The smallest positive period of the function $f(x) = \sin\left(2x + \frac{\pi}{3}\right)$ is A. $4\pi$ B. $2\pi$ C. $\pi$ D. $\frac{\pi}{2}$
Q4
Vectors Introduction & 2DVector Properties and Identities (Conceptual)View
Let non-zero vectors $\boldsymbol{a}, \boldsymbol{b}$ satisfy $|\boldsymbol{a}+\boldsymbol{b}| = |\boldsymbol{a}-\boldsymbol{b}|$, then A. $\boldsymbol{a} \perp \boldsymbol{b}$ B. $|\boldsymbol{a}| = |\boldsymbol{b}|$ C. $\boldsymbol{a} \parallel \boldsymbol{b}$ D. $|\boldsymbol{a}| > |\boldsymbol{b}|$
Q5
InequalitiesSolve Polynomial/Rational Inequality for Solution SetView
If $a > 1$, then the range of values of $x$ satisfying $\log_a(x^2 - 2) < \log_a x$ is A. $(\sqrt{2}, +\infty)$ B. $(\sqrt{2}, 2)$ C. $(1, \sqrt{2})$ D. $(1, 2)$
Q7
InequalitiesLinear Programming (Optimize Objective over Linear Constraints)View
Let $x, y$ satisfy the linear constraints $\left\{\begin{array}{l} 2x - 3y + 3 \geq 0, \\ y + 3 \geq 0, \\ 3x - 3 \leq 0 \end{array}\right.$ and let $z = 2x + y$. The minimum value of $z$ is A. $-15$ B. $-9$ C. $1$ D. $9$
Q8
Laws of LogarithmsAnalyze a Logarithmic Function (Limits, Monotonicity, Zeros, Extrema)View
The monotone increasing interval of the function $f(x) = \ln(x^2 - 2x - 9)$ is A. $(-\infty, -2)$ B. $(-\infty, 1)$ C. $(1, +\infty)$ D. $(4, +\infty)$
Five cards numbered $1, 2, 3, 4, 5$ are shuffled and three are drawn in order. The probability that the number on the first card is greater than the number on the third card is A. $\dfrac{1}{10}$ B. $\dfrac{1}{5}$ C. $\dfrac{3}{10}$ D. $\dfrac{2}{5}$
The maximum value of the function $f(x) = 2\cos x + \sin x$ is \_\_\_\_
Q14
Composite & Inverse FunctionsRecover a Function from a Composition or Functional EquationView
It is known that the function $f(x)$ is an odd function defined on $\mathbb{R}$. When $x \in (-\infty, 0)$, $f(x) = 2x^3 + x^2$. Then $f(2) = $ \_\_\_\_
Q16
Sine and Cosine RulesDetermine an angle or side from a trigonometric identity/equationView
In $\triangle ABC$, the interior angles $A$, $B$, $C$ have opposite sides $a$, $b$, $c$ respectively. If $2b\cos B = a\cos C + c\cos A$, then $B = $ \_\_\_\_
Q17
12 marksArithmetic Sequences and SeriesFind General Term FormulaView
(12 points) Let $\{a_n\}$ be a sequence with $a_1 + a_2 = 2$. (1) If $\{a_n\}$ is an arithmetic sequence and $a_1 + a_3 = 5$, find the general formula for $\{a_n\}$. (2) If $\{a_n\}$ is a geometric sequence and $T_n$ denotes the sum of the first $n$ terms of another related sequence with $T_n = 21$, find $S_n$.
(12 points) To compare the yields of an old and a new breeding method, a survey was conducted on 100 aquaculture farms, recording the yield (in kg) of a certain aquatic product. The frequency distribution histogram is given. (1) Let $A$ denote the event ``the yield using the old breeding method is less than 50 kg''. Estimate the probability of $A$. (2) Complete the contingency table below, and use the chi-squared test to determine whether we can be 99\% confident that yield is related to breeding method.
Q20
12 marksConic sectionsLocus and Trajectory DerivationView
(12 points) Let $O$ be the origin of coordinates. Point $M$ is on the ellipse $C: \dfrac{x^2}{2} + y^2 = 1$. The perpendicular from $M$ to the $x$-axis intersects the $x$-axis at $N$. Point $P$ satisfies $\overrightarrow{NP} = \sqrt{2}\,\overrightarrow{NM}$. (1) Find the trajectory equation of point $P$. (2) Let point $Q$ be on the line $x = -3$, and $\overrightarrow{OP} \cdot \overrightarrow{PQ} = 1$. Prove that the line $l$ passing through point $P$ and perpendicular to $OQ$ passes through the right focus $F$ of $C$.
Q21
12 marksApplied differentiationInequality proof via function studyView
(12 points) Let the function $f(x) = (1-x^2)e^x$. (1) Discuss the monotonicity of $f(x)$. (2) When $x \geq 0$, $f(x) \leq ax + 1$. Find the range of values of $a$.