Determinants, Functions, Limits, Continuity , Differentiability - Test paper, Mathematics Notes | EduRev

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: Determinants, Functions, Limits, Continuity , Differentiability - Test paper, Mathematics Notes | EduRev

 Page 1


Classes in Mathematics for IIT-JEE/AIEEE                       +2                                   Sapra Classes 
                                                                                                                                                          M a t h e m a t i c s    
  
Determinants, Functions, Limits, Continuity , Differentiability                       Class Test  - 7 B 
 
Time  :  2hr 30 minutes                M.M.   132 
   
Read the carefully the instructions given below :  
1. The test consists of 37 questions. 
The test contains four sections.  
Section A : Contains 16 questions & only one of the four choices is the correct answer. Each question  carries 3 marks and -1 
will be awarded for every wrong answer. 
Section B :   Contains 8 problems & these are multiple correct answer(s) type problems. Each question has four choices out of 
which one or more is / are correct. Each question carries 4.5 marks, however there is no negative marking.  
Section C :   Contains 10 questions in two passages. Each question carries 3 marks and -1 will be awarded for every wrong 
answer. 
  Section D :   Contains 3 matching problems. Each problems carries 6 marks and there is no negative marking 
 
  Student Name : __________________________________  Class : _____________________________ 
  Marks Obtained : ________________________________   Date of Test  : _________________________ 
 
Part A 
Q.1 The no. of non-positive integers satisfying the inequality ||x - 1| - x| = 4 are 
  (a) five   (b) two    (c) three  (d) infinite 
 
Q.2 All solutions of the equation  |x
2
 – x - 6| = x + 2  are 
  (a) natural nos. (b) negative integer  (c) rational nos. (d) irrational nos. 
 
Q.3 If in a triangle two angles are  Tan
-1
 2 & Tan
-1
 3  then the measure of the 3
rd
 angle is 
  (a) p / 4  (b) p / 2   (c) p / 3  (d) 3p / 4 
                        _____   _____ 
Q.4 The identity  Cos
-1
 x  –  Cos
-1
 y  =  Cos
-1
 (xy + v1 – x
2
  v1 – y
2
)  holds  if 
  (a) x
2
 + y
2
 = 1  (b) xy = 1   (c) x = y  (d) x + y = 0 
 
Q.5 If A, B, P & Q are sq. matrices of same order such that   adj B = A & |P| = |Q| = 1  then   
adj (Q
-1
 B P
-1
)   is given as 
 (a) QAP  (b) PAQ   (c) P
-1
 B Q
-1
  (d) PBQ 
 
Q.6 If  f(x) = [x] [Sin px], x ? (-1, 1) the f(x) is 
  (a) differentiable at x = 0 (b) continuous in (-1, 0)    (c) differentiable in (-1, 1)    (d) none 
 
Q.7 Let  f(x) = [tan
2
 x], where [.] denotes greatest integer function then 
  (a) lim f(x) does not exist     (b) f(x) is continuous at x = 0 
        x?0 
(c) f(x) is not differentiable at x = 0    (d)  f(0) = 1 
Q.8 The function f(x) = max {1 – x, 1 + x, 2}, x ? R   is 
  (a) continuous at all points except at one point  (b) differentiable at all x 
(c) diff. at all points except two points in domain            (d) discontinuous at x = 1& x = -1 
                                         SAPRA CLASSES  (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS)  
             SCO- 43 , SEC – 20 C , CHANDIGARH. 9041960872,       SCF. 18 , SEC 15 , PANCHKULA,  98720-27106 
Page 2


Classes in Mathematics for IIT-JEE/AIEEE                       +2                                   Sapra Classes 
                                                                                                                                                          M a t h e m a t i c s    
  
Determinants, Functions, Limits, Continuity , Differentiability                       Class Test  - 7 B 
 
Time  :  2hr 30 minutes                M.M.   132 
   
Read the carefully the instructions given below :  
1. The test consists of 37 questions. 
The test contains four sections.  
Section A : Contains 16 questions & only one of the four choices is the correct answer. Each question  carries 3 marks and -1 
will be awarded for every wrong answer. 
Section B :   Contains 8 problems & these are multiple correct answer(s) type problems. Each question has four choices out of 
which one or more is / are correct. Each question carries 4.5 marks, however there is no negative marking.  
Section C :   Contains 10 questions in two passages. Each question carries 3 marks and -1 will be awarded for every wrong 
answer. 
  Section D :   Contains 3 matching problems. Each problems carries 6 marks and there is no negative marking 
 
  Student Name : __________________________________  Class : _____________________________ 
  Marks Obtained : ________________________________   Date of Test  : _________________________ 
 
Part A 
Q.1 The no. of non-positive integers satisfying the inequality ||x - 1| - x| = 4 are 
  (a) five   (b) two    (c) three  (d) infinite 
 
Q.2 All solutions of the equation  |x
2
 – x - 6| = x + 2  are 
  (a) natural nos. (b) negative integer  (c) rational nos. (d) irrational nos. 
 
Q.3 If in a triangle two angles are  Tan
-1
 2 & Tan
-1
 3  then the measure of the 3
rd
 angle is 
  (a) p / 4  (b) p / 2   (c) p / 3  (d) 3p / 4 
                        _____   _____ 
Q.4 The identity  Cos
-1
 x  –  Cos
-1
 y  =  Cos
-1
 (xy + v1 – x
2
  v1 – y
2
)  holds  if 
  (a) x
2
 + y
2
 = 1  (b) xy = 1   (c) x = y  (d) x + y = 0 
 
Q.5 If A, B, P & Q are sq. matrices of same order such that   adj B = A & |P| = |Q| = 1  then   
adj (Q
-1
 B P
-1
)   is given as 
 (a) QAP  (b) PAQ   (c) P
-1
 B Q
-1
  (d) PBQ 
 
Q.6 If  f(x) = [x] [Sin px], x ? (-1, 1) the f(x) is 
  (a) differentiable at x = 0 (b) continuous in (-1, 0)    (c) differentiable in (-1, 1)    (d) none 
 
Q.7 Let  f(x) = [tan
2
 x], where [.] denotes greatest integer function then 
  (a) lim f(x) does not exist     (b) f(x) is continuous at x = 0 
        x?0 
(c) f(x) is not differentiable at x = 0    (d)  f(0) = 1 
Q.8 The function f(x) = max {1 – x, 1 + x, 2}, x ? R   is 
  (a) continuous at all points except at one point  (b) differentiable at all x 
(c) diff. at all points except two points in domain            (d) discontinuous at x = 1& x = -1 
                                         SAPRA CLASSES  (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS)  
             SCO- 43 , SEC – 20 C , CHANDIGARH. 9041960872,       SCF. 18 , SEC 15 , PANCHKULA,  98720-27106 
Classes in Mathematics for IIT-JEE/AIEEE                       +2                                   Sapra Classes 
                                                                                                                                                          M a t h e m a t i c s    
 
Q.9 The domain of the derivative of the function   f(x) =    tan
-1
 x    if  |x| = 1 
           (|x| - 1)   if  |x| > 1  is 
                2 
  (a) R – {0}  (b) R – {1}   (c) R – {0, -1, 1} (d) R – {-1, 1} 
 
Q.10 Given  f ' (2) = 6 & f ' (1) = 4  ,  lim   f(2h + 2 + h
2
) – f(2)      is equal to  
         
h?0
    f(h – h
2
 + 1) – f(1) 
  (a) 3 / 2  (b) 5 / 2   (c) 3   (d) – 3 
 
Q.11 Range of the expression  f(x) =  3 – 2 Cos x    ? x ? R 
           Cos x + 2 
  (a) [1/3, 1/5]  (b) [-5, 3]   (c) (0, 2/3]  (d) [1/3, 5] 
 
Q.12 Let  f(x) =      Cos x   x 1 
           2 Sin x    x
2
 2x then  lim   f ' (x)   =  
            Tan x       x 1          
x?0        
 x 
  (a) 4   (b) 3    (c) 0    (d) -2 
 
Q.13 If  lim   [(a - n)nx – tan x] Sin nx    = 0 , where n is a non-zero real no. then a is equal to 
       x?0
                    x
2
 
 (a)     n      (b)   n + 1   (c) n + 1/n  (d) n 
      n + 1             n 
Q.14 If  f(x) is differentiable and strictly increasing function, then the value of  lim   f(x
2
) – f(x)     is 
                     
x?0
    f(x) – f(0) 
  (a) -1   (b) 1    (c) 0   (d) 2 
 
Q.15 If  y
2
 = P(x) is a polynomial of degree 3 then  2   d  (y
2
  d
2
 y )  equals 
                       dx         dx
2
 
  (a) p''' (x) + p' (x) (b) p'' (x) p''' (x)  (c) p(x) p''' (x)  (d) a constant 
 
Q.16 If   ? (x) =     x
2
 – 5x + 3 2x – 5        3 
            3x
2
 + x + 4 6x + 1        9  
           7x
2
 – 6x + 9 14x – 6      21  =   ax
3
 + bx
2
 + cx + d   then 
  (a) a ? 0, d = 141  (b) a = b = c = 0  (c) c = 7  (d) d = 29 
 
 
PART B :-  4½ Marks Questions 
 
Q.17 The integer n for which   lim  (Cos x - 1) (Cos x - e
x
)   is equal to ? (a finite non zero no.) then is 
equal to      
x?0
                   x
n
 
  (a) n = 4  (b) n = 3   (c) ? = -1/2  (d) ? =  1/2 
 
Q.18 If  |C| = 1 / 2 and f(x) is diff. at x = 0 given by  f(x) =    b Sin
-1
 (c + x)  , -1/2 < x < 0 
          2 
         1 / 2       ,         x = 0 
               e
ax / 2
 – 1       ,  0 < x < 1 / 2 ,    then 
         x 
  (a) a = 1
  
(b) 16 b
2
 = 4 – c
2  
(c) a = -1
  
(d) 64 b
2
 = 4 – c
2
 
 
                                         SAPRA CLASSES  (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS)  
             SCO- 43 , SEC – 20 C , CHANDIGARH. 9041960872,       SCF. 18 , SEC 15 , PANCHKULA,  98720-27106 
Page 3


Classes in Mathematics for IIT-JEE/AIEEE                       +2                                   Sapra Classes 
                                                                                                                                                          M a t h e m a t i c s    
  
Determinants, Functions, Limits, Continuity , Differentiability                       Class Test  - 7 B 
 
Time  :  2hr 30 minutes                M.M.   132 
   
Read the carefully the instructions given below :  
1. The test consists of 37 questions. 
The test contains four sections.  
Section A : Contains 16 questions & only one of the four choices is the correct answer. Each question  carries 3 marks and -1 
will be awarded for every wrong answer. 
Section B :   Contains 8 problems & these are multiple correct answer(s) type problems. Each question has four choices out of 
which one or more is / are correct. Each question carries 4.5 marks, however there is no negative marking.  
Section C :   Contains 10 questions in two passages. Each question carries 3 marks and -1 will be awarded for every wrong 
answer. 
  Section D :   Contains 3 matching problems. Each problems carries 6 marks and there is no negative marking 
 
  Student Name : __________________________________  Class : _____________________________ 
  Marks Obtained : ________________________________   Date of Test  : _________________________ 
 
Part A 
Q.1 The no. of non-positive integers satisfying the inequality ||x - 1| - x| = 4 are 
  (a) five   (b) two    (c) three  (d) infinite 
 
Q.2 All solutions of the equation  |x
2
 – x - 6| = x + 2  are 
  (a) natural nos. (b) negative integer  (c) rational nos. (d) irrational nos. 
 
Q.3 If in a triangle two angles are  Tan
-1
 2 & Tan
-1
 3  then the measure of the 3
rd
 angle is 
  (a) p / 4  (b) p / 2   (c) p / 3  (d) 3p / 4 
                        _____   _____ 
Q.4 The identity  Cos
-1
 x  –  Cos
-1
 y  =  Cos
-1
 (xy + v1 – x
2
  v1 – y
2
)  holds  if 
  (a) x
2
 + y
2
 = 1  (b) xy = 1   (c) x = y  (d) x + y = 0 
 
Q.5 If A, B, P & Q are sq. matrices of same order such that   adj B = A & |P| = |Q| = 1  then   
adj (Q
-1
 B P
-1
)   is given as 
 (a) QAP  (b) PAQ   (c) P
-1
 B Q
-1
  (d) PBQ 
 
Q.6 If  f(x) = [x] [Sin px], x ? (-1, 1) the f(x) is 
  (a) differentiable at x = 0 (b) continuous in (-1, 0)    (c) differentiable in (-1, 1)    (d) none 
 
Q.7 Let  f(x) = [tan
2
 x], where [.] denotes greatest integer function then 
  (a) lim f(x) does not exist     (b) f(x) is continuous at x = 0 
        x?0 
(c) f(x) is not differentiable at x = 0    (d)  f(0) = 1 
Q.8 The function f(x) = max {1 – x, 1 + x, 2}, x ? R   is 
  (a) continuous at all points except at one point  (b) differentiable at all x 
(c) diff. at all points except two points in domain            (d) discontinuous at x = 1& x = -1 
                                         SAPRA CLASSES  (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS)  
             SCO- 43 , SEC – 20 C , CHANDIGARH. 9041960872,       SCF. 18 , SEC 15 , PANCHKULA,  98720-27106 
Classes in Mathematics for IIT-JEE/AIEEE                       +2                                   Sapra Classes 
                                                                                                                                                          M a t h e m a t i c s    
 
Q.9 The domain of the derivative of the function   f(x) =    tan
-1
 x    if  |x| = 1 
           (|x| - 1)   if  |x| > 1  is 
                2 
  (a) R – {0}  (b) R – {1}   (c) R – {0, -1, 1} (d) R – {-1, 1} 
 
Q.10 Given  f ' (2) = 6 & f ' (1) = 4  ,  lim   f(2h + 2 + h
2
) – f(2)      is equal to  
         
h?0
    f(h – h
2
 + 1) – f(1) 
  (a) 3 / 2  (b) 5 / 2   (c) 3   (d) – 3 
 
Q.11 Range of the expression  f(x) =  3 – 2 Cos x    ? x ? R 
           Cos x + 2 
  (a) [1/3, 1/5]  (b) [-5, 3]   (c) (0, 2/3]  (d) [1/3, 5] 
 
Q.12 Let  f(x) =      Cos x   x 1 
           2 Sin x    x
2
 2x then  lim   f ' (x)   =  
            Tan x       x 1          
x?0        
 x 
  (a) 4   (b) 3    (c) 0    (d) -2 
 
Q.13 If  lim   [(a - n)nx – tan x] Sin nx    = 0 , where n is a non-zero real no. then a is equal to 
       x?0
                    x
2
 
 (a)     n      (b)   n + 1   (c) n + 1/n  (d) n 
      n + 1             n 
Q.14 If  f(x) is differentiable and strictly increasing function, then the value of  lim   f(x
2
) – f(x)     is 
                     
x?0
    f(x) – f(0) 
  (a) -1   (b) 1    (c) 0   (d) 2 
 
Q.15 If  y
2
 = P(x) is a polynomial of degree 3 then  2   d  (y
2
  d
2
 y )  equals 
                       dx         dx
2
 
  (a) p''' (x) + p' (x) (b) p'' (x) p''' (x)  (c) p(x) p''' (x)  (d) a constant 
 
Q.16 If   ? (x) =     x
2
 – 5x + 3 2x – 5        3 
            3x
2
 + x + 4 6x + 1        9  
           7x
2
 – 6x + 9 14x – 6      21  =   ax
3
 + bx
2
 + cx + d   then 
  (a) a ? 0, d = 141  (b) a = b = c = 0  (c) c = 7  (d) d = 29 
 
 
PART B :-  4½ Marks Questions 
 
Q.17 The integer n for which   lim  (Cos x - 1) (Cos x - e
x
)   is equal to ? (a finite non zero no.) then is 
equal to      
x?0
                   x
n
 
  (a) n = 4  (b) n = 3   (c) ? = -1/2  (d) ? =  1/2 
 
Q.18 If  |C| = 1 / 2 and f(x) is diff. at x = 0 given by  f(x) =    b Sin
-1
 (c + x)  , -1/2 < x < 0 
          2 
         1 / 2       ,         x = 0 
               e
ax / 2
 – 1       ,  0 < x < 1 / 2 ,    then 
         x 
  (a) a = 1
  
(b) 16 b
2
 = 4 – c
2  
(c) a = -1
  
(d) 64 b
2
 = 4 – c
2
 
 
                                         SAPRA CLASSES  (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS)  
             SCO- 43 , SEC – 20 C , CHANDIGARH. 9041960872,       SCF. 18 , SEC 15 , PANCHKULA,  98720-27106 
Classes in Mathematics for IIT-JEE/AIEEE                       +2                                   Sapra Classes 
                                                                                                                                                          M a t h e m a t i c s    
Q.19 If  x ? [-p, p] then solution set of the inequality  Sin x + Sin 2x > 0   is 
  (a) (-p, -2p / 3)   (b) (p, 3p / 4)   (c) (0, 2p / 3)               (d) (-3p / 4, -p / 2) 
Q.20 Solution set of the inequality  |8x
2
 + 25x + 12| = | |x
3
 + 6x
2
 + 5x - 12| - |x
3
 – 2x
2
 – 20x - 24| |   is 
  (a) [-4, -3]  (b) [-3, -6]  (c) [1, 6]       (d) holds free for exactly 9 integers 
 
Q.21 The no. of possible values of  t  for which the system of equations  (a - t)x + by + cz = 0,  
bx + (c - t)y + az = 0, cx + ay + (b - t)z = 0, has a non-trivial solution are ________.(say ?) and the 
product of these values is ____________ (say µ), then 
 (a) ? = 2  (b) ? = 3  (c) µ =  a    b   c  (d) µ =   a   b   c 
         b   c   a     c   a   b 
         c   a    b      b  c   a 
Q.22 Consider the f(x) = Sin
-1
 (Cot 
-1
 x)  then  
  (a) domain of f(x) is [Cot 1, 8)    (b) domain of f(x) is [0, Cot 1] 
(c) range of f(x) is (0, p / 2]     (d) range of f(x) is (p / 4, p / 2] 
        ___________ 
Q.23 Consider the definition  f(x) = v3 Sec
-1
 x – p                                                                       __ 
  (a) domain of f(x) is {-1} ? [1, 2]           (b) range of f(x) is [0, v2p] – {p / 2} 
                __ 
(c) domain of f(x) is (- 8, -1] ? [2, 8)          (d) range of f(x) is (p / 2, v2p] 
 
Q.24 The system of equations ax + by + cz = q - r ; bx + cy + az = r – p; cx + ay + bz = p – q   is 
  (a) inconsistent if a = b = c & p , q, r are distinct  (b) consistent if p = q = r 
(c) consistent if a, b, c are distinct & a + b + c ? 0  (d) inconsistent if p = q = r 
 
PART  C :-  Passage I 
 
  Consider the function f(x) =   2x + 1 , x < 0   g(x) =     3 – 4x  , x < 1 
      3x – 4 , x = 0    ,       5 + x   ,  x = 1 
 
Q.25 for  x ? [1, 8)  fog (x) =  
  (a) 8x – 7   (b) 11 + 3x   (c) 7 – 8x  (d) 12 – 5x 
 
Q.26 x ?(- 8, 3/4]   fog(x)  is defined as 
  (a) 3x – 11   (b) 7 – 8x   (c) 11 + 3x  (d) 5 – 12x 
 
Q.27 Range of  fog(x) =  
  (a) [-4, 8)   (b) [3, 8)   (c) R   (d) [-4, 3] 
 
Q.28 for  x ? [0, 5/3) , gof (x) = 
  (a) 2x + 6   (b) 3x + 1   (c) 19 – 12x  (d) -1 – 8x 
 
Q.29 Range of  gof(x) is 
  (a) [4, 8)   (b) [-1, 8)   (c) [-1, 19]  (d)  [19, 8) 
 
Passage II :- 
 
 We define 6 bijection as follows :- 
1. f : [p / 2, 3p / 2] ? [-1, 1]  defines as f(x) = Sin x   
2.    g : [-p , 0] ? [-1, 1] , g(x) = Cos x 
3. h : [0, p] – {p / 2} ?R ,  h(x) = tan x     
                                         SAPRA CLASSES  (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS)  
             SCO- 43 , SEC – 20 C , CHANDIGARH. 9041960872,       SCF. 18 , SEC 15 , PANCHKULA,  98720-27106 
Page 4


Classes in Mathematics for IIT-JEE/AIEEE                       +2                                   Sapra Classes 
                                                                                                                                                          M a t h e m a t i c s    
  
Determinants, Functions, Limits, Continuity , Differentiability                       Class Test  - 7 B 
 
Time  :  2hr 30 minutes                M.M.   132 
   
Read the carefully the instructions given below :  
1. The test consists of 37 questions. 
The test contains four sections.  
Section A : Contains 16 questions & only one of the four choices is the correct answer. Each question  carries 3 marks and -1 
will be awarded for every wrong answer. 
Section B :   Contains 8 problems & these are multiple correct answer(s) type problems. Each question has four choices out of 
which one or more is / are correct. Each question carries 4.5 marks, however there is no negative marking.  
Section C :   Contains 10 questions in two passages. Each question carries 3 marks and -1 will be awarded for every wrong 
answer. 
  Section D :   Contains 3 matching problems. Each problems carries 6 marks and there is no negative marking 
 
  Student Name : __________________________________  Class : _____________________________ 
  Marks Obtained : ________________________________   Date of Test  : _________________________ 
 
Part A 
Q.1 The no. of non-positive integers satisfying the inequality ||x - 1| - x| = 4 are 
  (a) five   (b) two    (c) three  (d) infinite 
 
Q.2 All solutions of the equation  |x
2
 – x - 6| = x + 2  are 
  (a) natural nos. (b) negative integer  (c) rational nos. (d) irrational nos. 
 
Q.3 If in a triangle two angles are  Tan
-1
 2 & Tan
-1
 3  then the measure of the 3
rd
 angle is 
  (a) p / 4  (b) p / 2   (c) p / 3  (d) 3p / 4 
                        _____   _____ 
Q.4 The identity  Cos
-1
 x  –  Cos
-1
 y  =  Cos
-1
 (xy + v1 – x
2
  v1 – y
2
)  holds  if 
  (a) x
2
 + y
2
 = 1  (b) xy = 1   (c) x = y  (d) x + y = 0 
 
Q.5 If A, B, P & Q are sq. matrices of same order such that   adj B = A & |P| = |Q| = 1  then   
adj (Q
-1
 B P
-1
)   is given as 
 (a) QAP  (b) PAQ   (c) P
-1
 B Q
-1
  (d) PBQ 
 
Q.6 If  f(x) = [x] [Sin px], x ? (-1, 1) the f(x) is 
  (a) differentiable at x = 0 (b) continuous in (-1, 0)    (c) differentiable in (-1, 1)    (d) none 
 
Q.7 Let  f(x) = [tan
2
 x], where [.] denotes greatest integer function then 
  (a) lim f(x) does not exist     (b) f(x) is continuous at x = 0 
        x?0 
(c) f(x) is not differentiable at x = 0    (d)  f(0) = 1 
Q.8 The function f(x) = max {1 – x, 1 + x, 2}, x ? R   is 
  (a) continuous at all points except at one point  (b) differentiable at all x 
(c) diff. at all points except two points in domain            (d) discontinuous at x = 1& x = -1 
                                         SAPRA CLASSES  (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS)  
             SCO- 43 , SEC – 20 C , CHANDIGARH. 9041960872,       SCF. 18 , SEC 15 , PANCHKULA,  98720-27106 
Classes in Mathematics for IIT-JEE/AIEEE                       +2                                   Sapra Classes 
                                                                                                                                                          M a t h e m a t i c s    
 
Q.9 The domain of the derivative of the function   f(x) =    tan
-1
 x    if  |x| = 1 
           (|x| - 1)   if  |x| > 1  is 
                2 
  (a) R – {0}  (b) R – {1}   (c) R – {0, -1, 1} (d) R – {-1, 1} 
 
Q.10 Given  f ' (2) = 6 & f ' (1) = 4  ,  lim   f(2h + 2 + h
2
) – f(2)      is equal to  
         
h?0
    f(h – h
2
 + 1) – f(1) 
  (a) 3 / 2  (b) 5 / 2   (c) 3   (d) – 3 
 
Q.11 Range of the expression  f(x) =  3 – 2 Cos x    ? x ? R 
           Cos x + 2 
  (a) [1/3, 1/5]  (b) [-5, 3]   (c) (0, 2/3]  (d) [1/3, 5] 
 
Q.12 Let  f(x) =      Cos x   x 1 
           2 Sin x    x
2
 2x then  lim   f ' (x)   =  
            Tan x       x 1          
x?0        
 x 
  (a) 4   (b) 3    (c) 0    (d) -2 
 
Q.13 If  lim   [(a - n)nx – tan x] Sin nx    = 0 , where n is a non-zero real no. then a is equal to 
       x?0
                    x
2
 
 (a)     n      (b)   n + 1   (c) n + 1/n  (d) n 
      n + 1             n 
Q.14 If  f(x) is differentiable and strictly increasing function, then the value of  lim   f(x
2
) – f(x)     is 
                     
x?0
    f(x) – f(0) 
  (a) -1   (b) 1    (c) 0   (d) 2 
 
Q.15 If  y
2
 = P(x) is a polynomial of degree 3 then  2   d  (y
2
  d
2
 y )  equals 
                       dx         dx
2
 
  (a) p''' (x) + p' (x) (b) p'' (x) p''' (x)  (c) p(x) p''' (x)  (d) a constant 
 
Q.16 If   ? (x) =     x
2
 – 5x + 3 2x – 5        3 
            3x
2
 + x + 4 6x + 1        9  
           7x
2
 – 6x + 9 14x – 6      21  =   ax
3
 + bx
2
 + cx + d   then 
  (a) a ? 0, d = 141  (b) a = b = c = 0  (c) c = 7  (d) d = 29 
 
 
PART B :-  4½ Marks Questions 
 
Q.17 The integer n for which   lim  (Cos x - 1) (Cos x - e
x
)   is equal to ? (a finite non zero no.) then is 
equal to      
x?0
                   x
n
 
  (a) n = 4  (b) n = 3   (c) ? = -1/2  (d) ? =  1/2 
 
Q.18 If  |C| = 1 / 2 and f(x) is diff. at x = 0 given by  f(x) =    b Sin
-1
 (c + x)  , -1/2 < x < 0 
          2 
         1 / 2       ,         x = 0 
               e
ax / 2
 – 1       ,  0 < x < 1 / 2 ,    then 
         x 
  (a) a = 1
  
(b) 16 b
2
 = 4 – c
2  
(c) a = -1
  
(d) 64 b
2
 = 4 – c
2
 
 
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Q.19 If  x ? [-p, p] then solution set of the inequality  Sin x + Sin 2x > 0   is 
  (a) (-p, -2p / 3)   (b) (p, 3p / 4)   (c) (0, 2p / 3)               (d) (-3p / 4, -p / 2) 
Q.20 Solution set of the inequality  |8x
2
 + 25x + 12| = | |x
3
 + 6x
2
 + 5x - 12| - |x
3
 – 2x
2
 – 20x - 24| |   is 
  (a) [-4, -3]  (b) [-3, -6]  (c) [1, 6]       (d) holds free for exactly 9 integers 
 
Q.21 The no. of possible values of  t  for which the system of equations  (a - t)x + by + cz = 0,  
bx + (c - t)y + az = 0, cx + ay + (b - t)z = 0, has a non-trivial solution are ________.(say ?) and the 
product of these values is ____________ (say µ), then 
 (a) ? = 2  (b) ? = 3  (c) µ =  a    b   c  (d) µ =   a   b   c 
         b   c   a     c   a   b 
         c   a    b      b  c   a 
Q.22 Consider the f(x) = Sin
-1
 (Cot 
-1
 x)  then  
  (a) domain of f(x) is [Cot 1, 8)    (b) domain of f(x) is [0, Cot 1] 
(c) range of f(x) is (0, p / 2]     (d) range of f(x) is (p / 4, p / 2] 
        ___________ 
Q.23 Consider the definition  f(x) = v3 Sec
-1
 x – p                                                                       __ 
  (a) domain of f(x) is {-1} ? [1, 2]           (b) range of f(x) is [0, v2p] – {p / 2} 
                __ 
(c) domain of f(x) is (- 8, -1] ? [2, 8)          (d) range of f(x) is (p / 2, v2p] 
 
Q.24 The system of equations ax + by + cz = q - r ; bx + cy + az = r – p; cx + ay + bz = p – q   is 
  (a) inconsistent if a = b = c & p , q, r are distinct  (b) consistent if p = q = r 
(c) consistent if a, b, c are distinct & a + b + c ? 0  (d) inconsistent if p = q = r 
 
PART  C :-  Passage I 
 
  Consider the function f(x) =   2x + 1 , x < 0   g(x) =     3 – 4x  , x < 1 
      3x – 4 , x = 0    ,       5 + x   ,  x = 1 
 
Q.25 for  x ? [1, 8)  fog (x) =  
  (a) 8x – 7   (b) 11 + 3x   (c) 7 – 8x  (d) 12 – 5x 
 
Q.26 x ?(- 8, 3/4]   fog(x)  is defined as 
  (a) 3x – 11   (b) 7 – 8x   (c) 11 + 3x  (d) 5 – 12x 
 
Q.27 Range of  fog(x) =  
  (a) [-4, 8)   (b) [3, 8)   (c) R   (d) [-4, 3] 
 
Q.28 for  x ? [0, 5/3) , gof (x) = 
  (a) 2x + 6   (b) 3x + 1   (c) 19 – 12x  (d) -1 – 8x 
 
Q.29 Range of  gof(x) is 
  (a) [4, 8)   (b) [-1, 8)   (c) [-1, 19]  (d)  [19, 8) 
 
Passage II :- 
 
 We define 6 bijection as follows :- 
1. f : [p / 2, 3p / 2] ? [-1, 1]  defines as f(x) = Sin x   
2.    g : [-p , 0] ? [-1, 1] , g(x) = Cos x 
3. h : [0, p] – {p / 2} ?R ,  h(x) = tan x     
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4.    u : [-p / 2, p / 2] - {0}? R , u(x) = Cot x 
5. v :  [0, p] – {p / 2} ? R – (-1, 1) , v(x) = Sec x 
6. w : [-p / 2 , p / 2] – {0} ? R – (-1, 1) ,  w(x) = Cosec x 
  
answer the following questions :- 
 
Q.30 Sin
-1
 x + Cos
-1
 x =   x ? (-1, 1] 
  (a) p   (b) –p / 2   (c) p / 2   (d) –p 
 
Q.31 Domain of  f 
-1
 [h
-1
 (x)] is, where [x] starts for greatest integer = x   
  (a) (- 8, tan 2) ? [0, 8) (b) (- 8, tan 1)   (c) (tan 2, 0)   (d) (tan 2, tan 1] 
 
Q.32 Sin
-1
 (Sin 16) + Tan
-1
 (Cot 4) 
  (a) 12 + p   (b) 16 - 4p  (c) 16 + 3p/2       (d) 12 - 5p/2 
 
Q.33 for  x ? [-1, 1] – {0}, Sin
-1
 x = ? + Sec
-1
 1 / x    then  ? = 
  (a) p    (b) –p    (c) p / 2  (d) –p / 2 
 
Q.34 Consider P(x) = h
-1
 (x) + f 
-1
 (x) + v 
-1
 (x)  then the difference between greatest & smallest value of  
p(x)  is 
  (a) 3p / 2   (b) p / 2   (c) 5p / 2  (d) p 
 
PART D :-  Match the columns  
 
Match the definitions of the function in the column I with set of points of continuity in the column II. 
 
Q.35  Column I     ______          Column II 
 (a) f(x) = [x] + vx – [x]               (p)  { vx ; x ? I} 
 (b) f(x) = [x] + [-x]               (q)   (0, 8)  
 (c) f(x) = Cos log x               (r)     R – Z 
 (d) f(x) = x
2
 + [x
2
]                 (s)     R 
 
Q.36  Column I           Column II 
 (a) The function f(x) =     x
2
 + 3x + 9 ;  x = 1     (p)   a = 3 
          bx + 2   ;  x > 1     is diff. ? x ? R,  then 
 (b) The function f(x) =   1 / |x|   ;  |x| = 1      (q)    b = 5 
             ax
2
 + b ;  |x| < 1,          is diff. everywhere then  (r)    a = 35 / 9 
 
 (c) The function f(x) =    ax
2
 – bx + 2  ,  x < 3     (s)    b = 3 / 2 
        bx
2
 – 3     ,   x = 3    is diff. everywhere then (t)    a = -1 / 2 
 
Q.37  Column I              Column II 
 (a) The total no. of elements in the range of      (p)         1 
f(x) = [1 + Sin x] + [Cos x - 1] + [Tan
-1
 x] x ?[0, 2p] where [.] denotes 
step function is / are        (q)   2 
 
 (b) only element which belongs to the range of  [(Sin x) + |Cos x|]  (r)    5 
 
 (c ) the total no. of solution of  8 Cos x = x   is / are    (s)    4 
 
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Page 5


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Determinants, Functions, Limits, Continuity , Differentiability                       Class Test  - 7 B 
 
Time  :  2hr 30 minutes                M.M.   132 
   
Read the carefully the instructions given below :  
1. The test consists of 37 questions. 
The test contains four sections.  
Section A : Contains 16 questions & only one of the four choices is the correct answer. Each question  carries 3 marks and -1 
will be awarded for every wrong answer. 
Section B :   Contains 8 problems & these are multiple correct answer(s) type problems. Each question has four choices out of 
which one or more is / are correct. Each question carries 4.5 marks, however there is no negative marking.  
Section C :   Contains 10 questions in two passages. Each question carries 3 marks and -1 will be awarded for every wrong 
answer. 
  Section D :   Contains 3 matching problems. Each problems carries 6 marks and there is no negative marking 
 
  Student Name : __________________________________  Class : _____________________________ 
  Marks Obtained : ________________________________   Date of Test  : _________________________ 
 
Part A 
Q.1 The no. of non-positive integers satisfying the inequality ||x - 1| - x| = 4 are 
  (a) five   (b) two    (c) three  (d) infinite 
 
Q.2 All solutions of the equation  |x
2
 – x - 6| = x + 2  are 
  (a) natural nos. (b) negative integer  (c) rational nos. (d) irrational nos. 
 
Q.3 If in a triangle two angles are  Tan
-1
 2 & Tan
-1
 3  then the measure of the 3
rd
 angle is 
  (a) p / 4  (b) p / 2   (c) p / 3  (d) 3p / 4 
                        _____   _____ 
Q.4 The identity  Cos
-1
 x  –  Cos
-1
 y  =  Cos
-1
 (xy + v1 – x
2
  v1 – y
2
)  holds  if 
  (a) x
2
 + y
2
 = 1  (b) xy = 1   (c) x = y  (d) x + y = 0 
 
Q.5 If A, B, P & Q are sq. matrices of same order such that   adj B = A & |P| = |Q| = 1  then   
adj (Q
-1
 B P
-1
)   is given as 
 (a) QAP  (b) PAQ   (c) P
-1
 B Q
-1
  (d) PBQ 
 
Q.6 If  f(x) = [x] [Sin px], x ? (-1, 1) the f(x) is 
  (a) differentiable at x = 0 (b) continuous in (-1, 0)    (c) differentiable in (-1, 1)    (d) none 
 
Q.7 Let  f(x) = [tan
2
 x], where [.] denotes greatest integer function then 
  (a) lim f(x) does not exist     (b) f(x) is continuous at x = 0 
        x?0 
(c) f(x) is not differentiable at x = 0    (d)  f(0) = 1 
Q.8 The function f(x) = max {1 – x, 1 + x, 2}, x ? R   is 
  (a) continuous at all points except at one point  (b) differentiable at all x 
(c) diff. at all points except two points in domain            (d) discontinuous at x = 1& x = -1 
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Q.9 The domain of the derivative of the function   f(x) =    tan
-1
 x    if  |x| = 1 
           (|x| - 1)   if  |x| > 1  is 
                2 
  (a) R – {0}  (b) R – {1}   (c) R – {0, -1, 1} (d) R – {-1, 1} 
 
Q.10 Given  f ' (2) = 6 & f ' (1) = 4  ,  lim   f(2h + 2 + h
2
) – f(2)      is equal to  
         
h?0
    f(h – h
2
 + 1) – f(1) 
  (a) 3 / 2  (b) 5 / 2   (c) 3   (d) – 3 
 
Q.11 Range of the expression  f(x) =  3 – 2 Cos x    ? x ? R 
           Cos x + 2 
  (a) [1/3, 1/5]  (b) [-5, 3]   (c) (0, 2/3]  (d) [1/3, 5] 
 
Q.12 Let  f(x) =      Cos x   x 1 
           2 Sin x    x
2
 2x then  lim   f ' (x)   =  
            Tan x       x 1          
x?0        
 x 
  (a) 4   (b) 3    (c) 0    (d) -2 
 
Q.13 If  lim   [(a - n)nx – tan x] Sin nx    = 0 , where n is a non-zero real no. then a is equal to 
       x?0
                    x
2
 
 (a)     n      (b)   n + 1   (c) n + 1/n  (d) n 
      n + 1             n 
Q.14 If  f(x) is differentiable and strictly increasing function, then the value of  lim   f(x
2
) – f(x)     is 
                     
x?0
    f(x) – f(0) 
  (a) -1   (b) 1    (c) 0   (d) 2 
 
Q.15 If  y
2
 = P(x) is a polynomial of degree 3 then  2   d  (y
2
  d
2
 y )  equals 
                       dx         dx
2
 
  (a) p''' (x) + p' (x) (b) p'' (x) p''' (x)  (c) p(x) p''' (x)  (d) a constant 
 
Q.16 If   ? (x) =     x
2
 – 5x + 3 2x – 5        3 
            3x
2
 + x + 4 6x + 1        9  
           7x
2
 – 6x + 9 14x – 6      21  =   ax
3
 + bx
2
 + cx + d   then 
  (a) a ? 0, d = 141  (b) a = b = c = 0  (c) c = 7  (d) d = 29 
 
 
PART B :-  4½ Marks Questions 
 
Q.17 The integer n for which   lim  (Cos x - 1) (Cos x - e
x
)   is equal to ? (a finite non zero no.) then is 
equal to      
x?0
                   x
n
 
  (a) n = 4  (b) n = 3   (c) ? = -1/2  (d) ? =  1/2 
 
Q.18 If  |C| = 1 / 2 and f(x) is diff. at x = 0 given by  f(x) =    b Sin
-1
 (c + x)  , -1/2 < x < 0 
          2 
         1 / 2       ,         x = 0 
               e
ax / 2
 – 1       ,  0 < x < 1 / 2 ,    then 
         x 
  (a) a = 1
  
(b) 16 b
2
 = 4 – c
2  
(c) a = -1
  
(d) 64 b
2
 = 4 – c
2
 
 
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Q.19 If  x ? [-p, p] then solution set of the inequality  Sin x + Sin 2x > 0   is 
  (a) (-p, -2p / 3)   (b) (p, 3p / 4)   (c) (0, 2p / 3)               (d) (-3p / 4, -p / 2) 
Q.20 Solution set of the inequality  |8x
2
 + 25x + 12| = | |x
3
 + 6x
2
 + 5x - 12| - |x
3
 – 2x
2
 – 20x - 24| |   is 
  (a) [-4, -3]  (b) [-3, -6]  (c) [1, 6]       (d) holds free for exactly 9 integers 
 
Q.21 The no. of possible values of  t  for which the system of equations  (a - t)x + by + cz = 0,  
bx + (c - t)y + az = 0, cx + ay + (b - t)z = 0, has a non-trivial solution are ________.(say ?) and the 
product of these values is ____________ (say µ), then 
 (a) ? = 2  (b) ? = 3  (c) µ =  a    b   c  (d) µ =   a   b   c 
         b   c   a     c   a   b 
         c   a    b      b  c   a 
Q.22 Consider the f(x) = Sin
-1
 (Cot 
-1
 x)  then  
  (a) domain of f(x) is [Cot 1, 8)    (b) domain of f(x) is [0, Cot 1] 
(c) range of f(x) is (0, p / 2]     (d) range of f(x) is (p / 4, p / 2] 
        ___________ 
Q.23 Consider the definition  f(x) = v3 Sec
-1
 x – p                                                                       __ 
  (a) domain of f(x) is {-1} ? [1, 2]           (b) range of f(x) is [0, v2p] – {p / 2} 
                __ 
(c) domain of f(x) is (- 8, -1] ? [2, 8)          (d) range of f(x) is (p / 2, v2p] 
 
Q.24 The system of equations ax + by + cz = q - r ; bx + cy + az = r – p; cx + ay + bz = p – q   is 
  (a) inconsistent if a = b = c & p , q, r are distinct  (b) consistent if p = q = r 
(c) consistent if a, b, c are distinct & a + b + c ? 0  (d) inconsistent if p = q = r 
 
PART  C :-  Passage I 
 
  Consider the function f(x) =   2x + 1 , x < 0   g(x) =     3 – 4x  , x < 1 
      3x – 4 , x = 0    ,       5 + x   ,  x = 1 
 
Q.25 for  x ? [1, 8)  fog (x) =  
  (a) 8x – 7   (b) 11 + 3x   (c) 7 – 8x  (d) 12 – 5x 
 
Q.26 x ?(- 8, 3/4]   fog(x)  is defined as 
  (a) 3x – 11   (b) 7 – 8x   (c) 11 + 3x  (d) 5 – 12x 
 
Q.27 Range of  fog(x) =  
  (a) [-4, 8)   (b) [3, 8)   (c) R   (d) [-4, 3] 
 
Q.28 for  x ? [0, 5/3) , gof (x) = 
  (a) 2x + 6   (b) 3x + 1   (c) 19 – 12x  (d) -1 – 8x 
 
Q.29 Range of  gof(x) is 
  (a) [4, 8)   (b) [-1, 8)   (c) [-1, 19]  (d)  [19, 8) 
 
Passage II :- 
 
 We define 6 bijection as follows :- 
1. f : [p / 2, 3p / 2] ? [-1, 1]  defines as f(x) = Sin x   
2.    g : [-p , 0] ? [-1, 1] , g(x) = Cos x 
3. h : [0, p] – {p / 2} ?R ,  h(x) = tan x     
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4.    u : [-p / 2, p / 2] - {0}? R , u(x) = Cot x 
5. v :  [0, p] – {p / 2} ? R – (-1, 1) , v(x) = Sec x 
6. w : [-p / 2 , p / 2] – {0} ? R – (-1, 1) ,  w(x) = Cosec x 
  
answer the following questions :- 
 
Q.30 Sin
-1
 x + Cos
-1
 x =   x ? (-1, 1] 
  (a) p   (b) –p / 2   (c) p / 2   (d) –p 
 
Q.31 Domain of  f 
-1
 [h
-1
 (x)] is, where [x] starts for greatest integer = x   
  (a) (- 8, tan 2) ? [0, 8) (b) (- 8, tan 1)   (c) (tan 2, 0)   (d) (tan 2, tan 1] 
 
Q.32 Sin
-1
 (Sin 16) + Tan
-1
 (Cot 4) 
  (a) 12 + p   (b) 16 - 4p  (c) 16 + 3p/2       (d) 12 - 5p/2 
 
Q.33 for  x ? [-1, 1] – {0}, Sin
-1
 x = ? + Sec
-1
 1 / x    then  ? = 
  (a) p    (b) –p    (c) p / 2  (d) –p / 2 
 
Q.34 Consider P(x) = h
-1
 (x) + f 
-1
 (x) + v 
-1
 (x)  then the difference between greatest & smallest value of  
p(x)  is 
  (a) 3p / 2   (b) p / 2   (c) 5p / 2  (d) p 
 
PART D :-  Match the columns  
 
Match the definitions of the function in the column I with set of points of continuity in the column II. 
 
Q.35  Column I     ______          Column II 
 (a) f(x) = [x] + vx – [x]               (p)  { vx ; x ? I} 
 (b) f(x) = [x] + [-x]               (q)   (0, 8)  
 (c) f(x) = Cos log x               (r)     R – Z 
 (d) f(x) = x
2
 + [x
2
]                 (s)     R 
 
Q.36  Column I           Column II 
 (a) The function f(x) =     x
2
 + 3x + 9 ;  x = 1     (p)   a = 3 
          bx + 2   ;  x > 1     is diff. ? x ? R,  then 
 (b) The function f(x) =   1 / |x|   ;  |x| = 1      (q)    b = 5 
             ax
2
 + b ;  |x| < 1,          is diff. everywhere then  (r)    a = 35 / 9 
 
 (c) The function f(x) =    ax
2
 – bx + 2  ,  x < 3     (s)    b = 3 / 2 
        bx
2
 – 3     ,   x = 3    is diff. everywhere then (t)    a = -1 / 2 
 
Q.37  Column I              Column II 
 (a) The total no. of elements in the range of      (p)         1 
f(x) = [1 + Sin x] + [Cos x - 1] + [Tan
-1
 x] x ?[0, 2p] where [.] denotes 
step function is / are        (q)   2 
 
 (b) only element which belongs to the range of  [(Sin x) + |Cos x|]  (r)    5 
 
 (c ) the total no. of solution of  8 Cos x = x   is / are    (s)    4 
 
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OBJECTIVE   RESPONSE   SHEET  
 
Name of the student :- _________________________ Class :- _________________  
 
Date :- _____________     Topic :-________________________________ 
 
Marks Obtained :- _________________________ 
 
Darken the bubble corresponding to correct choice from Q.1 – Q.34 
  
  A    B    C     D   A   B    C     D 
1 O    O    O    O  18 O   O    O    O 
2 O    O    O    O  19 O   O    O     O 
3 O    O    O    O  20 O   O    O     O 
4  O    O    O    O  21 O   O    O     O 
5 O    O    O    O  22 O   O    O     O 
 
6 O    O    O     O  23 O   O     O    O 
7 O    O    O     O  24 O   O     O    O 
8 O    O    O     O  25 O   O     O    O 
9 O    O    O     O  26 O   O     O    O 
 10 O    O    O     O  27 O   O     O    O 
 
11 O    O    O     O  28 O   O     O    O 
12 O    O    O     O  29 O   O     O    O 
13 O    O    O     O  30 O   O     O    O 
14 O    O    O     O  31 O   O     O    O 
15 O    O    O     O  32 O   O     O    O 
         
16 O    O    O     O  33 O   O     O     O 
17 O    O    O     O  34 O   O     O     O 
 
 
Match the following :- 
 
Q.35 p    q     r     s   Q.36 p     q    r    s    t   Q.37 p     q    r    s     
 
A O    O    O    O   A O    O   O    O   O      A O    O   O    O       
B O    O    O    O   B O    O   O    O   O   B O    O   O    O      
C O    O    O    O   C O    O   O   O   O   C O    O   O    O      
D O    O    O    O   D O    O   O    O   O   D O    O   O    O      
_____________________________________________________________________________________ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                         SAPRA CLASSES  (PREMIER INSTITUTE FOR IIT- JEE, MATHEMATICS)  
             SCO- 43 , SEC – 20 C , CHANDIGARH. 9041960872,       SCF. 18 , SEC 15 , PANCHKULA,  98720-27106 
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