Answer :
Sure, let's go through each question step-by-step.
### Question 2
Which one of the following is true about the quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex], where [tex]\( a, b, c \in \mathbb{R} \)[/tex] and [tex]\( a > 0 \)[/tex]?
When analyzing quadratic functions:
- [tex]\( f(x) = ax^2 + bx + c \)[/tex] is a parabola that opens upwards since [tex]\( a > 0 \)[/tex].
- The vertex of the parabola gives the minimum value because the parabola opens upwards.
The vertex of the parabola [tex]\( ax^2 + bx + c \)[/tex] occurs at:
[tex]\[ x = -\frac{b}{2a} \][/tex]
At this point, the value of the function is:
[tex]\[ f\left(-\frac{b}{2a}\right) \][/tex]
Thus, the minimum value of the function [tex]\( f(x) \)[/tex] is [tex]\( f\left(-\frac{b}{2a}\right) \)[/tex], and the axis of symmetry is [tex]\( x = -\frac{b}{2a} \)[/tex].
Let's look at each statement:
A. [tex]\( f(0) \)[/tex] is the minimum value of the function.
- This is incorrect because the minimum value occurs at [tex]\( f\left(-\frac{b}{2a}\right) \)[/tex], not at [tex]\( f(0) \)[/tex].
B. The range of [tex]\( f \)[/tex] is [tex]\( \left\{ y : y \geq f\left( -\frac{b}{2a} \right) \right\} \)[/tex].
- This is correct because for [tex]\( a > 0 \)[/tex], the parabola opens upwards, and the minimum value is [tex]\( f\left(-\frac{b}{2a}\right) \)[/tex], so the range is all values greater than or equal to this minimum value.
C. The axis of symmetry of [tex]\( f \)[/tex] is [tex]\( x = \frac{b}{2a} \)[/tex].
- This is incorrect. The correct formula for the axis of symmetry is [tex]\( x = -\frac{b}{2a} \)[/tex].
D. The graph of [tex]\( f \)[/tex] has two [tex]\( x \)[/tex]-intercepts.
- This is not necessarily true for all quadratic functions with [tex]\( a > 0 \)[/tex]. The number of [tex]\( x \)[/tex]-intercepts depends on the discriminant [tex]\( b^2 - 4ac \)[/tex]. If the discriminant is positive, there are two [tex]\( x \)[/tex]-intercepts; if zero, there is one; if negative, there are no [tex]\( x \)[/tex]-intercepts.
Thus, the correct answer is:
B. The range of [tex]\( f \)[/tex] is [tex]\( \left\{ y : y \geq f\left( -\frac{b}{2a} \right) \right\} \)[/tex].
### Question 3
A regular hexagon is inscribed in a circle whose area is [tex]\( 16\pi \)[/tex] sq. units. What is the area of this hexagon?
Given:
- The area of the circle [tex]\( A_{\text{circle}} = 16\pi \)[/tex] sq. units.
First, find the radius of the circle:
[tex]\[ A_{\text{circle}} = \pi r^2 = 16\pi \][/tex]
[tex]\[ r^2 = 16 \][/tex]
[tex]\[ r = 4 \][/tex]
For a regular hexagon inscribed in a circle:
- The radius of the circumscribed circle is equal to the side length [tex]\( s \)[/tex] of the hexagon.
[tex]\[ s = 4 \][/tex]
The formula for the area of a regular hexagon with side length [tex]\( s \)[/tex] is:
[tex]\[ A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} s^2 \][/tex]
Substitute [tex]\( s = 4 \)[/tex]:
[tex]\[ A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} (4)^2 \][/tex]
[tex]\[ A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} \times 16 \][/tex]
[tex]\[ A_{\text{hexagon}} = 24\sqrt{3} \text{ sq. units} \][/tex]
Thus, the correct answer is:
C. [tex]\( 24\sqrt{3} \)[/tex] sq. units
### Question 4
Which of the following defines a prime number in the natural number system?
A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. Thus, it must have exactly two distinct positive factors.
Let's review each statement:
A. It has exactly two distinct factors.
- This is correct. A prime number has exactly two distinct factors: 1 and the number itself.
B. It has at least one more factor other than itself and 1.
- This is incorrect because a prime number cannot have more than two factors.
C. It is divisible by 2.
- This is incorrect as it only applies to one specific prime number, 2.
D. It is divisible only by 1.
- This is also incorrect as it misses the distinction that it is also divisible by the prime number itself.
Thus, the correct answer is:
A. It has exactly two distinct factors.
### Question 2
Which one of the following is true about the quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex], where [tex]\( a, b, c \in \mathbb{R} \)[/tex] and [tex]\( a > 0 \)[/tex]?
When analyzing quadratic functions:
- [tex]\( f(x) = ax^2 + bx + c \)[/tex] is a parabola that opens upwards since [tex]\( a > 0 \)[/tex].
- The vertex of the parabola gives the minimum value because the parabola opens upwards.
The vertex of the parabola [tex]\( ax^2 + bx + c \)[/tex] occurs at:
[tex]\[ x = -\frac{b}{2a} \][/tex]
At this point, the value of the function is:
[tex]\[ f\left(-\frac{b}{2a}\right) \][/tex]
Thus, the minimum value of the function [tex]\( f(x) \)[/tex] is [tex]\( f\left(-\frac{b}{2a}\right) \)[/tex], and the axis of symmetry is [tex]\( x = -\frac{b}{2a} \)[/tex].
Let's look at each statement:
A. [tex]\( f(0) \)[/tex] is the minimum value of the function.
- This is incorrect because the minimum value occurs at [tex]\( f\left(-\frac{b}{2a}\right) \)[/tex], not at [tex]\( f(0) \)[/tex].
B. The range of [tex]\( f \)[/tex] is [tex]\( \left\{ y : y \geq f\left( -\frac{b}{2a} \right) \right\} \)[/tex].
- This is correct because for [tex]\( a > 0 \)[/tex], the parabola opens upwards, and the minimum value is [tex]\( f\left(-\frac{b}{2a}\right) \)[/tex], so the range is all values greater than or equal to this minimum value.
C. The axis of symmetry of [tex]\( f \)[/tex] is [tex]\( x = \frac{b}{2a} \)[/tex].
- This is incorrect. The correct formula for the axis of symmetry is [tex]\( x = -\frac{b}{2a} \)[/tex].
D. The graph of [tex]\( f \)[/tex] has two [tex]\( x \)[/tex]-intercepts.
- This is not necessarily true for all quadratic functions with [tex]\( a > 0 \)[/tex]. The number of [tex]\( x \)[/tex]-intercepts depends on the discriminant [tex]\( b^2 - 4ac \)[/tex]. If the discriminant is positive, there are two [tex]\( x \)[/tex]-intercepts; if zero, there is one; if negative, there are no [tex]\( x \)[/tex]-intercepts.
Thus, the correct answer is:
B. The range of [tex]\( f \)[/tex] is [tex]\( \left\{ y : y \geq f\left( -\frac{b}{2a} \right) \right\} \)[/tex].
### Question 3
A regular hexagon is inscribed in a circle whose area is [tex]\( 16\pi \)[/tex] sq. units. What is the area of this hexagon?
Given:
- The area of the circle [tex]\( A_{\text{circle}} = 16\pi \)[/tex] sq. units.
First, find the radius of the circle:
[tex]\[ A_{\text{circle}} = \pi r^2 = 16\pi \][/tex]
[tex]\[ r^2 = 16 \][/tex]
[tex]\[ r = 4 \][/tex]
For a regular hexagon inscribed in a circle:
- The radius of the circumscribed circle is equal to the side length [tex]\( s \)[/tex] of the hexagon.
[tex]\[ s = 4 \][/tex]
The formula for the area of a regular hexagon with side length [tex]\( s \)[/tex] is:
[tex]\[ A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} s^2 \][/tex]
Substitute [tex]\( s = 4 \)[/tex]:
[tex]\[ A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} (4)^2 \][/tex]
[tex]\[ A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} \times 16 \][/tex]
[tex]\[ A_{\text{hexagon}} = 24\sqrt{3} \text{ sq. units} \][/tex]
Thus, the correct answer is:
C. [tex]\( 24\sqrt{3} \)[/tex] sq. units
### Question 4
Which of the following defines a prime number in the natural number system?
A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. Thus, it must have exactly two distinct positive factors.
Let's review each statement:
A. It has exactly two distinct factors.
- This is correct. A prime number has exactly two distinct factors: 1 and the number itself.
B. It has at least one more factor other than itself and 1.
- This is incorrect because a prime number cannot have more than two factors.
C. It is divisible by 2.
- This is incorrect as it only applies to one specific prime number, 2.
D. It is divisible only by 1.
- This is also incorrect as it misses the distinction that it is also divisible by the prime number itself.
Thus, the correct answer is:
A. It has exactly two distinct factors.
