Given that sin 2x = 1/2, then the value of sin 3x is

The corresponding sides are proportional.

$AB = 9 \text{ cm}$, $BC = 5 \text{ cm}$, $PR = 2 \text{ cm}$.

The length of side QR is $\boxed{\text{d) *3.6 cm}}$.

2. Length of tangent PQ:

The line segment OP bisects $\angle QPR$.
Given $\angle QPR = 60^\circ$, so $\angle QPO = \frac{1}{2} \times 60^\circ = 30^\circ$.

In the right-angled triangle $\triangle OQP$:

The length of the tangent is \boxed{\text{a) *3\sqrt{3} cm}}.

3. Value of $\tan^2 45^\circ - \cos^2 60^\circ$:

$\tan 45^\circ = 1$ and $\cos 60^\circ = \frac{1}{2}$.

The value is \boxed{\text{c) *\frac{3}{4}}}.

4. Value of $\sin 3x$:

We know that $\sin 60^\circ = \frac{\sqrt{3}}{2}$.

The value of $\sin 3x$ is $\boxed{\text{d) *1}}$.

5. Altitude of the sun:

Given $s = \sqrt{3}h$.

In the right-angled triangle formed by the tower and its shadow, the tangent of the angle of elevation is the ratio of the height to the shadow length.

We know that $\tan 30^\circ = \frac{1}{\sqrt{3}}$.

The altitude of the sun is \boxed{\text{b) *30^\circ}}.

6. Definition of figures with same shape but not necessarily same size:
Figures having the same shape and not necessarily the same size are called similar figures.
The correct option is $\boxed{\text{c) *similar figures}}$.

7. Value of x:

By the Basic Proportionality Theorem (Thales's Theorem), the line divides the sides proportionally.

$PM = x$, $MQ = 10 \text{ cm}$, $PN = (x-2) \text{ cm}$, $NR = 6 \text{ cm}$.

The value of x is $\boxed{\text{b) *5 cm}}$.

8. Type of triangle formed by the points (-4,0), (4,0), and (0,3):

Calculate the length of each side using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
Length of AB:

Length of BC:

Length of AC:

Since $BC = AC = 5$ units, two sides of the triangle are equal.
Therefore, the triangle is an isosceles triangle.
The triangle is a/an $\boxed{\text{b) *isosceles triangle}}$.

9. Value of p:

Use the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

The value of p is $\boxed{\text{b) *\pm 4 only}}$.

10. Value of a:

Assuming the question intended for the x-coordinate of Q to be 'a', i.e., Q(a,5), and P(3,4) is the midpoint of QR.

$P_x = \frac{Q_x + R_x}{2}$.

The value of a is $\boxed{\text{c) *8}}$.

11. Value of $9\sec^2 A - 9\tan^2 A$:

The value is $\boxed{\text{b) *9}}$.

12. Value of $\frac{1-\tan^2 45^\circ}{1+\tan^2 45^\circ}$:

The value is $\boxed{\text{d) *0}}$.

13. Sun's altitude when shadow length equals height:

Given $s = h$.

In the right-angled triangle formed by the tower and its shadow:

We know that $\tan 45^\circ = 1$.

The sun's altitude is \boxed{\text{d) *45^\circ}}.

14. Value of $\angle POA$:

The line segment OP bisects the angle between the tangents, so $\angle APO = \frac{1}{2} \angle APB = \frac{1}{2} \times 80^\circ = 40^\circ$.

The value of $\angle POA$ is \boxed{\text{a) *50^\circ}}.

15. Common point of a tangent and a circle:
The common point of a tangent to a circle and the circle is called the point of contact.
The correct option is $\boxed{\text{c) *point of contact}}$.

16. Area of sector of angle P (in degrees) of a circle with radius R:
The formula for the area of a sector with angle $\theta$ (in degrees) and radius R is $\frac{\theta}{360^\circ} \pi R^2$.
Given the angle is P, the area is $\frac{P}{360^\circ} \pi R^2$.
Option d) is $\frac{P}{720^\circ} \times 2\pi R^2 = \frac{P \times 2\pi R^2}{720^\circ} = \frac{P \pi R^2}{360^\circ}$.
This matches the formula.
The area of the sector is \boxed{\text{d) *\frac{P}{720^\circ} \times 2\pi R^2}}.

17. Relationship between height and radius of a cylinder:

Divide both sides by $2\pi r$.

The relationship is $\boxed{\text{r = 2h}}$.