Quadratic Equations and Inequalities · Mathematics · NDA

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MCQ (Single Correct Answer)

1
The number of real roots of the equation $${x^2} - 3\left| x \right| + 2 = 0$$ is
NDA 2015 Paper 2
2
If the sum of the roots of the equation $$a{x^2} + bx + c = 0$$ is equal to the sum of their squares, then
NDA 2015 Paper 2
3
If the roots of the equation x2 $$-$$ nx + m = 0 differ by 1, then
NDA 2015 Paper 2
4
If a, b and c are the sides of a $$\Delta$$ABC, then $${a^{{1 \over p}}} + {b^{{1 \over p}}} - {c^{{1 \over p}}}$$, where p > 1, is
NDA 2015 Paper 2
5
Under what condition does the above equation have real roots?
NDA 2016 Paper 2
6
Under what condition is $${1 \over {{\alpha ^2}}} + {1 \over {{\beta ^2}}} < 1$$?
NDA 2016 Paper 2
7
What is the value of $$\alpha$$?
NDA 2016 Paper 2
8
If $$\beta$$, 2, 2 m are in GP, then what is the value of $$\beta$$$$\sqrt m $$?
NDA 2016 Paper 2
9
If c > 0 and 4a + c < 2b, then ax2 $$-$$ bx + c = 0 has a root in which one of the following intervals?
NDA 2016 Paper 2
10
Consider the following

1. $$\beta$$ < $$-$$ $$\alpha$$

2. $$\beta$$ < | $$\alpha$$ |

Which of the above is/are correct?
NDA 2016 Paper 1
11
Consider the following

1. $$\alpha$$ + $$\beta$$ + $$\alpha$$$$\beta$$ > 0

2. $$\alpha$$2$$\beta$$ + $$\beta$$2$$\alpha$$ > 0

Which of the above is/are correct?
NDA 2016 Paper 1
12
If x2 $$-$$ px + 4 > 0 for all real values of x, then which one of the following is correct?
NDA 2016 Paper 1
13
If one root of the equation (l $$-$$ m) x2 + lx + 1 = 0 is double the other and l is real, then what is the greatest value of m?
NDA 2016 Paper 1
14
The roots of the equation $$(q - r){x^2} + (r - p)x + (p - q) = 0$$ are
NDA 2017 Paper 2
15
The sum of all real roots of the equation $${\left| {x - 3} \right|^2} + \left| {x - 3} \right| - 2 = 0$$ is
NDA 2017 Paper 2
16
It is given that the roots of the equation $${x^2} - 4x - {\log _3}P = 0$$ are real. For this the minimum value of P is
NDA 2017 Paper 2
17
If $$\alpha$$ and $$\beta$$ are the roots of the equation 3x2 + 2x + 1 = 0, then the equation whose roots are $$\alpha$$ + $$\beta$$$$-$$1 and $$\beta$$ + $$\alpha$$$$-$$1 is
NDA 2017 Paper 2
18
In $$\Delta$$PQR, $$\angle$$R = $${\pi \over 2}$$. If $$\tan \left( {{P \over 2}} \right)$$ and $$\tan \left( {{Q \over 2}} \right)$$ are the roots of the equation ax2 + bx + c = 0, then which one of the following is correct?
NDA 2017 Paper 2
19
If the difference between the roots of the equation $${x^2} + kx + 1 = 0$$ is strictly less than $$\sqrt 5 $$, where $$\left| k \right| \ge 2$$, then k can be any element of the interval
NDA 2017 Paper 1
20
If the graph of a quadratic polynomial lies entirely above X-axis, then which one of the following is correct?
NDA 2017 Paper 1
21
If cot $$\alpha$$ and cot $$\beta$$ are the roots of the equation x2 + bx + c = 0 with b $$\ne$$ 0, then the value of cot ($$\alpha$$ + $$\beta$$) is
NDA 2017 Paper 1
22
The sum of the roots of the equation x2 + bx + c = 0 (where, b and c are non-zero) is equal to the sum of the reciprocals of their squares. Then, $${1 \over c},b,{c \over b}$$ are in
NDA 2017 Paper 1
23
The sum of the roots of the equation ax2 + x + c = 0 (where, a and c are non-zero) is equal to the sum of the reciprocals of their squares. Then, a, ca2, c2 are in
NDA 2017 Paper 1
24
If S = {x : x2 + 1 = 0, x is real}, then S is
NDA 2017 Paper 1
25
If $$\alpha$$ and $$\beta$$ ($$\ne$$ 0) are the roots of the quadratic equation x2 + $$\alpha$$x $$-$$ $$\beta$$ = 0, then the quadratic expression $$-$$x2 + $$\alpha$$x + $$\beta$$, where x $$\in$$ R has
NDA 2018 Paper 2
26
Let $$\alpha$$ and $$\beta$$ be real number and z be a complex number. If z2 + $$\alpha$$z + $$\beta$$ = 0 has two distinct non-real roots with Re(z) = 1, then it is necessary that
NDA 2018 Paper 1
27
The equation $$\left| {1 - x} \right| + {x^2} = 5$$ hs
NDA 2018 Paper 1
28
Let $$[x]$$ denote the greatest integer function. What is the number of solutions of the equation $${x^2} - 4x + [x] = 0$$ in the interval [0, 2] ?
NDA 2018 Paper 1
29
If p and q are the roots of the equation $${x^2} - 30x + 221 = 0$$, what is the value of p3 + q3?
NDA 2019 Paper 2
30
Consider the following statements in respect of the quadratic equation $$4(x - p)(x - q) - {r^2} = 0$$, where p, q and r are real numbers.

1. The roots are real.

2. The roots are equal, if p = q and r = 0.

Which of the above statements is/are correct?
NDA 2019 Paper 2
31
How many real roots does the equation $${x^2} + 3\left| x \right| + 2 = 0$$ have?
NDA 2019 Paper 2
32
What is the solution of x $$\le$$ 4, y $$\ge$$ 0 and x $$\le$$ $$-$$4, y $$\le$$ 0?
NDA 2019 Paper 2
33
If $$\alpha$$ and $$\beta$$ are the roots of x2 + x + 1 = 0, then what is $$\sum\limits_{j = 0}^3 {({\alpha ^j} + {\beta ^j})} $$ equal to?
NDA 2019 Paper 2
34
Under which one of the following conditions will be quadratic equation x2 + mx + 2 = 0 always have real roots?
NDA 2019 Paper 2
35
If both p and q belong to the set {1, 2, 3, 4}, then how many equations of the form px2 + qx + 1 = 0 will have real roots?
NDA 2019 Paper 2
36
What is the value of k for which the sum of the squares of the roots of $$2{x^2} - 2(k - 2)x - (k + 1) = 0$$ is minimum?
NDA 2019 Paper 2
37
If the roots of the equation $$a(b - c){x^2} + b(c - a)x + c(a - b) = 0$$ are equal, then which one of the following is correct?
NDA 2019 Paper 2
38
If $$\left| {{x^2} - 3x + 2} \right| > {x^2} - 3x + 2$$, then which one of the following is correct?
NDA 2019 Paper 2
39
What re the roots of the equation $$\left| {{x^2} - x - 6} \right| = x + 2$$?
NDA 2019 Paper 1
40
The equation px2 + qx + r = 0 (where p, q, r all are positive) has distinct real roots a and b. Which one of the following is correct?
NDA 2019 Paper 1
41
The number of real roots for the equation $${x^2} + 9\left| x \right| + 20 = 0$$ is
NDA 2019 Paper 1
42

If $\sqrt{2}$ and $\sqrt{3}$ are roots of the equation $a_0 + a_1 x + a_2 x^2 + a_3 x^3 + x^4 = 0$ where $a_0, a_1, a_2, a_3$ are integers, then which one of the following is correct?

NDA Mathematics 21 April 2024
43

If $a$, $b$, and $c$ $(a > 0, c > 0)$ are in GP, then consider the following in respect of the equation $ax^2 + bx + c = 0$:

1. The equation has imaginary roots.

2. The ratio of the roots of the equation is $1 : \omega$ where $\omega$ is a cube root of unity.

3. The product of roots of the equation is $\left(\frac{b^2}{a^2}\right)$.

Which of the statements given above are correct?

NDA Mathematics 21 April 2024
44

If $x^2 + mx + n$ is an integer for all integral values of $x$, then which of the following is/are correct?

1. $m$ must be an integer

2. $n$ must be an integer

Select the correct answer using the code given below:

NDA Mathematics 21 April 2024
45

Under which one of the following conditions does the equation $\left(\cos \beta-1\right)x^2+(\cos \beta)x+\sin \beta=0$ in $x$ have a real root for $\beta \in [0, \pi]$?

NDA Mathematics 21 April 2024
46
If $\rm {k}=\frac{{c}}{2},({c} \neq 0)$, then the roots of the equation are :
NDA Mathematics 3 September 2023
47
If k = c, then the roots of the equation are:
NDA Mathematics 3 September 2023
48
For how many integral values of k, the equation x2 - 4x + k = 0, where k is an integer has real roots and both of them lie in the interval (0, 5) ? 
NDA Mathematics 16 April 2023
49
How many real numbers satisfy the equation |x - 4| + |x - 7| = 15 ?
NDA Mathematics 16 April 2023
50

α and β are distinct real roots of the quadratic equation x2 + ax + b = 0. Which of the following statements is/are sufficient to find α ? 

1. α + β = 0, α2 + β2 = 2

2. αβ2 = -1, a = 0

Select the correct answer using the code given below :

NDA Mathematics 16 April 2023
51
What is the HM of the roots of the equation ? 
NDA Mathematics 16 April 2023
52
What is the GM of the roots of the equation ?
NDA Mathematics 16 April 2023
53
For how many quadratic equations, the sum of roots is equal to the product of roots?
NDA Mathematics 4 September 2022
54
Let p, q(p > q) be the roots of the quadratic equation x2 + bx + c = 0 where c > 0. If p2 + q2 − 11pq = 0, then what is p − q equal to ?
NDA Mathematics 4 September 2022
55
Let α and β be the roots of the equation x2 + px + q = 0. If α3 and β3 are the roots of the equation x+ mx + n = 0, then what is the value of m + n ?
NDA Mathematics 10 April 2022
56
Let α and β be the roots of the equation x2 - ax - bx + ab - c = 0. What is the quadratic equation whose roots are a and b?
NDA Mathematics 10 April 2022
57
If the roots of the equation x2 - ax - bx - cx + bc + ca = 0 are equal, then which one of the following is correct?
NDA Mathematics 10 April 2022
58
Let α and β (α > β) be the roots of the equation x2 - 8x + q = 0. If α2 - β2 = 16, then what is the value of q?
NDA Mathematics 10 April 2022
59
If α and β are the roots of the equation 4x2 + 2x - 1 = 0, then which one of the following is correct?
NDA Mathematics 18 April 2021
60
If one root of 5x2 + 26x + k = 0 is reciprocal of the other, then what is the value of k?
NDA Mathematics 18 April 2021
61
If the roots of the quadratic equation x2 + 2x + k = 0 are real, then
NDA Mathematics 18 April 2021
62
If k is one of the roots of the equation x(x + 1) + 1 = 0, then what is its other root?
NDA Mathematics 18 April 2021
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