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RD Sharma Solution for Class 12 Science Mathematics Chapter 6 - Determinants

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Our RD Sharma Textbook Solutions are considered extremely helpful for solving the tough questions which are asked in the CBSE Class 12 exam. TopperLearning Textbook Solutions are compiled by subject experts. Herein, you can find all the answers to the textbook questions for Chapter 6 - Determinants.

All our solutions are created in accordance with the latest CBSE syllabus, and they are amended from time to time to provide the most relevant answers. Our free RD Sharma Solutions for CBSE Class 12 Mathematics will strengthen your fundamentals in Mathematics and will help you in your attempts to score more marks in the final examination. CBSE Class 12 students can refer to our solutions any time — while doing their homework and while preparing for the exam.

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RD Sharma Solution for Class 12 Science Mathematics Chapter 6 - Determinants Page/Excercise 6.1

Solution 1(i)

Solution 1(ii)

Solution 1(iii)

Solution 1(iv)

Solution 1(v)

Solution 1(vi)

Solution 1(vii)

Solution 2 (i)

Solution 2 (ii)

Solution 2 (iii)

Solution 2 (iv)

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 10(v)

Solution 10(vi)

Solution 11

Solution 12(i)

Solution 12(ii)

RD Sharma Solution for Class 12 Science Mathematics Chapter 6 - Determinants Page/Excercise 6.2

Solution 1(i)

Solution 1(ii)

C o n s i d e r space t h e space d e t e r m i n a n t
capital delta equals open vertical bar table row 67 19 21 row 39 13 14 row 81 24 26 end table close vertical bar
A p p l y i n g space C subscript 1 rightwards arrow C subscript 1 minus 4 C subscript 3 comma space w e space g e t comma
capital delta equals open vertical bar table row 4 19 21 row cell minus 3 end cell 13 14 row cell minus 3 end cell 24 26 end table close vertical bar
rightwards double arrow capital delta equals open vertical bar table row 4 19 21 row cell minus 3 end cell 13 14 row cell minus 3 end cell 24 26 end table close vertical bar space space space space space space space space space
rightwards double arrow capital delta equals open vertical bar table row 1 32 35 row cell minus 3 end cell 13 14 row 0 11 12 end table close vertical bar space space space open square brackets A p p l y i n g space R subscript 3 rightwards arrow R subscript 3 minus R subscript 2 space a n d space R subscript 1 rightwards arrow R subscript 1 plus R subscript 2 close square brackets
rightwards double arrow capital delta equals open vertical bar table row 1 32 35 row 0 109 119 row 0 11 12 end table close vertical bar space space space open square brackets A p p l y i n g space R subscript 2 rightwards arrow 3 R subscript 1 plus R subscript 2 space close square brackets
rightwards double arrow capital delta equals 1 open parentheses 109 cross times 12 minus 119 cross times 11 close parentheses
rightwards double arrow capital delta equals minus 1

Solution 1(iii)

Solution 1(iv)

Solution 1(v)

L e t space capital delta space b e space t h e space d e t e r m i n a n t.
capital delta equals open vertical bar table row 1 4 9 row 4 9 16 row 9 16 25 end table close vertical bar
A p p l y i n g space R subscript 3 rightwards arrow R subscript 3 minus R subscript 2 comma space w e space g e t comma
capital delta equals open vertical bar table row 1 4 cell 9 minus 4 end cell row 4 9 cell 16 minus 9 end cell row 9 16 cell 25 minus 16 end cell end table close vertical bar
rightwards double arrow capital delta equals open vertical bar table row 1 4 5 row 4 9 7 row 9 16 9 end table close vertical bar
rightwards double arrow capital delta equals open vertical bar table row 1 5 5 row 4 13 7 row 9 25 9 end table close vertical bar space space space space space space space space space open square brackets A p p l y i n g space C subscript 2 rightwards arrow C subscript 1 plus C subscript 2 close square brackets
rightwards double arrow capital delta equals open vertical bar table row 1 0 0 row 4 cell minus 7 end cell cell minus 13 end cell row 9 cell minus 20 end cell cell minus 36 end cell end table close vertical bar space space space space space space space space space open square brackets A p p l y i n g space C subscript 2 rightwards arrow 5 C subscript 1 minus C subscript 2 space a n d space C subscript 3 rightwards arrow 5 C subscript 1 minus C subscript 3 close square brackets
rightwards double arrow capital delta equals 1 open parentheses 7 cross times 36 minus 13 cross times 20 close parentheses equals 252 minus 260 equals minus 8

Solution 1(vi)

Solution 1(vii)

Solution 1(viii)

L e t space capital delta equals space open vertical bar table row 102 18 36 row 1 3 4 row 17 3 6 end table close vertical bar
A p p l y i n g space R subscript 3 rightwards arrow 17 R subscript 2 minus R subscript 3 comma space w e space g e t comma
capital delta equals space open vertical bar table row 102 18 36 row 1 3 4 row 0 48 62 end table close vertical bar
A p p l y i n g space R subscript 2 rightwards arrow 102 R subscript 2 minus R subscript 1 comma space w e space g e t comma
capital delta equals space open vertical bar table row 102 18 36 row 0 288 372 row 0 48 62 end table close vertical bar
T h u s comma
capital delta equals 102 open parentheses 288 cross times 62 minus 372 cross times 48 close parentheses
rightwards double arrow capital delta equals 0

Solution 2(i)

Solution 2(ii)

Solution 2(iii)

Solution 2(iv)

Solution 2(v)

Solution 2(vi)

Solution 2(vii)

Solution 2(viii)

Solution 2(ix)

Solution 2(x)

Solution 2(xi)

Solution 2(xii)

Solution 2(xiii)

C o n s i d e r space t h e space d e t e r m i n a n t space open vertical bar table row cell sin alpha end cell cell cos alpha end cell cell cos open parentheses alpha plus delta close parentheses end cell row cell sin beta end cell cell cos beta end cell cell cos open parentheses beta plus delta close parentheses end cell row cell sin gamma end cell cell cos gamma end cell cell cos open parentheses gamma plus delta close parentheses end cell end table close vertical bar
L e t space capital delta equals space open vertical bar table row cell sin alpha end cell cell cos alpha end cell cell cos open parentheses alpha plus delta close parentheses end cell row cell sin beta end cell cell cos beta end cell cell cos open parentheses beta plus delta close parentheses end cell row cell sin gamma end cell cell cos gamma end cell cell cos open parentheses gamma plus delta close parentheses end cell end table close vertical bar
A p p l y i n g space C subscript 1 rightwards arrow C subscript 1 sin delta space a n d space C subscript 2 rightwards arrow C subscript 2 cos delta comma space w e space h a v e comma
capital delta equals space open vertical bar table row cell sin alpha sin delta end cell cell cos alpha cos delta end cell cell cos open parentheses alpha plus delta close parentheses end cell row cell sin beta sin delta end cell cell cos beta cos delta end cell cell cos open parentheses beta plus delta close parentheses end cell row cell sin gamma sin delta end cell cell cos gamma cos delta end cell cell cos open parentheses gamma plus delta close parentheses end cell end table close vertical bar
A p p l y i n g space C subscript 2 rightwards arrow C subscript 2 minus C subscript 1 comma space w e space h a v e
capital delta equals space open vertical bar table row cell sin alpha sin delta end cell cell cos alpha cos delta minus sin alpha sin delta end cell cell cos open parentheses alpha plus delta close parentheses end cell row cell sin beta sin delta end cell cell cos beta cos delta minus sin beta sin delta end cell cell cos open parentheses beta plus delta close parentheses end cell row cell sin gamma sin delta end cell cell cos gamma cos delta minus sin gamma sin delta end cell cell cos open parentheses gamma plus delta close parentheses end cell end table close vertical bar
rightwards double arrow capital delta equals space open vertical bar table row cell sin alpha sin delta end cell cell cos open parentheses alpha plus delta close parentheses end cell cell cos open parentheses alpha plus delta close parentheses end cell row cell sin beta sin delta end cell cell cos open parentheses beta plus delta close parentheses end cell cell cos open parentheses beta plus delta close parentheses end cell row cell sin gamma sin delta end cell cell cos open parentheses gamma plus delta close parentheses end cell cell cos open parentheses gamma plus delta close parentheses end cell end table close vertical bar
A p p l y i n g space C subscript 3 rightwards arrow C subscript 3 minus C subscript 2 comma space w e space h a v e comma
capital delta equals space open vertical bar table row cell sin alpha sin delta end cell cell cos open parentheses alpha plus delta close parentheses end cell 0 row cell sin beta sin delta end cell cell cos open parentheses beta plus delta close parentheses end cell 0 row cell sin gamma sin delta end cell cell cos open parentheses gamma plus delta close parentheses end cell 0 end table close vertical bar
rightwards double arrow capital delta equals 0

Solution 2(xiv)

Solution 2(xv)

Solution 2(xvi)

Solution 2(xvii)

Solution 3

Solution 4

L e t space capital delta equals space open vertical bar table row 1 a cell b c end cell row 1 b cell c a end cell row 1 c cell a b end cell end table close vertical bar
A p p l y i n g space R subscript 2 rightwards arrow R subscript 2 minus R subscript 1 space a n d space R subscript 3 rightwards arrow R subscript 3 minus R subscript 1 w e space g e t comma
capital delta equals space open vertical bar table row 1 a cell b c end cell row 0 cell b minus a end cell cell c a minus b c end cell row 0 cell c minus a end cell cell a b minus b a end cell end table close vertical bar
rightwards double arrow capital delta equals space open vertical bar table row 1 a cell b c end cell row 0 cell b minus a end cell cell c open parentheses a minus b close parentheses end cell row 0 cell c minus a end cell cell b open parentheses a minus c close parentheses end cell end table close vertical bar
T a k i n g space open parentheses a minus b close parentheses space a n d space open parentheses a minus c close parentheses space c o m m o n comma space w e space h a v e
capital delta equals open parentheses a minus b close parentheses open parentheses a minus c close parentheses space open vertical bar table row 1 a cell b c end cell row 0 cell minus 1 end cell c row 0 cell minus 1 end cell b end table close vertical bar
rightwards double arrow capital delta equals open parentheses a minus b close parentheses open parentheses c minus a close parentheses open parentheses b minus c close parentheses

Solution 5

L e t space capital delta equals space open vertical bar table row cell x plus lambda end cell x x row x cell x plus lambda end cell x row x x cell x plus lambda end cell end table close vertical bar
A p p l y i n g space C subscript 1 rightwards arrow C subscript 1 plus C subscript 2 plus C subscript 3 space w e space g e t comma
capital delta equals space open vertical bar table row cell 3 x plus lambda end cell x x row cell 3 x plus lambda end cell cell x plus lambda end cell x row cell 3 x plus lambda end cell x cell x plus lambda end cell end table close vertical bar
T a k i n g space open parentheses 3 x plus lambda close parentheses space c o m m o n comma space w e space h a v e
capital delta equals open parentheses 3 x plus lambda close parentheses space open vertical bar table row 1 x x row 1 cell x plus lambda end cell x row 1 x cell x plus lambda end cell end table close vertical bar
A p p l y i n g space R subscript 2 rightwards arrow R subscript 2 minus R subscript 1 comma space R subscript 3 rightwards arrow R subscript 3 minus R subscript 1 comma space w e space g e t comma
capital delta equals open parentheses 3 x plus lambda close parentheses space open vertical bar table row 1 x x row 0 lambda 0 row 0 0 lambda end table close vertical bar
rightwards double arrow capital delta equals lambda squared open parentheses 3 x plus lambda close parentheses

Solution 6

L e t space capital delta equals space open vertical bar table row a b c row c a b row b c a end table close vertical bar
A p p l y i n g space C subscript 1 rightwards arrow C subscript 1 plus C subscript 2 plus C subscript 3 space w e space g e t comma
capital delta equals space open vertical bar table row cell a plus b plus c end cell b c row cell a plus b plus c end cell a b row cell a plus b plus c end cell c a end table close vertical bar
T a k i n g space open parentheses a plus b plus c close parentheses space c o m m o n comma space w e space h a v e
capital delta equals open parentheses a plus b plus c close parentheses space open vertical bar table row 1 b c row 1 a b row 1 c a end table close vertical bar
A p p l y i n g space R subscript 2 rightwards arrow R subscript 2 minus R subscript 1 comma space R subscript 3 rightwards arrow R subscript 3 minus R subscript 1 comma space w e space g e t comma
capital delta equals open parentheses a plus b plus c close parentheses space open vertical bar table row 1 b c row 0 cell a minus b end cell cell b minus c end cell row 0 cell c minus b end cell cell a minus c end cell end table close vertical bar
rightwards double arrow capital delta equals open parentheses a plus b plus c close parentheses open square brackets open parentheses a minus b close parentheses open parentheses a minus c close parentheses minus open parentheses b minus c close parentheses open parentheses c minus b close parentheses close square brackets
rightwards double arrow capital delta equals open parentheses a plus b plus c close parentheses open square brackets a squared minus a c minus a b plus b c plus b squared plus c squared minus 2 b c close square brackets
rightwards double arrow capital delta equals open parentheses a plus b plus c close parentheses open square brackets a squared plus b squared plus c squared minus a c minus a b minus b c close square brackets

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

W e space n e e d space t o space p r o v e space t h e space f o l l o w i n g space i d e n t i t y :
open vertical bar table row cell a plus b plus 2 c end cell a b row c cell b plus c plus 2 a end cell b row c a cell c plus a plus 2 b end cell end table close vertical bar equals 2 open parentheses a plus b plus c close parentheses cubed
A p p l y i n g space C subscript 1 rightwards arrow C subscript 1 plus C subscript 2 plus C subscript 3 comma space w e space h a v e comma
L. H. S equals open vertical bar table row cell 2 a plus 2 b plus 2 c end cell a b row cell 2 a plus 2 b plus 2 c end cell cell b plus c plus 2 a end cell b row cell 2 a plus 2 b plus 2 c end cell a cell c plus a plus 2 b end cell end table close vertical bar
T a k i n g space t h e space t e r m space 2 a plus 2 b plus 2 space a s space c o m m o n comma space w e space h a v e
L. H. S equals open parentheses 2 a plus 2 b plus 2 c close parentheses open vertical bar table row 1 a b row 1 cell b plus c plus 2 a end cell b row 1 a cell c plus a plus 2 b end cell end table close vertical bar
rightwards double arrow L. H. S equals 2 open parentheses a plus b plus c close parentheses open vertical bar table row 1 a b row 1 cell b plus c plus 2 a end cell b row 1 a cell c plus a plus 2 b end cell end table close vertical bar
A p p l y i n g space R subscript 2 rightwards arrow R subscript 2 minus R subscript 1 space a n d space R subscript 3 rightwards arrow R subscript 3 minus R subscript 1 space w e space h a v e
L. H. S equals 2 open parentheses a plus b plus c close parentheses open vertical bar table row 1 a b row 0 cell a plus b plus c end cell 0 row 0 0 cell a plus b plus c end cell end table close vertical bar
T h u s comma space w e space h a v e comma space
L. H. S equals 2 open parentheses a plus b plus c close parentheses open square brackets 1 cross times open parentheses a plus b plus c close parentheses squared close square brackets
space space space space space space space space space space space equals 2 open parentheses a plus b plus c close parentheses open parentheses a plus b plus c close parentheses squared
space space space space space space space space space space space equals 2 open parentheses a plus b plus c close parentheses cubed

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

 

Solution 25

W e space n e e d space t o space p r o v e space t h e space f o l l o w i n g space i d e n t i t y :
open vertical bar table row cell x plus 4 end cell x x row x cell x plus 4 end cell x row x x cell x plus 4 end cell end table close vertical bar equals 16 open parentheses 3 x plus 4 close parentheses
L e t space u s space c o n s i d e r space t h e space L. H. S space o f space t h e space a b o v e space e q u a t i o n.
capital delta equals open vertical bar table row cell x plus 4 end cell x x row x cell x plus 4 end cell x row x x cell x plus 4 end cell end table close vertical bar
A p p l y i n g space C subscript 1 rightwards arrow C subscript 1 plus C subscript 2 plus C subscript 3 comma space w e space g e t comma
capital delta equals open vertical bar table row cell 3 x plus 4 end cell x x row cell 3 x plus 4 end cell cell x plus 4 end cell x row cell 3 x plus 4 end cell x cell x plus 4 end cell end table close vertical bar
T a k i n g space t h e space c o m m o n space t e r m space 3 x plus 4 comma space w e space g e t comma
capital delta equals open parentheses 3 x plus 4 close parentheses open vertical bar table row 1 x x row 1 cell x plus 4 end cell x row 1 x cell x plus 4 end cell end table close vertical bar
A p p l y i n g space R subscript 2 rightwards arrow R subscript 2 minus R subscript 1 space a n d space R subscript 3 rightwards arrow R subscript 3 minus R subscript 1 comma space w e space g e t comma
capital delta equals open parentheses 3 x plus 4 close parentheses open vertical bar table row 1 x x row 0 4 0 row 0 0 4 end table close vertical bar
rightwards double arrow capital delta equals 16 open parentheses 3 x plus 4 close parentheses

 

 

Solution 26

W e space n e e d space t o space p r o v e space t h e space f o l l o w i n g space i d e n t i t y :
open vertical bar table row 1 cell 1 plus p end cell cell 1 plus p plus q end cell row 2 cell 3 plus 2 p end cell cell 4 plus 3 p plus 2 q end cell row 3 cell 6 plus 3 p end cell cell 10 plus 6 p plus 3 q end cell end table close vertical bar equals 1
L e t space u s space c o n s i d e r space t h e space L. H. S space o f space t h e space a b o v e space e q u a t i o n.
A p p l y i n g space C subscript 2 rightwards arrow C subscript 2 minus p C subscript 1 space a n d space C subscript 3 rightwards arrow C subscript 3 minus q C subscript 1 comma space w e space g e t
capital delta equals open vertical bar table row 1 1 cell 1 plus p end cell row 2 3 cell 4 plus 3 p end cell row 3 6 cell 10 plus 6 p end cell end table close vertical bar
A p p l y i n g space C subscript 3 rightwards arrow C subscript 3 minus p C subscript 2 space comma space w e space g e t
rightwards double arrow capital delta equals open vertical bar table row 1 1 1 row 2 3 4 row 3 6 10 end table close vertical bar
A p p l y i n g space C subscript 2 rightwards arrow C subscript 2 minus p C subscript 1 space a n d space C subscript 3 rightwards arrow C subscript 3 minus q C subscript 1 comma space w e space g e t
rightwards double arrow capital delta equals open vertical bar table row 1 0 0 row 2 1 2 row 3 3 7 end table close vertical bar
rightwards double arrow capital delta equals 1 open square brackets 7 minus 6 close square brackets equals 1

 

 

Solution 27

Solution 28

Solution 29

W e space n e e d space t o space p r o v e space t h e space f o l l o w i n g space i d e n t i t y :
open vertical bar table row cell a squared plus 1 end cell cell a b end cell cell a c end cell row cell a b end cell cell b squared plus 1 end cell cell b c end cell row cell c a end cell cell c b end cell cell c squared plus 1 end cell end table close vertical bar equals 1 plus a squared plus b squared plus c squared
L e t space u s space c o n s i d e r space t h e space L. H. S space o f space t h e space a b o v e space e q u a t i o n.
capital delta equals open vertical bar table row cell a squared plus 1 end cell cell a b end cell cell a c end cell row cell a b end cell cell b squared plus 1 end cell cell b c end cell row cell c a end cell cell c b end cell cell c squared plus 1 end cell end table close vertical bar
A p p l y i n g space R subscript 1 rightwards arrow R subscript 1 open parentheses a close parentheses comma space R subscript 2 rightwards arrow R subscript 2 open parentheses b close parentheses space a n d space R subscript 3 rightwards arrow R subscript 3 open parentheses c close parentheses comma space w e space g e t
capital delta equals fraction numerator 1 over denominator a b c end fraction open vertical bar table row cell a open parentheses a squared plus 1 close parentheses end cell cell a squared b end cell cell a squared c end cell row cell a b squared end cell cell b open parentheses b squared plus 1 close parentheses end cell cell b squared c end cell row cell c squared a end cell cell c squared b end cell cell c open parentheses c squared plus 1 close parentheses end cell end table close vertical bar
T a k i n g space a comma b comma space a n d space c space c o m m o n space f r o m space C subscript 1 comma C subscript 2 space a n d space C subscript 3 comma space r e s p e c t i v e l y comma space w e space g e t comma
capital delta equals fraction numerator a b c over denominator a b c end fraction open vertical bar table row cell open parentheses a squared plus 1 close parentheses end cell cell a squared end cell cell a squared end cell row cell b squared end cell cell open parentheses b squared plus 1 close parentheses end cell cell b squared end cell row cell c squared end cell cell c squared end cell cell open parentheses c squared plus 1 close parentheses end cell end table close vertical bar
A p p l y i n g space R subscript 1 rightwards arrow R subscript 1 plus R subscript 2 plus R subscript 3 comma space w e space g e t comma
capital delta equals fraction numerator a b c over denominator a b c end fraction open vertical bar table row cell open parentheses a squared plus b squared plus c squared plus 1 close parentheses end cell cell open parentheses a squared plus b squared plus c squared plus 1 close parentheses end cell cell open parentheses a squared plus b squared plus c squared plus 1 close parentheses end cell row cell b squared end cell cell open parentheses b squared plus 1 close parentheses end cell cell b squared end cell row cell c squared end cell cell c squared end cell cell open parentheses c squared plus 1 close parentheses end cell end table close vertical bar
T a k i n g space t h e space t e r m comma space open parentheses a squared plus b squared plus c squared plus 1 close parentheses space c o m m o n space f r o m space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma
capital delta equals open parentheses a squared plus b squared plus c squared plus 1 close parentheses open vertical bar table row 1 1 1 row cell b squared end cell cell open parentheses b squared plus 1 close parentheses end cell cell b squared end cell row cell c squared end cell cell c squared end cell cell open parentheses c squared plus 1 close parentheses end cell end table close vertical bar
A p p l y i n g space C subscript 2 rightwards arrow C subscript 2 minus C subscript 1 comma space C subscript 3 rightwards arrow C subscript 3 minus C subscript 1 comma space w e space g e t comma
capital delta equals open parentheses a squared plus b squared plus c squared plus 1 close parentheses open vertical bar table row 1 0 1 row cell b squared end cell 1 0 row cell c squared end cell 0 1 end table close vertical bar
rightwards double arrow capital delta equals open parentheses a squared plus b squared plus c squared plus 1 close parentheses

 

 

Solution 30

 

 

Solution 31

Solution 32

W e space n e e d space t o space p r o v e space t h e space f o l l o w i n g space i d e n t i t y :
open vertical bar table row cell b plus c end cell a a row b cell c plus a end cell b row c c cell a plus b end cell end table close vertical bar equals 4 a b c
L e t space u s space c o n s i d e r space t h e space L. H. S space o f space t h e space a b o v e space e q u a t i o n.
capital delta equals open vertical bar table row cell b plus c end cell a a row b cell c plus a end cell b row c c cell a plus b end cell end table close vertical bar
A p p l y i n g space R subscript 1 rightwards arrow R subscript 1 plus R subscript 2 plus R subscript 3 comma space w e space h a v e comma
capital delta equals open vertical bar table row cell 2 open parentheses b plus c close parentheses end cell cell 2 open parentheses a plus c close parentheses end cell cell 2 open parentheses a plus b close parentheses end cell row b cell c plus a end cell b row c c cell a plus b end cell end table close vertical bar
T a k i n g space 2 space c o m m o n space f r o m space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma
capital delta equals 2 open vertical bar table row cell open parentheses b plus c close parentheses end cell cell open parentheses a plus c close parentheses end cell cell open parentheses a plus b close parentheses end cell row b cell c plus a end cell b row c c cell a plus b end cell end table close vertical bar
A p p l y i n g space R subscript 2 rightwards arrow R subscript 2 minus R subscript 1 space a n d space R subscript 3 rightwards arrow R subscript 3 minus R subscript 1 comma space w e space h a v e comma
capital delta equals 2 open vertical bar table row cell open parentheses b plus c close parentheses end cell cell open parentheses a plus c close parentheses end cell cell open parentheses a plus b close parentheses end cell row cell minus c end cell 0 cell minus a end cell row cell minus b end cell cell minus a end cell 0 end table close vertical bar
A p p l y i n g space R subscript 1 rightwards arrow R subscript 1 plus R subscript 2 plus R subscript 3 comma space w e space h a v e comma
capital delta equals 2 open vertical bar table row 0 c b row cell minus c end cell 0 cell minus a end cell row cell minus b end cell cell minus a end cell 0 end table close vertical bar
rightwards double arrow capital delta equals 2 open parentheses 0 plus 2 a b c plus a b c close parentheses
rightwards double arrow capital delta equals 4 a b c

 

 

Solution 33

Solution 34

Solution 35

Solution 36

Solution 37

Solution 38

Solution 39

Solution 40

Solution 41

Solution 42

Solution 43

Solution 44

Solution 45

Solution 46

Solution 47

 C o n s i d e r space t h e space d e t e r m i n a n t space open vertical bar table row cell x plus 1 end cell cell x plus 2 end cell cell x plus a end cell row cell x plus 2 end cell cell x plus 3 end cell cell x plus b end cell row cell x plus 3 end cell cell x plus 4 end cell cell x plus c end cell end table close vertical bar space comma space w h e r e space a comma space b comma space c space a r e space i n space A. P.
L e t space capital delta equals open vertical bar table row cell x plus 1 end cell cell x plus 2 end cell cell x plus a end cell row cell x plus 2 end cell cell x plus 3 end cell cell x plus b end cell row cell x plus 3 end cell cell x plus 4 end cell cell x plus c end cell end table close vertical bar space
A p p l y i n g space C subscript 1 rightwards arrow C subscript 1 plus C subscript 2 plus C subscript 3 comma space w e space h a v e comma
capital delta equals open vertical bar table row cell 3 x plus 1 plus 2 plus a end cell cell x plus 2 end cell cell x plus a end cell row cell 3 x plus 2 plus 3 plus b end cell cell x plus 3 end cell cell x plus b end cell row cell 3 x plus 3 plus 4 plus c end cell cell x plus 4 end cell cell x plus c end cell end table close vertical bar
rightwards double arrow capital delta equals open vertical bar table row cell 3 x plus 3 plus a end cell cell x plus 2 end cell cell x plus a end cell row cell 3 x plus 5 plus b end cell cell x plus 3 end cell cell x plus b end cell row cell 3 x plus 7 plus c end cell cell x plus 4 end cell cell x plus c end cell end table close vertical bar
A p p l y i n g space R subscript 2 rightwards arrow R subscript 2 minus R subscript 1 space a n d space R subscript 3 rightwards arrow R subscript 3 minus R subscript 2 comma space w e space h a v e comma
rightwards double arrow capital delta equals open vertical bar table row cell 3 x plus 3 plus a end cell cell x plus 2 end cell cell x plus a end cell row cell 2 plus b minus a end cell 1 cell b minus a end cell row cell 2 plus c minus b end cell 1 cell c minus b end cell end table close vertical bar
S i n c e space a comma b space a n d space c space a r e space i n space a r i t h m e t i c space p r o g r e s s i o n comma space w e space h a v e
b minus a equals c minus b equals k left parenthesis s a y right parenthesis.
T h u s comma
capital delta equals open vertical bar table row cell 3 x plus 3 plus a end cell cell x plus 2 end cell cell x plus a end cell row cell 2 plus k end cell 1 k row cell 2 plus k end cell 1 k end table close vertical bar
S i n c e space t h e space s e c o n d space r o w space a n d space t h e space t h i r d space r o w space a r e space i d e n t i c a l comma space w e space h a v e
capital delta equals 0

 

Solution 48

Solution 49

Solution 50

Solution 51

Solution 52(i)

Solution 52(ii)

Solution 52(iii)

Solution 52(iv)

Solution 52(v)

Solution 52(vi)

Solution 52(vii)

Solution 52(viii)

Solution 52(ix)

RD Sharma Solution for Class 12 Science Mathematics Chapter 6 - Determinants Page/Excercise 6.3

Solution 1 (i)

Solution 1 (ii)

Solution 1 (iii)

Solution 1 (iv)

Solution 2 (i)

Solution 2 (iii)

Solution 2 (iv)

Solution 2(ii)

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12 (i)

Solution 12 (ii)

Solution 13 (i)

Solution 13 (ii)

RD Sharma Solution for Class 12 Science Mathematics Chapter 6 - Determinants Page/Excercise 6.4

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

RD Sharma Solution for Class 12 Science Mathematics Chapter 6 - Determinants Page/Excercise 6.5

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

RD Sharma Solution for Class 12 Science Mathematics Chapter 6 - Determinants Page/Excercise MCQ

Solution 1

Correct option: (d)

Solution 2

Correct option: (d)

Solution 3

Correct option: (d)

If A is a square matrix of order n then det(A) = a11C11+a21C21+a31C31

Solution 4

Correct option: (b)

Minor of an element can never be equal to cofactor of the same element.

Cij=(-1)i+jMij

Solution 5

Correct option: (e)

Solution 6

Correct option: (a)

Solution 7

Correct option: (a)

Solution 8

 

Solution 9

Correct option: (a)

 

Solution 10

Correct option: (a)

 

Solution 11

Correct option: (d)

Solution 12

Correct option: (b)

Solution 13

Correct option: (a)

Solution 14

Correct option: (c)

Solution 15

Correct option: (c)

Solution 16

Correct option: (b)

Solution 17

Correct option: (b)

Solution 18

Correct option:(a)

Solution 19

Correct option: (a)

Solution 20

Correct option: (b)

Solution 21

Correct option: (d)

Solution 22

Correct option: (c)

Solution 23

Correct option: (a)

Solution 24

Correct option: (c )

Solution 25

Correct option: (d)

Solution 26

Correct option: (d)

Solution 27

Correct option: (a)

 

Solution 28

Correct option: (a)

Solution 29

Correct option: (b)

Solution 30

Correct option: (a)

Solution 31

Correct option: (c)

Solution 32

Correct option: (d)

 

Solution 33

Correct option: (c)

RD Sharma Solution for Class 12 Science Mathematics Chapter 6 - Determinants Page/Excercise 6VSAQ

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36

Solution 37

Solution 38

Solution 39

Solution 40

Solution 41

L e t space A space b e space t h e space g i v e n space m a t r i x.
A equals open square brackets table row cell 6 minus x end cell 4 row cell 3 minus x end cell 1 end table close square brackets
T h e space m a t r i x space A space i s space sin g u l a r space i f space open vertical bar A close vertical bar equals 0.
T h u s comma space open vertical bar table row cell 6 minus x end cell 4 row cell 3 minus x end cell 1 end table close vertical bar equals 0
rightwards double arrow open parentheses 6 minus x close parentheses minus 4 open parentheses 3 minus x close parentheses equals 0
rightwards double arrow 6 minus x minus 12 plus 4 x equals 0
rightwards double arrow minus 6 plus 3 x equals 0
rightwards double arrow 3 x equals 6
rightwards double arrow x equals 2

 

 

Solution 42

C o n s i d e r space t h e space f o l l o w i n g space p r o p e r t y.
L e t space A equals open square brackets a subscript i j end subscript close square brackets space b e space a space s q u a r e space m a t r i x space o f space o r d e r space n space t h e n space open vertical bar k A close vertical bar equals k to the power of n open vertical bar A close vertical bar comma space
w h e r e space k space i s space a n y space c o n s tan t.
H e r e comma space n equals 3 space a n d space k equals 2.
T h u s comma space w e space h a v e comma space open vertical bar 2 A close vertical bar equals 2 cubed open vertical bar A close vertical bar equals 8 cross times 4 equals 32

 

 

Solution 43

C o n s i d e r space t h e space g i v e n space m a t r i x.
L e t space A equals space open vertical bar table row cell cos 15 degree end cell cell sin 15 degree end cell row cell sin 75 degree end cell cell cos 75 degree end cell end table close vertical bar
rightwards double arrow open vertical bar A close vertical bar equals open vertical bar table row cell cos 15 degree end cell cell sin 15 degree end cell row cell sin open parentheses 90 degree minus 15 degree close parentheses end cell cell cos open parentheses 90 degree minus 15 degree close parentheses end cell end table close vertical bar space
rightwards double arrow open vertical bar A close vertical bar equals space open vertical bar table row cell cos 15 degree end cell cell sin 15 degree end cell row cell cos 15 degree end cell cell sin 15 degree end cell end table close vertical bar
rightwards double arrow open vertical bar A close vertical bar equals space cos 15 degree cross times sin 15 degree minus cos 15 degree cross times sin 15 degree equals 0

 

 

Solution 44

C o n s i d e r space t h e space g i v e n space m a t r i x space A equals open square brackets table row 5 3 8 row 2 0 1 row 1 2 3 end table close square brackets.
W e space n e e d space t o space f i n d space t h e space c o f a c t o r space o f space t h e space e l e m e n t space a subscript 32.
W e space k n o w space t h a t space C subscript i j end subscript equals C o f a c t o r space o f space a subscript i j end subscript space i n space A equals open parentheses minus 1 close parentheses to the power of i plus j end exponent M subscript i j end subscript comma space w h e r e space M subscript i j end subscript space i s space m i n o r space o f space a subscript i j end subscript
T h u s comma space C subscript 32 equals C o f a c t o r space o f space a subscript 32 space i n space A equals open parentheses minus 1 close parentheses to the power of 3 plus 2 end exponent M subscript 32
rightwards double arrow C subscript 32 equals open parentheses minus 1 close parentheses to the power of 5 open vertical bar table row 5 8 row 2 1 end table close vertical bar equals minus open parentheses 5 minus 16 close parentheses equals 11

 

 

Solution 45

Solution 46

Solution 47

Solution 48

Solution 49

Solution 50

Solution 51

Solution 52

TopperLearning provides step-by-step solutions for each question in each chapter in the RD Sharma textbook for class 9. Access the CBSE Class 9 Mathematics Chapter 6 - Determinants for free. The textbook questions have been solved by our subject matter experts to help you understand how to answer them. Our RD Sharma Textbook Solutions will help you to study and revise, and you can easily clear your fundamentals of Chapter 6 - Determinants.

Text Book Solutions

CBSE XII Science - Mathematics

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