1 fx 1 f 1 - mnracademy.com
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DIFFERENTIATION mnracademy.com 1. Let [.] denote the greatest integer function and f ( x ) = tan 2 x ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ then 1) 1 (0) 1 f = 2) () fx is not continuous at x = 0 3) () fx is not differentiable at x = 0 4) 1 (0) 0 f = 2. Let ( ) ().() fx y fx fy + = for all x and y. If (5) 2 f = and 1 (0) 3 f = then 1 (5) f = 1) 5 2) 8 3) 0 4) 6 3. If 1 (0) 0, (0) 2 f f = = then the derivative of ( ( ( ))) y f f fx = at x = 0 is 1) 2 2) 8 3) 16 4) 4 4. Suppose that f is a differentiable function with the property that ( ) () () fx y fx fy xy + = + + and 0 1 () 3 h Lt fh h Æ = , then 1) f is a linear function 2) 2 () 3 fx x x = + 3) 2 () 3 2 x fx x = + 4) None 5. The right hand derivative of () [ ]sin fx x x = p at x = k, k is an integer is 1) ( 1) ( 1) k k - - p 2) 1 ( 1) ( 1) k k - - - p 3) ( 1) k k - p 4) 1 ( 1) k k - - p 6. If x x x x e e y e e - - - = + then dy dx = 1) 2 1 y + 2) 2 1 y - 3) 2 1 y - 4) 1 y 7. If () fx mx c = + and if 1 (0) (0) 1 f f = = , then (1) f = 1) 1 2 ) – 1 3) 2 4) – 2 8. If y = cos −1 5cos x −12sin x 13 ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ , x ∈ 0, π 2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ , then dy dx is equal to 1) 1 2) – 1 3) 0 4) 2 9. If d dx 1+ x 2 + x 4 1− x + x 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ax + b then (a, b) = 1) (2, 1) 2) (-2,1) 3) (2,-1) 4) (1,2) 10. If d dx 1+ x 2 + x 4 1− x + x 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ax + b then 1 (1) f = 1) 1 2) – 1 3) log 2 4) – log 2 11. The function 3 () | | fx x = 1) differentiable at x = 0 2) continuous but not differentiable at x = 0 3) discontinuous at x = 0 4) a function with range −∞, ∞ ( ) 12. If 1 1 sinh sinh 1 x y - - + = then
Transcript of 1 fx 1 f 1 - mnracademy.com
DIFFERENTIATION mnracademy.com 1. Let [.] denote the greatest integer function and
f (x) = tan2 x⎡
⎣⎢⎤⎦⎥ then
1) 1(0) 1f = 2) ( )f x is not continuous at x = 0 3) ( )f x is not differentiable at x = 0 4) 1(0) 0f = 2. Let ( ) ( ). ( )f x y f x f y+ = for all x and y. If (5) 2f = and 1(0) 3f = then 1(5)f = 1) 5 2) 8 3) 0 4) 6 3. If 1(0) 0, (0) 2f f= = then the derivative of ( ( ( )))y f f f x= at x = 0 is 1) 2 2) 8 3) 16 4) 4 4. Suppose that f is a differentiable function with the property that
( ) ( ) ( )f x y f x f y xy+ = + + and 0
1 ( ) 3hLt f hhÆ
= , then
1) f is a linear function 2) 2( ) 3f x x x= +
3) 2
( ) 32xf x x= + 4) None
5. The right hand derivative of ( ) [ ]sinf x x x= p at x = k, k is an integer is 1) ( 1) ( 1)k k- - p 2) 1( 1) ( 1)k k-- - p 3) ( 1)k k- p 4) 1( 1)k k-- p
6. If x x
x xe eye e
-
--
=+
then dydx=
1) 21 y+ 2) 2 1y - 3) 21 y- 4) 1y
7. If ( )f x mx c= + and if 1(0) (0) 1f f= = , then (1)f = 1) 1 2 ) – 1 3) 2 4) – 2
8. If y = cos−1 5cos x−12sin x
13
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟,x∈ 0,π
2
⎛
⎝⎜⎜⎜⎞
⎠⎟⎟⎟⎟ , then dy
dx is equal to
1) 1 2) – 1 3) 0 4) 2
9. If
ddx
1+ x2 + x4
1− x + x2
⎡
⎣⎢⎢
⎤
⎦⎥⎥= ax + b then (a, b) =
1) (2, 1) 2) (-2,1) 3) (2,-1) 4) (1,2)
10. If
ddx
1+ x2 + x4
1− x + x2
⎡
⎣⎢⎢
⎤
⎦⎥⎥= ax + b then 1(1)f =
1) 1 2) – 1 3) log 2 4) – log 2 11. The function 3( ) | |f x x= 1) differentiable at x = 0 2) continuous but not differentiable at x = 0 3) discontinuous at x = 0 4) a function with range −∞,∞( )
12. If 1 1sinh sinh 1x y- -+ = then
1) 2
211
dy ydx x
+=
+ 2)
2
21 01
dy ydx x
++ =
+ 3)
2
21 01
dx ydy x
++ =
+ 4)
2
211
dy ydx x
-+
-
13. If tan2xy x= , then (1 ) sindycosx x
dx+ - =
1) xy 2) y 3) 0 4) x 14. If ye xy e+ = , then 2(0)y =
1) 31e
2) 21e
3) 1e
4) 1
15. 2 22 2tan ,sin1 1t ty xt t
= =- +
then dydx
=
1) 0 2) cosx 3) tanx 4) 1
16.
ddx
sin−1(3x−4x3), 12
< x≤1⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟=
1)2
11 x-
2)2
31 x-
3)2
31 x--
4)2
11 x--
17. Derivativeof 100logx w.r.t 2x is
1) 21
2 log10x 2) 2
14 log10x
3) 12x
4) 14x
18. If 1f isdifferentiablefunctionand 11( )f x iscontinuousatx=0and 11(0)f a= ,thenthe
valueof 20
2 ( ) 3 (2 ) (4 )x
f x f x f xLtxÆ
- + is
1)a 2)2a 3)3a 4)None
19. If 1 1tan tan2
x y- - p+ = then
2
2d ydx=
1) 21x- 2) 3
2x
3) 32x- 4) 2
2x
20. If ( )f x y+ , ( ) ( )f x f y and ( )f x y- areinA.P, " x&yand (0) 0f π ,then1 1(7) ( 7)f f+ - =
1)7 2)0 3)6 4)1
21. Let ( ) [ ]22g x x x⎡ ⎤= −⎣ ⎦ ,where[x]denotesthegreatestintegerfunctionthen
1)g(x)iscontinuousonlyatx=1
2)g(x)isdiscontinuousonlyatx=0,1
3)g(x)isdiscontinuousforallintegralvaluesofx
4)Noneofthese
22. Thevalueoff(0),sothatthefunction ( )2 2 2 2a ax x x ax af x
a x a x− + − + +=
+ − −iscontinuousfor
allx,isgivenby
1) 1/2a− 2) 3/2a− 3) 1/2a 4) 3/2a
23. Inorderthatthefunction ( ) ( )cot xf x x 1= + iscontinuousatx=0,f(0)mustbedefinedas:
1) ( ) 1f 0e
= 2) ( )f 0 0= 3) ( )f 0 e= 4)Noneofthese
24. Thefunction { }f : R 0 R− → givenby ( ) 2x
1 2f xx e 1
= −−
canbemadecontinuousatx=0
bydefiningf(0)as:
1)-1 2)0 3)1 4)2
25.
2/22 3sin cos , 0
3 0( )
ab xx x xa b
e xf x
⎛ ⎞⎛ ⎞⎜ ⎟+ ≠⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠=
⎧⎪⎪⎪= ⎨⎪⎪⎪⎩
iscontinuousat 0,x b R= ∀ ∈ then mina is
1)-1/8 2)-1/4 3)-1/24)0
26. If ( )1/4
11 1log1 2xy Tan xx
−⎧ ⎫+⎪ ⎪⎛ ⎞= −⎨ ⎬⎜ ⎟−⎝ ⎠⎪ ⎪⎩ ⎭ then dy
dx=
(1) 21xx−
(2) 2
41xx−
(3) 41xx+
(4) 41xx−
27. If 3 33cos 2cos , 3sin 2sinx yθ θ θ θ= − = − then dydx
=
(1) tanθ (2) cotθ (3) cot2θ (4) tan
2θ
28. ( )( )( )( ){ }2 2 4 4 8 8d x a x a x a x adx
+ + + + =
(1) ( )
16 15 16
215 16x x a a
x a− +
− (2)
( )
16 15 16
2x x a a
x a− +
−
(3) 16 16x ax a−−
(4) none
29. 1 1 0 dyx y y xdx
+ + + = ⇒ =
(1) ( )21
1 x+ (2)
( )21
1 x−
+ (3) 2
11 x+
(4) 21
1 x−
30. If ( )2
1 12
log / 3 2log1 6loglog
e x xy Tan Tanxex
− −⎛ ⎞ ⎛ ⎞+⎜ ⎟= + ⎜ ⎟⎜ ⎟ −⎝ ⎠⎜ ⎟⎝ ⎠
then dydx
=
(1) 0 (2) 1 (3) 21
1 x+ (4) 2
31 x+
31. If ....
. 0y toy ex e x+ ∞+= > then dy
dx is
(1) 1xx+
(2) 1 xx+ (3) 1 x
x− (4) 1
x
32. If ( )2cos 1 01 2cos 10 1 2cos
xf x x
x= then 1
3f π⎛ ⎞ =⎜ ⎟⎝ ⎠
(1) 5 (2)- 4 (3) 3− (4) -2 1)4 2) 4 3) 2 4) 3 5) 3 6) 3 7) 3 8) 19)1 10) 2 11) 4 12) 2 13) 4 14) 2 15) 4
16) 3 17) 2 18) 3 19) 2 20) 221)122)123)3 24)325)2(26) 2 (27) 2 (28) 1 (29) 2 (30) 1 (31)3(32)3 (33) 3
