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Posted By: RAFI Member Level: Gold Posted Date: 17 Jun 2008
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2007 Indira Gandhi National Open University (IGNOU) B.Com Computer Science CS-60: FOUNDATION COURSE IN MATHEMATICS, December, 2006 Question paper
CS-60: FOUNDATION COURSE IN MATHEMATICS
IN COMPUTING
December, 2006
Time: 3 hrs Maximum Marks: 75
Note: Question No. 1 is compulsory. Attempt any three questions from Questions No. 2 to 5. Calculators are not allowed.
1. (a) Check whether the function, f, given by
f(x) = cos x – sin x, x?[0, ?/2]
is monotonic or not.
(b) If f and g are functions defined no [a, b] such that f.g is continuous on [a, b], must f and g be continuous? Give reasons for your answer.
(c) If y = sin-1x , prove that
?1-x2
(1-x2) yn+2 – (2n+3) x yn+1 – (n+1)2 yn = 0.
(d) Prove that
Cos 50= cos5? (1-10 tan2? + 5tan4?).
(e) Does the line x+y+z = 0 touch the parabola y2=8x? If yes, find the point of contact.
(f) Taking 6 subdivisions of the interval [1, 7], find an approximate value of
? dx, using the Trapezoidal Rule.
(g) Can the following system of equations be solved by Cramer’s Rule? If yes, apply this rule to solve it. Otherwise, apply the Gaussian method to solve the system.
2x-y+z = 0
3x+y-3z = 7
x-3y+5z = -7
(h) Evaluate:
? dx
2. (a) Find the upper and lower product sums of f defined by f(x) = 4/(x-1) w.r.t. the partition P = [2, 4, 6, 8]of [2, 8].
(b) Use the Mean Value Theorem, to prove that | sin(a-b) sin (a+b) = |a-b|, where a, b ?R.
(c) Solve the equation,
4x4-11x3+21x2-10x-4 = 0
given that one of the roots is 1 - i?3.
(d) Prove that
? dx = ?2/4
0
3. (a) Find the tangent planes to the sphere
x2+y2+z2-4x+2y-6z+5 = 0
which are parallel to the plane 2x+2y –z =5
(b) Describe the sets, P ?R, Q?R and Qc, where
P = {x?N| x is a factor of 50}
Q = {x?Z| x is a multiple of 5 and |x| =25}
R = {0, 3, -5, -15} and Universal set U = {x ?I | I is the set of integers}.
(c) Find all the asymptotes of the curve.
y2 = 4x2+3x+2
x2-7x-8
(d) Find the derivative of
(sin x)logx + (log x)tanx
w.r.t.x.
4. (a) Find the points of intersection of the line
x+5 = y-4 = z-11 with the conicoid 12x2-y2 = 8, xy = 3.
-3
12x2-17y2+7z2 = 7.
(b) Find the angle of intersection of the curves.
x2 – y2 = 8, xy = 3.
(c) Reduce the equation
5x2-6xy+5y2+22x – 26y + 29 = 0
to canonical form. Hence identify the conic it represents and draw its rough sketch.
5. (a) Show that _____
3?1+3 + 3?2 + 3?3 + …. + 3?n < n.3?(n+1)/2
(b) Find lim [1/(x-3) – 2/(x2-4x+3)] if it exists.
(c) Find the equation of the cone, whose vertex is at (3, 2, 1) and the guiding curve is y2 =4x, z = 0
(d) Find the length of an arch of the cycloid
x = a (?+sin ?), y = a(1+cos?).
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