LFM Pure

gaokao 2015 Q19 View

19. The graph of function $f ( x )$ is obtained from the graph of function $g ( x ) = \cos x$ by the following transformations: first, stretch the vertical coordinates of all points on the graph of $g ( x )$ to 2 times their original length (horizontal coordinates unchanged), then shift the resulting graph to the right by $\frac { \pi } { 2 }$ units.
(1) Find the analytical expression of function $f ( x )$ and the equation of its axis of symmetry;
(2) Given that the equation $f ( x ) + g ( x ) = m$ has two distinct solutions $a$ and $b$ in $[ 0,2 \pi )$:
1) Find the range of the real number $m$;
2) Prove that $\cos ( a - b ) = \frac { 2 m ^ { 2 } } { 5 } - 1$.

gaokao 2017 Q3 View

The smallest positive period of the function $f(x) = \sin\left(2x + \frac{\pi}{3}\right)$ is
A. $4\pi$
B. $2\pi$
C. $\pi$
D. $\frac{\pi}{2}$

gaokao 2017 Q14 View

14. The maximum value of the function $f ( x ) = \sin ^ { 2 } x + \sqrt { 3 } \cos x - \frac { 3 } { 4 } \left( x \in \left[ 0 , \frac { \pi } { 2 } \right] \right)$ is ______

gaokao 2018 Q8 5 marks Function Analysis via Identity Transformation View

Given the function $f ( x ) = 2 \cos ^ { 2 } x - \sin ^ { 2 } x + 2$, then
A. The minimum positive period of $f ( x )$ is $\pi$, and the maximum value is 3
B. The minimum positive period of $f ( x )$ is $\pi$, and the maximum value is 4
C. The minimum positive period of $f ( x )$ is $2 \pi$, and the maximum value is 3
D. The minimum positive period of $f ( x )$ is $2 \pi$, and the maximum value is 4

gaokao 2018 Q10 5 marks Function Analysis via Identity Transformation View

If $f ( x ) = \cos x - \sin x$ is decreasing on $[ 0 , a ]$, then the maximum value of $a$ is
A. $\frac { \pi } { 4 }$
B. $\frac { \pi } { 2 }$
C. $\frac { 3 \pi } { 4 }$
D. $\pi$

gaokao 2018 Q10 5 marks Function Analysis via Identity Transformation View

If $f ( x ) = \cos x - \sin x$ is an even function on $[ - a , a ]$, then the maximum value of $a$ is
A. $\frac { \pi } { 4 }$
B. $\frac { \pi } { 2 }$
C. $\frac { 3 \pi } { 4 }$
D. $\pi$

gaokao 2018 Q15 5 marks Count zeros or intersection points involving trigonometric curves View

The number of zeros of the function $f ( x ) = \cos \left( 3 x + \frac { \pi } { 6 } \right)$ on $[ 0, \pi ]$ is $\_\_\_\_$.

gaokao 2018 Q16 5 marks Function Analysis via Identity Transformation View

Given the function $f ( x ) = 2 \sin x + \sin 2 x$, the minimum value of $f ( x )$ is $\_\_\_\_$

gaokao 2019 Q5 View

5. The number of zeros of the function $f ( x ) = 2 \sin x - \sin 2 x$ on $[ 0,2 \pi ]$ is
A. 2
B. 3
C. 4
D. 5

gaokao 2019 Q6 View

6. The monotonically increasing interval of the function $f ( x ) = \cos \left( 3 x + \frac { \pi } { 2 } \right)$ is
A. $\left[ \frac { \pi } { 6 } + \frac { 2 k \pi } { 3 } , \frac { \pi } { 2 } + \frac { 2 k \pi } { 3 } \right] ( k \in \mathbb{Z} )$
[Figure]
Front View
[Figure]
Top View
B. $\left[ \frac { \pi } { 6 } + \frac { k \pi } { 3 } , \frac { \pi } { 2 } + \frac { k \pi } { 3 } \right] ( k \in \mathbb{Z} )$
C. $\left[ \frac { \pi } { 6 } + \frac { k \pi } { 3 } , \frac { \pi } { 6 } + \frac { k \pi } { 3 } \right] ( k \in \mathbb{Z} )$
D. $\left[ - \frac { \pi } { 6 } + \frac { 2 k \pi } { 3 } , \frac { \pi } { 6 } + \frac { 2 k \pi } { 3 } \right] ( k \in \mathbb{Z} )$

gaokao 2019 Q8 View

8. If $x _ { 1 } = \frac { \pi } { 4 } , x _ { 2 } = \frac { 3 \pi } { 4 }$ are two adjacent extreme points of the function $f ( x ) = \sin \omega x ( \omega > 0 )$, then $\omega =$
A. 2
B. $\frac { 3 } { 2 }$
C. 1
D. $\frac { 1 } { 2 }$

gaokao 2019 Q9 5 marks View

Among the following functions, which one has period $\frac { \pi } { 2 }$ and is monotonically increasing on the interval $\left( \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$?
A．$f ( x ) = | \cos 2 x |$
B．$f ( x ) = | \sin 2 x |$
C．$f ( x ) = \cos | x |$
D．$f ( x ) = \sin | x |$

gaokao 2019 Q9 5 marks View

The minimum positive period of the function $f ( x ) = \sin ^ { 2 } 2 x$ is $\_\_\_\_$.

gaokao 2019 Q9 View

9. Among the following functions, which one has period $\frac { \pi } { 2 }$ and is monotonically increasing on the interval $\left( \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$?
A. $f ( x ) = | \cos 2 x |$
B. $f ( x ) = | \sin 2 x |$
C. $f ( x ) = \cos | x |$
D. $f ( x ) = \sin | x |$

gaokao 2019 Q12 5 marks Count zeros or intersection points involving trigonometric curves View

Let the function $f ( x ) = \sin \left( \omega x + \frac { \pi } { 5 } \right) ( \omega > 0 )$. It is known that $f ( x )$ has exactly 5 zeros on $[ 0,2 \pi ]$. The following are four conclusions:
(1) $f ( x )$ has exactly 3 local maximum points on $( 0,2 \pi )$
(2) $f ( x )$ has exactly 2 local minimum points on $( 0,2 \pi )$
(3) $f ( x )$ is monotonically increasing on $\left( 0 , \frac { \pi } { 10 } \right)$
(4) The range of $\omega$ is $\left[ \frac { 12 } { 5 } , \frac { 29 } { 10 } \right)$
The numbers of all correct conclusions are
A. (1)(4)
B. (2)(3)
C. (1)(2)(3)
D. (1)(3)(4)

gaokao 2019 Q12 View

12. Let $f ( x ) = \sin \left( \omega x + \frac { \pi } { 5 } \right) ( \omega > 0 )$ . Given that $f ( x )$ has exactly 5 zeros on $[ 0,2 \pi ]$ , consider the following four conclusions:
(1) $f ( x )$ has exactly 3 local maximum points on $( 0,2 \pi )$
(2) $f ( x )$ has exactly 2 local minimum points on $( 0,2 \pi )$
(3) $f ( x )$ is monotonically increasing on $\left( 0 , \frac { \pi } { 10 } \right)$
(4) The range of $\omega$ is $\left[ \frac { 12 } { 5 } , \frac { 29 } { 10 } \right)$ The numbers of all correct conclusions are
A. (1)(4)
B. (2)(3)
C. (1)(2)(3)
D. (1)(3)(4)
II. Fill-in-the-Blank Questions: This section has 4 questions, each worth 5 points, for a total of 20 points.

gaokao 2020 Q7 5 marks View

The function $f ( x ) = \cos \left( \omega x + \frac { \pi } { 6 } \right)$ has a graph on $[ - \pi , \pi ]$ as shown. The minimum positive period of $f ( x )$ is
A. $\frac { 10 \pi } { 9 }$
B. $\frac { 7 \pi } { 6 }$
C. $\frac { 4 \pi } { 3 }$
D. $\frac { 3 \pi } { 2 }$

gaokao 2020 Q7 5 marks View

Let the function $f ( x ) = \cos \left( \omega x + \frac { \pi } { 6 } \right)$ on $[ - \pi , \pi ]$ have a graph as shown. Then the smallest positive period of $f ( x )$ is
A. $\frac { 10 \pi } { 9 }$
B. $\frac { 7 \pi } { 6 }$
C. $\frac { 4 \pi } { 3 }$
D. $\frac { 3 \pi } { 2 }$

gaokao 2020 Q12 5 marks View

Given the function $f ( x ) = \sin x + \frac { 1 } { \sin x }$, then
A. the minimum value of $f ( x )$ is 2
B. the graph of $f ( x )$ is symmetric about the $y$-axis
C. the graph of $f ( x )$ is symmetric about the line $x = \pi$
D. the graph of $f ( x )$ is symmetric about the line $x = \frac { \pi } { 2 }$

gaokao 2021 Q4 View

4. A

Solution: Based on the graph and properties of translation, the function $f ( x )$ is monotonically increasing on $\left( - \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } \right)$ and monotonically decreasing on $\left( - \frac { 2 \pi } { 3 } , \frac { 5 \pi } { 3 } \right)$.

gaokao 2021 Q15 View

15. The function $f ( x ) = 2 \cos ( \omega x + \varphi )$ has a partial graph shown in the figure. Then $f \left( \frac { \pi } { 2 } \right) =$ $\_\_\_\_$ . [Figure]

gaokao 2021 Q16 View

16. Given that the function $f(x) = 2\cos(\omega x + \varphi)$ has a partial graph as shown in the figure, then the minimum positive integer $x$ satisfying the condition $\left(f(x) - f\left(-\frac{7\pi}{4}\right)\right)\left(f(x) - f\left(\frac{4\pi}{3}\right)\right) > 0$ is $\_\_\_\_$.

III. Solution Questions (70 points total. Show all work, proofs, and calculations. Questions 17-21 are required for all students. Questions 22-23 are optional; students should answer according to requirements.)

gaokao 2022 Q5 5 marks Graph transformation and phase shift View

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. $\frac { 1 } { 2 }$

gaokao 2022 Q6 View

6. Let the function $f ( x ) = \sin \left( \omega x + \frac { \pi } { 4 } \right) + b$ ( $\omega > 0$ ) have minimum positive period $T$. If $\frac { 2 \pi } { 3 } < T < \pi$ and the graph of $y = f ( x )$ is symmetric about the point $\left( \frac { 3 \pi } { 2 } , 2 \right)$, then $f \left( \frac { \pi } { 2 } \right) =$
A. $1$
B. $\frac { 3 } { 2 }$
C. $\frac { 5 } { 2 }$
D. $3$

gaokao 2022 Q11 5 marks Find absolute extrema on a closed interval or domain View

The function $f ( x ) = \cos x + ( x + 1 ) \sin x + 1$ on the interval $[ 0,2 \pi ]$ has minimum and maximum values respectively
A. $- \frac { \pi } { 2 } , \frac { \pi } { 2 }$
B. $- \frac { 3 \pi } { 2 } , \frac { \pi } { 2 }$
C. $- \frac { \pi } { 2 } , \frac { \pi } { 2 } + 2$
D. $- \frac { 3 \pi } { 2 } , \frac { \pi } { 2 } + 2$

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LFM Pure

Straight Lines & Coordinate Geometry 242

Circles 702

Simultaneous equations 134

Proof 868

Proof by induction 36

Sine and Cosine Rules 185

Radians, Arc Length and Sector Area 22

Trig Graphs & Exact Values 95

4. A

III. Solution Questions (70 points total. Show all work, proofs, and calculations. Questions 17-21 are required for all students. Questions 22-23 are optional; students should answer according to requirements.)

Standard trigonometric equations 142

Trig Proofs 92

Quadratic trigonometric equations 8

Trigonometric equations in context 11

Modulus function 37

Matrices 1085

Linear transformations 72

Invariant lines and eigenvalues and vectors 339

Reciprocal Trig & Identities 63

Addition & Double Angle Formulae 88

Harmonic Form 14

Small angle approximation 14

Chain Rule 281

Product & Quotient Rules 13

Differentiating Transcendental Functions 160

Connected Rates of Change 48

Standard Integrals and Reverse Chain Rule 82

Integration by Substitution 213

Integration by Parts 44

Implicit equations and differentiation 27

Differential equations 308

3x3 Matrices 164