As shown in the figure, three cylinders with radius $\sqrt{3}$ and different heights are mutually externally tangent and placed on a plane $\alpha$. Let $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$ be the centers of the bases of the three cylinders that do not meet plane $\alpha$. Triangle $\mathrm{QPR}$ is an isosceles triangle, and the angle between plane $\mathrm{QPR}$ and plane $\alpha$ is $60°$. If the heights of the three cylinders are $8$, $a$, and $b$ respectively, find the value of $a + b$. (Given: $8 < a < b$) [4 points]

Given non-zero vectors $\vec { a } , \vec { b }$ satisfying $| \vec { b } | = 4 | \vec { a } |$ and $\vec { a } \perp ( 2 \vec { a } + \vec { b } )$, the angle between $\vec { a }$ and $\vec { b }$ is
(A) $\frac { \pi } { 3 }$
(B) $\frac { \pi } { 2 }$
(C) $\frac { 2 \pi } { 3 }$
(D) $\frac { 5 \pi } { 6 }$

6. If non-zero vectors $\mathbf { a } , \mathbf { b }$ satisfy $| \mathbf { a } | = \frac { 2 \sqrt { 2 } } { 3 } | \mathbf { b } |$ and $(\mathbf{a}-\mathbf{b})\perp(3\mathbf{a}+2\mathbf{b})$, then the angle between $\mathbf { a }$ and $\mathbf { b }$ is
A. $\frac { \pi } { 4 }$
B. $\frac { \pi } { 2 }$
C. $\frac { 3 \pi } { 4 }$
D. $\pi$

Given that $\boldsymbol { a } , \boldsymbol { b }$ are unit vectors and $\boldsymbol { a } \cdot \boldsymbol { b } = 0$ , if $\boldsymbol { c } = 2 \boldsymbol { a } - \sqrt { 5 } \boldsymbol { b }$ , then $\cos \langle \boldsymbol { a } , \boldsymbol { c } \rangle =$ \_\_\_\_\_\_.

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

7. Given non-zero vectors $a , b$ satisfying $| a | = 2 | b |$ and $( a - b ) \perp b$, the angle between $a$ and $b$ is
A. $\frac { \pi } { 6 }$
B. $\frac { \pi } { 3 }$
C. $\frac { 2 \pi } { 3 }$
D. $\frac { 5 \pi } { 6 }$

7. Given non-zero vectors $\boldsymbol { a } , \boldsymbol { b }$ satisfying $| \boldsymbol { a } | = 2 | \boldsymbol { b } |$ and $( \boldsymbol { a } - \boldsymbol { b } ) \perp \boldsymbol { b }$ , the angle between $\boldsymbol { a }$ and $\boldsymbol { b }$ is
A. $\frac { \pi } { 6 }$
B. $\frac { \pi } { 3 }$
C. $\frac { 2 \pi } { 3 }$
D. $\frac { 5 \pi } { 6 }$

13. Given vectors $\boldsymbol { a } = ( 2,2 ) , \boldsymbol { b } = ( - 8,6 )$ , then $\cos \langle \boldsymbol { a } , \boldsymbol { b } \rangle =$

13. Given that $\boldsymbol { a } , \boldsymbol { b }$ are unit vectors and $\boldsymbol { a } \cdot \boldsymbol { b } = 0$ , if $\boldsymbol { c } = 2 \boldsymbol { a } - \sqrt { 5 } \boldsymbol { b }$ , then $\cos \langle \boldsymbol { a } , \boldsymbol { c } \rangle =$ $\_\_\_\_$ .

Given vectors $\boldsymbol { a } , \boldsymbol { b }$ satisfying $| \boldsymbol { a } | = 5 , | \boldsymbol { b } | = 6 , \boldsymbol { a } \cdot \boldsymbol { b } = - 6$ , then $\cos \langle \boldsymbol { a } , \boldsymbol { a } + \boldsymbol { b } \rangle =$
A. $- \frac { 31 } { 35 }$
B. $- \frac { 19 } { 35 }$
C. $\frac { 17 } { 35 }$
D. $\frac { 19 } { 35 }$

9. Given a cube $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, then
A. The angle between lines $B C _ { 1 }$ and $D A _ { 1 }$ is $90 ^ { \circ }$
B. The angle between lines $B C _ { 1 }$ and $C A _ { 1 }$ is $90 ^ { \circ }$
C. The angle between line $B C _ { 1 }$ and plane $B B _ { 1 } D _ { 1 } D$ is $45 ^ { \circ }$
D. The angle between line $B C _ { 1 }$ and plane $A B C D$ is $45 ^ { \circ }$

Vectors $|\boldsymbol{a}| = |\boldsymbol{b}| = 1 , |\boldsymbol{c}| = \sqrt{2}$ and $\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c} = 0$ , then $\cos \langle \boldsymbol{a} - \boldsymbol{b} , \boldsymbol{b} - \boldsymbol{c} \rangle =$
A. $-\frac{1}{5}$
B. $-\frac{2}{5}$
C. $\frac{2}{5}$
D. $\frac{4}{5}$

Let $\vec { a }$ and $\vec { b }$ be two unit vectors such that $\vec { a } \cdot \vec { b } = 0$. For some $x , y \in \mathbb { R }$, let $\vec { c } = x \vec { a } + y \vec { b } + ( \vec { a } \times \vec { b } )$. If $| \vec { c } | = 2$ and the vector $\vec { c }$ is inclined at the same angle $\alpha$ to both $\vec { a }$ and $\vec { b }$, then the value of $8 \cos ^ { 2 } \alpha$ is $\_\_\_\_$.

If $\hat { u }$ and $\hat { v }$ are unit vectors and $\theta$ is the acute angle between them, then $2 \hat { u } \times 3 \hat { v }$ is a unit vector for
(1) exactly two values of $\theta$
(2) more than two values of $\theta$
(3) no value of $\theta$
(4) exactly one value of $\theta$

Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three non-zero vectors such that no two of them are collinear and $( \vec { a } \times \vec { b } ) \times \vec { c } = \frac { 1 } { 3 } | \vec { b } | | \vec { c } | \vec { a }$. If $\theta$ is the angle between vectors $\vec { b }$ and $\vec { c }$, then a value of $\sin \theta$ is
(1) $\frac { - 2 \sqrt { 3 } } { 3 }$
(2) $\frac { 2 \sqrt { 2 } } { 3 }$
(3) $\frac { - \sqrt { 2 } } { 3 }$
(4) $\frac { 2 } { 3 }$

If a unit vector $\vec { a }$ makes angles $\frac { \pi } { 3 }$ with $\hat { i } , \frac { \pi } { 4 }$ with $\hat { j }$ and $\theta \in ( 0 , \pi )$ with $\widehat { k }$, then a value of $\theta$ is:
(1) $\frac { 5 \pi } { 6 }$
(2) $\frac { 5 \pi } { 12 }$
(3) $\frac { \pi } { 4 }$
(4) $\frac { 2 \pi } { 3 }$

For $p > 0$, a vector $\vec { v } _ { 2 } = 2 \hat { i } + ( p + 1 ) \hat { j }$ is obtained by rotating the vector $\vec { v } _ { 1 } = \sqrt { 3 } p \hat { i } + \hat { j }$ by an angle $\theta$ about origin in counter clockwise direction. If $\tan \theta = \frac { ( \alpha \sqrt { 3 } - 2 ) } { ( 4 \sqrt { 3 } + 3 ) }$, then the value of $\alpha$ is equal to $\underline{\hspace{1cm}}$.

Two vectors $\vec { A }$ and $\vec { B }$ have equal magnitudes. If magnitude of $\vec { A } + \vec { B }$ is equal to two times the magnitude of $\vec { A } - \vec { B }$, then the angle between $\vec { A }$ and $\vec { B }$ will be
(1) $\cos ^ { - 1 } \left( \frac { 3 } { 5 } \right)$
(2) $\cos ^ { - 1 } \left( \frac { 1 } { 3 } \right)$
(3) $\sin ^ { - 1 } \left( \frac { 1 } { 3 } \right)$
(4) $\sin ^ { - 1 } \left( \frac { 3 } { 5 } \right)$

Let $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$ and let $\vec{b}$ be a vector such that $\vec{a} \times \vec{b} = 2\hat{i} - \hat{k}$ and $\vec{a} \cdot \vec{b} = 3$. Then the projection of $\vec{b}$ on the vector $\vec{a} - \vec{b}$ is:
(1) $\frac{2}{\sqrt{21}}$
(2) $2\sqrt{\frac{3}{7}}$
(3) $\frac{2}{3}\sqrt{\frac{7}{3}}$
(4) $\frac{2}{3}$

Let $S$ be the set of all $a \in R$ for which the angle between the vectors $\vec { u } = a \left( \log _ { e } b \right) \hat { i } - 6 \hat { j } + 3 \hat { k }$ and $\vec { v } = \left( \log _ { e } b \right) \hat { i } + 2 \hat { j } + 2 a \left( \log _ { e } b \right) \hat { k } , ( b > 1 )$ is acute. Then $S$ is equal to
(1) $\left( - \infty , - \frac { 4 } { 3 } \right)$
(2) $\Phi$
(3) $\left( - \frac { 4 } { 3 } , 0 \right)$
(4) $\left( \frac { 12 } { 7 } , \infty \right)$

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + 2 \widehat { \mathrm { k } }$ and $\alpha \hat { \mathrm { i } } + 2 \alpha \hat { \mathrm { j } } - 2 \widehat { \mathrm { k } }$ is acute, is $\_\_\_\_$.

The set of all $\alpha$, for which the vectors $\vec { a } = \alpha t \hat { i } + 6 \hat { j } - 3 \hat { k }$ and $\vec { b } = t \hat { i } - 2 \hat { j } - 2 \alpha t \hat { k }$ are inclined at an obtuse angle for all $t \in \mathbb { R }$, is
(1) $\left( - \frac { 4 } { 3 } , 1 \right)$
(2) $[ 0,1 )$
(3) $\left( - \frac { 4 } { 3 } , 0 \right]$
(4) $( - 2,0 ]$

Let $P ( x , y , z )$ be a point in the first octant, whose projection in the $x y$-plane is the point $Q$. Let $O P = \gamma$; the angle between $O Q$ and the positive $x$-axis be $\theta$; and the angle between $O P$ and the positive $z$-axis be $\phi$, where $O$ is the origin. Then the distance of $P$ from the $x$-axis is
(1) $\gamma \sqrt { 1 - \sin ^ { 2 } \phi \cos ^ { 2 } \theta }$
(2) $\gamma \sqrt { 1 - \sin ^ { 2 } \theta \cos ^ { 2 } \phi }$
(3) $\gamma \sqrt { 1 + \cos ^ { 2 } \phi \sin ^ { 2 } \theta }$
(4) $\gamma \sqrt { 1 + \cos ^ { 2 } \theta \sin ^ { 2 } \phi }$

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}| = 1$, $|\vec{b}| = 2$ and the angle formed by $\vec{a}$ and $\vec{b}$ is $60^\circ$. Set $\vec{u} = x\vec{a} + \vec{b}$ and $\vec{v} = x\vec{a} - \vec{b}$ for a real number $x$. When $x > 1$, we are to find the value of $x$ such that the angle formed by $\vec{u}$ and $\vec{v}$ is $30^\circ$. In the following, $\vec{u} \cdot \vec{v}$ denotes the inner product of $\vec{u}$ and $\vec{v}$, and $\vec{a} \cdot \vec{b}$ denotes the inner product of $\vec{a}$ and $\vec{b}$.
First of all, since the angle formed by $\vec{u}$ and $\vec{v}$ is $30^\circ$, we obtain $$(\vec{u} \cdot \vec{v})^2 = \frac{\mathbf{A}}{\mathbf{B}}|\vec{u}|^2|\vec{v}|^2.$$
When we express this equation in terms of $x$, noting $\vec{a} \cdot \vec{b} = \mathbf{C}$, we have $$x^4 - \mathbf{DE}x^2 + \mathbf{FG} = 0.$$
By transforming this, we also have $$\left(x^2 - \mathbf{H}\right)^2 = (\mathbf{I}x)^2.$$
When this is solved for $x$, we obtain $$x = \mathbf{J} + \sqrt{\mathbf{KL}},$$ noting $x > 1$.

Let $a , b$ be real numbers, and let $O$ be the origin of the coordinate plane. It is known that the graph of the quadratic function $f ( x ) = a x ^ { 2 }$ and the circle $\Omega : x ^ { 2 } + y ^ { 2 } - 3 y + b = 0$ both pass through point $P \left( 1 , \frac { 1 } { 2 } \right)$, and let point $C$ be the center of $\Omega$.
Find the cosine of the angle between vectors $\overrightarrow { C O }$ and $\overrightarrow { C P }$.