# Maths-1999-Set II

**Q1) If y = cot x, show that d2y/dx2 + 2y dy/dx = 0 (Marks 2)**

Ans1) Here y = cot x

differentiating both sides w.r.t. x:-

dy/dx = -cosec2x

differentiating again w.r.t. x:-

d2y/dx2 = -d/dx (cosec2x)

d2y/dx2 = -2cosecx(-cosec x . cot x ) = +2cosec2x . cot x

or d2y/dx2 = -2 dy/dx . y

or d2y/dx2 + 2y dy/dx = 0.

**Q3) If
A = 2 2
-3 1
4 0 B = 6 2
1 3
0 4 **

**find the matrix C such that A + B + C is a zero matrix. (Marks 2)**

Ans3) Here

A = 2 2

-3 1

4 0 B = 6 2

1 3

0 4

we want to find C such that A + B + C = 0

< => C = 0 – A – B = -A – B

= – 2 2

-3 1

4 0 – 6 2

1 3

0 4

= -2 -2

3 -1

-4 0 + -6 -2

-1 -3

0 -4

= -8 -4

2 -4

-4 -4

**Q5) Construct a 3 x 2 matrix whose elements in the ith row and the jth coloumn are given by:-
aij = (3i + j)/2 (Marks 2)**

Ans5) We want a 3 x 2 matrix

=> 1 i 3 and 1 j 2

where aij = (3i + j)/2

… a11 = (3 + 1)/2 = 2; a12 = (3 + 2)/2 = 5/2;

a21 = (6 + 1)/2 = 7/2; a22 = (6 + 2)/2 = 4;

a31 = (9 + 1)/2 = 5; a32 = (9 + 2)/2 = 11/2

… The reqd matrix is :-

a11 a12

a21 a22

a31 a32

=

2 5/2

7/2 4

5 11/2