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DIFFERENTIATION JEE TYPE QUESTIONS
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146 DifferentiationDerivative at a Point
Basic Level
1. If f (x ) | x | , then f (0) [MNR 1982]
(a) 0 (b) 1 (c) x (d) None of these
1 ,x 0
2. If f (x ) then f (0) [MP PET 1994]
1 sin x , 0 x
2
(a) 1 (b) 0 (c) (d) Does not exist
ax b; x 0
2
3. If f (x ) possesses derivative at x = 0, then
x 2; x 0
(a) a = 0, b = 0 (b) a > 0, b = 0 (c) a R , b = 0 (d) None of these
4. The derivative of f (x ) 3 | 2 x | at the point x 0 3 is [Orissa JEE 2002]
(a) 3 (b) – 3 (c) 0 (d) Does not exist
5. The derivative of y = 1 – |x | at x = 0 is [SCRA 1996]
(a) 0 (b) 1 (c) – 1 (d) Does not exist
6. The derivative of f (x ) | x x | at x = 2 is
2
[AMU 1999]
(a) – 3 (b) 0 (c) 3 (d) Not defined
d
7. The value of [| x 1 | | x 5 |] at x = 3 is [MP PET 2000]
dx
(a) – 2 (b) 0 (c) 2 (d) 4
xf (a) af (x )
8. If f(x) has a derivative at x = a, then lim is equal to
x a x a
(a) f (a) af (a) (b) af (a) f (a) (c) f (a) f (a) (d) af (a) f (a)
9. If f (x ) x 2 , then f ( f ( x )) at x = 4 is [DCE 2001]
(a) 8 (b) 1 (c) 4 (d) 5
10. Let 3f(x) – 2f(1/x) = x, then f (2) is equal to [MP PET 2000]
(a) 2/7 (b) 1/2 (c) 2 (d) 7/2
af (x ) xf (a)
11. If f(x) is a differentiable function, then lim is [UPSEA
x a x a
(a) af (a) f (a) (b) af (a) f (a) (c) af (a) f (a) (d) af (a) f (a)
12. The differential coefficient of the function |x – 1|+ |x – 3| at the point x = 2 is [Rajasthan PET 2002]
(a) – 2 (b) 0 (c) 2 (d) Undefined
13. If f (x ) | x 3 | , then f (3) [Rajast
, Differentiation 147
(a) 0 (b) 1 (c) –1 (d) Does not exist
Advance Level
dy
14. If y cot 1 (cos 2 x ) , then the value of at x will be [IIT 19
dx 6
1/2 1/2
2 1
(a) (b) (c) (3) (d) (6)
3 3
2
1 3
15. The values of x, at which the first derivative of the function x w.r.t. x is , are
x 4
1 3 2
(a) 2 (b) (c) (d)
2 2 3
16. The number of points at which the function f (x ) | x 0.5 | | x 1| tan x does not have a derivative in the
interval (0, 2), is
[MNR 1995]
(a) 1 (b) 2 (c) 3 (d) 4
x
17. The set of all those points, where the function f (x ) is differentiable, is
1 | x |
(a) (–, ) (b) [0, ) (c) (–, 0) (0, ) (d) (0, )
18. Let f (x y ) f (x ) f (y ) and f (x ) 1 xg (x )G(x ) where lim g(x ) a and lim G(x ) b then f (x ) is equal to
x 0 x 0
(a) 1+ ab (b) ab (c) a/b (d) None of these
19. f(x) is a function such that f (x ) f (x ) and f (x ) g(x ) and h(x) is a function such that h(x ) [ f (x )]2 [g(x )]2 and
h(5) = 11, then the value of h(10) is
(a) 0 (b) 1 (c) 10 (d) None of these
20. Let f (x y ) f (x ) f (y ) for all x and y. Suppose that f (3) 3 and f (0) 11, then f (3 ) is given by
(a) 22 (b) 33 (c) 28 (d) None of these
Some Standard Differentiation
Basic Level
(1 x )2 dy
21. If y 2
, then is [MP PET 1999]
x dx
2 2 2 2 2 2 2 2
(a) 2 3 (b) 2 3 (c) (d)
x x x x x2 x3 x3 x2
dv
22. If 2 t v 2 , then is equal to [MP PET 1992]
dt
(a) 0 (b) 1/4 (c) 1/2 (d) 1/v
dy
23. If x y 1 y , then
2
[MP PET 2001]
dx
1 y2 1 y2
(a) 0 (b) x (c) (d)
1 2y 2
1 2y 2
dp
24. If pv = 81, then is at v = 9 equal to [MP PET 1999]
dv
(a) 1 (b) –1 (c) 2 (d) None of these
,148 Differentiation
1 x dy
25. If y , then [AISSE 1981; Rajasthan PET
1x dx
1995]
2 1 1 2
(a) (b) (c) (d)
(1 x ) (1 x ) (1 x ) (1 x ) 2(1 x ) (1 x ) (1 x ) (1 x )
26. The derivative of f (x ) x | x | is [SCRA
(a) 2x (b) –2x (c) 2 x 2 (d) 2|x |
27. The derivative of F[ f { (x )}] is [AMU 2001]
(a) F[ f { (x )}] (b) F[ f { (x )}] f { (x )} (c) F[ f { (x )}] f { (x )} (d) F[ f { (x )}] f { (x )} (x )
d
28. (sin 2 x 2 ) equals [Rajasthan PET 1996]
dx
(a) 4x cos (2 x 2 ) (b) 2 sin x 2 . cos x 2 (c) 4x sin (x 2 ) (d) 4x sin (x 2 ). cos( x 2 )
dy
29. If y sec x 0 , then [MP PET 1997]
dx
180
(a) secx tanx (b) sec x 0 tan x 0 (c) sec x 0 tan x 0 (d) sec x 0 tan x 0
180
dy
30. If sin y e x cos y e , then at (1, ) is [Kerala (Engg.) 2002]
dx
(a) sin y (b) –x cos y (c) e (d) sin y – x cos y
2
dy
31. If y a sin x b cos x , then y 2 is a
dx
(a) Function of x (b) Function of y (c) Function of x and y (d) Constant
d
32. [cos(1 x 2 )2 ] [AISSE 1981; AI CBSE 1979]
dx
(a) 2 x (1 x 2 ) sin(1 x 2 )2 (b) 4 x (1 x 2 ) sin(1 x 2 )2 (c) 4 x (1 x 2 ) sin(1 x 2 )2 (d) 2(1 x 2 ) sin(1 x 2 )2
dy
33. If y cos(sin x 2 ) , then at x ,
2 dx
(a) – 2 (b) 2 (c) 2 (d) 0
2
d
34. [sin n x cos nx ] []
dx
(a) n sin n 1 x cos(n 1)x (b) n sin n 1 x cos nx (c) n sinn 1 x cos(n 1)x (d) n sinn 1 x sin(n 1)x
d
35. cos(sin x 2 ) [DSSE 1979]
dx
(a) sin(sin x 2 ). cos x 2 .2 x (b) sin(sin x 2 ). cos x 2 .2 x (c) sin(sin x 2 ). cos 2 x . 2 x (d) None of these
dy
36. If y sin( sin x cos x ) , then [DSSE 1987]
dx
1 cos sin x cos x cos sin x cos x
(a) (b)
2 sin x cos x sin x cos x
1 cos sin x cos x
(c) (cos x sin x ) (d) None of these
2 sin x cos x
1 x2
37. If y sin , then dy [AISSE 1987]
1 x
2
dx
, Differentiation 149
4x 1 x2 x 1 x2 x 1 x2 4x 1 x2
(a) . cos
(b) . cos
(c) . cos
(d) . cos
1x 2
1 x
2
(1 x )
2 2
1 x
2
(1 x )
2
1 x
2
(1 x )
2 2
1 x
2
d 2
38. (x cos x )4 [DSSE 1987]
dx
(a) 4(x 2 cos x )(2 x sin x ) (b) 4(x 2 cos x )(2 x sin x ) (c) 4(x 2 cos x )3 (2 x sin x ) (d) 4(x 2 cos x )3 (2 x sin x )
d cot 2 x 1
39.
dx cot 2 x 1
(a) – sin 2x (b) 2 sin 2x (c) 2 cos 2x (d) – 2 sin 2x
d 1 sin 2 x
40. [AISSE 1985; DSSE 1986]
dx 1 sin 2 x
(a) sec 2 x (b) sec 2 x (c) sec 2 x (d) sec 2 x
4 4 4
tan x cot x dy
41. If y , then
tan x cot x dx
(a) 2 tan 2x sec 2x (b) tan 2x sec 2x (c) – tan 2x sec 2x (d) – 2 tan 2x sec 2x
d
42. sec 2 x cos ec 2 x [DSSE 1981]
dx
(a) 4 cosec 2x . cot 2x (b) – 4 cosec 2x . cot 2x (c) – 4 cosec x . cot 2x (d) None of these
5x dy
43. If y cos 2 (2 x 1) , then [IIT 1980]
3
(1 x ) 2 dx
5(3 x ) 5(3 x ) 5(3 x )
(a) 2 sin(4 x 2) (b) 2 sin(4 x 4 ) (c) 2 sin(2 x 1) (d) None of these
3(1 x ) 3(1 x ) 3(1 x )
dy
44. If y sin x y , then equals to [Rajasthan PET 2001]
dx
sin x cos x sin x cos x
(a) (b) (c) (d)
2y 1 2y 1 2y 1 2y 1
d
45. log | x | .....( x 0)
dx
1 1
(a) (b) (c) x (d) – x
x x
d
46. log x
(1 / x ) is equal to [AMU 1999]
dx
1 1
(a) (b) – 2 (c) (d) 0
2 x x2 x
d
47. log(log x ) [IIT 1985]
dx
x log x
(a) (b) (c) (x log x )1 (d) None of these
log x x
d
48. (log tan x ) [MNR 1986]
dx
(a) 2 sec 2x (b) 2 cosec 2x (c) sec 2x (d) cosec 2x
dy
49. If y log x x , then [MNR 1978]
dx
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