Nature of Roots

Q.
Find the values of p if the equation $x^2-x+p=0$ does not possess real roots.
(A) p > $\frac{1}{8}$
(B) p > $\frac{1}{4}$
(C) p > $\frac{1}{2}$
(D) p > 1
$\left[1\right]$

Q.

The condition for the equation $x^2+bx+c=0$ to have real roots is ____.

1. $b^2-4c\le 0$

2. $b^2-4c<0$

3. $b^2-4c\ge 0$

4. $b^2+4c>0$

Q.

Vijay inherited a large amount of wealth from his father and by establishing a soft drinks company, he squared his wealth. He then started a company which manufactured airplane toys and incurred huge losses amounting to k (>0) times the wealth he inherited from his father. His total assets now stand at a debt of 4 crore. Comment on the value of k.

1. Any value of “k” is possible

2. k is any value greater than or equal to 4

3. None of the above

4. Depends on what amount is inherited initially - “x”

Q.

Comment on the nature of the roots of the equation $2x^2–6x+8=0$

1. real, unequal and rational

2. real and equal

3. real, unequal and irrational

4. imaginary

Q.

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

1. -4

2. 0

3. -2

4. 4

Q.

The equation $x^2+x-5=0$ has two distinct real roots.

1. True

2. False

Q.

Solve for x if $4(2x+3{)}^2-(2x+3)-14=0$.

1. $x=-\frac{1}{2},-\frac{19}{8}$

2. $x=\frac{1}{2},\frac{19}{8}$

3. $x=\frac{1}{2},-\frac{19}{8}$

4. $x=-\frac{1}{2},\frac{19}{8}$

Q.

In the figure given above, PA and PB are tangents to the circle from point P. How many real values of $x$ exist such that the length of PA is $x$ and length of PB is $x^2+1$?

1. 2

2. 3

3. 0

4. 1

Q.

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

1. 1

2. 2

3. Infinite

4. 0

Q.

A bus covers a distance of 240 km at a uniform speed. Due to heavy rain its speed gets reduced by 10 km/h and takes two hours longer to cover the distance. Assuming the uniform speed to be 'x' km/h, form an equation and solve it to evaluate 'x'.

1. 50 kmph

2. 40 kmph

3. 90 kmph

4. 20 kmph