BACHELOR IN COMPUTER APPLICATIONS
Term-End Examination

December, 2005

CS60 : FOUNDATION COURSE lN MATHEMATICS IN COMPUTING

Time: 3 hours
Maximum Marks: 75

Note : Question number 1 is compulsory. Solve any three questions from Q.no. 2 to 5.1. (a) Find f(2), f(1/2), f(-3.5), if f(x) = [x] Further for which values of x is f(x) = 3 ? (3)

(b) Find 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 given by
s = 4t³-6t²-24t
Show that the direction of motion is reversed after 2 seconds. (4)

(e) Reduce the equation
x²-4xy+4y²+5√y-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 ∩ B = Φ = A ∩ C, then prove that B = C. (3)

(g) Obtain the maximum possible domain and range of the function defined by f(x) = √2-x/x. (3)

Obtain d/dt (∫^sint_-t dx/x²(x-1)²). (3)

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

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

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

9			, x = ±3

(b) If α, β, γ are the roots of the equation
x³+6x²+6x+8 = 0, find the equation whose roots are (α + β), (β + γ), (γ + α). (6)

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

3. (a) Find 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 ∫^π/2₀ √sinx/√sinx + √cosx dx = π/4. (5)

(c) Find all z ∈ C such that z⁴ = -8(1+i√3). (4)

(d) Which of the following conicoids has a centre at the origin ? Give reasons for your answer. (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) √x+2. (4)

(b) Solve the following system of equtions using Cramer’s rule : (4)
2x + y – z = 1 x – 2y + 3z = 6 x – y + 2z = 9

(c) Check whether ayz + bzx + cxy = 0 and √ax+√by+√cz = 0 are reciprocal cones.

(d) If y = e^mtan-1_x, show that
(1+x²)y_n+1 + (2nx-m)y_n + n(n-1)y_n-2 = 0. (3)

5. (a) Sketch the surface defined 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) Compute the area bounded by y² = 9x and x² = 9y. (3)

(c) Use the Cauchy – Schwarz inequality to solve x³-25x²-4x+100 = 0, given 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. Give reasons for your answer. (2)

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