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Principle: Modulus / absolute value behaviour
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Set · 2 questions
Consider the following for the next two (02) items that follow: Let
f
(
x
)
=
sin
[
π
2
]
x
+
cos
[
−
π
2
]
x
f(x)=\sin[\pi^{2}]x+\cos[-\pi^{2}]x
f
(
x
)
=
sin
[
π
2
]
x
+
cos
[
−
π
2
]
x
where
[
⋅
]
[\cdot]
[
⋅
]
is a greatest integer function.
Q51
#51
NDA → Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Moderate
What is
f
(
π
2
)
f\!\left(\dfrac{\pi}{2}\right)
f
(
2
π
)
equal to?
Add
Lever: Modulus / absolute value behaviour
A
-1
B
0
C
1
D
2
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[Q79 · Apr · 2023]
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Q52
#52
NDA → Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Moderate
What is
f
(
π
4
)
f\!\left(\dfrac{\pi}{4}\right)
f
(
4
π
)
equal to?
Add
Lever: Modulus / absolute value behaviour
A
−
1
2
-\dfrac{1}{\sqrt{2}}
−
2
1
B
-1
C
1
D
1
2
\dfrac{1}{\sqrt{2}}
2
1
Tap an option to check your answer.
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[Q80 · Apr · 2023]
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Set · 2 questions
Consider the following for the next two (02) items that follow: Let
I
=
∫
a
b
∣
x
∣
x
d
x
I=\int_{a}^{b}\dfrac{|x|}{x}\,dx
I
=
∫
a
b
x
∣
x
∣
d
x
,
a
<
b
a<b
a
<
b
.
Q53
#53
NDA → Mathematics → Definite Integration → Integration of Absolute Value, Piecewise, and Greatest Integer Functions
·
Easy
What is
I
I
I
equal to when
a
<
0
<
b
a<0<b
a
<
0
<
b
?
Add
Lever: Modulus / absolute value behaviour
A
a
+
b
a+b
a
+
b
B
a
−
b
a-b
a
−
b
C
b
−
a
b-a
b
−
a
D
(
a
+
b
)
2
\dfrac{(a+b)}{2}
2
(
a
+
b
)
Tap an option to check your answer.
Show solution
[Q84 · Apr · 2023]
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Q54
#54
NDA → Mathematics → Definite Integration → Integration of Absolute Value, Piecewise, and Greatest Integer Functions
·
Easy
What is
I
I
I
equal to when
a
<
b
<
0
a<b<0
a
<
b
<
0
?
Add
Lever: Modulus / absolute value behaviour
A
a
+
b
a+b
a
+
b
B
a
−
b
a-b
a
−
b
C
b
−
a
b-a
b
−
a
D
(
a
+
b
)
2
\dfrac{(a+b)}{2}
2
(
a
+
b
)
Tap an option to check your answer.
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[Q85 · Apr · 2023]
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Set · 3 questions
Consider the following for the next three (03) items that follow: Let
f
(
x
)
=
∣
ln
x
∣
,
x
≠
1
f(x)=|\ln x|,\, x\neq1
f
(
x
)
=
∣
ln
x
∣
,
x
=
1
.
Q55
#55
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Easy
What is the derivative of
f
(
x
)
f(x)
f
(
x
)
at
x
=
0.5
x=0.5
x
=
0.5
?
Add
Lever: Modulus / absolute value behaviour
A
-2
B
-1
C
1
D
2
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[Q86 · Apr · 2023]
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Q56
#56
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Easy
What is the derivative of
f
(
x
)
f(x)
f
(
x
)
at
x
=
2
x=2
x
=
2
?
Add
Lever: Modulus / absolute value behaviour
A
−
1
2
-\dfrac{1}{2}
−
2
1
B
-1
C
1
2
\dfrac{1}{2}
2
1
D
2
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Show solution
[Q87 · Apr · 2023]
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Q57
#57
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Hard
What is the derivative of
f
∘
f
(
x
)
f\circ f(x)
f
∘
f
(
x
)
, where
1
<
x
<
2
1<x<2
1
<
x
<
2
?
Add
Lever: Modulus / absolute value behaviour
A
1
ln
x
\dfrac{1}{\ln x}
ln
x
1
B
1
x
ln
x
\dfrac{1}{x\ln x}
x
ln
x
1
C
−
1
ln
x
-\dfrac{1}{\ln x}
−
ln
x
1
D
−
1
x
ln
x
-\dfrac{1}{x\ln x}
−
x
ln
x
1
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[Q88 · Apr · 2023]
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Set · 2 questions
Consider the following for the next two (02) items that follow: Let
f
(
x
)
=
{
x
+
6
,
x
≤
1
p
x
+
q
,
1
<
x
<
2
5
x
,
x
≥
2
f(x)=\begin{cases}x+6, & x\leq1\\px+q, & 1<x<2\\5x, & x\geq2\end{cases}
f
(
x
)
=
⎩
⎨
⎧
x
+
6
,
p
x
+
q
,
5
x
,
x
≤
1
1
<
x
<
2
x
≥
2
and
f
(
x
)
f(x)
f
(
x
)
is continuous.
Q58
#58
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Moderate
What is the value of
p
p
p
?
Add
Lever: Modulus / absolute value behaviour
A
2
B
3
C
4
D
5
Tap an option to check your answer.
Show solution
[Q89 · Apr · 2023]
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Q59
#59
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Moderate
What is the value of
q
q
q
?
Add
Lever: Modulus / absolute value behaviour
A
2
B
3
C
4
D
5
Tap an option to check your answer.
Show solution
[Q90 · Apr · 2023]
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Q60
#60
NDA → Mathematics → Functions → Domain, Range, and Function Properties
·
Easy
What is the range of the function
f
(
x
)
=
x
+
∣
x
∣
f(x)=x+|x|
f
(
x
)
=
x
+
∣
x
∣
if the domain is the set of real numbers?
Add
Lever: Modulus / absolute value behaviour
A
(
0
,
∞
)
(0,\infty)
(
0
,
∞
)
B
[
0
,
∞
)
[0,\infty)
[
0
,
∞
)
C
(
−
∞
,
∞
)
(-\infty,\infty)
(
−
∞
,
∞
)
D
[
1
,
∞
)
[1,\infty)
[
1
,
∞
)
Tap an option to check your answer.
Show solution
[Q97 · Apr · 2023]
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Q61
#61
NDA → Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Easy
What is
lim
x
→
5
5
−
x
∣
x
−
5
∣
\lim_{x\to5}\dfrac{5-x}{|x-5|}
lim
x
→
5
∣
x
−
5∣
5
−
x
equal to?
Add
Lever: Modulus / absolute value behaviour
A
-1
B
0
C
1
D
Limit does not exist
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Show solution
[Q99 · Apr · 2023]
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Q62
#62
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Easy
What is the derivative of
x
∣
x
∣
\dfrac{x}{|x|}
∣
x
∣
x
with respect to
x
x
x
, where
x
<
0
x<0
x
<
0
?
Add
Lever: Modulus / absolute value behaviour
A
-1
B
0
C
1
D
x
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Show solution
[Q74 · Apr · 2026]
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Set · 2 questions
For the next two (02) items that follow: Let
f
(
x
)
=
{
a
x
(
x
−
1
)
,
x
<
1
x
−
1
,
1
≤
x
≤
3
p
x
2
+
q
x
+
2
,
x
>
3
f(x)=\begin{cases}ax(x-1), & x<1\\ x-1, & 1\leq x\leq3\\ px^2+qx+2, & x>3\end{cases}
f
(
x
)
=
⎩
⎨
⎧
a
x
(
x
−
1
)
,
x
−
1
,
p
x
2
+
q
x
+
2
,
x
<
1
1
≤
x
≤
3
x
>
3
. Given that
f
(
x
)
f(x)
f
(
x
)
is continuous for all x but not differentiable at x=1. Further
f
′
(
x
)
f'(x)
f
′
(
x
)
is continuous at x=3.
Q63
#63
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Hard
What is the value of p?
Add
Lever: Differentiability at a point
A
-1
B
−
1
3
-\dfrac{1}{3}
−
3
1
C
1
3
\dfrac{1}{3}
3
1
D
1
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Show solution
[Q95 · Apr · 2026]
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Q64
#64
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Easy
What is the value of q?
Add
Lever: Differentiability at a point
A
-1
B
−
1
3
-\dfrac{1}{3}
−
3
1
C
1
3
\dfrac{1}{3}
3
1
D
1
Tap an option to check your answer.
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[Q96 · Apr · 2026]
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Set · 2 questions
For the next two (02) items that follow: Let
f
(
x
)
=
tan
(
x
2
)
f(x)=\tan(x^2)
f
(
x
)
=
tan
(
x
2
)
and
g
(
x
)
=
x
∣
x
∣
g(x)=x|x|
g
(
x
)
=
x
∣
x
∣
for
∣
x
∣
<
π
/
2
|x|<\sqrt{\pi/2}
∣
x
∣
<
π
/2
.
Q65
#65
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Moderate
If
p
(
x
)
=
f
(
x
)
g
(
x
)
p(x)=f(x)g(x)
p
(
x
)
=
f
(
x
)
g
(
x
)
, then which of the following statements is/are correct? I.
p
(
x
)
p(x)
p
(
x
)
is continuous at
x
=
0
x=0
x
=
0
. II.
p
(
x
)
p(x)
p
(
x
)
is differentiable at
x
=
0
x=0
x
=
0
. Select the answer using the code given below.
Add
Lever: Differentiability at a point
A
I only
B
II only
C
Both I and II
D
Neither I nor II
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Show solution
[Q99 · Apr · 2026]
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Q66
#66
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Moderate
If
q
(
x
)
=
f
∘
g
(
x
)
q(x)=f\circ g(x)
q
(
x
)
=
f
∘
g
(
x
)
, then which of the following statements is/are correct? I.
q
(
x
)
q(x)
q
(
x
)
is continuous at
x
=
0
x=0
x
=
0
. II.
q
(
x
)
q(x)
q
(
x
)
is differentiable at
x
=
0
x=0
x
=
0
. Select the answer using the code given below.
Add
Lever: Differentiability at a point
A
I only
B
II only
C
Both I and II
D
Neither I nor II
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Show solution
[Q100 · Apr · 2026]
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Q67
#67
NDA → Mathematics → Sets & Relations → Relations — Properties, Cartesian Product, and Counting
·
Moderate
Let
A
=
{
−
3
,
−
2
,
−
1
,
0
,
1
,
2
,
3
}
A=\{-3,-2,-1,0,1,2,3\}
A
=
{
−
3
,
−
2
,
−
1
,
0
,
1
,
2
,
3
}
and
B
=
{
0
,
1
,
4
,
9
}
B=\{0,1,4,9\}
B
=
{
0
,
1
,
4
,
9
}
. How many elements does the subset of
A
×
B
A\times B
A
×
B
corresponding to the relation
R
=
{
(
x
,
y
)
:
∣
x
∣
<
y
}
R=\{(x,y):|x|<y\}
R
=
{(
x
,
y
)
:
∣
x
∣
<
y
}
have, where
x
∈
A
x\in A
x
∈
A
and
y
∈
B
y\in B
y
∈
B
?
Add
Lever: Modulus / absolute value behaviour
A
9
B
12
C
15
D
16
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[Q43 · Sep · 2025]
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Set · 2 questions
For the following two (02) items: Let
f
(
x
)
=
{
1
−
cos
2
x
x
2
,
x
<
0
9
,
x
=
0
x
(
16
+
x
)
−
4
,
x
>
0
f(x)=\begin{cases}\dfrac{1-\cos2x}{x^2}, & x<0\\ 9, & x=0\\ \dfrac{\sqrt{x}}{\sqrt{(16+\sqrt{x})}-4}, & x>0\end{cases}
f
(
x
)
=
⎩
⎨
⎧
x
2
1
−
cos
2
x
,
9
,
(
16
+
x
)
−
4
x
,
x
<
0
x
=
0
x
>
0
Q68
#68
NDA → Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Moderate
What is
lim
x
→
0
−
f
(
x
)
\displaystyle\lim_{x\to 0^-}f(x)
x
→
0
−
lim
f
(
x
)
equal to?
Add
Lever: Modulus / absolute value behaviour
A
2
B
4
C
6
D
8
Tap an option to check your answer.
Show solution
[Q71 · Sep · 2025]
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Q69
#69
NDA → Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Moderate
What is
lim
x
→
0
+
f
(
x
)
\displaystyle\lim_{x\to 0^+}f(x)
x
→
0
+
lim
f
(
x
)
equal to?
Add
Lever: Modulus / absolute value behaviour
A
6
B
7
C
8
D
9
Tap an option to check your answer.
Show solution
[Q72 · Sep · 2025]
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Set · 2 questions
For the following three (03) items: Consider the function
f
(
x
)
=
x
∣
x
∣
f(x)=x|x|
f
(
x
)
=
x
∣
x
∣
.
Q70
#70
NDA → Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Easy
What is
lim
x
→
−
1
f
(
x
)
\displaystyle\lim_{x\to-1}f(x)
x
→
−
1
lim
f
(
x
)
equal to?
Add
Lever: Modulus / absolute value behaviour
A
-1
B
0
C
1
D
Limit does not exist
Tap an option to check your answer.
Show solution
[Q73 · Sep · 2025]
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Q71
#71
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Moderate
Consider the following statements: I. The function is increasing in the interval
(
−
∞
,
∞
)
(-\infty,\infty)
(
−
∞
,
∞
)
. II. The function is differentiable at
x
=
0
x=0
x
=
0
. Which of the statements given above is/are correct?
Add
Lever: Differentiability at a point
A
I only
B
II only
C
Both I and II
D
Neither I nor II
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[Q75 · Sep · 2025]
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Set · 2 questions
For the following two (02) items: Consider the function
f
(
x
)
=
{
4
(
5
x
)
,
x
<
0
8
k
+
x
,
x
≥
0
f(x)=\begin{cases}4(5^x), & x<0\\ 8k+x, & x\geq0\end{cases}
f
(
x
)
=
{
4
(
5
x
)
,
8
k
+
x
,
x
<
0
x
≥
0
.
Q72
#72
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Easy
If the function is continuous, then what is the value of
k
k
k
?
Add
Lever: Modulus / absolute value behaviour
A
0.5
B
1
C
1.5
D
2
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[Q83 · Sep · 2025]
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Q73
#73
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Easy
What is
f
′
(
−
1
)
f'(-1)
f
′
(
−
1
)
equal to?
Add
Lever: Modulus / absolute value behaviour
A
2
5
ln
5
\dfrac{2}{5}\ln5
5
2
ln
5
B
3
5
ln
5
\dfrac{3}{5}\ln5
5
3
ln
5
C
4
5
ln
5
\dfrac{4}{5}\ln5
5
4
ln
5
D
20
ln
5
20\ln5
20
ln
5
Tap an option to check your answer.
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[Q84 · Sep · 2025]
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Set · 2 questions
For the following two (02) items: Let the function
f
(
x
)
=
∣
x
−
3
∣
+
∣
x
−
4
∣
f(x)=|x-3|+|x-4|
f
(
x
)
=
∣
x
−
3∣
+
∣
x
−
4∣
be defined on the interval
[
0
,
5
]
[0,5]
[
0
,
5
]
.
Q74
#74
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Easy
What is
d
y
d
x
\dfrac{dy}{dx}
d
x
d
y
at
x
=
3.5
x=3.5
x
=
3.5
equal to?
Add
Lever: Differentiability at a point
A
0
B
1
C
2
D
3.5
Tap an option to check your answer.
Show solution
[Q87 · Sep · 2025]
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Q75
#75
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Moderate
Consider the following statements: I. The function is differentiable at
x
=
3
x=3
x
=
3
. II. The function is differentiable at
x
=
4
x=4
x
=
4
. Which of the statements given above is/are correct?
Add
Lever: Differentiability at a point
A
I only
B
II only
C
Both I and II
D
Neither I nor II
Tap an option to check your answer.
Show solution
[Q88 · Sep · 2025]
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