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

Created by: Nimish Aren

## : 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
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
-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
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
-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
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
-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
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
-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)
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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

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
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
-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

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
_____________________________________________________________________________________

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