Question Bank
Bank
Guides
Notes
NDA
Sign in
Filtered questions
49 questions match
Filters
1
Download
· 49
Filtering by:
Principle: Vieta — sum and product of roots
Clear all
Copy link
Q26
#26
NDA → Mathematics → Trigonometric Equations → Solving Specific Forms — Double-Angle, Product, Logarithmic, and Vieta
·
Moderate
If
sin
θ
\sin\theta
sin
θ
and
cos
θ
\cos\theta
cos
θ
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: Compound angle: sin/cos/tan(A ± B)
A
a
2
+
b
2
−
2
a
c
=
0
a^{2}+b^{2}-2ac=0
a
2
+
b
2
−
2
a
c
=
0
B
−
a
2
+
b
2
+
2
a
c
=
0
-a^{2}+b^{2}+2ac=0
−
a
2
+
b
2
+
2
a
c
=
0
C
a
2
−
b
2
+
2
a
c
=
0
a^{2}-b^{2}+2ac=0
a
2
−
b
2
+
2
a
c
=
0
D
a
2
+
b
2
+
2
a
c
=
0
a^{2}+b^{2}+2ac=0
a
2
+
b
2
+
2
a
c
=
0
Tap an option to check your answer.
Show solution
[Q16 · Sep · 2021]
Report
Q27
#27
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Hard
If the roots of the equation
4
x
2
−
(
5
k
+
1
)
x
+
5
k
=
0
4x^{2}-(5k+1)x+5k=0
4
x
2
−
(
5
k
+
1
)
x
+
5
k
=
0
differ by unity, then which one of the following is a possible value of
k
k
k
?
Add
Lever: Vieta — sum and product of roots
A
−
3
-3
−
3
B
−
1
-1
−
1
C
−
1
5
-\frac{1}{5}
−
5
1
D
−
3
5
-\frac{3}{5}
−
5
3
Tap an option to check your answer.
Show solution
[Q45 · Sep · 2021]
Report
Q28
#28
NDA → Mathematics → Quadratic Equations → Special Quadratics — Parametric, Logarithmic, Constructed
·
Hard
Let
α
\alpha
α
and
β
\beta
β
be the roots of the equation
x
2
+
p
x
+
q
=
0
x^2+px+q=0
x
2
+
p
x
+
q
=
0
. If
α
3
\alpha^3
α
3
and
β
3
\beta^3
β
3
are the roots of the equation
x
2
+
m
x
+
n
=
0
x^2+mx+n=0
x
2
+
m
x
+
n
=
0
, then what is the value of
m
+
n
m+n
m
+
n
?
Add
Lever: Vieta — sum and product of roots
A
p
3
+
q
3
+
p
q
p^3+q^3+pq
p
3
+
q
3
+
pq
B
p
3
+
q
3
−
p
q
p^3+q^3-pq
p
3
+
q
3
−
pq
C
p
3
+
q
3
+
3
p
q
p^3+q^3+3pq
p
3
+
q
3
+
3
pq
D
p
3
+
q
3
−
3
p
q
p^3+q^3-3pq
p
3
+
q
3
−
3
pq
Tap an option to check your answer.
Show solution
[Q17 · Apr · 2022]
Report
Q29
#29
NDA → Mathematics → Quadratic Equations → Special Quadratics — Parametric, Logarithmic, Constructed
·
Hard
Let
α
\alpha
α
and
β
\beta
β
be the roots of the equation
x
2
−
a
x
−
b
x
+
a
b
−
c
=
0
x^2-ax-bx+ab-c=0
x
2
−
a
x
−
b
x
+
ab
−
c
=
0
. What is the quadratic equation whose roots are
a
a
a
and
b
b
b
?
Add
Lever: Vieta — sum and product of roots
A
x
2
−
α
x
−
β
x
+
α
β
+
c
=
0
x^2-\alpha x-\beta x+\alpha\beta+c=0
x
2
−
α
x
−
β
x
+
α
β
+
c
=
0
B
x
2
−
α
x
−
β
x
+
α
β
−
c
=
0
x^2-\alpha x-\beta x+\alpha\beta-c=0
x
2
−
α
x
−
β
x
+
α
β
−
c
=
0
C
x
2
+
α
x
+
β
x
+
α
β
+
c
=
0
x^2+\alpha x+\beta x+\alpha\beta+c=0
x
2
+
α
x
+
β
x
+
α
β
+
c
=
0
D
x
2
+
α
x
+
β
x
+
α
β
−
c
=
0
x^2+\alpha x+\beta x+\alpha\beta-c=0
x
2
+
α
x
+
β
x
+
α
β
−
c
=
0
Tap an option to check your answer.
Show solution
[Q18 · Apr · 2022]
Report
Q30
#30
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Easy
Let
α
\alpha
α
and
β
\beta
β
(
α
>
β
\alpha>\beta
α
>
β
) be the roots of the equation
x
2
−
8
x
+
q
=
0
x^2-8x+q=0
x
2
−
8
x
+
q
=
0
. If
α
2
−
β
2
=
16
\alpha^2-\beta^2=16
α
2
−
β
2
=
16
, then what is the value of
q
q
q
?
Add
Lever: Vieta — sum and product of roots
A
-15
B
-10
C
10
D
15
Tap an option to check your answer.
Show solution
[Q20 · Apr · 2022]
Report
Q31
#31
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Easy
For how many quadratic equations, the sum of roots is equal to the product of roots?
Add
Lever: Vieta — sum and product of roots
A
0
B
1
C
2
D
Infinitely many
Tap an option to check your answer.
Show solution
[Q8 · Sep · 2022]
Report
Q32
#32
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Moderate
Let
p
,
q
p, q
p
,
q
(
p
>
q
p > q
p
>
q
) be the roots of the quadratic equation
x
2
+
b
x
+
c
=
0
x^2 + bx + c = 0
x
2
+
b
x
+
c
=
0
where
c
>
0
c > 0
c
>
0
. If
p
2
+
q
2
−
11
p
q
=
0
p^2 + q^2 - 11pq = 0
p
2
+
q
2
−
11
pq
=
0
, then what is
p
−
q
p - q
p
−
q
equal to?
Add
Lever: Vieta — sum and product of roots
A
3
c
3\sqrt{c}
3
c
B
3
c
3c
3
c
C
9
c
9\sqrt{c}
9
c
D
9
c
9c
9
c
Tap an option to check your answer.
Show solution
[Q11 · Sep · 2022]
Report
Q33
#33
NDA → Mathematics → Complex Numbers → Modulus, Argument, and Conjugate
·
Hard
If
α
\alpha
α
and
β
\beta
β
are the distinct roots of equation
x
2
−
x
+
1
=
0
x^{2}-x+1=0
x
2
−
x
+
1
=
0
, then what is the value of
∣
α
100
+
β
100
α
100
−
β
100
∣
\left|\frac{\alpha^{100}+\beta^{100}}{\alpha^{100}-\beta^{100}}\right|
α
100
−
β
100
α
100
+
β
100
?
Add
Lever: Cube roots of unity (1 + ω + ω² = 0, ω³ = 1)
A
3
\sqrt{3}
3
B
2
\sqrt{2}
2
C
1
D
1
3
\frac{1}{\sqrt{3}}
3
1
Tap an option to check your answer.
Show solution
[Q5 · Apr · 2023]
Report
Q34
#34
NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
·
Hard
What is the sum of the roots of the equation
∣
0
x
−
a
x
−
b
0
0
x
−
c
x
+
b
x
+
c
1
∣
=
0
\begin{vmatrix}0&x-a&x-b\\0&0&x-c\\x+b&x+c&1\end{vmatrix}=0
0
0
x
+
b
x
−
a
0
x
+
c
x
−
b
x
−
c
1
=
0
?
Add
Lever: Vieta — sum and product of roots
Concept: Singular matrices and determinant equations
A
a
+
b
+
c
a+b+c
a
+
b
+
c
B
a
−
b
+
c
a-b+c
a
−
b
+
c
C
a
+
b
−
c
a+b-c
a
+
b
−
c
D
a
−
b
−
c
a-b-c
a
−
b
−
c
Tap an option to check your answer.
Show solution
[Q10 · Apr · 2023]
Report
Q35
#35
NDA → Mathematics → Complex Numbers → Modulus, Argument, and Conjugate
·
Moderate
If
2
−
i
3
2-i\sqrt{3}
2
−
i
3
where
i
=
−
1
i=\sqrt{-1}
i
=
−
1
is a root of the equation
x
2
+
a
x
+
b
=
0
x^{2}+ax+b=0
x
2
+
a
x
+
b
=
0
, then what is the value of
(
a
+
b
)
(a+b)
(
a
+
b
)
?
Add
Lever: Vieta — sum and product of roots
A
-11
B
-3
C
0
D
3
Tap an option to check your answer.
Show solution
[Q11 · Apr · 2023]
Report
Q36
#36
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Hard
α
\alpha
α
and
β
\beta
β
are distinct real roots of the quadratic equation
x
2
+
a
x
+
b
=
0
x^{2}+ax+b=0
x
2
+
a
x
+
b
=
0
. Which of the following statements is/are sufficient to find
α
\alpha
α
? 1.
α
+
β
=
0
\alpha+\beta=0
α
+
β
=
0
,
α
2
+
β
2
=
2
\alpha^{2}+\beta^{2}=2
α
2
+
β
2
=
2
2.
α
β
2
=
−
1
\alpha\beta^{2}=-1
α
β
2
=
−
1
,
a
=
0
a=0
a
=
0
Add
Lever: Vieta — sum and product of roots
A
1 only
B
2 only
C
Both 1 and 2
D
Neither 1 nor 2
Tap an option to check your answer.
Show solution
[Q23 · Apr · 2023]
Report
Set · 2 questions
Consider the following for the next two (02) items that follow: A quadratic equation is given by
(
3
+
2
2
)
x
2
−
(
4
+
2
3
)
x
+
(
8
+
4
3
)
=
0
(3+2\sqrt{2})x^{2}-(4+2\sqrt{3})x+(8+4\sqrt{3})=0
(
3
+
2
2
)
x
2
−
(
4
+
2
3
)
x
+
(
8
+
4
3
)
=
0
.
Q37
#37
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Hard
What is the HM of the roots of the equation?
Add
Lever: Vieta — sum and product of roots
A
2
B
4
C
2
2
2\sqrt{2}
2
2
D
2
3
2\sqrt{3}
2
3
Tap an option to check your answer.
Show solution
[Q49 · Apr · 2023]
Report
Q38
#38
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Hard
What is the GM of the roots of the equation?
Add
Lever: Vieta — sum and product of roots
A
2
(
6
−
3
+
2
−
1
)
\sqrt{2}(\sqrt{6}-\sqrt{3}+\sqrt{2}-1)
2
(
6
−
3
+
2
−
1
)
B
2
(
6
+
3
−
2
−
1
)
\sqrt{2}(\sqrt{6}+\sqrt{3}-\sqrt{2}-1)
2
(
6
+
3
−
2
−
1
)
C
(
6
−
3
+
2
−
1
)
(\sqrt{6}-\sqrt{3}+\sqrt{2}-1)
(
6
−
3
+
2
−
1
)
D
(
6
+
3
+
2
−
1
)
(\sqrt{6}+\sqrt{3}+\sqrt{2}-1)
(
6
+
3
+
2
−
1
)
Tap an option to check your answer.
Show solution
[Q50 · Apr · 2023]
Report
Q39
#39
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Hard
If the highest degree coefficient is equal to 1, then what is the total number of quadratic equations which are unchanged on squaring their roots?
Add
Lever: Cube roots of unity (1 + ω + ω² = 0, ω³ = 1)
A
6
B
4
C
2
D
None
Tap an option to check your answer.
Show solution
[Q11 · Apr · 2026]
Report
Q40
#40
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Moderate
Let
α
\alpha
α
and
β
\beta
β
be the roots of the quadratic equation
x
2
−
2
b
x
+
c
2
=
0
x^2-2bx+c^2=0
x
2
−
2
b
x
+
c
2
=
0
where
b
b
b
,
c
c
c
are positive real numbers. Let A be the arithmetic mean of
α
\alpha
α
and
β
\beta
β
; and G be the geometric mean of
α
\alpha
α
and
β
\beta
β
. What are the roots of the quadratic equation
x
2
−
(
b
+
c
)
x
+
b
c
=
0
x^2-(b+c)x+bc=0
x
2
−
(
b
+
c
)
x
+
b
c
=
0
?
Add
Lever: AM-GM / mean inequalities (incl. x + 1/x ≥ 2)
A
A, G
B
2A, G
C
A, 2G
D
2A, 2G
Tap an option to check your answer.
Show solution
[Q12 · Apr · 2026]
Report
Set · 1 question
For the following two (02) items: Consider the equation
a
b
x
2
+
b
c
x
+
c
a
=
c
a
x
2
+
a
b
x
+
b
c
abx^2+bcx+ca = cax^2+abx+bc
ab
x
2
+
b
c
x
+
c
a
=
c
a
x
2
+
ab
x
+
b
c
.
Q41
#41
NDA → Mathematics → Quadratic Equations → Nature of Roots and Boundary Conditions
·
Moderate
If the roots of the equation are equal, then
a
a
a
,
b
b
b
,
c
c
c
are in
Add
Lever: Vieta — sum and product of roots
A
AP
B
GP
C
HP
D
None of the above
Tap an option to check your answer.
Show solution
[Q32 · Sep · 2025]
Report
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
.
Q42
#42
NDA → 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
Tap an option to check your answer.
Show solution
[Q31 · Sep · 2023]
Report
Q43
#43
NDA → 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
Tap an option to check your answer.
Show solution
[Q32 · Sep · 2023]
Report
Q44
#44
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Hard
If
−
2
-2
−
2
and
3
3
3
are roots of the equation
a
0
+
a
1
x
+
a
2
x
2
+
a
3
x
3
+
x
4
=
0
a_0+a_1 x+a_2 x^2+a_3 x^3+x^4=0
a
0
+
a
1
x
+
a
2
x
2
+
a
3
x
3
+
x
4
=
0
where
a
0
,
a
1
,
a
2
,
a
3
a_0,a_1,a_2,a_3
a
0
,
a
1
,
a
2
,
a
3
are integers, then which one of the following is correct?
Add
Lever: Vieta — sum and product of roots
A
a
2
=
a
3
=
0
a_2=a_3=0
a
2
=
a
3
=
0
B
a
2
=
0
a_2=0
a
2
=
0
and
a
3
=
−
5
a_3=-5
a
3
=
−
5
C
a
0
=
6
,
a
3
=
0
a_0=6,\;a_3=0
a
0
=
6
,
a
3
=
0
D
a
1
=
0
a_1=0
a
1
=
0
and
a
2
=
5
a_2=5
a
2
=
5
Tap an option to check your answer.
Show solution
[Q8 · Apr · 2024]
Report
Q45
#45
NDA → Mathematics → Quadratic Equations → Nature of Roots and Boundary Conditions
·
Hard
If
a
,
b
a,b
a
,
b
and
c
(
a
>
0
,
c
>
0
)
c\;(a>0,\,c>0)
c
(
a
>
0
,
c
>
0
)
are in GP, then consider the following in respect of the equation
a
x
2
+
b
x
+
c
=
0
ax^2+bx+c=0
a
x
2
+
b
x
+
c
=
0
: (A) The equation has imaginary roots. (B) The ratio of the roots of the equation is
1
:
ω
1:\omega
1
:
ω
where
ω
\omega
ω
is a cube root of unity. (C) The product of roots of the equation is
b
2
a
2
\dfrac{b^2}{a^2}
a
2
b
2
. Which of the statements given above are correct?
Add
Lever: Cube roots of unity (1 + ω + ω² = 0, ω³ = 1)
A
(A) and (B) only
B
(B) and (C) only
C
(A) and (C) only
D
(A), (B) and (C)
Tap an option to check your answer.
Show solution
[Q32 · Apr · 2024]
Report
Q46
#46
NDA → Mathematics → Trigonometric Identities → Compound Angle Formulas
·
Moderate
If
tan
α
\tan\alpha
tan
α
and
tan
β
\tan\beta
tan
β
are the roots of the equation
x
2
−
6
x
+
8
=
0
x^2-6x+8=0
x
2
−
6
x
+
8
=
0
, then what is the value of
cos
(
2
α
+
2
β
)
\cos(2\alpha+2\beta)
cos
(
2
α
+
2
β
)
?
Add
Lever: Compound angle: sin/cos/tan(A ± B)
A
13
75
\dfrac{13}{75}
75
13
B
13
85
\dfrac{13}{85}
85
13
C
17
85
\dfrac{17}{85}
85
17
D
19
85
\dfrac{19}{85}
85
19
Tap an option to check your answer.
Show solution
[Q48 · Apr · 2024]
Report
Q47
#47
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Moderate
If
n
n
n
is a root of the equation
x
2
+
p
x
+
m
=
0
x^2+px+m=0
x
2
+
p
x
+
m
=
0
and
m
m
m
is a root of the equation
x
2
+
p
x
+
n
=
0
x^2+px+n=0
x
2
+
p
x
+
n
=
0
, where
m
≠
n
m\neq n
m
=
n
, then what is the value of
p
+
m
+
n
p+m+n
p
+
m
+
n
?
Add
Lever: Vieta — sum and product of roots
A
−
1
-1
−
1
B
0
0
0
C
1
1
1
D
2
2
2
Tap an option to check your answer.
Show solution
[Q16 · Sep · 2024]
Report
Q48
#48
NDA → Mathematics → Properties of Triangle → Triangle Identities — A+B+C=π, Half-Angle, and Double-Angle
·
Moderate
The roots of the equation
7
x
2
−
6
x
+
1
=
0
7x^2-6x+1=0
7
x
2
−
6
x
+
1
=
0
are
tan
α
\tan\alpha
tan
α
and
tan
β
\tan\beta
tan
β
, where
2
α
2\alpha
2
α
and
2
β
2\beta
2
β
are the angles of a triangle. Which one of the following is correct?
Add
Lever: Double / half-angle formulas
A
The triangle is equilateral
B
The triangle is isosceles but not right-angled
C
The triangle is right-angled
D
The triangle is right-angled isosceles
Tap an option to check your answer.
Show solution
[Q24 · Sep · 2024]
Report
Q49
#49
NDA → Mathematics → Quadratic Equations → Vieta's Relations and Root-Coefficient Identities
·
Moderate
If one root of the equation
x
2
−
k
x
+
k
=
0
x^2 - kx + k = 0
x
2
−
k
x
+
k
=
0
exceeds the other by
2
3
\frac{2}{\sqrt{3}}
3
2
, then which one of the following is a value of
k
k
k
?
Add
Lever: Vieta — sum and product of roots
A
3
3
3
B
6
6
6
C
9
9
9
D
12
12
12
Tap an option to check your answer.
Show solution
[Q3 · Apr · 2025]
Report
Prev
Page 2 of 2
Next
1
2
Question Bank
Bank
Guides
Notes
NDA
Sign in