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# Fundamental of Mathematics Code-6372 1st year (Hon.) Test Examination-2009

Govt. M. M. college, Jessore

Sub-Fundamental of Mathematics  Code-6372

Full Marks-50   Time-2.5 hours

[All questions are of equal value. Answer any five questions ]

1. (a) Define Rational number and Irrational number. Show that between two rational numbers

there exists at least one rational number.

(b) If x= cosq + i sinq and then prove that ,1+ cosq =.

1. (a) State the theorem of Descarte’s rule of signs. Find the nature of the roots of the equation

. (b) If the roots of are a, b, c then find an equation whose roots are .

1. (a) Determine the values of l such that the following system of linear equations have

(i) no solution (ii) more then one solution (iii) a unique solution x + ly + z =1

x + y + l z =1

l x + y + z =1

(b) If A = then prove that .

1. (a) If S and T are two subspaces of a vector space V(F), then prove that S + T is a subspace of V(F). (b) Define linear independence of a set of vectors. If the vectors are linearly  independent

then show that the vectors are linearly independent.

5.(a) Define Eigen values and Eigen vectors. Find the Eigen values of the matrix A=    (b) If A = then find using Cayley-Hamilton theorem.

1. (a) Show that the equation represents a pair of straight lines.(b) Show that the equation of bisectors of the angles between the lines represented by is .

7.(a) A line makes angles α, β, γ, δ with the four diagonals of a cube. Prove that

(b) Find the equation of the plane which passes through the points (2, 2, 1) and is

perpendicular to the plane 2x + 6y + 6z + 9=0 .

1. (a) Find the projection of the vector on a straight line joining the points

A(2, 3, -1) and B(-2, -4, 3).

(b) Show that the three vectors are

perpendicular to each other.