LFM Pure and Mechanics

gaokao 2015 Q6 5 marks Perpendicularity or Parallel Condition View

Let $a, b$ be non-zero vectors. ``$a \cdot b = | a | | b |$'' is ``$a \parallel b$'' a

gaokao 2015 Q6 Angle or Cosine Between Vectors View

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. $\pi$

gaokao 2015 Q6 Expressing a Vector as a Linear Combination View

6. Given vectors $\vec { a } = ( 2,1 ) , \vec { a } = ( 1 , - 2 )$, if $m \vec { a } + n \vec { b } = ( 9 , - 8 ) ( m n \in R )$, then the value of $\mathrm { m } - \mathrm { n }$ is $\_\_\_\_$ .

gaokao 2015 Q7 5 marks Angle or Cosine Between Vectors View

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) $\frac { 5 \pi } { 6 }$

gaokao 2015 Q7 Inequality or Proof Involving Vectors View

7. For any vectors $\vec { a } , \vec { b }$, which of the following relations always holds?
A. $| \vec { a } \bullet \vec { b } | \leq | \vec { a } | | \vec { b } |$
B. $| \vec { a } - \vec { b } | \leq | | \vec { a } | - | \vec { b } | |$
C. $( \vec { a } + \vec { b } ) ^ { 2 } = | \vec { a } + \vec { b } | ^ { 2 }$
D. $( \vec { a } + \vec { b } ) ( \vec { a } - \vec { b } ) = \vec { a } ^ { 2 } - \vec { b } ^ { 2 }$

gaokao 2015 Q7 Dot Product Computation View

7. Let quadrilateral $A B C D$ be a parallelogram, $| \overrightarrow { A B } | = 6$, $| \overrightarrow { A D } | = 4$. If points $\mathrm { M }$ and $\mathrm { N }$ satisfy $\overrightarrow { B M } = 3 \overrightarrow { M C }$, $\overrightarrow { D N } = 2 \overrightarrow { N C }$, then $\overrightarrow { A M } \cdot \overrightarrow { N M } =$
(A) $20$
(B) $15$
(C) $9$
(D) $6$

gaokao 2015 Q9 Optimization of a Vector Expression View

9. Given $\overrightarrow { A B } \perp \overrightarrow { A C }$, $| \overrightarrow { A B } | = \frac { 1 } { t }$, $| \overrightarrow { A C } | = t$, if point $P$ is a point in the plane of $\triangle A B C$, and $\overrightarrow { A P } = \frac { \overrightarrow { A B } } { | \overrightarrow { A B } | } + \frac { \overrightarrow { A C } } { | \overrightarrow { A C } | }$, then the maximum value of $\overrightarrow { P B } \cdot \overrightarrow { P C }$ equals
A. 13
B. 15
C. 19
D. 21

gaokao 2015 Q9 Optimization of a Vector Expression View

9. Points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ move on the circle $\chi ^ { 2 } + y ^ { 2 } = 1$, and $\mathrm { AB } \perp \mathrm { BC }$. If point P has coordinates $( 2,0 )$, then the maximum value of $| \overrightarrow { P A } + \overrightarrow { P B } + \overrightarrow { P C } |$ is
A. 6
B. 7
C. 8
D. 9

gaokao 2015 Q11 Dot Product Computation View

11. Given that vectors $\overrightarrow { O A } \perp \overrightarrow { O B } , | \overrightarrow { O A } | = 3$, then $\overrightarrow { O A } \bullet \overrightarrow { O B } =$ $\_\_\_\_$.

gaokao 2015 Q11 Dot Product Computation View

11. Given vectors $\overrightarrow{OA} \perp \overrightarrow{AB}$ and $|\overrightarrow{OA}| = 3$, then $\overrightarrow{OA} \cdot \overrightarrow{OB} = $ $\_\_\_\_$ .

gaokao 2015 Q13 Expressing a Vector as a Linear Combination View

13. In $\triangle A B C$, points $M , N$ satisfy $\overrightarrow { A M } = 2 \overrightarrow { M C } , \overrightarrow { B N } = \overrightarrow { N C }$. If $\overrightarrow { M N } = x \overrightarrow { A B } + y \overrightarrow { A C }$, then $x =$ $\_\_\_\_$ $\_\_\_\_$ ; $y =$ $\_\_\_\_$.

gaokao 2015 Q13 Perpendicularity or Parallel Condition View

Let vectors $\mathrm { a }$ and $\mathrm { b }$ be non-parallel. If vector $\lambda a + b$ is parallel to $\boldsymbol { a } + 2 b$, then the real number $\lambda = $ $\_\_\_\_$.

gaokao 2015 Q13 Dot Product Computation View

13. In isosceles trapezoid $ABCD$, $AB \parallel DC$, $AB = 2$, $BC = 1$, $\angle ABC = 60 ^ { \circ }$. Points $E$ and $F$ are on segments $BC$ and $CD$ respectively, with $\overrightarrow { BE } = \frac { 2 } { 3 } \overrightarrow { BC }$, $\overrightarrow { DF } = \frac { 1 } { 6 } \overrightarrow { DC }$. Then the value of $\overrightarrow { AE } \cdot \overrightarrow { AF }$ is $\_\_\_\_$.

gaokao 2015 Q13 Magnitude of Vector Expression View

13. Given that $\vec { e } _ { 1 } , \vec { e } _ { 2 }$ are unit vectors in the plane, and $\vec { e } _ { 1 } \cdot \vec { e } _ { 2 } = \frac { 1 } { 2 }$ . If the plane vector $\vec { b }$ satisfies $\vec { b } \cdot \vec { e } _ { 1 } = \vec { b } \cdot \vec { e } _ { 2 } = 1$ , then $| \vec { b } | =$ $\_\_\_\_$.

gaokao 2015 Q14 Dot Product Computation View

14. Let vectors $a _ { k } = \left( \cos \frac { k \pi } { 6 } , \sin \frac { k \pi } { 6 } + \cos \frac { k \pi } { 6 } \right) ( k = 0,1,2 , \cdots , 12 )$, then the value of $\sum _ { k = 0 } ^ { 12 } \left( a _ { k } \cdot a _ { k + 1 } \right)$ is $\_\_\_\_$.

gaokao 2015 Q14 5 marks Optimization of a Vector Expression View

In isosceles trapezoid ABCD, $\mathrm{AB} \parallel \mathrm{DC}$, $\mathrm{AB} = 2$, $\mathrm{BC} = 1$, $\angle \mathrm{ABC} = 60°$. Moving points E and F are on segments BC and DC respectively, with $\overrightarrow{\mathrm{BE}} = \lambda\overrightarrow{\mathrm{BC}}$, $\overrightarrow{\mathrm{DF}} = \frac{1}{9\lambda}\overrightarrow{\mathrm{DC}}$. Then the minimum value of $\overrightarrow{\mathrm{AE}} \cdot \overrightarrow{\mathrm{AF}}$ is .

gaokao 2015 Q15 Expressing a Vector as a Linear Combination View

15. Given that $\vec { e } _ { 1 } , \vec { e } _ { 2 }$ are unit vectors in space with $\vec { e } _ { 1 } \cdot \vec { e } _ { 2 } = \frac { 1 } { 2 }$ . If the space vector $\vec { b }$ satisfies $\vec { b } \cdot \vec { e } _ { 1 } = 2 , \vec { b } \cdot \vec { e } _ { 2 } = \frac { 5 } { 2 }$ , and for all $x , y \in \mathbb{R}$ , $\left| \vec { b } - \left( x \vec { e } _ { 1 } + y \vec { e } _ { 2 } \right) \right| \geq \left| \vec { b } - \left( x _ { 0 } \vec { e } _ { 1 } + y _ { 0 } \vec { e } _ { 2 } \right) \right| = 1$ ( $x _ { 0 } , y _ { 0 } \in \mathbb{R}$ ), then $x _ { 0 } =$ $\_\_\_\_$ , $y _ { 0 } =$ $\_\_\_\_$ , $| \vec { b } | =$ $\_\_\_\_$ . III. Solution Questions: This section contains 5 questions, for a total of 74 points. Solutions should include explanations, proofs, or calculation steps.

gaokao 2017 Q4 Vector 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. $|\boldsymbol{a}| > |\boldsymbol{b}|$

gaokao 2017 Q12 Optimization of a Vector Expression View

12. Given that $\triangle A B C$ is an equilateral triangle with side length 2, and $P$ is a point in the plane $A B C$, the minimum value of $\overrightarrow { P A } \cdot ( \overrightarrow { P B } + \overrightarrow { P C } )$ is
A. $- 2$
B. $- \frac { 3 } { 2 }$
C. $- \frac { 4 } { 3 }$
D. $- 1$
II. Fill in the Blank: This section has 4 questions, each worth 5 points.

gaokao 2017 Q13 5 marks Magnitude of Vector Expression View

Given vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ with an angle of $60 ^ { \circ }$ between them, $| \boldsymbol{a} | = 2$, and $| \boldsymbol{b} | = 1$, then $| \boldsymbol{a} + 2 \boldsymbol{b} | = $ \_\_\_\_

gaokao 2018 Q4 5 marks Dot Product Computation View

Vectors $\boldsymbol { a } , \boldsymbol { b }$ satisfy $| \boldsymbol { a } | = 1 , \boldsymbol { a } \cdot \boldsymbol { b } = - 1$, then $\boldsymbol { a } \cdot ( 2 \boldsymbol { a } - \boldsymbol { b } ) =$
A. 4
B. 3
C. 2
D. 0

gaokao 2018 Q4 5 marks Dot Product Computation View

Given vectors $\boldsymbol { a } , \boldsymbol { b }$ satisfying $| \boldsymbol { a } | = 1 , \boldsymbol { a } \cdot \boldsymbol { b } = - 1$, then $\boldsymbol { a } \cdot ( 2 \boldsymbol { a } - \boldsymbol { b } ) =$
A. 4
B. 3
C. 2
D. 0

gaokao 2018 Q6 5 marks Expressing a Vector as a Linear Combination View

In $\triangle A B C$, $A D$ is the median to side $B C$, and $E$ is the midpoint of $A D$. Then $\overrightarrow { E B } =$
A. $\frac { 3 } { 4 } \overrightarrow { A B } - \frac { 1 } { 4 } \overrightarrow { A C }$
B. $\frac { 1 } { 4 } \overrightarrow { A B } - \frac { 3 } { 4 } \overrightarrow { A C }$
C. $\frac { 3 } { 4 } \overrightarrow { A B } + \frac { 1 } { 4 } \overrightarrow { A C }$
D. $\frac { 1 } { 4 } \overrightarrow { A B } + \frac { 3 } { 4 } \overrightarrow { A C }$

gaokao 2018 Q7 5 marks Expressing a Vector as a Linear Combination View

In $\triangle A B C$, $AD$ is the median to side $BC$, and $E$ is the midpoint of $AD$. Then $\overrightarrow { E B } =$
A. $\frac { 3 } { 4 } \overrightarrow { A B } - \frac { 1 } { 4 } \overrightarrow { A C }$
B. $\frac { 1 } { 4 } \overrightarrow { A B } + \frac { 3 } { 4 } \overrightarrow { A C }$
C. $\frac { 3 } { 4 } \overrightarrow { A B } + \frac { 1 } { 4 } \overrightarrow { A C }$
D. $\frac { 1 } { 4 } \overrightarrow { A B } + \frac { 3 } { 4 } \overrightarrow { A C }$

gaokao 2018 Q13 5 marks Perpendicularity or Parallel Condition View

Given vectors $a = ( 1,2 ) , b = ( 2 , - 2 ) , c = ( 1 , \lambda )$. If $c \parallel ( 2a + b )$, then $\lambda = $ $\_\_\_\_$.

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LFM Pure and Mechanics

Indices and Surds 107

Exponential Functions 100

Laws of Logarithms 200

Exponential Equations & Modelling 58

Arithmetic Sequences and Series 271

Geometric Sequences and Series 203

Differentiation from First Principles 37

Tangents, normals and gradients 93

Stationary points and optimisation 384

Applied differentiation 164

Indefinite & Definite Integrals 533

Areas Between Curves 83

Areas by integration 109

Volumes of Revolution 61

Numerical integration 32

Vectors Introduction & 2D 204

Vectors 3D & Lines 233

Travel graphs 11

SUVAT in 2D & Gravity 6

Constant acceleration (SUVAT) 32

Variable acceleration (1D) 40

Variable acceleration (vectors) 26

Forces, equilibrium and resultants 9

Newton's laws and connected particles 18

Pulley systems 5

Motion on a slope 4

Friction 10

Momentum and Collisions 15

Moments 13

Parametric curves and Cartesian conversion 11

Parametric differentiation 15

Parametric integration 4

Projectiles 40