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                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 1  
 
Date: 9
th
 January 2020 (Shift 2) 
Time: 2:30 P.M. to 5:30 P.M. 
Subject: Mathematics 
 
1. If  ?? (?? )= |
?? +?? ?? +2 ?? +1
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| and ?? -2?? +?? =1 then 
a. ?? (-50)=-1                                                            b. ?? (50)= 1              
c. ?? (50)=-501                                                                                        d. ?? (50)= 501 
       Answer: (?? ) 
       Solution:  
       Given ?? (?? )= |
?? +?? ?? +2 ?? +1
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| 
       ?? -2?? +?? =1 
       Applying ?? 1
??? 1
-2?? 2
+?? 3
 
       ?? (?? )= |
?? -2?? +?? 0 0
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| 
       Using ?? -2?? +?? =1 
       ??? (?? )= (?? +3)
2
-(?? +2)(?? +4) 
       ??? (?? )=1 
       ??? (50)=1  
       ??? (-50)=1 
 
 
2. If   ?? (?? )= 
{
 
 
 
 
   ??                    0<?? < 
1
2
1
2
                         ?? = 
1
2
1-??                  
1
2
<?? <1
 
       ?? (?? )= (?? - 
1
2
)
2
 then find the area bounded by ?? (?? ) and ?? (?? ) from ?? =  
1
2
  to ?? =  
v3
2
. 
a. 
v3
2
 
-
1
3
 
                                                                     b. 
v3
4
 
+
1
3
 
               
c. 2v3                                                                                        d. 3v3
Page 2


                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 1  
 
Date: 9
th
 January 2020 (Shift 2) 
Time: 2:30 P.M. to 5:30 P.M. 
Subject: Mathematics 
 
1. If  ?? (?? )= |
?? +?? ?? +2 ?? +1
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| and ?? -2?? +?? =1 then 
a. ?? (-50)=-1                                                            b. ?? (50)= 1              
c. ?? (50)=-501                                                                                        d. ?? (50)= 501 
       Answer: (?? ) 
       Solution:  
       Given ?? (?? )= |
?? +?? ?? +2 ?? +1
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| 
       ?? -2?? +?? =1 
       Applying ?? 1
??? 1
-2?? 2
+?? 3
 
       ?? (?? )= |
?? -2?? +?? 0 0
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| 
       Using ?? -2?? +?? =1 
       ??? (?? )= (?? +3)
2
-(?? +2)(?? +4) 
       ??? (?? )=1 
       ??? (50)=1  
       ??? (-50)=1 
 
 
2. If   ?? (?? )= 
{
 
 
 
 
   ??                    0<?? < 
1
2
1
2
                         ?? = 
1
2
1-??                  
1
2
<?? <1
 
       ?? (?? )= (?? - 
1
2
)
2
 then find the area bounded by ?? (?? ) and ?? (?? ) from ?? =  
1
2
  to ?? =  
v3
2
. 
a. 
v3
2
 
-
1
3
 
                                                                     b. 
v3
4
 
+
1
3
 
               
c. 2v3                                                                                        d. 3v3
                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 2  
 
       Answer: (?? ) 
       Solution: 
       Given ?? (?? )= 
{
 
 
 
 
   ??                    0<?? < 
1
2
1
2
                         ?? = 
1
2
1-??                  
1
2
<?? <1
 
       ?? (?? )= (?? - 
1
2
)
2
 
       The area between ?? (?? ) and ?? (?? ) from ?? =
1
2
  to =
v3
2
 : 
        
       Points of intersection of ?? (?? ) and (?? ) : 
 
1-?? = (?? - 
1
2
)
2
 
       ??? = 
v3
2
 ,-
v3
2
  
       Required area =?
(?? (?? )-?? (?? ))????
v3
2
1
2
 
                            =? (1-?? - (?? - 
1
2
)
2
)????
v3
2
1
2
 
                                    =?? - 
?? 2
2
-
1
3
(?? - 
1
2
)
3
|
1
2
v3
2
  
                                    = 
v3
4
- 
1
3
 
 
Page 3


                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 1  
 
Date: 9
th
 January 2020 (Shift 2) 
Time: 2:30 P.M. to 5:30 P.M. 
Subject: Mathematics 
 
1. If  ?? (?? )= |
?? +?? ?? +2 ?? +1
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| and ?? -2?? +?? =1 then 
a. ?? (-50)=-1                                                            b. ?? (50)= 1              
c. ?? (50)=-501                                                                                        d. ?? (50)= 501 
       Answer: (?? ) 
       Solution:  
       Given ?? (?? )= |
?? +?? ?? +2 ?? +1
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| 
       ?? -2?? +?? =1 
       Applying ?? 1
??? 1
-2?? 2
+?? 3
 
       ?? (?? )= |
?? -2?? +?? 0 0
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| 
       Using ?? -2?? +?? =1 
       ??? (?? )= (?? +3)
2
-(?? +2)(?? +4) 
       ??? (?? )=1 
       ??? (50)=1  
       ??? (-50)=1 
 
 
2. If   ?? (?? )= 
{
 
 
 
 
   ??                    0<?? < 
1
2
1
2
                         ?? = 
1
2
1-??                  
1
2
<?? <1
 
       ?? (?? )= (?? - 
1
2
)
2
 then find the area bounded by ?? (?? ) and ?? (?? ) from ?? =  
1
2
  to ?? =  
v3
2
. 
a. 
v3
2
 
-
1
3
 
                                                                     b. 
v3
4
 
+
1
3
 
               
c. 2v3                                                                                        d. 3v3
                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 2  
 
       Answer: (?? ) 
       Solution: 
       Given ?? (?? )= 
{
 
 
 
 
   ??                    0<?? < 
1
2
1
2
                         ?? = 
1
2
1-??                  
1
2
<?? <1
 
       ?? (?? )= (?? - 
1
2
)
2
 
       The area between ?? (?? ) and ?? (?? ) from ?? =
1
2
  to =
v3
2
 : 
        
       Points of intersection of ?? (?? ) and (?? ) : 
 
1-?? = (?? - 
1
2
)
2
 
       ??? = 
v3
2
 ,-
v3
2
  
       Required area =?
(?? (?? )-?? (?? ))????
v3
2
1
2
 
                            =? (1-?? - (?? - 
1
2
)
2
)????
v3
2
1
2
 
                                    =?? - 
?? 2
2
-
1
3
(?? - 
1
2
)
3
|
1
2
v3
2
  
                                    = 
v3
4
- 
1
3
 
 
                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 3  
 
3. If ?? ?(?? ?~ ?? ) is false. Truth value of ?? and ?? will be 
a. TF  b. FT 
c. TT d. FF 
      Answer: (?? ) 
Solution:  
      Given ?? ?(?? ?~ ?? ) 
      Truth table: 
?? ?? ~?? (?? ?~ ?? ) ?? ?(?? ?~ ?? ) 
T T F F F 
T F T T T 
F T F F T 
F F T F T 
        ?? ?(?? ?~ ?? ) is false when ?? is true and ?? is true. 
 
4. ?
????
cos
2
?? 
 (sec2?? +tan2?? ) 
 =?? tan?? +2log?? (?? )+?? then ordered pair (?? , ?? (?? )) is
a. (1,1+tan?? ) b. (1,1-tan?? )
c. (-1,1+tan?? )                                                                   d.     (-1,1-tan?? ) 
 
Answer: (?? ) 
      Solution:  
       Let ?? = ?
????
cos
2
?? (sec2?? +tan2?? ) 
 
      ?? = ?
sec
2
?? ????
 (
1+tan
2
?? 1-tan
2
?? )+ (
2tan?? 1- tan
2
?? )
 
       ?? = ?
(1- tan
2
?? )(sec
2
?? )????
(1+tan?? )
2
 
       Let tan?? =??  ? sec
2
?? ???? =d?? 
       ?? = ?
(1- ?? 2
)
(1+?? )
2
 ???? =  ?
(1- ?? )
(1+?? )
 ???? 
       ?? = (
2
1+?? -1)???? 
       ?? =2ln|1+?? |-?? +?? 
       ?? =2ln|1+tan?? |-tan?? +?? 
Page 4


                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 1  
 
Date: 9
th
 January 2020 (Shift 2) 
Time: 2:30 P.M. to 5:30 P.M. 
Subject: Mathematics 
 
1. If  ?? (?? )= |
?? +?? ?? +2 ?? +1
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| and ?? -2?? +?? =1 then 
a. ?? (-50)=-1                                                            b. ?? (50)= 1              
c. ?? (50)=-501                                                                                        d. ?? (50)= 501 
       Answer: (?? ) 
       Solution:  
       Given ?? (?? )= |
?? +?? ?? +2 ?? +1
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| 
       ?? -2?? +?? =1 
       Applying ?? 1
??? 1
-2?? 2
+?? 3
 
       ?? (?? )= |
?? -2?? +?? 0 0
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| 
       Using ?? -2?? +?? =1 
       ??? (?? )= (?? +3)
2
-(?? +2)(?? +4) 
       ??? (?? )=1 
       ??? (50)=1  
       ??? (-50)=1 
 
 
2. If   ?? (?? )= 
{
 
 
 
 
   ??                    0<?? < 
1
2
1
2
                         ?? = 
1
2
1-??                  
1
2
<?? <1
 
       ?? (?? )= (?? - 
1
2
)
2
 then find the area bounded by ?? (?? ) and ?? (?? ) from ?? =  
1
2
  to ?? =  
v3
2
. 
a. 
v3
2
 
-
1
3
 
                                                                     b. 
v3
4
 
+
1
3
 
               
c. 2v3                                                                                        d. 3v3
                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 2  
 
       Answer: (?? ) 
       Solution: 
       Given ?? (?? )= 
{
 
 
 
 
   ??                    0<?? < 
1
2
1
2
                         ?? = 
1
2
1-??                  
1
2
<?? <1
 
       ?? (?? )= (?? - 
1
2
)
2
 
       The area between ?? (?? ) and ?? (?? ) from ?? =
1
2
  to =
v3
2
 : 
        
       Points of intersection of ?? (?? ) and (?? ) : 
 
1-?? = (?? - 
1
2
)
2
 
       ??? = 
v3
2
 ,-
v3
2
  
       Required area =?
(?? (?? )-?? (?? ))????
v3
2
1
2
 
                            =? (1-?? - (?? - 
1
2
)
2
)????
v3
2
1
2
 
                                    =?? - 
?? 2
2
-
1
3
(?? - 
1
2
)
3
|
1
2
v3
2
  
                                    = 
v3
4
- 
1
3
 
 
                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 3  
 
3. If ?? ?(?? ?~ ?? ) is false. Truth value of ?? and ?? will be 
a. TF  b. FT 
c. TT d. FF 
      Answer: (?? ) 
Solution:  
      Given ?? ?(?? ?~ ?? ) 
      Truth table: 
?? ?? ~?? (?? ?~ ?? ) ?? ?(?? ?~ ?? ) 
T T F F F 
T F T T T 
F T F F T 
F F T F T 
        ?? ?(?? ?~ ?? ) is false when ?? is true and ?? is true. 
 
4. ?
????
cos
2
?? 
 (sec2?? +tan2?? ) 
 =?? tan?? +2log?? (?? )+?? then ordered pair (?? , ?? (?? )) is
a. (1,1+tan?? ) b. (1,1-tan?? )
c. (-1,1+tan?? )                                                                   d.     (-1,1-tan?? ) 
 
Answer: (?? ) 
      Solution:  
       Let ?? = ?
????
cos
2
?? (sec2?? +tan2?? ) 
 
      ?? = ?
sec
2
?? ????
 (
1+tan
2
?? 1-tan
2
?? )+ (
2tan?? 1- tan
2
?? )
 
       ?? = ?
(1- tan
2
?? )(sec
2
?? )????
(1+tan?? )
2
 
       Let tan?? =??  ? sec
2
?? ???? =d?? 
       ?? = ?
(1- ?? 2
)
(1+?? )
2
 ???? =  ?
(1- ?? )
(1+?? )
 ???? 
       ?? = (
2
1+?? -1)???? 
       ?? =2ln|1+?? |-?? +?? 
       ?? =2ln|1+tan?? |-tan?? +?? 
                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 4  
 
       Given ?? =?? tan?? +2log?? (?? )+?? 
       ??? = -1 , ?? (?? )=|1+???????? | 
 
5. Let ?? ?? is a positive term of GP and ? ?? 2?? +1
=200
100
?? =1
 , ? ?? 2?? =100
100
?? =1
, then the value of ? ?? ?? 200
?? =1
 
      is
a. 150 b. 225 
c. 300 d. 175 
  Answer: (?? ) 
       Solution:  
       ?? ?? is a positive term of GP. 
       Let GP be ?? ,???? ,????
2
,..... 
       ? ?? 2?? +1
100
?? =1
 =?? 3
+ ?? 5
+ .......+?? 201
 
       200=????
2
+ ????
4
+ .......+ ????
201
 
       200= 
????
2
(?? 200
-1)
?? 2
-1
     . . . (1) 
       Also,       ? ?? 2?? 100
?? =1
 =100 
       100=?? 2
+ ?? 4
+  ...  .....+ ?? 200
 
       100=???? + ????
3
+  ........+????
199
 
       100= 
???? (?? 200
-1)
?? 2
-1
     . . . (2) 
       From (1) and (2), r = 2 
       And  ? ?? 2?? +1
100
?? =1
+ ? ?? 2?? 100
?? =1
 =300  
       ? ?? 2
+ ?? 3 
+ ?? 4
  ...  .....+ ?? 200
+?? 201
=300 
       ? ???? + ????
2
+ ????
3
+ .......+ ????
200
=300 
       ??? (?? +???? + ?? ?? 2
+ .....+ ????
199
)=300 
       ?2(?? 1
+ ?? 2
+ ?? 3
+  ......+ ?? 200
 )=300 
       ? ?? ?? 200
?? =1
 =150 
 
6. ?? is a complex number such that |Re(?? )|+|Im(?? )|=4, then |?? | cannot be equal to 
 
a. v8 b. v7 
c. v
17
2
 d. v10 
Page 5


                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 1  
 
Date: 9
th
 January 2020 (Shift 2) 
Time: 2:30 P.M. to 5:30 P.M. 
Subject: Mathematics 
 
1. If  ?? (?? )= |
?? +?? ?? +2 ?? +1
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| and ?? -2?? +?? =1 then 
a. ?? (-50)=-1                                                            b. ?? (50)= 1              
c. ?? (50)=-501                                                                                        d. ?? (50)= 501 
       Answer: (?? ) 
       Solution:  
       Given ?? (?? )= |
?? +?? ?? +2 ?? +1
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| 
       ?? -2?? +?? =1 
       Applying ?? 1
??? 1
-2?? 2
+?? 3
 
       ?? (?? )= |
?? -2?? +?? 0 0
?? +?? ?? +3 ?? +2
?? +?? ?? +4 ?? +3
| 
       Using ?? -2?? +?? =1 
       ??? (?? )= (?? +3)
2
-(?? +2)(?? +4) 
       ??? (?? )=1 
       ??? (50)=1  
       ??? (-50)=1 
 
 
2. If   ?? (?? )= 
{
 
 
 
 
   ??                    0<?? < 
1
2
1
2
                         ?? = 
1
2
1-??                  
1
2
<?? <1
 
       ?? (?? )= (?? - 
1
2
)
2
 then find the area bounded by ?? (?? ) and ?? (?? ) from ?? =  
1
2
  to ?? =  
v3
2
. 
a. 
v3
2
 
-
1
3
 
                                                                     b. 
v3
4
 
+
1
3
 
               
c. 2v3                                                                                        d. 3v3
                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 2  
 
       Answer: (?? ) 
       Solution: 
       Given ?? (?? )= 
{
 
 
 
 
   ??                    0<?? < 
1
2
1
2
                         ?? = 
1
2
1-??                  
1
2
<?? <1
 
       ?? (?? )= (?? - 
1
2
)
2
 
       The area between ?? (?? ) and ?? (?? ) from ?? =
1
2
  to =
v3
2
 : 
        
       Points of intersection of ?? (?? ) and (?? ) : 
 
1-?? = (?? - 
1
2
)
2
 
       ??? = 
v3
2
 ,-
v3
2
  
       Required area =?
(?? (?? )-?? (?? ))????
v3
2
1
2
 
                            =? (1-?? - (?? - 
1
2
)
2
)????
v3
2
1
2
 
                                    =?? - 
?? 2
2
-
1
3
(?? - 
1
2
)
3
|
1
2
v3
2
  
                                    = 
v3
4
- 
1
3
 
 
                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 3  
 
3. If ?? ?(?? ?~ ?? ) is false. Truth value of ?? and ?? will be 
a. TF  b. FT 
c. TT d. FF 
      Answer: (?? ) 
Solution:  
      Given ?? ?(?? ?~ ?? ) 
      Truth table: 
?? ?? ~?? (?? ?~ ?? ) ?? ?(?? ?~ ?? ) 
T T F F F 
T F T T T 
F T F F T 
F F T F T 
        ?? ?(?? ?~ ?? ) is false when ?? is true and ?? is true. 
 
4. ?
????
cos
2
?? 
 (sec2?? +tan2?? ) 
 =?? tan?? +2log?? (?? )+?? then ordered pair (?? , ?? (?? )) is
a. (1,1+tan?? ) b. (1,1-tan?? )
c. (-1,1+tan?? )                                                                   d.     (-1,1-tan?? ) 
 
Answer: (?? ) 
      Solution:  
       Let ?? = ?
????
cos
2
?? (sec2?? +tan2?? ) 
 
      ?? = ?
sec
2
?? ????
 (
1+tan
2
?? 1-tan
2
?? )+ (
2tan?? 1- tan
2
?? )
 
       ?? = ?
(1- tan
2
?? )(sec
2
?? )????
(1+tan?? )
2
 
       Let tan?? =??  ? sec
2
?? ???? =d?? 
       ?? = ?
(1- ?? 2
)
(1+?? )
2
 ???? =  ?
(1- ?? )
(1+?? )
 ???? 
       ?? = (
2
1+?? -1)???? 
       ?? =2ln|1+?? |-?? +?? 
       ?? =2ln|1+tan?? |-tan?? +?? 
                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 4  
 
       Given ?? =?? tan?? +2log?? (?? )+?? 
       ??? = -1 , ?? (?? )=|1+???????? | 
 
5. Let ?? ?? is a positive term of GP and ? ?? 2?? +1
=200
100
?? =1
 , ? ?? 2?? =100
100
?? =1
, then the value of ? ?? ?? 200
?? =1
 
      is
a. 150 b. 225 
c. 300 d. 175 
  Answer: (?? ) 
       Solution:  
       ?? ?? is a positive term of GP. 
       Let GP be ?? ,???? ,????
2
,..... 
       ? ?? 2?? +1
100
?? =1
 =?? 3
+ ?? 5
+ .......+?? 201
 
       200=????
2
+ ????
4
+ .......+ ????
201
 
       200= 
????
2
(?? 200
-1)
?? 2
-1
     . . . (1) 
       Also,       ? ?? 2?? 100
?? =1
 =100 
       100=?? 2
+ ?? 4
+  ...  .....+ ?? 200
 
       100=???? + ????
3
+  ........+????
199
 
       100= 
???? (?? 200
-1)
?? 2
-1
     . . . (2) 
       From (1) and (2), r = 2 
       And  ? ?? 2?? +1
100
?? =1
+ ? ?? 2?? 100
?? =1
 =300  
       ? ?? 2
+ ?? 3 
+ ?? 4
  ...  .....+ ?? 200
+?? 201
=300 
       ? ???? + ????
2
+ ????
3
+ .......+ ????
200
=300 
       ??? (?? +???? + ?? ?? 2
+ .....+ ????
199
)=300 
       ?2(?? 1
+ ?? 2
+ ?? 3
+  ......+ ?? 200
 )=300 
       ? ?? ?? 200
?? =1
 =150 
 
6. ?? is a complex number such that |Re(?? )|+|Im(?? )|=4, then |?? | cannot be equal to 
 
a. v8 b. v7 
c. v
17
2
 d. v10 
                                           9
th
 January 2020 (Shift 2), Mathematics                                                        Page | 5  
 
Answer: (?? ) 
Solution:  
       |Re(?? )|+|Im(?? )|=4 
       Let ?? =?? +???? 
       ?|?? |+|?? |=4 
     
       ??? lies on the rhombus.  
       Maximum value of |?? |=4 when ?? =4,-4,4?? ,-4?? 
       Minimum value of |?? |=2v2 when ?? =2±2??, ±2+2?? 
       |?? |?[2v2 ,4] 
       |?? |?[v8 ,v16] 
       |?? |?v7 
 
7. ?? (?? ):[0,5]??? ,?? (?? )=? ?? 2
 ?? (?? )???? ,?? (1)=3,
?? 0
?? (?? )=? ?? (?? )????
?? 1
 then correct choice is 
a. ?? (?? ) has no critical point 
b. ?? (?? ) has local minimum at ?? =1 
c. ?? (?? ) has local maximum at ?? =1 
d. ?? (?? ) has point of inflection at ?? =1 
Answer: (b) 
Solution: 
?? (?? )=?? 2
?? (?? )  
Put ?? =1 
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FAQs on JEE Main 2020 January 9 Question Paper Shift 2 - JEE Main & Advanced Previous Year Papers

1. क्या JEE Main 2020 January 9 Question Paper Shift 2 में किसी खास विषय पर प्रश्न पूछे गए थे?
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Ans. हां, JEE Main 2020 January 9 Question Paper Shift 2 में गणित विषय पर अधिक प्रश्न पूछे गए थे।
4. JEE Main 2020 January 9 Question Paper Shift 2 की तैयारी के लिए सर्वश्रेष्ठ संदेश क्या होगा?
Ans. JEE Main 2020 January 9 Question Paper Shift 2 की तैयारी के लिए प्रैक्टिस सेट्स और मॉक टेस्ट करना बेहतर हो सकता है।
5. JEE Main 2020 January 9 Question Paper Shift 2 के नतीजे कब जारी होंगे?
Ans. JEE Main 2020 January 9 Question Paper Shift 2 के नतीजे जल्द ही ऑफिशियल वेबसाइट पर जारी किए जाएंगे।
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