MODULE -1 (INVERSE TRIGONOMETRIC FUNCTION) EVOLVE BATCH TIMING 9-10:30 P.
JEE MAINS -SINGLE CHOICE CORRECT (EX-1)
1. cos −1 1/ 2 + 2sin −1 1/ 2 is equal to
(a) / 4 (b) /6 (c) /3 (d) 2 / 3
2. The principal value of sin −1 sin ( 2 / 3) is
(a) −2 / 3 (b) 2 / 3 (c) 4 / 3 (d) None of
these
3. If A = tan −1 x, then the value of sin 2 A is
2x 2x 2x
(a) (b) (c) (d) None of
1 − x2 (1 − x ) 2 1 + x2
these
4. The value of tan sin −1 ( 3 / 5 ) + cot −1 ( 3 / 2 ) is
(a) 6 /17 (b) 7 /16 (c) 16 / 7 (d) 17 / 6
5. The value of
2− 3
−1 12
sin cot sin −1
−1
+ cos + sec −1
2 is
4 4
(a) 0 (b) /4 (c) /6 (d) / 2
6. sin −1 x + cos −1 x = / 2 is true for
(a) all real x (b) positive real x only (c) x −1, 1 only (d)
x 0, / 2 only
7. cos cos −1
( ) ( )
3 / 2 + sin −1 1/ 2 is equal to
(a) ( 3 + 1) / 2 2 (b) ( )
3 −1 / 2 2 (c) (− )
3 +1 / 2 2 (d)
( − 3 − 1) / 2 2
8. tan sin −1 ( 3 / 5 ) − cos −1 ( −4 / 5 ) is equal to
(a) 24 / 7 (b) −24 / 7 (c) 7 / 24 (d) −7 / 24
If sin sin (1/ 5 ) + cos x = 1 then x is equal to
−1 −1
9.
(a) 1 (b) 0 (c) 4/5 (d) 1/ 5
MODULE -1 (INVERSE TRIGONOMETRIC FUNCTION) EVOLVE BATCH TIMING 9-10:30 P.M
10. Indicate the relation which is not true
(a) tan tan −1 x = x (b) cot cot −1 x = x (c) tan −1 tan x = x (d)
sin sin −1 x = x
11. cos −1 ( cos 5 / 4 ) is given by
(a) 5 / 4 (b) 3 / 4 (c) − / 4 (d) None of
these
12. sin cot −1 cos tan −1 x is equal to
x2 −1 x2 − 2 x2 + 1
(a) 2 (b) 2 (c) 2 (d) None of
x +2 x +1 x +2
these
13. The value of sin sin −1
( )
5 / 4 + tan −1 ( ( 5 /11) ) is
(a) 5/ 4 (11) (b) 4/ ( 35 ) (c) ( 55 ) / 8 (d) None of
these
14. If sin −1 (1/ 3) + sin −1 ( 2 / 3 ) = sin −1 x, then value of x is
(a) 0 (b) ( 5 −4 2 /9 ) (c) ( 5 +4 2 /9 ) (d) / 2
15. The value of x satisfying sin −1 x + sin −1 (1 − x ) = cos −1 x are
(a) 1, 0 (b) 1, − 1 (c) 0, 1/ 2 (d) None of
these
16. If x = tan −1 (1/ 7 ) and y = tan −1 (1/ 3) , then
(a) cos 2 x = sin 4 y (b) cos 4 y = cos 2 x (c) cos 2 y = sin 4 x (d) None of
these
( )
2
17. The solution of the inequality cot −1 x − 5cot −1 x + 6 0 is
(a) ( cot 3, cot 2 ) (b) ( −, cot 3) ( cot 2, ) (c) ( cot 2, ) (d) None of
these
18. The value of cos −1 ( cos12 ) − sin −1 ( sin12 ) is
(a) 0 (b) (c) 8 − 24 (d) None of
these
3
If sin −1 x + sin −1 y + sin −1 z = 3 / 2 then the value of x + y100 + z100 −
1000
19. is
101
x + y101 + z101
MODULE -1 (INVERSE TRIGONOMETRIC FUNCTION) EVOLVE BATCH TIMING 9-10:30 P.M
(a) 0 (b) 1 (c) 2 (d) 3
20. ( ) (
If sin cot −1 ( x + 1) = cos tan −1 x , then x = )
1 1 9
(a) − (b) (c) 0 (d)
2 2 4
4 2
21. The value of tan cos −1 + tan −1 is
5 3
(a) 6 /17 (b) 7 /16 (c) 16 / 7 (d) None of
these
22. The value of sin cot −1 x = ( )
(1 + x )
−3/ 2
(a) 1 + x2 (b) x (c) 2
(d)
(1 + x )
−1/ 2
2
2
23. The principal value of sin −1 sin is
3
(a) −2 / 3 (b) 2 / 3 (c) 4 / 3 (d) None of
these
1
24. The numerical value of tan 2 tan −1 − is equal to
5 4
7 6 5 4
(a) − (b) − (c) − (d) −
17 17 17 17
1− x 1
25. If tan −1 = tan −1 x, then values of x is :
1+ x 2
1 1
(a) (b) (c) 3 (d) 2
2 3
MODULE -1 (INVERSE TRIGONOMETRIC FUNCTION) EVOLVE BATCH TIMING 9-10:30 P.M
EXERCISE – II
IIT-JEE SINGLE CHOICE CORRECT
17
1. cos −1 cos − is equal to
15
17 17 2 13
(a) − (b) (c) (d)
15 15 15 15
1
2. If x 0, then tan −1 equals
x
(a) cot −1 x (b) − cot −1 x (c) − + cot −1 x (d) None of
these
3.
1 −1
(
If −1 x − , then sin 3 x − 4 x equals
2
3
)
(a) 3sin −1 x (b) − 3sin −1 x (c) − − 3sin −1 x (d) None of
these
4. The number of triplets satisfying sin −1 x + cos −1 y + sin −1 z = 2 , is
(a) 0 (b) 2 (c) 1 (d) Infinite
5. Which of the following is not true?
2x 1
(a) 2 tan −1 x = − sin −1 if x 1 (b) tan −1 = − + cot −1 x if x 0
1 + x2 x
x2 −1
(c) sec−1 x = sec −1 if x 1 (d) None of these
x
xy −1 yz −1 zx
6. If x 2 + y 2 + z 2 = r 2 , then tan −1 + tan + tan is equal to if x, y, z , r 0
zr xr yr
(a) (b) /2 (c) 0 (d) None of
these
1− x
7. The smallest and the largest values of tan −1 , 0 x 1 are
1+ x
(a) 0, (b) 0, / 4 (c) − / 4, / 4 (d)
/ 4, / 2
−1 3 −1 2 3 −1 2 3
Sum infinite terms of the series cot 1 + + cot 2 + + cot 3 + + .... is
2
8.
4 4 4
MODULE -1 (INVERSE TRIGONOMETRIC FUNCTION) EVOLVE BATCH TIMING 9-10:30 P.M
(a) / 4 (b) tan −1 2 (c) tan −1 3 (d) None of
these
9. If tan −1 x + tan −1 2 x + tan −1 3 x = , then :
(a) x = 0 (b) x = −1 (c) x =1 (d) x
10. If = sin −1 x + cos −1 x − tan −1 x, x 0, then the smallest interval in which lies is
3
(a) (b) 0 (c) − 0 (d)
2 4 4 4
4 2
1
( ) (
1 + x 2 x cos cot −1 x + sin cot −1 x ) − 1 =
2 2
11. If 0 x 1, then
x
(a) (b) x (c) x 1 + x2 (d) 1 + x2
1+ x 2
12. (
The number of solution of the equation; 1 + x 2 + 2 x sin cos −1 y = 0 is : )
(a) 1 (b) 2 (c) 3 (d) 4
1 + cos
13. tan 2 tan −1 + tan is equal to, ( 0, )
1 − cos
(a) / 4 (b) /2 (c) (d) 0
1 + x2 −1
14. If tan −1 = 4, then x is equal to
x
(a) tan 2 (b) tan 4 (c) tan (1/ 4 ) (d) tan 8
1 3 1
15. If x − , 1 , then sin −1 x− 1 − x 2 is equal to
2
2 2
1
(a) sin −1 − sin −1 x (b) sin −1 x − (c) sin −1 x + (d) None of
2 6 6
these
Let f ( x ) = cosec 1 + sin x , where [.] denotes the greatest integer function. Then f ( x )
−1 2
16.
is equals to
−1
(a)
2
(b) , cosec 2
2
(c) cosec 2 −1
(d) None of
these
17. The number of real solution of ( x, y ) , where y = sin x, y = cos −1 ( cos x ) , − 2 x 2 , is
MODULE -1 (INVERSE TRIGONOMETRIC FUNCTION) EVOLVE BATCH TIMING 9-10:30 P.M
(a) 2 (b) 1 (c) 3 (d) 4
18. If is the only real root of the equation x3 + bx 2 + cx + 1 = 0 ( b c ) , then the value of
1
tan −1 + tan −1 is equal to
(a) (b) − (c) 0 (d) Non-
2 2
existent
n
19. If cot −1 , n being a natural number, then maximum value of n is
6
(a) 1 (b) 5 (c) 9 (d) None of
these
20. If sin −1 a + sin −1 b + sin −1 c = , then the value of a (1 − a ) + b (1 − b ) + c (1 − c )
2 2 2
will
be
1 1
(a) 2abc (b) abc (c) abc (d) abc
2 3
n n
21. If cos−1 i = 0, then
i =1
i =1
i is equal to
(a) n (b) −n (c) 0 (d) None of
these
22. sin −1 ( sin 5 ) x 2 − 4 x holds if
(a) x = 2 − 9 − 2 (b) x = 2 + 9 − 2
(c) x 2 + 9 − 2 (d) (
x 2 − 9 − 2 , 2 + 9 − 2 )
23. If , , are the roots of the equation x 3 + px 2 + 2 x + p = 0, then general value of
tan −1 + tan −1 + tan −1 is
n
(a) n (b)
2
(c) ( 2n + 1) (d) depend upon value of p
2
If sin cos sin tan x = 1, where [.] denotes the greatest integer function, then x is
−1 −1 −1 −1
24.
given by the interval
(a) tan sin cos1, tan sin cos sin1 (b) ( tan sin cos1, tan sin cos sin1)
(c) −1, 1 (d) sin cos tan1,sin cos sin tan1
MODULE -1 (INVERSE TRIGONOMETRIC FUNCTION) EVOLVE BATCH TIMING 9-10:30 P.M
10 10
25. If sin
i =1
−1
xi = 5 , then x
i =1
2
i is equal to
(a) 0 (b) 5 (c) 10 (d) None of
these MORE THAN ONE CHOICE CORRECT
EXERCISE – III
1. If and are the roots of the equation 6 x 2 + 11x + 3 = 0, then
(a) both cos −1 and cos −1 are real
(b) exactly one out of cos −1 and cos −1 is real
(c) exactly one out of cosec −1 and cosec −1 is real
(d) both cot −1 and cot −1 are real
1
2. If x 1, then which of the following are real
2
(a) sec −1 x (b) cosec −1 x (c) tan −1 x (d) cos −1 x
3.
4 4 4 2 2 2 2 2 2 2 2 2 2
(
If sin −1 x + sin −1 y + sin −1 z = and x + y + z + 4 x y z = k x y + y z + z x , then )
the value of k is
(a) 1 (b) 2 (c) − 2 (d) −1
5 −1 2
4. Let tan −1 tan = , tan − tan = . Then
4 3
7
(a) (b) 4 − 3 = 0 (c) + = (d)
12
+ =
( cos x ) − ( sin x )
4 4
−1 −1
5. 0. Then x may belongs to
1
(a) x (b) −1 x 0 (c) 0 x 1 (d)
2
1
0 x
2
6. , and are the angles given by = 2 tan −1 ( ) ( )
2 − 1 , = 3sin −1 1/ 2 + sin −1 ( −1/ 2 )
and = cos −1 (1/ 3) then
(a) (b) (c) (d)
= =
7. Indicate the relation which is/are not true
MODULE -1 (INVERSE TRIGONOMETRIC FUNCTION) EVOLVE BATCH TIMING 9-10:30 P.M
(a) tan tan −1 x = x (b) cot cot −1 x = x
(c) tan −1 tan x = x (d) sin sin −1 x = x
3
8. The value(s) of x of the equation 2sin −1 x 2 − x + 1 + cos −1 x 2 − x = is / are
2
(a) 1 (b) 2 (c) 0 (d) −3
9. 3cos −1 x = 2 x + has
(a) x = −1 as the only one solution (b) x = −1 as one of the solution
(c) no solution (d) more than one solution
1
10. The integral solution of the equation tan −1 x + tan −1 −1
= tan 3 is
y
(a) ( 2, 7 ) (b) ( 4, − 13) (c) ( 5, − 8 ) (d) (1, 2 )
11. (
sin sin −1 x ) x
2
holds, then the value of x may belongs to
1
(a) 0, 1 (b) −1, 0 (c) 0, (d) −1, 1
2
x 1
12. If f ( x ) = cos −1 x + cos −1 + 3 − 3x 2 , then
2 2
2 2 2
(a) f = (b) f = 2 cos −1 −
3 3 3 3 3
1 1 1
(c) f = (d) f = 2 cos −1 −
3 3 3 3 3
13. (
If cos 2sin −1 x = ) 1
9
, then the value(s) of x is / are
2 2 5 5
(a) (b) − (c) (d) −
3 3 3 3
Let f ( x ) = e
cos −1 sin ( x + / 3)
14. then
8 5 /18 8 13 /18 7 /12
(a) f =e (b) f =e (c) f − =e (d)
9 9 4
7 11 /12
f − =e
4
MODULE -1 (INVERSE TRIGONOMETRIC FUNCTION) EVOLVE BATCH TIMING 9-10:30 P.M
a+b a +b
15. If a, b are such that 0 a b and a1 = , b1 = a1b , a2 = 1 1 , b2 = a2b1 and so on,
2 2
then
b2 − a 2 (b 2
− a2 ) (b 2
+ a2 )
(a) a = (b) b = (c) b = (d)
cos −1 ( a / b ) cos −1
(a / b) cos −1
(a / b)
(b 2
− a2 )
a =
cos −1
(a / b)
JEE MAINS-SINGLE CHOICE CORRECT
1. (d) 2. (d) 3. (c) 4. (d) 5. (a)
6. (c) 7. (b) 8. (a) 9. (d) 10. (c)
11. (b) 12. (c) 13. (c) 14. (c) 15. (c)
16. (a) 17. (b) 18. (c) 19. (c) 20. (a)
21. (d) 22. (d) 23. (d) 24. (a) 25. (b)
EXERCISE – II
IIT-JEE SINGLE CHOICE CORRECT
1. (d) 2. (c) 3. (c) 4. (c) 5. (c)
6. (b) 7. (b) 8. (b) 9. (c) 10. (d)
11. (c) 12. (a) 13. (d) 14. (d) 15. (b)
16. (b) 17. (c) 18. (b) 19. (b) 20. (a)
21. (a) 22. (d) 23. (a) 24. (a) 25. (c)
EXERCISE – III
MORE THAN ONE CHOICE CORRECT
1. (b, c, d) 2. (c, d) 3. (b, c) 4. (b, c) 5. (b, d)
MODULE -1 (INVERSE TRIGONOMETRIC FUNCTION) EVOLVE BATCH TIMING 9-10:30 P.M
6. (b, c) 7. (b, c) 8. (a, c) 9. (b, d) 10. (a, b, c, d)
11. (a, c) 12. (a, d) 13. (a, b) 14. (b, c) 15. (a, b)
