Question

Choose the correct answers for the following questions.
(1) Which one is the quadratic equation ?

(A) $\frac{5}{x}-3=x^2$

(B) x(x + 5) = 2

(C) n – 1 = 2n

(D) $\frac{1}{x^2}\left(x+2\right)=x$

(2) Out of the following equations which one is not a quadratic equation ?

(A) x2 + 4x = 11 + x2

(B) x2 = 4x

(C) 5x2 = 90

(D) 2x – x2 = x2 + 5

(3) The roots of x² + kx + k = 0 are real and equal, find k.

(A) 0

(B) 4

(C) 0 or 4

(D) 2

(4) For $\sqrt{2}{x}^2-5x+\sqrt{2}=0$ find the value of the discriminant.

(A) –5

(B) 17

(C) $\sqrt{2}$

(D) $2\sqrt{2}-5$

(5) Which of the following quadratic equations has roots 3,5 ?

(A) x2 – 15x + 8 = 0	(B) x2 – 8x + 15 = 0
(C) x2 + 3x + 5 = 0	(D) x2 + 8x – 15 = 0

(6) Out of the following equations, find the equation having the sum of its roots –5.

(A) 3x2 – 15x + 3 = 0	(B) x2 – 5x + 3 = 0
(C) x2 + 3x – 5 = 0	(D) 3x2 + 15x + 3 = 0

(7) $\sqrt{5}{m}^2-\sqrt{5}m+\sqrt{5}=0$ which of the following statement is true for this given equation ?

(A) Real and uneual roots	(B) Real and equal roots
(C) Roots are not real	(D) Three roots.

(8) One of the roots of equation x² + mx – 5 = 0 is 2; find m.

(A) –2

(B) $-\frac{1}{2}$

(C) $\frac{1}{2}$

(D) 2

Answer 1

(1) The general form of a quadratic equation is $a{x}^2+bx+c=0$.
For (A), $\frac{5}{x}-3=x^2$
$5-3x=x^3$
Thus, (A) is not quadratic equation.
For (B) x(x + 5) = 2
$x^2+5x-2=0$
Thus, (B) is a qudratic equation.
(C) is also not a quadratic equation.
(D) $\frac{1}{x^2}\left(x+2\right)=x$
$x+2=x^3$ which is also not quadratic.
Hence, the correct answer is option (B).

(2) (A) x² + 4x = 11 + x²
$\Rightarrow 4x=11$
Thus, (A) is not a quadratic equation.
(B) and (C) can be written as $x^2-4x+0=0$ and $5x^2-90+0=0$ respectively.
So, B and C are quadratic equations.
(D) 2x – x² = x² + 5 can be written as $2x^2-2x+5=0$.
So, it also forms a quadratic equation.
Hence, the correct answer is option (A).

(3) Given quadratic equation is x² + kx + k = 0
For real and equal roots, D = 0
$$\begin{gathered} b^2-4ac=0 \\ \Rightarrow k^2-4\times 1\times k=0 \\ \Rightarrow k^2-4k=0 \\ \Rightarrow k\left(k-4\right)=0 \\ \Rightarrow k=0,k=4 \end{gathered}$$
Hence, the correct answer is option C.

(4) $\sqrt{2}{x}^2-5x+\sqrt{2}=0$
$$\begin{gathered} a=\sqrt{2},b=-5,c=\sqrt{2} \\ D=b^2-4ac \\ ={\left(-5\right)}^2-4\times \sqrt{2}\times \sqrt{2} \\ =25-8 \\ =17 \end{gathered}$$
Hence, the correct answer is option B.

(5) The roots of the quadratic equation x² – 8x + 15 = 0 are
$$\begin{gathered} x^2-5x-3x+15=0 \\ \Rightarrow x\left(x-5\right)-3\left(x-5\right)=0 \\ \Rightarrow \left(x-3\right)\left(x-5\right)=0 \\ \Rightarrow x=3,5 \end{gathered}$$
Hence, the correct answer is option B.

(6) For the equation 3x² + 15x + 3 = 0
$\mathrm{Sum}\mathrm{of}\mathrm{roots}=\frac{-b}{a}=\frac{-15}{3}=-5$
Hence, the correct answer is option D.

(7) For the given quadratic equation $\sqrt{5}{m}^2-\sqrt{5}m+\sqrt{5}=0$
$D=b^2-4ac={\left(-\sqrt{5}\right)}^2-4\times \sqrt{5}\times \sqrt{5}=5-20=-15$
Since D < 0 so, the roots are not real.
Hence, the correct answer is option C.

(8) For the equation, x² + mx – 5 = 0
one of the roots is 2. So, 2 should satisfy the given equation.
$$\begin{gathered} {\left(2\right)}^2+m\times \left(2\right)-5=0 \\ \Rightarrow 4+2m-5=0 \\ \Rightarrow 2m-1=0 \\ \Rightarrow m=\frac{1}{2} \end{gathered}$$
Hence, the correct answer is option C.