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Q4401
#4401
Mathematics → Sets & Relations → Relations — Properties, Cartesian Product, and Counting
·
Moderate
The Cartesian product
A
×
A
A\times A
A
×
A
has 16 elements among which are
(
0
,
2
)
(0,2)
(
0
,
2
)
and
(
1
,
3
)
(1,3)
(
1
,
3
)
. Which of the following statements is/are correct? (1) It is possible to determine set
A
A
A
. (2)
A
×
A
A\times A
A
×
A
contains the element
(
3
,
2
)
(3,2)
(
3
,
2
)
.
Add
NDA Maths strategy
A
1 only
B
2 only
C
Both 1 and 2
D
Neither 1 nor 2
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[Q21 · Sep · 2023]
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Q4402
#4402
Mathematics → Sets & Relations → Relations — Properties, Cartesian Product, and Counting
·
Moderate
Let
A
=
{
1
,
2
,
…
,
20
}
A=\{1,2,\ldots,20\}
A
=
{
1
,
2
,
…
,
20
}
. Define
R
=
{
(
x
,
y
)
:
4
x
−
3
y
=
1
}
,
x
,
y
∈
A
R=\{(x,y):4x-3y=1\},\ x,y\in A
R
=
{(
x
,
y
)
:
4
x
−
3
y
=
1
}
,
x
,
y
∈
A
. Which statements are correct? (1) Domain of
R
R
R
is
{
1
,
4
,
7
,
10
,
13
,
16
}
\{1,4,7,10,13,16\}
{
1
,
4
,
7
,
10
,
13
,
16
}
. (2) Range of
R
R
R
is
{
1
,
5
,
9
,
13
,
17
}
\{1,5,9,13,17\}
{
1
,
5
,
9
,
13
,
17
}
. (3) Range of
R
R
R
equals codomain.
Add
NDA Maths strategy
A
1 only
B
2 only
C
1 and 2
D
2 and 3
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[Q22 · Sep · 2023]
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Q4403
#4403
Mathematics → Functions → Function Definition and Classification — Injectivity, Surjectivity, Bijectivity
·
Moderate
Consider the statements: (1)
f
(
x
)
=
x
3
,
0
≤
x
≤
2
f(x)=x^3,\ 0\le x\le2
f
(
x
)
=
x
3
,
0
≤
x
≤
2
and
4
x
,
2
≤
x
≤
8
4x,\ 2\le x\le8
4
x
,
2
≤
x
≤
8
is a function. (2)
g
(
x
)
=
x
2
,
0
≤
x
≤
4
g(x)=x^2,\ 0\le x\le4
g
(
x
)
=
x
2
,
0
≤
x
≤
4
and
3
x
,
4
≤
x
≤
8
3x,\ 4\le x\le8
3
x
,
4
≤
x
≤
8
is a function. Which is/are correct?
Add
NDA Maths strategy
A
1 only
B
2 only
C
Both 1 and 2
D
Neither 1 nor 2
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[Q23 · Sep · 2023]
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Q4404
#4404
Mathematics → Sets & Relations → Set Operations, Identities, and Cartesian Products of Sets
·
Hard
Consider: (1)
A
=
(
A
∪
B
)
∪
(
A
−
B
)
A=(A\cup B)\cup(A-B)
A
=
(
A
∪
B
)
∪
(
A
−
B
)
. (2)
A
∪
(
B
−
A
)
=
(
A
∪
B
)
A\cup(B-A)=(A\cup B)
A
∪
(
B
−
A
)
=
(
A
∪
B
)
. (3)
B
=
(
A
∪
B
)
−
(
A
−
B
)
B=(A\cup B)-(A-B)
B
=
(
A
∪
B
)
−
(
A
−
B
)
. Which are correct?
Add
NDA Maths strategy
A
1 and 2 only
B
2 and 3 only
C
1 and 3 only
D
1, 2 and 3
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[Q24 · Sep · 2023]
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Q4405
#4405
Mathematics → Functions → Functional Equations
·
Moderate
A function satisfies
f
(
x
−
y
)
=
f
(
x
)
f
(
y
)
f(x-y)=f(x)f(y)
f
(
x
−
y
)
=
f
(
x
)
f
(
y
)
, where
f
(
y
)
≠
0
f(y)\neq0
f
(
y
)
=
0
. If
f
(
1
)
=
0.5
f(1)=0.5
f
(
1
)
=
0.5
, then what is
f
(
2
)
+
f
(
3
)
+
f
(
4
)
+
f
(
5
)
+
f
(
6
)
f(2)+f(3)+f(4)+f(5)+f(6)
f
(
2
)
+
f
(
3
)
+
f
(
4
)
+
f
(
5
)
+
f
(
6
)
equal to?
Add
NDA Maths strategy
A
15
32
\frac{15}{32}
32
15
B
17
32
\frac{17}{32}
32
17
C
29
64
\frac{29}{64}
64
29
D
31
64
\frac{31}{64}
64
31
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[Q25 · Sep · 2023]
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Q4406
#4406
Mathematics → Inverse Trigonometry → Evaluation of Composite Inverse Trigonometric Expressions
·
Hard
What is
2
cot
(
1
2
cos
−
1
5
3
)
2\cot\!\left(\tfrac{1}{2}\cos^{-1}\!\tfrac{5}{3}\right)
2
cot
(
2
1
cos
−
1
3
5
)
equal to?
Add
NDA Maths strategy
A
-1
B
1
C
3
+
5
\sqrt{3}+\sqrt{5}
3
+
5
D
3
−
5
\sqrt{3}-\sqrt{5}
3
−
5
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[Q26 · Sep · 2023]
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Q4407
#4407
Mathematics → Inverse Trigonometry → Identities, Properties, and Sum-Difference Formulas
·
Moderate
If
sec
−
1
p
−
csc
−
1
q
=
0
\sec^{-1}p - \csc^{-1}q = 0
sec
−
1
p
−
csc
−
1
q
=
0
, where
p
>
0
,
q
>
0
p>0,q>0
p
>
0
,
q
>
0
, then what is the value of
p
−
2
+
q
−
2
p^{-2}+q^{-2}
p
−
2
+
q
−
2
?
Add
NDA Maths strategy
A
1
B
2
C
1
2
\frac{1}{2}
2
1
D
1
2
2
\frac{1}{2\sqrt{2}}
2
2
1
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[Q27 · Sep · 2023]
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Q4408
#4408
Mathematics → Inverse Trigonometry → Evaluation of Composite Inverse Trigonometric Expressions
·
Hard
What is
1
+
sin
(
2
cos
−
1
3
17
)
\sqrt{1+\sin\!\left(2\cos^{-1}\!\frac{3}{\sqrt{17}}\right)}
1
+
sin
(
2
cos
−
1
17
3
)
equal to?
Add
NDA Maths strategy
A
25
17
\frac{25}{17}
17
25
B
8
17
\frac{8}{17}
17
8
C
9
17
\frac{9}{17}
17
9
D
47
17
\frac{47}{17}
17
47
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[Q28 · Sep · 2023]
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Q4409
#4409
Mathematics → Trigonometric Equations → General Solutions and Counting Solutions of Trigonometric Equations
·
Hard
If
tan
(
π
cos
θ
)
=
cot
(
π
sin
θ
)
,
0
<
θ
<
π
/
2
\tan(\pi\cos\theta)=\cot(\pi\sin\theta),\ 0<\theta<\pi/2
tan
(
π
cos
θ
)
=
cot
(
π
sin
θ
)
,
0
<
θ
<
π
/2
; then what is the value of
8
sin
2
(
θ
+
π
4
)
8\sin^2\!\left(\theta+\frac{\pi}{4}\right)
8
sin
2
(
θ
+
4
π
)
?
Add
NDA Maths strategy
A
16
B
2
C
1
D
1
2
\frac{1}{2}
2
1
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[Q29 · Sep · 2023]
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Q4410
#4410
Mathematics → Trigonometric Identities → Compound Angle Formulas
·
Hard
If
tan
α
=
1
7
,
sin
β
=
1
10
;
0
<
α
,
β
<
π
2
\tan\alpha=\frac{1}{7},\ \sin\beta=\frac{1}{\sqrt{10}};\ 0<\alpha,\beta<\frac{\pi}{2}
tan
α
=
7
1
,
sin
β
=
10
1
;
0
<
α
,
β
<
2
π
, then what is the value of
cos
(
α
+
2
β
)
\cos(\alpha+2\beta)
cos
(
α
+
2
β
)
?
Add
Lever: Compound angle: sin/cos/tan(A ± B)
A
−
1
2
-\frac{1}{\sqrt{2}}
−
2
1
B
−
1
2
-\frac{1}{2}
−
2
1
C
1
2
\frac{1}{2}
2
1
D
1
2
\frac{1}{\sqrt{2}}
2
1
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[Q30 · Sep · 2023]
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Set · 2 questions
Consider the equation
(
1
−
x
)
4
+
(
5
−
x
)
4
=
82
(1-x)^4+(5-x)^4=82
(
1
−
x
)
4
+
(
5
−
x
)
4
=
82
.
Q4411
#4411
Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Moderate
What is the number of real roots of the equation?
Add
Lever: Vieta — sum and product of roots
A
0
B
2
C
4
D
8
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[Q31 · Sep · 2023]
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Q4412
#4412
Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Moderate
What is the sum of all the roots of the equation?
Add
Lever: Vieta — sum and product of roots
A
24
B
12
C
10
D
6
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[Q32 · Sep · 2023]
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Set · 3 questions
Consider equation-I:
z
3
+
2
z
2
+
2
z
+
1
=
0
z^3+2z^2+2z+1=0
z
3
+
2
z
2
+
2
z
+
1
=
0
and equation-II:
z
1985
+
z
100
+
1
=
0
z^{1985}+z^{100}+1=0
z
1985
+
z
100
+
1
=
0
.
Q4413
#4413
Mathematics → Complex Numbers → Cube Roots of Unity
·
Moderate
What are the roots of equation-I?
Add
Lever: Cube roots of unity (1 + ω + ω² = 0, ω³ = 1)
A
1
,
ω
,
ω
2
1,\omega,\omega^2
1
,
ω
,
ω
2
B
−
1
,
ω
,
ω
2
-1,\omega,\omega^2
−
1
,
ω
,
ω
2
C
1
,
−
ω
,
ω
2
1,-\omega,\omega^2
1
,
−
ω
,
ω
2
D
−
1
,
−
ω
,
−
ω
2
-1,-\omega,-\omega^2
−
1
,
−
ω
,
−
ω
2
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[Q33 · Sep · 2023]
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Q4414
#4414
Mathematics → Complex Numbers → Cube Roots of Unity
·
Moderate
Which one of the following is a root of equation-II?
Add
Lever: Cube roots of unity (1 + ω + ω² = 0, ω³ = 1)
A
−
1
-1
−
1
B
−
ω
-\omega
−
ω
C
−
ω
2
-\omega^2
−
ω
2
D
ω
\omega
ω
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[Q34 · Sep · 2023]
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Q4415
#4415
Mathematics → Complex Numbers → Cube Roots of Unity
·
Moderate
What is the number of common roots of equation-I and equation-II?
Add
Lever: Cube roots of unity (1 + ω + ω² = 0, ω³ = 1)
A
0
B
1
C
2
D
3
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[Q35 · Sep · 2023]
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Set · 2 questions
A quadratic equation is given by
(
a
+
b
)
x
2
−
(
a
+
b
+
c
)
x
+
k
=
0
(a+b)x^2-(a+b+c)x+k=0
(
a
+
b
)
x
2
−
(
a
+
b
+
c
)
x
+
k
=
0
, where
a
,
b
,
c
a,b,c
a
,
b
,
c
are real.
Q4416
#4416
Mathematics → Quadratic Equations → Special Quadratics — Parametric, Logarithmic, Constructed
·
Hard
If
k
=
c
/
2
(
c
≠
0
)
k=c/2\ (c\neq0)
k
=
c
/2
(
c
=
0
)
, then the roots of the equation are:
Add
NDA Maths strategy
A
Real and equal
B
Real and unequal
C
Real iff
a
>
c
a>c
a
>
c
D
Complex but not real
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[Q36 · Sep · 2023]
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Q4417
#4417
Mathematics → Quadratic Equations → Special Quadratics — Parametric, Logarithmic, Constructed
·
Moderate
If
k
=
c
k=c
k
=
c
, then the roots of the equation are:
Add
NDA Maths strategy
A
a
+
c
a
+
b
\frac{a+c}{a+b}
a
+
b
a
+
c
and
b
a
+
b
\frac{b}{a+b}
a
+
b
b
B
a
+
c
a
+
b
\frac{a+c}{a+b}
a
+
b
a
+
c
and
−
b
a
+
b
\frac{-b}{a+b}
a
+
b
−
b
C
1
1
1
and
c
a
+
b
\frac{c}{a+b}
a
+
b
c
D
−
1
-1
−
1
and
−
c
a
+
b
\frac{-c}{a+b}
a
+
b
−
c
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[Q37 · Sep · 2023]
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Set · 3 questions
Let
(
1
+
x
)
n
=
1
+
T
1
x
+
T
2
x
2
+
⋯
+
T
n
x
n
(1+x)^n = 1 + T_1 x + T_2 x^2 + \cdots + T_n x^n
(
1
+
x
)
n
=
1
+
T
1
x
+
T
2
x
2
+
⋯
+
T
n
x
n
.
Q4418
#4418
Mathematics → Binomial Theorem → Sums of Binomial Coefficients — Alternating, Weighted, and Symmetric
·
Moderate
What is
T
1
+
2
T
2
+
3
T
3
+
⋯
+
n
T
n
T_1+2T_2+3T_3+\cdots+nT_n
T
1
+
2
T
2
+
3
T
3
+
⋯
+
n
T
n
equal to?
Add
Lever: Pascal / binomial-coefficient identities
A
0
B
1
C
2
n
2^n
2
n
D
n
⋅
2
n
−
1
n\cdot2^{n-1}
n
⋅
2
n
−
1
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[Q38 · Sep · 2023]
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Q4419
#4419
Mathematics → Binomial Theorem → Sums of Binomial Coefficients — Alternating, Weighted, and Symmetric
·
Moderate
What is
1
−
T
1
+
2
T
2
−
3
T
3
+
⋯
+
(
−
1
)
n
n
T
n
1-T_1+2T_2-3T_3+\cdots+(-1)^n nT_n
1
−
T
1
+
2
T
2
−
3
T
3
+
⋯
+
(
−
1
)
n
n
T
n
equal to?
Add
Lever: Pascal / binomial-coefficient identities
A
0
B
−
2
n
−
1
-2^{n-1}
−
2
n
−
1
C
n
⋅
2
n
−
1
n\cdot2^{n-1}
n
⋅
2
n
−
1
D
1
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[Q39 · Sep · 2023]
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Q4420
#4420
Mathematics → Binomial Theorem → Sums of Binomial Coefficients — Alternating, Weighted, and Symmetric
·
Easy
What is
T
1
+
T
2
+
T
3
+
⋯
+
T
n
T_1+T_2+T_3+\cdots+T_n
T
1
+
T
2
+
T
3
+
⋯
+
T
n
equal to?
Add
Lever: Pascal / binomial-coefficient identities
A
2
n
2^n
2
n
B
2
n
−
1
2^n-1
2
n
−
1
C
2
n
−
1
2^{n-1}
2
n
−
1
D
2
n
+
1
2^n+1
2
n
+
1
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[Q40 · Sep · 2023]
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Set · 2 questions
Let
f
(
x
)
=
x
2
−
1
f(x)=x^2-1
f
(
x
)
=
x
2
−
1
and
g
∘
f
(
x
)
=
x
−
x
+
1
g\circ f(x)=x-\sqrt{x+1}
g
∘
f
(
x
)
=
x
−
x
+
1
.
Q4421
#4421
Mathematics → Functions → Composition and Inverse of Functions
·
Hard
Which one of the following is a possible expression for
g
(
x
)
g(x)
g
(
x
)
?
Add
NDA Maths strategy
A
x
+
1
−
x
+
1
4
\sqrt{x+1}-\sqrt[4]{x+1}
x
+
1
−
4
x
+
1
B
x
+
1
−
x
+
1
4
+
1
\sqrt{x+1}-\sqrt[4]{x+1}+1
x
+
1
−
4
x
+
1
+
1
C
x
+
1
+
x
+
1
4
\sqrt{x+1}+\sqrt[4]{x+1}
x
+
1
+
4
x
+
1
D
x
+
1
−
x
+
1
+
1
\sqrt{x+1}-\sqrt{x+1}+1
x
+
1
−
x
+
1
+
1
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[Q41 · Sep · 2023]
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Q4422
#4422
Mathematics → Functions → Composition and Inverse of Functions
·
Easy
What is
g
(
15
)
g(15)
g
(
15
)
equal to?
Add
NDA Maths strategy
A
1
B
2
C
3
D
4
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[Q42 · Sep · 2023]
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Set · 2 questions
Let
f
f
f
be defined on
R
∖
{
0
}
\mathbb{R}\setminus\{0\}
R
∖
{
0
}
and
2
f
(
x
)
+
f
(
1
x
)
=
x
+
3
2f(x)+f\!\left(\frac{1}{x}\right)=x+3
2
f
(
x
)
+
f
(
x
1
)
=
x
+
3
.
Q4423
#4423
Mathematics → Functions → Functional Equations
·
Moderate
What is
f
(
0.5
)
f(0.5)
f
(
0.5
)
equal to?
Add
NDA Maths strategy
A
1
2
\frac{1}{2}
2
1
B
2
3
\frac{2}{3}
3
2
C
1
D
2
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[Q43 · Sep · 2023]
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Q4424
#4424
Mathematics → Differentiation → Differentiation Techniques — Chain Rule, Logarithmic, Composite Functions
·
Moderate
If
f
f
f
is differentiable, then what is
f
′
(
0.5
)
f'(0.5)
f
′
(
0.5
)
equal to?
Add
NDA Maths strategy
A
1
4
\frac{1}{4}
4
1
B
2
3
\frac{2}{3}
3
2
C
2
D
4
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[Q44 · Sep · 2023]
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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
)
.
Q4425
#4425
Mathematics → Application of Derivatives → Monotonicity, Extrema, and Critical Points
·
Hard
The function is decreasing on:
Add
NDA Maths strategy
A
(
−
28
3
,
0
)
\left(-\frac{28}{3},0\right)
(
−
3
28
,
0
)
B
(
0
,
28
3
)
\left(0,\frac{28}{3}\right)
(
0
,
3
28
)
C
(
0
,
50
3
)
\left(0,\frac{50}{3}\right)
(
0
,
3
50
)
D
(
0
,
56
3
)
\left(0,\frac{56}{3}\right)
(
0
,
3
56
)
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[Q45 · Sep · 2023]
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