Final Step-B Booklet.pdf

Vidyamandir Classes VMC | Final Step - B 1 C lass XII | Math ematics Final Step - B | Mathematics Functions CHOOSE THE CORRECT ALTERNATIVE. ONLY ONE CHOICE IS CORRECT. HOWEVER , QUESTIONS MARKED ‘*’ MAY HAVE MORE THAN ONE CORRECT OPTION. 1. Let   2 3 4 f x x x x       and     1 g x f x   Then : (A) g ( x ) i s an even function (B) g ( x ) is an odd function (C) g ( x ) is neither even nor odd (D) g ( x ) is periodic 2. Period of     f x x x a b,     where a, b R   and [.] denotes the greatest integer function is : (A) a (B) b (C) a b  (D) 1 3. The domain of the function   2 1 2 5 6 x f x log x x           is : (where[.] den otes the greatest integer function) (A)     3 2 2 3 3 2 x , , ,           (B) 3 2 x ,         (C) 1 2 x ,         (D) No ne of these 4. The function   2 1 f x sec log x x                is : (A) Even (B) Odd (C) Constant (D) None of thes e 5. If   1 1 x f x log , x          1 1 x ,    then 3 2 2 3 2 1 3 1 x x x f f x x                   is : (A)   3 f x     (B)   2 f x     (C)   f x  (D)   f x 6. The range of the fu nction   2 2 1 1 f x x x    is : (A)   1 ,  (B)   2 ,  (C) 3 2 ,        (D) N on e o f these 7. If   4 4 2 x x f x ,   then 1 2 96 97 97 97 f f ..... f                      is equal to : (A) 1 (B) 48 ( C) 48  (D) 1  8. Let           2 and 0 f x y f x y f x f y , y R f k        , then : I f ( x )is even, if 1 k = II f ( x ) is odd, if 0 k  III f ( x ) is always odd IV f ( x ) is neither even nor odd for any value of k The correct choice is : (A) I, III (B) II, III (C) I, II (D) III, IV Vidyamandir Classes VMC | Final Step - B 2 C lass XII | Math ematics 9. The period of the function   f x which satisfies the relation         4 2 6 f x f x f x f x       is : (A) 6 (B) 7 (C) 8 (D) None of t hese 10. Let   6 5 25 25 3 1 3 2 9 6 7 81 3 7 125 409 log log log log x                          , then value of 2 log x is equal to : (A) 0 (B) 1 (C) 1  (D) None of these 11. Which of the following when simplified reduces to unity ? I. 1 5 4 3 81 log log log II. 2 2 2 6 3 log log  III. 3 2 1 64 6 27 log        IV.   3 5 1 2 3 6 log    The correct choice is : (A) I only (B) I I a nd IV only (C) I and III only (D) All the above 12. Given,   ; ; 1 a b c d log x log x log x & log x x          ,   , , , 1 a b c d R    then log abcd x has the value equal to : (A) 1  (B) 1 1 1 1        (C) 1 1 1 1 1        (D) None of these 13. The number of integral solutions of the equation       2 3 2 4 2 4 2 3 x / x x log x log x log x   is : (A) 0 (B) 1 (C) 2 (D) None of these 14. If   2 1    x x f x , x then   1  f x equals : (A) 1  x x (B)     1 x sgn x x   (C) 1   x x (D)     1  x sgn x x 15. If             5 f x sin x x x x      for 0 4 x ,         is invertible, where {.} and [.] represent fractional part and great est in teger functions respectively, then   1 f x  is : I. 1 sin x  II. 1 2 cos x    III.   1 sin x  IV.   1 cos x  The correct choice is : (A) I, II, III (B) II, II I (C) III, IV (D) None of these 16. Let     1 1 2 1 x G x F x a          , where ‘ a ’ is a positive real number not equal to 1 and   F x is an odd function. Which of the following statements is true ? (A)   G x is an od d f unction (B)   G x is an even function (C)   G x is neither even nor odd function (D) Whether   G x is an odd or even function depends on the value of ‘ a ’. Vidyamandir Classes VMC | Final Step - B 3 C lass XII | Math ematics 17. If 2 2 8 8 y y f x , x xy          , then     f m, n f n, m  is : (A) Depends over m and n both (B) Periodic and odd function (C) Constant number (D) None of these 18. If 21 0 2 11          r r f x = constant   x R and f ( x ) is periodic, then p er iod of f ( x ) is : (A) 1 (B) 1/11 (C) 2 (D) 4 19. Suppose f is a real function         4 and 1 4 f x f x f x f    . Then the value of   21 f is : (A) 16 (B) 21 (C) 64 (D) 105 20. Let f be a real valued function such that for any real x         15 15 and 30 30 f x f x f x f x        Then which of the following statements is true ? (A) f is odd and periodic (B) f is odd but not periodic (C) f is even and periodic (D) f is even nut not periodic 21. For , 0, 1,    x R x x let   0 1 1 f x x   and       1 0 , 0, 1, 2,..... n n f x f f x n    Then the value of   100 1 2 2 3 3 3 2 f f f               is equal to : (A) 8 3 (B) 5 3 (C) 4 3 (D) 1 3 22. Let : f R R  be defined by   1 1 x f x x    then f is : (A) both one - one and onto (B) one - one but not onto (C) onto but not one - one (D) neither one - one nor onto 23. Let   1 3 , 3 100 n f n n         where [ n ] denotes the greatest inte ge r l ess than or equal to n . Then   56 1 n f n   is equal to : (A) 56 (B) 689 (C) 1287 (D) 1399 *24. The given function     1 1 x n x a f x x a    is : (A) Odd, if n is even (B) Even, if n is odd (C) Odd, if n is odd (D) Even, if n is even 25. The domain of   2 1/ 2 1/ 4 1 log log 1 f x x                  is : (A)   0, 1 (B)   0, 1 (C)   1,  (D)   1,  Vidyamandir Classes VMC | Final Step - B 4 C lass XII | Math ematics Integer Answer Type Questions The Answer to the following questions are p osi tive integers of 1/2/3 digits and zero 26 Let f be a real valued function such that   2002 2 3 , 0           f x f x x x The value of   2 2000       f is _ __ ____ 27 Let a and b be constants and   2 sin cos 2    f x a x bx x x . If   2 15  f , then the value of   2  f is ________. 28 The range of       2 2 2 log 2 log 16sin 1    f x x is   ,  a , then the value of a is _________. 29 The period of f ( x ) sati sfying         1 3 1 5        f x f x f x f x , is ________ _. 30 The period of     sin 2         f x x (where, [.] denotes greatest integer function), is ________. 31 If     3 2 4 7 3 5 3             a f x x a x x is a one - one function, where   ,    a , then the value of    i s ____ 32 If f ( x ) be a polynomial of degree 4 with leading coefficient 1 satisfying f (1) = 10,     2 20, 3 30   f f , where       12 8 3968     f f , then the value of  is ________. 33 Let f : 2 , 6 3          B Defined by   2 2 cos 3 sin 2 1    f x x x , where   ,  B a b such that   1  f x exists, then ( )  a b is ________. 34 If f ( x ) is real valued function satisfying       1, ,        f x y f x f y xy x y R suc h that f (1 ) = 1, then the number of solution s , o f   ,   f n n n N , is ________. 35 If [.] denotes the greatest integer function and  n N , then period of     tan 2       x f x nx n nx n , is _ ___ 36 Let for     2 2 1 1 1 1 0, ,         a a f x ax bx c g x a x b x c and         p x f x g x . If   0  p x only for 1   x and   2 2   p , then t he value of p (2) is ________. 37 Let ( ) f x be a polynomial fu nction satisfying the relati on 1 1 ( ) ( )                f x f f x f x x , {0} and (3) 26      x R f , then (2) f is ___ . 38 If ( ) ( ) ( ), ,      f x y f x f y x y N and (1) 2  f , then 10 1 ( ) 2050    n f n is ___ . 39 Let ( ) f x be an in ver t ible function such tha t the graph ( )  y f x intersects the coordinate axes in point s A and B r espectively , s uch that Area of 16   OAB square units. If the graph of 1 ( )   y f x intersects the coordinate axe s in points ' A and ' B respecti vely, find the minimum length of diagonal OC of rectangle ' ' OA CB 40 If ( ) , 1 1      x f x x x , then for wha t value of  is { ( )}  f f x x ? Vidyamandir Classes VMC | Final Step - B 5 Class XII | Mathematics Differential Calculus - 1 CHOOSE THE CORRECT ALTERNATIVE. ONLY ONE CHOICE IS CORRECT. HOWEVER, QUESTIONS MARKED ‘*’ MAY HAVE MORE THAN ONE CORRECT OPTION. 1. If 2 3 2 4 lim 1 , x x a e x x           then ‘ a ’ is equal to : (A) 2 (B) 3 2 (C ) 2 3 (D) 1 2 2. If the function     1 , 1 cos , 1 2 x x f x a x b x              is differentiable at 1, x  then a b is equal to : (A) 2 2   (B) 2 2    (C) 2 2   (D)   1 1 cos 2    3. If   f x is differentiable function in the interval   0,  such that   1 1 f  and     2 2 lim 1, t x t f x x f t t x     for each 0, x  then   3/ 2 f is equal to : (A) 13 6 (B) 23 18 (C) 25 9 (D) 31 18 4.   2 0 1 cos 2 lim 2 tan tan 2 x x x x x x    is : (A ) 2  (B) 1 2  (C) 1 2 (D) 2 5. Let   , , 0 . a b R a   If the function f defined as   2 2 3 2 , 0 1 , 1 2 2 4 , 2 x x a f x a x b b x x                   is continuous in the interval   0, ,  the n an ordered pair   , a b is : (A)   2, 1 3  (B)   2, 1 3   (C)   2, 1 3   (D)   2, 1 3   6. 2 2 0 cos lim sin x x e x x   is equal to : (A) 3 (B) 3 2 (C) 5 4 (D) 2 7. Let k be a non - zero real number. If     2 1 , 0 sin log 1 4 12 , 0 x e x x x f x k x                         is a continuous function, then the value of k is : (A) 1 (B) 2 (C) 3 (D) 4 Vidyamandir Classes VMC | Final Step - B 6 Class XII | Mathematics 8. If   f x is continuous and   9 / 2 2 / 9, f  then 2 0 1 cos 3 lim x x f x         is equal to : (A) 9 / 2 (B) 2 / 9 (C) 0 (D) 8 / 9 9. If , nx y e  then 2 2 2 2 d y d x dx dy             is equal to : (A) nx n e (B) nx n e  (C) 1 (D) nx n e   10. If       2 2 2 tan 2 2 2 lim 5, 4 4 x x x k x k x x         then k is equal to : (A) 0 (B) 1 (C) 2 (D) 3 11. Let           , sin and f x x x g x x h x gof x    Then : (A)   h x is not differentiable at 0. x  (B)   h x is differentiable at 0, x  but   h x  is not continuous at 0. x  (C)   h x  is conti nuous at 0 x  but it is not differentiable at 0. x  (D)   h x  is differentiable at 0. x  12. If   2 1 5, , 2 f x x x x     and   g x is its inve rse function, then   7 g  equals : (A) 1 13  (B) 1 13 (C) 1 3 (D) 1 3  13. Let , : f g R R  be two function defined by       1 sin , 0 , and 0, 0 x x f x g x xf x x x                Statement I : f is a continuous function at 0 x  Statement II : g is a differentiable function at 0 x  (A) Statement I is true, statement II is false (B) Both statements I and II are tru e (C) Statement I is false, statement II is true (D) Both statements I and II are false 14. If the function     2 2 cos 1 , , x x f x x k x               is continuous at , x   then k equals : (A) 0 (B) 1 2 (C) 2 (D) 1 4 15. Let : f R R  be a function such that   2 , f x x  for all x R  Then, at 0, x  f is : (A) continuous but not differentiable (B) continuous as well as dif ferentiable (C) neither continuous nor differentiable (D) differentiable but not continuous 16. The value of 1 0 1 1 lim tan 2 1 4 x x x x                   is : (A) 1 (B) 1 2  (C) 2 (D) 0 Vidyamandir Classes VMC | Final Step - B 7 Class XII | Mathematics 17. For 0, 0, , 2 a t          let 1 sin t x a   and 1 cos t y a   Then, 2 1 dy dx        equals : (A) 2 2 x y (B) 2 2 y x (C) 2 2 2 x y y  (D) 2 2 2 x y x  18. Let   1 2 , f x x     and   1 g x x   ; then the set of all points where fog is discontinuous is : (A)   0, 2 (B)   0, 1, 2 (C)   0 (D) an empty set 19. If           sin sin and tan 0, f x x f x x f x g x       then   g x is : (A)   2 cos cos sin x x (B)   2 sin cos cos x x (C)   2 sin sin cos x x (D)   2 cos sin sin x x 20. Let f be a composite function of x defined by     2 1 1 , 1 2 f u u x x u u      Then the number of points x where f is discontinuous is : (A) 4 (B) 3 (C) 2 (D) 1 21. The value of     1 1 nx x lim x    , is : (where {.} denotes the fractional part function) (A) 1 (B) – 1 (C) 0 (D) None of these 22. The value of     1 x lim cos x cos x    is : (A) 1 (B) – 1 (C) 0 (D) None of these 23. The value of       2 3 2 3 1 1 x lim x x     is : (A) 1 (B) – 1 (C) 0 (D) None of these 24. The value of 3 3 2 4 5 4 6 5 7 3 2 1 1 6 2 3 1 n n n n lim , n n n n           n N  is : (A) 1 (B) – 1 (C) 0 (D)  25. The value of           1 4 1 2 3 4 x lim x x x x x             is : (A) 1 2 (B) 5 2 (C) 3 2 (D) 1 2  26. The value of     1 1 x x lim x    , is : (where [.] deno tes greatest integer function) (A) 0 (B) 1 (C) Not defined (D) None of these 27. The value of       1 2 1 5 1 4 2 2 15 2 7 2 x x x lim x x       is : (A) 1 5  (B) 1 2  (C) 2 25  (D) None of these 28. S uppose that   3 2 3 4 12 f x x x x     and     3 3 3 f x , x h x x K , x          , then : For that value of K that makes h ( x ) continuous at x = 3 then h ( x ) is : (A) Even (B) Odd (C)   0 1 f  (D) Neither even nor odd Vidyamandir Classes VMC | Final Step - B 8 Class XII | Mathematics 29. If   2 if 1 1 if 1 ax b x f x x x           is derivable at x = 1. Then the values of a and b are : (A) 1 3 2 2 a , b   (B) 1 1 2 2 a , b    (C) 1 3 2 2 a , b     (D) None of these 30. Let a function f : R R  be given by       f x y f x f y   for all x, y R  and   0 f x  for any x R  . If the function f ( x ) is differentiable at x = 0. (A)   0 x . f e  (B)   0 x . f  (C)   2 2 x f x  (D) None of these 31. The value of   3 2 0 4 1 1 3 x x lim x x sin n p                   is : (A)   9 4 p n  (B)   3 3 4 p n  (C)   3 12 4 p n  (D)   2 27 4 p n  32. 100 1 1 100 1 k k x x lim x              is : (A) 0 (B) 5050 (C) 4550 (D) 5050  33. The value of 2 2 2 2 3 4 x lim tan x sin x sin x        2 6 2 sin x sin x    is : (A) 1 / 10 (B) 1 / 11 (C) 1 / 12 (D) 1 / 8 34. 1 2 2 3 5 3 2 5 2 3 n n n n n n n lim , n N         is : (A) 5 (B) 3 (C) 1 (D) Zero * 35. If   2 2 3 6 x f x x    , then : (A)   1 3 x lim f x    (B)   1 3 x lim f x   (C)   1 3 x lim f x   (D)   1 3 x lim f x    36. 2 1 1 x lim x x n x                 is equal to : (A) 1/2 (B) 3/2 (C) 1/3 (D) 1 37. 2 2 3 0 x x e cos x lim x sin x    is equal to : (A) 1/4 (B) 1/6 (C) 1/12 (D) 1/8 Vidyamandir Classes VMC | Final Step - B 9 Class XII | Mathematics 38 Consider the following statements : S 1 :   0 x x lim x   is an indeterminate form (where [.] denotes greatest integer function). S 2 :   3 0 3 x x x sin lim   S 3 : 2 x x sin x lim x cos x    does not exist. S 4 :         2 ! 1 ! 0 3 ! n n n lim n N n        Which of the statements S 1 , S 2 , S 3 , S 4 are true or false : (A) FTFT (B) FTTT (C) FTFF (D) TTFT * 39. If   3 0 1 x x a cos x b sin x lim x      2 3 0 0 1 x x a cos x b sin x lim lim x x      , where R   , then : (A)     1 0 a, b ,   (B) a and b are any real numbers (C) 0   (D) 1 2   40. The value of       1 4 1 n tan x x lim x     is equal to : (where [.] denotes the integer function) (A) 0 (B) 1 (C) e (D) 1 e  41. The value of  3 3 4 1 [1 ] [2 ] n lim x x n     3 [ ] . . . . n x  is : [where [ x ] denotes greatest integer less than or equal to x ] (A) 2 x (B) 3 x (C) 6 x (D) 4 x 42. The value of 2 2 2 1 4 2 x x x x lim x x              is equal to : (A) 1 (B) 2 (C) 2 e (D) e 43. Let     1 2 f x x x               , when 2 2 x    where [.] represents greatest integer function. Then : (A) f ( x ) is continuous at x = 2 (B) f ( x ) is continuous at x = 1 (C) f ( x ) is continuous at x = – 1 (D) f ( x ) is discontinuous at x = 0 44. Let     f x sgn x  and     2 5 6 g x x x x    . Then function f ( g ( x )) is discontinuous at : (A) Infinitely many points (B) Exactly one point (C) Exactly three points (D) No point 45. If     1 1 3 4 0 2 0 0 / x x x e , x f x e , x             , then f ( x ) is : (A) Continuous as well differentiab le at x = 0 (B) Continuous but not differentiable at x = 0 (C) Neither differentiable at x = 0 nor continuous at x = 0 (D) None of these Vidyamandir Classes VMC | Final Step - B 10 Class XII | Mathematics 46. If       for 0 0 for 0 x x x sin x x f x x         , (where {.} denotes the fractional part function), then : (A) f is continuous and differentiable at x = 0 (B) f is continuous but not differentiable at x = 0 (C) f is continuous and differentiable at x = 2 (D) None of these 47. Given             2 5 1 for 0 1 3 0 for 0 x x x x a x a f x log a x x x , a a x                                     where [.] represents the integral part function, then : (A) f is conti nuous but not differentiable at x = 0 (B) f is continuous and differentiable at x = 0 (C) The differentiability of ‘ f ’ at x = 0 depends on the value of a (D) f is continuous and differentiable at x = 0 and for a = e only * 48. The function   2 3 1 3 13 1 4 2 4 x , x f x x x , x                                  is : (A) Continuous at x = 1 (B) Differentiable at x = 1 (C) Continuous at x = 3 (D) Differentiable at x = 3 49. If       2 2 3 2 1 , 0 2 1 1 , 2 3 4 9 4 2 , 3 4 4                         x x x f x x x x x x x , then : (A) f ( x ) is differentiable at x = 2 and x = 3 (B) f ( x ) is non - differentiab le at x = 2 and x = 3 (C) f ( x ) is differentiable at x = 3 but not at x = 2 (D) f ( x ) is differentiable at x = 2 but not at x = 3 50. If f ( x ) is differentiable everywhere, then : (A) f is differentiable everywhere (B) 2 f is differentiable everywhere (C) f f is not differentiable at some point (D) f f  is differentiable everywhere * 51.     2 1 1 sin . cos , 0 0 , 0                x x f x x x then : (A) Continuous no where in 1 1 x    (B) Continuous everywhere in 1 1 x    (C) Differentiable no where in 1 1 x    (D) Differentiable everywhere in 1 1 x    52. Let f ( x ) be defined in   2 2 ,  by   2 2 2 2 max( 4 , 1 ) , 2 0 min( 4 , 1 ) , 0 2                x x x f x x x x , then : (A) is continuous at all points (B) is not continuous at more than one point (C) is not differentiable only at one point (D) is not differentiable at more than one points Vidyamandir Classes VMC | Final Step - B 11 Class XII | Mathematics 53. The number of points at w hich the function     0 f x max, a x, a x, b , x , a b           can not be differentiable is : (A) 1 (B) 2 (C) 3 (D) None of these 54. Let   2 f x x x   and     0 0 1 1 max f t , t x, x g x sin x , x           , then in the interval   0 ,  (A) g ( x ) is everywhere continuous except at two points (B) g ( x ) is everywhere differentiable except at two points (C) g ( x ) is everywhere differentiable except at x = 1 (D) None of these 55. If f : R R  be a differentiable function, such that       2 2 4 ,       f x y f x f y xy x y R , then : (A)     1 0 1     f f (B)     1 0 1 f f     (C)     0 1 2 f f     (D)     0 1 2 f f     56. Consider the following statements : S 1 : Number of points where     2 sgn 1   f x x x is non - differentiable is 3. S 2 : Defined     3 sin 1 , 0 2 tan sin , 0             a x x f x x x x x , if f ( x ) be continuous at x = 0, then a = 1 2 S 3 : The set of all points, where the function 2 3 x x is differentiable is     0 0 , ,    S 4 : Number of number where     1 1 sin sin   f x x is non - differentiable in the interval   0 3 ,  is 3. Which of the statements S 1 , S 2 , S 3 , S 4 are true or false : (A) TTTF (B) TTTT (C) FTTF (D) TFTT For Quest ions 57 - 59 (A) Statement - 1 is True, Statement - 2 is True and Statement - 2 is a correct explanation for Statement - 1 (B) Statement - 1 is True, Statement - 2 is True and Statement - 2 is NOT a correct explanation for Statement - 1 (C) Statement - 1 is True, Statement - 2 is False (D) Statement - 1 is False, Statement - 2 is True 57. Statement 1 : 0 1 2 2 x cos x lim x   does not exist Statement 2 :   sin 2 sin sin lim 2 1 sin sin x x x x x n x        58. Statement 1 : 4 3 4 2 2 3 7 2 lim 3 3 2 3 x x x x x x x       Statement 2 : If P( x ) and Q( x ) are two polyn omials with rational coefficients, then :         coefficient of highest power of in P coefficient of highest power of in Q x P x x x lim Q x x x   59. Statement 1 :   2 5 6 2 1 x x f x x tan x x        is continuous function within the domain of f ( x ). Statement 2 : All absolute valued polynomial function, rational polynomial function , trigonometric functions are continuous within their domain. Vidyamandir Classes VMC | Final Step - B 12 Class XII | Mathematics 60. The value of a and b so that   4 3 2 4 3 2 lim 3 2 2 3 4 x x ax x bx x x cx x d            (A) 2 5 a , b R, c , d R     (B) 0 a , b R, c R, d R     (C) 2 2 3 a , b , c , d R     (D) None of these 61. The value of a and b so that   2 0 1 lim 2 sin x x x axe b n x cxe x x        is : (A) 3 12 9 a , b , c     (B) 2 12 9 a , b , c    (C) 0 1 9 a , b , c    (D) None of these 62. The value of 2 sec 2 0 lim sin 2 bx x ax                        is : (A) a b e (B) 2 2 a b e (C) 2 2 a b e  (D) None of these 63 The function     tan 6 tan 5 tan 6 , 0 5 2 2 , 2 1 cos , 2 x x a x b x f x b x x x                                   . Determine the values of ‘ a ’ and ‘ b ’, if f is continuous at 2 x   (A) 0 a  , 1 b   (B) 0 0 a , b   (C) 1 1 a , b    (D) None of these 64 Let                 1 1 sin 1 . cos 1 2 1 x x f x x x       , then the   0 lim x f x   and   0 lim x f x   is : (where {. } denotes the fractional part function) (A) 2 ,   (B) 2 2 2 ,   (C) 2 2 2 ,     (D) 2  , does not exist 65 If   2 3 2 1 sin , 0 1 3 8 2cos tan , 1 2 x ax b x f x x x x x x                         is differentiable in [0, 2], then ‘ a ’ and ‘ b ’ are : (where [.] stands greatest integer function) (A) 1 4 a , b    (B) 1 13 6 4 6 a , b     (C) 13 6 6 a , b    (D) None of these 66 If   1 sec tan y x   , then dy dx at x = 1 is equal to : (A) 1 2 (B) 1 2 (C) 1 (D) 2 67 If     2 2 1 , 1 1 , 0 , 2 2 , 1 1 , 0 x x x x f x g x x x x x                      and   h x x  then       0 x lim f g h x  is equal to : (A) 1 (B) 0 (C) 1  (D) Does not exist Vidyamandir Classes VMC | Final Step - B 13 Class XII | Mathematics 68 Let   2 1 1 sin sin , 0 0 , 0 x x f x x x x                      , then   x lim f x  is equal to : (A) 0 (B) 1 2  (C) 1 (D) None of these 69 3 3 lim x a x x a a                  ( a > 0), where [ x ] d enotes the greatest integer less than or equal to x , is equal to : (A) 2 1 a  (B) 2 1 a   (C) 2 a (D) 2 a  70 0 sin lim (1 ) x x x e x         , where [.] represents gre atest integer function, is equal to : (A) 1  (B) 1 (C) 0 (D) Does not exist 71. If   lim sin 1 sin x x x      and lim sin 1 sin x m x x          , where [.] denotes the greatest integer function, then : (A) 0 m    (B) 0 , m   is undefined (C) , m  both do not exist (D) 0 0 , m    (although m exist) 72. If   1 1 1 1 n f x x x                   , then   0 n lim f  is equal to : (A) 1 ( B) 1  (C) 2 (D) None of these 73 Given a real valued function f such that       2 2 2 [ ] , 0 ( [ ] ) 1 , 0 cot , 0 tan x x x x f x x x x x               where [.] represents greatest integer function and {.} represents fractional part function, then : (A)   0 1 x lim f x   (B)   0 1 x lim f x cot    (C)   2 1 0 1 x cot lim f x           (D)   0 1 x lim f x    74 The graph of the function   1 2 0 2 t x x f x lim cot t           is : (A) (B) (C) (D) Vidyamandir Classes VMC | Final Step - B 14 Class XII | Mathematics 75 If   x lim f x  exists and is finite and non - zero and       2 3 1 lim 3 x f x f x f x             , then the value of   x lim f x  is equal to: (A) 1 (B) – 1 (C) 2 (D) None of these 76 1 1 2 3 lim , n n x x x x x e e n x e n N x                          , is equal to : (A) 0 (B) 2 3 n        (C) 3 2 n        (D) None of these 77 0 exp 1 exp 1 lim lim y x ay by x n x n x x y                                              is equal to : (A) a b  (B) a b  (C) b a  (D)   a b   78 The function f ( x ) is defined       2 4 3 3 log 2 5 , if 1 1 4 4 , if 1 x x x x or x f x x              (A) is continuous at x = 1 (B) is discontinuous at x = 1 since (1 ) f  does not exist though (1 ) f  exists (C) is discontinuous at x = 1 since (1 ) f  doe s not exist though (1 ) f  exists (D) is discontinuous since neither (1 ) f  nor (1 ) f  exists 79 If   1 1 0 1 1 0 0 0 x x sin , x x f x x x sin , x x , x                                , then f ( x ) is : (A) Continuous as well as diff. at x = 0 (B ) Continuous at x = 0, but not diff. at x = 0 (C) Neither continuous at x = 0 nor diff. at x = 0 (D) None of these 80 Given           2 1 2 2 for 0 1 1 2 2 for 1 2 x x e x f x . f x a sgn x cos x bx x               is differentiable at x = 1 provided : (A) 1 2 a , b    (B) 1 2 a , b    (C) 3 4 a , b    (D) 3 4 a , b    * 81 Let   x f x sin x    , then : (A) ( ) 1 f     (B) ( ) 1 f    (C)   x lim f x   does not exist (D)   x lim f x   does not exist Vidyamandir Classes VMC | Final Step - B 15 Class XII | Mathematics * 82   1 lim n n x ax x A    is equal to : (A) n a if n N  (B)  if n Z   and 0 a A   (C) 1 1 A  if n = 0 (D) n a if 0 n Z , A    and 0 a  83 Let 2 2 2 4 0 4 lim , 0 x x a a x L a x       . If L is finite, then : (A) 2 a  (B) 1 a  (C) 1 64 L  (D) 1 32 L  84 If   1 2 2 0 lim 1 1 2 sin , 0 x x x n b b b            and   ,      , then the value of  is : (A) 4   (B) 3   (C) 6   (D) 2   85 If 2 1 lim 4 1 x x x ax b x                , then : (A) 1 4 a , b   (B) 1 4 a , b    (C) 2 3 a , b    (D) 2 3 a , b   86 Let   a  and   a  be the roots of the equation       2 3 6 1 1 1 1 1 1 0 a x a x a          where a  – 1. Then   0 a lim a    and   0 a lim a    are : (A) 5 2  and 1 (B) 1 2  and 1  (C) 7 2  and 2 (D) 9 2  and 3 87 Let     f a g a k   and their n th derivatives     n n f a , g a exist and are not equal for some n . Further if                 4 x a f a g x f a g a f x g a lim g x f x       , then the value of k is equal to : (A) 4 (B) 2 (C) 1 (D) 0 88. If   1 1 0 0 0 x x f x xe , x , x                   , then f ( x ) is : (A) Continuous as well as differentiable for all x (B) Continuous for all x but not differentiable at x = 0 (C) Neither diff erentiable nor continuous at x = 0 (D) Discontinuous everywhere 89. The function f :   0 R R   given by   2 1 2 1 x f x x e    can be made continuous at x = 0 by defining f (0) as : (A) 2 (B) 1  (C) 0 (D) 1 90. Let     1 1 1 1 0 1 x sin , x f x x , x           . Then which one of the following is true ? (A) f is differentiable at x = 0 and at x = 1 (B) f is differentiable at x = 0 but not at x = 1 (C) f is differentiable at x = 1 but not at x = 0 (D) f is neither differentiable at x = 0 nor at x = 1 Vidyamandir Classes VMC | Final Step - B 16 Class XII | Mathematics 91. Let f : R R  be a positive increasing function with     3 1 x f x lim f x   . Then     2 x f x lim f x  (A) 2/3 (B) 3/2 (C) 3 (D) 1 92. Let f :   0 R ,   be s uch that   5 x lim f x  exists and     2 5 9 0 5 x f x lim x     . Then   5 x lim f x  equals : (A) 0 (B) 1 (C) 2 (D) 3 93. The value of p and q for which the function     2 3 2 1 0 0 0                  sin p x sin x , x x f x q , x x x x , x x is continuous for all x  R are : (A) 1 3 2 2 p , q    (B) 5 1 2 2 p , q   (C) 3 1 2 2 p , q    (D) 1 3 2 2 p , q   94.     3 2 1 1 2 1 2 2 x x tan sin x lim x tan x                    is equal to : (A) 1 16 (B) 1 16  (C) 1 32 (D) 1 32  95. If     0 3 3 x n x n x lim k x        , then the value of k is : (A) 0 (B) 1 3  (C) 2 3 (D) 2 3  96. If 2 2 2 1 x x a b lim e x x           , then the value of a and b are : (A) a R, b R   (B) 1 a , b R   (C) 2 a R, b   (D) 1 2 a , b   97. Let  and  be the distinct roots of 2 0 ax bx c    , then     2 2 1 x cos ax bx c lim x        is equal to : (A)   2 1 2    (B)   2 2 2 a     (C) 0 (D)   2 2 2 a    98. The set of points, where   1 x f x x   is differentiable, is : (A)     1 1 , ,      (B)   ,   (C)   0 ,  (D)     0 0 , ,    99. If function f ( x ) is differentiable at x = a , then     2 2 x a x f a a f x lim x a    is : (A)   2 a f a   (B)     2 af a a f a   (C)     2 2 af a a f a   (D)     2 2 af a a f a   Vidyamandir Classes VMC | Final Step - B 17 Class XII | Mathematics 100. If f : R R  is a function defined by     2 1 2 x f x x cos          , where [ x ] denotes the greatest integer function then f is: (A) Continuous for every real x (B) Discontinuous only at x = 0 (C) Discontinuous only at non - zero integral values of x (D) Continuous only at x = 0 101. Consider the function,   2 5 f x x x , x R      Statement 1 :   4 0 f   Statement 2 : f is continuous in [2, 5], differentiable in (2, 5) and     2 5 f f  (A) Statement - 1 is True, Statement - 2 is True and Statement - 2 is a correct explanation for Statement - 1 (B) Statement - 1 is True, Statement - 2 is True a nd Statement - 2 is NOT a correct explanation for Statement - 1 (C) Statement - 1 is True, Statement - 2 is False (D) Statement - 1 is False, Statement - 2 is True 102.     0 1 2 3 4 x cos x cos x lim x tan x    is equal to : (A) 1 4  (B) 1 2 (C) 1 (D) 2 103. If   2 2 1 f x x    and   2 y f x  , then dy dx at x = 1 is equal to : (A) 2 (B) 1 (C) – 2 (D) – 1 104. If     x f x log n x ,   then   f x  at x e  is equal to : (A) 1 e (B) e (C) 1 (D) Zero 105. If 3 2 1 3 2 2 t x , y t t t     , then 3 dy dy x dx dx        is equal to : (A) 0 (B) 1  (C) 1 (D) 2 106. If   sin x f x x  , then 4 f         is equal to : (A) 1 2 2 4 2 2 4 2                 n    (B) 1 2 2 4 2 2 4 2                 n    (C) 1 2 2 2 2 4 2 4                 n    (D) 1 2 2 2 2 4 2 4 n                    107. Let f ( x ) be a polynomial in x . Then the second deriva tive of   x f e w.r.t. x is : (A)     x x x f e . e f e    (B)     2 2 x x x x f e . e f e . e    (C)   x x f e e  (D)     2 x x x x f e . e f e . e    108. If 1 1 0 x y y x     , then dy dx is equal to : (A)   2 1 1 x  (B)   2 1 1 x   (C)   2 1 1 x  (D)   1 1 x  Vidyamandir Classes VMC | Final Step - B 18 Class XII | Mathematics 109. Let g is the inverse function of f and     10 2 1 x f x x    . If   2 g a  , then   2 g  is equal to : (A) 10 2 a (B) 2 10 1 a a  (C) 10 2 1 a a  (D) 10 2 1 a a  110. If 1 2 1 1 y sin x x x x            and   1 2 1 dy p dx x x    , then p is equa l to : (A) 0 (B) 1 1 x  (C) 1 sin x  (D) 2 1 1 x  111. If 2 4 2 1 1 d x x ax b dx x x               , then the value of ‘ a ’ and ‘ b ’ are respectively : (A) 2 and 1 (B) – 2 and 1 (C) 2 and – 1 (D) None of the se 112. If 2 2 1 1 x y f x          and   f x sin x   , then dy dx is equal to : (A)   2 2 2 2 1 2 1 1 1 x x x sin x x            (B)     2 2 2 2 2 1 2 1 1 1 x x x sin x x            (C)   2 2 2 2 1 2 1 1 1 x x x sin x x            (D) None of these * 113. If 2 2   t x y e where 1 2 2 y t sin x y            , then dy dx is equal to : (A) x y x y   (B) x y x y   (C) y x y x   (D) 2 2 y x y   * 114. If f is twice differentiable such that     f x f x    and     f x g x   . If h ( x ) is a twice differentiable function such that           2 2 h' x f x g x   . If     0 2 1 4 h , h   , then the equation   y h x  represent s : (A) a curve of degree 2 (B) a curve passing through the origin (C) a straight line with slope 2 (D) a straight line with y intercept equal to 2 * 115. 2 1 2 1 1 2 1 x y cos x      , then dy dx is equal to : (A)   2 1 2 1 , x R x   (B)   2 1 0 2 1 , x x   (C)   2 1 0 2 1 , x x    (D)   2 1 0 2 1 , x x   116. Let f ( x ) be a polynomial function of second degree. If     1 1 f f   and a , b , c are in AP, then     f a , f b   and   f c  are in : (A) AP (B) GP (C) HP (D) Arithmetico - Geometric Progression Vidyamandir Classes VMC | Final Step - B 19 Class XII | Mathematics Integer Answer Type Questions The Answer to the following questions are positive integers of 1/2/3 digits and zero 117 Total number of integr al points where the function 2 2 ( ) [ ] [ ] f x x x   is continuous is ____ . [ Where [ x ] represents greatest integer less than or equal to x ] 118 If 2 sec 2 2 2 4 ( ) lim 16 x x f t dt x      equals 8 ( ) f r   , then r e qual ______ . 119 sin 1/ 0 1 lim (sin ) x x x x x                  is equal to _____ . 120 Total number of points of discontinuity of the function 2 3 ( ) min{1, , } f x x x  is ____ . 121 The set of all points where the function sin ( ) 1 | | x f x x   is non - differentia ble is____ . 1 22 Let  be the set of real numbers and : f    be such that for all x and y in  , 2 2 | ( ) ( ) | ( ) f x f y x y    . If (0) 1 f  , then find (10) f ____ . 123 T h e total number of points wh ere the function 2 1 ( ) [ ]cos 2 x f x x          is disco ntinuous is____. [ Where [ x ] represents greatest integer l ess than or equal to x ] 124 1 lim(1 ) tan 2 x x x            is equal to _____ . 125 2 2 sin lim cos x x x x x    is eq u al to ____. 126 2 2 0 0 cos lim sin x x t dt x x   is equal to _____ . 127 The total number of points where the fu nction   2 2 ( ) ( 1) 3 2 cos f x x x x x      is not differentiable is _____ 128 The total number of po ints of non - differentiability of ( ) 1 f x x   is _____ . 129 Let f be a twice differ entiable function "( ) ( ) f x f x   and ( ) '( ) g x f x  Given 2 2 ( ) ( ( )) ( ( )) F x f x g x   and (5) 5 F  , find (10) F 130 If 2 2 1 x y   and ' 0 yy ax   , then the value of a is___ . 131 The total number of points of discon tinuity of   2 tan ( ) ( ) 1 [ ] x f x x      is (where [ x ] represents the g reatest integer less than or equal x ) V idyamandi r Classes V MC | Final Step - B 20 Class XII | Mathematics Differential Calculus - 2 CHOOSE THE CORRECT ALTERNATIVE. ONLY ONE CHOICE IS CORRECT. HOWEVER, QUESTIONS MARKED ‘*’ MAY HAVE MORE THAN ONE CORRECT OPTION. 1. The minimum distance of a point on the curve 2 4 y x   from the origin is : (A ) 19 2 (B) 15 2 (C) 15 2 (D) 19 2 2. Let C be a curve given by   3 1 4 3, 4 y x x x     If P is a point on C , such that the tangent at P has slope 2 , 3 then a point through which the normal at P passes, is : (A)   2, 3 (B)   4, 3  (C)   1, 7 (D)   3, 4  3. The distance, from the origin, of the normal to the curve, 2cos 2 sin , 2sin 2 cos x t t t y t t t     at , 4 t   is : (A) 4 (B) 2 2 (C) 2 (D) 2 4. If Rolle’s theorem holds for the function     3 2 2 , 1, 1 , f x x bx cx x      at the point 1 , 2 x  then 2 b c  equals (A) 1 (B) 1  (C) 2 (D) 3  5. Let the tangents drawn to the circle, 2 2 16 x y   from the point   0, P h meet the x - ax is at points A and B . If the area of APB  is minimum, then h is equal to : (A) 4 3 (B) 3 3 (C) 3 2 (D) 4 2 6. The equation of a normal to the curve, sin sin 3 y x y          at 0, x  is : (A) 2 3 0 x y   (B) 2 3 0 y x   (C) 2 3 0 y x   (D) 2 3 0 x y   7. Let k and K be the minimum and the maximum val ues of the function   0.6 0.6 (1 ) 1 x f x x    in   0, 1 respectively, then the ordered pair ( k , K ) is equal to : (A)   0.6 1, 2 (B)   0.4 0.6 2 , 2  (C)   0.6 2 , 1  (D)   0.4 2 , 1  8. If the Rolle’s theorem holds for the function   3 2 2 f x x ax bx    in the interval   1, 1  for the point 1 , 2 c  then the value of 2 a b  is : (A) 1 (B) 1  (C) 2 (D) 2  9. For the curve 3sin cos , sin , 0 , y x e           the tangent is parallel to x - axis when  is : (A) 3 / 4  (B) / 2  (C) / 4  (D) / 6  10. Two ships A and B are sailing away from a fixed point O along routes such that AOB  is always 120°. At a certain instance, 8 , 6 OA km OB km   and the ship A is sailing at the rate of 20 km / hr while the ship B sailing at the rate of 30 km / hr . Then the distance between A and B is changing at the rate in ( km / hr ) : (A) 260 / 37 (B) 260 / 37 (C) 80 / 37 (D) 80 / 37