Set 2

If the tangent line to the curve $y = x^3 - 3x^2 - 9x$ at point $(a, b)$ has a gradient of $15$, then the possible value of $a + b$ is....

Given $x^2 + 2xy + 4x = -3$ and $9y^2 + 4xy + 12y = -1$. The value of $x + 3y$ is....

If an integer $p$ is a root of $f(x) = 0$ with $f(x) = px^2 - 3x - p - 3$, then the gradient of the tangent line to the curve $y = f(x)$ at the point with abscissa $x = p$ is....

If $(p, q)$ is the vertex point of the graph of function $f(x) = ax^2 + 2ax + a + 1$, with $f(a) = 19$, then $p + 2q + 3a = ....$

Given a straight line passing through $(0, -2)$ and $\left(\frac{3}{2}, 0\right)$. The distance from the parabola $y = x^2 - 1$ to that line is....

Given a sequence $-\frac{1}{2}, \frac{3}{4}, -\frac{1}{8}, \frac{3}{16}, ...$, the $12$th term of this sequence is....

Given a sequence $0, \frac{3}{4}, \frac{3}{16}, \frac{9}{64}, ...$, then the $12$th term of this sequence is....

Given a sequence $0, \frac{5}{6}, \frac{5}{36}, \frac{35}{216}, ...$, the $12$th term of this sequence is....

A geometric sequence has $3$ first terms $a, b, b^2$. If $a$ and $b$ are roots of the quadratic equation $2x^2 + kx + 6 = 0$, then the fourth term of the sequence and the value of $k$ respectively are....

Suppose $x_1$ and $x_2$ are integers that are roots of the quadratic equation $x^2 - (2k + 4)x + (3k + 4) = 0$. If $x_1, k, x_2$ form the first three terms of a geometric series, then the formula for the $n$th term of the series is....

Given $f(x) = \sin^2 x$. If $f'(x)$ represents the first derivative of $f(x)$, then

Given $f(x) = \sqrt{1 + x}$. The value of $\lim_{h \to 0} \frac{f(3 + 2h^2) - f(3 - 3h^2)}{h^2}$ is....

If $\lim_{x \to -3} \frac{\frac{1}{ax} + \frac{1}{3}}{bx^3 + 27} = -\frac{1}{3^5}$, the value of $a + b$ for $a$ and $b$ positive integers is....

If $\log_{a^2}(3^a - 8)^{-4} \cdot \log_3 \sqrt{a} = a - 2$, then $\log_a\left(\frac{1}{8}\right) = ....$

If $(\log_2 x)^2 - (\log_2 y)^2 = \log_2 256$ and $\log_2 x^2 - \log_2 y^2 = \log_2 16$. Then the value of $\log_2 x^6 y^{-2}$ is....

If $2 \log_4 x - \log_4(4x + 3) = -1$, then $\log_2 x = ....$

If $a$ satisfies the equation $\log_2 2x + \log_3 3x = \log_4 4x^2$, then the value of $\log_a 3 = ....$

If $\alpha$ and $\beta$ are roots of the equation $\log_3 x - \log_x\left(2x - 4 + \frac{4}{x}\right) = 1$, then $\alpha + \beta = ....$

If $b > a$, the value of $x$ that satisfies $|x - 2a| + a \leq b$ is....

The solution set of $9 - x^2 \geq |x + 3|$ is....

The solution set of $16 - x^2 \leq |x + 4|$ is....

The solution set of inequality $\log|x + 1| \geq \log 3 + \log|2x - 1|$ is....

The number of real numbers $x$ that satisfy the equation $|x^2 - 4| = x + |x - 2|$ 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 $x, y, z$ satisfy the system of equations

Then the value of $2x + 2y - 3z =$....

If the roots of the equation $x^2 - ax + b = 0$ satisfy the equation $2x^2 - (a + 3)x + (3b - 2) = 0$, then....

1. $a = 3$
2. $b = 2$
3. $2a - 2ab + 3b = 0$
4. $ab = 5$

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 with the tangent line at $x = 4$
4. $y = \frac{4}{3}x - \frac{25}{3}$ is the tangent line to the curve at $(4, -3)$

If $k$ is the smallest positive integer such that two quadratic functions $f(x) = (k - 1)x^2 + kx - 1$ and $g(x) = (k - 2)x^2 + x + 2k$ intersect at two different points $(x_1, y_1)$ and $(x_2, y_2)$, then the quadratic equation with roots $x_1 + x_2$ and $y_1 + y_2$ is....

If the polynomial $f(x)$ is divisible by $(x - 1)$, then the remainder when $f(x)$ is divided by $(x - 1)(x + 1)$ is....

If the polynomial $ax^3 + 2x^2 + 5x + b$ is divided by $(x^2 - 1)$ and gives a remainder of $(6x + 5)$, then $a + 3b$ equals....

Given that $p(x)$ and $g(x)$ are two different polynomials, with $p(10) = m$ and $g(10) = n$. If $p(x)h(x) = \left(\frac{p(x)}{g(x)} - 1\right)(p(x) + g(x))$, $h(10) = -\frac{16}{15}$, then the maximum value of $|m + n| =$....

Given that the polynomial $f(x)$ divided by $2x^2 - x - 1$ has a remainder of $4ax - b$ and divided by $2x^2 + 3x + 1$ has a remainder of $-2bx + a - 11$. If $f(x - 2)$ is divisible by $x - 3$, then $a + 2b + 6 =$....

Given that the polynomial $f(x)$ divided by $x^2 + 3x + 2$ has a remainder of $3bx + a - 2$ and divided by $x^2 - 2x - 3$ has a remainder of $ax - 2b$. If $f(3) + f(-2) = 6$, then $a + b =$....

If angles $A$ and $B$ satisfy the system of equations

Then $\tan(2A + B)$ equals....

Given a rectangular prism $ABCD.EFGH$ where $AB = 6$ cm, $BC = 8$ cm, and $BF = 4$ cm. If $\alpha$ is the angle between $AH$ and $BD$, then $\cos 2\alpha =$....

The function $f(x) = 3 \sin x + 3 \cos x$ defined on the interval $(0, 2\pi)$ reaches its maximum value at $x =$....

If $\begin{bmatrix} \tan x & 1 \\ 1 & \tan x \end{bmatrix} \begin{bmatrix} \cos^2 x \\ \sin x \cos x \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix} \frac{1}{2}$ where $b = 2a$, then $0 \leq x \leq \pi$ that satisfies is....

1. $\frac{\pi}{6}$
2. $\frac{\pi}{12}$
3. $\frac{5\pi}{6}$
4. $\frac{5\pi}{12}$

If $\cos(A + B) = \frac{2}{5}$, $\cos A \cos B = \frac{3}{4}$, then the value of $\tan A \tan B =$....