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Q201
#201
Maths → Complex Numbers → Geometry of Complex Numbers
·
Moderate
Add
A point
z
z
z
moves in the complex plane such that
a
r
g
(
z
−
2
z
+
2
)
=
π
4
arg\left( \frac{z - 2}{z + 2} \right)=\frac{\pi}{4}
a
r
g
(
z
+
2
z
−
2
)
=
4
π
, then the minimum value of
∣
z
−
9
2
−
2
i
∣
2
|z- 9\sqrt{2}- 2i|^{2}
∣
z
−
9
2
−
2
i
∣
2
is equal to
Show answer
[Q82 · Paper 22 · 2021]
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Q202
#202
Maths → Three Dimensional Geometry → Line and Plane
·
Moderate
Add
line
x
−
1
2
=
y
−
2
3
=
z
+
1
6
\frac{x- 1}{2}=\frac{y- 2}{3}=\frac{z+ 1}{6}
2
x
−
1
=
3
y
−
2
=
6
z
+
1
and the plane
2
x
−
y
+
z
=
6
2x - y + z = 6
2
x
−
y
+
z
=
6
from the point
(
−
1
,
−
1
,
2
)
( - 1, - 1,2)
(
−
1
,
−
1
,
2
)
is
Show answer
[Q83 · Paper 22 · 2021]
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Q203
#203
Maths → Application of Derivatives → Increasing and Decreasing Functions
·
Moderate
Add
If '
R
R
R
' is the least value of '
a
a
a
' such that the function
f
(
x
)
=
x
2
+
a
x
+
1
f(x) =x^{2}+ax+ 1
f
(
x
)
=
x
2
+
a
x
+
1
is increasing on [1, 2] and 'S' is the greatest value of '
a
a
a
' such that the function
f
(
x
)
=
x
2
+
a
x
+
1
f(x) =x^{2}+ax+ 1
f
(
x
)
=
x
2
+
a
x
+
1
is decreasing on
[
1
,
2
]
\lbrack 1,2\rbrack
[
1
,
2
]
, then the value of
∣
R
−
S
∣
|R - S|
∣
R
−
S
∣
is
Show answer
[Q84 · Paper 22 · 2021]
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Q204
#204
Maths → Sequences and Series → Special Sums
·
Moderate
Add
The mean of 10 numbers
7
×
8
,
10
×
10
,
13
×
12
,
16
×
14
,
…
7 \times 8,10 \times 10,13 \times 12,16 \times 14,\ldots
7
×
8
,
10
×
10
,
13
×
12
,
16
×
14
,
…
. Is
Show answer
[Q85 · Paper 22 · 2021]
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Q205
#205
Maths → Conic Sections → Circle
·
Moderate
Add
If the variable line
3
x
+
4
y
=
α
3x + 4y = \alpha
3
x
+
4
y
=
α
lies between the two circles
(
x
−
1
)
2
+
(
y
−
1
)
2
=
1
(x - 1)^{2}+ (y - 1)^{2}= 1
(
x
−
1
)
2
+
(
y
−
1
)
2
=
1
and
(
x
−
9
)
2
+
(
y
−
1
)
2
=
4
(x - 9)^{2}+ (y - 1)^{2}= 4
(
x
−
9
)
2
+
(
y
−
1
)
2
=
4
, without intercepting a chord on either circle, then the sum of all the integral values of
α
\alpha
α
is
Show answer
[Q86 · Paper 22 · 2021]
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Q206
#206
Maths → Permutations and Combinations → Combinations
·
Moderate
Add
The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is
Show answer
[Q87 · Paper 22 · 2021]
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Q207
#207
Maths → Integrals → Fundamental Theorem of Calculus
·
Moderate
Add
If
x
ϕ
(
x
)
=
∫
5
x
(
3
t
2
−
2
ϕ
′
(
t
)
)
d
t
,
x
>
−
2
x\phi(x) =\int_{5}^{x} \left( 3t^{2}- 2\phi^{'}(t) \right)dt,x> - 2
x
ϕ
(
x
)
=
∫
5
x
(
3
t
2
−
2
ϕ
′
(
t
)
)
d
t
,
x
>
−
2
, and
ϕ
(
0
)
=
4
\phi(0) = 4
ϕ
(
0
)
=
4
, then
ϕ
(
2
)
\phi(2)
ϕ
(
2
)
is
Show answer
[Q88 · Paper 22 · 2021]
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Q208
#208
Maths → Binomial Theorem → General Term and Irrational Terms
·
Moderate
Add
If
(
3
6
4
4
)
k
\left( \frac{3^{6}}{4^{4}} \right)k
(
4
4
3
6
)
k
is the term, independent of
x
x
x
, in the binomial expansion of
(
x
4
−
12
x
2
)
12
\left( \frac{x}{4}-\frac{12}{x^{2}} \right)^{12}
(
4
x
−
x
2
12
)
12
, then
k
k
k
is equal to
Show answer
[Q89 · Paper 22 · 2021]
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Q209
#209
Maths → Probability → Independent Events
·
Moderate
Add
An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8 . The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is p, then
98
p
98p
98
p
is equal to
Show answer
[Q90 · Paper 22 · 2021]
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Q210
#210
Maths → Trigonometry → Trigonometric Equations
·
Moderate
Add
Let
S
S
S
be the sum of all solutions (in radians) of the equation
sin
4
θ
+
cos
4
θ
−
sin
θ
cos
θ
=
0
\sin^{4}\theta+\cos^{4}\theta- \sin\theta \cos\theta= 0
sin
4
θ
+
cos
4
θ
−
sin
θ
cos
θ
=
0
in
[
0
,
4
π
]
\lbrack 0,4\pi\rbrack
[
0
,
4
π
]
. Then
8
S
π
\frac{8\text{ }S}{\pi}
π
8
S
is equal to
Show answer
[Q81 · Paper 21 · 2021]
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Q211
#211
Maths → Three Dimensional Geometry → The Plane
·
Moderate
Add
Let
S
S
S
be the mirror image of the point
Q
(
1
,
3
,
4
)
Q(1,3,4)
Q
(
1
,
3
,
4
)
with respect to the plane
2
x
−
y
+
z
+
3
=
0
2x - y + z + 3 = 0
2
x
−
y
+
z
+
3
=
0
and let
R
(
3
,
5
,
γ
)
R(3,5,\gamma)
R
(
3
,
5
,
γ
)
be a point of this plane. Then the square of the length of the line segment
S
R
SR
S
R
is
Show answer
[Q82 · Paper 21 · 2021]
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Q212
#212
Maths → Probability → Classical Probability
·
Moderate
Add
The probability distribution of random variable
X
X
X
is given by: ----------------------------------------------------------------------------------------------------------------------------------------------------------------------
X
X
X
1 2 3 4 5 ------------------- ---------------- ---------------------------------- ---------------------------------- ---------------------------------- ------------------------
P
(
X
)
P(X)
P
(
X
)
K
K
K
2
K
2\text{ }K
2
K
2
K
2\text{ }K
2
K
3
K
3\text{ }K
3
K
K
\text{ }K
K
---------------------------------------------------------------------------------------------------------------------------------------------------------------------- Let
p
=
P
(
1
<
X
<
4
∣
X
<
3
)
p = P(1 < X < 4 \mid X < 3)
p
=
P
(
1
<
X
<
4
∣
X
<
3
)
. If
5
p
=
λ
K
5p =\lambda K
5
p
=
λ
K
, then
λ
\lambda
λ
equal to
Show answer
[Q83 · Paper 21 · 2021]
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Q213
#213
Maths → Complex Numbers → Geometry of Complex Numbers
·
Moderate
Add
Let
z
1
z_{1}
z
1
and
z
2
z_{2}
z
2
be two complex numbers such that
a
r
g
(
z
1
−
z
2
)
=
π
4
arg\left( z_{1}-z_{2} \right)=\frac{\pi}{4}
a
r
g
(
z
1
−
z
2
)
=
4
π
and
z
1
,
z
2
z_{1},z_{2}
z
1
,
z
2
satisfy the equation
∣
z
−
3
∣
=
R
e
(
z
)
|z- 3| = Re(z)
∣
z
−
3∣
=
R
e
(
z
)
. Then the imaginary part of
z
1
+
z
2
z_{1}+z_{2}
z
1
+
z
2
is equal to
Show answer
[Q84 · Paper 21 · 2021]
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Q214
#214
Maths → Permutations and Combinations → Combinations
·
Moderate
Add
Let
S
=
{
1
,
2
,
3
,
4
,
5
,
6
,
9
}
S\mathbf{= \{ 1,2,3,4,5,6,9\}}
S
=
{
1
,
2
,
3
,
4
,
5
,
6
,
9
}
. Then the number of elements in the set
T
=
{
A
⊆
S
:
A
≠
ϕ
\mathbf{T = \{ A \subseteq S:A \neq}\phi
T
=
{
A
⊆
S
:
A
=
ϕ
and the sum of all the elements of
A
A
A
is not a multiple of 3
}
\}
}
is
Show answer
[Q85 · Paper 21 · 2021]
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Q215
#215
Maths → Conic Sections → Hyperbola
·
Moderate
Add
Let
A
(
sec
θ
,
2
tan
θ
)
A(\sec\theta,\ 2\tan\theta)
A
(
sec
θ
,
2
tan
θ
)
and
B
(
sec
ϕ
,
2
tan
ϕ
)
B(\sec\phi,\ 2\tan\phi)
B
(
sec
ϕ
,
2
tan
ϕ
)
, where
θ
+
ϕ
=
π
2
\theta+\phi=\dfrac{\pi}{2}
θ
+
ϕ
=
2
π
, be two points on the hyperbola
2
x
2
−
y
2
=
2
2x^{2}-y^{2}=2
2
x
2
−
y
2
=
2
. If
(
α
,
β
)
(\alpha,\beta)
(
α
,
β
)
is the point of intersection of the normals to the hyperbola at
A
A
A
and
B
B
B
, then
(
2
β
)
2
(2\beta)^{2}
(
2
β
)
2
is equal to ____.
Show answer
[Q86 · Paper 21 · 2021]
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Q216
#216
Maths → Conic Sections → Circle
·
Moderate
Add
Two circles each of radius 5 units touch each other at the point
(
1
,
2
)
(1,2)
(
1
,
2
)
. If the equation of their common tangent is
4
x
+
3
y
=
10
4x + 3y = 10
4
x
+
3
y
=
10
, and
C
1
(
α
,
β
)
C_{1}(\alpha,\beta)
C
1
(
α
,
β
)
and
C
2
(
γ
,
δ
)
C_{2}(\gamma,\delta)
C
2
(
γ
,
δ
)
,
C
1
≠
C
2
C_{1}\neq C_{2}
C
1
=
C
2
are their centres, then
∣
(
α
+
β
)
(
γ
+
δ
)
∣
|(\alpha + \beta)(\gamma + \delta)|
∣
(
α
+
β
)
(
γ
+
δ
)
∣
is equal to
Show answer
[Q87 · Paper 21 · 2021]
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Q217
#217
Maths → Binomial Theorem → Binomial Coefficients and Series
·
Moderate
Add
3
×
7
22
+
2
×
10
22
−
44
3 \times7^{22}+ 2 \times10^{22}- 44
3
×
7
22
+
2
×
1
0
22
−
44
when divided by 18 leaves the remainder
Show answer
[Q88 · Paper 21 · 2021]
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Q218
#218
Maths → Statistics → Standard Deviation
·
Moderate
Add
An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2 . The variance of marks obtained by 30 girls is also 2 . The average marks of all 50 candidates is 15. If
μ
\mu
μ
is the average marks of girls and
σ
2
\sigma^{2}
σ
2
is the variance of marks of 50 candidates, then
μ
+
σ
2
\mu+\sigma^{2}
μ
+
σ
2
is equal to
Show answer
[Q89 · Paper 21 · 2021]
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Q219
#219
Maths → Integrals → Integration by Substitution
·
Moderate
Add
If
∫
2
e
x
+
3
e
−
x
4
e
x
+
7
e
−
x
d
x
=
1
14
(
u
x
+
v
log
e
(
4
e
x
+
7
e
−
x
)
)
+
C
\int\frac{2e^{x}+ 3e^{-x}}{4e^{x}+ 7e^{-x}}dx=\frac{1}{14}\left( ux+v\log_{e}\left( 4e^{x}+ 7e^{-x} \right) \right)+C
∫
4
e
x
+
7
e
−
x
2
e
x
+
3
e
−
x
d
x
=
14
1
(
ux
+
v
lo
g
e
(
4
e
x
+
7
e
−
x
)
)
+
C
, where
C
C
C
is a constant of integration, then
u
+
v
u+v
u
+
v
is equal to
Show answer
[Q90 · Paper 21 · 2021]
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Q220
#220
Maths → Vector Algebra → Dot Product and Projection
·
Moderate
Add
Let
a
→
=
i
^
+
5
j
^
+
α
k
^
,
b
→
=
i
^
+
3
j
^
+
β
k
^
\overrightarrow{a}=\widehat{i}+ 5\widehat{j}+\alpha\widehat{k},\overrightarrow{b}=\widehat{i}+ 3\widehat{j}+\beta\widehat{k}
a
=
i
+
5
j
+
α
k
,
b
=
i
+
3
j
+
β
k
and
c
→
=
−
i
^
+
2
j
^
−
3
k
^
\overrightarrow{c}= -\widehat{i}+ 2\widehat{j}- 3\widehat{k}
c
=
−
i
+
2
j
−
3
k
be three vectors such that,
∣
b
→
×
c
→
∣
=
5
3
|\overrightarrow{b}\times\overrightarrow{c}| = 5\sqrt{3}
∣
b
×
c
∣
=
5
3
and
a
→
\overrightarrow{a}
a
is perpendicular to
b
→
\overrightarrow{b}
b
. Then the greatest amongst the values of
∣
a
→
∣
2
|\overrightarrow{a}|^{2}
∣
a
∣
2
is
Show answer
[Q81 · Paper 20 · 2021]
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Q221
#221
Maths → Application of Derivatives → Maxima and Minima
·
Moderate
Add
The number of distinct real roots of the equation
3
x
4
+
4
x
3
−
12
x
2
+
4
=
0
3x^{4}+ 4x^{3}- 12x^{2}+ 4 = 0
3
x
4
+
4
x
3
−
12
x
2
+
4
=
0
is
Show answer
[Q82 · Paper 20 · 2021]
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Q222
#222
Maths → Conic Sections → Circle
·
Moderate
Add
Let the equation
x
2
+
y
2
+
p
x
+
(
1
−
p
)
y
+
5
=
0
x^{2}+y^{2}+px+ (1 -p)y+ 5 = 0
x
2
+
y
2
+
p
x
+
(
1
−
p
)
y
+
5
=
0
represent circles of varying radius
r
∈
(
0
,
5
]
r\in (0,5\rbrack
r
∈
(
0
,
5
]
. Then the number of elements in the set
S
=
{
q
:
q
=
p
2
and
q
is an integer
}
S=\{q : q=p^{2} \text{ and } q \text{ is an integer}\}
S
=
{
q
:
q
=
p
2
and
q
is an integer
}
is:
Show answer
[Q83 · Paper 20 · 2021]
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Q223
#223
Maths → Relations and Functions → Sets and Venn Diagrams
·
Moderate
Add
If
A
=
{
x
∈
R
:
∣
x
−
2
∣
>
1
}
,
B
=
{
x
∈
R
:
x
2
−
3
>
1
}
\mathbf{A = \{ x \in R:|x - 2| > 1\},B =}\left\{ x \in R:\sqrt{x^{2}- 3}> 1 \right\}
A
=
{
x
∈
R
:
∣x
−
2∣
>
1
}
,
B
=
{
x
∈
R
:
x
2
−
3
>
1
}
,
C
=
{
x
∈
R
:
∣
x
−
4
∣
≥
2
}
C\mathbf{= \{}x\in R:|x- 4| \geq 2\}
C
=
{
x
∈
R
:
∣
x
−
4∣
≥
2
}
and
Z
Z
Z
is the set of all integers, then the number of subsets of the set
(
A
∩
B
∩
C
)
C
∩
Z
(A \cap B \cap C)^{C}\cap Z
(
A
∩
B
∩
C
)
C
∩
Z
is
Show answer
[Q84 · Paper 20 · 2021]
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Q224
#224
Maths → Integrals → Integration by Substitution
·
Moderate
Add
If
∫
d
x
(
x
2
+
x
+
1
)
2
=
a
tan
−
1
(
2
x
+
1
3
)
+
b
(
2
x
+
1
x
2
+
x
+
1
)
+
C
\int\frac{dx}{\left( x^{2}+x+ 1 \right)^{2}}=a\tan^{- 1}\left( \frac{2x+ 1}{\sqrt{3}} \right)+b\left( \frac{2x+ 1}{x^{2}+x+ 1} \right)+C
∫
(
x
2
+
x
+
1
)
2
d
x
=
a
tan
−
1
(
3
2
x
+
1
)
+
b
(
x
2
+
x
+
1
2
x
+
1
)
+
C
,
x
>
0
x > 0
x
>
0
where
C
C
C
is the constant of integration, then the value of
9
(
3
a
+
b
)
9(\sqrt{3}a+b)
9
(
3
a
+
b
)
is equal to
Show answer
[Q85 · Paper 20 · 2021]
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Q225
#225
Maths → Determinants → System of Linear Equations
·
Moderate
Add
If the system of linear equations
2
x
+
y
−
z
=
3
2x + y - z = 3
2
x
+
y
−
z
=
3
x
−
y
−
z
=
α
x - y - z =\alpha
x
−
y
−
z
=
α
3
x
+
3
y
+
β
z
=
3
3x+ 3y+\beta z= 3
3
x
+
3
y
+
β
z
=
3
has infinitely many solution, then
α
+
β
−
α
β
\alpha+\beta-\alpha\beta
α
+
β
−
α
β
is equal to
Show answer
[Q86 · Paper 20 · 2021]
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