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Principle: Modulus / absolute value behaviour
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Set · 2 questions
Let
f
(
x
)
=
{
x
3
,
x
2
<
1
x
2
,
x
2
≥
1
f(x) = \begin{cases} x^3, & x^2 < 1 \\ x^2, & x^2 \geq 1 \end{cases}
f
(
x
)
=
{
x
3
,
x
2
,
x
2
<
1
x
2
≥
1
Q101
#101
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Easy
Add
What is
lim
x
→
0
f
′
(
x
)
\lim_{x \to 0} f'(x)
lim
x
→
0
f
′
(
x
)
equal to?
Lever: Modulus / absolute value behaviour
Concept: Continuity vs differentiability
A
2
2
2
B
1
1
1
C
0
0
0
D
Limit does not exist
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[Q79 · Apr · 2025]
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Q102
#102
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Moderate
Add
Consider the following statements: (I). The function is continuous at
x
=
−
1
x = -1
x
=
−
1
. (II). The function is differentiable at
x
=
1
x = 1
x
=
1
. Which of the statements given above is/are correct?
Lever: Differentiability at a point
Concept: Continuity vs differentiability
A
I only
B
II only
C
Both I and II
D
Neither I nor II
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[Q80 · Apr · 2025]
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Set · 2 questions
Let the function
f
(
x
)
=
sin
[
x
]
f(x) = \sin[x]
f
(
x
)
=
sin
[
x
]
, where
[
⋅
]
[\cdot]
[
⋅
]
is the greatest integer function and
g
(
x
)
=
∣
x
∣
g(x) = |x|
g
(
x
)
=
∣
x
∣
.
Q103
#103
NDA → Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Moderate
Add
What is
lim
x
→
0
{
f
(
x
)
⋅
g
(
x
)
}
\lim_{x\to 0}\{f(x) \cdot g(x)\}
lim
x
→
0
{
f
(
x
)
⋅
g
(
x
)}
equal to?
Lever: Modulus / absolute value behaviour
Concept: Left-hand and right-hand limits
A
−
1
-1
−
1
B
0
0
0
C
1
1
1
D
Limit does not exist
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[Q83 · Apr · 2025]
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Q104
#104
NDA → Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Moderate
Add
What is
lim
x
→
0
f
(
x
)
g
(
x
)
\lim_{x\to 0} \frac{f(x)}{g(x)}
lim
x
→
0
g
(
x
)
f
(
x
)
equal to?
Lever: Modulus / absolute value behaviour
Concept: Left-hand and right-hand limits
A
−
sin
1
-\sin 1
−
sin
1
B
sin
1
\sin 1
sin
1
C
0
0
0
D
Limit does not exist
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[Q84 · Apr · 2025]
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Set · 2 questions
Let
f
(
x
)
=
[
x
2
]
f(x)=[x^2]
f
(
x
)
=
[
x
2
]
, where
[
⋅
]
[\cdot]
[
⋅
]
is the greatest integer function.
Q105
#105
NDA → Mathematics → Definite Integration → Integration of Absolute Value, Piecewise, and Greatest Integer Functions
·
Moderate
Add
What is
∫
2
3
f
(
x
)
d
x
\int_{\sqrt{2}}^{\sqrt{3}} f(x)\,dx
∫
2
3
f
(
x
)
d
x
equal to?
Lever: Modulus / absolute value behaviour
Concept: Integrating greatest-integer (floor) functions
A
3
−
2
\sqrt{3}-\sqrt{2}
3
−
2
B
2
(
3
−
2
)
2(\sqrt{3}-\sqrt{2})
2
(
3
−
2
)
C
3
−
2
3-\sqrt{2}
3
−
2
D
1
1
1
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[Q99 · Apr · 2025]
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Q106
#106
NDA → Mathematics → Definite Integration → Integration of Absolute Value, Piecewise, and Greatest Integer Functions
·
Hard
Add
What is
∫
2
2
f
(
x
)
d
x
\int_{\sqrt{2}}^{2} f(x)\,dx
∫
2
2
f
(
x
)
d
x
equal to?
Lever: Modulus / absolute value behaviour
Concept: Integrating greatest-integer (floor) functions
A
6
−
3
−
2
2
6-\sqrt{3}-2\sqrt{2}
6
−
3
−
2
2
B
6
−
3
−
2
6-\sqrt{3}-\sqrt{2}
6
−
3
−
2
C
6
−
3
+
2
2
6-\sqrt{3}+2\sqrt{2}
6
−
3
+
2
2
D
6
+
3
−
2
2
6+\sqrt{3}-2\sqrt{2}
6
+
3
−
2
2
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[Q100 · Apr · 2025]
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