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Principle: Modulus / absolute value behaviour
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Q76
#76
NDA → Mathematics → Definite Integration → Integration of Absolute Value, Piecewise, and Greatest Integer Functions
·
Easy
What is
∫
n
n
+
1
(
x
−
[
x
]
)
d
x
\int_n^{n+1}(x-[x])\,dx
∫
n
n
+
1
(
x
−
[
x
])
d
x
, where
[
⋅
]
[\cdot]
[
⋅
]
is the greatest integer function and
n
n
n
is natural number?
Add
Lever: Modulus / absolute value behaviour
A
4
n
+
1
2
\dfrac{4n+1}{2}
2
4
n
+
1
B
2
n
+
1
2
\dfrac{2n+1}{2}
2
2
n
+
1
C
1
2
\dfrac{1}{2}
2
1
D
1
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[Q92 · Sep · 2025]
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Q77
#77
NDA → Mathematics → Applications of Integration → Area Bounded by a Curve, Lines, and Axes
·
Easy
What is the area of the region bounded by
∣
x
∣
≤
2
k
|x|\leq2k
∣
x
∣
≤
2
k
and
∣
y
∣
≤
k
|y|\leq k
∣
y
∣
≤
k
, where
k
k
k
is a positive real number?
Add
Lever: Modulus / absolute value behaviour
A
2
k
2
2k^2
2
k
2
B
4
k
2
4k^2
4
k
2
C
5
k
2
5k^2
5
k
2
D
8
k
2
8k^2
8
k
2
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[Q98 · Sep · 2025]
Report
Set · 2 questions
Let
f
(
x
)
=
∣
x
∣
f(x)=|x|
f
(
x
)
=
∣
x
∣
and
g
(
x
)
=
[
x
]
−
1
g(x)=[x]-1
g
(
x
)
=
[
x
]
−
1
. Let
h
(
x
)
=
f
(
g
(
x
)
)
g
(
f
(
x
)
)
h(x)=\frac{f(g(x))}{g(f(x))}
h
(
x
)
=
g
(
f
(
x
))
f
(
g
(
x
))
.
Q78
#78
NDA → Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Hard
What is
lim
x
→
0
+
h
(
x
)
\lim_{x\to0^+}h(x)
lim
x
→
0
+
h
(
x
)
equal to?
Add
Lever: Modulus / absolute value behaviour
A
-2
B
-1
C
0
D
1
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[Q57 · Sep · 2023]
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Q79
#79
NDA → Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Hard
What is
lim
x
→
0
−
h
(
x
)
\lim_{x\to0^-}h(x)
lim
x
→
0
−
h
(
x
)
equal to?
Add
Lever: Modulus / absolute value behaviour
A
-2
B
-1
C
0
D
2
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[Q58 · Sep · 2023]
Report
Set · 2 questions
Let
f
(
x
)
=
x
−
3
∣
x
−
3
∣
+
a
;
x
<
3
f(x)=\frac{x-3}{|x-3|}+a;\ x<3
f
(
x
)
=
∣
x
−
3∣
x
−
3
+
a
;
x
<
3
,
a
−
b
;
x
=
3
a-b;\ x=3
a
−
b
;
x
=
3
,
x
−
3
∣
x
−
3
∣
+
b
;
x
>
3
\frac{x-3}{|x-3|}+b;\ x>3
∣
x
−
3∣
x
−
3
+
b
;
x
>
3
.
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
3
x=3
x
=
3
.
Q80
#80
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Moderate
What is the value of
a
a
a
?
Add
Lever: Modulus / absolute value behaviour
A
-1
B
1
C
2
D
3
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[Q59 · Sep · 2023]
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Q81
#81
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Moderate
What is the value of
b
b
b
?
Add
Lever: Modulus / absolute value behaviour
A
-1
B
1
C
2
D
3
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[Q60 · Sep · 2023]
Report
Set · 2 questions
Let
f
(
x
)
=
a
x
x
+
1
+
b
,
x
<
1
f(x)=\frac{ax}{x+1}+b,\ x<1
f
(
x
)
=
x
+
1
a
x
+
b
,
x
<
1
and
x
−
1
,
1
≤
x
≤
2
\sqrt{x-1},\ 1\leq x\leq2
x
−
1
,
1
≤
x
≤
2
.
Q82
#82
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Hard
If
f
(
x
)
f(x)
f
(
x
)
is differentiable at
x
=
1
x=1
x
=
1
, then what is the value of
(
a
+
b
)
(a+b)
(
a
+
b
)
?
Add
Lever: Differentiability at a point
A
−
1
3
-\frac{1}{3}
−
3
1
B
-1
C
0
D
1
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[Q63 · Sep · 2023]
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Q83
#83
NDA → Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Moderate
What is
lim
x
→
0
f
(
x
)
\lim_{x\to0}f(x)
lim
x
→
0
f
(
x
)
equal to?
Add
Lever: Modulus / absolute value behaviour
A
−
1
3
-\frac{1}{3}
−
3
1
B
−
2
3
-\frac{2}{3}
−
3
2
C
0
D
1
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Show solution
[Q64 · Sep · 2023]
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Q84
#84
NDA → Mathematics → Differentiation → Differentiation Techniques — Chain Rule, Logarithmic, Composite Functions
·
Easy
If
f
(
x
)
=
∣
ln
∣
x
∣
∣
f(x)=|\ln|x||
f
(
x
)
=
∣
ln
∣
x
∣∣
where
0
<
x
<
1
0<x<1
0
<
x
<
1
, then what is
f
′
(
0.5
)
f'(0.5)
f
′
(
0.5
)
equal to?
Add
Lever: Modulus / absolute value behaviour
A
-2
B
-1
C
0
D
2
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[Q65 · Sep · 2023]
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Q85
#85
NDA → Mathematics → Applications of Integration → Area Bounded by a Curve, Lines, and Axes
·
Easy
What is the area between
f
(
x
)
=
x
∣
x
∣
f(x)=x|x|
f
(
x
)
=
x
∣
x
∣
and the
x
x
x
-axis for
x
∈
[
−
1
,
1
]
x\in[-1,1]
x
∈
[
−
1
,
1
]
?
Add
Lever: Modulus / absolute value behaviour
A
2
3
\frac{2}{3}
3
2
B
1
2
\frac{1}{2}
2
1
C
1
4
\frac{1}{4}
4
1
D
1
3
\frac{1}{3}
3
1
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[Q68 · Sep · 2023]
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Q86
#86
NDA → Mathematics → Functions → Function Definition and Classification — Injectivity, Surjectivity, Bijectivity
·
Moderate
Let
A
=
{
x
∈
R
:
−
1
<
x
<
1
}
A=\{x\in\mathbb{R}:-1<x<1\}
A
=
{
x
∈
R
:
−
1
<
x
<
1
}
. Which of the following is/are bijective functions from
A
A
A
to itself? (A)
f
(
x
)
=
x
∣
x
∣
f(x)=x|x|
f
(
x
)
=
x
∣
x
∣
(B)
g
(
x
)
=
cos
(
π
x
)
g(x)=\cos(\pi x)
g
(
x
)
=
cos
(
π
x
)
Select the correct answer using the code given below:
Add
Lever: Modulus / absolute value behaviour
A
A only
B
B only
C
Both A and B
D
Neither A nor B
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[Q22 · Apr · 2024]
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Q87
#87
NDA → Mathematics → Sets & Relations → Relations — Properties, Cartesian Product, and Counting
·
Moderate
Let
R
R
R
be a relation on the open interval
(
−
1
,
1
)
(-1,1)
(
−
1
,
1
)
and is given by
R
=
{
(
x
,
y
)
:
∣
x
+
y
∣
<
2
}
R=\{(x,y):|x+y|<2\}
R
=
{(
x
,
y
)
:
∣
x
+
y
∣
<
2
}
. Then which of the following is correct?
Add
Lever: Modulus / absolute value behaviour
A
R is reflexive but neither symmetric nor transitive
B
R is reflexive and symmetric but not transitive
C
R is reflexive and transitive but not symmetric
D
R is an equivalence relation
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[Q23 · Apr · 2024]
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Q88
#88
NDA → Mathematics → Definite Integration → Integration of Absolute Value, Piecewise, and Greatest Integer Functions
·
Easy
Let
p
=
∫
a
b
f
(
x
)
d
x
p=\displaystyle\int_a^b f(x)\,dx
p
=
∫
a
b
f
(
x
)
d
x
and
q
=
∫
a
b
∣
f
(
x
)
∣
d
x
q=\displaystyle\int_a^b|f(x)|\,dx
q
=
∫
a
b
∣
f
(
x
)
∣
d
x
. If
f
(
x
)
=
e
−
x
f(x)=e^{-x}
f
(
x
)
=
e
−
x
, then which one of the following is correct?
Add
Lever: Modulus / absolute value behaviour
A
p
=
2
q
p=2q
p
=
2
q
B
p
=
−
q
p=-q
p
=
−
q
C
4
p
=
q
4p=q
4
p
=
q
D
p
=
q
p=q
p
=
q
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[Q77 · Apr · 2024]
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Q89
#89
NDA → Mathematics → Functions → Domain, Range, and Function Properties
·
Moderate
Which one of the following is correct in respect of
f
(
x
)
=
1
∣
x
∣
−
x
f(x)=\dfrac{1}{|x|-x}
f
(
x
)
=
∣
x
∣
−
x
1
and
g
(
x
)
=
1
x
−
∣
x
∣
g(x)=\dfrac{1}{x-|x|}
g
(
x
)
=
x
−
∣
x
∣
1
?
Add
Lever: Modulus / absolute value behaviour
A
f(x) has some domain and g(x) has no domain
B
f(x) has no domain and g(x) has some domain
C
f(x) and g(x) have the same domain
D
f(x) and g(x) do not have any domain
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[Q80 · Apr · 2024]
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Q90
#90
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Moderate
Let
f
(
x
)
=
∣
x
∣
+
1
f(x)=|x|+1
f
(
x
)
=
∣
x
∣
+
1
and
g
(
x
)
=
[
x
]
−
1
g(x)=[x]-1
g
(
x
)
=
[
x
]
−
1
, where
[
⋅
]
[\cdot]
[
⋅
]
is the greatest integer function. Let
h
(
x
)
=
f
(
x
)
⋅
g
(
x
)
h(x)=f(x)\cdot g(x)
h
(
x
)
=
f
(
x
)
⋅
g
(
x
)
. Consider the following statements: (A)
f
(
x
)
f(x)
f
(
x
)
is differentiable for all
x
<
0
x<0
x
<
0
. (B)
g
(
x
)
g(x)
g
(
x
)
is continuous at
x
=
0.0001
x=0.0001
x
=
0.0001
. (C) The derivative of
g
(
x
)
g(x)
g
(
x
)
at
x
=
2.5
x=2.5
x
=
2.5
is 1. Which of the statements given above are correct?
Add
Lever: Differentiability at a point
A
A and B only
B
B and C only
C
A and C only
D
A, B and C
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[Q87 · Apr · 2024]
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Q91
#91
NDA → Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Hard
Let
f
(
x
)
=
∣
x
∣
+
1
f(x)=|x|+1
f
(
x
)
=
∣
x
∣
+
1
,
g
(
x
)
=
[
x
]
−
1
g(x)=[x]-1
g
(
x
)
=
[
x
]
−
1
,
h
(
x
)
=
f
(
x
)
⋅
g
(
x
)
h(x)=f(x)\cdot g(x)
h
(
x
)
=
f
(
x
)
⋅
g
(
x
)
. What is
lim
x
→
0
−
h
(
x
)
+
lim
x
→
0
+
h
(
x
)
\displaystyle\lim_{x\to0^-}h(x)+\lim_{x\to0^+}h(x)
x
→
0
−
lim
h
(
x
)
+
x
→
0
+
lim
h
(
x
)
equal to?
Add
Lever: Modulus / absolute value behaviour
A
−
3
2
-\dfrac{3}{2}
−
2
3
B
−
1
2
-\dfrac{1}{2}
−
2
1
C
1
2
\dfrac{1}{2}
2
1
D
3
2
\dfrac{3}{2}
2
3
Tap an option to check your answer.
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[Q88 · Apr · 2024]
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Set · 2 questions
Let
f
(
x
)
=
∣
x
−
1
∣
f(x)=|x-1|
f
(
x
)
=
∣
x
−
1∣
,
g
(
x
)
=
[
x
]
g(x)=[x]
g
(
x
)
=
[
x
]
and
h
(
x
)
=
f
(
x
)
⋅
g
(
x
)
h(x)=f(x)\cdot g(x)
h
(
x
)
=
f
(
x
)
⋅
g
(
x
)
where
[
⋅
]
[\cdot]
[
⋅
]
is the greatest integer function.
Q92
#92
NDA → Mathematics → Definite Integration → Integration of Absolute Value, Piecewise, and Greatest Integer Functions
·
Hard
What is
∫
−
1
0
h
(
x
)
d
x
\displaystyle\int_{-1}^{0}h(x)\,dx
∫
−
1
0
h
(
x
)
d
x
equal to?
Add
Lever: Modulus / absolute value behaviour
A
−
3
2
-\dfrac{3}{2}
−
2
3
B
−
1
-1
−
1
C
0
0
0
D
1
2
\dfrac{1}{2}
2
1
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[Q93 · Apr · 2024]
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Q93
#93
NDA → Mathematics → Definite Integration → Integration of Absolute Value, Piecewise, and Greatest Integer Functions
·
Moderate
What is
∫
0
2
h
(
x
)
d
x
\displaystyle\int_{0}^{2}h(x)\,dx
∫
0
2
h
(
x
)
d
x
equal to?
Add
Lever: Modulus / absolute value behaviour
A
−
3
2
-\dfrac{3}{2}
−
2
3
B
−
1
-1
−
1
C
0
0
0
D
1
2
\dfrac{1}{2}
2
1
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[Q94 · Apr · 2024]
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Q94
#94
NDA → Mathematics → Functions → Greatest Integer Function
·
Moderate
Let
z
=
[
y
]
z=[y]
z
=
[
y
]
and
y
=
[
x
]
−
x
y=[x]-x
y
=
[
x
]
−
x
, where
[
⋅
]
[\cdot]
[
⋅
]
is the greatest integer function. If
x
x
x
is not an integer but positive, then what is the value of
z
z
z
?
Add
Lever: Modulus / absolute value behaviour
A
−
1
-1
−
1
B
0
0
0
C
1
1
1
D
2
2
2
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[Q71 · Sep · 2024]
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Q95
#95
NDA → Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Easy
Which one of the following is correct regarding
lim
x
→
3
∣
x
−
3
∣
x
−
3
\displaystyle\lim_{x\to3}\dfrac{|x-3|}{x-3}
x
→
3
lim
x
−
3
∣
x
−
3∣
?
Add
Lever: Modulus / absolute value behaviour
A
Limit exists and is equal to 1
B
Limit exists and is equal to 0
C
Limit exists and is equal to -1
D
Limit does not exist
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[Q74 · Sep · 2024]
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Q96
#96
NDA → Mathematics → Functions → Greatest Integer Function
·
Moderate
If
f
(
x
)
=
[
x
]
2
−
30
[
x
]
+
221
=
0
f(x)=[x]^2-30[x]+221=0
f
(
x
)
=
[
x
]
2
−
30
[
x
]
+
221
=
0
, where
[
x
]
[x]
[
x
]
is the greatest integer function, then what is the sum of all integer solutions?
Add
Lever: Modulus / absolute value behaviour
A
13
13
13
B
17
17
17
C
27
27
27
D
30
30
30
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[Q77 · Sep · 2024]
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Set · 1 question
Let
f
(
x
)
=
[
x
]
2
−
x
2
f(x)=[x]^2-x^2
f
(
x
)
=
[
x
]
2
−
x
2
.
Q97
#97
NDA → Mathematics → Functions → Greatest Integer Function
·
Hard
What is
f
(
0.999
)
+
f
(
1.001
)
f(0.999)+f(1.001)
f
(
0.999
)
+
f
(
1.001
)
equal to?
Add
Lever: Modulus / absolute value behaviour
A
−
1
-1
−
1
B
0
0
0
C
1
1
1
D
2
2
2
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[Q83 · Sep · 2024]
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Q98
#98
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Hard
Let
f
(
x
)
=
[
x
]
2
−
[
x
2
]
f(x) = [x]^2 - [x^2]
f
(
x
)
=
[
x
]
2
−
[
x
2
]
, where
[
⋅
]
[\cdot]
[
⋅
]
is the greatest integer function. Consider the following statements: A.
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
0
x = 0
x
=
0
. B.
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
1
x = 1
x
=
1
. Which of the statements given above is/are correct?
Add
Lever: Modulus / absolute value behaviour
A
A only
B
B only
C
Both A and B
D
Neither A nor B
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[Q84 · Sep · 2024]
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Set · 2 questions
Let
f
(
x
)
=
∣
x
2
−
x
−
2
∣
f(x)=|x^2-x-2|
f
(
x
)
=
∣
x
2
−
x
−
2∣
.
Q99
#99
NDA → Mathematics → Definite Integration → Integration of Absolute Value, Piecewise, and Greatest Integer Functions
·
Moderate
What is
∫
0
2
f
(
x
)
d
x
\displaystyle\int_0^2 f(x)\,dx
∫
0
2
f
(
x
)
d
x
equal to?
Add
Lever: Modulus / absolute value behaviour
A
0
0
0
B
1
1
1
C
5
3
\dfrac{5}{3}
3
5
D
10
3
\dfrac{10}{3}
3
10
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[Q91 · Sep · 2024]
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Q100
#100
NDA → Mathematics → Definite Integration → Integration of Absolute Value, Piecewise, and Greatest Integer Functions
·
Moderate
What is
∫
1
3
f
(
x
)
d
x
\displaystyle\int_1^3 f(x)\,dx
∫
1
3
f
(
x
)
d
x
equal to?
Add
Lever: Modulus / absolute value behaviour
A
2
2
2
B
3
3
3
C
4
4
4
D
5
5
5
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[Q92 · Sep · 2024]
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