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Q4576
#4576
Mathematics → Differential Equations → Solving and Verifying ODEs — Separable, IVP, and Applications
·
Moderate
If
d
y
d
x
=
2
e
x
y
3
,
y
(
0
)
=
1
2
\dfrac{dy}{dx}=2e^x y^3,\;y(0)=\dfrac{1}{2}
d
x
d
y
=
2
e
x
y
3
,
y
(
0
)
=
2
1
, then what is
4
y
2
(
2
−
e
x
)
4y^2(2-e^x)
4
y
2
(
2
−
e
x
)
equal to?
Add
NDA Maths strategy
A
1
1
1
B
2
2
2
C
3
3
3
D
4
4
4
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[Q76 · Apr · 2024]
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Q4577
#4577
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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Q4578
#4578
Mathematics → Definite Integration → Properties of Definite Integrals — Symmetry, King's, Odd/Even
·
Moderate
What is
∫
0
π
/
2
a
+
sin
x
2
(
a
+
sin
x
+
cos
x
)
d
x
\displaystyle\int_0^{\pi/2}\dfrac{a+\sin x}{2(a+\sin x+\cos x)}\,dx
∫
0
π
/2
2
(
a
+
sin
x
+
cos
x
)
a
+
sin
x
d
x
equal to?
Add
NDA Maths strategy
A
π
4
\dfrac{\pi}{4}
4
π
B
π
2
\dfrac{\pi}{2}
2
π
C
1
1
1
D
0
0
0
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[Q78 · Apr · 2024]
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Q4579
#4579
Mathematics → Application of Derivatives → Monotonicity, Extrema, and Critical Points
·
Moderate
The non-negative values of
b
b
b
for which the function
16
x
3
3
−
4
b
x
2
+
x
\dfrac{16x^3}{3}-4bx^2+x
3
16
x
3
−
4
b
x
2
+
x
has neither maximum nor minimum in the range
x
>
0
x>0
x
>
0
is
Add
NDA Maths strategy
A
0
<
b
<
1
0<b<1
0
<
b
<
1
B
1
<
b
<
2
1<b<2
1
<
b
<
2
C
b
>
2
b>2
b
>
2
D
0
≤
b
<
1
0\leq b<1
0
≤
b
<
1
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[Q79 · Apr · 2024]
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Q4580
#4580
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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Set · 2 questions
Given that
∫
3
cos
x
+
4
sin
x
2
cos
x
+
5
sin
x
d
x
=
α
x
29
+
β
29
ln
∣
2
cos
x
+
5
sin
x
∣
+
c
\displaystyle\int\dfrac{3\cos x+4\sin x}{2\cos x+5\sin x}\,dx=\dfrac{\alpha x}{29}+\dfrac{\beta}{29}\ln|2\cos x+5\sin x|+c
∫
2
cos
x
+
5
sin
x
3
cos
x
+
4
sin
x
d
x
=
29
α
x
+
29
β
ln
∣2
cos
x
+
5
sin
x
∣
+
c
.
Q4581
#4581
Mathematics → Indefinite Integration → Integration by Partial Fractions
·
Moderate
What is the value of
α
\alpha
α
?
Add
NDA Maths strategy
Concept: Express the Numerator via the Denominator and Its Derivative
A
7
7
7
B
13
13
13
C
17
17
17
D
26
26
26
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[Q81 · Apr · 2024]
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Q4582
#4582
Mathematics → Indefinite Integration → Integration by Partial Fractions
·
Easy
What is the value of
β
\beta
β
?
Add
NDA Maths strategy
Concept: Express the Numerator via the Denominator and Its Derivative
A
7
7
7
B
13
13
13
C
17
17
17
D
26
26
26
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[Q82 · Apr · 2024]
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Q4583
#4583
Mathematics → Application of Derivatives → Monotonicity, Extrema, and Critical Points
·
Hard
Let
f
(
x
)
=
x
ln
x
(
x
>
1
)
f(x)=\dfrac{x}{\ln x}\;(x>1)
f
(
x
)
=
ln
x
x
(
x
>
1
)
. Consider the following statements: (A)
f
(
x
)
f(x)
f
(
x
)
is increasing in the interval
(
e
,
∞
)
(e,\infty)
(
e
,
∞
)
. (B)
f
(
x
)
f(x)
f
(
x
)
is decreasing in the interval
(
1
,
e
)
(1,e)
(
1
,
e
)
. (C)
9
ln
7
>
7
ln
9
9\ln7>7\ln9
9
ln
7
>
7
ln
9
. Which of the statements given above are correct?
Add
NDA Maths strategy
A
(A) and (B) only
B
(B) and (C) only
C
(A) and (C) only
D
(A), (B) and (C)
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[Q83 · Apr · 2024]
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Q4584
#4584
Mathematics → Application of Derivatives → Monotonicity, Extrema, and Critical Points
·
Hard
Let
f
(
x
)
=
x
ln
x
(
x
>
1
)
f(x)=\dfrac{x}{\ln x}\;(x>1)
f
(
x
)
=
ln
x
x
(
x
>
1
)
. Consider the following statements: (A)
f
′
′
(
e
)
=
1
e
f''(e)=\dfrac{1}{e}
f
′′
(
e
)
=
e
1
. (B)
f
(
x
)
f(x)
f
(
x
)
attains a local minimum value at
x
=
e
x=e
x
=
e
. (C) A local minimum value of
f
(
x
)
f(x)
f
(
x
)
is
e
e
e
. Which of the statements given above are correct?
Add
NDA Maths strategy
A
(A) and (B) only
B
(B) and (C) only
C
(A) and (C) only
D
(A), (B) and (C)
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[Q84 · Apr · 2024]
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Set · 2 questions
Let
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
be two functions such that
g
(
x
)
=
x
−
1
x
g(x)=x-\dfrac{1}{x}
g
(
x
)
=
x
−
x
1
and
f
∘
g
(
x
)
=
x
3
−
1
x
3
f\!\circ\! g(x)=x^3-\dfrac{1}{x^3}
f
∘
g
(
x
)
=
x
3
−
x
3
1
.
Q4585
#4585
Mathematics → Functions → Composition and Inverse of Functions
·
Hard
What is
g
[
f
(
x
)
−
3
x
]
g[f(x)-3x]
g
[
f
(
x
)
−
3
x
]
equal to?
Add
NDA Maths strategy
A
x
3
−
1
x
3
x^3-\dfrac{1}{x^3}
x
3
−
x
3
1
B
x
3
+
1
x
3
x^3+\dfrac{1}{x^3}
x
3
+
x
3
1
C
x
2
−
1
x
2
x^2-\dfrac{1}{x^2}
x
2
−
x
2
1
D
x
2
+
1
x
2
x^2+\dfrac{1}{x^2}
x
2
+
x
2
1
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[Q85 · Apr · 2024]
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Q4586
#4586
Mathematics → Differentiation → Parametric, Implicit, and Higher-Order Derivatives
·
Easy
What is
f
′
′
(
x
)
f''(x)
f
′′
(
x
)
equal to?
Add
NDA Maths strategy
A
−
2
x
3
-\dfrac{2}{x^3}
−
x
3
2
B
2
x
+
2
x
3
2x+\dfrac{2}{x^3}
2
x
+
x
3
2
C
6
x
+
3
6x+3
6
x
+
3
D
6
x
6x
6
x
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[Q86 · Apr · 2024]
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Q4587
#4587
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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Q4588
#4588
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
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[Q88 · Apr · 2024]
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Set · 2 questions
Let
ϕ
(
a
)
=
∫
a
a
+
100
π
∣
sin
x
∣
d
x
\phi(a)=\displaystyle\int_a^{a+100\pi}|\sin x|\,dx
ϕ
(
a
)
=
∫
a
a
+
100
π
∣
sin
x
∣
d
x
.
Q4589
#4589
Mathematics → Definite Integration → Fundamental Theorem, Periodic Integrals, and Leibniz Rule
·
Easy
What is
ϕ
(
a
)
\phi(a)
ϕ
(
a
)
equal to?
Add
NDA Maths strategy
A
0
0
0
B
a
a
a
C
100
a
100a
100
a
D
200
200
200
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[Q89 · Apr · 2024]
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Q4590
#4590
Mathematics → Definite Integration → Fundamental Theorem, Periodic Integrals, and Leibniz Rule
·
Easy
What is
ϕ
′
(
a
)
\phi'(a)
ϕ
′
(
a
)
equal to?
Add
NDA Maths strategy
A
0
0
0
B
π
\pi
π
C
100
100
100
D
200
200
200
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[Q90 · Apr · 2024]
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Q4591
#4591
Mathematics → Application of Derivatives → Monotonicity, Extrema, and Critical Points
·
Easy
A differentiable function
f
(
x
)
f(x)
f
(
x
)
has a local maximum at
x
=
0
x=0
x
=
0
. Let
y
=
2
f
(
x
)
+
a
x
−
b
y=2f(x)+ax-b
y
=
2
f
(
x
)
+
a
x
−
b
. Which of the following is/are correct? (A)
f
′
(
0
)
=
0
f'(0)=0
f
′
(
0
)
=
0
(B)
f
′
′
(
0
)
<
0
f''(0)<0
f
′′
(
0
)
<
0
Select the correct answer using the code given below:
Add
NDA Maths strategy
A
A only
B
B only
C
Both A and B
D
Neither A nor B
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[Q91 · Apr · 2024]
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Q4592
#4592
Mathematics → Application of Derivatives → Monotonicity, Extrema, and Critical Points
·
Moderate
A differentiable function
f
(
x
)
f(x)
f
(
x
)
has a local maximum at
x
=
0
x=0
x
=
0
. Let
y
=
2
f
(
x
)
+
a
x
−
b
y=2f(x)+ax-b
y
=
2
f
(
x
)
+
a
x
−
b
. The function
y
y
y
has a relative maxima at
x
=
0
x=0
x
=
0
for
Add
NDA Maths strategy
A
a
>
0
,
b
=
0
a>0,b=0
a
>
0
,
b
=
0
B
for all
b
b
b
and
a
=
0
a=0
a
=
0
C
for all
b
>
0
b>0
b
>
0
only
D
for all
a
a
a
and
b
=
0
b=0
b
=
0
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[Q92 · 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.
Q4593
#4593
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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Q4594
#4594
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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Set · 2 questions
Let
∫
d
x
x
+
1
−
x
−
1
=
α
(
x
+
1
)
3
/
2
+
β
(
x
−
1
)
3
/
2
+
c
\displaystyle\int\dfrac{dx}{\sqrt{x+1}-\sqrt{x-1}}=\alpha(x+1)^{3/2}+\beta(x-1)^{3/2}+c
∫
x
+
1
−
x
−
1
d
x
=
α
(
x
+
1
)
3/2
+
β
(
x
−
1
)
3/2
+
c
.
Q4595
#4595
Mathematics → Indefinite Integration → Integration by Substitution — Algebraic, Trigonometric, and Composite Forms
·
Moderate
What is the value of
α
\alpha
α
?
Add
NDA Maths strategy
Concept: Rationalising a Surd Denominator
A
1
3
\dfrac{1}{3}
3
1
B
2
3
\dfrac{2}{3}
3
2
C
1
1
1
D
4
3
\dfrac{4}{3}
3
4
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[Q95 · Apr · 2024]
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Q4596
#4596
Mathematics → Indefinite Integration → Integration by Substitution — Algebraic, Trigonometric, and Composite Forms
·
Moderate
What is the value of
β
\beta
β
?
Add
NDA Maths strategy
Concept: Rationalising a Surd Denominator
A
−
2
3
-\dfrac{2}{3}
−
3
2
B
−
1
3
-\dfrac{1}{3}
−
3
1
C
1
3
\dfrac{1}{3}
3
1
D
2
3
\dfrac{2}{3}
3
2
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[Q96 · Apr · 2024]
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Set · 2 questions
The circle
x
2
+
y
2
−
2
x
=
0
x^2+y^2-2x=0
x
2
+
y
2
−
2
x
=
0
is partitioned by line
y
=
x
y=x
y
=
x
in two segments. Let
A
1
,
A
2
A_1,A_2
A
1
,
A
2
be the areas of major and minor segments respectively.
Q4597
#4597
Mathematics → Applications of Integration → Area Bounded by a Curve, Lines, and Axes
·
Hard
What is the value of
A
1
A_1
A
1
?
Add
NDA Maths strategy
A
π
−
2
4
\dfrac{\pi-2}{4}
4
π
−
2
B
π
+
2
4
\dfrac{\pi+2}{4}
4
π
+
2
C
3
π
−
2
4
\dfrac{3\pi-2}{4}
4
3
π
−
2
D
3
π
+
2
4
\dfrac{3\pi+2}{4}
4
3
π
+
2
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[Q97 · Apr · 2024]
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Q4598
#4598
Mathematics → Applications of Integration → Area Bounded by a Curve, Lines, and Axes
·
Hard
What is the value of
2
A
1
+
A
2
A
1
−
3
A
2
\dfrac{2A_1+A_2}{A_1-3A_2}
A
1
−
3
A
2
2
A
1
+
A
2
?
Add
NDA Maths strategy
A
π
\pi
π
B
1
1
1
C
−
1
-1
−
1
D
−
π
-\pi
−
π
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[Q98 · Apr · 2024]
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Set · 2 questions
Let
3
f
(
x
)
+
f
(
1
x
)
=
1
x
+
1
3f(x)+f\!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}+1
3
f
(
x
)
+
f
(
x
1
)
=
x
1
+
1
.
Q4599
#4599
Mathematics → Functions → Functional Equations
·
Hard
What is
f
(
x
)
f(x)
f
(
x
)
equal to?
Add
NDA Maths strategy
A
1
8
x
−
x
8
+
1
4
\dfrac{1}{8x}-\dfrac{x}{8}+\dfrac{1}{4}
8
x
1
−
8
x
+
4
1
B
3
8
x
+
x
8
+
3
4
\dfrac{3}{8x}+\dfrac{x}{8}+\dfrac{3}{4}
8
x
3
+
8
x
+
4
3
C
3
8
x
+
x
8
+
1
4
\dfrac{3}{8x}+\dfrac{x}{8}+\dfrac{1}{4}
8
x
3
+
8
x
+
4
1
D
3
8
x
−
x
8
+
1
4
\dfrac{3}{8x}-\dfrac{x}{8}+\dfrac{1}{4}
8
x
3
−
8
x
+
4
1
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[Q99 · Apr · 2024]
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Q4600
#4600
Mathematics → Definite Integration → Properties of Definite Integrals — Symmetry, King's, Odd/Even
·
Hard
What is
8
∫
1
2
f
(
x
)
d
x
8\displaystyle\int_{1}^{2}f(x)\,dx
8
∫
1
2
f
(
x
)
d
x
equal to?
Add
NDA Maths strategy
A
ln
(
8
e
)
\ln(8e)
ln
(
8
e
)
B
ln
(
4
e
)
\ln(4e)
ln
(
4
e
)
C
ln
2
\ln2
ln
2
D
ln
2
−
1
\ln2-1
ln
2
−
1
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[Q100 · Apr · 2024]
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