Questions for math interview

1..Define projective geometry.
ANS..There are two geometries between them: similarity and affine. To see the relationships between these different geometries, consult Figure 1. Projective geometry models well the imaging process of a camera because it allows a much larger class of transformations than just translations and rotations, a class which includes perspective projections. Of course, the drawback is that fewer measures are preserved -- certainly not lengths, angles, or parallelism. Projective transformations preserve type (that is, points remain points and lines remain lines), incidence (that is, whether a point lies on a line), and a measure known as the cross ratio.
2..if the value of a third order determinent is 11,then the square of the determinent formed by the cofactors will be
ANS..121
3..Find the interval for theinitial value problem dy/dx=y*2 ,y(1)=-1.
ANS..(1/4, 5/4).
4..Define singular solution of a diffential equation.
ANS..A singular solution ys(x) of an ordinary differential equation is a solution that is tangent to every solution from the family of general solutions. By tangent we mean that there is a point x where ys(x) = yc(x) and y's(x) = y'c(x) where yc is any general solution.
5..A group G has subgroups of order 4 and 10.order of G is less than 50.what you conclude about order of G.
ANS..IT IS EITHER 40 or 2o.
6..If D IS AN INTERGRAL domain, what we can say about characteristic of D.
ANS...Eithre 0 or a prime.
7..HOW MANY homomorphism possible from Z[2]to Z[3].
ans..only one.
8..How many subgroups are possible for Z[2]*Z[4].
ANS..EIGHT
9.Find the soution of sin(dy/dx)=a with y(0)=1.
ANS..sin{(y-1)/x}=a
10..THE numbers of all possible triplets(a,b,c), such that a+bcos2x+csin*2(x)=0.
ANS--INFINITE.
11..Provide geometrical interpretation of all two-rowed orthogonal matrices.
ANS..Every two rowed orthogonal matrix corresponds either to a rotation about origin or to a reglection in a line through the origin.
12..provide some contexts in which the quardatic form appear.
ANS..1..STUDY OF CONICS AND QUADRICS.
2.investigation of max. and mini..of serval variable.
3.stability of equilibrium in mechanics.
13..LET A be 2-rowed square matrix and det(I+A)=1+detA. WHAT IS TRACE OF A,\.
ANS..zero.
14.if completeness preserved under homeomorphism.
ANS...No.we have R and ]0,1[, which are homeomorphic to each other.one of which complete and other is not.