating candidates for backend engineering roles.
 Designers: NIL
Math Program:
We are currently hiring Math Graduates as our backend engineers. Both
undergraduates and postgraduates in Math or Equivalent subjects are
eligible to apply for this program. An entrance exam that’s conducted and
proctored by Zemuria Team at our full discretion is part of our application
process.
Evaluation
Process:
 Level 01: Must score a minimum of 90% to proceed to Level 02
 Level 02: Must secure a minimum of 60% to sit for an HR interview.
 HR Interview
Post Program schedule:
All
selected candidates will be given free training along with stipend in any of
our operational offices. Post training period, you’ll be inducted as an
Intern and post successful completion of internship, will join as a Full
time Employee.
Syllabus:
Level 01 (For UG and PG)
Graph Theory,
Ordinary Differential Equation & Multivariate Calculus,
Partial Differential Equations,
Discrete Mathematics,
Matrix,
Integral Calculus and Fourier series,
Differential Equation,
Probability & Statistics,
Linear Algebra,
Vector Calculus
Level 02 (For UG)
Metric Space & Complex Analysis,
Statistics,
Real Analysis,
Linear Differential Equations,
Differential Geometry,
Fluid Statics,
Dynamics,
Astronomy & Space Science,
Applied Mathematics,
Financial Mathematics,
Stochastic Processes,
Classical Dynamics,
Applied Statistics,
Operations Research,
Vector Algebra
Level 02 (For PG)
Fuzzy graph,
Topology,
Sequences and Series,
Complex Analysis,
Analytic Number Theory,
Non linear differential equations,
Complex Variable,
Functional Analysis,
Integral Equations, Calculus of Variations and Fourier Transform,
Fuzzy sets and their applications ,
Theory of Numbers,
Mathematical Modelling,
Measure Theory and Integartion,
Stochastic Differential Equations,
Analytical Geometry 3D,
Algebra(All topics)
March
 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.

Let X be a square matrix. Consider the following two
statements on X.
I. X is invertible.
II. Determinant of X is nonzero.
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  Find the value of ∫2x cos (x^{2} – 5).
 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).
 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 )
 State order completeness property of real numbers. show that the set of rational number is not order complete.
 Let V be ndimensional vector space over the filed F and w be mdimensional vector space over F. Then the L(v,w) is finite dimensional and has dimension mn.

Differentiate the following expressions (with respect to x):
(4x^{2}−3x−7)log(x)
Integrate the following expressions (with respect to x): sinxln(cosx).  The sum of two numbers is 82 and their product is 1456, find the two numbers.
 Show that the sum 1 + 2 + · · · + n of the first n positive integers is O(n2).
 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).
 (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.
 Solve the ODE combined with initial condition:dx/dt = 5x−3, x(2)=1
 Let F be a field and T be the linear equation of 𝐹^{2} defined by T(x,y) = ( x+y,x) then T is a non singular and T is onto and also find 𝑇^{1}
 Find the P.I of (𝐷^{2}− 2𝐷𝐷^{1}+ 𝐷^{︲})𝑦 = 𝑐𝑜𝑠(𝑥 − 3𝑦)

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 
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 → ∞.  Solve (𝐷^{3} − 8𝐷 + 4)𝑦 = 𝑒^{2𝑥} + 𝑐𝑜𝑠𝑒𝑐2𝑥
 If the graph is semiEulerian, 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
 Let G be a simple graph with n > 3 vertices. Suppose that every pair of distinct nonadjacentvertices (u, v) satisfies the inequality d(u) + d(v) > n. Then G is Hamiltonian
April

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} 
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. 
Derive Bernoulli's solution of the wave
equation.
(or)
Let y(x) be a nontrivial 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.  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 XY(R,R)0, then prove that the power series expansion of this solution is also valid on the same interval.

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 edgedisjoint union of cycles.  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

Reduce to canonical form and find the general solution:
u_{xx} + 5u_{xy} + 6u_{yy} = 0.  Find the Fundamental Solution of the Laplace Operator for n = 3.

Solve the following initialboundary 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} 
State and prove Taylor's theorem.
(or)
State and prove root test and ratio test.
 Solve y^{2}pxyq=x(z2y).

(a) Find the general solution of
x(y^{2}z^{2})p+y(z^{2}x^{2})q=z(x^{2}y^{2})
(b)Consider the logistic equation,
du/dt = ru(1 − u) with u(t = 0) = u_{0}. Find the solution of the initial value problem. 
Abel’s differential equation of the first kind is written in
the form
du/dt = a_{0}(t) + a_{1}(t)u +
a_{2}(t)u^{2} +
a_{3}(t)u^{3}
where aj (j = 0, 1, 2, 3) are known smooth functions of
t.
(i) Show that du/dt = a_{0}(t) + a_{1}(t)u + a_{2}(t)u^{2} + a_{3}(t)u^{3} can be put into the standard form as dz/dx = z^{3} + p(t)
(ii) Show that if a3 = 0, (1) reduces to the Ricatti equation.
Show that for a^{0} = 0, a^{1} 6= 0 and either a^{2} = 0, a^{3} 6= 0 or a^{2} 6= 0, a^{3} = 0, it becomes the nonlinear differential equation of Bernoulli type, which has an explicit general solution.  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.

Solve the following PDE for f(x, y, t):
f_{t} + xf_{x} + 3t^{2} f^{y} = 0
f(x, y, 0) = x^{2} + y^{2}.  Solve:(3z4y)p+(4x2z)q=2y3x.
 Solve the following initialboundary value problem in the domain x > 0, t > 0, for the unknown vector U = u(1) / u(2).

What is the solution with initial value y(0)=−1?
y︲+(tanx)y=cos^{2}x  Find the second order derivative of x^{20}.

A trigonometric curve C satisfies the differential equation
dy/dx (cosx + ysinx) = cos^{3}x .
(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.
May
 A equilibrium point u^{⋆} of an autonomous scalar differential equation is asymptotically stable if and only if F(u) > 0 for u^{⋆} − δ < u < u^{⋆} and F(u) < 0 for u^{⋆} < u < u^{⋆} + δ, for some δ > 0.
 If f, g are continuous on [0, ∞) of exponential order, then L[f] = L[g] ⇒ f = g.

Let U(x, t) ∈ C∞ be 2πperiodic in x. Consider the linear
equation ut + Uux + u_{xx}+ u_{xxxx} = 0,
u(x, 0) = f(x), f(x) = f(x + 2π) ∈ C^{∞}.
(a) Derive an energy estimate for u.
(b) Prove that one can estimate all derivatives ∂pu/∂xp.
(c) Indicate how to prove existence of solutions.  Solve the Cauchy problem (a) u_{tt} = a^{2}u_{xx} + cos x (b) u(x, 0) = sin x ut(x, 0) = 1 + x.
 Find the equation of a curve passing through the point (1, 1) if the perpendicular distance of the origin from the normal at any point P(x, y) of the curve is equal to the distance of P from the x – axis.
 Solve x^{2} (dy/dx) − xy = 1 + cos( y/x ) , x ≠ 0 and x = 1, y = π/2

State the type of the differential equation for the
equation.
xdy – ydx = (x^{2} + y^{2})^{1/2}and solve it.  What is the probability that Ram will choose a marble at random and that it is not black if the bowl contains 3 red, 2 black and 5 green marbles.
 Find the equation of a curve whose tangent at any point on it, different from origin, has slope y +( y/x).
 Use the proof of PicardLindelof’s theorem to find the solution to y^{︲}= 2 y + 3 y(0) = 1 .