Mathematics Times – July 2019

Sequence and Series

1.For each positive real number , let A be

the set of all natural numbers n such that

sin n 1 sin( ) n .

Let Ac be the complement of Ain the set of

all natural numbers. Then [2016]

(a) A A A^1213 ,^25 are all finite sets

(b) 1

3

A is a finite set but 1 2

2 5

A A, are infinite

sets

(c) , 1 1 2

2 3 5

A A Ac, ,c c are all finite sets

(d) 1 2

3 5

A A, are finite sets and 1

2

A is an infinite

set
2.Let a,b,c,d be real numbers such that

^32 ^4

1

n

k

ak bk ck d n



    

for every natural numbers n. Then
a b c d   is equal to [2013]
(a) 15 (b) 16 (c) 31 (d) 32
3.The sum of

1 1 1 (1!) (2 2 1)(2!) ... (^2      ^2    n n^2 1)

n! is [2011]

(a) ( 2)!n (b)    n (^1) n 1! 1
(c) ( 2)! 1n  (d) n n 1! 1
4.The arithmetic mean and the geometric mean
of two distinct 2-digit numbers x and y are
two integers one of which can be obtained by
reversing the digits of the other (in base 10
representation). Then x y equals [2011]
(a) 82 (b) 116 (c) 130 (d) 148
5.Suppose the sides of a triangle form a
geometric progression with common ratio r.
Then r lies in the interval [2010]
(a)
1 5
0,
2
   
 
 
(b)
1 5 2 5
,
2 2
   
 
 
(c)
1 5 1 5
,
2 2
   
 
 
(d)
2 5
,
2
  
 
 
6.Let a 0  0 and an 3 an 1  1 for n 1. Then
the remainder obtained dividing a 2010 by 11 is
[2010]
(a) 0 (b) 7 (c) 3 (d) 4
Vectors
1.Let ABC be an acute scalene triagle, and O
and H be its circumcentre and Ortho-center
respectively. Further let N be the midpoint of
OH. The value of the vector sum
NA NB
 
NC

is [2017]

(a) 0 (Zero vector) (b) HO



(c)

1

2

HO



(d)

1

2

OH



2.Let v be a vector in the plane such that

v i v i v j     2

     

. Then v



lies in the

interval [2016]

(a) (0,1] (b) (1,2] (c) (2,3] (d) (3,4]

Sequence and Series

1. 
   

[2016]

2. 
   

[2013]

3. 
   

[2011]1]

4. 
   

[2011]

5.

[2010]

6.

[2010]

Vectors

1.

[2017]

2.

[2016]