Let's solve the questions one by one from the image.

Question 31

Problem: Find the value of $x$ in the diagram.

Steps:

1. The given figure has a quadrilateral where the angles are given by $(2x)^o$, $(x + 10)^o$, $(4x + 50)^o$, and $(3x)^o$.
2. In any quadrilateral, the sum of all interior angles is $360^\circ$.

$(2x) + (x + 10) + (4x + 50) + (3x) = 360^\circ$

$2x + x + 10 + 4x + 50 + 3x = 360$

$10x + 60 = 360$

$10x = 360 - 60 = 300$

$x = \frac{300}{10} = 30$

So, the value of $x$ is $\boxed{30}$.

Question 32

Problem: The interior angles of a triangle are $(y + 10)^\circ$, $(2y - 40)^\circ$, and $(3y - 90)^\circ$. Which of the following accurately describes the triangle?

Steps:

1. The sum of the interior angles of a triangle is $180^\circ$.
2. Set up the equation:

$(y + 10) + (2y - 40) + (3y - 90) = 180^\circ$

$y + 10 + 2y - 40 + 3y - 90 = 180$

$6y - 120 = 180$

$6y = 300$

$y = \frac{300}{6} = 50$

3. Substitute $y = 50$ back into the expressions for the angles:

• $y + 10 = 50 + 10 = 60^\circ$
• $2y - 40 = 100 - 40 = 60^\circ$
• $3y - 90 = 150 - 90 = 60^\circ$

All angles are $60^\circ$, which indicates that the triangle is an equilateral triangle.

Thus, the correct answer is $\boxed{C}$ (equilateral triangle).

Question 33

Problem: Given that $\sin A = \frac{3}{5}$ and $0^\circ \leq A \leq 90^\circ$, find the value of $(\tan A - \cos A)$.

Steps:

1. First, find $\cos A$ using the identity:

$\sin^2 A + \cos^2 A = 1$

$\left(\frac{3}{5}\right)^2 + \cos^2 A = 1$

$\frac{9}{25} + \cos^2 A = 1$

$\cos^2 A = 1 - \frac{9}{25} = \frac{16}{25}$

$\cos A = \frac{4}{5}$

2. Next, find $\tan A$:

$\tan A = \frac{\sin A}{\cos A} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}$

3. Now, calculate $(\tan A - \cos A)$:

$\tan A - \cos A = \frac{3}{4} - \frac{4}{5}$

To subtract, find a common denominator (20):

$\tan A - \cos A = \frac{15}{20} - \frac{16}{20} = \frac{-1}{20}$

Thus, the correct answer is $\boxed{D}$ $\frac{-1}{20}$.

Do you need further explanations or have any questions?

Here are five related questions you might find useful:

1. How can you determine the angles in a cyclic quadrilateral?
2. What is the relationship between the interior and exterior angles of polygons?
3. How would you solve for $y$ if given different expressions for the triangle's angles?
4. How can you verify trigonometric identities involving sine and cosine?
5. What other trigonometric functions can be derived using sine and cosine?

Tip: When dealing with problems involving triangles, always remember to use the sum of angles rule to set up your equations.