Mock Test 1 — Quadratic Equations

If the equations $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ have exactly one common root, then the value of $(a+b)$ is:

The range of the function $f(x) = \frac{x^2 - 3x + 4}{x^2 + 3x + 4}$ for all $x \in \mathbb{R}$ is:

Let $\alpha, \beta$ be the roots of $x^2 - 5x - 1 = 0$. If $a_n = \alpha^n + \beta^n$, then the value of $\frac{a_{12} - a_{10}}{a_{11}}$ is equal to:

The number of real solutions of the equation $x^2 - 3|x| + 2 = 0$ is exactly:

If both roots of the quadratic equation $x^2 - 4mx + 4m^2 - 1 = 0$ are strictly greater than $-1$, then the parameter $m$ must satisfy:

The maximum value of the expression $y = 3 + 4x - 2x^2$ for real values of $x$ is:

The total number of positive real roots of the higher-degree equation $x^7 - x^4 + x^2 - 1 = 0$ is at most:

If $a, b, c$ are rational numbers and the discriminant of $ax^2 + bx + c = 0$ is $D = 20$, then the roots are always:

The solution set of the inequality $\log_{2}(x^2 - 3x) \le 2$ has how many integers?

If the roots of $(q-r)x^2 + (r-p)x + (p-q) = 0$ are real and equal, then $p, q, r$ form an:

Find the number of real solutions to the radical equation $\sqrt{x^2 + 1} - x = 2$.

Evaluate the value of $m$ for which the expression $mx^2 - 2(m+2)x + m + 5 = 0$ has equal roots.

Find the minimum integer value of $k$ for which the equation $x^2 - 2kx + 16 = 0$ has real and distinct roots.

Assertion (A): The quadratic equation $x^2 + 2ix - 1 = 0$ has equal roots, but its roots are not real.
Reason (R): If the discriminant $D = 0$, the roots of a quadratic equation are always real, regardless of whether the coefficients are real or imaginary.

Assertion (A): The cubic equation $x^3 - 3x + 1 = 0$ must have at least one real root inside the open interval $(0, 2)$.
Reason (R): If a continuous polynomial function satisfies $P(a) \cdot P(b) < 0$, there exists at least one real solution to $P(x) = 0$ inside the open interval $(a, b)$.