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Principle: Vieta — sum and product of roots
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Q1
#1
NDA → Mathematics → Quadratic Equations → Nature of Roots and Boundary Conditions
·
Hard
If the difference between the roots of the equation
x
2
+
k
x
+
1
=
0
x^2 + kx + 1 = 0
x
2
+
k
x
+
1
=
0
is strictly less than
5
\sqrt{5}
5
, where
∣
k
∣
≥
2
|k| \geq 2
∣
k
∣
≥
2
, then
k
k
k
can be any element of the interval
Add
Lever: Vieta — sum and product of roots
A
(
−
3
,
−
2
]
∪
[
2
,
3
)
(-3, -2] \cup [2, 3)
(
−
3
,
−
2
]
∪
[
2
,
3
)
B
(
−
3
,
3
)
(-3, 3)
(
−
3
,
3
)
C
[
−
3
,
−
2
]
∪
[
2
,
3
]
[-3, -2] \cup [2, 3]
[
−
3
,
−
2
]
∪
[
2
,
3
]
D
None of the above
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[Q4 · Apr · 2017]
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Q2
#2
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Moderate
If the roots of the equation
x
2
+
p
x
+
q
=
0
x^2 + px + q = 0
x
2
+
p
x
+
q
=
0
are in the same ratio as those of the equation
x
2
+
l
x
+
m
=
0
x^2 + lx + m = 0
x
2
+
l
x
+
m
=
0
, then which one of the following is correct?
Add
Lever: Vieta — sum and product of roots
A
p
2
m
=
l
2
q
p^2 m = l^2 q
p
2
m
=
l
2
q
B
m
2
p
=
l
2
q
m^2 p = l^2 q
m
2
p
=
l
2
q
C
m
2
p
−
q
2
l
m^2 p - q^2 l
m
2
p
−
q
2
l
D
m
2
p
2
=
l
2
q
m^2 p^2 = l^2 q
m
2
p
2
=
l
2
q
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[Q5 · Apr · 2017]
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Q3
#3
NDA → Mathematics → Trigonometric Identities → Compound Angle Formulas
·
Moderate
If
cot
α
\cot\alpha
cot
α
and
cot
β
\cot\beta
cot
β
are the roots of the equation
x
2
+
b
x
+
c
=
0
x^2 + bx + c = 0
x
2
+
b
x
+
c
=
0
with
b
≠
0
b \neq 0
b
=
0
, then the value of
cot
(
α
+
β
)
\cot(\alpha+\beta)
cot
(
α
+
β
)
is
Add
Lever: Compound angle: sin/cos/tan(A ± B)
A
c
−
1
b
\dfrac{c-1}{b}
b
c
−
1
B
1
−
c
b
\dfrac{1-c}{b}
b
1
−
c
C
b
c
−
1
\dfrac{b}{c-1}
c
−
1
b
D
b
1
−
c
\dfrac{b}{1-c}
1
−
c
b
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[Q15 · Apr · 2017]
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Q4
#4
NDA → Mathematics → Sequence & Series → Interrelating AP, GP and HP
·
Hard
The sum of the roots of the equation
x
2
+
b
x
+
c
=
0
x^2 + bx + c = 0
x
2
+
b
x
+
c
=
0
(where
b
b
b
and
c
c
c
are non-zero) is equal to the sum of the reciprocals of their squares. Then
1
c
,
b
,
c
b
\dfrac{1}{c},\, b,\, \dfrac{c}{b}
c
1
,
b
,
b
c
are in
Add
Lever: Vieta — sum and product of roots
Concept: Roots, coefficients, and progression conditions
A
AP
B
GP
C
HP
D
None of the above
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[Q16 · Apr · 2017]
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Q5
#5
NDA → Mathematics → Sequence & Series → Interrelating AP, GP and HP
·
Hard
The sum of the roots of the equation
a
x
2
+
x
+
c
=
0
ax^2 + x + c = 0
a
x
2
+
x
+
c
=
0
(where
a
a
a
and
c
c
c
are non-zero) is equal to the sum of the reciprocals of their squares. Then
a
,
c
a
2
,
c
2
a,\, ca^2,\, c^2
a
,
c
a
2
,
c
2
are in
Add
Lever: Vieta — sum and product of roots
Concept: Roots, coefficients, and progression conditions
A
AP
B
GP
C
HP
D
None of the above
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[Q17 · Apr · 2017]
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Q6
#6
NDA → Mathematics → Quadratic Equations → Special Quadratics — Parametric, Logarithmic, Constructed
·
Moderate
The roots of the equation
(
q
−
r
)
x
2
+
(
r
−
p
)
x
+
(
p
−
q
)
=
0
(q-r)x^2 + (r-p)x + (p-q) = 0
(
q
−
r
)
x
2
+
(
r
−
p
)
x
+
(
p
−
q
)
=
0
are
Add
Lever: Vieta — sum and product of roots
A
r
−
p
q
−
r
,
1
2
\dfrac{r-p}{q-r},\ \dfrac{1}{2}
q
−
r
r
−
p
,
2
1
B
p
−
q
q
−
r
,
1
\dfrac{p-q}{q-r},\ 1
q
−
r
p
−
q
,
1
C
q
−
r
p
−
q
,
1
\dfrac{q-r}{p-q},\ 1
p
−
q
q
−
r
,
1
D
r
−
p
p
−
q
,
1
2
\dfrac{r-p}{p-q},\ \dfrac{1}{2}
p
−
q
r
−
p
,
2
1
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[Q5 · Sep · 2017]
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Q7
#7
NDA → Mathematics → Matrices & Determinants → Matrix Operations, Polynomials, and Equations
·
Hard
If
α
\alpha
α
and
β
\beta
β
are the roots of the equation
1
+
x
+
x
2
=
0
1 + x + x^2 = 0
1
+
x
+
x
2
=
0
, then the matrix product
(
1
β
α
α
)
(
α
β
1
β
)
\begin{pmatrix}1 & \beta \\ \alpha & \alpha\end{pmatrix}\begin{pmatrix}\alpha & \beta \\ 1 & \beta\end{pmatrix}
(
1
α
β
α
)
(
α
1
β
β
)
is equal to
Add
Lever: Cube roots of unity (1 + ω + ω² = 0, ω³ = 1)
Concept: Matrix polynomials and equations
A
(
1
1
1
2
)
\begin{pmatrix}1 & 1 \\ 1 & 2\end{pmatrix}
(
1
1
1
2
)
B
(
−
1
−
1
−
1
2
)
\begin{pmatrix}-1 & -1 \\ -1 & 2\end{pmatrix}
(
−
1
−
1
−
1
2
)
C
(
1
−
1
−
1
2
)
\begin{pmatrix}1 & -1 \\ -1 & 2\end{pmatrix}
(
1
−
1
−
1
2
)
D
(
−
1
−
1
−
1
−
2
)
\begin{pmatrix}-1 & -1 \\ -1 & -2\end{pmatrix}
(
−
1
−
1
−
1
−
2
)
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[Q8 · Sep · 2017]
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Q8
#8
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Hard
If
α
\alpha
α
and
β
\beta
β
are the roots of the equation
3
x
2
+
2
x
+
1
=
0
3x^2 + 2x + 1 = 0
3
x
2
+
2
x
+
1
=
0
, then the equation whose roots are
α
+
β
−
1
\alpha + \beta^{-1}
α
+
β
−
1
and
β
+
α
−
1
\beta + \alpha^{-1}
β
+
α
−
1
is
Add
Lever: Vieta — sum and product of roots
A
3
x
2
+
8
x
+
16
=
0
3x^2 + 8x + 16 = 0
3
x
2
+
8
x
+
16
=
0
B
3
x
2
−
8
x
−
16
=
0
3x^2 - 8x - 16 = 0
3
x
2
−
8
x
−
16
=
0
C
3
x
2
+
8
x
−
16
=
0
3x^2 + 8x - 16 = 0
3
x
2
+
8
x
−
16
=
0
D
x
2
+
8
x
+
16
=
0
x^2 + 8x + 16 = 0
x
2
+
8
x
+
16
=
0
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[Q24 · Sep · 2017]
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Q9
#9
NDA → Mathematics → Properties of Triangle → Sine and Cosine Rules — Solving Triangles
·
Moderate
In
△
P
Q
R
\triangle PQR
△
P
QR
,
∠
R
=
π
2
\angle R = \dfrac{\pi}{2}
∠
R
=
2
π
. If
tan
(
P
2
)
\tan\!\left(\dfrac{P}{2}\right)
tan
(
2
P
)
and
tan
(
Q
2
)
\tan\!\left(\dfrac{Q}{2}\right)
tan
(
2
Q
)
are the roots of the equation
a
x
2
+
b
x
+
c
=
0
ax^2 + bx + c = 0
a
x
2
+
b
x
+
c
=
0
, then which one of the following is correct?
Add
Lever: Sine rule + Cosine rule
A
a
=
b
+
c
a = b + c
a
=
b
+
c
B
b
=
c
+
a
b = c + a
b
=
c
+
a
C
c
=
a
+
b
c = a + b
c
=
a
+
b
D
b
=
c
b = c
b
=
c
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[Q29 · Sep · 2017]
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Q10
#10
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Hard
If
α
\alpha
α
and
β
\beta
β
(
≠
0
)
(\neq 0)
(
=
0
)
are the roots of the quadratic equation
x
2
+
α
x
−
β
=
0
x^2 + \alpha x - \beta = 0
x
2
+
α
x
−
β
=
0
, then the quadratic expression
−
x
2
+
α
x
+
β
-x^2 + \alpha x + \beta
−
x
2
+
α
x
+
β
where
x
∈
R
x \in \mathbb{R}
x
∈
R
has
Add
Lever: Vieta — sum and product of roots
A
Least value
−
1
4
-\frac{1}{4}
−
4
1
B
Least value
−
9
4
-\frac{9}{4}
−
4
9
C
Greatest value
1
4
\frac{1}{4}
4
1
D
Greatest value
9
4
\frac{9}{4}
4
9
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[Q10 · Sep · 2018]
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Q11
#11
NDA → Mathematics → Trigonometric Equations → Solving Specific Forms — Double-Angle, Product, Logarithmic, and Vieta
·
Moderate
If
cos
α
\cos\alpha
cos
α
and
cos
β
\cos\beta
cos
β
(
0
<
α
<
β
<
π
)
(0 < \alpha < \beta < \pi)
(
0
<
α
<
β
<
π
)
are the roots of the quadratic equation
4
x
2
−
3
=
0
4x^2 - 3 = 0
4
x
2
−
3
=
0
, then what is the value of
sec
α
×
sec
β
\sec\alpha \times \sec\beta
sec
α
×
sec
β
?
Add
Lever: Compound angle: sin/cos/tan(A ± B)
A
−
4
3
-\frac{4}{3}
−
3
4
B
4
3
\frac{4}{3}
3
4
C
3
4
\frac{3}{4}
4
3
D
−
3
4
-\frac{3}{4}
−
4
3
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[Q44 · Sep · 2018]
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Q12
#12
NDA → Mathematics → Quadratic Equations → Nature of Roots and Boundary Conditions
·
Hard
The ratio of roots of the equations
a
x
2
+
b
x
+
c
=
0
ax^2 + bx + c = 0
a
x
2
+
b
x
+
c
=
0
and
p
x
2
+
q
x
+
r
=
0
px^2 + qx + r = 0
p
x
2
+
q
x
+
r
=
0
are equal. If
D
1
D_1
D
1
and
D
2
D_2
D
2
are respective discriminants, then what is
D
1
D
2
\frac{D_1}{D_2}
D
2
D
1
equal to?
Add
Lever: Vieta — sum and product of roots
A
a
2
p
2
\frac{a^2}{p^2}
p
2
a
2
B
b
2
q
2
\frac{b^2}{q^2}
q
2
b
2
C
c
2
r
2
\frac{c^2}{r^2}
r
2
c
2
D
None of the above
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[Q49 · Sep · 2018]
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Q13
#13
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Easy
The equation
p
x
2
+
q
x
+
r
=
0
px^2 + qx + r = 0
p
x
2
+
q
x
+
r
=
0
(where p, q, r, all are positive) has distinct real roots a and b. Which one of the following is correct?
Add
Lever: Vieta — sum and product of roots
A
a
>
0
,
b
>
0
a > 0,\ b > 0
a
>
0
,
b
>
0
B
a
<
0
,
b
<
0
a < 0,\ b < 0
a
<
0
,
b
<
0
C
a
>
0
,
b
<
0
a > 0,\ b < 0
a
>
0
,
b
<
0
D
a
<
0
,
b
>
0
a < 0,\ b > 0
a
<
0
,
b
>
0
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[Q14 · Apr · 2019]
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Q14
#14
NDA → Mathematics → Trigonometric Equations → Solving Specific Forms — Double-Angle, Product, Logarithmic, and Vieta
·
Moderate
If the roots of the equation
x
2
+
p
x
+
q
=
0
x^2 + px + q = 0
x
2
+
p
x
+
q
=
0
are
tan
19
°
\tan19°
tan
19°
and
tan
26
°
\tan26°
tan
26°
, then which one of the following is correct?
Add
Lever: Compound angle: sin/cos/tan(A ± B)
A
q
−
p
=
1
q - p = 1
q
−
p
=
1
B
p
−
q
=
1
p - q = 1
p
−
q
=
1
C
p
+
q
=
2
p + q = 2
p
+
q
=
2
D
p
+
q
=
3
p + q = 3
p
+
q
=
3
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[Q46 · Apr · 2019]
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Q15
#15
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Moderate
What is the value of
k
k
k
for which the sum of the squares of the roots of
2
x
2
−
2
(
k
−
2
)
x
−
(
k
+
1
)
=
0
2x^{2} - 2(k-2)x - (k+1) = 0
2
x
2
−
2
(
k
−
2
)
x
−
(
k
+
1
)
=
0
is minimum ?
Add
Lever: Vieta — sum and product of roots
A
-1
B
1
C
3
2
\frac{3}{2}
2
3
D
2
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[Q5 · Sep · 2019]
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Q16
#16
NDA → Mathematics → Sequence & Series → Arithmetic Progressions
·
Moderate
Let
m
m
m
and
n
n
n
(
m
<
n
)
(m < n)
(
m
<
n
)
be the roots of the equation
x
2
−
16
x
+
39
=
0
x^{2} - 16x + 39 = 0
x
2
−
16
x
+
39
=
0
. If four terms
p
p
p
,
q
q
q
,
r
r
r
and
s
s
s
are inserted between
m
m
m
and
n
n
n
to form an AP, then what is the value of
p
+
q
+
r
+
s
p + q + r + s
p
+
q
+
r
+
s
?
Add
Lever: AP three-term: 2b = a + c
Concept: The arithmetic mean and symmetric terms
A
29
B
30
C
32
D
35
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[Q15 · Sep · 2019]
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Q17
#17
NDA → Mathematics → Complex Numbers → Cube Roots of Unity
·
Moderate
If
α
\alpha
α
and
β
\beta
β
are the roots of
x
2
+
x
+
1
=
0
x^{2} + x + 1 = 0
x
2
+
x
+
1
=
0
, then what is
∑
j
=
0
3
(
α
j
+
β
j
)
\sum_{j=0}^{3}\left(\alpha^{j} + \beta^{j}\right)
∑
j
=
0
3
(
α
j
+
β
j
)
equal to ?
Add
Lever: Cube roots of unity (1 + ω + ω² = 0, ω³ = 1)
A
8
B
6
C
4
D
2
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[Q18 · Sep · 2019]
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Q18
#18
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Moderate
If
p
p
p
and
q
q
q
are the roots of the equation
x
2
−
30
x
+
221
=
0
x^{2} - 30x + 221 = 0
x
2
−
30
x
+
221
=
0
, what is the value of
p
3
+
q
3
p^{3} + q^{3}
p
3
+
q
3
?
Add
Lever: Vieta — sum and product of roots
A
7010
B
7110
C
7210
D
7240
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[Q105 · Sep · 2019]
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Q19
#19
NDA → Mathematics → Trigonometric Equations → Solving Specific Forms — Double-Angle, Product, Logarithmic, and Vieta
·
Moderate
If
cot
α
\cot\alpha
cot
α
and
cot
β
\cot\beta
cot
β
are the roots of the equation
x
2
−
3
x
+
2
=
0
x^2-3x+2=0
x
2
−
3
x
+
2
=
0
, then what is
cot
(
α
+
β
)
\cot(\alpha+\beta)
cot
(
α
+
β
)
equal to?
Add
Lever: Compound angle: sin/cos/tan(A ± B)
A
1
2
\frac{1}{2}
2
1
B
1
3
\frac{1}{3}
3
1
C
2
D
3
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[Q43 · Apr · 2020]
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Q20
#20
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Hard
The roots
α
\alpha
α
and
β
\beta
β
of a quadratic equation satisfy the relations
α
+
β
=
α
2
+
β
2
\alpha+\beta=\alpha^2+\beta^2
α
+
β
=
α
2
+
β
2
and
α
β
=
α
2
β
2
\alpha\beta=\alpha^2\beta^2
α
β
=
α
2
β
2
. What is the number of such quadratic equations?
Add
Lever: Vieta — sum and product of roots
A
0
B
2
C
3
D
4
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[Q44 · Apr · 2020]
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Q21
#21
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Hard
If
α
\alpha
α
and
β
\beta
β
are the roots of the equation
4
x
2
+
2
x
−
1
=
0
4x^2+2x-1=0
4
x
2
+
2
x
−
1
=
0
, then which one of the following is correct?
Add
Lever: Vieta — sum and product of roots
A
β
=
−
2
α
2
−
2
α
\beta = -2\alpha^2 - 2\alpha
β
=
−
2
α
2
−
2
α
B
β
=
4
α
2
−
3
α
\beta = 4\alpha^2 - 3\alpha
β
=
4
α
2
−
3
α
C
β
=
α
2
−
3
α
\beta = \alpha^2 - 3\alpha
β
=
α
2
−
3
α
D
β
=
−
2
α
2
+
2
α
\beta = -2\alpha^2 + 2\alpha
β
=
−
2
α
2
+
2
α
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[Q42 · Apr · 2021]
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Q22
#22
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Easy
If one root of
5
x
2
+
26
x
+
k
=
0
5x^2+26x+k=0
5
x
2
+
26
x
+
k
=
0
is reciprocal of the other, then what is the value of
k
k
k
?
Add
Lever: Vieta — sum and product of roots
A
2
B
3
C
5
D
8
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[Q43 · Apr · 2021]
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Q23
#23
NDA → Mathematics → Quadratic Equations → Special Quadratics — Parametric, Logarithmic, Constructed
·
Moderate
If
k
k
k
is one of the roots of the equation
x
(
x
+
1
)
+
1
=
0
x(x+1)+1=0
x
(
x
+
1
)
+
1
=
0
, then what is its other root?
Add
Lever: Cube roots of unity (1 + ω + ω² = 0, ω³ = 1)
A
1
B
−
k
-k
−
k
C
k
2
k^2
k
2
D
−
k
2
-k^2
−
k
2
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[Q113 · Apr · 2021]
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Q24
#24
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Hard
The quadratic equation
3
x
2
−
(
k
2
+
5
k
)
x
+
3
k
2
−
5
k
=
0
3x^{2}-(k^{2}+5k)x+3k^{2}-5k=0
3
x
2
−
(
k
2
+
5
k
)
x
+
3
k
2
−
5
k
=
0
has real roots of equal magnitude and opposite sign. Which one of the following is correct?
Add
Lever: Vieta — sum and product of roots
A
0
<
k
<
5
3
0<k<\frac{5}{3}
0
<
k
<
3
5
B
0
<
k
<
3
5
0<k<\frac{3}{5}
0
<
k
<
5
3
only
C
3
5
<
k
<
5
3
\frac{3}{5}<k<\frac{5}{3}
5
3
<
k
<
3
5
D
No such value of
k
k
k
exists
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[Q6 · Sep · 2021]
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Q25
#25
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Easy
If
p
p
p
and
q
q
q
are the non-zero roots of the equation
x
2
+
p
x
+
q
=
0
x^{2}+px+q=0
x
2
+
p
x
+
q
=
0
, then how many possible values can
q
q
q
have?
Add
Lever: Vieta — sum and product of roots
A
Nil
B
One
C
Two
D
Three
Tap an option to check your answer.
Show solution
[Q8 · Sep · 2021]
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