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Page 1
Solved Examples on Trigonometric
Ratios, Functions & Equations
JEE Mains
Q1. The angles of a quadrilateral are in A.P. and the greatest angle is ??????
°
, the
angles in radians are
(a)
?? ?? ,
?? ?? ?? ,
?? ?? ?? ,
?? ?? ??
(b)
?? ?? ,
?? ?? ,
?? ?? ?? ,
?? ?? ??
(c)
?? ?? ????
,
?? ?? ????
,
???? ?? ????
,
???? ?? ????
(d) None of these
Ans: (a) Let the angles in degrees be ?? - 3?? , ?? - ?? , ?? + ?? , ?? + 3??
Sum of the angles = 4?? = 360
°
? ?? = 90
°
Also greatest angle = ?? + 3?? = 120
°
, Hence, 3?? = 120
°
- ?? = 120
°
- 90
°
= 30
°
? ?? =
10
°
Hence the angles are 90
°
- 30
°
, 90
°
- 10
°
, 90
°
+ 10
°
and 90
°
+ 30
°
That is, the angles in degrees are 60
°
, 80
°
, 100
°
and 120
°
? In terms of radians the angles are 60 ×
?? 180
, 80 ×
?? 180
, 100 ×
?? 180
and 120×
?? 180
that is
?? 3
,
4?? 9
,
5?? 9
and
2?? 3
.
Q2. If ?????? ?? =
?? ?? +?? and ?????? ?? =
?? ?? ?? +?? then ?? + ?? =
(a)
?? ??
(b)
?? ??
(c)
?? ??
(d) None of these
Ans: (b) We have tan ?? =
?? ?? +1
and tan ?? =
1
2?? +1
tan ( ?? + ?? ) =
?? ?? + 1
+
1
2?? + 1
1 -
?? ( ?? + 1)
·
1
( 2?? + 1)
=
2?? 2
+ ?? + ?? + 1
2?? 2
+ ?? + 2?? + 1 - ?? [? tan ( ?? + ?? )=
tan ?? + tan ?? 1 - tan ?? tan ?? ]
=
2?? 2
+ 2?? + 1
2?? 2
+ 2?? + 1
= 1 ? tan ( ?? + ?? )= tan
?? 4
Hence ?? + ?? =
?? 4
Q3. The value of ?????? ????
°
+ ???? ???? ????
°
is
(a)
?? v ??
(b) v ??
Page 2
Solved Examples on Trigonometric
Ratios, Functions & Equations
JEE Mains
Q1. The angles of a quadrilateral are in A.P. and the greatest angle is ??????
°
, the
angles in radians are
(a)
?? ?? ,
?? ?? ?? ,
?? ?? ?? ,
?? ?? ??
(b)
?? ?? ,
?? ?? ,
?? ?? ?? ,
?? ?? ??
(c)
?? ?? ????
,
?? ?? ????
,
???? ?? ????
,
???? ?? ????
(d) None of these
Ans: (a) Let the angles in degrees be ?? - 3?? , ?? - ?? , ?? + ?? , ?? + 3??
Sum of the angles = 4?? = 360
°
? ?? = 90
°
Also greatest angle = ?? + 3?? = 120
°
, Hence, 3?? = 120
°
- ?? = 120
°
- 90
°
= 30
°
? ?? =
10
°
Hence the angles are 90
°
- 30
°
, 90
°
- 10
°
, 90
°
+ 10
°
and 90
°
+ 30
°
That is, the angles in degrees are 60
°
, 80
°
, 100
°
and 120
°
? In terms of radians the angles are 60 ×
?? 180
, 80 ×
?? 180
, 100 ×
?? 180
and 120×
?? 180
that is
?? 3
,
4?? 9
,
5?? 9
and
2?? 3
.
Q2. If ?????? ?? =
?? ?? +?? and ?????? ?? =
?? ?? ?? +?? then ?? + ?? =
(a)
?? ??
(b)
?? ??
(c)
?? ??
(d) None of these
Ans: (b) We have tan ?? =
?? ?? +1
and tan ?? =
1
2?? +1
tan ( ?? + ?? ) =
?? ?? + 1
+
1
2?? + 1
1 -
?? ( ?? + 1)
·
1
( 2?? + 1)
=
2?? 2
+ ?? + ?? + 1
2?? 2
+ ?? + 2?? + 1 - ?? [? tan ( ?? + ?? )=
tan ?? + tan ?? 1 - tan ?? tan ?? ]
=
2?? 2
+ 2?? + 1
2?? 2
+ 2?? + 1
= 1 ? tan ( ?? + ?? )= tan
?? 4
Hence ?? + ?? =
?? 4
Q3. The value of ?????? ????
°
+ ???? ???? ????
°
is
(a)
?? v ??
(b) v ??
(c) ?? v ??
(d)
?? ??
Ans: (b) cot 70
°
+ 4cos 70
°
=
cos 70
°
+4sin 70
°
·cos 70
°
sin 70
°
=
cos 70
°
+2sin 140
°
sin 70
°
=
cos 70
°
+ 2sin ( 180 - 40
°
)
sin 70
°
=
sin 20
°
+ sin 40
°
+ sin 40
°
sin 70
°
=
2sin 30
°
cos 10
°
+ sin 40
°
sin 70
°
=
sin 80
°
+ sin 40
°
sin 70
°
=
2sin 60
°
cos 20
°
sin 70
°
= v 3.
Q4. The value of
?????? ?? ?????? ?? ?? when ever defined never lie between
(a)
?? ?? and 3
(b)
?? ?? and 4
(c)
?? ?? and 5
(d) 5 and 6
Ans: (a) Let, ?? =
tan ?? tan 3?? =
tan ?? 3tan ?? -tan
3
?? 1-3tan
2
??
?? =
1 - 3tan
2
?? 3 - tan
2
?? =
1
3
- tan
2
?? 1 -
1
3
tan
2
??
Hence, ?? should never lie between
1
3
and 3 whenever defined.
Q5. If ?? + ?? + ?? = ?? , then ?????? ?? ?? + ?????? ?? ?? - ?????? ?? ?? equal to
(a) ?? - ???????? ?? ?????? ?? ?????? ??
(b) ?? - ???????? ?? ?????? ?? ?????? ??
(c) ?? + ???????? ?? ?????? ?? ?????? ??
(d) ?? + ???????? ?? ?????? ?? ?????? ??
Ans: (a)
cos
2
?? + cos
2
?? - cos
2
?? = cos
2
?? + ( 1 - sin
2
?? )- cos
2
?? = 1 + [cos
2
?? - sin
2
?? ] - cos
2
?? = 1 + cos ( ?? + ?? ) cos ( ?? - ?? )- cos
2
?? = 1 + cos ( ?? - ?? ) cos ( ?? - ?? )- cos
2
?? = 1 - cos ?? [cos ( ?? - ?? )+ cos ?? ]
= 1 - cos ?? [cos ( ?? - ?? )+ cos {?? - ( ?? + ?? ) }] = 1 - cos ?? [cos ( ?? - ?? )- cos ( ?? + ?? ) ]
= 1 - cos ?? [2sin ?? sin ?? ] = 1 - 2sin ?? sin ?? cos ?? .
Q6. If ?????? ( ?????? ?? )= ?????? ( ?????? ?? ) , then ?????? ?? ?? equal to
(a) ( ?? ?? + ?? )
?? ??
(b)
?? ( ?? ?? +?? ) ??
Page 3
Solved Examples on Trigonometric
Ratios, Functions & Equations
JEE Mains
Q1. The angles of a quadrilateral are in A.P. and the greatest angle is ??????
°
, the
angles in radians are
(a)
?? ?? ,
?? ?? ?? ,
?? ?? ?? ,
?? ?? ??
(b)
?? ?? ,
?? ?? ,
?? ?? ?? ,
?? ?? ??
(c)
?? ?? ????
,
?? ?? ????
,
???? ?? ????
,
???? ?? ????
(d) None of these
Ans: (a) Let the angles in degrees be ?? - 3?? , ?? - ?? , ?? + ?? , ?? + 3??
Sum of the angles = 4?? = 360
°
? ?? = 90
°
Also greatest angle = ?? + 3?? = 120
°
, Hence, 3?? = 120
°
- ?? = 120
°
- 90
°
= 30
°
? ?? =
10
°
Hence the angles are 90
°
- 30
°
, 90
°
- 10
°
, 90
°
+ 10
°
and 90
°
+ 30
°
That is, the angles in degrees are 60
°
, 80
°
, 100
°
and 120
°
? In terms of radians the angles are 60 ×
?? 180
, 80 ×
?? 180
, 100 ×
?? 180
and 120×
?? 180
that is
?? 3
,
4?? 9
,
5?? 9
and
2?? 3
.
Q2. If ?????? ?? =
?? ?? +?? and ?????? ?? =
?? ?? ?? +?? then ?? + ?? =
(a)
?? ??
(b)
?? ??
(c)
?? ??
(d) None of these
Ans: (b) We have tan ?? =
?? ?? +1
and tan ?? =
1
2?? +1
tan ( ?? + ?? ) =
?? ?? + 1
+
1
2?? + 1
1 -
?? ( ?? + 1)
·
1
( 2?? + 1)
=
2?? 2
+ ?? + ?? + 1
2?? 2
+ ?? + 2?? + 1 - ?? [? tan ( ?? + ?? )=
tan ?? + tan ?? 1 - tan ?? tan ?? ]
=
2?? 2
+ 2?? + 1
2?? 2
+ 2?? + 1
= 1 ? tan ( ?? + ?? )= tan
?? 4
Hence ?? + ?? =
?? 4
Q3. The value of ?????? ????
°
+ ???? ???? ????
°
is
(a)
?? v ??
(b) v ??
(c) ?? v ??
(d)
?? ??
Ans: (b) cot 70
°
+ 4cos 70
°
=
cos 70
°
+4sin 70
°
·cos 70
°
sin 70
°
=
cos 70
°
+2sin 140
°
sin 70
°
=
cos 70
°
+ 2sin ( 180 - 40
°
)
sin 70
°
=
sin 20
°
+ sin 40
°
+ sin 40
°
sin 70
°
=
2sin 30
°
cos 10
°
+ sin 40
°
sin 70
°
=
sin 80
°
+ sin 40
°
sin 70
°
=
2sin 60
°
cos 20
°
sin 70
°
= v 3.
Q4. The value of
?????? ?? ?????? ?? ?? when ever defined never lie between
(a)
?? ?? and 3
(b)
?? ?? and 4
(c)
?? ?? and 5
(d) 5 and 6
Ans: (a) Let, ?? =
tan ?? tan 3?? =
tan ?? 3tan ?? -tan
3
?? 1-3tan
2
??
?? =
1 - 3tan
2
?? 3 - tan
2
?? =
1
3
- tan
2
?? 1 -
1
3
tan
2
??
Hence, ?? should never lie between
1
3
and 3 whenever defined.
Q5. If ?? + ?? + ?? = ?? , then ?????? ?? ?? + ?????? ?? ?? - ?????? ?? ?? equal to
(a) ?? - ???????? ?? ?????? ?? ?????? ??
(b) ?? - ???????? ?? ?????? ?? ?????? ??
(c) ?? + ???????? ?? ?????? ?? ?????? ??
(d) ?? + ???????? ?? ?????? ?? ?????? ??
Ans: (a)
cos
2
?? + cos
2
?? - cos
2
?? = cos
2
?? + ( 1 - sin
2
?? )- cos
2
?? = 1 + [cos
2
?? - sin
2
?? ] - cos
2
?? = 1 + cos ( ?? + ?? ) cos ( ?? - ?? )- cos
2
?? = 1 + cos ( ?? - ?? ) cos ( ?? - ?? )- cos
2
?? = 1 - cos ?? [cos ( ?? - ?? )+ cos ?? ]
= 1 - cos ?? [cos ( ?? - ?? )+ cos {?? - ( ?? + ?? ) }] = 1 - cos ?? [cos ( ?? - ?? )- cos ( ?? + ?? ) ]
= 1 - cos ?? [2sin ?? sin ?? ] = 1 - 2sin ?? sin ?? cos ?? .
Q6. If ?????? ( ?????? ?? )= ?????? ( ?????? ?? ) , then ?????? ?? ?? equal to
(a) ( ?? ?? + ?? )
?? ??
(b)
?? ( ?? ?? +?? ) ??
(c) ?? ?? ( ?? ?? + ?? )
(d) None of these
Ans:
(b)
tan ( cot ?? )= cot ( tan ?? )? tan ( cot ?? )= tan (
?? 2
- tan ?? )
cot ?? = ???? +
?? 2
- tan ?? tan ?? + cot ?? =
( 2?? + 1) ?? 2
?
sin ?? cos ?? +
cos ?? sin ?? =
( 2?? + 1) ?? 2
1
sin ?? · cos ?? =
( 2?? + 1) ?? 2
? sin 2?? =
4
( 2?? + 1) ?? .
Q7. The only value of ?? for which ?? ?????? ?? + ?? ?? ???? ?? > ?? ?? -( ?? /v ?? )
hold is
(a)
?? ?? ??
(b)
?? ?? ??
(c)
?? ??
(d) All values of ?? .
Ans: (a) Since A.M. = G.M.
1
2
( 2
sin ?? + 2
cos ?? )=
v
2
sin ?? · 2
cos ?? ? 2
sin ?? + 2
cos ?? = 2.2
sin ?? +cos ?? 2
? 2
sin ?? + 2
cos ?? = 2
1+
sin ?? +cos ?? 2
And, we know that sin ?? + cos ?? = -v 2
? 2
sin ?? + 2
cos ?? > 2
1-( 1/v 2)
for ?? =
5?? 4
.
Q8. The period of ?????? ?? ?? + ?????? ?? ?? is
(a)
?? ??
(b) ??
(c) ?? ??
(d)
?? ?? ??
Ans: (a) ( sin
2
?? )
2
+ ( cos
2
?? )
2
= 1 - 2sin
2
?? cos
2
?? = 1 -
1
4
2sin
2
2?? = 1 - (
1-cos 4?? 4
)=
3
4
+
1
4
cos 4??
Therefore period is
2?? 4
=
?? 2
.
Trick: ?? ( ?? )= sin
4
?? + cos
4
?? ? ?? (
?? 2
+ ?? )= sin
4
(
?? 2
+ ?? )+ cos
4
(
?? 2
+ ?? )? ?? (
?? 2
+ ?? )=
cos
4
?? + sin
4
?? = ?? ( ?? )
Hence the period is
?? 2
.
Page 4
Solved Examples on Trigonometric
Ratios, Functions & Equations
JEE Mains
Q1. The angles of a quadrilateral are in A.P. and the greatest angle is ??????
°
, the
angles in radians are
(a)
?? ?? ,
?? ?? ?? ,
?? ?? ?? ,
?? ?? ??
(b)
?? ?? ,
?? ?? ,
?? ?? ?? ,
?? ?? ??
(c)
?? ?? ????
,
?? ?? ????
,
???? ?? ????
,
???? ?? ????
(d) None of these
Ans: (a) Let the angles in degrees be ?? - 3?? , ?? - ?? , ?? + ?? , ?? + 3??
Sum of the angles = 4?? = 360
°
? ?? = 90
°
Also greatest angle = ?? + 3?? = 120
°
, Hence, 3?? = 120
°
- ?? = 120
°
- 90
°
= 30
°
? ?? =
10
°
Hence the angles are 90
°
- 30
°
, 90
°
- 10
°
, 90
°
+ 10
°
and 90
°
+ 30
°
That is, the angles in degrees are 60
°
, 80
°
, 100
°
and 120
°
? In terms of radians the angles are 60 ×
?? 180
, 80 ×
?? 180
, 100 ×
?? 180
and 120×
?? 180
that is
?? 3
,
4?? 9
,
5?? 9
and
2?? 3
.
Q2. If ?????? ?? =
?? ?? +?? and ?????? ?? =
?? ?? ?? +?? then ?? + ?? =
(a)
?? ??
(b)
?? ??
(c)
?? ??
(d) None of these
Ans: (b) We have tan ?? =
?? ?? +1
and tan ?? =
1
2?? +1
tan ( ?? + ?? ) =
?? ?? + 1
+
1
2?? + 1
1 -
?? ( ?? + 1)
·
1
( 2?? + 1)
=
2?? 2
+ ?? + ?? + 1
2?? 2
+ ?? + 2?? + 1 - ?? [? tan ( ?? + ?? )=
tan ?? + tan ?? 1 - tan ?? tan ?? ]
=
2?? 2
+ 2?? + 1
2?? 2
+ 2?? + 1
= 1 ? tan ( ?? + ?? )= tan
?? 4
Hence ?? + ?? =
?? 4
Q3. The value of ?????? ????
°
+ ???? ???? ????
°
is
(a)
?? v ??
(b) v ??
(c) ?? v ??
(d)
?? ??
Ans: (b) cot 70
°
+ 4cos 70
°
=
cos 70
°
+4sin 70
°
·cos 70
°
sin 70
°
=
cos 70
°
+2sin 140
°
sin 70
°
=
cos 70
°
+ 2sin ( 180 - 40
°
)
sin 70
°
=
sin 20
°
+ sin 40
°
+ sin 40
°
sin 70
°
=
2sin 30
°
cos 10
°
+ sin 40
°
sin 70
°
=
sin 80
°
+ sin 40
°
sin 70
°
=
2sin 60
°
cos 20
°
sin 70
°
= v 3.
Q4. The value of
?????? ?? ?????? ?? ?? when ever defined never lie between
(a)
?? ?? and 3
(b)
?? ?? and 4
(c)
?? ?? and 5
(d) 5 and 6
Ans: (a) Let, ?? =
tan ?? tan 3?? =
tan ?? 3tan ?? -tan
3
?? 1-3tan
2
??
?? =
1 - 3tan
2
?? 3 - tan
2
?? =
1
3
- tan
2
?? 1 -
1
3
tan
2
??
Hence, ?? should never lie between
1
3
and 3 whenever defined.
Q5. If ?? + ?? + ?? = ?? , then ?????? ?? ?? + ?????? ?? ?? - ?????? ?? ?? equal to
(a) ?? - ???????? ?? ?????? ?? ?????? ??
(b) ?? - ???????? ?? ?????? ?? ?????? ??
(c) ?? + ???????? ?? ?????? ?? ?????? ??
(d) ?? + ???????? ?? ?????? ?? ?????? ??
Ans: (a)
cos
2
?? + cos
2
?? - cos
2
?? = cos
2
?? + ( 1 - sin
2
?? )- cos
2
?? = 1 + [cos
2
?? - sin
2
?? ] - cos
2
?? = 1 + cos ( ?? + ?? ) cos ( ?? - ?? )- cos
2
?? = 1 + cos ( ?? - ?? ) cos ( ?? - ?? )- cos
2
?? = 1 - cos ?? [cos ( ?? - ?? )+ cos ?? ]
= 1 - cos ?? [cos ( ?? - ?? )+ cos {?? - ( ?? + ?? ) }] = 1 - cos ?? [cos ( ?? - ?? )- cos ( ?? + ?? ) ]
= 1 - cos ?? [2sin ?? sin ?? ] = 1 - 2sin ?? sin ?? cos ?? .
Q6. If ?????? ( ?????? ?? )= ?????? ( ?????? ?? ) , then ?????? ?? ?? equal to
(a) ( ?? ?? + ?? )
?? ??
(b)
?? ( ?? ?? +?? ) ??
(c) ?? ?? ( ?? ?? + ?? )
(d) None of these
Ans:
(b)
tan ( cot ?? )= cot ( tan ?? )? tan ( cot ?? )= tan (
?? 2
- tan ?? )
cot ?? = ???? +
?? 2
- tan ?? tan ?? + cot ?? =
( 2?? + 1) ?? 2
?
sin ?? cos ?? +
cos ?? sin ?? =
( 2?? + 1) ?? 2
1
sin ?? · cos ?? =
( 2?? + 1) ?? 2
? sin 2?? =
4
( 2?? + 1) ?? .
Q7. The only value of ?? for which ?? ?????? ?? + ?? ?? ???? ?? > ?? ?? -( ?? /v ?? )
hold is
(a)
?? ?? ??
(b)
?? ?? ??
(c)
?? ??
(d) All values of ?? .
Ans: (a) Since A.M. = G.M.
1
2
( 2
sin ?? + 2
cos ?? )=
v
2
sin ?? · 2
cos ?? ? 2
sin ?? + 2
cos ?? = 2.2
sin ?? +cos ?? 2
? 2
sin ?? + 2
cos ?? = 2
1+
sin ?? +cos ?? 2
And, we know that sin ?? + cos ?? = -v 2
? 2
sin ?? + 2
cos ?? > 2
1-( 1/v 2)
for ?? =
5?? 4
.
Q8. The period of ?????? ?? ?? + ?????? ?? ?? is
(a)
?? ??
(b) ??
(c) ?? ??
(d)
?? ?? ??
Ans: (a) ( sin
2
?? )
2
+ ( cos
2
?? )
2
= 1 - 2sin
2
?? cos
2
?? = 1 -
1
4
2sin
2
2?? = 1 - (
1-cos 4?? 4
)=
3
4
+
1
4
cos 4??
Therefore period is
2?? 4
=
?? 2
.
Trick: ?? ( ?? )= sin
4
?? + cos
4
?? ? ?? (
?? 2
+ ?? )= sin
4
(
?? 2
+ ?? )+ cos
4
(
?? 2
+ ?? )? ?? (
?? 2
+ ?? )=
cos
4
?? + sin
4
?? = ?? ( ?? )
Hence the period is
?? 2
.
Q9. In a ? ?????? , ?? : ?? : ?? = ?? : ?? : ?? . Then [?? + ?? + ?? v ?? ] is equal to
(a) ?? ??
(b) ?? ??
(c) ?? ??
(d) ?? ??
Ans: (c)
?? : ?? : ?? = 3: 5: 4 ? ?? + ?? + ?? = 12?? = 180
°
? ?? = 15
°
? ?? = 45
°
, ?? = 75
°
, ?? = 60
°
?? sin 45
°
=
?? sin 75
°
=
?? sin 60
°
= ?? (say)
i.e., ?? =
1
v 2
?? , ?? =
v 3 + 1
2v 2
?? , ?? =
v 3
2
?? . Hence [?? + ?? + ?? v 2] = 3?? .
Q10. If in a triangle ?????? , ?? ?????? ??
?? ?? + ?? ?????? ??
?? ?? =
?? ?? ?? , then a, c, b are:
(A) In A.P.
(B) In G.P.
(C) In H.P.
(D) None
Ans: (A) ?? cos
2
?? 2
+ ?? cos
2
?? 2
=
3
2
??
b
( cos ?? + 1)
2
+ ?? ( cos ?? + 1)
2
=
3
2
?? ?? cos ?? + ?? + ?? cos ?? + ?? = 3?? ? ?? + ?? + ( ?? cos ?? + ?? cos ?? )= 3?? ? ?? cos ?? + ?? cos ?? = ?? ? ?? + ?? = 2?? ? ?? , ?? , ?? are in A.P.
? a+ b = 2c ? a , b, c are in A.P.
Q11. Given ?? ?? + ?? ?? + ?????????? ?? (
?? ?? ( ?? + ?? ) )= ?? then, which of the following holds
good?
(A) ?? = ?? ;
?? ?? ? ??
(B) ?? = -?? ;
?? ?? ? ??
(C) ?? ? ?? ; ?? ? ??
(D) ?? , ?? are finite but not possible to find
Ans: (B) ?? 2
+ 2?? + cosec
2
(
?? 2
( ?? + ?? ) )= 0
Page 5
Solved Examples on Trigonometric
Ratios, Functions & Equations
JEE Mains
Q1. The angles of a quadrilateral are in A.P. and the greatest angle is ??????
°
, the
angles in radians are
(a)
?? ?? ,
?? ?? ?? ,
?? ?? ?? ,
?? ?? ??
(b)
?? ?? ,
?? ?? ,
?? ?? ?? ,
?? ?? ??
(c)
?? ?? ????
,
?? ?? ????
,
???? ?? ????
,
???? ?? ????
(d) None of these
Ans: (a) Let the angles in degrees be ?? - 3?? , ?? - ?? , ?? + ?? , ?? + 3??
Sum of the angles = 4?? = 360
°
? ?? = 90
°
Also greatest angle = ?? + 3?? = 120
°
, Hence, 3?? = 120
°
- ?? = 120
°
- 90
°
= 30
°
? ?? =
10
°
Hence the angles are 90
°
- 30
°
, 90
°
- 10
°
, 90
°
+ 10
°
and 90
°
+ 30
°
That is, the angles in degrees are 60
°
, 80
°
, 100
°
and 120
°
? In terms of radians the angles are 60 ×
?? 180
, 80 ×
?? 180
, 100 ×
?? 180
and 120×
?? 180
that is
?? 3
,
4?? 9
,
5?? 9
and
2?? 3
.
Q2. If ?????? ?? =
?? ?? +?? and ?????? ?? =
?? ?? ?? +?? then ?? + ?? =
(a)
?? ??
(b)
?? ??
(c)
?? ??
(d) None of these
Ans: (b) We have tan ?? =
?? ?? +1
and tan ?? =
1
2?? +1
tan ( ?? + ?? ) =
?? ?? + 1
+
1
2?? + 1
1 -
?? ( ?? + 1)
·
1
( 2?? + 1)
=
2?? 2
+ ?? + ?? + 1
2?? 2
+ ?? + 2?? + 1 - ?? [? tan ( ?? + ?? )=
tan ?? + tan ?? 1 - tan ?? tan ?? ]
=
2?? 2
+ 2?? + 1
2?? 2
+ 2?? + 1
= 1 ? tan ( ?? + ?? )= tan
?? 4
Hence ?? + ?? =
?? 4
Q3. The value of ?????? ????
°
+ ???? ???? ????
°
is
(a)
?? v ??
(b) v ??
(c) ?? v ??
(d)
?? ??
Ans: (b) cot 70
°
+ 4cos 70
°
=
cos 70
°
+4sin 70
°
·cos 70
°
sin 70
°
=
cos 70
°
+2sin 140
°
sin 70
°
=
cos 70
°
+ 2sin ( 180 - 40
°
)
sin 70
°
=
sin 20
°
+ sin 40
°
+ sin 40
°
sin 70
°
=
2sin 30
°
cos 10
°
+ sin 40
°
sin 70
°
=
sin 80
°
+ sin 40
°
sin 70
°
=
2sin 60
°
cos 20
°
sin 70
°
= v 3.
Q4. The value of
?????? ?? ?????? ?? ?? when ever defined never lie between
(a)
?? ?? and 3
(b)
?? ?? and 4
(c)
?? ?? and 5
(d) 5 and 6
Ans: (a) Let, ?? =
tan ?? tan 3?? =
tan ?? 3tan ?? -tan
3
?? 1-3tan
2
??
?? =
1 - 3tan
2
?? 3 - tan
2
?? =
1
3
- tan
2
?? 1 -
1
3
tan
2
??
Hence, ?? should never lie between
1
3
and 3 whenever defined.
Q5. If ?? + ?? + ?? = ?? , then ?????? ?? ?? + ?????? ?? ?? - ?????? ?? ?? equal to
(a) ?? - ???????? ?? ?????? ?? ?????? ??
(b) ?? - ???????? ?? ?????? ?? ?????? ??
(c) ?? + ???????? ?? ?????? ?? ?????? ??
(d) ?? + ???????? ?? ?????? ?? ?????? ??
Ans: (a)
cos
2
?? + cos
2
?? - cos
2
?? = cos
2
?? + ( 1 - sin
2
?? )- cos
2
?? = 1 + [cos
2
?? - sin
2
?? ] - cos
2
?? = 1 + cos ( ?? + ?? ) cos ( ?? - ?? )- cos
2
?? = 1 + cos ( ?? - ?? ) cos ( ?? - ?? )- cos
2
?? = 1 - cos ?? [cos ( ?? - ?? )+ cos ?? ]
= 1 - cos ?? [cos ( ?? - ?? )+ cos {?? - ( ?? + ?? ) }] = 1 - cos ?? [cos ( ?? - ?? )- cos ( ?? + ?? ) ]
= 1 - cos ?? [2sin ?? sin ?? ] = 1 - 2sin ?? sin ?? cos ?? .
Q6. If ?????? ( ?????? ?? )= ?????? ( ?????? ?? ) , then ?????? ?? ?? equal to
(a) ( ?? ?? + ?? )
?? ??
(b)
?? ( ?? ?? +?? ) ??
(c) ?? ?? ( ?? ?? + ?? )
(d) None of these
Ans:
(b)
tan ( cot ?? )= cot ( tan ?? )? tan ( cot ?? )= tan (
?? 2
- tan ?? )
cot ?? = ???? +
?? 2
- tan ?? tan ?? + cot ?? =
( 2?? + 1) ?? 2
?
sin ?? cos ?? +
cos ?? sin ?? =
( 2?? + 1) ?? 2
1
sin ?? · cos ?? =
( 2?? + 1) ?? 2
? sin 2?? =
4
( 2?? + 1) ?? .
Q7. The only value of ?? for which ?? ?????? ?? + ?? ?? ???? ?? > ?? ?? -( ?? /v ?? )
hold is
(a)
?? ?? ??
(b)
?? ?? ??
(c)
?? ??
(d) All values of ?? .
Ans: (a) Since A.M. = G.M.
1
2
( 2
sin ?? + 2
cos ?? )=
v
2
sin ?? · 2
cos ?? ? 2
sin ?? + 2
cos ?? = 2.2
sin ?? +cos ?? 2
? 2
sin ?? + 2
cos ?? = 2
1+
sin ?? +cos ?? 2
And, we know that sin ?? + cos ?? = -v 2
? 2
sin ?? + 2
cos ?? > 2
1-( 1/v 2)
for ?? =
5?? 4
.
Q8. The period of ?????? ?? ?? + ?????? ?? ?? is
(a)
?? ??
(b) ??
(c) ?? ??
(d)
?? ?? ??
Ans: (a) ( sin
2
?? )
2
+ ( cos
2
?? )
2
= 1 - 2sin
2
?? cos
2
?? = 1 -
1
4
2sin
2
2?? = 1 - (
1-cos 4?? 4
)=
3
4
+
1
4
cos 4??
Therefore period is
2?? 4
=
?? 2
.
Trick: ?? ( ?? )= sin
4
?? + cos
4
?? ? ?? (
?? 2
+ ?? )= sin
4
(
?? 2
+ ?? )+ cos
4
(
?? 2
+ ?? )? ?? (
?? 2
+ ?? )=
cos
4
?? + sin
4
?? = ?? ( ?? )
Hence the period is
?? 2
.
Q9. In a ? ?????? , ?? : ?? : ?? = ?? : ?? : ?? . Then [?? + ?? + ?? v ?? ] is equal to
(a) ?? ??
(b) ?? ??
(c) ?? ??
(d) ?? ??
Ans: (c)
?? : ?? : ?? = 3: 5: 4 ? ?? + ?? + ?? = 12?? = 180
°
? ?? = 15
°
? ?? = 45
°
, ?? = 75
°
, ?? = 60
°
?? sin 45
°
=
?? sin 75
°
=
?? sin 60
°
= ?? (say)
i.e., ?? =
1
v 2
?? , ?? =
v 3 + 1
2v 2
?? , ?? =
v 3
2
?? . Hence [?? + ?? + ?? v 2] = 3?? .
Q10. If in a triangle ?????? , ?? ?????? ??
?? ?? + ?? ?????? ??
?? ?? =
?? ?? ?? , then a, c, b are:
(A) In A.P.
(B) In G.P.
(C) In H.P.
(D) None
Ans: (A) ?? cos
2
?? 2
+ ?? cos
2
?? 2
=
3
2
??
b
( cos ?? + 1)
2
+ ?? ( cos ?? + 1)
2
=
3
2
?? ?? cos ?? + ?? + ?? cos ?? + ?? = 3?? ? ?? + ?? + ( ?? cos ?? + ?? cos ?? )= 3?? ? ?? cos ?? + ?? cos ?? = ?? ? ?? + ?? = 2?? ? ?? , ?? , ?? are in A.P.
? a+ b = 2c ? a , b, c are in A.P.
Q11. Given ?? ?? + ?? ?? + ?????????? ?? (
?? ?? ( ?? + ?? ) )= ?? then, which of the following holds
good?
(A) ?? = ?? ;
?? ?? ? ??
(B) ?? = -?? ;
?? ?? ? ??
(C) ?? ? ?? ; ?? ? ??
(D) ?? , ?? are finite but not possible to find
Ans: (B) ?? 2
+ 2?? + cosec
2
(
?? 2
( ?? + ?? ) )= 0
[cosec
2
?? = 1 + cot
2
?? ]
? ?? 2
+ 2?? + 1 + cot
2
(
?? 2
( ?? + ?? ) )= 0
? ( ?? + 1)
2
+ cot
2
(
?? 2
( ?? + ?? ) )= 0
For the above equation to be valid
?? + 1 = 0 and cot (
?? 2
( ?? + ?? ) )= 0
? ?? = -1 and
?? 2
( ?? + ?? )= ( 2?? + 1)
?? 2
? ?? + ?? = 2?? + 1 ? ?? = 2?? + 2
? ?? = -1 and
?? 2
? 1
Q12. In any triangle ?????? , ( ?? + ?? )
?? ?????? ??
?? ?? + ( ?? - ?? )
?? ?????? ??
?? ?? =
(A) ?? ( ?? + ?? )
(B) ?? ( ?? + ?? )
(C) ?? ( ?? + ?? )
(D) ?? ??
Ans: (D) ( ?? + ?? )
2
sin
2
?? 2
+ ( ?? - ?? )
2
cos
2
?? 2
= ( ?? 2
+ ?? 2
+ 2???? ) sin
2
?? 2
+ ( ?? 2
+ ?? 2
- 2???? ) cos
2
?? 2
= ( ?? 2
+ ?? 2
)
?? 3
+ 2???? ( sin
2
?? 2
- cos
2
?? 2
)
= ?? 2
+ ?? 2
+ 2???? ( -cos ?? )
= ?? 2
+ ?? 2
- 2????
?? 2
+ ?? 2
- ?? 2
2????
= ?? 2
Q13. If in a ? ?????? , ?????? ?? ?? + ?????? ?? ?? + ?????? ?? ?? = ???????? ?? · ?????? ?? · ?????? ?? then
(A) ? ?????? may be a scalene triangle
(B) ? ?????? is a right triangle
(C) ? ?????? is an obtuse angled triangle
(D) ? ?????? is an equilateral triangle
Ans: (D) sin
3
?? + sin
3
?? + sin
3
?? = 3sin ?? sin ?? sin ??
? In triangle
?? sin ?? =
?? sin ?? =
?? sin ?? = ??
i.e. sin ?? ? ?? , sin ?? ? ?? , sin ?? ? ??
? a
3
+ b
3
+ c
3
= 3abc
?? 3
+ ?? 3
+ ?? 3
- 3?????? = ( ?? + ?? + ?? ) ( ?? 2
+ ?? 2
+ ?? 2
- ???? - ???? - ???? )= 0
? ?? 2
+ ?? 2
+ ?? 2
- ???? - ???? - ???? = 0
? ( ?? - ?? )
2
+ ( ?? - ?? )
2
+ ( ?? - ?? )
2
= 0 ? ?? = ?? = ??
? Triangle should be equilateral
Q14. ?????? ?? ?? = ???????? ?? ?????? ?? ?? ?????? ?? ?? in ?? = ?? = ?? has:
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