class: sz1-a(revision) jee-main model date: 02-01-21 time: … · 2021. 1. 4. · narayana co...

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Class: SZ1-A(Revision) JEE-MAIN MODEL Date: 02-01-21

Time: 3hrs RTM-04 Max. Marks: 300

Jee-Main_Model

SZ1-A Physics Initial Key Dt. 02-01-2021

Q.No. 01 02 03 04 05 06 07 08 09 10

Ans. 1 3 3 1 4 2 4 2 4 1

Q.No. 11 12 13 14 15 16 17 18 19 20

Ans. 3 2 1 2 1 3 2 2 1 1

Q.No. 21 22 23 24 25

Ans. 6.00 10.55 to 10.60

34.28 to 34.30

10.00 40.00

SZ1-A Chemistry Initial Key Dt. 02-01-2021

Q.No. 26 27 28 29 30 31 32 33 34 35

Ans. 3 3 4 1 4 4 2 1 4 1

Q.No. 36 37 38 39 40 41 42 43 44 45

Ans. 4 1 3 4 1 3 1 1 3 2

Q.No. 46 47 48 49 50

Ans. 6.46 21.37 1.20 to

1.25 124.50 -0.90

SZ1-A Mathematics Initial Key Dt. 02-01-2021

Q.No. 51 52 53 54 55 56 57 58 59 60

Ans. 3 2 4 1 3 4 3 4 1 4

Q.No. 61 62 63 64 65 66 67 68 69 70

Ans. 2 2 1 4 4 1 2 2 4 4

Q.No. 71 72 73 74 75

Ans. 0.00 3.00 0.00 1.00 1.00

SZ1-A_JEE-MAIN_RTM-04_Key&Solutions_Exam.Dt.02-01-21

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01. 1 2p p p= +

Powers added up individual powers will be 2D

02. Increases

03. Power of the lenses does not depend on its size

04. The ciliary muscles do not contract and the eye is least strained

05. size of object directly depends

08. 0 10 , 5f cm Fe cm= =

0 0 0

1 1 1

v u f− =

0

1 11 1

0.5v= − = −

Object distance of eyepiece ( )1x cm= +

1 1 1

30 1 5x+ =

+

6x =

1 6x+ =

5x cm=

13.


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14.

15.


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16.

19.


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23.

25.


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30.

31. q and w represents path function

32.

42.

43.


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44.

45.

46.

47.


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48.

49.

50.


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51. System has unique solution if det. of coefficinet 0=

1 1 1

2 1 1 0 0

3 2

−

52. AX B=

2 3 6 4

5 7 14 , , 1

3 2 4 0

x

A X y B

z

−

= − = = −

now 0A = .

The equations either have no solution or an infinite number of

solutions to decide about this we proceed to find ( ).adj A B

0 0 0

22 26 58

11 13 29

adjA

= − − − −

( )

0

. 114 0

57

adj A B

= − −

no solution

53. AX B=

1 1 1 6

0 1 2 8

0 3 6 24

A B

=

2 2 1R R R⎯⎯→ −

3 3 1R R R⎯⎯→ −

1 1 1 6

0 1 2 8

0 0 0 0

= =

Rank A = rank [A][B] = 2

System has infinitely many solution.

System is equivalent to 6, 2 8x y z y z+ + = + =

8 2 , 2z y x= = − = − +


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54. 0A =

Now

4 6 2 4 6 2

6 9 3 6 9 3

2 3 1 2 3 1

T

adj A

− −

= − − = − − − −

( )

2

2

2 2

4 6 2 1 4 6 2

6 9 3 6 9 3 0

2 3 1 2 3

adj A B

− − +

= − − = − + − = − − +

Since system is consistent

22 6 4 0 1,2 − + = =

55. Augumented martrix

1 1 1 5 1 1 1 5

1 2 0 1 1 5

1 2 3 9 0 0 3 9

A B

= = − − − −

2 2 1

3 3 1

R R R

R R R

⎯⎯→ −

⎯⎯→ −

Rank A = Rank [AB] if 3

The system has unique solution if 3

If 3, 9 = = , rank A = rank [AB] = 2 < no. of unknowns

The system has infinitely many solution if 3, 9 = =

If 3, 9 = , rank A = 2, rank [AB] = 3 no solution

56. System has non-trival solution 0

a b c

b c a

c a b

=

3 3 3 3 0a b c abc + + − =

( )( )( )2 2 0a b c c bw aw c bw aw + + + + + + =

Since , ,a b c are positive 0a b c+ + , either 2 0aw bw c+ + = or

4 2 0aw bw c+ + =

Hence roots of the equation 2 0at bt c+ + = are 2,w w

57.

1 1 1

1 1 1 0

1 1 1

a

b

c

−

− =

−


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ab bc ca abc + + =

( )1

2 2 2 3

3

ab bc caa b c

+ +

( )1

3 3abc

( )min 27abc =

59. The system has infinite solutions if

1 0

0 1 0 1,1

0 1

a

a a

a

= = −

For 1a = we have the system 0, 0, 0x y y z z x+ = + = + = which have

only one solution 0x y z= = =

1a = −

60. Non-trival solution if

1 1

1 1 0

1 1

−

− − =

−

( )2 1 0 − =

0,1, 1 = −

61. 1 2A A A I− = + +

3 2I A A A= + +

33 6 19

35 4 30

30 13 22

A

− −

= − −

3 26 11A A A I− + =

multiplying both the sides by 1A−

1 2 6 11A A A I− = − +

6, 11 = − =

5+ =

62. ( ) ( )( ) ( ) 1 1T

TTBB I A I A I A I A− −

= − + + −

( ) ( )( )1 TI A I A I A−

= − + +


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( ) ( )( ) ( )1

1 TI A I A I A I A

−− = − + − −

A is skew-symmetric TA A= −

( ) ( ) ( ) ( )11 TI A I A I A I A−−

− − + −

( ) ( )( )( )1 1

I A I A I A I A− −

= − − + +

.TBB I I I = =

63. ( ) ( )1

*A I A I A−

= + −

( ) ( )1

** * *A I A I A−

= + −

( ) ( ) ( ) ( ) ( )1 1 1

* 2I A I A I A I A I A I A I− − −

+ = + + + + − = + ( )1

2 I A−

= +

( ) ( )1 1

*2

I A I A−

+ = +

( ) ( ) ( ) ( ) ( )1 1 1

* 2I A I A I A I A I A I A A− − −

− = + + − + − = +

( ) ( )11

** 22

A I A I A A A−

= + + =

64. A has rank 3 0A

( )

1

1 0

1

b c

y a b c y b c

b y c

+ + + +

+

( )

1

0 0 0

0 0

b c

y a b c y

y

+ + + ( )2 0y y a b c + + +

65. matrix A has all integer entries

cofactor of each entry is integer

adjoine A has integers entries

( ) ( )11

A adj A adj AA

− = =

66. T TPP I P P= =

TQ PAP=

T T T TP Q P PAP AP = =


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2005 2004 2004 2 2003 ...T T T TP Q P P Q Q P AP Q P A P Q P= = =

( ) ( )2004 2004 2005T TA P QP A P PA A= = =

21 2 1

...0 1 0 1

nn

A A

= =

20051 2005

0 1A

=

20051 2005

0 1

TP Q P

=

67. TA A I=

2 2 2 1a b c + + =

0ab bc ca+ + =

Since , , 0a b c

0ab bc ca+ +

3 3 3a b c + + is exists

( )( )3 3 3 2 2 2 4a b c a b c a b c ab bc ca+ + = + + + + − − − =

68. characteristic equation of A is

0A I− =

3 26 11 6 0 − + − =

A satisfies characteristic equation 3 26 11 6 0A A A I− + − =

1 26 6 11A A A I− = − +

6, 11C d= − =

69. 3 3ij

P a

=

2i jij ijb a+=

3 3ij

Q b

=

11 12 13

21 22 23

31 32 33

, 2

a a a

P a a a P

a a a

= =


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11 12 13 11 12 13

21 22 23 21 22 23

31 32 33 31 32 33

4 8 16

8 16 32

16 32 64

b b b a a a

Q b b b a a a

c c c a a a

= =

Determinant of

11 12 13 11 12 13

21 22 33 21 22 23

31 32 33 31 32 33

4 8 16

8 16 32 4 8 16 2 2 2

16 32 64 4 4 4

a a a a a a

Q a a a a a a

a a a a a a

= =

11 12 13

2 3 4 1 2 1 13

21 22 23

31 32 33

4 8 16 2 4 2 2 2 2 2 2 2

a a a

a a a

a a a

= = =

70. 2TP P I= +

2 TP P I= +

2 2P P I I= + +

4 3P P I= +

3 3P I= −

PX X= −

71. 3 3 2 2,P Q P Q Q P= =

P Q

Subtracting 3 2 3 2P P Q Q Q P− = −

( ) ( )2 2 0P P Q Q P Q− + − =

( )( )2 2 0P Q P Q+ − =

If ( )2 2 0P Q P Q+ − =

If 2 2 0P Q+ , then 2 2P Q+ is invertible

0P Q − =

Controduction

Hence 2 2 0P Q+ =

72. for no solution

1 8 4

3 3 1

k k

k k k

+=

+ −


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2 4 3 8k k k+ + =

2 4 3 0, 1,3k k k− + = =

If 8 4 1

1 false1 3 2

k

= +

If 8 4 3

3 true6 9 1

k

= =−

3k =

class: sz1-a(revision) jee-main model date: 02-01-21 time: … · 2021. 1. 4. · narayana co...

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