Filed under:
IGNOU BCA
CS-610: Foundation Course In Mathematics In Computing of December 1999
Question No 1. (a) Test whether the following function is 1-1 and/or onto: f : Rï?§R such that f(x) = 3×2-5 [2]
(b) Differentiate y w.r.t. x, where y = log(3×2.sin(x + x2)) [3]
(c) Compute the value of following definite integral
(d) Find area of the region bounded by y = 5x – x2, x = 0, x = 5 and lying above the x-axis. [3]
(e) Find length of the cycloid x = a (ï?± - sinï?±) and y = a (1- cos ï?±) [4]
(f) Find whether the following matrix is non-singular or not? [3]
(g) For the sets
A = {x | x = 5n, n Є N, and x  32},
B = { x | x = 3n, n Є N, and x  41},
C = { x | x = 3+2n, n Є N, and x  15},
Where N denotes the set of natural numbers, find out (A ∩ B) x (A\C), where ‘x’ denotes the cross-product of two sets and X \Y = {x Є X |x Є Y }
(h) Write the following expression in the form a + ib where a, b Є R and i = -1 [3]
(j) Find the equation of st. line parallel to the x + y +1 = 0 and passing through point (2,3) [2]
(k) Find the equation of the tangent at the point (2,1) to the ellipse x2 + 4y2 = 8 [2]
(l) Find the point of intersection of the line x = y = z and the plane x + 2y + 3z = 3 [3]
Question No 2. (a) Find the smallest interval of real numbers that contains the following sets of numbers as subsets; {- 1, 7, 3.5,4}, ] -2,6],
{x | x = 3m, m Є N, n  4} [2]
(b) Evalute : (i) (ii) [4]
(c) Find where: (i) y = cos-1(4×3 – 3x) and (ii) [5]
Question No 3. Evaluate the following integrals: [3+4+4]
(i) (ii) (iii)
Question No4. (a) Prove that the function f(x) = sinx + cosx is monotonically increasing on the interval [0, ï?°/4]
(b) Evaluate the following definite integer using Trapezoidal rule or Simpson’s rule by taking n=4:
Question No 5. (a) For sets P, Q and R prove that (( P U Q)\R)C = (PC U R) ∩(QC U R), where X\Y is as defined in Q. No.1(g) and Xc denotes complement of X in the Universal set. [3]
(b) Apply De Moivre’s formula to prove that cos 2� = cos2� and sin2� = 2sin�cos� [3]
(c) Find all the four real/complex roots of the biquadratic equation z4 – 4z2 + 4 –2i = 0 where i = -1
Question No 6. (a) Using Cramer’s rule, solve the following system of linear equations : [7]
x + y – z + 2 = 0
2x + 5 – y + z = 0
-4 + 3z – 2y + x = 0
(b) For positive real numbers a, b, x and y prove or disprove (ab + xy)(ax + by)  4abxy [4]
Question No 7. (a) What is the normal at the point of contact of the tangent y = mx + a\m to the curve y2 = 4ax ?
(b) Find the equation of the cone with vertex at the origin, and whose base curve is the circle x2 + y2 + z2 = 16 and x + 2y + 3z = 9 [5]
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