The document provides information about a test on monotonic functions. It is divided into 4 levels with varying difficulty of questions. Level 1 contains 48 questions on basic monotonic properties of functions like polynomials, rational functions, and absolute value functions. Level 2 has 16 questions involving more complex functions like exponentials, logarithms, and trigonometric functions. Level 3 has 12 questions of greater difficulty, and Level 4 contains the most challenging questions. In total there are 92 questions on the test covering monotonicity of different function types.

1. MONOTONICITY
Total No.of questions in Function are -
Level # 1 ........................................ 48
Level # 2 ........................................ 16
Level # 3 ........................................ 12
Level # 4 ........................................ 16
Total No. of questions......................................................... 92

2. LEVEL # 1
Monotonicity in algebric function
Q.1 When x < 0, function f(x) = x2 is
(A) Decreasing
(B) Increasing
(C) Constant
(D) Not monotonic
Q.2 When x > 1, function f(x) = x3 is
(A) Increasing (B) Decreasing
(C) Constant (D) not monotonic
Q.3 In the interval (0, 1), f(x) = x2 – x + 1 is
(A) Monotonic (B) Not monotonic
(C) Decreasing (D) Increasing
Q.4 f(x) = x + 1/x, x  0 is increasing when
(A) | x | < 1 (B) | x | > 1
(C) | x | < 2 (D) | x | > 2
Q.5 The function f(x) =
| |
x
x
(x  0), x > 0 is
(A) Decreasing
(B) Increasing
(C) Constant function
(D) None of these
Q.6 When x  (0, 1), function f(x) = 1/ x is
(A) Increasing
(B) Decreasing
(C) Neither increasing nor decreasing
(D) Constant
Q.7 Function f(x) = 3x4 + 7x2 + 3 is
(A) Monotonically increasing
(B) Monotonically decreasing
(C) Not monotonic
(D) Odd function
Q.8 For what values of x, the function
f(x) = x +
4
2
x
is monotonically decreasing
(A) x < 0 (B) x > 2
(C) x < 2 (D) 0 < x < 2
Q.9 If f(x) =
x
x
2
2
 for –7  x  7, then f(x) is
increasing function of x in the interval
(A) [7, 0] (B) (2, 7]
(C) [–2, 2] (D) [0, 7]
Q.10 The function y =
x
x
1 2

decreases in the
interval
(A) (–, –) (B) (–1, –1)
(C) (0, ) (D) (–, –1)
Q.11 For which value of x, the function
f(x) = x2 –2x is decreasing
(A) x > 1 (B) x > 2
(C) x < 1 (D) x < 2
Q.12 Function f(x) =
x
x


2
1
, x  –1 is
(A) Increasing
(B) Decreasing
(C) Not monotonic
(D) None of these
Q.13 Function f(x) = x3 is
(A) Increasing in (0, ) and decreasing in (–, 0)
(B) Decreasing in (0, ) and increasing in (–, 0)
(C) Decreasing throughout
(D) Increasing throughout
Q.14 Function f(x) = x | x | is
(A) Monotonic increasing
(B) Monotonic decreasing
(C) Not monotonic
(D) None of these
Q.15 If f and g are two decreasing functions such
that fog is defined then fog is
(A) Decreasing (B) Increasing
(C) Can't say (D) None of these
Q.16 For the function f(x) = | x |, x > 0 is
(A) Decreasing
(B) Increasing
(C) Constant function
(D) None of these
Q.17 In the following , monotonic increasing
fucntion is
(A) x + | x | (B) x – | x |
(C) | x | (D) x | x |
Q.18 At x = 0, f(x) =
x
x


1
2
is
(A) Increasing (B)Decreasing
(C) Not monotonic (D) Constant
Q.19 If f(x) = 2x3 – 9x2 + 12x – 6, then in which
interval f(x) is monotonically increasing
(A) (1, 2) (B) (–, 1)
(C) (2, ) (D) (–, 1) or (2, )

3. Q.20 For the function f(x) = x3 – 6x2 – 36x + 7
which of the following statement is false
(A) f(x) is decareasing, if –2 < x < 6
(B) f(x) is increasing, if –3 < x < 5
(C) f(x) is increasing, if x < –2
(D) f(x) is increasing, if x > 6
Q.21 In which interval the function f(x) =
x
x
2
2
 ,
– 6  x  6 ( x  0) is decreasing
(A) (6, 0) (B) ( –2, 2)
(C) (2, 6) (D) None of these
Q.22 Function f(x) = x2(x –2)2 is
(A) Increasing in (0, 1)  (2, )
(B) Decreasing in (0, 1)  (2, )
(C) Decreasing function
(D) Increasing function
Q.23 For 0  x  1, the function
f(x) = |x| + |x – 1| is
(A) Monotonically increasing
(B) Monotonically decreasing
(C) Constant function
(D) Identity function
Q.24 If f and g are two increasing function such
that fog is defined then fog is
(A) Increasing
(B) Decreasing
(C) Neither increasing nor decreasing
(D) None of these
Monotonicity in exponential
function
Q.25 Function f(x) = ax is monotonically increasing
if
(A) a < 0 (B) a > 0
(C) a < 1 (D) a > 1
Q.26 The function f(x) = ex, –1  x < 0 is
(A) Decreasing
(B) Increasing
(C) Constant function
(D) Neither increasing, nor decreasing
Q.27 Function f(x) = e–1/x (x > 0) is
(A) Increasing
(B) Decreasing
(C) Not monotonic
(D) None of these
Q.28 Which of the following function is not
monotonic
(A) ex – e–x (B) ex + e–x
(C) e–1/x (D) None of these
Q.29 In the following, decreasing function is
(A) In x (B)
1
| |
x
(C) e1/x (D) None of these
Q.30 For every value of x of the function
f(x) =
1
5x is
(A) Decreasing
(B) Increasing
(C) Neither increasing nor decreasing
(D) Increasing for x > 0 and decreasing for x < 0
Q.31 The interval in which the function f(x)=xe4–x
decreases is
(A) (–, 1) (B) (1, )
(C) (0, 4) (D) None of these
Q.32 Function
e
e
x
x
2
2
1
1


is
(A) Increasing
(B) Decreasing
(C) Neither increasing nor decreasing
(D) Even function
Q.33 Which of the following functions is a
monotonic increasing function for all values
of x
(A) sin x (B) cos x
(C) ex (D) 2–x
Q.34 Which of the following function is
monotonically decreasing for all real values
of x
(A) e–x (B) x2
(C) tan x (D) | x |
Monotonicity in logrithmic function
Q.35 Function f(x) = 2x2 – log x is increasing when
(A) x  (0, 1/2)
(B) x  (1/2, )
(C) x  (–1/2, 1/2)
(D) x  (–, –1/2)  (1/2, )
Q.36 Function f(x) = x – log x decreasing, when
(A) x  (0, 1) (B) x  (–1, 1)
(C) x  (1, ) (D) None of these

4. Q.37 For x > 0, the function f(x) = log x, x > 0 is-
(A) Decreasing
(B) Increasing
(C) Constant function
(D) Odd function
Q.38 Function f(x) =
logx
x
is increasing in
(A) (1, 2e) (B) (0, e)
(C) (2, 2e) (D) (1/e, 2e)
Q.39 Function f(x) = log sin x is monotonic in
creasing when
(A) x (/2, ) (B) x (–/2, 0)
(C) x (0, ) (D) x (0, /2)
Monotonicity in trignometric
function
Q.40 Function f(x) = kx + 3 sin x is decreasing if-
(A) k < –3 (B) k > –3
(C) k < 3 (D) k > 3
Q.41 When   x < 3/2, tan x is
(A) Increasing
(B) Decreasing
(C) Not monotonic
(D) Constant
Q.42 Function f(x) =
 sin cos
sin cos
x x
x x


3
2 6
is increasing
when
(A)  < 1 (B)  > 1
(C)  < 2 (D)  > 2
Q.43 The function f(x) = x + sin x is monotonically
increasing for
(A) x > 0 (B) x < 0
(C) All values of x (D) No value of x
Q.44 The function f(x) = x + cos x is
(A) Always monotonocally increasing
(B) Always monotonocally decreasing
(C) Increasing for certain range of x
(D) None of these
Q.45 For what value of 'a' the function
f(x) = x + cos x – a increases
(A) 0 (B) 1
(C) –1 (D) Any value
Q.46 Function f(x) = cot–1 x + x is increasing in-
(A) (1, ) (B) (–1, )
(C) (–, ) (D) (0, )
Q.47 For x > 0, which of the following function
is not monotonic
(A) x + | x | (B) ex
(C) log x (D) sin x
Q.48 f(x) = x – 3cos x is monotonic increasing if
(A)  > 3 (B)  > –3
(C)  < 3 (D)  < –3

5. LEVEL # 2
Q.1 If f(x) = x5 – 20x3 + 240 x, then f(x) is
(A) Monotonic increasing everywhere
(B) Monotonic decreasing only in (0, )
(C) Monotonic decreasing everywhere
(D) Monotonic increasing only in (–, 0)
Q.2 The function y =
x
x
log
increases in the interval
(A) (–, 0) (B) (e, )
(C) (0, ) (D) (–, e)
Q.3 For x > 0, which of the following statement
is true
(A) x < log (1 + x) (B) x > log (1 + x)
(C) x  log (1 + x) (D) None of these
Q.4 The function f(x) = 2log (x – 2) – x2 + 4x + 1
increases in the interval
(A) (1, 2) (B) (2, 3)
(C) (–, –1) (D) (2, 4)
Q.5 If a < 0 then function (eax + e–ax) is monotonic
decreasing when
(A) x < 0 (B) x > 0
(C) x > 1 (D) x < 1
Q.6 Function f(x) = x100 + sin x – 1 is increasing
in the interval
(A) (0, 1) (B) (–/2, /2)
(C) (–1, 1) (D) None of these
Q.7 In which interval f(x) = 2x2 – log | x|, (x  0)
is monotonically decreasing
(A) (–1/2, 1/2)
(B) (–, –1/2)
(C) (–, –1/2)  (0, 1/2)
(D) (–, –1/2)  (1/2, )
Q.8 If f(x) = x3 – 10x2 + 200x – 10, then f(x) is-
(A) Decreasing in (–, 10] and increasing in (10, )
(B) Increasing in (–, 10] and decreasing in (10, )
(C) Increasing for every value of x
(D) Decreasing for every value of x
Q.9 If the domain of f(x) = sin x is
D = {x : 0  x  }, then f(x) is
(A) Increasing in D
(B) Decreasing in D
(C) Decreasing in [0,/2] and increasing in [/2,]
(D) Neither increasing nor decreasing
Q.10 Function f(x) = log (1 + x) –
2
1
x
x

is
monotonic increasing when
(A) x < 0 (B) x > 1
(C) x  R (D) x  R0
Q.11 f(x) = 2x – tan–1 x – log (x + 1 2
 x ) is
monotonic increasing when
(A) x > 0 (B) x < 0
(C) x  R (D) x  R0
Q.12 If f'(x) = g(x)(x – )2 where g()  0 and g(x)
is continuous at x =  then function f(x)
(A) increasing near to  if g() > 0
(B) decreasing near to  if g() > 0
(C) increasing near to  if g() < 0
(D) increasing near to  for every value of g()
Q.13 Function cos2 x + cos2 (/3 +x) – cos x cos
(/3 +x) for all real values of x will be
(A) Increasing
(B) Constant
(C) Decreasing
(D) None of these
Q.14 Let f(x) > 0 and g(x) < 0 for all ,
R
x 
then-
(A) f{g(x)} > f{g(x + 1)}
(B) f{g(x–1)} < f{g(x + 1)}
(C) g{f(x–1)} < g{f(x + 1)}
(D) g{f(x)} > g{f(x – 1)}
Q.15 The interval of increases of the function given
by )
7
/
2
tan(
e
x
)
x
(
f x



 is
(A) (0,) (B) (–0)
(C) (1, ) (D) None
Q.16 x
sin
5
bx
ax
x
)
x
(
f 2
2
3



 is an increasing
function in the set of real numbers if a and
b satisfy the condition
(A) 0
15
b
3
a2


 (B) 0
15
b
3
a2



(C) 0
15
b
3
a2


 (D) 0
b
,
0
a 


6. LEVEL # 3
Q.1 The function ƒ(x) = cos(  /x) is increasing in
the interval -
(A) (2n+1, 2n), nN
(B) N
n
,
n
2
,
1
n
2
1








(C) N
n
,
1
n
2
1
,
2
n
2
1









(D) none of these
Q.2 Let y = x2e–x, then the interval in which y
increases with respect to x is -
(A) )
,
( 
 (B) (–2, 0)
(C) (2,  ) (D) (0, 2)
Q.3 The function y = x3 – 3x2 + 6x – 17
(A) Increaseseverywhere
(B) Decreases everywhere
(C) Increases for positive x and decreses for
negative x
(D) Increases for negative x and decreases for
positive x
Q.4 The function ƒ(x) = xx decrease on the interval -
(A) (0, e) (B) (0, 1)
(C) (0, 1/e) (D) none of these
Assertion & Reason Type Question :-
All questions are Assertion & Reason type
questions. Each of these questions contains
two statements : Statement-I (Assertion) and
Statement-II (Reason). Answer these ques
tions from the following four option.
(A) If both Statement- I and Statement- II are
true, and Statement - II is the correct
explanation of Statement– I.
(B) If both Statement - I and Statement - II
are true but Statement - II is not the
correct explanation of Statement – I.
(C) If Statement - I is true but Statement -
II is false.
(D) If Statement - I is false but Statement -
II is true.
Q.5 Statement I : Both sin x and cos x are
decreasing function in (/2, ).
Statement II : If a differentiable function
decreasing in an interval (a, b) then it's
derivative also decreases in (a, b).
Q.6 Statement I : If f(x) and g(x) are
monotonically (or strictly) increasing (or
decreasing) functions on [a, b] then gof(x) is
always monotonically (or strictly) incseasing
function in [a, b].
Statement II : If one of the two functions
f(x) and g(x) is strictly (or monotonically)
increasing and other strictly (monotonically)
decreasing, then (gof) (x) is sometimes strictly
(monotonically) decreasing on [a, b].
Q.7 Statement I : f(x) = )
x
e
log(
)
x
log(



is increasing
on 




 
 e
,
e
.
Statement II : x log x is increasing for
x > 1/e.
Passage :
If f(x) = | x – 2| + |x – 4| + |x – 6|
On the basis of above information, answer the
following questions-
Q.8 Find the set of values of x such that f(x) is in-
creases-
(A) [2, 4] (B) [4,6]
(C) [4, ) (D) None of these
Q.9 Find the set of values of x such that f(x) is de-
creases–
(A) (– ) (B) (– ]
(C) (– ,6] (D) None of these
Q.10 If f(x) is symmetrical about the line x = K, then-
(A) K = 2 (B) K = 4
(C) K = 6 (D) None of these
Q.11 Find the set of values of x such that f(x) is invert-
ible–
(A) [4, ) (B) ( – , 4]
(C) [2,) (D)(A)&(B)arecorrect
Q.12 Find the set of values of a such that equation
f(x) – a = 0 has no solution-
(A) [4,5] (B) [2, 4)
(C) (– , 4) (D) None of these

7. LEVEL # 4
(Questions asked in Previous AIEEE & IIT-JEE)
SECTION - A
Q.1 A function is matched below against an interval
whre it is supposed to be increasing. Which of
the follownig pairs is incorrectly matched ?
interval function
(1) (–, ) x3 – 3x2 + 3x + 3
(2) [2, ) 2x3 – 3x2 – 12x + 6
(3) 







3
1
, 3x2 – 2x + 1
(4) (–, –4] x3 + 6x2 + 6
Q.2 The function f(x) = tan-1 (sinx + cosx) is an
increasing function in-
(1) (/4,  /2)
(2) (–/2,  /4)
(3) (0,  /2)
(4) (–/2,  /2)
SECTION - B
Q.1 If f(x) =
3 12 1 1 2
37 2 3
2
x x x
x x
    
  
R
S
T
,
, then f(x) is
(A) Increasing in [ –1, 2]
(B) Continuous in [–1, 3]
(C) Greatest at x = 2
(D) All above correct
Q.2 The function f defined by f(x) = (x + 2)e–x is -
(A) Decreasing for all x
(B) Decreasing in (–, –1) and increasing
(–1, )
(C) Increasing for all x
(D) Decreasing in (–1, ) and increasing in
(–, –1)
Q.3 Function f(x) =
log( )
log( )
 

x
e x
is decreasing in
the interval -
(A) (–, ) (B) (0, )
(C) (–, 0) (D) No where
Q.4 If f(x) =
x
x
sin
and g(x) =
x
x
tan
, where 0 <
x 1, then in this interval -
(A) Both f(x) and g(x) are increasing func-
tions
(B) Both f(x) and g(x) are decreasing function
(C) f(x) is an increasing function
(D) g(x) is an increasing function
Q.5 Let h(x) = f(x) – (f(x))2
+ (f(x))3
for every real
number x. Then-
(A) h is increasing whenever f is increasing
(B) h is increasing whenever f is decreasing
(C) h is decreasing whenever f is increasing
(D) nothing can be said in general
Q.6 The function f(x) = sin4 x + cos4 x increases
if -
(A) 0 < x <

8
(B)

4
< x <
3
8

(C)
3
8

< x <
5
8

(D)
5
8

< x <
3
4

Q.7 Let f(x) = e x x dx
x
( )( )
 
z 1 2 . Then f de-
creases in the interval -
(A) (–, –2) (B) ( –2, –1)
(C) (1, 2) (D) (2, + )
Q.8 Consider the following statement S and R -
S : Both sin x and cos x are decreasing
function in the interval


2
,
F
H
G I
K
J
R : If a differentiable function decreases in an
interval (a, b), then its derivative also decreases
in (a, b)
Which of the following is true ?
(A) Both S and R are wrong
(B) Both S and R are correct, but R is not
the correct explanation for S
(C) S is correct and R is the correct expla
nation for S
(D) S is correct and R is wrong

8. Q.9 Let f (x) = x ex(1 – x), then f (x) is -
(B) Decreasing on R
(C) Increasing on R
(D) Decreasing on [–1/2, 1]
Q.10 The length of a longest interval in which the
function 3 sin x – 4 sin3 x is increasing, is
-
(A) /3 (B) /2
(C) 3/2 (D) 
Q.11 f(x) = x2
– 2bx + 2c2
& g(x) = – x2
– 2cx +
b2
if the minimum value of f(x) is always
greater than maximum value of g(x) then.
(A) |
b
|
2
|
c
|  (B) b
2
c 
(C) b
2
c 
 (D) |
b
|
2
|
c
| 
Q.12 Let f(x) = 
 1
x
x
2
2 dt
e
2
t
 , x  (– , ) then the
interval for which f(x) is increasing is
(A) ]
0
,
( (B) )
,
0
[ 
(C) [–2, 2) (D) no where
Q.13 Let f(x) = x3
+ bx2
+ cx + d ; 0 < b2
< c then
f(x)-
(A) is strictly increasing
(B) has local maxima
(C) has local minima
(D) is bounded curve
Q.14 Let the function g : (–)  




 


2
,
2
be
given by g(u) = 2 tan–1
(eu
) –
2

. Then g is-
(A) even and is strictly increasing in (0, )
(B) odd and is strictly decreasing in (–)
(C) odd and is strictly increasing in (–)
(D) neither even nor odd, but is strictly in
creasing in (–)

9. LEVEL # 2
LEVEL # 1
ANSWER KEY
LEVEL # 3
LEVEL # 4
SECTION - A
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ans. A A B B C B C D B D C A D A B B D A D B
Q.No. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Ans. B A C A D B A B C A B A C A B A B B D A
Q.No. 41 42 43 44 45 46 47 48
Ans. A B C A D C D A
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Ans. A B B B A A C C D B C A B A B C
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12
Ans. D D A C C C D C B B D C
Q.No. 1 2
Ans. C B
SECTION - B
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Ans. D D B C A B C D A A A A A C