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Jee 2014 Booklet5 Hwt Differential Calculus 1

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(1)

Choose the correct alternative. Only one choice is correct. 1. Statement : 1 The domain of

 

2 4 2 x f x x     

  , where, [x] is the greatest integer function is x  

, 2

 

1 2,

Statement : 2 The domain of f x

 

  x24  x 3, where, [x] represents the greatest integer function x 

,2

Statement : 3 The domain of f x

 

sin log log x

2 3

 

2 1 2 2 2 3 3 n n x , n I          

Choose the correct choice :

(A) FTF (B) TFT (C) TTT (D) FFF 2. Iff x

2

 

x3

22x, then f (x) = (A) x22 (B) x25 (C) x24x9 (D)

x5

22

x2

3. If 1 1 1 1 1 1 1 x x y sec sin , x x x                 , then dy dx is : (A) 0 (B) 1 (C) 1 (D) 2

4. If

tan x

 

ytan y

x, then dy

dx(A)

log tan y log tan x (B)

1 2 log tan y log tan x(C) ( ) 2 2 y log tan y sin x(D)

22 22

log tan y y cos ec x log tan x x cos ec y

 

5. Iflog4log3log x2  1, then x is :

(A) 234 (B) 9 (C) 24 (D) 432 6. If f : R is defined byR

 

2 1 3 5 2 1 2 2 4 2 x x f x logx x x x          | | , , , The value of 1 1 2 3

 

3 3 3 3 2 f f  f  f  f         is : (A) 0 (B) 1 (C) 19 (D) 1 + log 2

7. Ifxa cos3, ya sin3, then

3 2 2 2 2 1 dy dx d y dx             / is equal to :

(2)

8. If

 

6 3 2 4 x f x x    , then

 

1 fx is : (A) 2 4 6 3 x x   (B) 6 4 2 3 x x   (C) 4 3 6 2 x x

(D) Does not exist

9.

 

0

2

f xsin x    sin x ,  x , where [ ] represents the greatest integer function can also be represented as :

(A)

 

0 0 1 1 1 1 2 , x f x sin , x           (B)

 

1 0 4 2 1 1 4 2 2 , x f x , x           (C)

 

0 0 1 1 1 2 , x f x sin , x          (D)

 

0 0 4 1 1 4 1 1 2 , x f x , x sin , x               10. 3 7 5 x x x lim x x    | | | | (A) 3/2 (B) 3/7 (C) 3/5 (D) 2

(3)

Choose the correct alternative. Only one choice is correct. 1. If 1 3 1 3 1 1 3 1 3 0 1 x a y tan , x x a           / / / / , then dy dx is : (A)

1 3 2 3 1 1 x /x / (B)

2 3 2 3 2 1 x /x / (C)

2 3 2 3 1 3x / 1x / (D)

1 3 2 3 1 3x / 1x / 2. If f x

2

x25x11, f x

 

f

 

 is :x (A) 2x2 + 11 (B) 2(x2 + 7) (C) 2(x2 + 11) (D) 2(x2 + 10) 3. If 2 2 1 1 f x x x x     ; then f (x) is : (A) x 1 x      (B) 2 2 x(C) x22 (D) x 1 x      4. The value of

0 1 2 1 x x x

a log x sin x cos x

lim e       where, a > 0, is : (A) 1 2 (B) 0 (C) 1 2  (D) 2 5. A function f is defined on1 1, as

 

1 1 0 2 1 0 1 2 x , x f x x , x              . Then value of f x

 

f

 

| |x is : (A) f (x) (B) 1 1 1 2 2 1 1 0 2 2 1 0 1 2 x , x x , x x , x                   (C) 2f (x) (D) 1 1 10 4 2 1 0 1 , x x , x       

6. Domain of the function f x

 

 5| |xx2 is :6

(A)

,2

 

 3,

(B)  3, 2  2 3, (C)

 , 2

  

 2 3, (D) R  

3, 2 2 3, ,

7. If [ ] denotes the greatest integer function,

2 5 2 x sin cos x lim cos x           is :

(A) 0 (B) 1 (C)(D) Does not exist

8. If0

n n

1 n n x y, lim y x     / is : (A) e (B) x (C) y (D) nxn1

(4)

9. If

 

2 3 1 3 4 5 0 5. log log x 1       / then| x | belongs to : (A) 1,

(B) 2 1 5,       (C) 2 5,        (D) 17 3 15, 5       10. 1 1 2 1 6 x x lim x sin x               is : (A) 1 2 3 (B) 2 3 (C) 3 2 (D) 2 3

(5)

Choose the correct alternative. Only one choice is correct. 1. A single formula that gives f (x) for all x where,0

 

3 0 3 3 3 3 x , x f x x , x        is : (A) f x

 

|2x 1| 4x (B) f x

 

|x 3| 2x (C) f x

 

|3x 9| x (D) f x

 

|x 3| 3x

2. If ysin

8sin1x

then

 

2 2 2 1 x d y xdy dx dx   equals : (A)64 y (B) 64 y (C) 64y (D) 64 y

3. Let f x

 

x2 x 1 where [ ] denotes the greatest integer function. Then, in (0, 2), f (x) is discontinuous at the point : (A) 1 5 2 x  (B) 1 5 2 x 

(C) x = 1 (D) Both (A) and (C)

4. 3 2 3 4 8 5 2 2 1 x x x x lim x       is : (A) 4 (B) 2 (C) 1 (D) 2

5. The domain of the function

 

    1 4 2 2 2 1 3 9x 27 x 219 3 x f x /              is x(A)3 3, (B) 3,

(C) 5 2,      (D) [0, 1] 6. If yu4 where u cos x,dy dx  is :

(A) 4u3 (B) 4 cos x sin x3

(C) 4 sin x cos x3 (D) u 7. If

 

1 1 2 2 x x log log f x x           forx0

andf

 

0 a and f (x) is continuous at x = 0, then a is :

(A) 0 (B) 1 (C) 1 (D) 1/2 8.

3 2 3 27 2 9 x x log x lim x     (A) 12 (B) 8 (C) 9 (D) e1 9. 1 0 1 x x x x e x e lim x                     (A) 12 (B) 8 (C) 9 (D) e1 10. If

0 1 x x a cos x b sin x lim x     then : (A) a = b (B) a = b + 1 (C) a = b – 1 (D) None of these

(6)

DATE : TIME : 30 Minutes MARKS : [ ___ /10] TEST CODE : DC-1 [4]

START TIME : END TIME : TIME TAKEN: PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct.

1. If f (x) is a thrice differentiable function such that

 

 

 

 

3 0 4 3 3 3 2 12 x f x f x f x f x lim x      , then f 

 

0 is equal to : (A) 12 (B) 8 (C) 9 (D) e1 2. Let

 

2 f x x cos x          

  where, [ ] denotes the greatest integer function. Then, the domain of f is :

(A) xR, x not an integer (B) x  

, 2

  

1,

(C) xR, x 2 (D) x  

, 1

3. If 2 2 7 4 xxyy  , then dy dx at x = 1 and 1 2 y is : (A) 3 4 (B) 5 4  (C) 21 8 (D) 21 8 

4. Let ysin xsin xsin x. . . , then dy

dx is : (A) 2 y cos x (B) 2 1 cos x y (C) 2 1 2 1 y y   (D) 2 1 cos x y

5. Iff x

 

sin log x

and 2 3 3 2 x y f x      , then dy dxat x is equal to :1

(A) cos log

5

(B) sin log

5

(C) 12

5

5 cos log (D)

5 5 12 sin log 6. If the function

 

3 4 2 3 3 8 3 x k x .k , x f x x , x       is continuous at x = 3, then k is : (A) 3 (B) 3 (C) 15 (D) 15 7. If 1

1

0 2

ysincos xcossin x ,  x , then dy

dx is :

(A) 2 (B) 2 (C) 1 (D) 0

8. Range of the function

 

1

3 5 f x sin x   is : (A)

 ,

(B) 111 3 ,       (C) 1 1 4 2,       (D)  5, 5

9. Range of the function f x

  

x1



x3



x5

is :

(A)

 ,

(B) 111 3 ,       (C) 1 1 4 2,       (D)  5, 5

10. Ifcos yx cos

y

then

1 x2 2x cos

.dy dx

  is :

(7)

Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than one correct option. 1. 4 1 2 1 4 4 x tan x log lim x sin x                         is :

(A) 4 (B) 1/4 (C) 0 (D) does not exist

*2. The function f x

 

 1 |tan |x is :

(A) continuous everywhere (B) discontinuous when xn , nZ

(C) not differentiable when

2 1

2

xn, nZ

(D) discontinuous at

2 1

2

xn , n and not differentiable atZ

2

n x, nZ

3. If xsin and ycos3 then

2 2 2 2yd y 4 dy dx dx      is :

(A) 6cos2

7sin2cos2

(B) cos2

13sin2cos2

(C) 3cos2

cos213sin2

(D) 3cos2

17sin2cos2

4. 2 4 9 1 1 n x x x x . . . x n lim x        is : (A)

1



2

6 n nn(B)

1 2



1

6 n nn(C)

1 2



1

6 n nn(D)

1 2



1

6 n nn5. 1 0 x x sin x lim x        / is : (A) 1 (B) 1 (C) 0 (D) e 6. If

 

0 1 0 sin x x , x f x x x , x              

, where [ ] denotes the greatest integer function, then :

(A)

 

0 1 x lim f x sin   (B) 0

 

1 x lim f x   (C)

 

0 x lim f x

 does not exist (D) x 0

 

lim f x

exists but f (x) is not continuous at x = 0

7. If f p

 

2, f

 

p  and6

       

0 x p g x f p g p f x lim x p     then

 

 

g p g p  is : (A) 3 : 1 (B) 1 : 3 (C) 1 : 12 (D) 12 : 1

(8)

8. Let

 

3 2 n n x , x x f x px qx r , x x      

 . If the two roots of

2

0

pxqx  are reciprocal to one another, then, the value of p, q, rr

for which f (x) is continuous and differentiable at x = x0 are respectively :

(A)

2 2 3 0 0 0 2 2 0 0 3 2 1 1 x x x p r , q x x       (B)

2 2 3 0 0 0 2 2 0 0 3 2 1 1 x x x p r , q x x       (C)

2 2 3 0 0 0 2 2 0 0 3 2 1 1 x x x p r , q x x       (D)

3 2 3 0 0 0 2 2 0 0 3 2 1 1 x x x p r , q x x       9. If

 

2 2 1 0 2 2 2 p |cos x| cot x cot m x | cos x | , x f x q , x e , x                           

is a continuous function on

0, then the value of p and q are respectively :

(A) m, em /  (B) m / m e,(C) m m , e  (D) em /, m

10. The set of all points of discontinuity of the inverse of

 

x x x x e e f x e e      is : (A) (B)

  , 1 (C) 1,

(D) R 

1 1,

(9)

Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than one correct option.

1. In x(0, 1),f x

 

3x21, where [x] stands for the greatest integer not exceeding x is :

(A) continuous

(B) continuous except at one point (C) continuous except at two points (D) continuous except at three points

*2. 2 02 25 5 xlimx x x       is equal to : (A)

2 2 0 2 1 5 e x x log x lim x    (B) 2 0 1 x x e x lim x     (C)

2 4 0 2 1 5 x cos x lim x   (D) 0 5 x x sin lim x3. If x2xy3y2  then1

2 3 2 6 d y x y dx  is : (A) 0 (B) 12 (C) 22 (D) 22 4. 1 n k k t

is equal to where tk=

 

2 1 1 1 2 2 1 cot k k k k        

(A) tan1

n1



n2

tan12 (B) tan1

n 1

tan12

(C) tan12 (D) ntan12 *5. Let f x

 

|2x 9| | | |2x  2x9|

Which of the following are true ? (A) f (x) is not differentiable at 9

2 x(B) f (x) is not differentiable at 9 2 x (C) f (x) is not differentiable at x = 0 (D) f (x) is not differentiable at 9 0 9 2 2 x , , 6. Let

 

 

 

4 0 3 0 x x e x , x f x x x , x          

Where [ ] denotes the greatest integer function. Then, (A) f (x) is discontinuous at x = 0

(B) f (x) is continuous at x = 0

(C)

 

0

xlim f x exists (D) None of these

7. If u3x12 and vx6, then du

dv is :

(A) 6x6 (B) 36x11

(C) 6x5 (D) 3x6

*8. Let f x

 

max x, x , x

2 3

in    . Then :2 x 2 (A) f (x) is continuous in   2 x 2 (B) f (x) is not differentiable at x = 1 (C)

 

1 3 35 2 8 f   f     (D)

 

1 3 27 2 2 f   f       9. 3xlog

1 3 x2x2

 for x0  (A) 0 1 2 ,       (B) (0, 1) (C) (0, 2) (D) (0, 3) 10. 3 5 2

1 3 2 2

3 2 xxlogxxx in : (A) 0 1 2 ,       (B) (0, 1) (C) (0, 2) (D) (0, 3)

(10)

DATE : TIME : 30 Minutes MARKS : [ ___ /10] TEST CODE : DC-1 [7]

START TIME : END TIME : TIME TAKEN: PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct.

1. If

 

11

2

3

1

4 e

f x cos | |xlogx  , then its domain is :

(A)2 6, (B)

6 2,

  

 2 3,

(C)6 2, (D) 2 2,

  

 2 3,

2. Domain of the function

 

16 3 1 1 4 2 x x f xcos  cos ec  , x , is :R (A) 2 10 3, 3       (B) 10 3 3 ,       (C) 2 10 3 3 , ,         (D) None of these 3. 1 1 2 1 x cos x lim x         is equal to : (A) (B) 2 (C) 4 (D) 0 4. If 5 3 243 1 x a x x lim x a    , then a is equal to : (A) 1 (B) 0 (C) e (D) None 5.

 

1 : 1 : 1 1 1 : 1 x f x x x x            . Then f (x) at x = 0 :

(A) is continuous (B) is discontinuous (C) is differentiable

(D) is non-differentiable

6. Let f x

 

|2sgn

 

2x | . Then f (x) has :2 (A) removable discontinuity (B) infinite discontinuity (C) no discontinuity (D) essential discontinuity 7. If yloge

 

x3 3sin1xkx2 and

1 2 3 2 y     , then k = (A) 6 (B) 6 (C) 2 3 (D) None 8. If xet2, ytan1

2t , then1

dy dx(A)

2 2 2 2 2 1 t e t t t    (B)

2 2 2 2 2 1 t e t t t    (C)

2 2 2 2 1 t e t t t    (D) None of these 9. If

 

2 4 1 e x f x sin log , x R x             

, then domain and range of f (x) are given by :

(A)

2 1,

,1 1, (B)

  , 1

  

1, ,0 1, (C)

 

0 1, ,1 1, (D) None of these 10. The function

 

! 1 ! x x f x sin cos n n             is :

(A) non periodic

(B) periodic with period 2 n ! (C) periodic with period 2 (n + 1) ! (D) periodic with period of 2(n + 1)

(11)

Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than one correct option. 1.

0 1 2 1 2 x cos x lim x       is : (A) 1 (B) 1

(C) Does not exist (D) None of these

2.

2

  

2 1 0 1 x b lim a x sin , a, b, R, a a x                 , is equal to : (A) b (B) a b2 (C) a2/b (D) None 3. If

 

2 2 : 0 : 0 ax b x f x x x      possesses derivative at x = 0, then : (A) a0, b0 (B) a0, b0 (C) aR, b0 (D) None of these *4. If

 

1 2 x f x    x     ; ([.] = G.I.F.), then at 1 2 x : (A) f (x) is continuous (B) f (x) is differentiable (C) Discontinuous (D) None of these

5. If ya

1cos

, xa

sin

, then

2 2 d y dx at 2   is : (A) 4 (B) 1 2a

(C) Does not exist (D) None of these 6. If xe sin t , yte cos tt

and y

xy

2 k xy

y

, then k =

(A) 1 (B) 1

(C) 2 (D) 2

7. If log0 3.

x 1

log0 09.

x , then x lies in the1

interval : (A)

2,

(B) (1, 2) (C)

 2, 1

(D) None of these 8. Let

 

2 2 4 sin x f x x x     

  , [.] G.I.F., then which one is not true :

(A) f is periodic (B) f is even

(C) f is many-one (D) f is onto 9. 1 0 x tan x lim x          , [.] = G.I.F. is : (A) 1 (B) 0

(C) Does not exist (D) None of these

10. If

 

0 0 a b c x x sin x lim , a, b, c R sin x

   , exists and is

non-zero, then :

(A) a  b c 0 (B) a  b c 0 (C) a  b c 0 (D) None of these

(12)

DATE : TIME : 30 Minutes MARKS : [ ___ /10] TEST CODE : DC-1 [9]

START TIME : END TIME : TIME TAKEN: PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than one correct option.

1. Let f

 

x be continuous at x = 0 andf

 

0  .k

The value of

 

 

 

2 0 2 3 2 4 x f x f x f x lim x    is : (A) k (B) 2k (C) 3k (D) None of these 2. If

 

2 1 17 66 f x x x    , then 2 2 f x       is discontinuous at x = (A) 2 (B) 2 7 24 3 11 , , (C) 7 24 3 11, (D) None of these 3. If x yy x , then dy/dx is :1 (A)

y y x log y x y log x x   (B)

y x y log x x y log x y   (C)

y y x log y x x y log x    (D) None of these 4. If x t 1, y t 1 t t

    , then dy/dx is equal to :

(A) 2 2 ( 1) t t(B) 2 2 ( 1) ( 1) t t   (C) 2 2 ( 1) ( 1) t t   (D) 2 2 (1 ) t t5. If f x

 

min x

| 1|,| |,|x x1|

, then :

(A) f is odd (B) f is even

(C) f is periodic (D) None of these

6. Which of the following functions is an even function ?

(A)

 

1 1 x x a f x a    (B)

 

1 1 x x a f x x a    (C)

 

x x x x a a f x a a      (D) f x

 

sin x 7. 0 1 1 cos ec x x tan x lim sin x          (A) 1/e (B) e (C) 1 (D) None 8.

  

2 ! 0 1 1 a n n sin n lim , a n       (A) 1 (B)  (C) 0 (D) None

*9. For a real number x, let [x] denote the greatest integer less than or equal to x. Then

 

2 1 tan x f x x      is :

(A) Continuous at some x

(B) Continuous at all x butf

 

x does not exist for some x

(C) f

 

x exists for all x butf x does not exist

 

for some x

(D) f

 

x exists for all x

10.

 

1 1 1 : 0 1 0 : 0 / x / x e x f x e x     has at x = 0 :

(A) Removable discontinuity (B) Non-removable discontinuity (C) No discontinuity

(13)

Choose the correct alternative. Only one choice is correct. 1. If 2 0 1 1 9 y x du u  

, then 2 2 d y dx is equal to : (A)

2

1 1 9 y(B)

1 9 y 2

(C) 9y (D) None of these

2. If g is the inverse of f andf

 

xsin x, then g x

 

(A)cos ec g x

 

(B) sin g x

 

(C) cos ec g x

 

(D) None of these 3. Period of the function f ( x )sin3

 

xtan

 

x ,

where [.] & {.} denote the integral part and fractional parts respectively, is given by :

(A) 1 (B) 2

(C) 3 (D)

4. Range of f x

 

 sin xcos x, where [.] denotes the greatest integer function, is :

(A) {0} (B) {0, 1} (C) {1} (D) None 5. If p, q0 and 0 x x px q px q lim a, lim b qx p qx p       , then a/b = (A) 1 (B) p2/q2 (C) q2/p2 (D) None 6. If 3 0 2 x sin x a sin x lim b x

 , then a and b, respectively, are :

(A) 2, 1 (B) 2 1,

(C)  2, 1 (D) 2,1

7. Let f x

 

max

2sin x,1cos x , x

0,

. Then set of points of non-differentiability is : (A) (B) 2       (C) 13 5 cos        (D) 13 5 cos       8. Let

 

 

3 1 0 0 1 x a cos x b sin x f x , x , f x      . If f

(x) is continuous at x = 0, a and b are given by :

(A) 5 3 2 2, (B)  5, 3 (C) 5 3 2, 2   (D) None of these 9. If xexy y sin x2 , then dy/dx at x = 0 is : (A) 0 (B) 1 (C) 1 (D) None 10. Let

 

 

   

 2 4 

 

2 2

 

2 2  u x    f x log , u , v , u , v x

 

2 1 v , f 

 

2 is equal to : (A) 0 (B) 1 (C) 1 (D) None

References

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