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Principle: AM-GM / mean inequalities (incl. x + 1/x ≥ 2)
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Set · 1 question
For the following two (02) items: Let
α
\alpha
α
and
β
\beta
β
be the roots of the quadratic equation
x
2
+
(
log
0.5
(
a
2
)
)
x
+
(
log
0.5
(
a
2
)
)
4
=
0
x^2 + (\log_{0.5}(a^2))x + (\log_{0.5}(a^2))^4 = 0
x
2
+
(
lo
g
0.5
(
a
2
))
x
+
(
lo
g
0.5
(
a
2
)
)
4
=
0
where
a
2
≠
1
a^2 \neq 1
a
2
=
1
and
log
0.5
(
a
2
)
>
0
\log_{0.5}(a^2) > 0
lo
g
0.5
(
a
2
)
>
0
. Further,
β
2
=
α
(
log
a
2
(
0.5
)
)
\beta^2 = \alpha(\log_{a^2}(0.5))
β
2
=
α
(
lo
g
a
2
(
0.5
))
.
Q26
#26
NDA → Mathematics → Sequence & Series → Interrelating AP, GP and HP
·
Moderate
If
(
10
+
log
10
x
)
(10 + \log_{10} x)
(
10
+
lo
g
10
x
)
,
(
10
+
log
10
y
)
(10 + \log_{10} y)
(
10
+
lo
g
10
y
)
and
(
10
+
log
10
z
)
(10 + \log_{10} z)
(
10
+
lo
g
10
z
)
are in AP, then consider the following statements: I. The GM of
x
x
x
and
z
z
z
is
y
2
y^2
y
2
. II. The AM of
log
10
x
\log_{10} x
lo
g
10
x
and
log
10
z
\log_{10} z
lo
g
10
z
is
log
10
y
\log_{10} y
lo
g
10
y
. Which of the statements given above is/are correct?
Add
Lever: AM-GM / mean inequalities (incl. x + 1/x ≥ 2)
Concept: The log bridge: a GP becomes an AP
A
I only
B
II only
C
Both I and II
D
Neither I nor II
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[Q5 · Sep · 2025]
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Q27
#27
NDA → Mathematics → Application of Derivatives → Optimisation — Geometric, Trigonometric, AM-GM
·
Easy
A wire of length 20 cm is to be bent into a rectangle. Which of the following statements is/are correct? I. The rectangle of the largest area is the square. II. It is possible to form a rectangle of an area of 27 cm
2
^2
2
. Select the answer using the code given below.
Add
Lever: AM-GM / mean inequalities (incl. x + 1/x ≥ 2)
A
I only
B
II only
C
Both I and II
D
Neither I nor II
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[Q96 · Sep · 2025]
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Q28
#28
NDA → Mathematics → Application of Derivatives → Monotonicity, Extrema, and Critical Points
·
Moderate
Consider the following statements: Statement-I: The function
f
(
x
)
=
x
3
+
128
x
f(x)=\dfrac{x^3+128}{x}
f
(
x
)
=
x
x
3
+
128
has a minimum value 48 at
x
=
4
x=4
x
=
4
. Statement-II: As
x
x
x
increases through 4,
f
′
(
x
)
f'(x)
f
′
(
x
)
changes sign from positive to negative. Which one of the following is correct?
Add
Lever: AM-GM / mean inequalities (incl. x + 1/x ≥ 2)
A
Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
B
Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
C
Statement-I is correct but Statement-II is not correct
D
Statement-I is not correct but Statement-II is correct
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[Q100 · Sep · 2025]
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Set · 1 question
Given that
4
x
2
+
y
2
=
9
4x^2+y^2=9
4
x
2
+
y
2
=
9
.
Q29
#29
NDA → Mathematics → Application of Derivatives → Optimisation — Geometric, Trigonometric, AM-GM
·
Moderate
What is the maximum value of
x
y
xy
x
y
?
Add
Lever: AM-GM / mean inequalities (incl. x + 1/x ≥ 2)
A
9
4
\frac{9}{4}
4
9
B
3
2
\frac{3}{2}
2
3
C
4
9
\frac{4}{9}
9
4
D
2
3
\frac{2}{3}
3
2
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[Q48 · Sep · 2023]
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Q30
#30
NDA → Mathematics → Logarithms → Solving Logarithmic Equations and Applications
·
Hard
For
x
≥
y
>
1
x\geq y>1
x
≥
y
>
1
, let
log
x
x
y
+
log
y
y
x
=
k
\log_x\!\dfrac{x}{y}+\log_y\!\dfrac{y}{x}=k
lo
g
x
y
x
+
lo
g
y
x
y
=
k
, then the value of
k
k
k
can never be equal to
Add
Lever: AM-GM / mean inequalities (incl. x + 1/x ≥ 2)
A
−
1
-1
−
1
B
−
1
2
-\dfrac{1}{2}
−
2
1
C
0
0
0
D
1
1
1
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[Q28 · Apr · 2024]
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Q31
#31
NDA → Mathematics → Sequence & Series → Interrelating AP, GP and HP
·
Moderate
If
p
,
1
,
q
p, 1, q
p
,
1
,
q
are in AP and
p
,
2
,
q
p, 2, q
p
,
2
,
q
are in GP, then which of the following statements is/are correct? I.
p
,
4
,
q
p, 4, q
p
,
4
,
q
are in HP. II.
1
p
,
1
4
,
1
q
\frac{1}{p}, \frac{1}{4}, \frac{1}{q}
p
1
,
4
1
,
q
1
are in AP. Select the answer using the code given below.
Add
Lever: AM-GM / mean inequalities (incl. x + 1/x ≥ 2)
Concept: Mixed and chained progression problems
A
I only
B
II only
C
Both I and II
D
Neither I nor II
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[Q6 · Apr · 2025]
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