Bharathidasan University 2006 B.Sc Mathematics Fundamentals in - Question Paper

Fundamentals in Mathematics

Time: three hours
Maximum Marks: 75

Note : ques. number one is compulsory. Solve any 3 ques. from Q.no. two to 5.
1. (a) obtain f(2), f(1/2), f(-3.5), if f(x) = [x] Further for which values of x is f(x) = three ? (3)

(b) obtain the complex conjugate of 3+5i/1+2i. (3)

(c) Differentiate (sin x)xw.r.t. x. (2)

(d) The distance s moved by a particle in time t is provided by
s = 4t3-6t2-24t
Show that the direction of motion is reversed after two seconds. (4)

(e) decrease the formula
x2-4xy+4y2+5vy-10 = 0
to standard form. Hence identify the conic represented by it. (7)

(f) lf A. B and C are sets such that A ? B = A U C and A n B = F = A n C, then prove that B = C. (3)

(g) find the maximum possible domain and range of the function described by f(x) = v2-x/x. (3)

find d/dt (?sint-t dx/x2(x-1)2). (3)

(i) provide an example, with justification, of a conicoid whose planar parts are either circles or parabolas. (2)

2. (a) obtain the points of continuity and discontinuity, if any, for


2x3-54
------------------- , x ? ±3
f(x) = x2-9

9 , x = ±3
(b) If a, ß, ? are the roots of the formula
x3+6x2+6x+8 = 0, obtain the formula whose roots are (a + ß), (ß + ?), (? + a). (6)

(c) Prove that y = mx + c is a tangent to x2 = 4ay if c=-am2. (4)

3. (a) obtain the length of the major and minor axes, the eccentricity, co-ordinates of the vertices and foci of 2x2 + 3y2 = 12. (4)

(b) Show that ?p/20 vsinx/vsinx + vcosx dx = &pi/4. (5)

(c) obtain all z ? C such that z4 = -8(1+iv3). (4)

(d) Which of the subsequent conicoids has a centre at the origin ? provide reasons for your ans. (2)
(i) x2+y2-z2+3xy-3yz+2zx+18x = 0
(ii) 2x2+y2+2z2+8x-4y+5 = 0

4. (a) Evaluate ? (x-1)dx/(x+1) vx+2. (4)

(b) Solve the subsequent system of equtions using Cramer's rule : (4)
2x + y - z = one x - 2y + 3z = six x - y + 2z = 9

(c) Check whether ayz + bzx + cxy = 0 and vax+vby+vcz = 0 are reciprocal cones.

(d) If y = emtan-1x, show that
(1+x2)yn+1 + (2nx-m)yn + n(n-1)yn-2 = 0. (3)

5. (a) Sketch the surface described by
x2/16 - y2/9 - z2/9 = 1.
Name the curves formed by intersecting this surface with (4)
(i) y = 3
(ii) x = 4.

(b) calculate the area bounded by y2 = 9x and x2 = 9y. (3)

(c) Use the Cauchy - Schwarz inequality to solve x3-25x2-4x+100 = 0, provided that all its roots are rational. (6)

(d) If f and g are functions over R, such that f + g is continuous, then must f be continuous. provide reasons for your ans. (2)

Earning:   Approval pending.