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Page 1
7
th
January 2020 (Shift 1), Mathematics Page | 1
Date: 7
th
January 2020 (Shift 1)
Time: 9:30 A.M. to 12:30 P.M.
Subject: Mathematics
1. The area of the region enclosed by the circle ?? 2
+ ?? 2
= 2 which is not common to the region
bounded by the parabola ?? 2
= ?? and the straight line ?? = ?? , is
a.
1
12
( 24?? - 1) b.
1
6
( 12?? - 1)
c.
1
12
( 6?? - 1) d.
1
12
( 12?? - 1)
Answer: ( b)
Solution:
Required area = area of the circle – area bounded by given line and parabola
Required area = ?? ?? 2
- ? ( ?? - ?? 2
) ????
1
0
Area = 2?? - (
?? 2
2
-
?? 3
3
)
0
1
= 2?? -
1
6
sq. units
2. If ?? ( ?? ) = ?? 2
+ ?? - 1 and ( ?????? ) ( ?? ) = 4?? 2
- 10?? + 5, then ?? (
5
4
) is
a.
1
2
b. -
1
2
c. -
1
3
d.
1
3
Answer: ( b)
Solution:
?? ( ?? ) = ?? 2
+ ?? - 1
Page 2
7
th
January 2020 (Shift 1), Mathematics Page | 1
Date: 7
th
January 2020 (Shift 1)
Time: 9:30 A.M. to 12:30 P.M.
Subject: Mathematics
1. The area of the region enclosed by the circle ?? 2
+ ?? 2
= 2 which is not common to the region
bounded by the parabola ?? 2
= ?? and the straight line ?? = ?? , is
a.
1
12
( 24?? - 1) b.
1
6
( 12?? - 1)
c.
1
12
( 6?? - 1) d.
1
12
( 12?? - 1)
Answer: ( b)
Solution:
Required area = area of the circle – area bounded by given line and parabola
Required area = ?? ?? 2
- ? ( ?? - ?? 2
) ????
1
0
Area = 2?? - (
?? 2
2
-
?? 3
3
)
0
1
= 2?? -
1
6
sq. units
2. If ?? ( ?? ) = ?? 2
+ ?? - 1 and ( ?????? ) ( ?? ) = 4?? 2
- 10?? + 5, then ?? (
5
4
) is
a.
1
2
b. -
1
2
c. -
1
3
d.
1
3
Answer: ( b)
Solution:
?? ( ?? ) = ?? 2
+ ?? - 1
7
th
January 2020 (Shift 1), Mathematics Page | 2
?????? ( ?? ) = 4?? 2
- 10?? + 5
?? ( ?? ( ?? ) ) = 4?? 2
- 10?? + 5
?? 2
( ?? )+ ?? ( ?? )- 1 = 4?? 2
- 10?? + 5
Putting ?? =
5
4
& ?? (
5
4
) = ??
?? 2
+ ?? +
1
4
= 0
?? = -
1
2
?? (
5
4
) = -
1
2
3. If ?? = ?? ( ?? ) is the solution of the differential equation ?? ?? (
????
????
- 1) = ?? ?? such that ?? ( 0) = 0, then
?? ( 1) is equal to
a. ln2 b. 2 + ln2
c. 1 + ln2 d. 3 + ln2
Answer: ( c)
Solution:
?? ?? ( ?? '
- 1) = ?? ??
?
????
????
= ?? ?? -?? + 1
Let ?? - ?? = ??
1 -
????
????
=
????
????
So, we can write
? 1 -
????
????
= ?? ?? + 1
? -?? -?? ???? = ????
? ?? -?? = ?? + ??
? ?? ?? -?? = ?? + ??
1 = 0 + ??
? ?? ?? -?? = ?? + 1
at ?? = 1
? ?? ?? -1
= 2
? ?? = 1 + ln2
4. If ?? = ???? + 4 is a tangent to both the parabolas, ?? 2
= 4?? and ?? 2
= 2???? , then ?? is equal to
a. -64 b. -32
c. -128 d. 16
Answer: ( c)
Page 3
7
th
January 2020 (Shift 1), Mathematics Page | 1
Date: 7
th
January 2020 (Shift 1)
Time: 9:30 A.M. to 12:30 P.M.
Subject: Mathematics
1. The area of the region enclosed by the circle ?? 2
+ ?? 2
= 2 which is not common to the region
bounded by the parabola ?? 2
= ?? and the straight line ?? = ?? , is
a.
1
12
( 24?? - 1) b.
1
6
( 12?? - 1)
c.
1
12
( 6?? - 1) d.
1
12
( 12?? - 1)
Answer: ( b)
Solution:
Required area = area of the circle – area bounded by given line and parabola
Required area = ?? ?? 2
- ? ( ?? - ?? 2
) ????
1
0
Area = 2?? - (
?? 2
2
-
?? 3
3
)
0
1
= 2?? -
1
6
sq. units
2. If ?? ( ?? ) = ?? 2
+ ?? - 1 and ( ?????? ) ( ?? ) = 4?? 2
- 10?? + 5, then ?? (
5
4
) is
a.
1
2
b. -
1
2
c. -
1
3
d.
1
3
Answer: ( b)
Solution:
?? ( ?? ) = ?? 2
+ ?? - 1
7
th
January 2020 (Shift 1), Mathematics Page | 2
?????? ( ?? ) = 4?? 2
- 10?? + 5
?? ( ?? ( ?? ) ) = 4?? 2
- 10?? + 5
?? 2
( ?? )+ ?? ( ?? )- 1 = 4?? 2
- 10?? + 5
Putting ?? =
5
4
& ?? (
5
4
) = ??
?? 2
+ ?? +
1
4
= 0
?? = -
1
2
?? (
5
4
) = -
1
2
3. If ?? = ?? ( ?? ) is the solution of the differential equation ?? ?? (
????
????
- 1) = ?? ?? such that ?? ( 0) = 0, then
?? ( 1) is equal to
a. ln2 b. 2 + ln2
c. 1 + ln2 d. 3 + ln2
Answer: ( c)
Solution:
?? ?? ( ?? '
- 1) = ?? ??
?
????
????
= ?? ?? -?? + 1
Let ?? - ?? = ??
1 -
????
????
=
????
????
So, we can write
? 1 -
????
????
= ?? ?? + 1
? -?? -?? ???? = ????
? ?? -?? = ?? + ??
? ?? ?? -?? = ?? + ??
1 = 0 + ??
? ?? ?? -?? = ?? + 1
at ?? = 1
? ?? ?? -1
= 2
? ?? = 1 + ln2
4. If ?? = ???? + 4 is a tangent to both the parabolas, ?? 2
= 4?? and ?? 2
= 2???? , then ?? is equal to
a. -64 b. -32
c. -128 d. 16
Answer: ( c)
7
th
January 2020 (Shift 1), Mathematics Page | 3
Solution:
Any tangent to the parabola ?? 2
= 4?? is ?? = ???? +
?? ??
Comparing it with ?? = ???? + 4, we get
1
?? = 4 ? ?? =
1
4
Equation of tangent becomes ?? =
?? 4
+ 4
?? =
?? 4
+ 4 is a tangent to ?? 2
= 2????
? ?? 2
= 2?? (
?? 4
+ 4)
Or 2?? 2
- ???? - 16?? = 0,
?? = 0
?? 2
+ 128?? = 0,
? ?? = 0 (not possible),
?? = -128
5. If ?? and ?? are two real roots of the equation ( ?? + 1)tan
2
?? - v 2 ?? tan?? = 1 - ?? , where ( ?? ? 1)
and ?? are real numbers. If tan
2
( ?? + ?? ) = 50, then value of ?? is
a. 10 b. 5
c. 7 d. 12
Answer: ( a)
Solution:
( ?? + 1)tan
2
?? - v 2?? tan?? = 1 - ??
tan
2
( ?? + ?? )= 50
Now,
tan?? + tan?? =
v 2?? ?? + 1
, tan?? tan?? =
?? - 1
?? + 1
? (
v 2?? ?? + 1
1 -
?? - 1
?? + 1
)
2
= 50
?
2?? 2
4
= 50
? ?? 2
= 100
? ?? = 10
6. If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then
the length of its latus rectum is
a. 4
b. 3v 2
c. 9
d. 2v 2
Page 4
7
th
January 2020 (Shift 1), Mathematics Page | 1
Date: 7
th
January 2020 (Shift 1)
Time: 9:30 A.M. to 12:30 P.M.
Subject: Mathematics
1. The area of the region enclosed by the circle ?? 2
+ ?? 2
= 2 which is not common to the region
bounded by the parabola ?? 2
= ?? and the straight line ?? = ?? , is
a.
1
12
( 24?? - 1) b.
1
6
( 12?? - 1)
c.
1
12
( 6?? - 1) d.
1
12
( 12?? - 1)
Answer: ( b)
Solution:
Required area = area of the circle – area bounded by given line and parabola
Required area = ?? ?? 2
- ? ( ?? - ?? 2
) ????
1
0
Area = 2?? - (
?? 2
2
-
?? 3
3
)
0
1
= 2?? -
1
6
sq. units
2. If ?? ( ?? ) = ?? 2
+ ?? - 1 and ( ?????? ) ( ?? ) = 4?? 2
- 10?? + 5, then ?? (
5
4
) is
a.
1
2
b. -
1
2
c. -
1
3
d.
1
3
Answer: ( b)
Solution:
?? ( ?? ) = ?? 2
+ ?? - 1
7
th
January 2020 (Shift 1), Mathematics Page | 2
?????? ( ?? ) = 4?? 2
- 10?? + 5
?? ( ?? ( ?? ) ) = 4?? 2
- 10?? + 5
?? 2
( ?? )+ ?? ( ?? )- 1 = 4?? 2
- 10?? + 5
Putting ?? =
5
4
& ?? (
5
4
) = ??
?? 2
+ ?? +
1
4
= 0
?? = -
1
2
?? (
5
4
) = -
1
2
3. If ?? = ?? ( ?? ) is the solution of the differential equation ?? ?? (
????
????
- 1) = ?? ?? such that ?? ( 0) = 0, then
?? ( 1) is equal to
a. ln2 b. 2 + ln2
c. 1 + ln2 d. 3 + ln2
Answer: ( c)
Solution:
?? ?? ( ?? '
- 1) = ?? ??
?
????
????
= ?? ?? -?? + 1
Let ?? - ?? = ??
1 -
????
????
=
????
????
So, we can write
? 1 -
????
????
= ?? ?? + 1
? -?? -?? ???? = ????
? ?? -?? = ?? + ??
? ?? ?? -?? = ?? + ??
1 = 0 + ??
? ?? ?? -?? = ?? + 1
at ?? = 1
? ?? ?? -1
= 2
? ?? = 1 + ln2
4. If ?? = ???? + 4 is a tangent to both the parabolas, ?? 2
= 4?? and ?? 2
= 2???? , then ?? is equal to
a. -64 b. -32
c. -128 d. 16
Answer: ( c)
7
th
January 2020 (Shift 1), Mathematics Page | 3
Solution:
Any tangent to the parabola ?? 2
= 4?? is ?? = ???? +
?? ??
Comparing it with ?? = ???? + 4, we get
1
?? = 4 ? ?? =
1
4
Equation of tangent becomes ?? =
?? 4
+ 4
?? =
?? 4
+ 4 is a tangent to ?? 2
= 2????
? ?? 2
= 2?? (
?? 4
+ 4)
Or 2?? 2
- ???? - 16?? = 0,
?? = 0
?? 2
+ 128?? = 0,
? ?? = 0 (not possible),
?? = -128
5. If ?? and ?? are two real roots of the equation ( ?? + 1)tan
2
?? - v 2 ?? tan?? = 1 - ?? , where ( ?? ? 1)
and ?? are real numbers. If tan
2
( ?? + ?? ) = 50, then value of ?? is
a. 10 b. 5
c. 7 d. 12
Answer: ( a)
Solution:
( ?? + 1)tan
2
?? - v 2?? tan?? = 1 - ??
tan
2
( ?? + ?? )= 50
Now,
tan?? + tan?? =
v 2?? ?? + 1
, tan?? tan?? =
?? - 1
?? + 1
? (
v 2?? ?? + 1
1 -
?? - 1
?? + 1
)
2
= 50
?
2?? 2
4
= 50
? ?? 2
= 100
? ?? = 10
6. If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then
the length of its latus rectum is
a. 4
b. 3v 2
c. 9
d. 2v 2
7
th
January 2020 (Shift 1), Mathematics Page | 4
Answer: ( b)
Solution:
Let the equation of ellipse be
?? 2
?? 2
+
?? 2
?? 2
= 1 ( ?? > ?? )
Now 2???? = 6 &
2?? ?? = 12
? ???? = 3 &
?? ?? = 6
? ?? 2
= 18
? ?? 2
?? 2
= ?? 2
= ?? 2
- ?? 2
= 9
? ?? 2
= 9
Length of latus rectum =
2?? 2
?? =
2×9
v 18
= 3v 2
7. The logical statement ( ?? ? ?? )? ( ?? ?~ ?? ) is equivalent to
a. ~ ?? b. ?? c. ?? ? ?? d. ?? ? ??
Answer: ( a)
Solution:
?? ?? ?? ? ?? ~ ?? ?? ?~ ?? ( ?? ? ?? )? ( ?? ?~ ?? )
T T T F F F
T F F F T F
F T T T T T
F F T T T T
Clearly ( ?? ? ?? )? ( ?? ?~ ?? ) is equivalent to ~ ??
8. If the system of equations
2?? + 2???? + ???? = 0
2?? + 3???? + ???? = 0
2?? + 4???? + ???? = 0,
where ?? , ?? , ?? ? ?? are non-zero and distinct, has non zero solution then
a. ?? + ?? + ?? = 0 b. ?? , ?? , ?? are in A.P
c.
1
?? ,
1
?? ,
1
?? are in A.P
d. ?? , ?? , ?? are in G.P
Answer: ( c)
Solution:
Page 5
7
th
January 2020 (Shift 1), Mathematics Page | 1
Date: 7
th
January 2020 (Shift 1)
Time: 9:30 A.M. to 12:30 P.M.
Subject: Mathematics
1. The area of the region enclosed by the circle ?? 2
+ ?? 2
= 2 which is not common to the region
bounded by the parabola ?? 2
= ?? and the straight line ?? = ?? , is
a.
1
12
( 24?? - 1) b.
1
6
( 12?? - 1)
c.
1
12
( 6?? - 1) d.
1
12
( 12?? - 1)
Answer: ( b)
Solution:
Required area = area of the circle – area bounded by given line and parabola
Required area = ?? ?? 2
- ? ( ?? - ?? 2
) ????
1
0
Area = 2?? - (
?? 2
2
-
?? 3
3
)
0
1
= 2?? -
1
6
sq. units
2. If ?? ( ?? ) = ?? 2
+ ?? - 1 and ( ?????? ) ( ?? ) = 4?? 2
- 10?? + 5, then ?? (
5
4
) is
a.
1
2
b. -
1
2
c. -
1
3
d.
1
3
Answer: ( b)
Solution:
?? ( ?? ) = ?? 2
+ ?? - 1
7
th
January 2020 (Shift 1), Mathematics Page | 2
?????? ( ?? ) = 4?? 2
- 10?? + 5
?? ( ?? ( ?? ) ) = 4?? 2
- 10?? + 5
?? 2
( ?? )+ ?? ( ?? )- 1 = 4?? 2
- 10?? + 5
Putting ?? =
5
4
& ?? (
5
4
) = ??
?? 2
+ ?? +
1
4
= 0
?? = -
1
2
?? (
5
4
) = -
1
2
3. If ?? = ?? ( ?? ) is the solution of the differential equation ?? ?? (
????
????
- 1) = ?? ?? such that ?? ( 0) = 0, then
?? ( 1) is equal to
a. ln2 b. 2 + ln2
c. 1 + ln2 d. 3 + ln2
Answer: ( c)
Solution:
?? ?? ( ?? '
- 1) = ?? ??
?
????
????
= ?? ?? -?? + 1
Let ?? - ?? = ??
1 -
????
????
=
????
????
So, we can write
? 1 -
????
????
= ?? ?? + 1
? -?? -?? ???? = ????
? ?? -?? = ?? + ??
? ?? ?? -?? = ?? + ??
1 = 0 + ??
? ?? ?? -?? = ?? + 1
at ?? = 1
? ?? ?? -1
= 2
? ?? = 1 + ln2
4. If ?? = ???? + 4 is a tangent to both the parabolas, ?? 2
= 4?? and ?? 2
= 2???? , then ?? is equal to
a. -64 b. -32
c. -128 d. 16
Answer: ( c)
7
th
January 2020 (Shift 1), Mathematics Page | 3
Solution:
Any tangent to the parabola ?? 2
= 4?? is ?? = ???? +
?? ??
Comparing it with ?? = ???? + 4, we get
1
?? = 4 ? ?? =
1
4
Equation of tangent becomes ?? =
?? 4
+ 4
?? =
?? 4
+ 4 is a tangent to ?? 2
= 2????
? ?? 2
= 2?? (
?? 4
+ 4)
Or 2?? 2
- ???? - 16?? = 0,
?? = 0
?? 2
+ 128?? = 0,
? ?? = 0 (not possible),
?? = -128
5. If ?? and ?? are two real roots of the equation ( ?? + 1)tan
2
?? - v 2 ?? tan?? = 1 - ?? , where ( ?? ? 1)
and ?? are real numbers. If tan
2
( ?? + ?? ) = 50, then value of ?? is
a. 10 b. 5
c. 7 d. 12
Answer: ( a)
Solution:
( ?? + 1)tan
2
?? - v 2?? tan?? = 1 - ??
tan
2
( ?? + ?? )= 50
Now,
tan?? + tan?? =
v 2?? ?? + 1
, tan?? tan?? =
?? - 1
?? + 1
? (
v 2?? ?? + 1
1 -
?? - 1
?? + 1
)
2
= 50
?
2?? 2
4
= 50
? ?? 2
= 100
? ?? = 10
6. If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then
the length of its latus rectum is
a. 4
b. 3v 2
c. 9
d. 2v 2
7
th
January 2020 (Shift 1), Mathematics Page | 4
Answer: ( b)
Solution:
Let the equation of ellipse be
?? 2
?? 2
+
?? 2
?? 2
= 1 ( ?? > ?? )
Now 2???? = 6 &
2?? ?? = 12
? ???? = 3 &
?? ?? = 6
? ?? 2
= 18
? ?? 2
?? 2
= ?? 2
= ?? 2
- ?? 2
= 9
? ?? 2
= 9
Length of latus rectum =
2?? 2
?? =
2×9
v 18
= 3v 2
7. The logical statement ( ?? ? ?? )? ( ?? ?~ ?? ) is equivalent to
a. ~ ?? b. ?? c. ?? ? ?? d. ?? ? ??
Answer: ( a)
Solution:
?? ?? ?? ? ?? ~ ?? ?? ?~ ?? ( ?? ? ?? )? ( ?? ?~ ?? )
T T T F F F
T F F F T F
F T T T T T
F F T T T T
Clearly ( ?? ? ?? )? ( ?? ?~ ?? ) is equivalent to ~ ??
8. If the system of equations
2?? + 2???? + ???? = 0
2?? + 3???? + ???? = 0
2?? + 4???? + ???? = 0,
where ?? , ?? , ?? ? ?? are non-zero and distinct, has non zero solution then
a. ?? + ?? + ?? = 0 b. ?? , ?? , ?? are in A.P
c.
1
?? ,
1
?? ,
1
?? are in A.P
d. ?? , ?? , ?? are in G.P
Answer: ( c)
Solution:
7
th
January 2020 (Shift 1), Mathematics Page | 5
|
2 2?? ?? 2 3?? ?? 2 4?? ?? | = 0
?? 2
? ?? 2
- ?? 1
, ?? 3
? ?? 3
- ?? 1
? |
2 2?? ?? 0 3?? - 2?? ?? - ?? 0 4?? - 2?? ?? - ?? | = 0
? ( 3?? - 2?? ) ( ?? - ?? )- ( 4?? - 2?? ) ( ?? - ?? ) = 0
? 3???? - 2???? - 3???? + 2?? 2
- [4???? - 4???? - 2???? + 2?? 2
] = 0
? -???? + 2???? - ???? = 0
? ???? + ???? = 2????
?
1
?? +
1
?? =
2
??
9. 5 numbers are in A.P whose sum is 25 and product is 2520. If one of these 5 numbers is -
1
2
, then the
greatest number amongst them is
a.
21
2
b. 16
c. 27 d. 7
Answer: ( b)
Solution:
Let 5 numbers be ?? - 2?? , ?? - ?? , ?? , ?? + ?? , ?? + 2??
5?? = 25
?? = 5
( ?? - 2?? ) ( ?? - ?? ) ?? ( ?? + ?? ) ( ?? + 2?? ) = 2520
( 25 - 4?? 2
) ( 25 - ?? 2
) = 504
4?? 4
- 125?? 2
+ 121 = 0
4?? 4
- 4?? 2
- 121 ?? 2
+ 121 = 0
?? 2
= 1 ???? ?? 2
=
121
4
?? = ±
11
2
For ?? =
11
2
, ?? + 2?? is the greatest term, ?? + 2?? = 5 + 11 = 16
10. If ?? is a root of the equation ?? 2
+ ?? + 1 = 0 and ?? =
1
v 3
[
1 1 1
1 ?? ?? 2
1 ?? 2
?? ] then ?? 31
equal to
a. ?? b. ?? 2
c. ?? 3
d. ?? 4
Answer: ( ?? )
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Dec 23, 2024 Last updated |
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JEE Main 2020 January 7 Question Paper Shift 1 Notes
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JEE Main 2020 January 7 Question Paper Shift 1 JEE Questions
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The content of JEE Main 2020 January 7 Question Paper Shift 1 is prepared as per the latest JEE syllabus.
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