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Set · 1 question
A function is defined by
f
(
x
)
=
(
x
+
1
)
2
/
3
(
2
x
+
4
)
6
/
3
(
6
x
+
9
)
f(x)=(x+1)^{2/3}(2x+4)^{6/3}(6x+9)
f
(
x
)
=
(
x
+
1
)
2/3
(
2
x
+
4
)
6/3
(
6
x
+
9
)
.
Q4426
#4426
Mathematics → Application of Derivatives → Monotonicity, Extrema, and Critical Points
·
Hard
The function attains local minimum value at:
Add
NDA Maths strategy
A
x
=
−
28
3
x=-\frac{28}{3}
x
=
−
3
28
B
x
=
−
1
x=-1
x
=
−
1
C
x
=
0
x=0
x
=
0
D
x
=
28
3
x=\frac{28}{3}
x
=
3
28
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[Q46 · Sep · 2023]
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Set · 2 questions
Given that
4
x
2
+
y
2
=
9
4x^2+y^2=9
4
x
2
+
y
2
=
9
.
Q4427
#4427
Mathematics → Application of Derivatives → Optimisation — Geometric, Trigonometric, AM-GM
·
Easy
What is the maximum value of
y
y
y
?
Add
NDA Maths strategy
A
3
2
\frac{3}{2}
2
3
B
3
C
4
D
6
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[Q47 · Sep · 2023]
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Q4428
#4428
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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Set · 2 questions
A function is defined by
f
(
x
)
=
π
+
sin
2
x
f(x)=\pi+\sin^2 x
f
(
x
)
=
π
+
sin
2
x
.
Q4429
#4429
Mathematics → Functions → Domain, Range, and Function Properties
·
Easy
What is the range of the function?
Add
NDA Maths strategy
A
[0,1]
B
[
π
,
π
+
1
]
[\pi,\pi+1]
[
π
,
π
+
1
]
C
[
π
−
1
,
π
+
1
]
[\pi-1,\pi+1]
[
π
−
1
,
π
+
1
]
D
[
π
−
1
,
π
−
1
]
[\pi-1,\pi-1]
[
π
−
1
,
π
−
1
]
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[Q49 · Sep · 2023]
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Q4430
#4430
Mathematics → Functions → Domain, Range, and Function Properties
·
Easy
What is the period of the function?
Add
NDA Maths strategy
A
2
π
2\pi
2
π
B
π
\pi
π
C
π
2
\frac{\pi}{2}
2
π
D
The function is non-periodic
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[Q50 · Sep · 2023]
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Set · 2 questions
A parabola passes through
(
1
,
2
)
(1,2)
(
1
,
2
)
and satisfies
d
y
d
x
=
2
y
x
,
x
>
0
,
y
>
0
\frac{dy}{dx}=\frac{2y}{x},\ x>0,y>0
d
x
d
y
=
x
2
y
,
x
>
0
,
y
>
0
.
Q4431
#4431
Mathematics → Conics → Parabola — Equation, Properties, and Latus Rectum
·
Moderate
What is the directrix of the parabola?
Add
NDA Maths strategy
A
y
=
−
1
8
y=-\frac{1}{8}
y
=
−
8
1
B
y
=
1
8
y=\frac{1}{8}
y
=
8
1
C
x
=
−
1
8
x=-\frac{1}{8}
x
=
−
8
1
D
x
=
1
8
x=\frac{1}{8}
x
=
8
1
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[Q51 · Sep · 2023]
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Q4432
#4432
Mathematics → Conics → Parabola — Equation, Properties, and Latus Rectum
·
Easy
What is the length of latus rectum of the parabola?
Add
NDA Maths strategy
A
1
B
1
2
\frac{1}{2}
2
1
C
1
4
\frac{1}{4}
4
1
D
1
8
\frac{1}{8}
8
1
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[Q52 · Sep · 2023]
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Set · 2 questions
Let
f
(
x
)
=
a
x
−
1
+
b
x
−
1
f(x)=a^{x-1}+b^{x-1}
f
(
x
)
=
a
x
−
1
+
b
x
−
1
and
g
(
x
)
=
x
−
1
g(x)=\sqrt{x-1}
g
(
x
)
=
x
−
1
.
Q4433
#4433
Mathematics → Limits & Continuity → Limit Evaluation Techniques — L'Hôpital, Rationalization, Standard Forms
·
Hard
What is
lim
x
→
1
f
(
x
)
−
1
g
(
x
)
\lim_{x\to1}\frac{f(x)-1}{g(x)}
lim
x
→
1
g
(
x
)
f
(
x
)
−
1
equal to?
Add
NDA Maths strategy
A
ln
(
a
b
)
4
\frac{\ln(ab)}{4}
4
l
n
(
ab
)
B
ln
(
a
b
)
2
\frac{\ln(ab)}{2}
2
l
n
(
ab
)
C
ln
(
a
b
)
\ln(ab)
ln
(
ab
)
D
2
ln
(
a
b
)
2\ln(ab)
2
ln
(
ab
)
Tap an option to check your answer.
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[Q53 · Sep · 2023]
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Q4434
#4434
Mathematics → Limits & Continuity → Limit Evaluation Techniques — L'Hôpital, Rationalization, Standard Forms
·
Hard
What is
lim
x
→
1
f
(
x
)
1
/
g
(
x
)
\lim_{x\to1}f(x)^{1/g(x)}
lim
x
→
1
f
(
x
)
1/
g
(
x
)
equal to?
Add
NDA Maths strategy
A
a
b
ab
ab
B
a
b
\sqrt{ab}
ab
C
2
a
b
2\sqrt{ab}
2
ab
D
a
b
2
\sqrt{ab}^2
ab
2
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[Q54 · Sep · 2023]
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Set · 2 questions
Let
f
(
x
)
=
2
−
x
+
2
+
x
f(x)=\sqrt{2-x}+\sqrt{2+x}
f
(
x
)
=
2
−
x
+
2
+
x
.
Q4435
#4435
Mathematics → Functions → Domain, Range, and Function Properties
·
Easy
What is the domain of the function?
Add
NDA Maths strategy
A
(-2,2)
B
[-2,2]
C
R
−
(
−
2
,
2
)
\mathbb{R}-(-2,2)
R
−
(
−
2
,
2
)
D
R
−
[
−
2
,
2
]
\mathbb{R}-[-2,2]
R
−
[
−
2
,
2
]
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[Q55 · Sep · 2023]
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Q4436
#4436
Mathematics → Functions → Domain, Range, and Function Properties
·
Moderate
What is the greatest value of the function?
Add
NDA Maths strategy
A
3
\sqrt{3}
3
B
6
\sqrt{6}
6
C
8
\sqrt{8}
8
D
4
\sqrt{4}
4
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[Q56 · Sep · 2023]
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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
))
.
Q4437
#4437
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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Q4438
#4438
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]
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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
.
Q4439
#4439
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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Q4440
#4440
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]
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Set · 2 questions
Let
I
=
∫
−
2
π
2
π
sin
4
x
+
cos
4
x
1
+
3
x
d
x
I=\int_{-2\pi}^{2\pi}\frac{\sin^4x+\cos^4x}{1+3^x}\,dx
I
=
∫
−
2
π
2
π
1
+
3
x
s
i
n
4
x
+
c
o
s
4
x
d
x
.
Q4441
#4441
Mathematics → Definite Integration → Properties of Definite Integrals — Symmetry, King's, Odd/Even
·
Moderate
What is
∫
0
π
(
sin
4
x
+
cos
4
x
)
d
x
\int_0^\pi(\sin^4x+\cos^4x)\,dx
∫
0
π
(
sin
4
x
+
cos
4
x
)
d
x
equal to?
Add
NDA Maths strategy
A
3
π
8
\frac{3\pi}{8}
8
3
π
B
3
π
4
\frac{3\pi}{4}
4
3
π
C
3
π
2
\frac{3\pi}{2}
2
3
π
D
3
π
3\pi
3
π
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[Q61 · Sep · 2023]
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Q4442
#4442
Mathematics → Definite Integration → Properties of Definite Integrals — Symmetry, King's, Odd/Even
·
Hard
What is
I
I
I
equal to?
Add
NDA Maths strategy
A
0
B
3
π
4
\frac{3\pi}{4}
4
3
π
C
3
π
2
\frac{3\pi}{2}
2
3
π
D
3
π
3\pi
3
π
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[Q62 · Sep · 2023]
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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
.
Q4443
#4443
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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Q4444
#4444
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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[Q64 · Sep · 2023]
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Q4445
#4445
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
Tap an option to check your answer.
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[Q65 · Sep · 2023]
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Q4446
#4446
Mathematics → Differentiation → Differentiation Techniques — Chain Rule, Logarithmic, Composite Functions
·
Moderate
If
f
′
(
x
)
=
cos
(
ln
x
)
f'(x)=\cos(\ln x)
f
′
(
x
)
=
cos
(
ln
x
)
and
y
=
f
(
2
x
−
3
x
)
y=f\!\left(\frac{2x-3}{x}\right)
y
=
f
(
x
2
x
−
3
)
, then what is
d
y
d
x
\frac{dy}{dx}
d
x
d
y
equal to?
Add
NDA Maths strategy
A
cos
(
ln
2
x
−
3
x
)
\cos\!\left(\ln\frac{2x-3}{x}\right)
cos
(
ln
x
2
x
−
3
)
B
−
3
x
2
sin
(
ln
2
x
−
3
x
)
-\frac{3}{x^2}\sin\!\left(\ln\frac{2x-3}{x}\right)
−
x
2
3
sin
(
ln
x
2
x
−
3
)
C
3
x
2
cos
(
ln
2
x
−
3
x
)
\frac{3}{x^2}\cos\!\left(\ln\frac{2x-3}{x}\right)
x
2
3
cos
(
ln
x
2
x
−
3
)
D
−
3
x
2
cos
(
ln
2
x
−
3
x
)
-\frac{3}{x^2}\cos\!\left(\ln\frac{2x-3}{x}\right)
−
x
2
3
cos
(
ln
x
2
x
−
3
)
Tap an option to check your answer.
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[Q66 · Sep · 2023]
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Q4447
#4447
Mathematics → Definite Integration → Fundamental Theorem, Periodic Integrals, and Leibniz Rule
·
Easy
What is
∫
0
8
π
∣
sin
x
∣
d
x
\int_0^{8\pi}|\sin x|\,dx
∫
0
8
π
∣
sin
x
∣
d
x
equal to?
Add
NDA Maths strategy
A
2
B
4
C
8
D
16
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[Q67 · Sep · 2023]
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Q4448
#4448
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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Q4449
#4449
Mathematics → Differential Equations → Order, Degree, and Solutions of ODE
·
Easy
What are the order and degree respectively of
x
2
(
d
3
y
d
x
3
)
2
+
(
d
y
d
x
)
4
+
sin
x
=
0
x^2\!\left(\frac{d^3y}{dx^3}\right)^2+\left(\frac{dy}{dx}\right)^4+\sin x=0
x
2
(
d
x
3
d
3
y
)
2
+
(
d
x
d
y
)
4
+
sin
x
=
0
?
Add
NDA Maths strategy
A
3 and 4
B
1 and 4
C
2 and 2
D
3 and 2
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[Q69 · Sep · 2023]
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Q4450
#4450
Mathematics → Differential Equations → Formation of ODE from Curves and General Solutions
·
Moderate
What is the differential equation of all parabolas of the type
y
2
=
4
a
(
x
−
b
)
y^2=4a(x-b)
y
2
=
4
a
(
x
−
b
)
?
Add
NDA Maths strategy
A
y
d
2
y
d
x
2
+
(
d
y
d
x
)
2
=
0
y\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^2=0
y
d
x
2
d
2
y
+
(
d
x
d
y
)
2
=
0
— wait, re-examine
B
d
2
y
d
x
2
+
x
2
(
d
y
d
x
)
2
=
0
\frac{d^2y}{dx^2}+x^2\left(\frac{dy}{dx}\right)^2=0
d
x
2
d
2
y
+
x
2
(
d
x
d
y
)
2
=
0
C
y
2
d
2
y
d
x
2
+
(
d
y
d
x
)
2
=
0
y^2\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^2=0
y
2
d
x
2
d
2
y
+
(
d
x
d
y
)
2
=
0
D
y
d
2
y
d
x
2
+
(
d
y
d
x
)
2
=
0
y\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^2=0
y
d
x
2
d
2
y
+
(
d
x
d
y
)
2
=
0
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[Q70 · Sep · 2023]
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