JEE Main Practice Sheet: Limits, Continuity, and Differentiability PDF

Lakshya JEE Main (2025) MANTHAN Mathematics Limits, Continuity and Differentiability Exercise - 1 ln ( 3 + x ) − ln ( 3 − x )...

Lakshya JEE Main (2025) MANTHAN Mathematics Limits, Continuity and Differentiability Exercise - 1 ln ( 3 + x ) − ln ( 3 − x ) 7. If lim = k , then the value of k is x →0 x EXISTENCE OF LIMIT (LHL/RHL) AND 1 2 2 INDETERMINATE FORMS (a) 0 (b) − (c) (d) − 3 3 3 x 1. lim = π x →0x −θ 8. lim 2 = (a) 1 (b) –1 θ→ cotθ π 2 (c) 0 (d) Does not exist (a) 0 (b) –1 (c) 1 (d) ∞ 2. Consider the following statements (where [.] denotes greatest integer function). log e (1 + x ) 9. lim S1 : lim [ x] is an indeterminate form. x→0 3x − 1 x → 0− x (a) loge3 (b) 0 (c) 1 (d) log3e ln x ∞ S2 : lim is an indeterminate form of type. x →∞ x ∞ ln x 10. lim = x →1 x −1 S3 : lim([1 + x ]) 1/ x is an indeterminate form of the type ∞ x →0 (a) 1 (b) –1 State, in order, whether S1 , S2 , S3 are true (T) or false (F) (c) 0 (d) ∞ (a) FTF (b) FFT x n − 2n (c) TFF (d) TTT 11. If lim = 80 , where n is a positive integer, then n = x →2 x − 2  x 2 + 1 x ≥ 1 (a) 3 (b) 5 3. If f ( x) =  then the value of lim f ( x) is 3 x − 1 x < 1 →1 (c) 2 (d) 1 (a) 1 (b) 2 1 − cos2 x 12. lim = (c) 3 (d) Does not exist x →0 x (a) 0 (b) 1 ALGEBRIC METHODS, SUBSTITUTION, (c) 2 (d) 4 FACTORIZATION, RATIONALIZATION, 13. If lim kx cosec x = lim x cosec kx , then k = STANDARD LIMIT, EXPANSION AND L-HOSPITAL x →0 x →0 (a) 1 (b) –1 x3 − 8 4. The value of the limit of as x tends to 2 is (c) ±1 (d) ±2 x2 − 4 3 x3cotx (a) 3 (b) (c) 1 (d) 0 14. lim = x →0 1 − cosx 2 x3 − x 2 − 18 (a) 0 (b) 1 5. The value of the limit of as x tends to 3 is x −3 (c) 2 (d) –2 (a) 3 (b) 9 x(e x − 1) (c) 18 (d) 21 15. lim = x →0 1 − cosx 3x /2 − 3 (a) 0 6. The value of lim is (b) ∞ x→ 2 3x − 9 (c) –2 (a) 0 (b) 1/3 (c) 1/6 (d) ln3 (d) 2 (1)  sin2 x + sin6 x x2 − 2 x 16. lim = 25. The value of lim is x →0 sin5 x − sin3 x x →0 2sinx (a) 1/2 (b) 1/4 (a) 1 (b) 2 (c) 2 (d) 4 (c) –1 (d) 3 1 + sinx − 1 − sinx 17. lim = 1∞, 0°, ∞° FORM x →0 x x (a) –1 (b) 1 1  26. lim 1 +  equal to (c) 2 (d) –2 x →∞  mx  ( x + 2)5/3 − (a + 2)5/3 (a) e1/m (b) e–1/m 18. lim = x→a x−a (c) em (d) me 5 5 (a) (a + 2) 2/3 (b) (a + 2)5/3 x +b 3 3  x+a 27. lim   = 5 2/3 5 5/3 x →∞  x + b  (c) a (d) a 3 3 (a) 1 (b) eb–a (a + h) 2 sin ( a + h ) − a 2 sina (c) ea–b (d) eb 19. lim = h →0 h 5/ x  ax + bx + cx  (a) a cos a + a2 sin a (b) a sin a + a2 cos a 28. lim   equals x →0  3  (c) 2a sin a + a2 cos a (d) 2a cos a + a2 sin a (a) (abc)1/5 (b) (abc)1/3  x−3  20. lim  = (c) (abc)5/3 (d) 0 x →3  x − 2 − 4 − x  (a) 1 (b) 2 cosec x  1 + tan x  (c) –1 (d) –2 29. lim   equals x→0  1 + sin x  2cosx − 1 (a) 0 (b) 1 (c) e (d) e–1 21. lim = x →π /4 cotx − 1 1 1 INFINITE LIMIT (a) (b) 2 2 4x + 3 30. Evaluate lim 1 x →∞ x − 8 (c) (d) 1 2 2 (a) 4 (b) 1 (c) 3 (d) 8 1 2 x 2 + 3x + 4 (1 − cos2 x ) 31. lim is equal to 22. lim 2 = x →∞ 3x 2 + 3x + 4 x →0 x 2 (a) 1 (b) –1 (a) (b) 1 (c) 0 (d) ∞ 3 (c) 0 (d) Limit does not exist ln x − [ x ] n 32. lim , n ∈ N, is ([∙] denotes greatest integer less 23. If f ( a ) = 2, f ′ ( a ) = 1, g ( a ) = −1 ; g ′ ( a ) = 2, then x →∞ [ x] than or equal to x) g ( x) f (a) − g (a) f ( x) lim = (a) –1 (b) 0 x→a x−a (c) 1 (d) Does not exist (a) 3 (b) 5 (c) 0 (d) –3 1− n 33. The value of lim n will be ax − bx n →∞ 24. The value of lim is ∑r x →0 x r= (a) log (a/b) (b) log (b/a) (a) –2 (b) –1 (c) log (ab) (d) 0 (c) 2 (d) 1 (2)  n2 n  r2 − r +1 1 + x, when x ≤ 2 34. lim ∏  2 = 41. If f ( x) =  , then n →∞ 3 r =2  r + r + 1  5 − x, when 2 < x < 3 1 3 (a) f(x) is continuous at x = 2 (a) 0 (b) 1 (c) (d) (b) f(x) is discontinuous at x = 2 2 2 (c) f(x) is continuous at x = 3 ∞ – ∞ FORM (d) f (x) is discontinuous at x = 3  1 ln (1 + x )  35. lim  − =  3π x →0 x x2   1, when 0 < x ≤  4 42. If f ( x) =  , then (a) 1/2 (b) –1/2 2 2sin x, when 3 π 3 1   − for x = 1  3 (a) 4 (b) 3 (c) 2 (d) 1 (a) –1/9 (b) –2/9 (c) –1/3 (d) 1/3 1 − sin x π  π − 2 x , x ≠ 2 1 52. If f ( x) =  , be continuous at x = π/2, then ) x 2 sin , x ≠ 0, f (0) 59. The function f ( x= = 0 at x = 0 π x  λ, x =  2 (a) Is continuous but not differentiable value of λ is (b) Is discontinuous (a) –1 (b) 1 (c) 0 (d) 2 (c) Is having continuous derivative  (d) Is continuous and differentiable  (1 + ax)1/ x x 0 (a) x = 0 only (b) x = π only  x (c) x = k π (k ∈ N) (d) x = k π (k ∈ I) only 3(ea + b + c) is equal to: 61. Which of the following is not true? (a) 3 (b) 6 (a) Every differentiable function is continuous. (c) 7 (d) 8 (b) If derivative of a function is zero at all points, then the 54. If f(x) = |x| + [x – 1], where [.] is greatest integer function, function is constant. then f(x) is: (c) If a function has maximum or minima at a point, then the (a) Continuous at x = 0 as well as at x = 1 function is differentiable at that point and its derivative (b) Continous at x = 0 but not at x = 1 is zero. (c) Continuous at x = 1 but not at x = 0 (d) If a function is constant, then its derivative is zero at all (d) Neither continuous at x = 0 nor at x = 1 points. (4)  62. If f is a real-valued differentiable function satisfying sin ( ln (1 + x ) ) |f(x) – f(y)| ≤ (x – y)2, x, y ∈ R and f(0) = 0, then f(1) equals 71. The value of lim is x →0 ln (1 + sinx ) (a) 1 (b) 2 (c) 0 (d) –1 (a) 0 (b) 1/2 (c) 1/4 (d) 1 63. The number of points of non-differentiability of 1 f(x) = || x | –1| + | cos πx| for – 2 < x < 2 is x3 ⋅ sin + x +1 72. lim x = (a) 7 (b) 9 (c) 5 (d) 6 x →∞ x2 + x + 1 (a) 0 (b) 1/2 FUNCTION BY SOLVING FUNCTIONAL (c) 1 (d) None of these EQUATION −3n + (−1) n 73. lim is 64. If f : R → R satisfies f(x + y) = f(x) + f(y), for all x, y ∈ R and n →∞ 4n − (−1) n 6 f(1) = 2, then ∑ f (k ) is k =1 (a) − 3 4 (a) 24 (b) 40 3 3 (c) 50 (d) 42 (b) − if n is even; if n is odd 4 4 65. Let f(x) be a polynomial function satisfying the relation 1 1 (c) Not exist if n is even − if n is odd f(x). f =  f ( x) + f   ∀× ∈ R − {0} and f(2) = –7. x x (d) +1 if n is even; does not exist if n is odd Determine f ′(1). 5n +1 + 3n − 22 n (a) –1 (b) –5 74. lim is equal to n →∞ 5n + 2 n + 32 n + 3 (c) –3 (d) 5 (a) 5 (b) 3 MISCELLANEOUS (c) 1 (d) Zero 66. lim sin −1 (sec x)  sin [ x − 3]  x →0 75. lim   (where [∙] denotes greatest integer x →0  [ x − 3]  π   (a) (b) 1 2 function) (c) Zero (d) Does not exist (a) 0 (b) 1  x  (c) Does not exist (d) sin 1 67. lim sec −1  = x →∞  x +1  x+2 x +1 76. lim   = (a) 0 (b) π x →∞  x − 2  (c) π (d) Does not exist (a) e4 (b) e–4 2 (c) e2 (d) None of these cosecx  π  ( ) 5/ x 68. lim  cot  + x   = 77. lim 1 + tan 2 x = x →0  4  x → 0+ (a) e–1 (b) e2 (a) e5 (b) e2 (c) e–2 (d) e1 (c) e (d) None of these  ( )( )  n 69. lim (1 + x ) 1 + x 2 1 + x 4 ….. 1 + x 2   if | x |< 1 has 78. lim sin(6 x 2 ) is equal to n →∞    x → 0 ln cos(2 x 2 − x ) the value equal to (a) 0 (b) 1 (a) 12 (b) –12 (c) 1 – x (d) (1 – x)–1 (c) 6 (d) –6 1 ( x 3 + 27) ln ( x − 2) 70. lim is equal to 79. The limiting value of (cosx) x sin as x → 0 is x →3 ( x 2 − 9) (a) – 8 (b) 8 (c) 9 (d) – 9 (a) 1 (b) e (c) 0 (d) None of these (5)  2x  a b   | tan −1 | x ||  80. If lim 1 + + 2  e 2, then the values of a and b are = 88. lim   (where [·] denotes greatest integer x →∞  x x  x →0  x  (a) a ∈ R, b ∈ R (b) a = 1, b ∈ R function) then sum of RHL + LHL (c) a ∈ R, b = 2 (d) a = 1, b = 2 (a) 0 (b) 1 (c) –1 (d) DNE 81. Which of the following is/are true. 2t 2 − t − 1 − t 2 − t + 1 (a) If lim { f ( x ) + g ( x )} exists, then both lim f ( x ) and 89. lim x→a x→a t →∞ t (tan π / 8) lim g ( x ) exist x→a (a) 0 (b) 1 (c) –1 (d) 2 (b) If lim f ( x ) and ü g ( x ) exist, then x→a x→a 1 n  r +1 lim { f ( x ) + g ( x )} exist. 90. lim n →∞ n 2 ∏  r − 1  r =2 x → a (a) 1 (b) 1/2 (c) If lim f ( x ) and lim g ( x ) exist, then lim f ( x ) g ( x ) (c) 2 (d) None of these x→a x→a x→ a 91. x5 + px + r = 0, x1, x2, x3, x4 and x5 are the roots of the equation does not exists then the value of lim( x − x2 )( x − x3 )( x − x4 )( x − x5 ) x → x1 (d) If lim { f ( x ) g ( x )} exists, then both lim f ( x ) and x→a x→a (a) 5x14 –p lim g ( x ) exist. (b) –5x14 +p x→a (c) –5x14 –p n p sin 2 (n !) 82. nlim , 0 < p < 1, is equal to (d) 5x14 +p →∞ n +1  a sin x   b tan x  (a) 0 (b) ∞ 92. Evalute lim  +  if a and b are positive x →0  x   x  (c) 1 (d) None of these integers (where [.] is GIF). 2 83. If lim( p tan qx 2 − 3cos 2 x + 4)1/ (3 x ) = e5/3 ; p, q ∈ R then (a) a + b – 1 (b) a + b + 1 x →0 (c) a + b (d) a – b – 1 1 1 tan πx (a)=p 2, q = = (b) p = ,q 2 2 sin(πx 4 ) + ( x + 2) n ⋅ 2 2 2 x +1 , 93. Let f ( x) = nlim then lim f ( x) (c) p = 2, q = 2 (d) p = 2, q = 4 →∞ 1 + ( x + 2) n − x 4 x →−1 f ( x) − 5 is equal to 84. If lim = 3 then lim f ( x) 22 x→4 x−2 x→4 (a) p (b) 7 (a) 10 (b) 11 (c) 7 (d) 8 2 (c) 1 (d) Non-existent e x − cos x 85. lim x 2 + ax + b x →0 x2 94. Let a, b be constants such that lim exist and x →1 (ln(2 − x)) 2 (a) 0.5 (b) 1 (c) 1.5 (d) Does not exist have the value equal to l. Find the value of (a + b + l). (a) 0 (b) 1 π − tan −1 x 1 (c) –1 (d) None of these 4 86. If lim sin(ln x ) exists and has the value equal to , then x →1 e −x n 8 1 1 95. If f ( x) = = and g ( x) 2 , then find the points of find n. x −1 x −1 (a) 2 (b) 3 (c) 4 (d) 5 discontinuities of fog(x) 87. If [.] is greatest integer function and n is a positive integer, (a) x =± 1, ± 2   nx   n tan x   then lim  +  = (b) x =± 2, ± 2 x → 0  sin x     x  (c) x =± 2, ± 3 (a) 2n (b) 2 (c) 2n – 1 (d) 2n + 2 (d) x =± 1, ± 3 (6)  104. Which of the following function(s) defined below is  (1 + px) − (1 − px)  , −1 ≤ x < 0 discontinuous everywhere? 96. f ( x) =  x is continuous 2x +1 1 if x ∈ Q  x if x ∈ Q  ,0 ≤ x ≤1 (a) f ( x) =  (b) g ( x) =   x−2  0 if x ∉ Q 1 − x if x ∉ Q in the interval [–1, 1], then ‘p’ is equal to  x if x ∈ Q  x if x ∈ Q (a) –1 (b) –1/2 (c) h( x) =  (d) k ( x) =   0 if x ∈ Q  − x if x ∉ Q (c) 1/2 (d) 1  e1/ x − 1  1/ x , x≠0 Exercise - 2 97. For the function f ( x) =  e + 1 , which of the 0 , x = 0   100 k  following is correct  ∑ x  − 100 1. lim  k =1  is equal to (a) lim f ( x) does not exist x →1 x −1 x →0 (a) 0 (b) 5050 (b) f(x) is continuous at x = 0 (c) 4550 (d) – 5050 (c) Exists but is not continuous at  π (d) None of these x− 2 2. lim   is (where [∙] denotes greatest integer function) 98. Function f(x) = (|x – 1| + |x – 2| + cos x) where x ∈ [0,4] is x →π /2 cosx   not continuous at number of points   (a) 3 (b) 2 (a) –1 (b) 0 (c) 1 (d) 0 (c) –2 (d) Does not exist 99. Let f(x) be a continuous function defined for 1 ≤ x ≤ 3. 3. lim+ (ln(sin 3 x) − ln(x 4 + ex 3 )) = x →0 If f(x) takes rational values of for all x and f(2) = 10 then the value of f(1.5) is (a) 0 (b) 1 (a) 7.5 (c) –1 (d) Does not exist (b) 10  π   π  (c) 8 4. lim ncos   sin   is equal to n →∞  4n   4n  (d) None of these π π  0, x < 0 (a) (b) 100. Let f ( x) =  2 , then for all values of x 3 4 x , x ≥ 0 π (c) (d) None of these (a) f is continuous but not differentiable 6 (b) f is differentiable but not continuous sinx 4 − x 4 cosx 4 + x 20 5. lim is equal to (c) f ′ is continuous but not differentiable x →0 4 x 4  e 2 x − 1 − 2 x 4  (d) f ′ is continuous and differentiable   1 101. If for all values of x and y ; f(x + y) = f(x).f(y) and f(5) = 2, (a) 0 (b) − 6 f ′(0) = 3, then f ′(5) is 1 (c) (d) Does not exist (a) 3 (b) 4 6 (c) 5 (d) 6 6. lim x →∞ ( ) x 4 − x 2 + 1 − ax 2 − b =0 then 102. Number of points where function is non-differentiable (a) a = 1, b = –2 (b) a = 1, b = 1 |(x – 1)3| + |(x + 2)3| (c) a = 1, b = –1/2 (d) None of these (a) 1 (b) 2  1  1  (c) 0 (d) 3  xsin   + sin  2  x ≠ 0 7. Let f ( x ) =  x x  then lim f ( x ) x →∞ 103. Number of points where function is non-differentiable  0 x=0  |(x – 1)3| + |x| equals (a) 1 (b) 2 (a) 0 (b) –1/2 (c) 0 (d) 3 (c) 1 (d) None of these (7)  2− cosx − 1 18. lim | x ( x − 1) |[cos2 x ] , where [∙] denotes greatest integer 8. lim = x →0 x →π /2 x ( x − π / 2 ) function, is equal to 2 ln 2 (a) 1 (b) 0 (a) (b) ln2 π (c) e (d) Does not exist 2 π  (c) (d) Does not exists ln cot  − qx  π  4  = 1 , and p = kq then 1 19. If lim x →0 tanpx 9. lim ( x) ln sin x is equal to (a) k = 1 (b) k = 2 x →0 (c) k = 3 (d) k = 4 (a) 1 (b) 0 (c) e (d) Does not exist 20. Let f(x) and g(x) be differentiable functions on (–∞, ∞) 2 and let f '(x)and g'(x) denote derivatives of f (x) and 10. lim (cosmx) n / x x →0 1 1 g(x) respectively. If= f (0) = , g ( 0) , f ′ ( 0 ) 1 and = (a) e–m n/4 2 2 (b) e–m n/2 2 3 ( ) 2 2 (c) e–mn /2 (d) e–mn /4 2 f 2 x 2 + 3x − 1 g'(0) = 2, then the value of lim is 1 − cosx cos2 x x →0 3g ( x ) − 1 11. The limit lim is equal to x →0 x2 1 (a) 1/2 (b) –1/2 (a) 1 (b) (c) 3/2 (d) 1 2 (c) –1 (d) 0 cos ( sinx ) − cosx 12. The value of lim is equal to 1 x →0 x4 21. The value of lim x 2  2  where [.] denotes G.I.F., is (a) 1/5 (b) 1/6 x→ 0 x  (c) 1/4 (d) 1/2 (a) 0 (b) – 2 x  x2 + 5x + 3  (c) –1 (d) 1 13. lim  2  is equal to x →∞ x + x + 3  22. If f is a positive function such that f(x + T) = f(x) (T > 0), ∀   (a) e4 (b) e2 (c) e3 (d) e x ∈ R, then  π   f ( x + T ) + 2 f ( x + 2T ) +... + nf ( x + nT )  14. lim  sec x  ( ln x ) is equal to lim n  = x →1  2  n →∞ 2 2  f ( x + T ) + 4 f ( x + 4T ) +... + n f ( x + n T )  2 2 (a) − (b) ln2 (a) 2 (b) 2ln 2 3 π 2 (c) (d) (c) 3 (d) None of these 2ln 2 πln 2 2 sin [ cosx ] sin x 15. lim equals to (where [∙] denotes greatest integer 23. Lt , (where [⋅] denotes greatest x →0 1 + [cosx ] x→ π 2 1  cos −1  (3sin x − sin 3 x)  function)  4  (a) Equal to 1 (b) sin 1 integer function) is (c) Equal to zero (d) Does not exist 2 (a) (b) 1  1 1 x π 16. lim  sin + cos  is x →∞  x x 4 (a) e (b) e2 (c) (d) Does not exist π (c) 1/e (d) Does not exist 24. If lim{ ln x} and lim{ ln x} exists finitely but they are not x2 − + x →c x →c − e 2 − cosx equal (where {⋅} denotes fractional part function), then 17. lim is equal to x →0 x3sinx (a) ‘c’ can take only rational values 1 1 1 1 (b) ‘c’ can take only irrational values (a) (b) (c) (d) 4 6 12 8 (c) ‘c’ can take infinite values in which only one is irrational (d) ‘c’ can take infinite values in which only one is rational (8)  f ( x) ax − sin 3 x 25. If f(x) be a bi-quadratic polynomial and lim =4 33. If lim = b and (a, b) ≠ (0, 0) then 2b/a is x →0 (tan −1 x)3 x →0 x3 then f(1) can not be equal to (a) 3 (b) –3 (a) 0 (b) – 4 (c) 4 (d) 3 (c) 0 (d) 1  x − [ x]; x ∉ I cos −1 ( x − a log a x) 26. Let f ( x) =  ; where I is the set of integers 34. lim (a > 1) is equal to 1; x ∈I x →∞ sec −1 ( a x log a ) x and [x] represents greatest integer ≤ x. (a) 2 (b) 1 ( f ( x)) 2 n − 1 (c) loga2 (d) 0 If g ( x) = nlim →∞ ( f ( x )) 2 n + 1 , then |g(x)| = f(x) is satisfied by  π (a) no real x (b) all integer values of x 35. lim x 2 sin  ln cos  x →∞  x (c) x = 0 only (d) x = 1 only (a) – p2/2 (b) p2/2 sin(sin x) − sin x 1 27. If lim x 0 ax 5 + bx 4 + cx 3 + dx 2 = − , then (c) p2/4 (d) –p2/4 12 x2 (a) a = 2, b = 0, c ∈ R, d = 2  e π / x + e −π / x  36. ü   (b) a = 0, b = 2, c ∈ R, d ∈ R x →∞  cos (π / x)  2 2 (c) a = 2, b ∈ R, c = 0, d = 0 (a) – ep (b) ep 2 (d) a ∈ R, b ∈ R, c = 2, d = 0 (c) e–p (d) ep   +λ π sin x − x 2 − {x} ⋅ {− x} 28. If lim  tan −1  −  = 1, then ordered pair(s) (l, m) 37. lim {∙} is fractional part of x →∞   +m 4 x →0 x cos x − x 2 − {x} ⋅ {− x} can be 1 1 (a) (2022, 2023) (b) (2023, 2022) (a) − (b) 3 3 (c) (2024, 2022) (d) (2022, 2024) (c) 1 (d) Does not exist 29. If a and b are roots of x 2 + 1 − cos 2θ x + θ = 0 where n Sr 38. Let Sn = 1 + 2 + 3 +... + n and Pn = ∏ , where r =2 Sr − 1 π  1 1 0  4 (a) e2 – 7 (b) e2 – 8 (c) e2 – 6 (d) None of these is (where [. ] denotes greatest integer function) π π 3 x + x + x x −3 (a) −1 (b) +1 42. The value of lim is 4 4 x →1 x3 − 1 1 1 π π (a) (b) (c) 1 − (d) −1 − 3 9 4 4 7 2 sin(ln | x |) ; x ≠ 0 (c) (d) 9 9 50. The function f ( x) =   1 ; x=0 1  x   x  (a) Is continuous at x = 0 43. lim    + x   tan −1 x  (where [∙] is G.I.F. and {∙} denotes x →0   sin x   (b) Has removable discontinuity at x = 0 fractional part of x) (c) Has jump discontinuity at x = 0 (a) 0 (b) 1 (c) e (d) DNE (d) Has discontinuity of IInd type at x = 0 1 1 51. If f ( x)= x( x − x + 1), then indicate the correct =44. If y = 2 where p , then the number of points p + p 2 − x −1 alternative of discontinuties of y = f(x), x ∈ R is (a) f(x) is continuous but not differentiable at x = 0 (a) 1 (b) 2 (b) f(x) is differentiable at x = 0 (c) 3 (d) Infinite (c) f(x) is not differentiable at x = 0 45. If the function f(x) = [sin [x]] in (0, 2π) is not continuous at (d) None of these (p, q) then p + q is ([·] denotes greatest. integer ≤ x). (a) 4 (b) –1  x(3e1/ x + 4)  , x≠0 (c) 2 (d) 3 52. If f ( x) =  2 − e1/ x , then f ( x) is  0 , x=0 46. Let f(x) = – 1 + |x – 2| and g(x) = 1 – |x| then set of all possible  value(s) of x for which (fog) (x) is discontinuous is: (a) Continuous as well differentiable at x = 0 (a) {0, 1, 2} (b) Continuous but not differentiable at x = 0 (b) {0, 2} (c) Neither differentiable at x = 0 not continuous at x = 0 (c) {0} (d) None of these (d) An empty set | x −3| 1   1 47. The set of all points for which = f ( x) + is  x 1 + sin x  , x>0 | x − 2 | [1 + x]      1 continuous is 53. If f ( x) = − − x 1 + sin  , x < 0, then f ( x) is (where [ ⋅ ] represents greatest integer function)   x (a) R  0 , x=0  (b) R – [–1, 0]  (c) R – ({2} ∪ [–1, 0]) (a) Continuous as well diff. at x = 0. (d) R – {(–1, 0) ∪ Ι} (b) Continuous at x = 0, but not diff. at = 0. 48. If [x] denotes the greatest integer less than or equal to x, (c) Neither continuous at x = 0 nor diff. at x = 0. consider the function f(x) = [x] [sin π x] in (–1, 1), then f(x) is: (d) None of these. (a) Continuous at x = 0 54. Number of points in f (x) = max. {|tan x|, cos|x|} where (b) Continuous in (–1, 0) function is not differentiable in [–π, π] ? (c) Differentiable in (–1, 1) (a) 2 (b) 4 (d) None of these (c) 6 (d) 8 (10)  61. Let f(x) = sgn(x) and h(x) = x (x2 – 5x + 6). The function  x 2 –1  , 03 be a continous function at x = 0. Then is equal to  a  b , x=3  [08 April, 2024 (Shift-II)] Where [x] denotes the greatest integer less than or equal to (a) 5 (b) 4 (c) 8 (d) 6 x. If S denotes the set of all ordered pairs (a, b) such that f(x) is continuous at x = 3, then the number of elements in S is: 8. Lef f : ( 0, π ) → R be a function given by (a) 2 (b) ∞ many (c) 4 (d) 1  tan 8 x  8 π tan 7 x 2. Let f : R → R be a function given by , 0< x<   7  2 1 − cos 2 x  ,x < 0  π  x2 f ( x) =  a − 8, x=   2 f ( x ) = a , x = 0, where a, b ∈ R.  b tan x π  (1+ | cot x) a , 0, ∀x ∈ R (a) (2,0) (b) (1,0) (b) f ( x ) = 0, ∀x ∈ R (c) (1,1) (d) (2,1) (c) f (x) can take any value in R 62. Let f, g : R → R be two real valued functions defined as (d) f ( x ) < 0, ∀x ∈ R − | x + 3 | , x < 0  x + k1 x , x < 0 2 68. Let the functions f: R → R and g : R → R be defined as: f ( x) =  and g ( x ) =  ,  e x , x≥0  4 x + k2 , x ≥ 0  x + 2, x < 0  x3 , x π (d) f ′′ ( x ) = 0 for some x ∈ ( 0, 2 )  following is true? [27 July, 2021 (Shift-II)]  1 ; | x | ≥1 (a) f is differentiable everywhere in (0, ∞) 75. If f ( x) =  | x | is differentiable at every point of (b) f is not continuous exactly at two points in (0, ∞) ax 2 + b ; | x | < 1  (c) f is continuous everywhere but not differentiable exactly the domain, then the values of a and b are respectively at two points in (0, ∞)  [18 March, 2021 (Shift-I)] (d) f is continuous everywhere but not differentiable exactly 1 1 at one point in (0, ∞) (a) , (b) 1 , − 3 2 2 2 2 70. Let [t] denote the greatest integer less than or equal to t. Let 5 3 f(x) = x – [x], g(x) = 1 – x + [x], and h(x) = min{f(x), g(x)}, (c) ,− (d) − 1 , 3 2 2 2 2 x ∈ [–2, 2]. Then h is: [26 Aug, 2021(Shift-II)] 76. The number of points, at which the function f (x) = |2x + (a) not continuous at exactly three points in [–2, 2] 1|–3| x + 2 | + | x2 + x – 2|, x∈R is not differentiable, is _______. (b) not continuous at exactly four points in [–2, 2]  [25 Feb, 2021 (Shift-I)] (c) Continuous in [–2, 2] but not differentiable at exactly 77. Let f: [0, 3] → R be defined by three points in (–2, 2) f (x) = min{x – [x], 1+ [x] – x} (d) Continuous in [–2, 2] but not differentiable at more than Where [x] is the greatest integer less than or equal to x. four points in (–2, 2) Let P denote the set containing all x ∈[0,3]where f is discontinuous, 71. The function f(x) = |x2 – 2x – 3|.e|9x2 – 12x + 4| is not differentiable at and Q denote the set containing all x ∈(0,3) where f is not exactly: [31 Aug, 2021 (Shift-I)] differentiable. Then the sum of number of elements in P and Q is (a) three points (b) four points equal to________. [27 July, 2021 (Shift-I)] (c) two points (d) one point 78. Let a function g: [0, 4] → R be defined as g(x) = 72. Let f (x) be a differentiable function at x = a with f’ (a) = 2 and f(a) max{t − 6t + 9t − 3}, 0 ≤ x ≤ 3 , then the number of points in 3 2 xf ( a ) − af ( x ) =  0≤ t ≤ x = 4. Then lim equals: 4 − x ,3 < x ≤ 4 x→a x−a  [26 Feb, 2021 (Shift-II)] the interval (0,4) where g(x) is NOT differentiable, is (a) 2a – 4 (b) 4 – 2a  [20 July, 2021 (Shift-II)] (c) a + 4 (d) 2a + 4 79. A function f is defined on [–3, 3] as f (x) = 73. Let f : [ 0, ∞ ) → [ 0, ∞ ) be defined as min {| x |, 2 − x 2 }, − 2 ≤ x ≤ 2  where [x] denotes the greatest x [| x |], 2 1  2 in [0, ∞).  (c) f is continuous at every point in [0, ∞) and differentiable  [4 Sep, 2020 (Shift-II)] except at the integer points. (a) Both continuous and differentiable on R – {–1}. (d) f is both continuous and differentiable except at the integer points in [0, ∞). (b) Both continuous and differentiable on R – {1}. 74. Let f be any continuous function on [0, 2] and twice differentiable (c) Continuous on R – {–1} and differentiable on R – {–1, 1}. on (0, 2). If f(0) = 0, f(1) = 1 and f(2) = 2, then (d) Continuous on R – {1} and differentiable on R – {–1, 1}.  [31 Aug, 2021 (Shift-II) 81. Let f : R → R be a function defined by f(x) = max {x, x2}. Let S (a) f ′′ ( x ) = 0 for all x ∈ ( 0, 2 ) denote the set of all points in R, where f is not differentiable. Then:  [6 Sep, 2020 (Shift-II)] (b) f ′′ ( x ) > 0 for all x ∈ ( 0, 2 ) (a) {0, 1} (b) {0} (c) φ ( an empty set) (d) {1} (23)  82. If a function f(x). defined by 89. Let f ( x) =  −1, − 2 ≤ x < 0 and g(x) = |f(x)| + f(|x|), Then in the ae x + be − x , −1 ≤ x < 1 2  x − 1, 0 ≤ x ≤ 2  = f ( x ) cx 2 1≤ x ≤ 3 interval (–2, 2), g(x) is [11 Jan, 2019 (Shift-I)]  2 ax + 2cx, 3 < x ≤ 4 (a) Differentiable at all points be continuous for some a, b, c ∈ R and f ′(0) + f ′(2) = e, then the (b) Not continuous value of a is [2 Sep, 2020 (Shift I)] (c) Not differentiable at two points 1 e (d) Not differentiable at one point (a) (b) e 2 − 3e + 13 e 2 − 3e + 13  90. Let f ( x) = max{| x |, x }, | x |≤ 2. Let S the set of points in (c) e (d) e  8 − 2 | x |, 2 π (a) Is an empty set (b) Equals {–2, – 1, 0, 1, 2} differentiable, then the ordered pair (k1, k2) is equal to: (c) Equals {–2, – 1, 1, 2} (d) Equals {– 2,2}  [5 Sep, 2020 (Shift-I)] 91. Let K be the set of all real values of x where the function. (a) ( 1, 0) (b) (1, 1) f(x) = sin |x| – |x| + 2 (x – π) cos |x| is not differentiable. Then 1  1  the set K is equal to [11 Jan, 2019 (Shift-II)] (c)  , −1 (d)  ,1  2  2  (a) φ (an empty set) (b) {π} 84. Let S be the set of points where the function, (c) {0} (d) {0, π} f(x) =|2 – | x – 3||, x ∈ R is not differentiable. Then ∑ f ( f ( x)) is 92. Let S be the set of all points in (– p, p) at which the function, f(x) = equal to ____________. x∈S min {sinx, cosx} is non-differentiable. Then S is a subset of which [7 Jan, 2020 (Shift-I)] of the following? [12 Jan, 2019 (Shift-I)] 85. Let f : R → R be defined as  π π  3π π 3π π  (a) − , 0,  (b) − , − , ,   5 1 2  4 4  4 4 4 4  x sin  x  + 5 x , x < 0     π π π π  3π π π 3π  (c) − , − , ,  (d) − , − , ,  = f ( x) = 0, x 0.  2 4 4 2  4 2 2 4   x 5 cos  1  + λx 2 , x > 0   x Differentiation of Function Satisfying The value of λ for which f ″(0) exists, is _______. the Given Rule  [6 Sep, 2020 (Shift-I)] 93. Let for a differentiable function f : (0, ∞) R, 86. Let f : (–1, 1) → R be a function defined by f (x) = max x {− | x |, − 1 − x }. if K be the set of all points at which f is not 2 f(x) – f(y) ≥ log e   + x – y, " x, y  (0, ∞).  y 20 differentiable, then K has exactly:  1  Then ∑ f '  2  is equal to _____. [10 Jan, 2019 (Shift-II)] n =1 n  (a) Five elements (b) One element [27 Jan, 2024 (Shift-I)] (c) Three elements (d) Two elements 94. If f (x) = x3 – x2 f ‘(1) + xf “(2) – f “’(3), x ∈ R, then 87. Let f(x) = 15 – |x – 10|; x ∈ R. Then the set of all values of x, at [24 Jan, 2023 (Shift-II)] which the function, g(x) = f ( f (x)) is not differentiable, is (a) 3f (1) + f (2) = f (3) (b) f (3) + f (2) = f (1)  [9 April, 2019 (Shift-I)] (c) 2f (0) – f (1) + f (3) = f (2) (d) f (1) + f (2) + f (3) = f (0) (a) {5, 10, 15, 20} (b) {10, 15} 10 (c) {5, 10, 15} (d) {10} 95. Let = f ( x) ∑ kx k , x ∈ R. If 2f(2) + f ‘(2) = 119(2)n +1 then n is 88. Let f: R → R be differentiable at c ∈ R and f(c) = 0. If g(x) = f(x) , k1 then at x = c, g is: [10 April, 2019 (Shift-I)] equal to [13 April, 2023 (Shift-II)] (a) differentiable if f ‘(c) = 0 96. If f ( x ) = x + g ′ (1) x + g ′′ ( 2 ) and 2 (b) differentiable if f ‘(c) ≠ 0 g ( x ) = f (1) x 2 + xf ′ ( x ) + f ′′ ( x ) , then the value of (c) not differentiable f ( 4 ) − g ( 4 ) is equal to [1 Feb, 2023 (Shift-I)] (d) not differentiable if f ‘(c) = 0 (24)  97. Let f :  →  be a differentiable function that satisfies the rela- L’ Hospital’s Rule, tion f ( x + = y) f ( x ) + f ( y ) − 1, ∀x , y ∈ . Higher Order Derivative If f ′ ( 0 ) = 2 , then f ( −2 ) is equal to  [29 Jan, 2023 (Shift-I)]  π π 98. Let f and g be twice differentiable functions on R such that 106. Let f :  − ,  → R be a differentiable function such that  2 2 f ”(x) = g”(x) + 6x 1 x ∫ x f ( t ) dt f ’ (1) = 4g’(1) – 3 = 9 f (0) = , If the lim x →0 0 x2 = a , then 8a2 is equal 2 e −1 f(2) = 3g(2) = 12 to: [30 Jan, 2024 (Shift-I)] Then which of the following is NOT true? (a) 16 (b) 2  [29 Jan, 2023 (Shift-II)] (c) 1 (d) 4 (a) g(–2) – f(–2) = 20 (b) If –1 < x < 2, then ǀf(x) – g(x)ǀ < 8 107. Let y ( x ) = ( )( )( (1 + x ) 1 + x 2 1 + x 4 1 + x8 1 + x16. )( ) (c) ǀf ’(x) – g’(x)ǀ < 6 ⇒ –1 < x < 1 Then y′ – y′′ at x = –1 is equal to[25 Jan, 2023 (Shift-I)] 3 (a) 976 (b) 464 (d) There exists x0 ∈ 1,  such that f(x0) = g(x0) (c) 496 (d) 944  2 99. Let f : R → R, satisfy f(x + y) = 2xf(y) + 4yf(x), ∀ x, y ∈ R. If f(2) = ( x ) x x , x > 0 , then y ′′ ( 2 ) − 2 y ′ ( 2 ) is equal to: 108. If y= [1 Feb, 2023 (Shift-II)] 3, then 14. f ′(4) is equal to _____. [26 June, 2022 (Shift-II)] f ′(2) (a) 8log e 2 − 2 (b) 4 log e 2 + 2 100. Let c, k ∈ R. If f (x) = (c + 1) x 2 + (1 – c 2)x + 2k and f (x 2 (c) 4 ( log e 2 )2 − 2 (d) 4 ( log e 2 ) + 2 + y) = f (x) + f (y) – xy, for all x, y ∈ R, then the value of 2( f (1) + f (2) + f (3) +... f (20)) is equal to ________. 109. For the curve C : (x2 + y2 – 3) + (x2 – y2 – 1)5 = 0, the value of 3y′ – y3y″, at the point (α, α), α > 0, on C, is equal to _______.  [29 June, 2022 (Shift-I)]  [27 July, 2022 (Shift-II)] 101. Let f : R → R satisfy the equation f(x + y) = f(x).f(y) for all x, y ∈ R  π and f(x) ≠ 0 for any x ∈ R. If the function f is differentiable at x = 0 t sin 2t and y ( t ) 2 2 sin t sin 2t , t ∈  0,  110. Let x ( t ) = 2 2 cos = 2   1 2 and f ′(0) = 3, then lim ( f (h) − 1) is equal to _______  dy  h →0 h 1+    [18 March, 2021 (Shift-II)] Then  dx  at t = π is equal to : d2y 4 102. Suppose f (x) is a polynomial of degree four, having critical points at –1, 0, 1. If T = {x ∈ R | f (x) = f (0)}, then the sum of squares of  dx 2 [28 July, 2022 (Shift-II)] all the elements of T is: [3 Sep, 2020 (Shift II)] (a) 4 (a) −2 2 (b) 2 3 3 (b) 2 (c) 6 (c) 1 (d) −2 3 3 (d) 8 111. Let f(x) be a polynomial function suchthat f(x) + f ′(x) + f ″(x) = x5 103. Let f and g be differentiable functions on R such that fog is the f ( x) identity function. If for some a, b ∈ R, g′ (a) = 5 and g (a) = b, + 64. Then, the value of ü  [25 June, 2022 (Shift-I)] x →1 x −1 then f ′(b) is equal to: [9 Jan, 2020 (Shift-II)] (a) 1 (b) 5 (a) –15 (b) –60 (c) 60 (d) 15 (c) 1 (d) 2 2 5 5 112. If y(x) = (xx)x, x > 0 then d x + 20 at x = 1 is equal to ________. 104. Suppose a differentiable function f(x) satisfies the identity f(x + y)  dy 2 [27 June, 2022 (Shift-II)] = f(x) + f(y) + xy2 + x2 y, for all real x and y. If lim f ( x) = 1, then 113. Let f be a twice differentiable function defined on R f ′(3) is equal to ____. x →0 x such that f(0) = 1, f  ‘ (0) = 2 and f(x) ≠ 0 for all x∈R. If  [4 Sep, 2020 (Shift-I)] f ( x) f ′ ( x) = 0 , for all x∈R, then the value of f(1) lies 105. If f(1) = 1, f ′(1) = 3, then the derivative of f ( f ( f (x))) + (f (x))2 at f ′ ( x) f ′′ ( x ) x = 1 is in the interval. [8 April, 2019 (Shift-II)]  [24 Feb, 2021 (Shift-II)] (a) 12 (b) 33 (a) (3, 6) (b) (0,3) (c) 9 (d) 15