If x=1/4 and x-2 when y=3, find the value of x when y=4.

The problem states that $x$ is inversely proportional to $y$, which can be written as $x \propto \frac{1}{y}$.
This means $x = \frac{k}{y}$ for some constant $k$.
Given $x=2$ when $y=3$:

$$2 = \frac{k}{3}$$

Multiply both sides by 3 to find $k$:

$$k = 2 \times 3$$ $$k = 6$$

Now, we need to find the value of $x$ when $y=4$. Use the constant $k=6$:

$$x = \frac{6}{y}$$ $$x = \frac{6}{4}$$ $$x = \frac{3}{2}$$

The correct option is D.
The final answer is $\boxed{\text{D. } \frac{3}{2}}$.

The diameter of the sphere is $d = 6.5$ cm.
The radius of the sphere is $r = \frac{d}{2} = \frac{6.5}{2} = 3.25$ cm.
The formula for the volume of a sphere is $V = \frac{4}{3}\pi r^3$.
Substitute the value of $r$:

$$V = \frac{4}{3} \times \pi \times (3.25)^3$$ $$V = \frac{4}{3} \times \pi \times 34.328125$$ $$V = \frac{137.3125}{3} \times \pi$$ $$V \approx 45.770833 \times 3.14159$$ $$V \approx 143.789 \text{ cm}^3$$

Rounding to one decimal place, $V \approx 143.8 \text{ cm}^3$.
The correct option is B.
The final answer is $\boxed{\text{B. } 143.8 \text{ cm}^3}$.

When two fair dice are thrown, the total number of possible outcomes is $6 \times 6 = 36$.
The odd numbers on a die are 1, 3, 5.
For the first die to show an odd number, there are 3 possibilities (1, 3, 5).
For the second die to show an odd number, there are also 3 possibilities (1, 3, 5).
The number of outcomes where both dice show an odd number is $3 \times 3 = 9$.
The probability of getting two odd numbers is $\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$.

$$P(\text{two odd numbers}) = \frac{9}{36}$$ $$P(\text{two odd numbers}) = \frac{1}{4}$$

The correct option is D.
The final answer is $\boxed{\text{D. } \frac{1}{4}}$.

We need to solve for $y$ in the equation $4y^3 = 500$.
Divide both sides by 4:

$$y^3 = \frac{500}{4}$$ $$y^3 = 125$$

Take the cube root of both sides:

$$y = \sqrt[3]{125}$$ $$y = 5$$

The correct option is B.
The final answer is $\boxed{\text{B. } 5}$.

The table shows the scores and the number of students who obtained each score.
Scores: 8, 10, 12, 15, 18, 20
No. of Students: 3, 5, 6, 3, 2, 1
Total number of students = $3+5+6+3+2+1 = 20$.
Since there are 20 students (an even number), the median is the average of the scores at the $\frac{N}{2}$-th and $(\frac{N}{2}+1)$-th positions.
Here, $N=20$, so the median is the average of the 10th and 11th scores.
Let's find the cumulative frequency:

• Score 8: 3 students (1st to 3rd scores)
• Score 10: 3 + 5 = 8 students (4th to 8th scores)
• Score 12: 8 + 6 = 14 students (9th to 14th scores) The 10th score falls within the "Score 12" group. The 11th score also falls within the "Score 12" group. Therefore, both the 10th and 11th scores are 12. The median score is $\frac{12+12}{2} = \frac{24}{2} = 12$. The correct option is A. The final answer is $\boxed{\text{A. } 12}$.

Let's analyze the given truth table for Y:

P	Q	Y
T	T	T
T	F	F
F	T	F
F	F	F
This truth table shows that Y is true only when both P and Q are true. This is the definition of the logical AND operation, denoted as $P \land Q$.
The correct option is C.
The final answer is $\boxed{\text{C. } P \land Q}$.

We need to find the gradient of a line joining the points $(2, 5)$ and $(3, 1)$.
Let $(x_1, y_1) = (2, 5)$ and $(x_2, y_2) = (3, 1)$.
The formula for the gradient $m$ is:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Substitute the coordinates:

$$m = \frac{1 - 5}{3 - 2}$$ $$m = \frac{-4}{1}$$ $$m = -4$$

None of the options A, B, C, D, E match -4. Let me recheck the options provided in the image. The options are A. 5, B. 6, C. 5, D. 7, E. 6. It seems there might be an error in the question's options or my interpretation of the image. Assuming the calculation is correct, the gradient is -4. If I must choose from the given options, there is no correct answer. However, if the question intended to ask for something else or if the options are for a different question, I cannot proceed. Given the options, I cannot select a correct answer.

Let's re-examine the image for question 19's options. The options are A. 5, B. 6, C. 5, D. 7, E. 6. There are no negative options. This indicates a potential error in the question or options provided. I will state the calculated answer.

The final answer is $\boxed{-4}$.