The parametric equation of line $l$ is $\left\{ \begin{array} { l } x = 1 + 3 t , \\ y = 2 + 4 t \end{array} \right.$ ($t$ is the parameter). The distance from point $(1,0)$ to line $l$ is (A) $\frac { 1 } { 5 }$ (B) $\frac { 2 } { 5 }$ (C) $\frac { 4 } { 5 }$ (D) $\frac { 6 } { 5 }$
In astronomy, the brightness of a celestial body can be described by magnitude or luminosity. The magnitude and luminosity of two stars satisfy $m _ { 2 } - m _ { 1 } = \frac { 5 } { 2 } \lg \frac { E _ { 1 } } { E _ { 2 } }$, where the luminosity of a star with magnitude $m _ { k }$ is $E _ { k } ( k = 1,2 )$. Given that the magnitude of the Sun is $- 26.7$ and the magnitude of Sirius is $- 1.45$, the ratio of the luminosity of the Sun to that of Sirius is (A) $10 ^ { 10.1 }$ (B) 10.1 (C) $\lg 10.1$ (D) $10 ^ { - 10.1 }$
Let points $A , B , C$ be non-collinear. Then ``the angle between $\overrightarrow { A B }$ and $\overrightarrow { A C }$ is acute'' is ``$| \overrightarrow { A B } + \overrightarrow { A C } | > | \overrightarrow { B C } |$'' a (A) sufficient but not necessary condition (B) necessary but not sufficient condition (C) necessary and sufficient condition (D) neither sufficient nor necessary condition
There are many beautifully shaped and meaningful curves in mathematics. The curve $C : x ^ { 2 } + y ^ { 2 } = 1 + | x | y$ is one of them (as shown in the figure). Three conclusions are given: (1) The curve $C$ passes through exactly 6 lattice points (points with both integer coordinates); (2) The distance from any point on curve $C$ to the origin does not exceed $\sqrt { 2 }$; (3) The area of the ``heart-shaped'' region enclosed by curve $C$ is less than 3. The sequence numbers of all correct conclusions are (A) (1) (B) (2) (C) (1)(2) (D) (1)(2)(3)
Let $\left\{ a _ { n } \right\}$ be an arithmetic sequence with the sum of the first $n$ terms being $S _ { n }$. If $a _ { 2 } = - 3 , S _ { 5 } = - 10$, then $a _ { 5 } =$ $\_\_\_\_$, and the minimum value of $S _ { n }$ is $\_\_\_\_$.
A certain geometric solid is obtained by removing a quadrangular prism from a cube. Its three-view drawing is shown in the figure. If the side length of each small square on the grid paper is 1, then the volume of this geometric solid is $\_\_\_\_$.
Let $l , m$ be two different lines outside plane $\alpha$. Three propositions are given: (1) $l \perp m$; (2) $m \| \alpha$; (3) $l \perp \alpha$. Using two of these propositions as conditions and the remaining one as the conclusion, write out a correct proposition: $\_\_\_\_$.
Let the function $f ( x ) = \mathrm { e } ^ { x } + a \mathrm { e } ^ { - x }$ ($a$ is a constant). If $f ( x )$ is an odd function, then $a =$ $\_\_\_\_$; if $f ( x )$ is an increasing function on $\mathbb { R }$, then the range of $a$ is $\_\_\_\_$.
In $\triangle A B C$, $a = 3 , \quad b - c = 2 , \quad \cos B = - \frac { 1 } { 2 }$. (I) Find the values of $b$ and $c$; (II) Find the value of $\sin ( B - C )$.
As shown in the figure, in the quadrangular pyramid $P - A B C D$, $P A \perp$ plane $A B C D$, $A D \perp C D$, $A D \| B C$, $P A = A D = C D = 2$, $B C =$ 3. $E$ is the midpoint of $P D$, and point $F$ is on $P C$ such that $\frac { P F } { P C } = \frac { 1 } { 3 }$. (I) Prove that: $C D \perp$ plane $P A D$; (II) Find the cosine of the dihedral angle $F - A E - P$; (III) Let point $G$ be on $P B$ such that $\frac { P G } { P B } = \frac { 2 } { 3 }$. Determine whether line $A G$ lies in plane $A E F$, and explain the reason.
The parabola $C : x ^ { 2 } = - 2 p y$ passes through the point $( 2 , - 1 )$. (I) Find the equation of parabola $C$ and the equation of its directrix; (II) Let $O$ be the origin. A line $l$ with non-zero slope passes through the focus of parabola $C$ and intersects parabola $C$ at two points $M , N$. The line $y = - 1$ intersects lines $O M$ and $O N$ at points $A$ and $B$ respectively. Prove that the circle with $A B$ as diameter passes through two fixed points on the $y$-axis.
Given the function $f ( x ) = \frac { 1 } { 4 } x ^ { 3 } - x ^ { 2 } + x$. (I) Find the equation of the tangent line to the curve $y = f ( x )$ with slope 1; (II) When $x \in [ - 2,4 ]$, prove that: $x - 6 \leqslant f ( x ) \leqslant x$; (III) Let $F ( x ) = | f ( x ) - ( x + a ) | ( a \in \mathbb { R } )$. Let $M ( a )$ denote the maximum value of $F ( x )$ on the interval $[ - 2,4 ]$. When $M ( a )$ is minimized, find the value of $a$.
Given a sequence $\left\{ a _ { n } \right\}$, if we select the $i _ { 1 }$-th term, the $i _ { 2 }$-th term, $\cdots$, the $i _ { m }$-th term $\left( i _ { 1 } < i _ { 2 } < \cdots < i _ { m } \right)$, and if $a _ { i _ { 1 } } < a _ { i _ { 2 } } < \cdots < a _ { i _ { m } }$, then the new sequence $a _ { i _ { 1 } } , a _ { i _ { 2 } } , \cdots, a _ { i _ { m } }$ is called an increasing subsequence of length $m$ of $\left\{ a _ { n } \right\}$. By convention, any single term of the sequence $\left\{ a _ { n } \right\}$ is an increasing subsequence of length 1 of $\left\{ a _ { n } \right\}$. (I) Write out an increasing subsequence of length 4