Quadratic Formula for Finding Roots

Q. The sum of all real values of x satisfying the equation $(x^2-5x+5{)}^{x^2+4x-60}=1$ is :

1. 6
2. 3
3. -4
4. 5

Q.

The solution of ${\sin }^{-1}\left(x\right)-{\sin }^{-1}\left(2x\right)=\pm \frac{\mathrm{\pi }}{3}$ is

1. $\pm \frac{1}{3}$

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

3. $\pm \frac{\sqrt{3}}{2}$

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

Q. The number of the real roots of the equation $(x+1{)}^2+|x-5|=\frac{27}{4}$ is

Q.

Using the quadratic formula solve ${(a+b)}^2{x}^2+8(a^2-b^2)x+16{(a-b)}^2=0$ for $x$.

Q. The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has

1. one purely imaginary root
2. all real roots
3. two real and two purely imaginary roots
4. neither real nor purely imaginary roots

Q. If x = $\sqrt{1+\sqrt{1+\sqrt{1+.....toinfinity}}},$ then x =

1. $\frac{1-\sqrt{5}}{2}$
2. $\frac{1+\sqrt{5}}{2}$
3. None of these
4. $\frac{1\pm \sqrt{5}}{2}$

Q. The value of $\sqrt{8+2\sqrt{8+2\sqrt{8+2\sqrt{8+...\infty }}}}$ is

1. $6$
2. $-2$ and $4$
3. $4$
4. $8$

Q. The sum of the roots of the equation $3^{{\log }_{\sqrt{2}}x}-12\cdot 3^{{\log }_{2}x}+27=0$ is

1. $6$
2. $8$
3. $12$
4. $4$

Q. If $a\in R$ and the equation $-3(x-[x]{)}^2+2(x-\left[x\right])+a^2=0$ (where, [x] denotes the greatest integer $\ge x)$ has no integral solution, then all possible values of a lie in the interval

1. $(-1,0)\cup (0,1)$
2. (-2, -1)
3. $(-\infty ,-2)\cup (2,\infty )$
4. (1, 2)

Q. The set of values of $a$ for which the function $f\left(x\right)=\frac{a{x}^3}{3}+(a+2)x^2+(a-1)x+2$ possesses a negative point of inflection is

1. $\{-4,5\}$
2. $(-\infty ,-2)\cup (0,\infty )$
3. $\varphi$
4. $(-2,0)$

Q. The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has

1. one purely imaginary root
2. all real roots
3. two real and two purely imaginary roots
4. neither real nor purely imaginary roots

Q. Let $\alpha$ and $\beta$ be the roots of equation $x^2-6x-2=0$ . If $a_n={\alpha }^n{\beta }^n$, for $n\le 1$, then the value of $\frac{a_{10}-2a_8}{2a_9}$ is equal to :

1. 3
2. -3
3. 6
4. -6

Q. if $\alpha ,\beta$ are the roots of the equation $x^2-x+1=0$, then ${\alpha }^{2009}+{\beta }^{2009}$ =

1. -1
2. 1
3. 2
4. -2

Q. The sum of the roots of the equation $3^{{\log }_{\sqrt{2}}x}-12\cdot 3^{{\log }_{2}x}+27=0$ is

1. $4$
2. $6$
3. $8$
4. $12$

Q. Let $-\frac{\pi }{6}<\theta <-\frac{\pi }{12}$. Suppose ${\alpha }_1$ and ${\beta }_1$ are the roots of the equation $x^2-2x\sec \theta +1=0$ and ${\alpha }_2$ and ${\beta }_2$ are the roots of the equation $x^2+2x\tan \theta -1=0$. If ${\alpha }_1>{\beta }_1$ and ${\alpha }_2>{\beta }_2$, then ${\alpha }_1+{\beta }_2$ equals:

1. $2\sec \theta$
2. $-2\tan \theta$
3. $2(\sec \theta -\tan \theta )$
4. $0$

Q. The value of $2+\frac{1}{2+\frac{1}{2+.....\infty }}$ is

1. $1-\sqrt{2}$
2. $1\pm \sqrt{2}$
3. $\sqrt{2}$
4. None of these

Q. Let $\alpha$ and $\beta$ be the roots of equation $x^2-6x-2=0$ . If $a_n={\alpha }^n{\beta }^n$, for $n\le 1$, then the value of $\frac{a_{10}-2a_8}{2a_9}$ is equal to :

1. 3
2. -3
3. 6
4. -6

Q. If x = $\sqrt{1+\sqrt{1+\sqrt{1+.....toinfinity}}},$ then x =

1. $\frac{1+\sqrt{5}}{2}$
2. $\frac{1-\sqrt{5}}{2}$
3. $\frac{1\pm \sqrt{5}}{2}$
4. None of these

Q.

If $a{x}^2+bx+c=0$, then x =

1. $\frac{b\pm \sqrt{b^2-4ac}}{2a}$

2. $\frac{-b\pm \sqrt{b^2-ac}}{2a}$

3. $\frac{2c}{-b\pm \sqrt{b^2-4ac}}$

4. None of these