EXERCISE – I JEE MAIN - HCL Learning

Page # 20 CONTINUITY & DIFFERENTIABILITY

EXERCISE – I JEE MAIN

1. A function f(x) is defined as below 4. Let f(x) = sgn (x) and g(x) = x (x2 – 5x + 6).
The function f(g(x)) is discontinuous at
f(x) = cos(sinx)  cos x (A) infinitely many points (B) exactly one point
x2 , x  0 and f(0) = a, f(x) is (C) exactly three points (D) no point
Sol.
continuous at x = 0 if a equals

(A) 0 (B) 4 (C) 5 (D) 6
Sol.

 (1 px)  (1 px) ,1 x  0
 ,0  x  1
2. f(x) =  x is 5. If y = 11
 2x  1 where t = , then the number
t2  t  2
 x  2 x 1

continuous in the interval [–1, 1], then ‘p’ is equal to: of points of discontinuities of y = f(x), x  R is

(A) –1 (B) – 1/2 (C) 1/2 (D) 1 (A) 1 (B) 2 (C) 3 (D) infinite

Sol. Sol.

3. Let f(x) =  x  1 [x] when – 2  x  2. Then 6. The equation 2 tan x + 5x – 2 = 0 has
 2 (A) no solution
(B) at least one real solution in [0, /4]
(where [ * ] represents greatest integer function) (C) two real solution in [0, /4]
(A) f(x) is continuous at x = 2 (D) None of these
Sol.
(B) f(x) is continuous at x = 1

(C) f(x) is continuous at x = –1

(D) f(x) is discontinuous at x = 0

Sol.

CONTINUITY & DIFFERENTIABILITY Page # 21

7. If f(x) = x ( x  x  1) , then indicate the correct 10. The function f(x) = sin–1 (cos x) is
(A) discontinuous at x = 0 (B) continuous at x = 0
alternative(s) (C) differentiable at x = 0 (D) none of these
(A) f(x) is continuous but not differentiable at x = 0 Sol.
(B) f(x) is differentiable at x = 0
(C) f(x) is not differentiable at x = 0
(D) None of these
Sol.

11. Let f(x) be defined in [–2, 2] by

max ( 4  x2 , 1 x2 ) ,2  x  0

 x(3e1/ x f(x) =  min ( 4  x2, 1 x2) , 0  x  2 then f(x)

 4) , x0
then f(x) is
8. If f(x) =  2  e1/ x (A) is continuous at all points
 0 , x0 (B) is not continuous at more than one point
(C) is not differentiable only at one point
(A) continuous as well differentiable at x = 0 (D) is not differentiable at more than one point.
(B) continuous but not differentiable at x = 0 Sol.
(C) neither differentiable at x = 0 not continuous at x = 0
(D) none of these
Sol.

x 12. If f(x) is differentiable everywhere, then
9. If f(x) = x  1  x be a real valued function then
(A) f(x) is continuous, but f(0) does not exist (A) | f | is differentiable everywhere
(B) f(x) is differentiable at x = 0 (B) | f |2 is differentiable everywhere
(C) f(x) is not continuous at x = 0
(D) f(x) is not differentiable at x = 0 (C) f | f | is not differentiable at some point
Sol.
(D) f + | f | is differentiable everywhere
Sol.

Page # 22 CONTINUITY & DIFFERENTIABILITY

13. Let f(x + y) = f(x) f(y) all x and y. Suppose that 16. Let [x] denote the integral part of x  R and
g(x) = x – [x]. Let f(x) be any continuous function
f(3) = 3 and f(0) = 11 then f(3) is given by with f(0) = f(1) then the function h(x) = f(g(x))
(A) has finitely many discontinuities
(A) 22 (B) 44 (C) 28 (D) 33 (B) is continuous on R
(C) is discontinuous at some x = c
Sol. (D) is a constant function.
Sol.

14. If f : R  R be a differentiable function, such that

f(x + 2y) = f(x) + f(2y) + 4xy  x, y  R, then 17. The function f defined by f(x)= lim  (1  sin x)t  1 is
 (1  sin x)t  1
(A) f (1) = f(0) + 1 (B) f(1) = f(0) – 1 t

(C) f(0) = f(1) + 2 (D) f(0) = f(1) – 2
(A) everywhere continuous
Sol.
(B) discontinuous at all integer values of x

(C) continuous at x = 0 (D) none of these

Sol.

max f(t),0  t  x,0  x 1  x 1 sin 1  , x0
sin x, x  1 .  x
15. Let f(x) = x – x2 and g(x) = 
  x 1 sin 1 
18. If f(x) =   x
Then in the interval [0, )  , x  0 , then f(x) is

(A) g(x) is everywhere continuous except at two points 0 , x0

(B) g(x) is everywhere differentiable except at two points 

(C) g(x) is everywhere differentiable except at x = 1

(D) none of these (A) continuous as well diff. at x = 0

Sol. (B) continuous at x = 0, but not diff. at = 0

(C) neither continuous at x=0 nor diff. at x=0

(D) none of these

Sol.

CONTINUITY & DIFFERENTIABILITY Page # 23

19. The functions defined by f(x) = max {x2, (x – 1)2, 22. Let f : R  R be a function such that
2x (1 – x)}, 0 x  1
(A) is differentiable for all x f  x  y   f(x)  f(y) , f(0) = 0 and f(0) = 3, then
(B) is differentiable for all x except at one point 3  3
(C) is differentiable for all x except at two points
(D) is not differentiable at more than two points f(x)
Sol. (A) x is differentiable in R

(B) f(x) is continuous but not differentiable in R

(C) f(x) is continuous in R (D) f(x) is bounded in R

Sol.

20. Let f(x) = x3 – x2 + x + 1 and 23. Suppose that f is a differentiable function with

max{f(t)} for 0  t  x for 0  x  1 the property that f(x + y) = f(x) + f(y) + xy and lim 1
g(x) =  h0 h
 3  x  x2 for 1  x  2 then

(A) g(x) is continuous & derivable at x = 1 f(h) = 3 then

(B) g(x) is continuous but not derivable at x = 1 (A) f is a linear function (B) f(x) = 3x + x2

(C) g(x) is neither continuous nor derivable at x = 1 x2 (D) none of these
(C) f(x) = 3x +
(D) g(x) is derivable but not continuous at x = 1
2
Sol. Sol.

21. Let f(x) be continuous at x = 0 and f(0) = 4 then 24. If a differentiable function f satisfies

value of lim 2f( x)  3f (2x)  f ( 4x) is  x  y   4  2(f(x)  f(y))
x2
x0 f  x, y  R, find f(x)

(A) 11 (B) 2 (C) 12 (D) none of these 3 3

Sol. (A) 1/7 (B) 2/7 (C) 8/7 (D) 4/7

Sol.

Page # 24 CONTINUITY & DIFFERENTIABILITY

25. Let f : R  R be a function defined by f(x) = Min 28. Let f(x + y) = f(x) f(y) for all x, y, where f(0)  0.
{x + 1, |x| + 1}. Then which of the following is true ?
(A) f(x)  1 for all x  R If f(0) = 2, then f(x) is equal to
(B) f(x) is not differentiable at x = 1 (A) Aex (B) e2x
(C) f(x) is differentiable everywhere (C) 2x (D) None of these
(D) f(x) is not differentiable at x = 0
Sol. Sol.

26. The function f : R /{0}  R given by 29. A function f : R  R satisfies the equation
f(x + y) = f(x) . f(y) for all x, y  R, f(x)  0. Suppose
f(x) = 1  2 can be made continuous at x = 0 by that the function is differentiable at x = 0 and f(0) = 2
x e2x  then f(x) =
(A) f(x) (B) 2 f(x) (C) – f(x) (D) – 2 f(x)
Sol.

1

defining f(0) as

(A) 2 (B) –1 (C) 0 (D) 1

Sol.

30. Let f(x) = [cos x + sin x], 0 < x < 2 where [x]

denotes the greatest integer less than or equal to x.

the number of points of discontinuity of f(x) is

27. Function f(x) = (|x – 1| + |x – 2| + cos x) where (A) 6 (B) 5 (C) 4 (D) 3

x  [0, 4] is not continuous at number of points Sol.

(A) 3 (B) 2 (C) 1 (D) 0

Sol.

CONTINUITY & DIFFERENTIABILITY Page # 25

34. Let f(x) be a continuous function defined for 1

x2  1  ; x0  x  3. If f(x) takes rational values of for all x and f(2)
  x2  , is [ x ]
31. The function f(x) = = 10 then the value of f(1.5) is

 0 ; x  0 (A) 7.5 (B) 10 (C) 8 (D) None of these

represents the greatest integer less than or equal to x Sol.

(A) continuous at x = 1 (B) continuous at x = –1

(C) continuous at x = 0 (D) continuous at x = 2

Sol.

35. If f(x) = p |sin x| + q . e|x| + r|x|3 and f(x) is

sin(ln | x |) x  0 differentiable at x = 0, then
32. The function f(x) = 
 1 x0 (A) p = q = r = 0 (B) p = 0, q = 0, r  R

(A) is continuous at x = 0 (C) q = 0, r = 0, p  R (D) p + q = 0, r  R

(B) has removable discontinuity at x = 0 Sol.

(C) has jump discontinuity at x = 0

(D) has discontinuity of IInd type at x = 0

Sol.

33. The set of all point for which f(x) = |x 3|  1 36. Let f(x) = sin x, g(x) = [x + 1] and g(f(x)) = h(x)
|x 2| [1 x]

is continuous is then h    is (where [*] is the greatest integer
2
(where [ * ] represents greatest integer function)
function)
(A) R (B) R – [–1, 0]

(C) R – ({2}  [–1, 0]) (D) R – {(–1, 0)  n, n} (A) nonexistent (B) 1

Sol. (C) –1 (D) None of these

Sol.

Page # 26 CONTINUITY & DIFFERENTIABILITY

37. If f(x) = [tan2 x] then 40. If f(x) = sgn (cos 2x – 2 sin x + 3) then f(x)
(where [ * ] denotes the greatest integer function) (where sgn ( ) is the signum function)
(A) is continuous over its domain
(A) Lim f(x) does not exist (B) has a missing point discontinuity
(C) has isolated point discontinuity
x0 (D) has irremovable discontinuity.
Sol.
(B) f(x) is continuous at x = 0 (D) f(0) = 1
(C) f(x) is non-differentiable at x = 0
Sol.

41. Let g(x) = tan–1|x| – cot–1|x|, f(x) = [x] {x},
[x  1]

38. If f(x) = [x]2 + {x}2 , then h(x) = |g (f (x) ) | then which of the following holds

(where, [ * ] and { * } denote the greatest integer good ?
and fractional part functions respectively)
(A) f(x) is continuous at all integral points (where { * } denotes fractional part and [ * ] denotes
(B) f(x) is continuous and differentiable at x = 0 the integral part)

(C) f(x) is discontinuous  x  – {1} (A) h is continuous at x = 0
(D) f(x) is differentiable  x .
Sol. (B) h is discontinuous at x = 0
(C) h(0–) = /2 (D) h(0+) = –/2

Sol.

39. If f is an even function such that Lim f(h)  f(0) 42. Consider f(x) = Limit xn  sin xn for x > 0, x  1,
h0 h f(1) = 0 then n xn  sin x n

has some finite non-zero value, then

(A) f is continuous and derivable at x = 0 (A) f is continuous at x = 1

(B) f is continuous but not derivable at x = 0 (B) f has a finite discontinuity at x = 1

(C) f may be discontinuous at x = 0 (C) f has an infinite or oscillatory discontinuity at x = 1.

(D) None of these (D) f has a removable type of discontinuity at x = 1.

Sol. Sol.

CONTINUITY & DIFFERENTIABILITY Page # 27

1 x  1 x
45. Consider f(x) = {x} , x  0 ; g(x) = cos
 [{| x |}]ex2 {[x  {x}]}
 for x  0
43. Given f(x)=  (e1/x2  1) sgn(sin x) then, f(x)  1 f(g(x))
  2
0 for x  0  for x0
 1
(where {x} is the fractional part function; [x] is the 2x, –  < x< 0, h(x) =  for x0
4 f(x) for x0
step up function and sgn(x) is the signum function of x)

(A) is continuous at x = 0 

(B) is discontinuous at x = 0

(C) has a removable discontinuity at x = 0 then, which of the following holds good

(D) has an irremovable discontinuity at x = 0 (where { * } denotes fractional part function)
(A) ‘h’ is continuous at x = 0
Sol.

(B) ‘h’ is discontinuous at x = 0

(C) f(g(x)) is an even function

(D) f(x) is an even function

Sol.

44. Consider f(x) =  x[x]2 log(1x) 2 for  1 x  0 46. Consider the function defined on [0, 1]  R,
ln(ex2  2 {x}) the
 sin x  x cos x
 for 0  x  1 f(x) = x2 if x  0 and f(0) = 0, then the
function f(x)
 tan x (A) has a removable discontinuity at x = 0
(B) has a non removable finite discontinuity at x = 0
(where [ * ] & { * } are the greatest integer function (C) has a non removable infinite discontinuity at x = 0
& fractional part function respectively) (D) is continuous at x = 0
Sol.
(A) f(0) = ln 2  f is continuous at x = 0

(B) f(0) = 2  f is continuous at x = 0
(C) f(0) = e2  f is continuous at x = 0

(D) f has an irremovable discontinuity at x = 0

Sol.

Page # 28 CONTINUITY & DIFFERENTIABILITY

|x| 49. The function f(x) is defined as follows
47. Let f(x) = sin x for x  0 & f(0) = 1 then ,
  x if x  0
(A) f(x) is conti. & diff. at x = 0
(B) f(x) is continuous & not derivable at x = 0  x2 if 0  x  1 then f(x) is
(C) f(x) is discont. & not diff. at x = 0 f(x) = 
(D) None of these x3  x  1 if x  1
Sol.
(A) derivable & cont. at x = 0

(B) derivable at x = 1 but not cont. at x = 1

(C) neither derivable nor cont. at x = 1

(D) not derivable at x = 0 but cont. at x = 1

Sol.

48. For what triplets of real number (a, b, c) with

a  0 the function f(x) = x x1 is x  {x}  x sin{x} for x  0
ax2  bx  c otherwise 50. If f(x) =  for x  0 then
 0
differentiable for all real x ?
(where { * } denotes the fractional part function)
(A) {(a, 1 – 2a, a)| a  R, a  0}
(A) ‘f’ is cont. & diff. at x = 0
(B) {(a, 1 – 2a, c)| a, c  R, a  0}
(B) ‘f’ is cont. but not diff. at x = 0
(C) {(a, b) | a, b, c  R, a + b + c = 1}
(C) ‘f’ is cont. & diff. at x = 2 (D) None of these
(D) {(a, 1 – 2a, 0)| a  R, a  0}
Sol.
Sol.

CONTINUITY & DIFFERENTIABILITY Page # 29

Answer Ex–I JEE MAIN

1. A 2. B 3. D 4. C 5. C 6. B 7. B 8. B
9. B 10. B 11. D 12. B 13. D 14. D 15. C 16. B
17. B 18. B 19. C 20. C 21. C 22. C 23. C 24. D
25. C 26. D 27. D 28. B 29. B 30. B 31. D 32. D
33. D 34. B 35. D 36. A 37. B 38. C 39. B 40. C
41. A 42. B 43. A 44. D 45. A 46. D 47. C 48. A
49. D 50. D