Set 3

Point $(a, b)$ on the curve $y = x^2 + 2$ has the closest distance to the line $y = x$. The value of $a + b$ that satisfies is....

If the quadratic equation $ax^2 + bx + c = 0$ has no real roots, then the graph of the function $y = ax^2 + bx + c$ is tangent to the line $y = -x$ when....

The equation of the tangent line to the parabola $y = \sqrt{x} + 1$ passing through the point $(-8, 0)$ is....

Given the function $mx^2 - 2x^2 + 2mx + m - 3$. For the function to always be below the $x$-axis, the possible value of $m$ is....

If a function $y = \sqrt{x^2 - 7}$, then

1. $y = \frac{4}{3}x - \frac{7}{3}$ is the equation of the tangent line at $x = 4$.
2. The curve is a circle centered at $(0, 0)$.
3. The line $y = -\frac{3}{4}x + 6$ intersects perpendicularly the tangent line at $x = 4$.
4. $y = \frac{4}{3}x - \frac{25}{3}$ is the tangent line of the curve at $(4, -3)$.

Given $a, \frac{1}{a}, \frac{1}{a^2 + 2a}$, $a \neq 0$, are respectively the 3rd, 4th, and 5th terms of a geometric sequence with ratio $r \neq 1$. The product of the first five terms of the geometric sequence is....

Let $u_n$ denote the $n$-th term of an arithmetic sequence. Given $u_1 \times u_2 = 10$ and $u_1 \times u_3 = 16$. If the terms of the arithmetic sequence are positive numbers, then $u_{10} = ....$

The roots of the equation $x^3 - 7x^2 + px + q = 0$ form a geometric sequence with ratio $2$. The value of $p + q$ is....

The sum of an infinite geometric series has a value of $\frac{9}{4}$. With the first term $u_1 = a$ and ratio $r = -\frac{1}{a}$. If $a > 0$, determine the value of $3u_6 - u_5$.

If the roots of the polynomial equation $x^3 - 12x^2 + (p + 4)x - (p + 8) = 0$ form an arithmetic sequence with common difference $2$, then $p - 36 = ....$

The value of $\lim_{x \to \frac{\pi}{2}} \frac{\sec 2x + 2}{\tan 2x}$ is....

If $p > 0$ and $\lim_{x \to p} \frac{x^3 + px^2 + qx}{x - p} = 12$, then the value of $p - q$ is....

The value of $\lim_{x \to -y} \frac{\tan x + \tan y}{\frac{x^2 - y^2}{-2y^2} \cdot (1 - \tan x \tan y)}$ is....

If $b, c \neq 0$ and $\lim_{x \to a} \frac{(x - a) \tan(b(a - x))}{\cos(c(x - a)) - 1} = d$, then $b = ....$

The value of $\lim_{x \to 3} \frac{(x + 6) \tan(2x - 6)}{x^2 - x - 6}$ is....

The inequality $\log_2(x^2 - x) \leq 1$ has a solution....

If $x > y \geq 1$ and $\log(x^2 + y^2 + 2xy) = 2 \log(x^2 - y^2)$, then $\log_x(1 + y) = ....$

If $\log_3 x + \log_4 y^2 = 5$, then the maximum value of $\log_3 x \cdot \log_2 y$ is....

The solution set of the inequality $\log_{\frac{1}{2}}(2x - 1) + \log_{\frac{1}{2}}(2 - x) \geq 2 \log_{\frac{1}{2}} x$ is....

If $x_1$ and $x_2$ satisfy the equation $(2 \log x - 1) \cdot \frac{1}{\log_x 10} = \log 10$, then $x_1x_2 = ....$

The solution set of the inequality $|x - 5|^2 - 3|x - 5| + 2 < 0$ is....

All values of $x$ that satisfy $|x| + |x - 2| > 3$ are....

All values of $x$ that satisfy $|x + 1| > x + 3$ and $|x + 2| < 3$ are....

All real numbers $x$ that satisfy $|2x + 1| < 5 - |2x|$ are....

The solution set of $\left|\frac{x}{3} - \frac{1}{4}\right| = \frac{1}{12}$ is....

Given the system of equations

The number of real pairs $(x, y)$ that satisfy the system of equations above is....

Let $\alpha$ and $\beta$ be the roots of the equation $x^2 - bx + 6 = 0$. If $\alpha + \beta$ and $\alpha - \beta$ are the roots of the equation $x^2 - 4x + c = 0$, the equation whose roots are $b$ and $c$ is....

What is the value of $a$ such that the solution $(x, y)$ of the system of equations

satisfies $x\sqrt{y} + 3 > 0$?

Given the quadratic equation $x^2 - 4(k + 1)x + k^2 - k + 7 = 0$ where one root is three times the other root and all roots are greater than $2$. The set of all values of $k$ that satisfy is....

Both roots of the quadratic equation $(m + 2)x^2 - (2m - 1)x + m + 1 = 0$ are negative. The range of values of $m$ that satisfies this is....

Given that the polynomial $f(x)$ when divided by $x^2 + x - 2$ leaves a remainder of $ax + b$, and when divided by $x^2 - 4x + 3$ leaves a remainder of $2bx + a - 1$. If $f(-2) = 7$, then $a^2 + b^2$ = ....

If $h(x)$ is the remainder of dividing $f(x) = 5x^4 - 2x^2 + 7x + 9$ by $x^2 - 5$. The value of $h(1)$ is....

Function $f(x)$ divided by $(x - 1)$ has a remainder of $3$, whereas if divided by $(x - 2)$ has a remainder of $4$. If $f(x)$ is divided by $x^2 - 3x + 2$, then the remainder is....

A third-degree polynomial $P(x) = x^3 + 2x^2 + mx + n$ divided by $x^2 - 4x + 3$ has a remainder of $3x + 2$. Then the value of $n$ = ....

Given the polynomial $g(x) = x^3 + x^2 - x + b$ is divisible by $(x - 1)$. If $g(x)$ is divided by $(x^2 - 1)$, then the remainder is....

The maximum value of the function $y = 4 \sin x \sin(x - 60°)$ is achieved when the value of $x$ = ....

The values of $x$, for $0° \leq x \leq 360°$ that satisfy $\sin x + \sin 2x > \sin 3x$ are....

If $\sin x - \sin y = -\frac{1}{3}$ and $\cos x - \cos y = \frac{1}{2}$, then the value of $\sin(x + y)$ = ....

If $3 \cos \theta - \sin \theta$ is expressed in the form $r \sin(\theta + \alpha)$ with $r > 0$ and $0° < \alpha < 360°$, then....

If $\sin 2x + \cos 2x = -16 \cos x + 8 \sin x + \cos^2 x$ with $0 \leq x \leq \frac{\pi}{2}$, then $\sin 2x$ = ....