Discriminant

Q.

Find the value of $k$ if the equation $x^2-2\left(k+1\right)x+k^2=0$ has equal roots?

Q. (α₁, α₂), (α₂, α3) and (α3, α₁) are respectively the roots of x² - 2ax + 2 = 0, x² -2bx + 3 = 0, x² - 2cx + 6 = O. If α₁, α₂, α3 € R+, then the value of a + b + c is equal to (1) 2 (3) 6 (2) 3 (4) 12

Q.

For what value of value of $k,$ does the quadratic equation $9x^2+8kx+16=0$ have equal roots.

Q.

The number of integral values of a for which the quadratic equation (x+a)(x+1991)+1 = 0 has integral roots are

1. 3

2. 0

3. 1

4. 2

Q. If the equation $a{x}^2+bx+6=0$ does not have two distinct real roots, then the least value of $3a+b$

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

Q.

If x is real, the expression $\frac{x+2}{2x^2+3x+6}$ takes all value in the interval

1. $(\frac{1}{13},\frac{1}{3})$

2. $(-\frac{1}{13},\frac{1}{3})$

3. $(-\frac{1}{3},\frac{1}{13})$

4. $[-\frac{1}{3},\frac{1}{13}]$

Q.

If $\alpha ,\beta$ are the roots of the equation $8x^2-3x+27=0$ then the value of ${\left(\frac{{\alpha }^2}{\beta }\right)}^{\frac{1}{3}}+{\left(\frac{{\beta }^2}{\alpha }\right)}^{\frac{1}{3}}$ is.

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

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

3. $4$

4. $\frac{7}{2}$

Q.

If a, b, c are distinct positive real numbers such that b(a + c) = 2ac then the roots of $a{x}^2+bx+c=0$ are

1. real and equal

2. real and distinct

3. Imaginary

4. data insufficient

Q. The discriminant of the quadratic equation $3x^2-4\sqrt{3}x+4=0$ is

1. $D=4$
2. $D=-2$
3. $D=0$
4. $D=96$

Q.

Discriminant of the quadratic equation $\sqrt{5}{x}^2-3\sqrt{3}x-2\sqrt{5}=0$ is

1. $67$

2. $76$

3. $-67$

4. $-76$

Q.

If $\omega$ is an imaginary cube root of unity and $\alpha$, $\beta$, $\gamma$ are the roots of real p, then for any x, y and z, the

values of $\frac{x\alpha +y\beta +z\gamma }{x\beta +y\gamma +z\alpha }$ are:

1. 

2. 

3. 

4. 

Q.

Solve

$-x^2+x-2=0$

Q.

Solve $x^2-2x+\frac{3}{2}=0$

Q. For any two complex number $z_1, z_2$ and any two real numbers $a$ and $b$,
$|az_1-bz_2|^2+|bz_1+az_2|^2=$

Q.

If both the roots of the quadratic equation $x^2$-2kx+$k^2$+k-5=0 are less than 5, then k lies in the interval.

1. 

2. [4, 5]

3. 

4. [5, 6]

Q.

The equation $x^2+px-q=0$ has one root as a square root of the other.
If $p^3+q^2=q(1+kp)$, find the value of k.

1. -3

2. 4

3. -1

4. -2

Q.

The values of a for which( $a^2-1$)$x^2$ + 2(a-1)x + 2 is positive for any x are

1. 

2. 

3. 

4. 

Q. Let $a,b,c,a_1,b_1,c_1,\in R$ and $a{x}^2+bx+c>0\forall x\in R$ and $a_1{x}^2+b_{1}x+c_1>0\forall x\in R.$ Then

1. $a{a}_1{x}^2+b{b}_{1}x+c{c}_1>0\forall x\in R$
2. $a{a}_1{x}^2+b{b}_{1}x+c{c}_1<0\forall x\in R$
3. $a{a}_1{x}^2+b{b}_{1}x+c{c}_1=0$, will have real roots.
4. Nothing can be said in general about the nature of roots of, $a{a}_1{x}^2+b{b}_{1}x+c{c}_1=0$

Q. If domain of $f\left(x\right)$ is $[0,1],$ then domain of $f\left(\right\{x{\}}^3+1),$ (where $\{.\}$ represents fractional part function) is

1. $\{x:x=k,k\in Q\}$
2. $\{x:k\le x<k+\frac{1}{2},k\in I\}$
3. $\{x:x=k,k\in I\}$
4. $(-\infty ,0)$

Q. Let $f\left(x\right)=\cos \left({\cos }^{-1}\left(\frac{2}{\left[x\right]-1}\right)\right)$ where $[\cdot ]$ is the greatest integer function, then

1. Range of $f\left(x\right)=[-1,1]-\left\{0\right\}$
2. Domain of $f\left(x\right)=R-\{0,1\}$
3. Domain of $f\left(x\right)$ excludes $3$ integral values of $x$
4. Range of $f\left(x\right)=\{\frac{2}{k}:k\in Z-\{-1,0,1\left\}\right\}$

Q. Let $a\in R$. For possible real values of $x$, $\frac{x^2+ax+3}{x^2+x+a}$ takes all real values, then

1. $4a^3+39<0$
2. $4a^3+39\ge 0$
3. $a\ge \frac{1}{4}$
4. $a<\frac{1}{4}$

Q. If the cube root of unity are $1,\omega ,{\omega }^2$, then the roots of the equation $(x+1{)}^3+8=0$ are

1. None of the above
2. 
3. 
4. -1, -1, -1

Q. If the equation $5x^2-10x+lo{g}_{\frac{1}{5}}a=0$ has real roots then the minimum value of a is

1. $\frac{1}{5^5}$
2. $\frac{1}{{10}^{10}}$
3. $\frac{1}{5^{10}}$
4. none of these

Q. p=3xsquare+px+3 , roots are reciprocal of another root find tje value of p

Q.

Find the value of $k$ for which the quadratic equation $\left(k+4\right)x^2+\left(k+1\right)x+1=0$ has equal roots.

Q.

Solve

$x^2-x+2=0$

Q.

Sum of the real values of 'a' for which the equation $(a^2-3a+2)x^2+(a^2-4a+3)x+(a^2-6a+5)=0$ has three distinct roots

1. 1

2. None

3. 11

4. 10

Q. Solve the equation 21 x 2 – 28 x + 10 = 0

Q.

If a < 0 and D < 0, what can be inferred about $f\left(x\right)$ = $a{x}^2+bx+c$?

1. f(x) is always negative

2. f(x) = is always positive

3. f(x) = 0 always

4. cannot determine nature of f(x)

Q. If the roots of the quadratic equation $a{x}^2+bx+c=0$ are opposite in sign and negative root has greater magnitude, then which of the following is/are correct?

1. $a>0,b>0\text{and}c<0$
2. $a<0,b<0\text{and}c>0$
3. $b>0,c>0\text{and}a<0$
4. $b<0,c<0\text{and}a>0$