Roots of a Quadratic Equation

Q. The number of real roots of the equation $5+|2^x-1|=2^x(2^x-2)$ is

1. $4$
2. $1$
3. $2$
4. $3$

Q.

If $[x{]}^2-5[x]+6=0$, where [.] denotes the greatest integer function, then

1. $xϵ[3,4]$

2. $xϵ(2,3]$

3. $xϵ[2,3]$

4. $xϵ[2,4]$

Q. If $\sqrt{\frac{2}{3}}$ is a solution of equation $3x^2+mx+2=0$. Find the value of $m$.

Q. If the roots of the equation $a{x}^2+bx+c=0$ are in the ratio m:n, then

1. 
2. 
3. 
4. 

Q. The largest integral value of $a$ for which every solution of the equation $x\left(\right[x]-5)+2\left\{x\right\}+6=0$ satisfies the inequality $(a-3)x^2+2(a+3)x-8a\le 0,$ where $\left[y\right]$ and $\left\{y\right\}$ denote the greatest integer and fractional part functions respectively, is

Q.

If $x=\frac{1}{2+\sqrt{3}}$, find the value of $x^3-x^2-11x+3$

1. $0$

2. $3$

3. $3x$

4. $4$

Q. If $f\left(x\right)$ is a quadratic expression such that $f\left(1\right)+f\left(2\right)=0$, and $-1$ is a root of $f\left(x\right)=0$ then the other root of $f\left(x\right)=0$ is:

1. $-\frac{5}{8}$
2. $-\frac{8}{5}$
3. $\frac{5}{8}$
4. $\frac{8}{5}$

Q. The sum of all the real roots of the equations $|x-2{|}^2+|x-2|-2=0$ is...

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

Q. If - 3 + ${ix}^{2}yand{x}^2+y+4i$ be conjugate of each other, then the value of x and y is

1. 
2. 
3. None of these
4. - 1 , 4

Q. If the difference of the roots of $x^2-px+q=0$ is unity, then the value of $p^2-4q$ is

Q. If the sum of a natural number and its reciprocal is $\frac{50}{7}$, then the number is

Q. The equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ has

1. no solution
2. one solution
3. more than two solutions
4. two solutions

Q. The number of non-zero solution(s) of the equation $x^2-5x-6\text{sgn}\left(x\right)=0$ is (where $\text{sgn}$ represents the signum function)

1. $2$
2. $3$
3. $1$
4. $0$

Q. Which one of the following is not a function?

1. $\left\{\right(x,y):x,yϵR,x^2=y\}$
2. $\left\{\right(x,y):x,yϵR,y^2=x\}$
3. $\left\{\right(x,y):x,yϵR,x^3=y\}$
4. $\left\{\right(x,y):x,yϵR,y^3=x\}$

Q.

The solution set of the equation $x^{lo{g}_x(1-x{)}^2}=9$ is

1. {0, - 2, 4}

2. None of these

3. {- 2, 4}

4. {4}

Q. The number of integral roots of the equation $x(x+1)(x+2)(x+3)=120$ is

Q.

If x is real and satisfies x + 2 > $\sqrt{x+4}$, then

1. 

2. 

3. 

4. 

Q. A quadratic polynomial $p\left(x\right)$ has $1+\sqrt{5}$ and $1-\sqrt{5}$ as its zeros and $p\left(1\right)=2$. Then the value of $p\left(0\right)$ is

1. $\frac{8}{5}$
2. $-\frac{4}{5}$
3. $\frac{4}{5}$
4. $-\frac{8}{5}$

Q. If the roots of $x^2+mx+n=0$ are twice the roots of $x^2+px+m=0,p\ne 0$, then the value of $\frac{n}{p}$ is

Q. The coefficient of $x$ in the equation $x^2+px+q=0$, was wrongly written as $17$ in place of $13$ and the roots thus found to be $-2$ and $-15$. The roots of the correct equation is/are

1. $-10$
2. $-3$
3. $-15$
4. $2$

Q.

The roots of the equation $x^4-8x^2-9$ = 0 are

1. 

2. 

3. 

4. 3i , 2

Q. The number of real solutions of the equation, $x^2-|x|-12=0$ is

1. $4$
2. $2$
3. $1$
4. $3$

Q. If the equation $a_n{x}^n+a_{n-1}{x}^{n-1}+.........+a_{1}x=0,a_1\ne 0,n\ge 2,$ has a positive root~ $x=\alpha$, then the equation $n{a}_n{x}^{n-1}+(n-1)a_{n-1}{x}^{n-2}+.......+a_1=0$ has a positive root, which is :

1. Greater than ∝
2. Smaller than ∝
3. Greater than or equal to $\propto$
4. Equal to ∝

Q.

Find the roots of the quadratic equation by using the quadratic formula the following: $-x^2+7x-10=0$.

Q.

How many roots, real or complex, does the polynomial $7+5x^4-3x^2=0$ have in all?

Q.

The complete solution of the in equation $x^2-4x<12$ is

1. x < -2 or x > 6

2. -6 < x < 2

3. 2 < x < 6

4. -2 < x < 6

Q. The quadratic equation whose roots are $-4$ and $6$ is given by

1. $x^2+2x-24=0$
2. $x^2-2x-24=0$
3. $x^2-2x+24=0$
4. $x^2-4x-24=0$

Q.

The number of real solution of:$|2x-x^2-3|=1$ is

1. 0

2. 2

3. 3

4. 4

Q. Solve the equation $x^2+3x+9=0$

Q. Let $\alpha ,\beta ,\gamma ,\delta$ be the roots of the equation $x^4-x^3-x^2-1=0.$ Also consider $p\left(x\right)=x^6-x^5-x^3-x^2-x,$ then the value of $p\left(\alpha \right)+p\left(\beta \right)+p\left(\gamma \right)+p\left(\delta \right)$ cannot be:

1. $4$
2. $5$
3. $6$
4. $7$