# Chapter 4 DETERMINANTS

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```					                                     Chapter 4
DETERMINANTS
1 Mark Questions

Q1        If A is a square matrix of order 3 and | A | = 5, find the value of  3 A
Q2        If 

1      2
=         2 
2          1    1
Q3        Find the value of determinant
1     2   4
 = 8 16 32
64 128 256
Q4        Find the value of determinant

2        2     2
=      x        y     z
yz      zx   x y
Q5        If a, b, care in A.P. find the value of determinant

x 1     x2 xa
 = x2 x3 xb
x3 x4 xc
Q6                                        2 k 
For what value of k, the matrix      has no inverse
3 5 
Q7        A B C are three non zero matrices of same order, then find the condition
on A such that AB = AC  B = C
Q8        Let Abe a non singular matrix of order 3 x 3, such that AdjA =100,
find A .
Q9        If A is a non singular matrix of order n, then write the value of Adj(AdjA) and
hence write the value of Adj(AdjA) if order of A is 3 and | A | = 5
Q10       Let A be a diagonal A = (d1, d2,             dn ) write the value of | A |.
……………,

Q11       Using determinants find the value of the line passing through the points (-1,3)
and (0,2).
Q12       Using determinants find the value of k for which the following system of
equations has unique solution,
2x – 5y = 26
3x + ky = 5

4 Mark Questions

Q1        If A = [aij] is a 3 x 3 matrix and Aij’s denote cofactors of the corresponding
elements aij’s then write the value of ,

( i)     a11 A11 + a 11A11 + a 11A11

(ii)    a12 A12 + a 22A22 + a 32A32
(iii)   a21 A11 + a 22A12 + a 23A13
(iv)    a11 A13 + a 21A23 + a 31A33

Q2                                               2 sin x    1         3     0
If x  R, /2  x  0 and                                =
1       sin x        4 sin x
find the values of x.

Q3                                             a a2 1 a3
If a ,b, c are all distinct and      b b 2 1  b 3 = 0, find the values of a, band
c c2 1  c3
c.

Q4        If x is a real number then show that if
1      sin x    1
 =  sin x     1    sin x   then ,                2  4
1      sin x   1

Q5        If x , y, z are real numbers such that x + y + z =  then find the value of ,
sin( x  y  z ) sin( x  z ) cos z
 sin y           0       tan x
cos(x  y )    tan( y  z )   0

Q6        Without expanding find the value of the following determinant,

sin    cos    cos(   )
 =        sin    cos    cos(   )
sin    cos    cos(   )

Q7        Find value of k, if area of the triangle with vertices P ( k ,0) , Q (4,0) and
R(0,2) is 4 square units
Q8                1       tan x               -1      cos 2 x  sin 2 x
If A =                show that A A =         sin 2 x cos 2 x 
 tan x    1                                           
Q9                      1 1 p       1 p  q
Prove that, 2 3  2 p 4  3 p  2q = 1
3 6  3 p 10  6 p  3q

Q10                                             x3 x4     x 
Using properties prove that,          x2  x3 x = 0
x 1 x  2 x  

where , ,  are in A.P.

Q11                           x y          x    x
Prove that,    5x  4 y  4 x 2 x = x3
10 x  8 y 8 x 3 x

6 Mark Questions

Q1                                          p b c
If a ≠ p b ≠ q    c ≠ r and       a   q c = 0 find the value of
a   b r

p     q      r
+     +
pa   qb   r c

Q2                           3 2
For the matrix A =     find the numbers a and b such that
1 1

A2 + aA + bI = O. Hence find A-1.

Q3        Solve the equation if a  0 and

xa      x         x
x    xa         x
x      x       xa

Q4                                                           a b c
Show that the value of the determinant  =         b c a is negative , It is
c a b
given that a , b , c , are positive and unequal.

Q5   Using matrix method, determine whether the following system of
equations is consistent or inconsistent.

3x – y - 2z = 2
2y – z = -1
3x – 5y = 3
Q6               3   1 1                   1   2  2
. If A-1.=  15 6  5
                    and B =  1 3
       0  find (AB)-1.

 5
     2 2 
                   0 2 1 
         

Q7               1  2 1
Let    A =   2 3 1
1
     1 5


1 Mark Questions

Q1   -135

Q2   0 (using properties)

Q3   0 (using properties)

Q4   0 (using properties)

Q5   0 (using properties)

Q6         10
k=
3

Q7   |A|  0

Q8   |A| =  10

Q9   For a non-singular matrix of order n > 1, Adj(AdjA) = (|A|)n-2. A

 if order of matrix A is 3 x 3 and |A| = 5, then
Q10   |A| = d1. d2 .d3……….dn

Q11   x+y=2

Q12          15                            15
k-       (all real numbers except -    )
2                             2

4 Mark Questions

Q1    (i) Value of |A|

(ii) Value of |A|

(iii)   0 (iv) 0
Q2          
x =     ,
6 2

Q3    abc = -1

Q5    0

Q6    0

Q7    k=8,0

6 Mark Questions

Q1    0

Q2    a = -4, b = 1

 1  2
A-1 =       
 1 3 

Q3               a               a
x =-           or   x =-
3               2

Q5    Inconsistent

Q6            9  3 5
(AB) =  2 1 0
-1
        
1
    0 2

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Description: CBSE BOARD EXAM AIEEE IIT-JEE STUDY MATERIAL MATHEMATICS CLASS XI XII SAMPLE PAPERS KEY SOLUTIONS ANSWERS QUESTIONS
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