সরকারি এম এম কলেজ, যশোর গনিত বিভাগ ৩য়  বর্ষ নির্বাচনী পরীক্ষা ২০১৩ বিষয়: লিনিয়ার প্রগ্রামিং  কোড: ৩৭৭৫

Linear Programming -2013

সরকারি এম এম কলেজ, যশোর গনিত বিভাগ ৩য়  বর্ষ নির্বাচনী পরীক্ষা ২০১৩ বিষয়: বলবিদ্যা কোড: ৩৭৭৭

Macnics -2013

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 =.

2. (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 .

3. (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 .

4. (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.

6. (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 .

8. (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.

Govt. M M College, Jessore

Department of Mathematics

M Sc. Preliminary Test Examination-2010

Subject – Real & Complex Analysis Code –3701

     Time – 4 hours             Full Marks – 100

(Answer any six questions)

Group-A: (Real Analysis)

(Answer Any Four)

1(a) Define supremum and infimum of a set of real numbers, state and prove the supremum and infimum principles for R and show that they are equivalent.

(b) Prove that if , then there exists a positive integer n such that

. If A= then find sup A, inf A, max A and min A.

• (a) Define open set and closed set of real number. Show that the union of two open

sets is open.

(b) Define limit point of a set of real number. State and prove the Bolzano–weierstrass theorem. If A ={(-1)ⁿ} then show that A has no limit point .

3 (a) Define countable set and uncountable set. Show that the interval [0, 1] is not countable.

(b) Define interior, exterior and boundary points of a set of real numbers. Prove that a subset is closed if it contains all of its limit points.

4.(a) Define continuity and differentiability of a function. Show that the function

is continuous at

(b) If then show that f(x) is not differentiable at x=0.

5. (a) State and prove Rolle’s theorem.

(b) Justify Rolle’s theorem for the function in the interval.

6. Justify the Mean-value theorem theorem for the function in the interval.

Group-B: (Complex Analysis)

(Answer Any Two)

7 (a) If be two complex numbers then prove that . (b) Sketch the region described by .

8.(a) If is analytic then find it interms of z when .

(b) State and prove Rouchy’s theorem.

9. (a) State and prove Cauchy’s integral formula the derivative of an analytic function.

(b) Use Cauchy’s integral formula to evaluate
where c is the circle

10. (a) Evaluate (any two) (i) , (ii) ,

Govt. M M College, Jessore

Department of Mathematics

3rd Year B. Sc. Honours Test Examination-2007

Subject – Real Analysis         Code – 3732

     Time – 4 hours             Full Marks – 100

1(a) Define supremum and infimum of a set of real numbers, state and prove the supremum and infimum principles for R and show that they are equivalent.

(b) Prove that if and, then there is a rational number such that

. If A= then find sup A, inf A, max A and min A.

2 (a) Define open set and closed set of real number. Show that the union of two open

sets is open.

(b) Define limit point and accumulation point of a set of real number. State and prove the Bolzano–weierstrass theorem. If A ={(-1)ⁿ} then show that A have two accumulation points but has no limit point .

3 (a) Define countable set and uncountable set. Show that the interval [0, 1] is not countable.

(b) Define interior, exterior and boundary points of a set of real numbers. Prove that a subset is closed if it contains all of its limit points.

4.(a) Define sequence and convergent sequence. Show that every convergent sequence is bounded. But the converse is not always true.

(b) Define limit of a sequence, show that every convergent sequence has a unique limit.

5.(a) Define Cauchy sequence, state and prove Cauchy’s general principle of convergence for a sequence.

(b) Prove that the sequence < x_n > is define by , converges to the positive roots of the equation converges the positive not of the equation

6.(a) Define convergent series and absolute convergent series. State and prove the Cauchy’s root test for convergence of infinite series.

(b) Test the convergence of (any two)

(i) ,     (ii) ,   (iii) , .

7 (a) Define uniformly continuous function. Show that is uniformly continuous on [0,1] but is not uniformly continuous on [0,1].

(b) State and prove the intermediate value theorem for continuous function.

8.(a) Define Riemann –integral. Write the necessary and sufficient condition for a function defined in the interval [a , b] is Riemann integrable and prove its sufficiency. .

(b) Prove that every monotonic function is Remann integrable. Show that in the interval [0,1] the function .

is Riemann integrable .

9.(a) Define point-wise convergent sequence and uniform convergence sequence of functions. State and prove weierstrass M-test for uniform convergence of a series of a function.

(b) If , where and then show that convergence point-wise but not uniformly convergence on [0, 1].

10.(a) Define metric and norm on R. Let a function be defined on a normed vector space .show that d is a metric on V.

(b) Define compact and complete metric space with examples. Show that Rⁿ is a metric space defined by.