Name of the University : Netaji Subhas Open University
Name of the Exam : BDP Term End Exam
Document Type : Sample Question Paper
Year: 2016
Subject: Mathematics

Website : http://www.wbnsou.ac.in/student_zone...c2015-June2016
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BDP Term End Mathematics Question Paper :
1st Paper : Differential Calculus and its Geometrical Applications )
Time : 2 Hours
Full Marks : 50

Answer any two questions. 10 × 2 = 20
1. a) Let A be a non-void bounded subset of IR. Let F = {|x - y|: x A,y A } . Obtain an expression of sup F in terms of bounds of A. 3
b) Let the sequence of real numbers {an }n converges to zero and the sequence of real numbers {bn }n be bounded in IR. Examine the convergence of {anbn }n .

c) Given a series Sn n1/x where x ? IR. Choose the correct statement from the following with reason : 4
i) The series is everywhere convergent
ii) The series is nowhere convergent
iii) The series is convergent when –1 < x < 0
iv) the series is convergent for x > 1.

2. a) Let the real-valued function f be continuous at p and in every neighbourhood of p, contained in the domain of f, f ( x ) assumes both positive and negative values.Find the value of f ( p ) .
b) Prove that there cannot be two different power series with the same interval of convergence and having the same sum function in the interval.

3. a) Prove that every bounded sequence in IR has at least one convergent sub-sequence.
b) Prove that every absolutely convergent series in IR can be expressed as the difference of two convergent series of positive terms and a conditionally convergent series in IR can be expressed as the difference of two divergent series.

4. a) Prove that the union of finite number of closed sets in IR is a closed set.
b) Prove that the set E = { x : 0 < x < 1} is not enumerable.

Mathematical Analysis-I :
1. a) If the pairs of straight lines x2 - 2pxy - y2 = 0 and x2 - 2qxy - y2 = 0 bisect the angles between each other, then prove that pq +1 = 0 .
b) Prove that the angle between two straight lines remains same due to change of coordinates for rotation of axes.

2. Find the equation of the sphere which passes through the points ( 1, 0, 0 ), ( 0, 1, 0 ), ( 0, 0, 1 ) and touches the plane 2x + 2y - z = 15 .
3. Show that the triangle bounded by the straight line px + qy +1 = 0 and the pair of straight lines ax2 + 2hxy + by2 = 0 will be equilateral if (3a + b)(a + 3b) - 4h2 = 0 and h (p2 - q2) - (a - b) pq = 0 .

4. Prove that x2 - 3xy + y2 +10x -10y + 21 = 0 represents a hyperbola with (-2,2) as centre.
6. Find the conjugate diameter of the diameter x = 2y of the hyperbola 16x2 - 9y2 = 144 .
7. Transform the equation 3x2 + 5y2 + 3z2 + 2yz + 2zx + 2xy - 4x - 8z + 5 = 0 to its canonical form and find the nature of the quadric represented by it.

8. Find the coordinates of the foot of the perpendicular from the point P ( 4, 3, 8 ) on the line joining the points A ( 2, 4, 6 ) and B (4, 0, 8).
9. If x2 + y2 + z2 + 7y - 2z + 2 = 0 , 2x + 3y + 4z = 8 is a great circle of a sphere, determine the equation of the sphere.

10. Prove that the generators of the hyperboloid x2 + y2 - z2 = 1 which intersects the plane z = 0 will be perpendicular to each other.
11. Find the distance of the point of intersection of the straight lines represented by 2x2 - 5xy + 3y2 - 2x + 3y = 0 from the origin.

12. Find the equation of the circle which passes through the point ( 1, 5 ) and touches the straight line 2x - 3y = 5 at the point ( 1, –1 ).
13. Find the nature of the conicoid 3x2 - 2y2 -12x -12y - 6z = 0 .