Mathematics Times – July 2019

17. Let AP ( ; )a d denote the set of all the term

of an infinite arithmetic progression with first

term a and common difference d > 0. If

AP(1;3)AP(2;5)AP(3, 7)AP a d( ; )

then a d equals.........

18. If

/ 4

sin

/ 4

2

(1 x)(2 cos 2 )

dx

I

e x







   then find

27 I^2 equals ....

SECTION-

1. b 2. b 3. c 4. c

SECTION-

5. (b,c) 6. (a,b,c) 7. (a,c,d) 8. (a,b,c)


6. (a,b,c) 10. (a,c) 11. (b,d) 12. (b,c,d)

SECTION-

13. (0.75) 14. (0.50) 15. (3.00)


14. (10.00) 17. (157.00) 18. (4.00)

SECTION-

SECTION-

SECTION-

SECTION-
1.Sol: Given that y = mx + 1 is a chord which
intersect the given circle. We know
perpendicular bisector of a chord in a circle
passes through the center of circle. That is
2 1
3

y

x m

 





 x my m    2 3 0 (1)

Substitute , the equation of a chord in eq(1),

we get x- coordinate of mid point of chord.

i.e.

x m mx (    1) 2m 3 0

x m m(1   ^2 ) 3 3 0

Since

3

5

x  , the above equation is rewritten

as

(^3) (1 (^2) ) 3 3 0
5
    m m
 (^) m m^2    5 6 0
i.e., m2,
Therefore 2  m 4
2.Sol: The problem asks for the minimum values
of two functions
SECTION-
1.Sol:
2.Sol:
 ( ) and  ( ) over the
semi - open interval 0, 2. But neither
^ nor  is given explicitly as a function of
. So we first have to determine them from
the given matrix equation,as follows.
M I M   ^1 (1)
We have a matrix of order 2. For such matrix,
there is an easy explicit formula for the inverse,
as follows
1
a b 1 d b
c d c a

    
    
    (2)
where   ad bc is the determinant. As a
result if we are given that
0
0
a b d b
c d c a
 

      
     
       (3)
then equating the entries in the first row and
the second column of the matrices on the two
sides gives
b
b

 

and hence
  (4)
and further , equating the entries in the first
row and the first column we get
d
a

 

,
which because of the last equation, becomes
d a d

   

(5)
Thus we have very simple way of determining
the functions  ( ) and  ( ) by merely
looking at the entries of M and doing a little
calculation to find its detreminant. Specifically,