LFM Pure

gaokao 2015 Q18 View

18. A student uses the ``five-point method'' to draw the graph of the function $f ( \mathrm { x } ) = \mathrm { A } \sin ( \omega \mathrm { x } + \varphi ) \left( \omega > 0 , \varphi < \frac { \pi } { 2 } \right)$ during a certain period, creates a table and fills in partial data as follows:

$\omega \mathrm { x } + \varphi$	0	$\frac { \pi } { 2 }$	$\pi$	$\frac { 3 \pi } { 2 }$	$2 \pi$
x		$\frac { \pi } { 3 }$		$\frac { 5 \pi } { 6 }$
$\mathrm {~A} \sin ( \omega \mathrm { x } + \varphi )$	0	5		- 5	0

(I) Please complete the above data, fill in the corresponding positions on the answer sheet, and directly write the analytical expression of the function $f ( \mathrm { x } )$; (II) Shift all points on the graph of $y = f ( \mathrm { x } )$ to the left by $\frac { \pi } { 6 }$ units to obtain the graph of $y = g ( \mathrm { x } )$. Find the center of symmetry of the graph of $y = g ( \mathrm { x } )$ that is closest to the origin O.

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 Q8 5 marks View

The partial graph of the function $y = \frac{\sin 2x}{1 - \cos x}$ is approximately (see options A, B, C, D in the figures provided).

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 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
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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 Q11 View

11. Regarding the function $f ( x ) = \sin | x | + | \sin x |$, there are four conclusions:
(1) $f ( x )$ is an even function
(2) $f ( x )$ is monotonically increasing on the interval $\left( \frac { \pi } { 2 } , \pi \right)$
(3) $f ( x )$ has 4 zeros on $[ - \pi , \pi ]$
(4) The maximum value of $f ( x )$ is 2
The numbers of all correct conclusions are
A. (1)(2)(4)
B. (2)(4)
C. (1)(4)
D. (1)(3)

gaokao 2019 Q11 View

11. Regarding the function $f ( x ) = \sin | x | + | \sin x |$ , there are four conclusions:
(1) $f ( x )$ is an even function
(2) $f ( x )$ is monotonically increasing on the interval $\left( \frac { \pi } { 2 } , \pi \right)$
(3) $f ( x )$ has 4 zeros on $[ - \pi , \pi ]$
(4) The maximum value of $f ( x )$ is 2
The numbers of all correct conclusions are
A. (1)(2)(4)
B. (2)(4)
C. (1)(4)
D. (1)(3)

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 2020 Q16 5 marks View

Regarding the function $f ( x ) = \sin x + \frac { 1 } { \sin x }$ , there are four propositions:
(1) The graph of $f ( x )$ is symmetric about the $y$-axis.
(2) The graph of $f ( x )$ is symmetric about the origin.
(3) The graph of $f ( x )$ is symmetric about the line $x = \frac { \pi } { 2 }$ .
(4) The minimum value of $f ( x )$ is 2 .
The sequence numbers of all true propositions are $\_\_\_\_$ .

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 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 2024 Q6 4 marks View

Given $f ( x ) = \sin \omega x , f \left( x _ { 1 } \right) = - 1 , f \left( x _ { 2 } \right) = 1 , \left| x _ { 1 } - x _ { 2 } \right| _ { \text {min} } = \frac { \pi } { 2 }$, then $\omega =$ \_\_\_\_

gaokao 2024 Q7 5 marks View

When $x \in [ 0,2 \pi ]$ , the number of intersection points of the curves $y = \sin x$ and $y = 2 \sin \left( 3 x - \frac { \pi } { 6 } \right)$ is
A. $3$
B. $4$
C. $6$
D. $8$

gaokao 2024 Q9 6 marks View

For functions $f ( x ) = \sin 2 x$ and $g ( x ) = \sin \left( 2 x - \frac { \pi } { 4 } \right)$, the correct statements are
A. $f ( x )$ and $g ( x )$ have the same zeros
B. $f ( x )$ and $g ( x )$ have the same maximum value
C. $f ( x )$ and $g ( x )$ have the same minimum positive period
D. The graphs of $f ( x )$ and $g ( x )$ have the same axes of symmetry

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LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

Simultaneous equations 79

Proof 711

Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

Trig Graphs & Exact Values 75

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 106

Trig Proofs 53

Quadratic trigonometric equations 36

Trigonometric equations in context 18

Modulus function 49

Matrices 1553

Linear transformations 26

Invariant lines and eigenvalues and vectors 188

Reciprocal Trig & Identities 44

Addition & Double Angle Formulae 98

Harmonic Form 12

Small angle approximation 20

Chain Rule 113

Product & Quotient Rules 18

Differentiating Transcendental Functions 183

Connected Rates of Change 56

Standard Integrals and Reverse Chain Rule 43

Integration by Substitution 116

Integration by Parts 74

Implicit equations and differentiation 43

Differential equations 352

3x3 Matrices 144