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Sagot :
Sure! Let's solve each problem step-by-step.
### Question 11
"Which one of the following is an exponential function?"
An exponential function is typically in the form [tex]\( f(x) = a b^x \)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants and [tex]\(b > 0\)[/tex].
Reviewing the provided options:
A. [tex]\( O = 2 B \cdot h \bigcirc^\prime = 3 \)[/tex] (not in an understandable form)
C. [tex]\( 0 = \)[/tex] (seems incomplete)
D. [tex]\( 0 = 2 \)[/tex] (not a valid equation)
None of these options clearly represent an exponential function, so the correct choice based on typical exponential form [tex]\(y = a e^{bx}\)[/tex] doesn't explicitly match. However, we may be dealing with a misprint or a non-traditional setup here.
### Question 12
"Which of the following is the radian measure of an angle [tex]\(330^\circ\)[/tex]?"
To convert degrees to radians:
[tex]\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \][/tex]
So for [tex]\(330^\circ\)[/tex]:
[tex]\[ 330^\circ \times \frac{\pi}{180} = \frac{330}{180} \pi = \frac{11}{6} \pi \][/tex]
Therefore, the radian measure of [tex]\(330^\circ\)[/tex] is:
[tex]\[ \boxed{\frac{11}{6} \text{ rad}} \][/tex]
### Question 13
"If an acute angle satisfies [tex]\(4 \sin x = 3 \cos x\)[/tex], then [tex]\(\sin x\)[/tex] is equal to:"
Given:
[tex]\[ 4 \sin x = 3 \cos x \][/tex]
Dividing both sides by [tex]\(\cos x\)[/tex]:
[tex]\[ 4 \tan x = 3 \Rightarrow \tan x = \frac{3}{4} \][/tex]
We need to find [tex]\(\sin x\)[/tex]. Recall the identity:
[tex]\[ \sin^2 x + \cos^2 x = 1 \][/tex]
We start by using [tex]\(\tan x = \frac{\sin x}{\cos x}\)[/tex]:
[tex]\[ \sin x = \cos x \cdot \tan x = \cos x \cdot \frac{3}{4} \][/tex]
Let's denote [tex]\(\cos x = \cos x = \cos x\)[/tex]. Using a 3-4-5 right triangle, where:
[tex]\[ \sin x = 3/5 \quad \text{and} \quad \cos x = 4/5 \][/tex]
Therefore:
[tex]\[ \sin x = \boxed{\frac{3}{5}} \][/tex]
### Question 14
"A function is said to be a logarithmic function, where"
A standard logarithmic function is given as:
[tex]\[ \text{A. } f(x) = \log_b x, b > 0, b \neq 1 \text{ and } x > 0 \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
### Question 15
"Which one of the following is not equal to [tex]\(\cos 72^\circ\)[/tex]?"
We need to check each option:
A. [tex]\(\cos(-648^\circ) = \cos(648^\circ) = \cos(360 \times 1 + 288^\circ) = \cos(288^\circ) = \cos(360^\circ - 72^\circ) = \cos(-72^\circ) = \cos(72^\circ)\)[/tex]
B. [tex]\(\sin(792^\circ) = \sin(792^\circ - 2 \times 360^\circ) = \sin(72^\circ)\)[/tex]
C. [tex]\(\sin(18^\circ)\)[/tex] – Not 72°; let's find out:
[tex]\(\sin(18^\circ)\)[/tex].
D. [tex]\(\cos(432^\circ) = \cos(360^\circ + 72^\circ) = \cos(72^\circ)\)[/tex]
Among these, [tex]\(\sin(18^\circ) = \cos(72^\circ)\)[/tex].
The answer here seems to revolve around precision, hence clear out options matching other than 18°.
Looking closely, based on identity checks:
[tex]\[ \boxed{\sin(18^\circ)} \][/tex] - double check for clear non-matching earlier checks.
Please verify ordered by close comparison to radians convention.
### Question 11
"Which one of the following is an exponential function?"
An exponential function is typically in the form [tex]\( f(x) = a b^x \)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants and [tex]\(b > 0\)[/tex].
Reviewing the provided options:
A. [tex]\( O = 2 B \cdot h \bigcirc^\prime = 3 \)[/tex] (not in an understandable form)
C. [tex]\( 0 = \)[/tex] (seems incomplete)
D. [tex]\( 0 = 2 \)[/tex] (not a valid equation)
None of these options clearly represent an exponential function, so the correct choice based on typical exponential form [tex]\(y = a e^{bx}\)[/tex] doesn't explicitly match. However, we may be dealing with a misprint or a non-traditional setup here.
### Question 12
"Which of the following is the radian measure of an angle [tex]\(330^\circ\)[/tex]?"
To convert degrees to radians:
[tex]\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \][/tex]
So for [tex]\(330^\circ\)[/tex]:
[tex]\[ 330^\circ \times \frac{\pi}{180} = \frac{330}{180} \pi = \frac{11}{6} \pi \][/tex]
Therefore, the radian measure of [tex]\(330^\circ\)[/tex] is:
[tex]\[ \boxed{\frac{11}{6} \text{ rad}} \][/tex]
### Question 13
"If an acute angle satisfies [tex]\(4 \sin x = 3 \cos x\)[/tex], then [tex]\(\sin x\)[/tex] is equal to:"
Given:
[tex]\[ 4 \sin x = 3 \cos x \][/tex]
Dividing both sides by [tex]\(\cos x\)[/tex]:
[tex]\[ 4 \tan x = 3 \Rightarrow \tan x = \frac{3}{4} \][/tex]
We need to find [tex]\(\sin x\)[/tex]. Recall the identity:
[tex]\[ \sin^2 x + \cos^2 x = 1 \][/tex]
We start by using [tex]\(\tan x = \frac{\sin x}{\cos x}\)[/tex]:
[tex]\[ \sin x = \cos x \cdot \tan x = \cos x \cdot \frac{3}{4} \][/tex]
Let's denote [tex]\(\cos x = \cos x = \cos x\)[/tex]. Using a 3-4-5 right triangle, where:
[tex]\[ \sin x = 3/5 \quad \text{and} \quad \cos x = 4/5 \][/tex]
Therefore:
[tex]\[ \sin x = \boxed{\frac{3}{5}} \][/tex]
### Question 14
"A function is said to be a logarithmic function, where"
A standard logarithmic function is given as:
[tex]\[ \text{A. } f(x) = \log_b x, b > 0, b \neq 1 \text{ and } x > 0 \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
### Question 15
"Which one of the following is not equal to [tex]\(\cos 72^\circ\)[/tex]?"
We need to check each option:
A. [tex]\(\cos(-648^\circ) = \cos(648^\circ) = \cos(360 \times 1 + 288^\circ) = \cos(288^\circ) = \cos(360^\circ - 72^\circ) = \cos(-72^\circ) = \cos(72^\circ)\)[/tex]
B. [tex]\(\sin(792^\circ) = \sin(792^\circ - 2 \times 360^\circ) = \sin(72^\circ)\)[/tex]
C. [tex]\(\sin(18^\circ)\)[/tex] – Not 72°; let's find out:
[tex]\(\sin(18^\circ)\)[/tex].
D. [tex]\(\cos(432^\circ) = \cos(360^\circ + 72^\circ) = \cos(72^\circ)\)[/tex]
Among these, [tex]\(\sin(18^\circ) = \cos(72^\circ)\)[/tex].
The answer here seems to revolve around precision, hence clear out options matching other than 18°.
Looking closely, based on identity checks:
[tex]\[ \boxed{\sin(18^\circ)} \][/tex] - double check for clear non-matching earlier checks.
Please verify ordered by close comparison to radians convention.
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