BACHELOR IN COMPUTER
APPLICATIONS
Term-End Examination
June, 2009
CS-60 : FOUNDATION COURSE IN MATHEMATICS IN COMPUTING
Time: 3 Hours Maximum Marks : 75
Note: Question number 1 is compulsory. Attempt any three question from Questions No. 2 to 6. Use of calculator is permitted.
1. / (a) / Find the equation of the line which passes through (-4,2)and (2,4).[15x3=45 Marks]
(b) / Find the equation of the circle whose radius is r=4 and whose centre is (0,0).
(c) / Find the co-ordinates of the vertex and the focus of the parabola y2=4(x+y).
(d) / Find the foci and vertices of the hyperbola 9x2-y2-36x+8y-5=0.
(e) / Evaluate:
d/dx(3x2+7)
(f) / Fill in the blanks:
(i) |2x+3y|<= ------
(ii)|2 x y| = ------
(g) / Evaluate ∫(3x2+14)dx
(h) / State for each, whether the following is true:
(i) 7 € [-8,5]
(ii)6 € [1,16]
(i) / Solve the following equations:
3x+5y=13
2x+9y=20
(j) / If U=[1,2,3,4,5,6,7,8,9,10]
A=[1,3,5] B=[2,4,6,8]
Find AUB.
(k) / Give the functions
f(x)=4x2+3
Find value of f(x)for each
(i)x=3 and (ii)x=7
(l) / Obtain 3+5i/2+i in the form of a+ib,a,b € R.
(m) / Solve the equation 2x2+5x+6=0
(n) / Show that for sets A,B and C:If A is subset of B and B is subset of C then A is subset of C.
(o) / Find the equation of the ellipse whose principle axis lies along the axis of co-ordinates, focus is (2,0)and length of latus rectum is 6.
2. / (a) / Evaluate: ∫(3sinx+11)dx.[3+4+3=10 Marks]
(b) / Evaluate:∫05(2x2+6)dx
(c) / Find the area of the smaller region lying above the x-axis and included between the circle x2+y2=2x and the parabola y2=x.
3. / (a) / Find the equation of the circle having radius 4 and as centre the point (3,4).[3+4+3=10 Marks]
(b) / Find the equation of the parabola whose vertex is (-2,2) and focus is (-6,6).
(c) / Find the eccentricity, the foci and equations of the directrix of the ellipse 9x2+4y2=36
4. / (a) / Draw the curve for the following equation:[3+4+3=10 Marks]
x2+4y2+4xy=0
(b) / Draw the curve for the following equation:
3x+7y=8
(c) / Find the value of the determinant:
|1 1 1|
|x x2 x3|
|y y2 y3|
5. / (a) / For any two sets A and B in a universal set U.Prove that:[3+4+3=10 Marks]
(i) (A∩B)' = A'U B'
(ii)(AUB)' = A'∩ B'
(b) / Apply De Moivoe's theorem, prove that
(i) cos2Ө = cos2Ө - sin2Ө.
(ii)sin2Ө = 2 sinӨ cosӨ.
(c) / Solve x2+7x+10=0
6. / (a) / Find the equation of the plane passing through (1,2,0) and the line
xcosα + ycosβ +zcosγ=1,x+y=z.[4+3+3=10 Marks]
(b) / Find the angle of intersection of the spheres
x2+y2+z2-2x+2y-4z+2=0 and x2+y2+z2=4
(c) / Find the equation of the tangent plane at the point (3,4,-1) to the cone 2yz-3zx-2xy=0.
BACHELOR IN COMPUTER
APPLICATIONS
Term-End Examination
December, 2008
CS-60 : FOUNDATION COURSE IN MATHEMATICS IN COMPUTING
Time: 3 Hours Maximum Marks : 75
Note: Question number 1 is compulsory. Attempt any three question from Questions No. 2 to 6. Use of calculator is permitted.
1. / (a) / Fill in the blanks : [3 marks]
  1. By commutativity of '.' in real numbers, we get x.y = ...... where x and y ae real numbers.
  2. By associative property of '.' in real numbers, we get (x.y).z = ...... for real numbers x,y and z.
  3. For real numbers x, y and z, then using transitivity of '>' in R, we get if x > y and y > z then......

(b) / For real numbers x and y, tell for each of the following, whether it is true or False : [3 marks]
  1. |x+y| always equals |x|+|y|
  2. |x.y| always equals |x|.|y|
  3. |x-y| always equals |x|-|y|,
where |x| = absolute value of x.
(c) / In each of the following, if f : R ~ {0}-> R is a function and is defined as : [3 marks]
  1. f(x)=5x, then tell wheter f is 1-1 or not and why.
  2. f(x)=2x4, then tell whether is 1-1 or not and why.
  3. f(x) = 2/x2, then tell whether f is onto or not andy why.

(d) / Given : f : R -> R and g : R -> R are two functions such that f(x) = 2x3 and g(x) = 7x+5, then find fog and gof. [3 marks]
(e) / Find dy/dx for each of the following : [3 marks]
  1. y=3 sin x
  2. y=17+5x
  3. y=x6

(f) / Evaluate each of the following : [3 marks]
  1. ∫ (3+x4) dx
  2. ∫ sin x dx
  3. ∫ 7 dx

(g) / Evaluate each of the following : [3 marks]
  1. 2∫3 (7+8x) dx
  2. 1∫2 e5x dx
  3. 0∫π/2 cos x dx

(h) / Solve the following system of linear equations : [3 marks]
  • 5x + 4y = 14
  • 3x + 7y = 13

(i) / Find the value of the determinant : [3 marks]
|2 3 6|
|4 1 12|
|3 2 9|
(j) / Find the arithmetic mean of the following numbers : 8, 15, 10, 12, 6. [3 marks]
(k) / Find the geometric mean of the following numbers : 2, 4, 8, 64. [3 marks]
(l) / For each of the following, tell whether it is true or false, where A, B and C are sets and U, ∩ denote respectively set union and set intersection : [3 marks]
  • A U B always equals B ∩ A
  • (A U B) U C = A U (B U C)
  • A ∩ ф = A, whether ф denotes empty set.

(m) / Draw a Venn diagram for sets A and B with universal set U such that A is a subset of B. [3 marks]
2. / (a) / State the following properties/laws of real number : [4 marks]
  1. Associative property of '+' in real numbers
  2. Distributivity of '.' over '+' in R
  3. Archimedean property
  4. Monotone property of '+' in R

(b) / Draw a graph for each of the following functions : [4 marks]
  1. f : R -> R such that f(x) = 7 for all x in R
  2. f : R -> R such that f(x) = 2x + 3 for alll x in R

(c) / Define each of the following concepts and give an example for each : [4 marks]
  • Odd function
  • composition of two functions

3. / (a) / Evaluate the following : [4 marks]
  1. 0∫π (x - sin x) dx
  2. ∫ ((2 / (3(1+x2))) dx

(b) / For each of the following function, find whether te function is monotonically increasing or monotonically decreasing or neigher , on given interval : [4 marks]
  1. f(x) = x2 - 1 on [0,2]
  2. f(x) = cos x on [0,π/2]

(c) / Prove the following inequality : ex > 1 + x2 / 2 + x 3 / 6. [2 marks]
(d) / Find the area of the region bounded by x = 0, x = 3, and y = 3. [2 marks]
4. / (a) / Do as directed : [4 marks]
  1. Describe the following set by listing method : { x|x is a divisor of 36}
  2. Describe the following by property method : {2, 4, 6, 8, ...}
  3. Show the following for any set A, ф sub set A, where ф denotes empty set.

(b) / Obtain conjugate of each of the following complex numbers : [3 marks]
  1. 3 + 5i
  2. 8i
  3. 12

(c) / Explain the following with suitable exmple : [5 marks]
  • Proof by counter-example
  • Proof by contradiction

5. / (a) / Solve the following : [6 marks]
  • x + 2y + 3z = 10
  • 2x + y + 2z = 10
  • 3x + 4y + z = 18

(b) / Find the value of the following determinant : [3 marks]
|1 2 3|
|1 2 4|
|3 1 2|
(c) / Explain each of the following concepts with one suitable example for each : [3 marks]
  • Harmonic Mean
  • Arithmethic Mean
  • Geometric Mean

6. / (a) / Find the mid point of the straight line joining the line segment A(2,3) and B(-5,7). [2 marks]
(b) / Find the equation of the straight line parallel to the line 2y + 3x + 1 = 0 and passing through the point (0,0). [3 marks]
(c) / Find the equation of a straight line in three-dimensional space joining the points (-1,0,1) and (2,1,4). [3 marks]
(d) / Let A(0,2,6), B(3,-4,7), C(6,3,2) and D(5,1,4) be four points in three-dimensional space. Find the projection of the line AB on CD. [4 marks]