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Page 1
7
th
January 2020 (Shift 2), Mathematics Page | 1
Date: 7
th
January 2020 (Shift 2)
Time: 2:30 P.M. to 5:30 P.M.
Subject: Mathematics
1. From any point ?? on the line ?? = 2?? , a perpendicular is drawn on ?? = ?? . Let the foot of
perpendicular be ?? . Find the locus of mid point of ???? .
a. 5?? = 7?? b. 2?? = 3?? c. 7?? = 5?? d. 3?? = 2??
Answer: ( a)
Solution:
Let ?? be the midpoint of ????
???? is perpendicular on line ?? = ??
? Equation of the line ???? can be written as ?? = -?? + ??
?? = -?? + ?? intersects ?? = ?? at ?? : (
?? 2
,
?? 2
)
?? = -?? + ?? intersects ?? = 2?? at ?? : (
2?? 3
,
?? 3
)
? Midpoint ?? : (
7?? 12
,
5?? 12
)
Locus of ?? : ?? =
7?? 12
Page 2
7
th
January 2020 (Shift 2), Mathematics Page | 1
Date: 7
th
January 2020 (Shift 2)
Time: 2:30 P.M. to 5:30 P.M.
Subject: Mathematics
1. From any point ?? on the line ?? = 2?? , a perpendicular is drawn on ?? = ?? . Let the foot of
perpendicular be ?? . Find the locus of mid point of ???? .
a. 5?? = 7?? b. 2?? = 3?? c. 7?? = 5?? d. 3?? = 2??
Answer: ( a)
Solution:
Let ?? be the midpoint of ????
???? is perpendicular on line ?? = ??
? Equation of the line ???? can be written as ?? = -?? + ??
?? = -?? + ?? intersects ?? = ?? at ?? : (
?? 2
,
?? 2
)
?? = -?? + ?? intersects ?? = 2?? at ?? : (
2?? 3
,
?? 3
)
? Midpoint ?? : (
7?? 12
,
5?? 12
)
Locus of ?? : ?? =
7?? 12
7
th
January 2020 (Shift 2), Mathematics Page | 2
?? =
5?? 12
? 5?? = 7??
2. Let ?? 1
and ?? 2
(where ?? 1
< ?? 2
) are two solutions of 2 cot
2
?? -
5
sin ?? +4 = 0, ?? ? [0,2?? ) then
? cos
2
3?? ????
?? 2
?? 1
is equal to
a.
?? 9
b.
2?? 3
c.
?? 3
+
1
6
d.
?? 3
Answer: ( d)
Solution:
2 cot
2
?? -
5
sin ?? + 4 = 0, ?? ? [0,2?? )
? 2cosec
2
?? - 2 - 5cosec ?? + 4 = 0
? 2cosec
2
?? - 4cosec ?? - cosec ?? + 2 = 0
? cosec ?? = 2 or
1
2
(Not possible)
As ?? ? [0,2?? ) ,
?? 1
=
?? 6
, ?? 2
=
5?? 6
? ? cos
2
3?????? = ?
( 1 + cos6?? )
2
5?? 6
?? 6
????
?? 2
?? 1
=
1
2
(
5?? 6
-
?? 6
) +
sin6?
12
|
?? 6
5?? 6
=
?? 3
Page 3
7
th
January 2020 (Shift 2), Mathematics Page | 1
Date: 7
th
January 2020 (Shift 2)
Time: 2:30 P.M. to 5:30 P.M.
Subject: Mathematics
1. From any point ?? on the line ?? = 2?? , a perpendicular is drawn on ?? = ?? . Let the foot of
perpendicular be ?? . Find the locus of mid point of ???? .
a. 5?? = 7?? b. 2?? = 3?? c. 7?? = 5?? d. 3?? = 2??
Answer: ( a)
Solution:
Let ?? be the midpoint of ????
???? is perpendicular on line ?? = ??
? Equation of the line ???? can be written as ?? = -?? + ??
?? = -?? + ?? intersects ?? = ?? at ?? : (
?? 2
,
?? 2
)
?? = -?? + ?? intersects ?? = 2?? at ?? : (
2?? 3
,
?? 3
)
? Midpoint ?? : (
7?? 12
,
5?? 12
)
Locus of ?? : ?? =
7?? 12
7
th
January 2020 (Shift 2), Mathematics Page | 2
?? =
5?? 12
? 5?? = 7??
2. Let ?? 1
and ?? 2
(where ?? 1
< ?? 2
) are two solutions of 2 cot
2
?? -
5
sin ?? +4 = 0, ?? ? [0,2?? ) then
? cos
2
3?? ????
?? 2
?? 1
is equal to
a.
?? 9
b.
2?? 3
c.
?? 3
+
1
6
d.
?? 3
Answer: ( d)
Solution:
2 cot
2
?? -
5
sin ?? + 4 = 0, ?? ? [0,2?? )
? 2cosec
2
?? - 2 - 5cosec ?? + 4 = 0
? 2cosec
2
?? - 4cosec ?? - cosec ?? + 2 = 0
? cosec ?? = 2 or
1
2
(Not possible)
As ?? ? [0,2?? ) ,
?? 1
=
?? 6
, ?? 2
=
5?? 6
? ? cos
2
3?????? = ?
( 1 + cos6?? )
2
5?? 6
?? 6
????
?? 2
?? 1
=
1
2
(
5?? 6
-
?? 6
) +
sin6?
12
|
?? 6
5?? 6
=
?? 3
7
th
January 2020 (Shift 2), Mathematics Page | 3
3. Coefficient of ?? 7
in ( 1 + ?? )
10
+ ?? ( 1 + ?? )
9
+ ?? 2
( 1 + ?? )
8
+ ? +?? 10
is
a. 260 b. 210
c. 420 d. 330
Answer: ( d)
Solution:
Coefficient of ?? 7
in ( 1 + ?? )
10
+ ?? ( 1 + ?? )
9
+ ?? 2
( 1 + ?? )
8
+ ? +?? 10
Applying sum of terms of G.P. =
( 1+?? )
10
(1- (
?? 1+?? )
11
)
(1-
?? 1+?? )
= ( 1 + ?? )
11
- ?? 11
Coefficient of ?? 7
?
11
?? 7
= 330
4. Let ?? and ?? are the roots of ?? 2
- ?? - 1 = 0 such that ?? ?? = ?? ?? + ?? ?? , ?? = 1 then which one is
incorrect?
a. ?? 5
= ?? 2
× ?? 3
b. ?? 1
+ ?? 2
+ ?? 3
+ ?? 4
+ ?? 5
= 26
c. ?? 5
= 11 d. ?? 3
= ?? 5
- ?? 4
Answer: ( a)
Solution:
Given ?? , ?? are the roots of ?? 2
- ?? - 1 = 0
? ?? + ?? = 1 & ???? = -1
? ?? 2
= ?? + 1 & ?? 2
= ?? + 1
?? ?? = ?? ?? -2
?? 2
+ ?? ?? -2
?? 2
?? ?? = ?? ?? -2
( ?? + 1)+ ?? ?? -2
( ?? + 1)
?? ?? = ?? ?? -1
+ ?? ?? -1
+ ?? ?? -2
+ ?? ?? -2
? ?? ?? = ?? ?? -1
+ ?? ?? -2
? ?? 3
= ?? 2
+ ?? 1
= 4
?? 4
= ?? 3
+ ?? 2
= 7
Page 4
7
th
January 2020 (Shift 2), Mathematics Page | 1
Date: 7
th
January 2020 (Shift 2)
Time: 2:30 P.M. to 5:30 P.M.
Subject: Mathematics
1. From any point ?? on the line ?? = 2?? , a perpendicular is drawn on ?? = ?? . Let the foot of
perpendicular be ?? . Find the locus of mid point of ???? .
a. 5?? = 7?? b. 2?? = 3?? c. 7?? = 5?? d. 3?? = 2??
Answer: ( a)
Solution:
Let ?? be the midpoint of ????
???? is perpendicular on line ?? = ??
? Equation of the line ???? can be written as ?? = -?? + ??
?? = -?? + ?? intersects ?? = ?? at ?? : (
?? 2
,
?? 2
)
?? = -?? + ?? intersects ?? = 2?? at ?? : (
2?? 3
,
?? 3
)
? Midpoint ?? : (
7?? 12
,
5?? 12
)
Locus of ?? : ?? =
7?? 12
7
th
January 2020 (Shift 2), Mathematics Page | 2
?? =
5?? 12
? 5?? = 7??
2. Let ?? 1
and ?? 2
(where ?? 1
< ?? 2
) are two solutions of 2 cot
2
?? -
5
sin ?? +4 = 0, ?? ? [0,2?? ) then
? cos
2
3?? ????
?? 2
?? 1
is equal to
a.
?? 9
b.
2?? 3
c.
?? 3
+
1
6
d.
?? 3
Answer: ( d)
Solution:
2 cot
2
?? -
5
sin ?? + 4 = 0, ?? ? [0,2?? )
? 2cosec
2
?? - 2 - 5cosec ?? + 4 = 0
? 2cosec
2
?? - 4cosec ?? - cosec ?? + 2 = 0
? cosec ?? = 2 or
1
2
(Not possible)
As ?? ? [0,2?? ) ,
?? 1
=
?? 6
, ?? 2
=
5?? 6
? ? cos
2
3?????? = ?
( 1 + cos6?? )
2
5?? 6
?? 6
????
?? 2
?? 1
=
1
2
(
5?? 6
-
?? 6
) +
sin6?
12
|
?? 6
5?? 6
=
?? 3
7
th
January 2020 (Shift 2), Mathematics Page | 3
3. Coefficient of ?? 7
in ( 1 + ?? )
10
+ ?? ( 1 + ?? )
9
+ ?? 2
( 1 + ?? )
8
+ ? +?? 10
is
a. 260 b. 210
c. 420 d. 330
Answer: ( d)
Solution:
Coefficient of ?? 7
in ( 1 + ?? )
10
+ ?? ( 1 + ?? )
9
+ ?? 2
( 1 + ?? )
8
+ ? +?? 10
Applying sum of terms of G.P. =
( 1+?? )
10
(1- (
?? 1+?? )
11
)
(1-
?? 1+?? )
= ( 1 + ?? )
11
- ?? 11
Coefficient of ?? 7
?
11
?? 7
= 330
4. Let ?? and ?? are the roots of ?? 2
- ?? - 1 = 0 such that ?? ?? = ?? ?? + ?? ?? , ?? = 1 then which one is
incorrect?
a. ?? 5
= ?? 2
× ?? 3
b. ?? 1
+ ?? 2
+ ?? 3
+ ?? 4
+ ?? 5
= 26
c. ?? 5
= 11 d. ?? 3
= ?? 5
- ?? 4
Answer: ( a)
Solution:
Given ?? , ?? are the roots of ?? 2
- ?? - 1 = 0
? ?? + ?? = 1 & ???? = -1
? ?? 2
= ?? + 1 & ?? 2
= ?? + 1
?? ?? = ?? ?? -2
?? 2
+ ?? ?? -2
?? 2
?? ?? = ?? ?? -2
( ?? + 1)+ ?? ?? -2
( ?? + 1)
?? ?? = ?? ?? -1
+ ?? ?? -1
+ ?? ?? -2
+ ?? ?? -2
? ?? ?? = ?? ?? -1
+ ?? ?? -2
? ?? 3
= ?? 2
+ ?? 1
= 4
?? 4
= ?? 3
+ ?? 2
= 7
7
th
January 2020 (Shift 2), Mathematics Page | 4
?? 5
= ?? 4
+ ?? 3
= 11
? ?? 5
? ?? 2
?? 3
& ?? 1
+ ?? 2
+ ?? 3
+ ?? 4
+ ?? 5
= 26
& ?? 3
= ?? 5
- ?? 4
5. The area bounded by 4?? 2
= ?? = 8?? + 12 is
a.
127
3
b.
125
3
c.
128
3
d.
124
3
Answer: ( c)
Solution:
For point of intersection,
4?? 2
= 8?? + 12
? ?? 2
- 2?? - 3 = 0
? ?? = 3, -1
Area bounded is given by
?? = ? ( 8?? + 12 - 4?? 2
) ????
3
-1
Page 5
7
th
January 2020 (Shift 2), Mathematics Page | 1
Date: 7
th
January 2020 (Shift 2)
Time: 2:30 P.M. to 5:30 P.M.
Subject: Mathematics
1. From any point ?? on the line ?? = 2?? , a perpendicular is drawn on ?? = ?? . Let the foot of
perpendicular be ?? . Find the locus of mid point of ???? .
a. 5?? = 7?? b. 2?? = 3?? c. 7?? = 5?? d. 3?? = 2??
Answer: ( a)
Solution:
Let ?? be the midpoint of ????
???? is perpendicular on line ?? = ??
? Equation of the line ???? can be written as ?? = -?? + ??
?? = -?? + ?? intersects ?? = ?? at ?? : (
?? 2
,
?? 2
)
?? = -?? + ?? intersects ?? = 2?? at ?? : (
2?? 3
,
?? 3
)
? Midpoint ?? : (
7?? 12
,
5?? 12
)
Locus of ?? : ?? =
7?? 12
7
th
January 2020 (Shift 2), Mathematics Page | 2
?? =
5?? 12
? 5?? = 7??
2. Let ?? 1
and ?? 2
(where ?? 1
< ?? 2
) are two solutions of 2 cot
2
?? -
5
sin ?? +4 = 0, ?? ? [0,2?? ) then
? cos
2
3?? ????
?? 2
?? 1
is equal to
a.
?? 9
b.
2?? 3
c.
?? 3
+
1
6
d.
?? 3
Answer: ( d)
Solution:
2 cot
2
?? -
5
sin ?? + 4 = 0, ?? ? [0,2?? )
? 2cosec
2
?? - 2 - 5cosec ?? + 4 = 0
? 2cosec
2
?? - 4cosec ?? - cosec ?? + 2 = 0
? cosec ?? = 2 or
1
2
(Not possible)
As ?? ? [0,2?? ) ,
?? 1
=
?? 6
, ?? 2
=
5?? 6
? ? cos
2
3?????? = ?
( 1 + cos6?? )
2
5?? 6
?? 6
????
?? 2
?? 1
=
1
2
(
5?? 6
-
?? 6
) +
sin6?
12
|
?? 6
5?? 6
=
?? 3
7
th
January 2020 (Shift 2), Mathematics Page | 3
3. Coefficient of ?? 7
in ( 1 + ?? )
10
+ ?? ( 1 + ?? )
9
+ ?? 2
( 1 + ?? )
8
+ ? +?? 10
is
a. 260 b. 210
c. 420 d. 330
Answer: ( d)
Solution:
Coefficient of ?? 7
in ( 1 + ?? )
10
+ ?? ( 1 + ?? )
9
+ ?? 2
( 1 + ?? )
8
+ ? +?? 10
Applying sum of terms of G.P. =
( 1+?? )
10
(1- (
?? 1+?? )
11
)
(1-
?? 1+?? )
= ( 1 + ?? )
11
- ?? 11
Coefficient of ?? 7
?
11
?? 7
= 330
4. Let ?? and ?? are the roots of ?? 2
- ?? - 1 = 0 such that ?? ?? = ?? ?? + ?? ?? , ?? = 1 then which one is
incorrect?
a. ?? 5
= ?? 2
× ?? 3
b. ?? 1
+ ?? 2
+ ?? 3
+ ?? 4
+ ?? 5
= 26
c. ?? 5
= 11 d. ?? 3
= ?? 5
- ?? 4
Answer: ( a)
Solution:
Given ?? , ?? are the roots of ?? 2
- ?? - 1 = 0
? ?? + ?? = 1 & ???? = -1
? ?? 2
= ?? + 1 & ?? 2
= ?? + 1
?? ?? = ?? ?? -2
?? 2
+ ?? ?? -2
?? 2
?? ?? = ?? ?? -2
( ?? + 1)+ ?? ?? -2
( ?? + 1)
?? ?? = ?? ?? -1
+ ?? ?? -1
+ ?? ?? -2
+ ?? ?? -2
? ?? ?? = ?? ?? -1
+ ?? ?? -2
? ?? 3
= ?? 2
+ ?? 1
= 4
?? 4
= ?? 3
+ ?? 2
= 7
7
th
January 2020 (Shift 2), Mathematics Page | 4
?? 5
= ?? 4
+ ?? 3
= 11
? ?? 5
? ?? 2
?? 3
& ?? 1
+ ?? 2
+ ?? 3
+ ?? 4
+ ?? 5
= 26
& ?? 3
= ?? 5
- ?? 4
5. The area bounded by 4?? 2
= ?? = 8?? + 12 is
a.
127
3
b.
125
3
c.
128
3
d.
124
3
Answer: ( c)
Solution:
For point of intersection,
4?? 2
= 8?? + 12
? ?? 2
- 2?? - 3 = 0
? ?? = 3, -1
Area bounded is given by
?? = ? ( 8?? + 12 - 4?? 2
) ????
3
-1
7
th
January 2020 (Shift 2), Mathematics Page | 5
?? = [
8?? 2
2
+ 12?? -
4?? 3
3
]
-1
3
?? = ( 36 + 36 - 36)- (4 - 12 +
4
3
)
?? = 44 -
4
3
=
128
3
6. Contrapositive of ?? ? ?? and ?? ? ?? then ?? ? ?? a. ?? ? ?? or ?? ? ?? or ?? ? ?? b. ?? ? ?? or ?? ? ?? and ?? ? ?? c. ?? ? ?? and ?? ? ?? or ?? ?C d. ?? ? ?? or ?? ? ?? or ?? ? ?? Answer: ( d)
Solution:
Given statements: ?? ? ?? and ?? ? ??
Let ?? ? ?? be ??
?? ? ?? be ??
?? ? ?? be ??
Modified statement: ( ?? ??? ) ? ??
Contrapositive: ~?? ? ~ ( ?? ??? )
? ?? ? ( ~?? ? ~?? )
? ?? ? ?? or ?? ? ?? or ?? ?C
7. Let 3 + 4 + 8 + 9 + 13 + 14 + 18 + … … … .40 terms = ?? . If ?? = ( 102 ) ?? then ?? =
a. 5 b. 10
c. 25 d. 20
Answer: ( d)
Solution:
?? = 3 + 4 + 8 + 9 + 13 + 14 + … … 40 terms
?? = 7 + 17 + 27 + 37 + ? … … .20 terms
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Oct 24, 2024 Last updated |
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