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Page 1
Solved Examples on Functions
JEE Mains
Q1. If ?? (?? )=
?? -|?? |
|?? |
, then ?? (-?? )=
(a) 1
(b) -2
(c) ??
(d) 2
Ans: (b) ?? (-1)=
-1-|-1|
|-1|
=
-1-1
1
=-2.
Q2. If ?? (?? )=?????? [
?? +?? ?? -?? ], then ?? [
?? ?? ?? +?? ?? ] is equal to
(a) [?? (?? )]
??
(b) [?? (?? )]
??
(c) ?? ?? (?? )
(d) ?? ?? (?? )
Ans: (c)
?? (?? )=log (
1+?? 1-?? )
??? (
2?? 1+?? 2
)=log [
1+
2?? 1+?? 2
1-
2?? 1+?? 2
]=log [
?? 2
+1+2?? ?? 2
+1-2?? ]=log [
1+?? 1-?? ]
2
=2log [
1+?? 1-?? ]=2?? (?? )
Q3. If ?? (?? )=?????? [?? ?? ]?? +?????? [-?? ?? ]?? , then
(a) ?? (
?? ?? )=??
(b) ?? (-?? )=??
(c) ?? (?? )=??
(d) ?? (
?? ?? )=-??
Ans: (d) ?? (?? )=cos [?? 2
]?? +cos [-?? 2
]??
?? (?? )=cos (9?? )+cos (-10?? )=cos (9?? )+cos (10?? )=2cos (
19?? 2
)cos (
?? 2
)
?? (
?? 2
)=2cos (
19?? 4
)cos (
?? 4
);?? (
?? 2
)=2×
-1
v2
×
1
v2
=-1.
Q4. If ?? :?? ??? satisfies ?? (?? +?? )=?? (?? )+?? (?? ) , for all ?? ,?? ??? and ?? (?? )=?? , then
?
?? =?? ?? ??? (?? ) is
(a)
?? ?? ??
(b)
?? (?? +?? )
??
(c) ?? ?? (?? +?? )
(d)
?? ?? (?? +?? )
??
Page 2
Solved Examples on Functions
JEE Mains
Q1. If ?? (?? )=
?? -|?? |
|?? |
, then ?? (-?? )=
(a) 1
(b) -2
(c) ??
(d) 2
Ans: (b) ?? (-1)=
-1-|-1|
|-1|
=
-1-1
1
=-2.
Q2. If ?? (?? )=?????? [
?? +?? ?? -?? ], then ?? [
?? ?? ?? +?? ?? ] is equal to
(a) [?? (?? )]
??
(b) [?? (?? )]
??
(c) ?? ?? (?? )
(d) ?? ?? (?? )
Ans: (c)
?? (?? )=log (
1+?? 1-?? )
??? (
2?? 1+?? 2
)=log [
1+
2?? 1+?? 2
1-
2?? 1+?? 2
]=log [
?? 2
+1+2?? ?? 2
+1-2?? ]=log [
1+?? 1-?? ]
2
=2log [
1+?? 1-?? ]=2?? (?? )
Q3. If ?? (?? )=?????? [?? ?? ]?? +?????? [-?? ?? ]?? , then
(a) ?? (
?? ?? )=??
(b) ?? (-?? )=??
(c) ?? (?? )=??
(d) ?? (
?? ?? )=-??
Ans: (d) ?? (?? )=cos [?? 2
]?? +cos [-?? 2
]??
?? (?? )=cos (9?? )+cos (-10?? )=cos (9?? )+cos (10?? )=2cos (
19?? 2
)cos (
?? 2
)
?? (
?? 2
)=2cos (
19?? 4
)cos (
?? 4
);?? (
?? 2
)=2×
-1
v2
×
1
v2
=-1.
Q4. If ?? :?? ??? satisfies ?? (?? +?? )=?? (?? )+?? (?? ) , for all ?? ,?? ??? and ?? (?? )=?? , then
?
?? =?? ?? ??? (?? ) is
(a)
?? ?? ??
(b)
?? (?? +?? )
??
(c) ?? ?? (?? +?? )
(d)
?? ?? (?? +?? )
??
Ans: (d) ?? (?? +?? )=?? (?? )+?? (?? )
put ?? =1,?? =0 ??? (1)=?? (1)+?? (0)=7
put ?? =1,?? =1 ??? (2)=2.?? (1)=2.7; similarly ?? (3)=3.7 and so on
??
?? =1
?? ??? (?? )=7(1+2+3+?..+?? )=
7?? (?? +1)
2
.
Q5. If ?? (?? )=
?? v
?? +?? v?? ?? -?? +
?? v
?? -?? v?? ?? -?? for ?? >?? , then ?? (???? )=
(a)
?? ??
(b)
?? ??
(c)
?? ??
(d)
?? ??
Ans: (c)
?? (?? )=
1
v
?? +2v2?? -4
+
1
v
?? -2v2?? -4
?? (11)=
1
v
11+2v18
+
1
v
11-2v18
=
1
3+v2
+
1
3-v2
=
3-v2
7
+
3+v2
7
=
6
7
.
Q6. Find the domain of definition of ?? (?? )=
?????? ?? (?? +?? )
?? ?? +?? ?? +??
(a) (-?? ,8)
(b) {-?? ,-?? }
(c) (-?? ,8)-{-?? ,-?? }
(d) (-8,8)
Ans: (c) Here ?? (?? )=
log
2
(?? +3)
?? 2
+3?? +2
=
log
2
(?? +3)
(?? +1)(?? +2)
exists if,
Numerator ?? +3>0 ??? >-3
and denominator (?? +1)(?? +2)?0??? ?-1,-2
Thus, from (i) and (ii); we have domain of ?? (?? ) is (-3,8)-{-1,-2}.
Q7. If the domain of function ?? (?? )=?? ?? -?? ?? +?? is (-8,8) , then the range of
function is
(a) (-8,8)
(b) [-?? ,8)
(c) (-?? ,?? )
(d) (-8,-?? )
Ans: (b) ?? 2
-6?? +7=(?? -3)
2
-2 Obviously, minimum value is -2 and maximum 8.
Q8. The domain of the function v?????? (?? ?? -?? ?? +?? ) is
(a) (-8,8)
(b) (-8,?? -v?? )?(?? +v?? ,8)
(c) (-8,?? ]?[?? ,8)
(d) [?? ,8)
Ans: (c) The function ?? (?? )=vlog (?? 2
-6?? +6) is defined when log (?? 2
-6?? +6)=0
??? 2
-6?? +6=1?(?? -5)(?? -1)=0
Page 3
Solved Examples on Functions
JEE Mains
Q1. If ?? (?? )=
?? -|?? |
|?? |
, then ?? (-?? )=
(a) 1
(b) -2
(c) ??
(d) 2
Ans: (b) ?? (-1)=
-1-|-1|
|-1|
=
-1-1
1
=-2.
Q2. If ?? (?? )=?????? [
?? +?? ?? -?? ], then ?? [
?? ?? ?? +?? ?? ] is equal to
(a) [?? (?? )]
??
(b) [?? (?? )]
??
(c) ?? ?? (?? )
(d) ?? ?? (?? )
Ans: (c)
?? (?? )=log (
1+?? 1-?? )
??? (
2?? 1+?? 2
)=log [
1+
2?? 1+?? 2
1-
2?? 1+?? 2
]=log [
?? 2
+1+2?? ?? 2
+1-2?? ]=log [
1+?? 1-?? ]
2
=2log [
1+?? 1-?? ]=2?? (?? )
Q3. If ?? (?? )=?????? [?? ?? ]?? +?????? [-?? ?? ]?? , then
(a) ?? (
?? ?? )=??
(b) ?? (-?? )=??
(c) ?? (?? )=??
(d) ?? (
?? ?? )=-??
Ans: (d) ?? (?? )=cos [?? 2
]?? +cos [-?? 2
]??
?? (?? )=cos (9?? )+cos (-10?? )=cos (9?? )+cos (10?? )=2cos (
19?? 2
)cos (
?? 2
)
?? (
?? 2
)=2cos (
19?? 4
)cos (
?? 4
);?? (
?? 2
)=2×
-1
v2
×
1
v2
=-1.
Q4. If ?? :?? ??? satisfies ?? (?? +?? )=?? (?? )+?? (?? ) , for all ?? ,?? ??? and ?? (?? )=?? , then
?
?? =?? ?? ??? (?? ) is
(a)
?? ?? ??
(b)
?? (?? +?? )
??
(c) ?? ?? (?? +?? )
(d)
?? ?? (?? +?? )
??
Ans: (d) ?? (?? +?? )=?? (?? )+?? (?? )
put ?? =1,?? =0 ??? (1)=?? (1)+?? (0)=7
put ?? =1,?? =1 ??? (2)=2.?? (1)=2.7; similarly ?? (3)=3.7 and so on
??
?? =1
?? ??? (?? )=7(1+2+3+?..+?? )=
7?? (?? +1)
2
.
Q5. If ?? (?? )=
?? v
?? +?? v?? ?? -?? +
?? v
?? -?? v?? ?? -?? for ?? >?? , then ?? (???? )=
(a)
?? ??
(b)
?? ??
(c)
?? ??
(d)
?? ??
Ans: (c)
?? (?? )=
1
v
?? +2v2?? -4
+
1
v
?? -2v2?? -4
?? (11)=
1
v
11+2v18
+
1
v
11-2v18
=
1
3+v2
+
1
3-v2
=
3-v2
7
+
3+v2
7
=
6
7
.
Q6. Find the domain of definition of ?? (?? )=
?????? ?? (?? +?? )
?? ?? +?? ?? +??
(a) (-?? ,8)
(b) {-?? ,-?? }
(c) (-?? ,8)-{-?? ,-?? }
(d) (-8,8)
Ans: (c) Here ?? (?? )=
log
2
(?? +3)
?? 2
+3?? +2
=
log
2
(?? +3)
(?? +1)(?? +2)
exists if,
Numerator ?? +3>0 ??? >-3
and denominator (?? +1)(?? +2)?0??? ?-1,-2
Thus, from (i) and (ii); we have domain of ?? (?? ) is (-3,8)-{-1,-2}.
Q7. If the domain of function ?? (?? )=?? ?? -?? ?? +?? is (-8,8) , then the range of
function is
(a) (-8,8)
(b) [-?? ,8)
(c) (-?? ,?? )
(d) (-8,-?? )
Ans: (b) ?? 2
-6?? +7=(?? -3)
2
-2 Obviously, minimum value is -2 and maximum 8.
Q8. The domain of the function v?????? (?? ?? -?? ?? +?? ) is
(a) (-8,8)
(b) (-8,?? -v?? )?(?? +v?? ,8)
(c) (-8,?? ]?[?? ,8)
(d) [?? ,8)
Ans: (c) The function ?? (?? )=vlog (?? 2
-6?? +6) is defined when log (?? 2
-6?? +6)=0
??? 2
-6?? +6=1?(?? -5)(?? -1)=0
This inequality hold if ?? =1 or ?? =5. Hence, the domain of the function will be (-8,1]?
[5,8) .
Q9. The domain of definition of the function ?? (?? ) given by ?? ?? +?? ?? =?? is
(a) (?? ,?? ]
(b) [?? ,?? ]
(c) (-8,?? ]
(d) (-8,?? )
Ans: (d) 2
?? =2-2
??
?? is real if 2-2
?? =0?2>2
?? ?1>??
? ?? ?(-8,1)
Q10. The domain of the function ?? (?? )=??????
?? +?? (?? ?? -?? ) is
(a) (-?? ,-?? )?(?? ,8)
(b) [-?? ,-?? )?[?? ,8)
(c) (-?? ,-?? )?(-?? ,-?? )?(?? ,8)
(d) [-?? ,-?? )?(-?? ,-?? )?[?? ,8]
Solution: (c) ?? (?? ) is to be defined when ?? 2
-1>0
??? 2
>1,??? <-1 or ?? >1 and 3+?? >0
??? >-3 and ?? ?-2
??? ?? =(-3,-2)?(-2,-1)?(1,8).
Q11. The range of
?? +?? ?? ?? ?? is
(a) (?? ,?? )
(b) (?? ,8)
(c) [?? ,?? ]
(d) [?? ,8)
Ans: (b) Let ?? =
1+?? 2
?? 2
??? 2
?? =1+?? 2
??? 2
(?? -1)=1??? 2
=
1
?? -1
Now since, ?? 2
>0?
1
?? -1
>0?(?? -1)>0??? >1??? ?(1,8)
Trick : ?? =
1+?? 2
?? 2
=1+
1
?? 2
. Now since,
1
?? 2
is always >0??? >1??? ?(1,8) .
Q12.If ?? (?? )=?? ?????? (???? +?? )+?? , then range of ?? (?? ) is
(a) [?? +?? ,?? +?? ?? ]
(b) [?? -?? ,?? +?? ]
(c) [?? +?? ,?? -?? ]
Ans: (a) [?? +?? ,?? +2?? ]
?? (?? )=?? cos (???? +?? )+??
For minimum cos (???? +?? )=-1
from (i), ?? (?? )=-?? +?? =(?? -?? ) ,
for maximum cos (???? +?? )=1
from (i), ?? (?? )=?? +?? =(?? +?? )
? Range of ?? (?? )=[?? -?? ,?? +?? ].
Q12. Range of the function ?? (?? )=
?? ?? +?? +?? ?? ?? +?? +?? ;?? ??? is
(a) (?? ,8)
(b) (?? ,???? /?? )
Page 4
Solved Examples on Functions
JEE Mains
Q1. If ?? (?? )=
?? -|?? |
|?? |
, then ?? (-?? )=
(a) 1
(b) -2
(c) ??
(d) 2
Ans: (b) ?? (-1)=
-1-|-1|
|-1|
=
-1-1
1
=-2.
Q2. If ?? (?? )=?????? [
?? +?? ?? -?? ], then ?? [
?? ?? ?? +?? ?? ] is equal to
(a) [?? (?? )]
??
(b) [?? (?? )]
??
(c) ?? ?? (?? )
(d) ?? ?? (?? )
Ans: (c)
?? (?? )=log (
1+?? 1-?? )
??? (
2?? 1+?? 2
)=log [
1+
2?? 1+?? 2
1-
2?? 1+?? 2
]=log [
?? 2
+1+2?? ?? 2
+1-2?? ]=log [
1+?? 1-?? ]
2
=2log [
1+?? 1-?? ]=2?? (?? )
Q3. If ?? (?? )=?????? [?? ?? ]?? +?????? [-?? ?? ]?? , then
(a) ?? (
?? ?? )=??
(b) ?? (-?? )=??
(c) ?? (?? )=??
(d) ?? (
?? ?? )=-??
Ans: (d) ?? (?? )=cos [?? 2
]?? +cos [-?? 2
]??
?? (?? )=cos (9?? )+cos (-10?? )=cos (9?? )+cos (10?? )=2cos (
19?? 2
)cos (
?? 2
)
?? (
?? 2
)=2cos (
19?? 4
)cos (
?? 4
);?? (
?? 2
)=2×
-1
v2
×
1
v2
=-1.
Q4. If ?? :?? ??? satisfies ?? (?? +?? )=?? (?? )+?? (?? ) , for all ?? ,?? ??? and ?? (?? )=?? , then
?
?? =?? ?? ??? (?? ) is
(a)
?? ?? ??
(b)
?? (?? +?? )
??
(c) ?? ?? (?? +?? )
(d)
?? ?? (?? +?? )
??
Ans: (d) ?? (?? +?? )=?? (?? )+?? (?? )
put ?? =1,?? =0 ??? (1)=?? (1)+?? (0)=7
put ?? =1,?? =1 ??? (2)=2.?? (1)=2.7; similarly ?? (3)=3.7 and so on
??
?? =1
?? ??? (?? )=7(1+2+3+?..+?? )=
7?? (?? +1)
2
.
Q5. If ?? (?? )=
?? v
?? +?? v?? ?? -?? +
?? v
?? -?? v?? ?? -?? for ?? >?? , then ?? (???? )=
(a)
?? ??
(b)
?? ??
(c)
?? ??
(d)
?? ??
Ans: (c)
?? (?? )=
1
v
?? +2v2?? -4
+
1
v
?? -2v2?? -4
?? (11)=
1
v
11+2v18
+
1
v
11-2v18
=
1
3+v2
+
1
3-v2
=
3-v2
7
+
3+v2
7
=
6
7
.
Q6. Find the domain of definition of ?? (?? )=
?????? ?? (?? +?? )
?? ?? +?? ?? +??
(a) (-?? ,8)
(b) {-?? ,-?? }
(c) (-?? ,8)-{-?? ,-?? }
(d) (-8,8)
Ans: (c) Here ?? (?? )=
log
2
(?? +3)
?? 2
+3?? +2
=
log
2
(?? +3)
(?? +1)(?? +2)
exists if,
Numerator ?? +3>0 ??? >-3
and denominator (?? +1)(?? +2)?0??? ?-1,-2
Thus, from (i) and (ii); we have domain of ?? (?? ) is (-3,8)-{-1,-2}.
Q7. If the domain of function ?? (?? )=?? ?? -?? ?? +?? is (-8,8) , then the range of
function is
(a) (-8,8)
(b) [-?? ,8)
(c) (-?? ,?? )
(d) (-8,-?? )
Ans: (b) ?? 2
-6?? +7=(?? -3)
2
-2 Obviously, minimum value is -2 and maximum 8.
Q8. The domain of the function v?????? (?? ?? -?? ?? +?? ) is
(a) (-8,8)
(b) (-8,?? -v?? )?(?? +v?? ,8)
(c) (-8,?? ]?[?? ,8)
(d) [?? ,8)
Ans: (c) The function ?? (?? )=vlog (?? 2
-6?? +6) is defined when log (?? 2
-6?? +6)=0
??? 2
-6?? +6=1?(?? -5)(?? -1)=0
This inequality hold if ?? =1 or ?? =5. Hence, the domain of the function will be (-8,1]?
[5,8) .
Q9. The domain of definition of the function ?? (?? ) given by ?? ?? +?? ?? =?? is
(a) (?? ,?? ]
(b) [?? ,?? ]
(c) (-8,?? ]
(d) (-8,?? )
Ans: (d) 2
?? =2-2
??
?? is real if 2-2
?? =0?2>2
?? ?1>??
? ?? ?(-8,1)
Q10. The domain of the function ?? (?? )=??????
?? +?? (?? ?? -?? ) is
(a) (-?? ,-?? )?(?? ,8)
(b) [-?? ,-?? )?[?? ,8)
(c) (-?? ,-?? )?(-?? ,-?? )?(?? ,8)
(d) [-?? ,-?? )?(-?? ,-?? )?[?? ,8]
Solution: (c) ?? (?? ) is to be defined when ?? 2
-1>0
??? 2
>1,??? <-1 or ?? >1 and 3+?? >0
??? >-3 and ?? ?-2
??? ?? =(-3,-2)?(-2,-1)?(1,8).
Q11. The range of
?? +?? ?? ?? ?? is
(a) (?? ,?? )
(b) (?? ,8)
(c) [?? ,?? ]
(d) [?? ,8)
Ans: (b) Let ?? =
1+?? 2
?? 2
??? 2
?? =1+?? 2
??? 2
(?? -1)=1??? 2
=
1
?? -1
Now since, ?? 2
>0?
1
?? -1
>0?(?? -1)>0??? >1??? ?(1,8)
Trick : ?? =
1+?? 2
?? 2
=1+
1
?? 2
. Now since,
1
?? 2
is always >0??? >1??? ?(1,8) .
Q12.If ?? (?? )=?? ?????? (???? +?? )+?? , then range of ?? (?? ) is
(a) [?? +?? ,?? +?? ?? ]
(b) [?? -?? ,?? +?? ]
(c) [?? +?? ,?? -?? ]
Ans: (a) [?? +?? ,?? +2?? ]
?? (?? )=?? cos (???? +?? )+??
For minimum cos (???? +?? )=-1
from (i), ?? (?? )=-?? +?? =(?? -?? ) ,
for maximum cos (???? +?? )=1
from (i), ?? (?? )=?? +?? =(?? +?? )
? Range of ?? (?? )=[?? -?? ,?? +?? ].
Q12. Range of the function ?? (?? )=
?? ?? +?? +?? ?? ?? +?? +?? ;?? ??? is
(a) (?? ,8)
(b) (?? ,???? /?? )
(c) (?? ,?? /?? ]
(d) (?? ,?? /?? ]
Ans: (c) ?? (?? )=1+
1
(?? +
1
2
)
2
+
3
4
? Range =(1,7/3].
Q13. Let ?? :?? ??? be a function defined by ?? (?? )=
?? -?? ?? -?? , where ?? ??? . Then
(a) ?? is one-one onto
(b) ?? is one-one into
(c) ?? is many one onto
(d) ?? is many one into
Ans: (b) For any ?? ,?? ??? , we have
?? (?? )=?? (?? )?
?? -?? ?? -?? =
?? -?? ?? -?? ??? =??
??? is one-one
Let ?? ??? such that ?? (?? )=?? ?
?? -?? ?? -?? =?? ??? =
?? -????
1-??
Clearly ?? ??? for ?? =1. So, ?? is not onto.
Q14. Let ?? (?? )=v?? ?? +???? , then the graph of the function ?? =?? (?? ) is symmetrical
about
(a) The ?? -axis
(b) The ?? -axis
(c) The origin
(d) The line ?? =??
Ans: (b) ?? (?? )=v?? 4
+15??? (-?? )=v(-?? )
4
+15=v?? 4
+15=?? (?? )
??? (-?? )=?? (?? )??? (?? ) is an even function ??? (?? ) is symmetric about y-axis.
Q15. The function ?? (?? )=?????? (?????? (?? +v?? ?? +?? )) is
(a) Even function
(b) Odd function
(c) Neither even nor odd
(d) Periodic function
Ans: (b) ?? (?? )=sin (log (?? +v1+?? 2
))
??? (-?? )=sin [log (-?? +v1+?? 2
)]??? (-?? )=sin log ((v1+?? 2
-?? )
(v1+?? 2
+?? )
(v1+?? 2
+?? )
)
??? (-?? )=sin log [
1
(?? +v1+?? 2
)
]??? (-?? )=sin [log (?? +v1+?? 2
)
-1
]
??? (-?? )=sin [-log (?? +v1+?? 2
)]??? (-?? )=-sin [log (?? +v1+?? 2
)]??? (-?? )=-?? (?? )
??? (?? ) is odd function.
Page 5
Solved Examples on Functions
JEE Mains
Q1. If ?? (?? )=
?? -|?? |
|?? |
, then ?? (-?? )=
(a) 1
(b) -2
(c) ??
(d) 2
Ans: (b) ?? (-1)=
-1-|-1|
|-1|
=
-1-1
1
=-2.
Q2. If ?? (?? )=?????? [
?? +?? ?? -?? ], then ?? [
?? ?? ?? +?? ?? ] is equal to
(a) [?? (?? )]
??
(b) [?? (?? )]
??
(c) ?? ?? (?? )
(d) ?? ?? (?? )
Ans: (c)
?? (?? )=log (
1+?? 1-?? )
??? (
2?? 1+?? 2
)=log [
1+
2?? 1+?? 2
1-
2?? 1+?? 2
]=log [
?? 2
+1+2?? ?? 2
+1-2?? ]=log [
1+?? 1-?? ]
2
=2log [
1+?? 1-?? ]=2?? (?? )
Q3. If ?? (?? )=?????? [?? ?? ]?? +?????? [-?? ?? ]?? , then
(a) ?? (
?? ?? )=??
(b) ?? (-?? )=??
(c) ?? (?? )=??
(d) ?? (
?? ?? )=-??
Ans: (d) ?? (?? )=cos [?? 2
]?? +cos [-?? 2
]??
?? (?? )=cos (9?? )+cos (-10?? )=cos (9?? )+cos (10?? )=2cos (
19?? 2
)cos (
?? 2
)
?? (
?? 2
)=2cos (
19?? 4
)cos (
?? 4
);?? (
?? 2
)=2×
-1
v2
×
1
v2
=-1.
Q4. If ?? :?? ??? satisfies ?? (?? +?? )=?? (?? )+?? (?? ) , for all ?? ,?? ??? and ?? (?? )=?? , then
?
?? =?? ?? ??? (?? ) is
(a)
?? ?? ??
(b)
?? (?? +?? )
??
(c) ?? ?? (?? +?? )
(d)
?? ?? (?? +?? )
??
Ans: (d) ?? (?? +?? )=?? (?? )+?? (?? )
put ?? =1,?? =0 ??? (1)=?? (1)+?? (0)=7
put ?? =1,?? =1 ??? (2)=2.?? (1)=2.7; similarly ?? (3)=3.7 and so on
??
?? =1
?? ??? (?? )=7(1+2+3+?..+?? )=
7?? (?? +1)
2
.
Q5. If ?? (?? )=
?? v
?? +?? v?? ?? -?? +
?? v
?? -?? v?? ?? -?? for ?? >?? , then ?? (???? )=
(a)
?? ??
(b)
?? ??
(c)
?? ??
(d)
?? ??
Ans: (c)
?? (?? )=
1
v
?? +2v2?? -4
+
1
v
?? -2v2?? -4
?? (11)=
1
v
11+2v18
+
1
v
11-2v18
=
1
3+v2
+
1
3-v2
=
3-v2
7
+
3+v2
7
=
6
7
.
Q6. Find the domain of definition of ?? (?? )=
?????? ?? (?? +?? )
?? ?? +?? ?? +??
(a) (-?? ,8)
(b) {-?? ,-?? }
(c) (-?? ,8)-{-?? ,-?? }
(d) (-8,8)
Ans: (c) Here ?? (?? )=
log
2
(?? +3)
?? 2
+3?? +2
=
log
2
(?? +3)
(?? +1)(?? +2)
exists if,
Numerator ?? +3>0 ??? >-3
and denominator (?? +1)(?? +2)?0??? ?-1,-2
Thus, from (i) and (ii); we have domain of ?? (?? ) is (-3,8)-{-1,-2}.
Q7. If the domain of function ?? (?? )=?? ?? -?? ?? +?? is (-8,8) , then the range of
function is
(a) (-8,8)
(b) [-?? ,8)
(c) (-?? ,?? )
(d) (-8,-?? )
Ans: (b) ?? 2
-6?? +7=(?? -3)
2
-2 Obviously, minimum value is -2 and maximum 8.
Q8. The domain of the function v?????? (?? ?? -?? ?? +?? ) is
(a) (-8,8)
(b) (-8,?? -v?? )?(?? +v?? ,8)
(c) (-8,?? ]?[?? ,8)
(d) [?? ,8)
Ans: (c) The function ?? (?? )=vlog (?? 2
-6?? +6) is defined when log (?? 2
-6?? +6)=0
??? 2
-6?? +6=1?(?? -5)(?? -1)=0
This inequality hold if ?? =1 or ?? =5. Hence, the domain of the function will be (-8,1]?
[5,8) .
Q9. The domain of definition of the function ?? (?? ) given by ?? ?? +?? ?? =?? is
(a) (?? ,?? ]
(b) [?? ,?? ]
(c) (-8,?? ]
(d) (-8,?? )
Ans: (d) 2
?? =2-2
??
?? is real if 2-2
?? =0?2>2
?? ?1>??
? ?? ?(-8,1)
Q10. The domain of the function ?? (?? )=??????
?? +?? (?? ?? -?? ) is
(a) (-?? ,-?? )?(?? ,8)
(b) [-?? ,-?? )?[?? ,8)
(c) (-?? ,-?? )?(-?? ,-?? )?(?? ,8)
(d) [-?? ,-?? )?(-?? ,-?? )?[?? ,8]
Solution: (c) ?? (?? ) is to be defined when ?? 2
-1>0
??? 2
>1,??? <-1 or ?? >1 and 3+?? >0
??? >-3 and ?? ?-2
??? ?? =(-3,-2)?(-2,-1)?(1,8).
Q11. The range of
?? +?? ?? ?? ?? is
(a) (?? ,?? )
(b) (?? ,8)
(c) [?? ,?? ]
(d) [?? ,8)
Ans: (b) Let ?? =
1+?? 2
?? 2
??? 2
?? =1+?? 2
??? 2
(?? -1)=1??? 2
=
1
?? -1
Now since, ?? 2
>0?
1
?? -1
>0?(?? -1)>0??? >1??? ?(1,8)
Trick : ?? =
1+?? 2
?? 2
=1+
1
?? 2
. Now since,
1
?? 2
is always >0??? >1??? ?(1,8) .
Q12.If ?? (?? )=?? ?????? (???? +?? )+?? , then range of ?? (?? ) is
(a) [?? +?? ,?? +?? ?? ]
(b) [?? -?? ,?? +?? ]
(c) [?? +?? ,?? -?? ]
Ans: (a) [?? +?? ,?? +2?? ]
?? (?? )=?? cos (???? +?? )+??
For minimum cos (???? +?? )=-1
from (i), ?? (?? )=-?? +?? =(?? -?? ) ,
for maximum cos (???? +?? )=1
from (i), ?? (?? )=?? +?? =(?? +?? )
? Range of ?? (?? )=[?? -?? ,?? +?? ].
Q12. Range of the function ?? (?? )=
?? ?? +?? +?? ?? ?? +?? +?? ;?? ??? is
(a) (?? ,8)
(b) (?? ,???? /?? )
(c) (?? ,?? /?? ]
(d) (?? ,?? /?? ]
Ans: (c) ?? (?? )=1+
1
(?? +
1
2
)
2
+
3
4
? Range =(1,7/3].
Q13. Let ?? :?? ??? be a function defined by ?? (?? )=
?? -?? ?? -?? , where ?? ??? . Then
(a) ?? is one-one onto
(b) ?? is one-one into
(c) ?? is many one onto
(d) ?? is many one into
Ans: (b) For any ?? ,?? ??? , we have
?? (?? )=?? (?? )?
?? -?? ?? -?? =
?? -?? ?? -?? ??? =??
??? is one-one
Let ?? ??? such that ?? (?? )=?? ?
?? -?? ?? -?? =?? ??? =
?? -????
1-??
Clearly ?? ??? for ?? =1. So, ?? is not onto.
Q14. Let ?? (?? )=v?? ?? +???? , then the graph of the function ?? =?? (?? ) is symmetrical
about
(a) The ?? -axis
(b) The ?? -axis
(c) The origin
(d) The line ?? =??
Ans: (b) ?? (?? )=v?? 4
+15??? (-?? )=v(-?? )
4
+15=v?? 4
+15=?? (?? )
??? (-?? )=?? (?? )??? (?? ) is an even function ??? (?? ) is symmetric about y-axis.
Q15. The function ?? (?? )=?????? (?????? (?? +v?? ?? +?? )) is
(a) Even function
(b) Odd function
(c) Neither even nor odd
(d) Periodic function
Ans: (b) ?? (?? )=sin (log (?? +v1+?? 2
))
??? (-?? )=sin [log (-?? +v1+?? 2
)]??? (-?? )=sin log ((v1+?? 2
-?? )
(v1+?? 2
+?? )
(v1+?? 2
+?? )
)
??? (-?? )=sin log [
1
(?? +v1+?? 2
)
]??? (-?? )=sin [log (?? +v1+?? 2
)
-1
]
??? (-?? )=sin [-log (?? +v1+?? 2
)]??? (-?? )=-sin [log (?? +v1+?? 2
)]??? (-?? )=-?? (?? )
??? (?? ) is odd function.
Q16. The function ?? (?? )=??????
????
?? +????????
????
?? -??????
????
?? is periodic with period
(a) 6
(b) 3
(c) 4
(d) 12
Ans: (d) ?sin ?? has period =2?? ?sin
????
2
has period =
2?? ?? 2
=4
?cos ?? has period =2?? ?cos
????
3
has period =
2?? ?? 3
=6?2cos
????
3
has period =6
?tan ?? has period =?? ?tan
????
4
has period =
?? ?? 4
=4.
L.C.M. of 4,6 and 4=12, period of ?? (?? )=12.
Q17. The period of the function ?? (?? )=?????? ?? ?? is
(a)
?? ??
(b) ??
(c) ?? ??
(d) None of these
Ans: (b) sin
2
?? =
1-cos 2?? 2
? Period =
2?? 2
=?? .
Q18. If ?? (?? )=?? ?? +?? -?? and
?? ?? (?? °?? )(?? )=?? ?? ?? -?? ?? +?? , then ?? (?? ) is equal to
(a) ?? ?? -??
(b) ?? ?? +??
(c) ?? ?? ?? +?? ?? +??
(d) ?? ?? ?? -?? ?? -??
Ans: (a) ?? (?? )=?? 2
+?? -2?(?? °?? )(?? )=?? [?? (?? )]=[?? (?? )]
2
+?? (?? )-2
Given,
1
2
(?? °?? )(?? )=2?? 2
-5?? +2 ?
1
2
[?? (?? )]
2
+
1
2
?? (?? )-1=2?? 2
-5?? +2
?[?? (?? )]
2
+?? (?? )=4?? 2
-10?? +6??? (?? )[?? (?? )+1]=(2?? -3)[(2?? -3)+1]??? (?? )=2?? -
3.
Q19.If ?? (?? )=
?? ?? -?? ?? -?? , then [?? {?? (?? )}] equals
(a) ??
(b)-??
(c)
?? ??
(d)-
?? ??
Ans: (a) ?? [?? (?? )]=
2(
2?? -3
?? -2
)-3
(
2?? -3
?? -2
)-2
=??
Q20. Suppose that ?? (?? )=?? +v?? and ?? (?? (?? ))=?? +?? v?? +?? , then ?? (?? ) is
(a) ?? +?? ?? ??
(b) ?? +?? ??
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