The graph of the function $f ( x ) = \sin \left( \omega x + \frac { \pi } { 3 } \right) ( \omega > 0 )$ is shifted left by $\frac { \pi } { 2 }$ units. If the minimum value of the resulting curve is $-1$ and the distance between two consecutive minimum points is $\pi$, then the minimum value of $\omega$ is A. $\frac { 1 } { 6 }$ B. $\frac { 1 } { 4 }$ C. $\frac { 1 } { 3 }$ D. $\frac { 1 } { 2 }$
From 6 cards labeled $1,2,3,4,5,6$ respectively, 2 cards are randomly drawn without replacement. The probability that one of the drawn numbers is a multiple of the other is A. $\frac { 1 } { 5 }$ B. $\frac { 1 } { 3 }$ C. $\frac { 2 } { 5 }$ D. $\frac { 2 } { 3 }$
The graph of the function $f ( x ) = \left( 3 ^ { x } - 3 ^ { - x } \right) \cos x$ on the interval $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is approximately [see figures A, B, C, D in the original paper].
Given vectors $\boldsymbol { a } = ( m , 3 ) , \boldsymbol { b } = ( 1 , m + 1 )$ . If $\boldsymbol { a } \perp \boldsymbol { b }$ , then $m =$ $\_\_\_\_$ .
Point $M$ lies on the line $2 x + y - 1 = 0$. Both points $( 3,0 )$ and $( 0,1 )$ lie on circle $\odot M$. Then the equation of $\odot M$ is $\_\_\_\_$ .
For the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with eccentricity $e$, write out a value of $e$ that satisfies the condition ``the line $y = kx$ intersects the hyperbola at four distinct points'' and give one such value $\_\_\_\_$ .
In $\triangle A B C$, point $D$ is on side $B C$, $\angle A D B = 120 ^ { \circ } , A D = 2 , C D = 2 B D$ . If $S_{\triangle ABD} : S_{\triangle ACD} = 1 : 2$, then $B D =$ $\_\_\_\_$ .
Let $S _ { n }$ denote the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given $\frac { 2 S _ { n } } { n } + n = 2 a _ { n } + 1$ . (1) Prove that $\left\{ a _ { n } \right\}$ is an arithmetic sequence; (2) If $a _ { 4 } , a _ { 7 } , a _ { 9 }$ form a geometric sequence, find the minimum value of $S _ { n }$ .
Given functions $f ( x ) = x ^ { 3 } - x , g ( x ) = x ^ { 2 } + a$. The tangent line to the curve $y = f ( x )$ at the point $\left( x _ { 1 } , f \left( x _ { 1 } \right) \right)$ is also tangent to the curve $y = g ( x )$ at some point. (1) If $x _ { 1 } = - 1$ , find $a$ ; (2) If $x_1 \neq 0$, prove that $a > \frac{1}{4}$ .