LFM Pure and Mechanics

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 = $ $\_\_\_\_$.

gaokao 2019 Q3 5 marks Dot Product Computation View

Given $\overrightarrow { A B } = ( 2,3 ) , \overrightarrow { A C } = ( 3 , t ) , | \overrightarrow { B C } | = 1$, then $\overrightarrow { A B } \cdot \overrightarrow { B C } =$
A． - 3
B． - 2
C． 2
D． 3

gaokao 2019 Q7 5 marks Angle or Cosine Between Vectors View

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

gaokao 2019 Q7 Angle or Cosine Between Vectors View

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

gaokao 2019 Q7 Angle or Cosine Between Vectors View

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

gaokao 2019 Q8 Volume of a Region Defined by Inequalities in 3D View

8. Two unit vectors $e _ { 1 } , e _ { 2 }$ have an angle of $60 ^ { \circ }$ between them. Vector $m = t e _ { 1 } + 2 e _ { 2 } ( t < 0 )$. Then
A. The maximum value of $\frac { | m | } { t }$ is $\frac { \sqrt { 3 } } { 2 }$
B. The minimum value of $\frac { | m | } { t }$ is $- 2$
C. The minimum value of $\frac { | m | } { t }$ is $\frac { \sqrt { 3 } } { 2 }$
D. The maximum value of $\frac { | m | } { t }$ is $- 2$

gaokao 2019 Q13 5 marks Angle or Cosine Between Vectors View

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 $\cos \langle \boldsymbol { a } , \boldsymbol { c } \rangle =$ \_\_\_\_\_\_.

gaokao 2019 Q13 Angle or Cosine Between Vectors View

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

gaokao 2019 Q13 Angle or Cosine Between Vectors View

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 $\cos \langle \boldsymbol { a } , \boldsymbol { c } \rangle =$ $\_\_\_\_$ .

gaokao 2020 Q6 5 marks Angle or Cosine Between Vectors View

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. $\frac { 19 } { 35 }$

gaokao 2020 Q12 5 marks Geometric Property Identification via Vectors View

Given vectors $\overrightarrow { a _ { 1 } } , \overrightarrow { a _ { 2 } } , \overrightarrow { b _ { 1 } } , \overrightarrow { b _ { 2 } } , \ldots , \overrightarrow { b _ { k } } \left( k \in \mathbb{N} ^ { * } \right)$ that are pairwise non-parallel in the plane, satisfying $\left| \overrightarrow { a _ { 1 } } - \overrightarrow { a _ { 2 } } \right| = 1$ and $\left| \overrightarrow { a _ { i } } - \overrightarrow { b _ { j } } \right| \in \{ 1,2 \}$ (where $i = 1,2$ and $j = 1,2 , \ldots , k$), find the maximum value of $k$ as $\_\_\_\_$

gaokao 2020 Q13 5 marks Perpendicularity or Parallel Condition View

Given unit vectors $\boldsymbol { a } , \boldsymbol { b }$ with an angle of $45 ^ { \circ }$ between them, and $k \boldsymbol { a } - \boldsymbol { b}$ is perpendicular to $\boldsymbol { a }$ , then $k = $ $\_\_\_\_$.

gaokao 2020 Q14 5 marks Perpendicularity or Parallel Condition View

Let vectors $\boldsymbol { a } = ( 1 , - 1 ) , \boldsymbol { b } = ( m + 1,2 m - 4 )$ . If $\boldsymbol { a } \perp \boldsymbol { b }$ , then $m =$ $\_\_\_\_$.

gaokao 2021 Q10 True/False or Multiple-Statement Verification View

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 $AC$.

gaokao 2021 Q13 Magnitude of Vector Expression View

13. If vectors $\vec { a } , \vec { b }$ satisfy $| \vec { a } | = 3 , | \vec { a } - \vec { b } | = 5 , \vec { a } \cdot \vec { b } = 1$, then $| \vec { b } | =$ $\_\_\_\_$ .

gaokao 2021 Q14 Perpendicularity or Parallel Condition View

14. Given vectors $\vec{a} = (3,1)$, $\vec{b} = (1,0)$, $\vec{c} = \vec{a} + k\vec{b}$. If $\vec{a} \perp \vec{c}$, then $k = \_\_\_\_$.

gaokao 2022 Q3 5 marks Magnitude of Vector Expression View

Given vectors $a = ( 2,1 ) , b = ( - 2,4 )$ , then $| a - b | =$
A. 2
B. 3
C. 4
D. 5

gaokao 2022 Q3 5 marks Dot Product Computation View

Given vectors $\boldsymbol{a}, \boldsymbol{b}$ satisfy $|\boldsymbol{a}| = 1, |\boldsymbol{b}| = \sqrt{3}, |\boldsymbol{a} - 2\boldsymbol{b}| = 3$, then $\boldsymbol{a} \cdot \boldsymbol{b} =$
A. $-2$
B. $-1$
C. $1$
D. $2$

gaokao 2022 Q3 Expressing a Vector as a Linear Combination View

3. In $\triangle A B C$, point $D$ is on side $A B$ with $B D = 2 D A$. Let $\overrightarrow { C A } = m$ and $\overrightarrow { C D } = n$. Then $\overrightarrow { C B } =$
A. $3 m - 2 n$
B. $- 2 m + 3 n$
C. $3 \boldsymbol { m } + 2 \boldsymbol { n }$
D. $2 m + 3 n$

gaokao 2022 Q13 5 marks Perpendicularity or Parallel Condition View

Given vectors $\boldsymbol { a } = ( m , 3 ) , \boldsymbol { b } = ( 1 , m + 1 )$ . If $\boldsymbol { a } \perp \boldsymbol { b }$ , then $m =$ $\_\_\_\_$ .

gaokao 2022 Q13 5 marks Dot Product Computation View

Let vectors $\boldsymbol { a }$ and $\boldsymbol { b }$ have an angle whose cosine is $\frac { 1 } { 3 }$, and $| \boldsymbol { a } | = 1$, $| \boldsymbol { b } | = 3$. Then $( 2 \boldsymbol { a } + \boldsymbol { b } ) \cdot \boldsymbol { b } =$ $\_\_\_\_$

gaokao 2023 Q4 5 marks Angle or Cosine Between Vectors View

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. $\frac{4}{5}$

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LFM Pure and Mechanics

Indices and Surds 96

Exponential Functions 197

Laws of Logarithms 242

Exponential Equations & Modelling 28

Arithmetic Sequences and Series 363

Geometric Sequences and Series 159

Differentiation from First Principles 45

Tangents, normals and gradients 140

Stationary points and optimisation 505

Applied differentiation 76

Indefinite & Definite Integrals 502

Areas Between Curves 71

Areas by integration 221

Volumes of Revolution 68

Numerical integration 49

Vectors Introduction & 2D 211

10. AC

Vectors 3D & Lines 524

Travel graphs 14

SUVAT in 2D & Gravity 31

Constant acceleration (SUVAT) 78

Variable acceleration (1D) 55

Variable acceleration (vectors) 32

Forces, equilibrium and resultants 24

Newton's laws and connected particles 21

Pulley systems 6

Motion on a slope 6

Friction 8

Momentum and Collisions 34

Moments 28

Parametric curves and Cartesian conversion 9

Parametric differentiation 13

Parametric integration 6

Projectiles 49