JEE Exam > JEE Notes > Mathematics (Maths) for JEE Main & Advanced > JEE Advanced Previous Year Questions (2021-2023): Trigonometric Ratios, Functions & Equations
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Page 1
JEE Advanced Previous Year
Questions (2 1 2 0 -2023):
Trigonometric Ratios, Functions &
Equations
Q1. Let ?? and ?? be real numbers such that -
?? ?? < ?? < ?? < ?? <
?? ?? .
If ?? ???? ? ( ?? + ?? ) =
?? ?? and ?? ???? ? ( ?? - ?? ) =
?? ?? , then the greatest integer less than or equal
to (
?? ?? ?? ? ?? ???? ?? ? ?? +
???? ?? ? ?? ?? ?? ?? ? ?? +
???? ?? ? ?? ?? ?? ?? ? ?? +
?? ?? ?? ? ?? ???? ?? ? ?? )
?? is [JEE Advanced 2022 Paper 2 Online]
Ans: 1
Given, sin ? ( ?? + ?? ) =
1
3
and c o s ? ( ?? - ?? ) =
2
3
Let, ?? =
s i n ? ?? c o s ? ?? +
c o s ? ?? s i n ? ?? +
c o s ? ?? s i n ? ?? +
s i n ? ?? c o s ? ??
=
sin ? ?? c o s ? ?? +
c o s ? ?? sin ? ?? +
c o s ? ?? sin ? ?? +
sin ? ?? c o s ? ??
? =
sin ? ?? sin ? ?? + c o s ? ?? c o s ? ?? sin ? ?? c o s ? ?? +
c o s ? ?? c o s ? ?? + sin ? ?? sin ? ?? sin ? ?? c o s ? ?? ? =
c o s ? ( ?? - ?? )
sin ? ?? c o s ? ?? +
c o s ? ( ?? - ?? )
sin ? ?? c o s ? ?? ? = c o s ? ( ?? - ?? ) [
2
2 s i n ? ?? c o s ? ?? +
2
2 s i n ? ?? c o s ? ?? ]
? =
2
3
[
2
sin ? 2 ?? +
2
sin ? 2 ?? ]
? =
4
3
[
1
sin ? 2 ?? +
1
sin ? 2 ?? ]
? =
4
3
[
sin ? 2 ?? + sin ? 2 ?? sin ? 2 ?? sin ? 2 ?? ]
? =
4 × 2
3
[
2 s i n ? (
2 ?? + 2 ?? 2
) c o s ? (
2 ?? - 2 ?? 2
)
2 s i n ? 2 ?? sin ? 2 ?? ]
=
16
3
[
sin ? ( ?? + ?? ) c o s ? ( ?? - ?? )
c o s ? ( 2 ?? - 2 ?? ) - c o s ? ( 2 ?? + 2 ?? )
]
Page 2
JEE Advanced Previous Year
Questions (2 1 2 0 -2023):
Trigonometric Ratios, Functions &
Equations
Q1. Let ?? and ?? be real numbers such that -
?? ?? < ?? < ?? < ?? <
?? ?? .
If ?? ???? ? ( ?? + ?? ) =
?? ?? and ?? ???? ? ( ?? - ?? ) =
?? ?? , then the greatest integer less than or equal
to (
?? ?? ?? ? ?? ???? ?? ? ?? +
???? ?? ? ?? ?? ?? ?? ? ?? +
???? ?? ? ?? ?? ?? ?? ? ?? +
?? ?? ?? ? ?? ???? ?? ? ?? )
?? is [JEE Advanced 2022 Paper 2 Online]
Ans: 1
Given, sin ? ( ?? + ?? ) =
1
3
and c o s ? ( ?? - ?? ) =
2
3
Let, ?? =
s i n ? ?? c o s ? ?? +
c o s ? ?? s i n ? ?? +
c o s ? ?? s i n ? ?? +
s i n ? ?? c o s ? ??
=
sin ? ?? c o s ? ?? +
c o s ? ?? sin ? ?? +
c o s ? ?? sin ? ?? +
sin ? ?? c o s ? ??
? =
sin ? ?? sin ? ?? + c o s ? ?? c o s ? ?? sin ? ?? c o s ? ?? +
c o s ? ?? c o s ? ?? + sin ? ?? sin ? ?? sin ? ?? c o s ? ?? ? =
c o s ? ( ?? - ?? )
sin ? ?? c o s ? ?? +
c o s ? ( ?? - ?? )
sin ? ?? c o s ? ?? ? = c o s ? ( ?? - ?? ) [
2
2 s i n ? ?? c o s ? ?? +
2
2 s i n ? ?? c o s ? ?? ]
? =
2
3
[
2
sin ? 2 ?? +
2
sin ? 2 ?? ]
? =
4
3
[
1
sin ? 2 ?? +
1
sin ? 2 ?? ]
? =
4
3
[
sin ? 2 ?? + sin ? 2 ?? sin ? 2 ?? sin ? 2 ?? ]
? =
4 × 2
3
[
2 s i n ? (
2 ?? + 2 ?? 2
) c o s ? (
2 ?? - 2 ?? 2
)
2 s i n ? 2 ?? sin ? 2 ?? ]
=
16
3
[
sin ? ( ?? + ?? ) c o s ? ( ?? - ?? )
c o s ? ( 2 ?? - 2 ?? ) - c o s ? ( 2 ?? + 2 ?? )
]
? =
16
3
[
1
3
×
2
3
( 2 c o s
2
? ( ?? - ?? ) - 1 ) - ( 1 - 2 sin
2
? ( ?? + ?? ) )
]
? =
32
27
[
1
2 ×
4
9
- 2 + 2 ×
1
9
]
? =
32
27
[
9
8 - 18 + 2
]
? =
32
27
[
9
- 8
]
? = -
4
3
? ? ?? 2
=
16
9
= 1 . 77
? ? [ ?? 2
] = 1
Q2. Consider the following lists :
List-I List-II
(I) { ?? ? [ -
?? ?? ?? ,
?? ?? ?? ] : ?? ???? ? ?? + ?? ???? ? ?? = ?? } (P) has two elements
(II) { ?? ? [ -
?? ?? ????
,
?? ?? ????
] : v ?? ?? ?? ?? ? ?? ?? = ?? } (Q) has three elements
(III) { ?? ? [ -
?? ?? ?? ,
?? ?? ?? ] : ?? ?? ???? ? ( ?? ?? ) = v ?? } (R) has four elements
(IV) { ?? ? [ -
?? ?? ?? ,
?? ?? ?? ] : ?? ???? ? ?? - ?? ???? ? ?? = ?? } (S) has five elements
(T) has six elements
The correct option is:
A (I) ? ( ?? ); (II) ? ( ?? ) ; ( ?????? ) ? ( ?? ); (IV) ? ( ?? )
(B) ( ?? ) ? ( ?? ) ; ( ???? ) ? ( ?? ) ; ( ?????? ) ? ( ?? ) ; ( ???? ) ? ( ?? )
(C) ( ?? ) ? ( ?? ) ; ( ???? ) ? ( ?? ); (III) ? ( ?? ); (IV) ? ( ?? )
(D) (I) ? ( ?? ) ; ( ???? ) ? ( ?? ) ; ( ?????? ) ? ( ?? ) ; ( ???? ) ? ( ?? )
[JEE Advanced 2022 Paper 1 Online]
Ans: (b)
Solving all question one by one we get,
( i ) { ?? ? [
- 2 ?? 3
,
2 ?? 3
] , c o s ? ?? + sin ? ?? = 1 }
c o s ? ?? + sin ? ?? = 1
Page 3
JEE Advanced Previous Year
Questions (2 1 2 0 -2023):
Trigonometric Ratios, Functions &
Equations
Q1. Let ?? and ?? be real numbers such that -
?? ?? < ?? < ?? < ?? <
?? ?? .
If ?? ???? ? ( ?? + ?? ) =
?? ?? and ?? ???? ? ( ?? - ?? ) =
?? ?? , then the greatest integer less than or equal
to (
?? ?? ?? ? ?? ???? ?? ? ?? +
???? ?? ? ?? ?? ?? ?? ? ?? +
???? ?? ? ?? ?? ?? ?? ? ?? +
?? ?? ?? ? ?? ???? ?? ? ?? )
?? is [JEE Advanced 2022 Paper 2 Online]
Ans: 1
Given, sin ? ( ?? + ?? ) =
1
3
and c o s ? ( ?? - ?? ) =
2
3
Let, ?? =
s i n ? ?? c o s ? ?? +
c o s ? ?? s i n ? ?? +
c o s ? ?? s i n ? ?? +
s i n ? ?? c o s ? ??
=
sin ? ?? c o s ? ?? +
c o s ? ?? sin ? ?? +
c o s ? ?? sin ? ?? +
sin ? ?? c o s ? ??
? =
sin ? ?? sin ? ?? + c o s ? ?? c o s ? ?? sin ? ?? c o s ? ?? +
c o s ? ?? c o s ? ?? + sin ? ?? sin ? ?? sin ? ?? c o s ? ?? ? =
c o s ? ( ?? - ?? )
sin ? ?? c o s ? ?? +
c o s ? ( ?? - ?? )
sin ? ?? c o s ? ?? ? = c o s ? ( ?? - ?? ) [
2
2 s i n ? ?? c o s ? ?? +
2
2 s i n ? ?? c o s ? ?? ]
? =
2
3
[
2
sin ? 2 ?? +
2
sin ? 2 ?? ]
? =
4
3
[
1
sin ? 2 ?? +
1
sin ? 2 ?? ]
? =
4
3
[
sin ? 2 ?? + sin ? 2 ?? sin ? 2 ?? sin ? 2 ?? ]
? =
4 × 2
3
[
2 s i n ? (
2 ?? + 2 ?? 2
) c o s ? (
2 ?? - 2 ?? 2
)
2 s i n ? 2 ?? sin ? 2 ?? ]
=
16
3
[
sin ? ( ?? + ?? ) c o s ? ( ?? - ?? )
c o s ? ( 2 ?? - 2 ?? ) - c o s ? ( 2 ?? + 2 ?? )
]
? =
16
3
[
1
3
×
2
3
( 2 c o s
2
? ( ?? - ?? ) - 1 ) - ( 1 - 2 sin
2
? ( ?? + ?? ) )
]
? =
32
27
[
1
2 ×
4
9
- 2 + 2 ×
1
9
]
? =
32
27
[
9
8 - 18 + 2
]
? =
32
27
[
9
- 8
]
? = -
4
3
? ? ?? 2
=
16
9
= 1 . 77
? ? [ ?? 2
] = 1
Q2. Consider the following lists :
List-I List-II
(I) { ?? ? [ -
?? ?? ?? ,
?? ?? ?? ] : ?? ???? ? ?? + ?? ???? ? ?? = ?? } (P) has two elements
(II) { ?? ? [ -
?? ?? ????
,
?? ?? ????
] : v ?? ?? ?? ?? ? ?? ?? = ?? } (Q) has three elements
(III) { ?? ? [ -
?? ?? ?? ,
?? ?? ?? ] : ?? ?? ???? ? ( ?? ?? ) = v ?? } (R) has four elements
(IV) { ?? ? [ -
?? ?? ?? ,
?? ?? ?? ] : ?? ???? ? ?? - ?? ???? ? ?? = ?? } (S) has five elements
(T) has six elements
The correct option is:
A (I) ? ( ?? ); (II) ? ( ?? ) ; ( ?????? ) ? ( ?? ); (IV) ? ( ?? )
(B) ( ?? ) ? ( ?? ) ; ( ???? ) ? ( ?? ) ; ( ?????? ) ? ( ?? ) ; ( ???? ) ? ( ?? )
(C) ( ?? ) ? ( ?? ) ; ( ???? ) ? ( ?? ); (III) ? ( ?? ); (IV) ? ( ?? )
(D) (I) ? ( ?? ) ; ( ???? ) ? ( ?? ) ; ( ?????? ) ? ( ?? ) ; ( ???? ) ? ( ?? )
[JEE Advanced 2022 Paper 1 Online]
Ans: (b)
Solving all question one by one we get,
( i ) { ?? ? [
- 2 ?? 3
,
2 ?? 3
] , c o s ? ?? + sin ? ?? = 1 }
c o s ? ?? + sin ? ?? = 1
? ? sin ? (
?? 4
+ ?? ) =
1
v 2
? ?
?? 4
+ ?? = ???? + ( - 1 )
?? ?? 4
? ? ?? = ???? + ( - 1 )
?? ?? 4
-
?? 4
So, ?? ? { 0 ,
?? 2
}
? ?? has 2 elements ? ??
(ii) { ?? ? (
- 5 ?? 18
,
5 ?? 18
) , v 3 t a n ? 3 ?? = 1 }
? t a n ? 3 ?? =
1
v 3
? ?? = ???? +
?? 6
? ?? =
????
3
+
?? 18
So ?? ? {
?? 18
,
- 5 ?? 18
}
? ?? has 2 elements ? ??
(iii) { ?? ? [
- 6 ?? 5
,
6 ?? 5
] , 2 c o s ? 2 ?? = v 3 }
? ? 2 c o s ? 2 ?? = v 3
? ? c o s ? 2 ?? =
v 3
2
? ? 2 ?? = 2 ???? ±
?? 6
? ? ?? = ???? ±
?? 12
So ?? ? { ±
?? 12
, ?? ±
?? 12
, - ?? ±
?? 12
}
So ?? ? { ±
?? 12
, ?? ±
?? 12
, - ?? ±
?? 12
}
? ?? has 6 elements ? T
( i v) { ?? ? [
- 7 ?? 4
,
7 ?? 4
] , sin ? ?? - c o s ? ?? = 1 }
? ? sin ? ?? - c o s ? ?? = 1
? ? sin ? ( ?? -
?? 4
) =
1
v 2
? ? ?? -
?? 4
= ???? + ( - 1 )
?? ?? 4
? ? ?? = ???? + ( - 1 )
?? ?? 4
+
?? 4
So,
?? ? {
?? 2
,
- 3 ?? 2
, ?? , - ?? }
? ?? has 4 elements ? R
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