LFM Pure

brazil-enem 2011 Q153 View

O valor de $\sin(30^\circ) + \cos(60^\circ) + \tan(45^\circ)$ é
(A) 1 (B) $\dfrac{3}{2}$ (C) 2 (D) $\dfrac{5}{2}$ (E) 3

brazil-enem 2015 Q153 View

QUESTION 153
The value of $\sin 30^\circ + \cos 60^\circ$ is
(A) $\frac{1}{2}$
(B) $\frac{\sqrt{2}}{2}$
(C) 1
(D) $\frac{\sqrt{3}}{2}$
(E) $\sqrt{2}$

brazil-enem 2015 Q166 View

QUESTION 166
The value of $\tan 45^\circ + \cos 0^\circ$ is
(A) 1
(B) 2
(C) $\sqrt{2}$
(D) $1 + \sqrt{2}$
(E) $2\sqrt{2}$

brazil-enem 2017 Q138 View

Rays of sunlight are hitting the surface of a lake forming an angle $x$ with its surface, as shown in the figure.
Under certain conditions, one can assume that the light intensity of these rays, on the lake surface, is given approximately by $I(x) = K \cdot \sin(x)$, where $k$ is a constant, and assuming that $x$ is between $0^{\circ}$ and $90^{\circ}$.
When $x = 30^{\circ}$, the light intensity is reduced to what percentage of its maximum value?
(A) $33\%$
(B) $50\%$
(C) $57\%$
(D) $70\%$
(E) $86\%$

brazil-enem 2024 Q156 View

In a right triangle, $\sin(\theta) = \dfrac{3}{5}$. What is the value of $\cos(\theta)$?
(A) $\dfrac{1}{5}$
(B) $\dfrac{2}{5}$
(C) $\dfrac{3}{5}$
(D) $\dfrac{4}{5}$
(E) $\dfrac{5}{4}$

brazil-enem 2024 Q172 View

The value of $\cos(60^\circ) + \sin(30^\circ)$ is:
(A) 0
(B) $\dfrac{1}{2}$
(C) 1
(D) $\dfrac{3}{2}$
(E) 2

cmi-entrance 2016 Q5 4 marks View

Find the value of the following sum of 100 terms. (Possible hint: also consider the same sum with $\sin^{2}$ instead of $\cos^{2}$.)
$$\cos^{2}\left(\frac{\pi}{101}\right) + \cos^{2}\left(\frac{2\pi}{101}\right) + \cos^{2}\left(\frac{3\pi}{101}\right) + \cdots + \cos^{2}\left(\frac{99\pi}{101}\right) + \cos^{2}\left(\frac{100\pi}{101}\right)$$

cmi-entrance 2022 QA4 4 marks View

Statements
(13) As $x \rightarrow - \infty$ the function $\cos \left( e ^ { x } \right)$ tends to a finite limit. (14) As $x \rightarrow \infty$ the function $\cos \left( e ^ { x } \right)$ changes sign infinitely many times. (15) As $x \rightarrow \infty$, the function $\sin ( \ln ( x ) )$ tends to a finite limit. (16) $\sin ( \ln ( x ) )$ changes sign only finitely many times as $x$ goes towards 0 from 1.

csat-suneung 2021 Q2 2 marks View

For $\theta$ satisfying $\frac { \pi } { 2 } < \theta < \pi$ and $\sin \theta = \frac { \sqrt { 21 } } { 7 }$, what is the value of $\tan \theta$? [2 points]
(1) $- \frac { \sqrt { 3 } } { 2 }$
(2) $- \frac { \sqrt { 3 } } { 4 }$
(3) 0
(4) $\frac { \sqrt { 3 } } { 4 }$
(5) $\frac { \sqrt { 3 } } { 2 }$

csat-suneung 2021 Q4 3 marks View

What is the maximum value of the function $f ( x ) = 4 \cos x + 3$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

csat-suneung 2024 Q2 3 marks View

For $\theta$ satisfying $\frac{3}{2}\pi < \theta < 2\pi$ and $\sin(-\theta) = \frac{1}{3}$, find the value of $\tan\theta$. [3 points]
(1) $-\frac{\sqrt{2}}{2}$
(2) $-\frac{\sqrt{2}}{4}$
(3) $-\frac{1}{4}$
(4) $\frac{1}{4}$
(5) $\frac{\sqrt{2}}{4}$

csat-suneung 2025 Q10 4 marks View

The function $f(x) = a\cos bx + 3$ is defined on the closed interval $[0, 2\pi]$ and has a maximum value of 13 at $x = \frac{\pi}{3}$. For the ordered pair $(a, b)$ of two natural numbers $a$ and $b$ satisfying this condition, what is the minimum value of $a + b$? [4 points]
(1) 12
(2) 14
(3) 16
(4) 18
(5) 20

gaokao None Q7 View

Calculate the exact value of: $\cos \left( \frac { \pi } { 11 } \right)$

gaokao 2015 Q3 View

3. As shown in the figure, the water depth change curve of a certain port from 6 to 18 o'clock approximately satisfies the function $y = 3 \sin \left( \frac { \pi } { 6 } x + \varphi \right) + k$. Based on this function, the maximum water depth (in meters) during this period is
A. 5
B. 6
C. 8
D. 10 [Figure]

gaokao 2015 Q3 View

3. Execute the program flowchart shown in the figure, the output value of $S$ is
A. $- \frac { \sqrt { 3 } } { 2 }$ B. $\frac { \sqrt { 3 } } { 2 }$ C. $- \frac { 1 } { 2 }$ D. $\frac { 1 } { 2 }$ [Figure]

gaokao 2015 Q4 View

4. Among the following functions, the one with minimum positive period $\pi$ and whose graph is symmetric about the origin is
A. $\mathrm { y } = \cos \left( 2 x + \frac { \pi } { 2 } \right)$
B. $y = \sin \left( 2 x + \frac { \pi } { 2 } \right)$
C. $y = \sin 2 x + \cos 2 x$
D. $y = \sin x + \cos x$

gaokao 2015 Q10 View

10. Let $x \in \mathbf{R}$ and $[x]$ denote the greatest integer not exceeding $x$. If there exists a real number $t$ such that $[t] = 1$, $[t^2] = 2$, $\ldots$, $[t^n] = n$ all hold simultaneously, then the maximum value of the positive integer $n$ is
A. 3
B. 4
C. 5
D. 6
II. Fill-in-the-Blank Questions: This section has 6 questions. Candidates must answer 5 of them, each worth 5 points, for a total of 25 points. Write your answers in the corresponding positions on the answer sheet. Answers in wrong positions, illegible writing, or ambiguous answers will receive no credit.

(A) Compulsory Questions (Questions 11-14)

gaokao 2015 Q11 View

11. The function $f ( x ) = \sin ^ { 2 } x + \sin x \cos x + 1$ has minimum positive period $\_\_\_\_$ and minimum value $\_\_\_\_$.

gaokao 2015 Q11 View

11. The minimum positive period of the function $f ( x ) = \sin ^ { 2 } x + \sin x \cos x + 1$ is $\_\_\_\_$ , and the decreasing interval is $\_\_\_\_$ .

gaokao 2015 Q14 View

14. Given the function $f ( x ) = \sin \omega x + \cos \omega x ( \omega > 0 ) , x \in \mathbb{R}$. If the function $f ( x )$ is monotonically increasing on the interval $( - \omega , \omega )$, and the graph of $f ( x )$ is symmetric about the line $x = \omega$, then the value of $\omega$ is $\_\_\_\_$.

III. Solution Questions: This section has 6 questions, for a total of 80 points.

gaokao 2015 Q15 13 marks View

Given the function $f ( x ) = \sin x - 2 \sqrt { 3 } \sin ^ { 2 } \frac { x } { 2 }$\n(I) Find the minimum positive period of $f ( x )$;\n(II) Find the minimum value of $f ( x )$ on the interval $\left[ 0 , \frac { 2 \pi } { 3 } \right]$.

gaokao 2015 Q15 View

15. (This question is worth 13 points) Given the function $f ( x ) = \sqrt { 2 } \sin \frac { x } { 2 } \cos \frac { x } { 2 } - \sqrt { 2 } \sin ^ { 2 } \frac { x } { 2 }$. (I) Find the minimum positive period of $f ( x )$; (II) Find the minimum value of $f ( x )$ on the interval $[ - \pi , 0 ]$.

gaokao 2015 Q15 13 marks View

Given the function $f(x) = \sin^2 x - \sin^2\left(x - \frac{\pi}{6}\right)$, $x \in \mathbb{R}$.
(I) Find the minimum positive period of $f(x)$;
(II) Find the maximum and minimum values of $f(x)$ on the interval $\left[-\frac{\pi}{3}, \frac{\pi}{4}\right]$.

gaokao 2015 Q16 14 marks View

16. (14 points) In $\triangle A B C$ , the sides opposite to angles $\mathrm { A } , \mathrm { B }$ , C are $a , b , c$ respectively. Given that $\tan \left( \frac { \pi } { 4 } + \mathrm { A } \right) = 2$ .
(1) Find the value of $\frac { \sin 2 A } { \sin 2 A + \cos ^ { 2 } A }$ ;
(2) If $\mathrm { B } = \frac { \pi } { 4 } , a = 3$ , find the area of $\triangle A B C$ .

gaokao 2015 Q18 13 marks View

Given the function $\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { 2 } \sin 2 \mathrm { x } - \sqrt { 3 } \cos ^ { 2 } x$ .
(I) Find the minimum positive period and minimum value of $\mathrm { f } ( \mathrm { x } )$;
(II) The graph of function $\mathrm { f } ( \mathrm { x } )$ is transformed by stretching each point's horizontal coordinate to twice the original length while keeping the vertical coordinate unchanged, resulting in the graph of function $\mathrm { g } ( \mathrm { x } )$. When $\mathrm { x } \in \left[ \frac { \pi } { 2 } , \pi \right]$, find the range of $\mathrm { g } ( \mathrm { x } )$.

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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

(A) Compulsory Questions (Questions 11-14)

III. Solution Questions: This section has 6 questions, for a total of 80 points.

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