* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 2011 - Bangabasi Evening College Library catalog
Survey
Document related concepts
Large numbers wikipedia , lookup
Infinitesimal wikipedia , lookup
Line (geometry) wikipedia , lookup
Bra–ket notation wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Series (mathematics) wikipedia , lookup
Real number wikipedia , lookup
Continuous function wikipedia , lookup
Karhunen–Loève theorem wikipedia , lookup
Collatz conjecture wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Elementary mathematics wikipedia , lookup
Transcript
BANGABASI EVENING COLLEGE MATHEMATICS -HONOURS (Module-III) denotes the set of real numbers and denotes the set of all positive integers Group-A (Real analysis-I) Answer any two questions 1. (a) State LUB Axiom of real numbers. Is the Axiom valid for the set of rational numbers?- Justify your answer. 2+1 (b) Prove that “Every infinite bounded set of real numbers has a limit point”. 3 (c) Prove that product of a bounded sequence and a null sequence is a null sequence. 4 2. (a) Define ‘ Denumerable set’. Prove that the intersection of an arbitrary collection of closed sets is closed. 1+3 n 13 ,n }. Give an example of a (b) Find the derived set of the following set: { set which has an infinite number of limit points. 2+1 1 2 (c) Examine whether 1 1 is a Cauchy sequence. The sequence 4 2n n {n 2 } is not a Cauchy sequence. Why? 2+1 3. (a) Let f : [a, b] be continuous in [a,b]. Prove that f is bounded on [a,b]. 3 11 1 1 does not converges. 3 (b) Prove that the sequence S n define by S n 23 n (c) If a function f (x) is continuous in [a, b] , then it attains its supremum and infimum at least once in [a, b] . Let f : be such that f (x) k (constant) for all x . Show that f is continuous on . 2+2 1/x e 4. (a) Discuss the continuity and discontinuity of the following function: f(x) , 1e1/x when x 0 and f (0) 1 . 3 (b) Give an example to show that a function continuous in an open interval may fail to be uniformly continuous in the interval. 3 f : (c) A function satisfies the condition f(x+y)=f(x)f(y) for all x, y . If f is continuous at x=0, prove that f is continuous on . 4 Group-B Full Marks-05 Answer one question cos n ( n 1 )( I I ) 2 sin( n 1 ) d 5. If In n n 2 cos , show that 2 3d. Hence evaluate 4cos 5 /2 2 n 1 6. If Jnsin xdx , n is positive integer, prove that 0 2n Jn Jn1. Use this to evaluate 2n1 /2 sin xdx. 7 5 0 x sin x dx . 2 0 1 cos x 7. Evaluate 5 (Module-IV) Group-A (Linear Algebra-I) Full Marks-20 Answer two questions 2 2 1 8. (a) Find the eigen values and all eigen vectors of A 1 3 1. 1 2 2 2 2 2 x y 10 z 4 yz 10 zx (b) Reduce the quadratic form 5 to the normal form and show that it is positive definite. 5+5 9. (a) Prove the following: (i) 0 is the eigen value of a singular matrix. (ii) The eigen values of a diagonal matrix are its diagonal elements. 1 (iii) If is the eigen value of a non-singular matrix A then is the eigen value of A 1 . (iv) If A and P be both n n matrices and P be non-singular, then A and P 1 AP have the same eigen value. 3 3 {( x , y , z ) : x y z 0 } (b) Show that W is a subspace of . Find a basis and the dimension of the subspace W of 3 . 5+5 2 0 1 10. (a) Show that the matrix 3 3 0 is nonsingular and express it as a product of 6 2 3 elementary matrices. (b). Expanding by Laplace’s method show that 0 a b a 0 d b d 0 c e f c e af 2. be cd f 0 5+5 A det B0 11. (a) Let A and B are real orthogonal matrices of the same order and det then show that A+B is singular matrix. (b) Determine the conditions for which the system xyz 1 x2yz b 5x7yazb2 admits of (i) only one solution (ii) no solution, (iii) many solutions. 3+7 12. (a) If , be any two vectors in a Euclidean space V, then prove that . (b) Use Gram-Schmidt process to obtain an orthogonal basis from the basis set {( 1 ,0 ,1 ), ( 1 ,1 ,1 ), ( 1 ,3 ,4 )} of the Euclidean space 3 with standard inner product. (c) State Cayley-Hamilton theorem. Verify Cayley-Hamilton theorem for the matrix 1 2 1 A1 1 1 .Express A 1 as a polynomial in A and hence compute A 1 . 2+4+4 2 3 1 Group-B (Vector Calculus-I) Full Marks-05 .Answer any one question 2 2 2 (,y ,) z 2 x 3 y zat the point P(2,1,3) 13. Find the Directional derivative of fx in the direction of the vector a iˆ 2kˆ . 5 A ) g r a d A divA . 14. Prove that div ( 5 2 2 2ˆ ˆ xi yˆ j z kis irrotational. Hence 15. Prove that the vector field defined by F find a scalar function (x, y, z) such that F . 5 Test Examination-2016 BANGABASI EVENING COLLEGE MATHEMATICS HONOURS First Paper Full Marks- 50 The figures in the margin indicate full marks (Module-I) Group-A (Classical Algebra) Full Marks-20 Answer any four questions ! 2 ! 3 ! 4 ! 100 !is divided by 15? 1. (i) What is the remainder when 1 3) , x 3(mod 5) and x 1(mod 7) . (ii) Solve the system of linear congruence’s: x 2(mod 2 .(i) Prove that there exists infinite number of prime integers. (ii) Find the remainder when 1920 is divided by 181. ib ab a 2 i log 2 2. 3. (i) Prove that sin ib b a a (ii) Find the principal value of 1 i . i 3+2 1 1 1 4. If cosh , where x, y, a are real and a 1 , then ( x iy ) cosh ( x iy ) cosh a prove that point (x,y) lies on the ellipse. 5 5. If x, y, x are positive real numbers and xyz 1, prove that 8 8 xyz ( 1 x )( 1 y )( 1 z ) . 27 5 12 12 6. (i) Show that a b is divisible by 91 if a and b are both prime to 91. (ii) Calculate (2048) and (5040) , where the function , is called Euler’s Phi function. 3+(1+1) 7. Solve the equation x 3 3x 2 3 0 by Cardan’s method. 8. If , , be the roots of the equation x 3 qx r 0 , find the equation whose roots are ( ) 2 , ( ) 2 , ( ) 2 . Group-B Full Marks-05 Answer any one question ( a , b ) : a , b Q ,a b is an ineger 9. Let be a relation defined by where Q is the set of all rational numbers. Is on Q is an equivalence relation? Justify. 10. Let S is the set of all 2x2 non-singular matrices whose elements are real numbers. S is a proper subset of M 2 ( R) . Prove that S forms a non commutative group under matrix multiplication “.”. 11. (a) Let A, B, C be non-empty sets. Prove that ( A B) (C D) ( A C ) ( B D) . (b) Let f : N N be the mapping given by f(x)=2x, where N+ is a set of all positive integers. Show that f is injective but not surjective. (Module-II) Group-A (Analytical Geometry of Two Dimensions) Full Marks-20 Answer any four questions 12. Show that the locus of the poles of the tangent to the director circle of the ellipse x2 y2 2 4 2 4 2 2 2 1 w.r.t this ellipse is x a y b ( a b )1 . 5 2 a b 2 13. If three normals from a point to the parabola y 4axcut the axis at the points whose distances from the vertex are in A.P, then show that the point lies on the curve 2 27 ay 2 (x 2 a )3. 5 2 14. Tangents are drawn to the parabola y 4ax at the points whose abscissae are in the ratio p:1. Show that the locus of their point of intersection is a parabola. 5 2 2 2 hxy by 2 gx 2 fy c 0 15. If the equation ax represents two straight lines 4 4 2 2 g c bf ag equidistant from the origin, show that f . 5 /r a cos b sin may touch the 16. Show that the condition that the straight line 1 2 2 1. circle r cos is bk 2ak 5 2 2 x 4 xy y 4 x 2 y a 0 17. Reduce the equation 4 to its canonical form, determine the nature of the conic for different values of a. 5 Group-B Full Marks-05 Answer any one question 18. If the external bisectors of the angles of a triangle intersect the opposite sides at the points P, Q, R, then prove that P, Q, R are collinear.. 19. (i) In any triangle ABC, with usual notations, prove that a cos B b cos A (i) c , (ii) a b c . sin A sin B sin C 20. If the diagonals of a quadrilateral bisect on another, then prove that the figure is a parallelogram.