On the coordinate plane, there are two arbitrary distinct vectors $\overrightarrow { \mathrm { OP } } , \overrightarrow { \mathrm { OQ } }$ with initial point at the origin O. When the endpoints $\mathrm { P } , \mathrm { Q }$ of the two vectors are translated 3 units in the $x$-direction and 1 unit in the $y$-direction to points $\mathrm { P } ^ { \prime } , \mathrm { Q } ^ { \prime }$ respectively, which of the following statements in are always true? [3 points]

ㄱ. $\left| \overrightarrow { \mathrm { OP } } - \overrightarrow { \mathrm { OP } ^ { \prime } } \right| = \sqrt { 10 }$ ㄴ. $| \overrightarrow { \mathrm { OP } } - \overrightarrow { \mathrm { OQ } } | = \left| \overrightarrow { \mathrm { OP } ^ { \prime } } - \overrightarrow { \mathrm { OQ } ^ { \prime } } \right|$ ㄷ. $\overrightarrow { \mathrm { OP } } \cdot \overrightarrow { \mathrm { OQ } } = \overrightarrow { \mathrm { OP } ^ { \prime } } \cdot \overrightarrow { \mathrm { OQ } ^ { \prime } }$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

In the plane, the pentagon ABCDE satisfies $$\overline { \mathrm { AB } } = \overline { \mathrm { BC } } , \overline { \mathrm { AE } } = \overline { \mathrm { ED } } , \angle \mathrm {~B} = \angle \mathrm { E } = 90 ^ { \circ }$$ Which of the following statements in are correct? [4 points]
Remarks ㄱ. For the midpoint M of segment BE, $\overrightarrow { \mathrm { AB } } + \overrightarrow { \mathrm { AE } }$ and $\overrightarrow { \mathrm { AM } }$ are parallel to each other. ㄴ. $\overrightarrow { \mathrm { AB } } \cdot \overrightarrow { \mathrm { AE } } = - \overrightarrow { \mathrm { BC } } \cdot \overrightarrow { \mathrm { ED } }$ ㄷ. $| \overrightarrow { \mathrm { BC } } + \overrightarrow { \mathrm { ED } } | = | \overrightarrow { \mathrm { BE } } |$
(1) ᄀ
(2) ᄃ
(3) ᄀ, ᄂ
(4) ㄴ,ㄷ
(5) ᄀ, ᄂ, ᄃ

10. AC

Solution: $\left| \overrightarrow { O P _ { 1 } } \right| = \left| \overrightarrow { O P _ { 2 } } \right| = 1$, so A is correct; $\left| \overrightarrow { A P _ { 1 } } \right| ^ { 2 } = 2 - 2 \cos \alpha , \left| \overrightarrow { A P _ { 2 } } \right| ^ { 2 } = 2 - 2 \cos \beta$, so B is incorrect; $\overrightarrow { O P _ { 1 } } \cdot \overrightarrow { O P _ { 2 } } \sin \alpha \sin \beta = \cos ( \alpha + \beta ) = \overrightarrow { O A } \cdot \overrightarrow { O P _ { 3 } }$, so C is correct; $\overrightarrow { O P _ { 2 } } \cdot \overrightarrow { O P _ { 3 } } = \cos ( \alpha + \beta ) \cos \beta - \sin ( \alpha + \beta ) \sin \beta \neq \overrightarrow { O P _ { 1 } } = \cos \alpha$, so D is incorrect. The answer is $AC$.

Let $\vec{x}, \vec{y}$ and $\vec{z}$ be three vectors each of magnitude $\sqrt{2}$ and the angle between each pair of them is $\frac{\pi}{3}$. If $\vec{a}$ is a nonzero vector perpendicular to $\vec{x}$ and $\vec{y} \times \vec{z}$ and $\vec{b}$ is a nonzero vector perpendicular to $\vec{y}$ and $\vec{z} \times \vec{x}$, then
(A) $\vec{b} = (\vec{b} \cdot \vec{z})(\vec{z} - \vec{x})$
(B) $\vec{a} = (\vec{a} \cdot \vec{y})(\vec{y} - \vec{z})$
(C) $\vec{a} \cdot \vec{b} = -(\vec{a} \cdot \vec{y})(\vec{b} \cdot \vec{z})$
(D) $\vec{a} = (\vec{a} \cdot \vec{y})(\vec{z} - \vec{y})$

Let $\triangle P Q R$ be a triangle. Let $\vec { a } = \overrightarrow { Q R } , \vec { b } = \overrightarrow { R P }$ and $\vec { c } = \overrightarrow { P Q }$. If $| \vec { a } | = 12 , | \vec { b } | = 4 \sqrt { 3 }$ and $\vec { b } \cdot \vec { c } = 24$, then which of the following is (are) true?
(A) $\frac { | \vec { c } | ^ { 2 } } { 2 } - | \vec { a } | = 12$
(B) $\frac { | \vec { c } | ^ { 2 } } { 2 } + | \vec { a } | = 30$
(C) $| \vec { a } \times \vec { b } + \vec { c } \times \vec { a } | = 48 \sqrt { 3 }$
(D) $\vec { a } \cdot \vec { b } = - 72$

Let $O$ be the origin and $\overrightarrow { O A } = 2 \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + \hat { \mathrm { k } } , \quad \overrightarrow { O B } = \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + 2 \hat { \mathrm { k } }$ and $\overrightarrow { O C } = \frac { 1 } { 2 } ( \overrightarrow { O B } - \lambda \overrightarrow { O A } )$ for some $\lambda > 0$. If $| \overrightarrow { O B } \times \overrightarrow { O C } | = \frac { 9 } { 2 }$, then which of the following statements is (are) TRUE ?
(A) Projection of $\overrightarrow { O C }$ on $\overrightarrow { O A }$ is $- \frac { 3 } { 2 }$
(B) Area of the triangle $O A B$ is $\frac { 9 } { 2 }$
(C) Area of the triangle $A B C$ is $\frac { 9 } { 2 }$
(D) The acute angle between the diagonals of the parallelogram with adjacent sides $\overrightarrow { O A }$ and $\overrightarrow { O C }$ is $\frac { \pi } { 3 }$

Let $\hat { \imath } , \hat { \jmath }$ and $\hat { k }$ be the unit vectors along the three positive coordinate axes. Let
$$\begin{aligned} & \vec { a } = 3 \hat { \imath } + \hat { \jmath } - \hat { k } , \\ & \vec { b } = \hat { \imath } + b _ { 2 } \hat { \jmath } + b _ { 3 } \hat { k } , \quad b _ { 2 } , b _ { 3 } \in \mathbb { R } , \\ & \vec { c } = c _ { 1 } \hat { \imath } + c _ { 2 } \hat { \jmath } + c _ { 3 } \hat { k } , \quad c _ { 1 } , c _ { 2 } , c _ { 3 } \in \mathbb { R } \end{aligned}$$
be three vectors such that $b _ { 2 } b _ { 3 } > 0 , \vec { a } \cdot \vec { b } = 0$ and
$$\left( \begin{array} { r c r } 0 & - c _ { 3 } & c _ { 2 } \\ c _ { 3 } & 0 & - c _ { 1 } \\ - c _ { 2 } & c _ { 1 } & 0 \end{array} \right) \left( \begin{array} { l } 1 \\ b _ { 2 } \\ b _ { 3 } \end{array} \right) = \left( \begin{array} { r } 3 - c _ { 1 } \\ 1 - c _ { 2 } \\ - 1 - c _ { 3 } \end{array} \right)$$
Then, which of the following is/are TRUE ?
(A) $\vec { a } \cdot \vec { c } = 0$
(B) $\vec { b } \cdot \vec { c } = 0$
(C) $| \vec { b } | > \sqrt { 10 }$
(D) $| \vec { c } | \leq \sqrt { 11 }$

Assertion $A$ : If $A , B , C , D$ are four points on a semi-circular arc with a centre at $O$ such that $| \overrightarrow { A B } | = | \overrightarrow { B C } | = | \overrightarrow { C D } |$. Then, $\overrightarrow { A B } + \overrightarrow { A C } + \overrightarrow { A D } = 4 \overrightarrow { A O } + \overrightarrow { O B } + \overrightarrow { O C }$
Reason $R$ : Polygon law of vector addition yields $\overrightarrow { A B } + \overrightarrow { B C } + \overrightarrow { C D } + \overrightarrow { A D } = 2 \overrightarrow { A O }$
In the light of the above statements, choose the most appropriate answer from the options given below.
(1) $A$ is correct but $R$ is not correct.
(2) $A$ is not correct but $R$ is correct.
(3) Both $A$ and $R$ are correct and $R$ is the correct explanation of $A$.
(4) Both $A$ and $R$ are correct but $R$ is not the correct explanation of $A$.

Let $a$ and $b$ be two unit vectors such that $| ( a + b ) + 2 ( a \times b ) | = 2$. If $\theta \in ( 0 , \pi )$ is the angle between $\hat { \mathrm { a } }$ and $\widehat { \mathrm { b } }$, then among the statements: $( S 1 ) : 2 | \widehat { a } \times \hat { b } | = | \widehat { a } - \hat { b } |$ $( S 2 )$ : The projection of $\widehat { a }$ on $( \widehat { a } + \widehat { b } )$ is $\frac { 1 } { 2 }$
(1) Only $( S 1 )$ is true.
(2) Only $( S 2 )$ is true.
(3) Both $( S 1 )$ and $( S 2 )$ are true.
(4) Both $( S 1 )$ and $( S 2 )$ are false.

In the coordinate plane, let $O$ be the origin, and let $A$ and $B$ be two distinct points different from $O$. Let $C_{1}$, $C_{2}$, $C_{3}$ be three points in the plane satisfying $\overrightarrow{OC}_{n} = \overrightarrow{OA} + n\overrightarrow{OB}$, $n = 1, 2, 3$. Select the correct options.
(1) $\overrightarrow{OC}_{1} \neq \overrightarrow{0}$
(2) $\overline{OC_{1}} < \overline{OC_{2}} < \overline{OC_{3}}$
(3) $\overrightarrow{OC}_{1} \cdot \overrightarrow{OA} < \overrightarrow{OC}_{2} \cdot \overrightarrow{OA} < \overrightarrow{OC}_{3} \cdot \overrightarrow{OA}$
(4) $\overrightarrow{OC_{1}} \cdot \overrightarrow{OB} < \overrightarrow{OC_{2}} \cdot \overrightarrow{OB} < \overrightarrow{OC_{3}} \cdot \overrightarrow{OB}$
(5) $C_{1}$, $C_{2}$, $C_{3}$ are collinear

On a plane, there is a trapezoid $A B C D$ with upper base $\overline { A B } = 10$, lower base $\overline { C D } = 15$, and leg length $\overline { A D } = \overline { B C } + 1$. Select the correct options.
(1) $\angle A > \angle B$
(2) $\angle B + \angle D < 180 ^ { \circ }$
(3) $\overrightarrow { B A } \cdot \overrightarrow { B C } < 0$
(4) The length of $\overline { B C }$ could be 2
(5) $\overrightarrow { C B } \cdot \overrightarrow { C D } < 30$

Let $P$ be a point inside $\triangle A B C$, and $\overrightarrow { A P } = a \overrightarrow { A B } + b \overrightarrow { A C }$ , where $a , b$ are distinct real numbers. Let $Q , R$ be on the same plane, with $\overrightarrow { A Q } = b \overrightarrow { A B } + a \overrightarrow { A C } , ~ \overrightarrow { A R } = a \overrightarrow { A B } + ( b - 0.05 ) \overrightarrow { A C }$ . Select the correct options.
(1) $Q , R$ are also both inside $\triangle A B C$
(2) $| \overrightarrow { A P } | = | \overrightarrow { A Q } |$
(3) Area of $\triangle A B P$ = Area of $\triangle A C Q$
(4) Area of $\triangle B C P$ = Area of $\triangle B C Q$
(5) Area of $\triangle A B P$ > Area of $\triangle A B R$