Nature of Roots of a Quadratic Equation

Q. If the equation $9x^2+6kx+4=0$ has equal roots, then the value of k = .

1. 2 or 0
2. 2 or 2
3. -2 or 2

Q.

If the equation $9x^2+6kx+4=0$ has equal roots, then find the value of k =

1. 2 , 0

2. -2, 0

3. 0

4. $\pm 2$

Q.

Which equation has imaginary roots?

Q.

Find the nature of roots of a quadratic equation, if its discriminant is equal to 4.

1. Two equal roots

2. No real roots

3. Two distinct real roots

4. Cannot be determined

Q. If the equation $9x^2+6kx+4=0$ has equal roots, then find the value of $k=$

1. 0
2. -2, 2
3. -2, 0
4. 2, 0

Q. If the equation $9x^2+6kx+4=0$ has equal roots, then the value of k = .

1. 2 or 0
2. -2 or 2
3. 2 or 2

Q.

If the roots of equation $x^2+4x+c=0$ are real then $c\ge 4$.

1. True

2. False

Q. Determine nature of roots of the quadratic equation $\sqrt{3}{x}^2+2\sqrt{3}x+\sqrt{3}=0$.
[2 Marks]

Q.

Let $f\left(x\right)=a{x}^2+bx+c$. Then, match the following.
$\begin{array}{cc}\text{a. Sum of roots of f(x) = 0}& 1.–\frac{b}{a}\\ \text{b. Product of roots of f(x) = 0}& 2.\frac{c}{a}\\ \text{c. Roots of f(x) = 0 are real and distinct}& 3.b^2–4ac=0\\ \text{d. Roots of f(x) = 0 are real and identical.}& 4.b^2–4ac>0\end{array}$

1. $a-2,b-1,c-3.d-4$

2. $a-1,b-2,c-4,d-3$

3. $a-3,c-4,b-2,d-1$

4. $a-1,b-2,c-3,d-4$

Q. For what value of 𝒌, the quadratic equation $x^2-kx+4=0$ has equal roots?

Q. (a) a, b, c are lengths of sides of an scalene triangle. If equation
$x^2+2(a+b+c)x+3\lambda (ab+bc+ac)=0$
has real and distinct roots, then the value of $\lambda$ is given by :
(a) $\lambda <\frac{4}{3}$
(b) $\frac{11}{3}<\lambda <\frac{17}{3}$
(c) $\frac{11}{6}<\lambda <\frac{15}{4}$
(d) $\lambda \ge 1$
(b) If a, b are the roots of $x^2-10cx-11d=0$ and c, d are the roots of $x^2-10ax-11b=0$, then find the value, of $a+b+c+d$. (a, b, c, d are distinct numbers).

Q. If the equation $9x^2+6kx+4=0$ has equal roots, then find the value of $k$.

Q. Find the values of $h$ for which the following equation has real and equal roots:
$x^2+2(h+2)x+9h=0$.

Q. If the equation $9x^2+6kx+4=0$ has equal roots, then the value of k = .

1. 2 or 0
2. 2 or 2
3. -2 or 2

Q.

The equation $x^2+4x+c=0$ has real roots, then

1. $c\ge 4$

2. $c\le 4$

3. $c\ge 6$

4. $c\le 8$

Q. The discriminant of the quadratic equation $x^2+5x+4=0$ is:

Q. If the roots of the given equation $2x^2+3(\lambda -2)x+\lambda +4=0$ be equal in magnitude but opposite in sign, then value of $\lambda$ is

1. $1$
2. $\frac{2}{3}$
3. $2$
4. $3$

Q. The number of real roots of the equation $x^2-3x+5=0$ is:

1. 0
2. 1
3. 2
4. Infinite

Q. Assertion (A): The roots of $(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$ are real
Reason (R): A quadratic equation with non-negative discriminant has real roots .

1. Both (A) and (R) are true and (R) is not the correct explanation of (A)
2. Both (A) and (R) are true and (R) is the correct explanation of (A)
3. (A) is true but (R) is false
4. (A) is false but (R) is true

Q. The quadratic equation which has D = 0 and a root as $\frac{1}{10}$ is

1. $x^2-20x-100=0$
2. $x^2-100x-100=0$
3. $100x^2-20x+1=0$
4. $100x^2-20x+100=0$

Q. The graph drawn below of a quadratic polynomial shows that it has no real roots.

1. True
2. False

Q.

With respect to the roots of $x^2–2x–3=0$, we can say that

1. roots are not real

2. one of the root is zero.

3. both of them are natural numbers

4. both of them are integers

Q. If the discriminant of a quadratic equation is 0, then it has real and _______ roots.

1. imaginary
2. unequal
3. equal
4. distinct

Q.

If the equation $9x^2+6kx+4=0$ has equal roots, then find the value of k =

1. 0

2. 2 , 0

3. -2, 0

4. $\pm 2$

Q. Let $a,b,c$ be the sides of a triangle, where $a\ne b\ne c$ and $\lambda \in R.$ If the roots of the equation $x^2+2(a+b+c)x+3\lambda (ab+bc+ca)=0$ are real. Then

1. $\lambda <\frac{4}{3}$
2. $\lambda \in (\frac{1}{3},\frac{5}{3})$
3. $\lambda >\frac{5}{3}$
4. $\lambda \in (\frac{4}{3},\frac{5}{3})$

Q. If the equation $9x^2+6kx+4=0$ has equal roots, then the value of k = .

1. 2 or 0
2. -2 or 2
3. 2 or 2

Q. Determine the set of values of $k$ for which the given quadratic equation has real roots:
$x^2-kx+9=0$

Q.

The nature of roots of $x^2-3x+2=0$ will be:

1. Two distinct real roots

2. No real roots

3. None of these

4. Two equal real roots

Q.

The roots of $2x^2–6x+8=0$ are ___________.

1. real, unequal and irrational

2. real and equal

3. imaginary

4. real, unequal and rational

Q.

Find the value of k for which the quadratic equation $x^2–4x+k=0$ has real and equal roots.

1. -2

2. 4

3. -4

4. 0