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

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