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
x24 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
x3
22x, then f (x) = (A) x22 (B) x25 (C) x24x9 (D)
x5
22
x2
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) 24. If
tan x
y tan y
x, then dydx (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 22log 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 27. Ifxa cos3, ya sin3, then
3 2 2 2 2 1 dy dx d y dx / is equal to :
8. If
6 3 2 4 x f x x , then
1 f x 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.
02
f x sin 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) 2Choose 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 / 1x / (D)
1 3 2 3 1 3x / 1x / 2. If f x
2
x25x11, 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) x22 (D) x 1 x 4. The value of
0 1 2 1 x x xa 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| |x x2 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) nxn19. 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 3Choose 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| 3x2. If ysin
8sin1x
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 forx0andf
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) e1 9. 1 0 1 x x x x e x e lim x (A) 12 (B) 8 (C) 9 (D) e1 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 theseDATE : 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) e1 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 x xyy , 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 y sin x sin x sin 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 2ysin cos x cos sin x , x , then dy
dx is :
(A) 2 (B) 2 (C) 1 (D) 0
8. Range of the function
13 5 f x sin x is : (A)
,
(B) 111 3 , (C) 1 1 4 2, (D) 5, 59. Range of the function f x
x1
x3
x5
is :(A)
,
(B) 111 3 , (C) 1 1 4 2, (D) 5, 510. Ifcos yx cos
y
then
1 x2 2x cos
.dy dx
is :
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
2x n , nZ
(D) discontinuous at
2 1
2x n , 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
cos213sin2
(D) 3cos2
17sin2cos2
4. 2 4 9 1 1 n x x x x . . . x n lim x is : (A)
1
2
6 n n n (B)
1 2
1
6 n n n (C)
1 2
1
6 n n n (D)
1 2
1
6 n n n 5. 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 : 18. Let
3 2 n n x , x x f x px qx r , x x . If the two roots of
2
0
px qx 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 /, m10. 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,
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
3x21, 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 xlim x 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 x 3. If x2xy3y2 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) tan1
n1
n2
tan12 (B) tan1
n 1
tan12(C) tan12 (D) ntan12 *5. Let f x
|2x 9| | | |2x 2x9|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 u3x12 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 x2x2
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 x x log x x x in : (A) 0 1 2 , (B) (0, 1) (C) (0, 2) (D) (0, 3)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
14 e
f x cos | |x log x , 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 x cos 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 3sin1xkx2 and1 2 3 2 y , then k = (A) 6 (B) 6 (C) 2 3 (D) None 8. If xet2, ytan1
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)
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) a0, b0 (B) a0, b0 (C) aR, b0 (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 these5. If ya
1cos
, xa
sin
, then2 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 , yt e 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
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 .kThe 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 t 5. If f x
min x
| 1|,| |,|x x1|
, 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 x10.
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
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 these2. If g is the inverse of f andf
x sin 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
x tan
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 x cos x , where [.] denotes the greatest integer function, is :(A) {0} (B) {0, 1} (C) {1} (D) None 5. If p, q0 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,1cos 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