Mock Test 2 — Quadratic Equations

Let $\alpha, \beta$ be the roots of $x^2 - x - 3 = 0$. If $S_n = \alpha^n + \beta^n$, then the value of $S_5$ is exactly:

The total number of real roots of the higher-degree equation $x^4 - 4x^3 + 6x^2 - 4x + 2 = 0$ is:

The range of the parameter $a$ for which both roots of the equation $x^2 - 2ax + a^2 - 1 = 0$ lie strictly inside the interval $(-3, 5)$ is:

The maximum value achieved by the rational expression $y = \frac{x+1}{x^2 + 4x + 7}$ for real $x$ is:

If the quadratic equations $x^2 + ax + b = 0$ and $x^2 + cx + d = 0$ share exactly one common root, and their other roots are non-zero integers, then the condition requires:

The number of real roots of the equation $\ln x = -x^2 + 4x - 4$ is exactly:

If $a, b, c$ are distinct real numbers, the roots of the equation $(x-a)(x-b) + (x-b)(x-c) + (x-c)(x-a) = 0$ are always:

The range of the function $f(x) = \log_3 (x^2 - 4x + 13)$ over its unrestricted domain is:

If the equation $x^2 - 2(a-1)x + (a+5) = 0$ has roots that are non-real complex numbers, the integer values that $a$ can take lie in the interval:

If $f(x) = a x^2 + b x + c$ satisfies $f(-1) < 1, f(1) > -1,$ and $f(3) < 1$, then the sign of the leading coefficient $a$ must be:

Find the number of distinct real solutions satisfying the equation $(x^2 - 5x + 5)^{x^2 - 9x + 20} = 1$.

Find the number of integer values of $x$ in the domain of the function $f(x) = \sqrt{\log_{10}(16 - x^2)}$.

Evaluate the value of the shared common root of the two cubic/quadratic patterns: $x^2 - 5x + 6 = 0$ and $x^3 - 3x^2 + 2x = 0$ that is greater than 1.

Assertion (A): The expression $f(x) = x^2 + 2x + 5$ is strictly positive ($f(x) > 0$) for all real values of $x$.
Reason (R): If the leading coefficient of a quadratic expression is positive ($a > 0$) and its discriminant is negative ($D < 0$), the corresponding parabola lies entirely above the x-axis.

Assertion (A): The equations $x^2 - 3x + 2 = 0$ and $2x^2 - 6x + 4 = 0$ have both roots in common.
Reason (R): Two quadratic equations share both roots if and only if their discriminants are equal ($D_1 = D_2$).