Mathematics Times – July 2019

3. Let a i j k  6 3 6
      
   and d i j k  

   

.

Suppose that a b c 

 

where b



is parallel to

d



and c



is perpendicular to d



. Then c



is

[2015]

(a) 5 4

4. What is the angle subtended by an edge of a
   regular tetrahedron at its center? [2014]

(a)

5. Let

i j k 

  

(b)7 2 5i j k 

  

(c)7 5i j k 

  

(d) 3 6 9i j k 

  

cos^11

2

 

 

 

(b)

cos^11

2

 

 

 

(c)

cos^11

3

 

 

 

(d)

cos^11

3

 

 

 

v v v v1 2 3 4, , ,

   

be unit vectors in the xy-plane,

one each in the interior of the four quadrants.

Which of the following statements is

necessarily true? [2013]

(a) v v v v    1     2 3 4 0

(b) There exist i j



with 1   i j 4

 

such

v vi j

 

is in the first quadrant

(c) There exist i j



with 1   i j 4

 

such that

v vi j 0



(d) There exist i j



with 1   i j 4

 

such that

v vi j 0

6. Let H be the orthocenter of an acute - angled
   triangle ABC and O be its circumcenter.
   Then





HA HB HC 

  

[2012]

(a) is equal to HO



(b) is equal to 3 HO



(c) is equal to 2 HO



(d) is not a scalar multiple of HO



in general

7. Let

respectively, then the number of possible

positions of the centriod of triangle ABC is

[2011]

(a) 1 (b) 2 (c) 3 (d) 6

8.Let

a b c, ,

 

, be three vectors in the xyz space

such that a b b c c a     ^0

    

. If A, B,C
are points with position vectors a b c, ,
 

u i j k v      2 , 3 2j k

     

be vectors in

R^3 and w be a unit vector in the xy-plane.

Then the maximum possible value of

u v w .

  

is [2010]

(a) 5 (b) 12 (c) 13 (d) 17

9.Let ABC be a triangle and P be a point inside

ABC such that PA PB PC 2  3  0

   

. The
ratio of the area of triangle ABC to that of
APC is [2010]

(a) 2 (b)

3

2

(c)

5

3 (d) 3

Statistics

1. Let n 3. A list of numbers x x 1 , ,......., 2 xnhas
   mean  and standard deviation . A new
   list of numbers y y 1 , ,......, 2 ynis made as

follows: 1 1 2

2

x x

y



 , 2 1 2

2

x x

y



 andy xi i

for i = 3,4,....,n. The mean and the standard

deviation of the new list areˆ. and ˆ. Then

which of the following is necessarily true?

[2014]

(a)

2.Let

 ˆ and  ˆ (b) ˆ and   ˆ

(c) ˆ (d)   ˆ

n 3. A list of numbers^0    x x 1 2 ... xn

has mean  and standard deviation .

A new list of numbers is made as follows

y 1  0 ,y x 2  2 ,..., ,y y x xn 1 n  1 n. The mean

and the standard deviation of the new list are

ˆ and ˆ. Which of the following is

necessarily true? [2013]

(a)

3. 
   

[2015]

4. 
   

   [2014]

   
   


5. 
   

[2013]

6. 
   

[2012]

7. 
   

[2011]

8.

[2010]

9.

[2010]

Statistics

1.

[2014]

2.

[2013]

   ˆ, ˆ

(b)    ˆ, ˆ