1
Seat No.: Enrolment
GUJARAT TECHNOLOGICAL UNIVERSITY
MCA Integrated SEMESTER- 1 EXAMINATION WINTER 2018
Subject Code: 4410604 Date: 08-01-2019
Subject Name: Basic Mathematics for IT
Time: 10.30 am to 1.00 pm Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q1

Define Following
I. Power Set
II. Existential Quantifier
III. Symmetric matrix
IV. Unit Vector
V. Transitive relation
VI. Locus of a Point
VII. Inverse of a function
07

If
Find

04
Difference between scalar and vector?
03
Q2

Define Scalar Matrix. Solve the following equation by using matrix inversion.
x 2y
3x y 2z
2x 2y +3z
07

How many positive integers between 1000 and 9999 inclusive
are divisible by
Are even?
Have distinct digits?
Are divisible by 5 or
Are not divisible by either 5 or
Are not divisible by
Have distinct digits and are even?
07
OR

Construct a truth table for each of these compound propositions.
V → q
V → Λ
→ ↔ q →
→ → →
01
02
02
02
Q3

Let p and q be the two propositions.
It is below freezing.
It is snowing.
Write these proposition using pp and q and logical connectives.
It is below freezing but not snowing.
It is not below freezing and it is not snowing.
It is either snowing or below freezing (or both).
01
01
01
2
It is either below freezing or it is snowing, but it is not snowing if it is below freezing.
That it is below freezing is necessary and sufficient for it to be snowing
02
02

What do you mean by basis step and inductive step? Let P is the statement that 12 22 n2 n for the positive integer n.
What is the statement P
(b)Show that P is true, completing the basis step of the proof.
What is inductive hypothesis?
(d)What do you do need to prove in the inductive step?
(e)Complete the inductive step.
07
OR
Q3

Solve the following system of linear equations by Gauss elimination method
5x-y+z=10
2x+4y=12

07

Find the equation of the circle passing through the points and
07
Q4

Find the first six terms of the sequence defined by following recurrence relations and initial conditions
an an-1 an-2 where a0 a1
07

Define bi-implication. Consider these statements. The first two are called premises and the third is called the conclusion. The entire set is called an argument. Express the statement using quantifier and logical connectives.
"All lions are fierce."
"Some lion do not drink coffee."
"Some fierce creatures do not drink coffee."
07
OR
Q4

Find the intercepts that the line 3x -2y makes on the axes. What is slope of line?
07

Translate the statement: "It is below freezing now.Therefor; it is either below freezing or raining now."
(ii)Translate the statement "You can access the Internet from campus only if you are a computer science major or you are not a freshman."
07
Q5

Find the equation of line joining the center of the two circles.
x2 y2 2x 4y
x2 y2 2x 4y 1=0
07

Show that the points and are the vertices of a square.
07
OR
Q5

Find the area of the triangle, the coordinates of whose vertices
07

(i)Find the locus of a point which is equidistant from the points and
03
(ii)Find the point which divide the join of externally in the ratio 2:3 and the point lying towards the point
04