Download answer all the questions (30 1 = 30)

XII – MATHEMATICS – UNIT TEST .1
ANSWER ALL THE QUESTIONS (301 = 30)
1  1 2 
1.
The rank of the matrix 2  2 4 is
4  4 8 
(a) 1
2.
3.
4.
5.
6.
7.
8.
9.
(b) 2
(c) 3
(d) 4
 1



2


 is
The rank of the diagonal matrix 
0


4 


0
(a) 0
(b) 2
(c) 3
If A = [2 0 1], then the rank of AAT is
(a) 1
(b) 2
(c) 3
1 
If A =  2  , then the rank of AAT is
 3 
(d) 5
(a) 3
(b) 0
(c) 1
  1 0 
If the rank of the matrix  0   1 is 2, then  is
 1 0  
(a) 1
(b) 2
(c) 3
If A is a scalar matrix with scalar k 0, of order3, then A–1 is
1
1
1
(a) 2 I
(b) 3 I
(c) I
k
k
k
 1 3 2 
If the matrix  1 k  3 has an inverse then the value of k is
 1 4 5 
(d) 2
(a) k is any real number
(b) k = – 4
(c) k  – 4
2 1 
If A = 
 , then (adjA) A =
1 4 
1 / 5 0 
1 0
(a) 
(b) 
(c)


 0 1 / 5
0 1 
If A is a square matrix of order n then adjA is
(a) A2
(b) An
(c) An – 1
(d) k  4
(d) 0
(d) any real number
(d) KI
5 0 
0  5


5 0 
(d) 

0 5 
(d) A
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
0 0 1 
The inverse of the matrix 0 1 0 is


1 0 0
 0 0 1
1 0 0
(a) 0 1 0
(b)  0 1 0


 1 0 0
0 0 1
0 0 1 
(c) 0 1 0


1 0 0
 1 0 0 
(d)  0  1 0 
 0 0  1
If A is a matrix of order 3, then det (KA) is
(a) k3 det (A)
(b) k2 det (A)
(c) k det (A)
(d) det (A)
If I is the unit matrix of order n, where k  0 is a constant, then adj (kI) is
(a) kn (adj I)
(b) k (adj I)
(c) k2 (adj I)
(d) kn – 1 (adj I)
0 0 
If A = 
, then A12 is

0 5 
0 0 
0 0 
0 0 
1 0
(a) 
(b) 
(c) 
(d) 



12 
0 5 
0 0 
0 1 
0 60
3 1
The inverse of 
 is
5 2
 2  1
 2 5 
 3  1
 3 5 
(a) 
(b) 
(c) 
(d) 




 5 3 
 1  3
 5  3
 1  2
In a system of 3 linear non-homogeneous equation with three unknowns, if  = 0 and
x = 0, y 0 and z = 0 then the system has
(a) unique solution (b) two solutions (c) infinitely many solution (d) no solution
The system of equations ax + y + z = 0; x + by + z = 0; x + y + cz = 0 has a non-trivial
1
1
1


solution then
=
1 a 1 b 1 c
(a) 1
(b) 2
(c) – 1
(d) 0
a b
c b
a c
If a ex + b ey = c; p ex + q ey = d and 1=
2 =
; 3 =
, then the value
p q
p d
d q
of (x, y) is


  

 


 

 
(a)  2 , 3 
(b)  log 2 , log 3  (c)  log 1 , log 1  (d)  log 1 , log 1 
1
1 
3
2 
2
3 
 1 1 



If the equation – 2x + y + z = l; x – 2y + z = m; x + y – 2z = n such that l + m + n = 0,
then the system has
(a) a non-zero unique solution
(b) trivial solution
(c) infinitely many solution
(d) no solution
(AT)– 1 is equal to
(a) A– 1
(b) AT
(c) A
(d) (A– 1)T
If (A) = r then which of the following is correct?
(a) all the minors of order r which do not vanish
(b) A has atleast one minor of r which does not vanish
(c) A has atleast one (r + 1) order minor which vanishes
(d) all (r + 1) and higher order minors should not vanish
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
In the system of 3 linear equations with three unknowns, if  = 0 and one of x, y or z
is non-zero then the system is
(a) consistent (b) inconsistent
(c) consistent and the system reduces to two
equations
(d) consistent and the system reduces to a single equation
In the system of 3 linear equations with three unknowns, if  = 0 and all 2  2 minors of
 = 0 and atleast one 2  2 minor of x or y or z is non-zero then the system is
(a) consistent (b) inconsistent
(c) consistent and the system reduces to two
equations
(d) consistent and the system reduces to a single equation
In the system of 3 linear equations with three unknowns, if  = 0, x = 0, y =0, z = 0
and atleast one 2  2 minor of   0 then the system is
(a) consistent (b) inconsistent
(c) consistent and the system reduces to two
equations
(d) consistent and the system reduces to a single equation
In the system of 3 linear equations with three unknowns, if  = 0 and all 2  2 minors of
, x, y, z are zeros and atleast one non-zero element is in  then the system is
(a) consistent (b) inconsistent
(c) consistent and the system reduces to two
equations
(d) consistent and the system reduces to a single equation
Every homogeneous system
(a) is always consistent
(b) has only trivial solution
(c) has infinitely many solution
(d) need not be consistent
If (A) = (A,B) = the number of unknowns then the system is
(a) consistent and has infinitely many solution
(b) consistent
(c) consistent and has a unique solution
(d) inconsistent
If (A)  (A,B) then the system is
(a) consistent and has infinitely many solution
(b) consistent
(c) consistent and has a unique solution
(d) inconsistent
In the system of 3 linear equations with three unknowns if (A) = (A,B) = 1
then the system is
(a) has a unique solution
(b) reduces to 2 equations and has infinitely many solutions
(c) reduces to a single equation and has infinitely many solution (d) inconsistent
In the system of 3 linear equations with three unknowns, in the non-homogeneous system
if (A) = (A,B) = 2 then the system has
(a) has a unique solution
(b) reduces to 2 equations and has infinitely many solutions
(c) reduces to a single equation and has infinitely many solution (d) inconsistent
In the homogeneous system with three unknowns if (A) < number of unknowns
then the system has
(a) only trivial solution
(b) trivial solution and infinitely many non-trivial solutions
(c) only non-trivial solutions (d) no solution
PART – II
ANSWER ANY FIVE (5) QUESTIONS (56 = 30)
31.
32.
33.
3 1  1
Find the inverse of 2  2 0 
1 2  1
5 2
 2  1
T
T T
If A = 
and B = 

 verify that (AB) = B A
7
3

1
1




Solve by matrix inversion method 2x – y + z = 7, 3x + y – 5z = 13, x + y +z = 5
34.
35.
36.
37.
1 2  1 3 
Find the rank of the matrices 2 4 1  2


3 6 3  7
Solve by Cramer’s rule 2x + y – z = 4; x + y – 2z = 0; 3x + 2y – 3z = 4
Solve by rank method 2x – y + z = 2; 6x – 3y + 3z = 6; 4x – 2y + 2z = 4
Solve the homogeneous system of linear equations 2x + y + z = 0; 5x + 3y + 2z = 0;
3x + 2y = –z using determinant method.
PART – III
ANSWER ANY FOUR (4) QUESTIONS (410 = 40)
38.
39.
40.
41.
42.
43.
1
1 1

If A = 1 2  3 , verify A (adjA) = (adjA) A = AI3
2  1 3 
Investigate for what values of ,  the simultaneous equation
x + y + z = 6; x + 2y + 3z = 0; x + 2y + z =  have (i) no solution (ii) a unique solution
and (iii) an infinite number of solutions.
For what value of  the equations x + y + 3z = 0; 4x + 3y + z = 0; 2x + y + 2z = 0 have
a (i) trivial solution (ii) non – trivial solution.
For what values of k, the system of equations kx + y + z = 1;x + ky + z =1; x + y + kz = 1
have (i) unique solution (ii) more than one solution (iii) no solution
A bag contains 3 types of coins namely Re. 1, Rs. 2 and Rs. 5. There are 30 coins
amounting to Rs. 100 in total. Find the number of each category.
Verify whether the given system of equations is consistent. If it is consistent, solve them.
2x + 5y + 7z = 52; x + y + z = 9; 2x + y – z = 0.