BACHELOR IN COMPUTER APPLICATIONS Term-End Examination December, 2OO8 COURSElN CS-60: FOUNDATION IN COMPUTING MATHEMATICS Time : 3 hours
MaximumMarks: 75
Nofe : Question No. 7 is compulsory. Attempt any three questionsfrom Questions No. 2 to 6. Use of calculator is permitted. 1.
(a) Fill in the blanks : (i)
'.' By commutativityof in real numbers,we getx.Y-...........
where x and y are real numbers. (ii) By associative property of '.' numbers,we get ( x . y ) . 2 : . . . , . . . . .,. . for real numbersx, y and z.
in real
(iii) For real numbers x, y and z, then using
transitivity of '>' in R, we get I f x > y a n dy > z t h e n . . . . . . . . . . . P.T,O.
(b) For real numbers x and y, tell for each of the following, whether it is True or Folse : ( i ) l * + y l a l w a y se q u a lls* l +
lVl
( i i ) l * . y l a l w a y s e q u a lls* l . l v l (iiill*-yl
a l w a y se q u a l sl * l - l y l ,
where l* | : absolutevalue of x. (c) In each of the following, if f : R - {0} -+ R is a function and is defined as (i)
f(x) : 5*, then tell whether f is 1 - 1 or not and why.
(ii) f(x) : 2*4, then tell whether f is 1 - 1 or not and why. (iii) I(x) = 2 x2, then tell whether f is onto or not and why. (d) G i v e n : f: R - + R
and g:R-+R
aretwo
functions such that f(x) = 2*3 and g(x) = 7x + 5, then find fog and gof. (e)
Find dy/dx for each of the following : (i) 9:
3 sinx
( i i )y = 1 7 + 5 x (iii) 9 = x6
cs-60
(0
3
Evaluateeach of the following :
(i) J la * *a1d* (r,
I
J
slnxox
(iii) | z a" J 3
(g) Evaluateeach of the following : (i)
3 |l
Q *8x) dx '
2 (ii)
2 [
J
n5* d*
1 (iii)
nl2 [ cosxdx J
0
t
(h) Solvethe followingsystemof linearequations:
3
5x+4Y:l{ 3x+7Y=lJ (i)
Find the valueof the determinant
12
3
3 6l
l+ 1 12|
ls 2 el cs-60
P.T.O.
Find the arithmetic mean of the following numbers : 8, 15,10,12,6
(k) Find the geometric mean of . the following numbers : 2, 4, g, 64 For each of the following, tell whether it is true or false, where A, B and C are sets and U, r^l denote respectively set union and set intersection: (i)
A U B always equalsB n A
( i i )( A u B )U c :
Au(BuC)
(iii) A n Q : A, where Q denotesempty set. (m) Draw g Venn diagram for sets A and B with
universalset U such that A is a subsetof B.
2. (a) State the following properties,/laws of real numbers : (i)
Associativepropeqtyof '+' in real numbers
(ii) Distributivityof '.' over '+' in R (iii) Archimedean property (iv) Monotone property of '+' in R
Draw
a
graph for each of the following
functions : (i) f :R-+Rsuchthat (x):Tforall xinR (ii) f :R-+Rsuchthat f ( x :) 2 * + 3 f o r a l l x i n R (c) Define each of the following concepts and give an example for each | (i)
Odd function
(ii) Composition of two functions 3.
(a) Evaluatethe following : fr
(i)
- sin x)dx [t (* ' 0
(ii) l:--:d" J 3 ( 1 +x ' ) For each of the following functions, find whether the function is monotonically increasing or monotonically decreasing or neither, on given interval : (i)
f(x) = 12 - 1 on [0, 2]
(ii) f(x) = sst x on 10, n/21
4
2
(c) Prov.ethe following inequality : e*>1+*2/z**g/6 Find the area of the region boundedby x=0, x=3
4.
a n dg = 3 .
(a) Do as directed : (i)
1+1+2
Describe the following set by listing method : {x lx is a divisorof 36}
(ii) Describe
the
following
by
method : { 2 ,4 , 6 , g , . . . } (iii) Show the following for any set A 0gA, where Q denotesempty set.
Obtain conjugate of each of complex numbers : (i) 3+5i (ii) 8i (ii0 L2
following
(c)
Explain the following with suitable example : (i)
5
Proof by counter-example
(ii) Proof by contradiction 5.
(a) Solve the following :
6
x+2y+32:10 2x+y+22=10 3x+4y*z=18 (b) Find the value of the following determinant :
3
7 2 rl 7 2 n l 3 1 2 1 (c)
Explain each of the following concepts with one suitable example for each i (i)
Harmonic Mean
(ii)
Arithmetic Mean
3
(iii) Geometric Mean
6.
(a) Find the mid point of thb straight line the line segmentA(2, 3) and B (-5, 7). (b) Find the equationof the straight line parallelto the line 2y + 3x + 1 : 0 and passingthrough the point (0, 0).
3
P.T.O.
(c)
Find the equations of a straight line in three-dimensional space joining the points (-1, 0, 1) and (2, l, 4). Let A (0, 2, 6), B (3, 4,7), C (6, 3, 2) and D (5, 1 , 4l be four points in three-dimensional space. Find the projection of the line AB on CD.
