Set 3

Explanation

Observe the pattern of the following number series.

The pattern formed is as follows.

1. $3 \xrightarrow{+3} 6$
2. $6 \xrightarrow{\times 3} 18$
3. $18 \xrightarrow{-3} 15$
4. $15 \xrightarrow{+3} m$
5. $m \xrightarrow{\times 3} 54$
6. $54 \xrightarrow{-3} 51$
7. $51 \xrightarrow{+3} 54$
8. $54 \xrightarrow{\times 3} 162$

It can be seen that the operation pattern repeats: $+3, \;\times 3, \; -3$.

To determine the value of $m$, we continue the pattern after $15$, which is adding $3$.

We can verify the value of $m$ with the next step (multiplying by $3$).

Thus, the value of $m$ is $18$.

Explanation

Given the operation $x \ast y = (x + y + 1)x + xy$.

First, we calculate the value of $-1 \ast 2$.

Next, substitute this result into the equation $a \ast (-1 \ast 2) = -10$, which becomes $a \ast (-4) = -10$.

We factor the quadratic equation to find the value of $a$.

Thus, the values of $a$ that satisfy the equation are $2$ or $5$.

Explanation

The relationship between speed ($v$) and time ($t$) is inversely proportional. This means the product of speed and time is constant (fixed distance).

From the table, we know:

• First speed $v_1 = 96 \text{ km/h}$
• First time $t_1 = 1 \text{ hour}$
• Second speed $v_2 = 72 \text{ km/h}$

We want to find the second time ($t_2$).

We convert the fraction of an hour into hours and minutes.

Thus, the time required is $1 \text{ hour } 20 \text{ minutes}$.

Explanation

Given:

• Initial jacket price = $\text{Rp}250.000,00$
• Discount = $25\%$
• Tax = $5\%$

First, we calculate the price after the discount. Since the discount is $25\%$, Mrs. Kartika pays $100\% - 25\% = 75\%$ of the initial price.

Next, a tax of $5\%$ is applied to the price after discount. Thus, the total to be paid is $100\% + 5\% = 105\%$ of the price after discount.

Directly, the calculation can be written as:

So, the price Mrs. Kartika has to pay is $\text{Rp}196.875,00$.

Explanation

Given:

• Perimeter ($K$) = $x \text{ cm}$
• Length ($p$) = $(8y + 5) \text{ cm}$
• Width ($l$) = $(6y + 1) \text{ cm}$

The formula for the perimeter of a rectangle is $K = 2(p + l)$. We substitute the known values into the formula.

Next, we rearrange the equation to solve for $y$.

Thus, the value of $y$ is $\left(\frac{x}{28} - \frac{3}{7}\right) \text{ cm}$.

Explanation

Let's analyze each statement for the line $5x - 10y = 3$.

Statement 1: Gradient

The equation of a line in the form $ax + by = c$ has a gradient $m = -\frac{a}{b}$. For $5x - 10y = 3$:

• $a = 5$
• $b = -10$

Statement 1 says the gradient is $-2$. So, Statement 1 is false.

Statement 2: X-axis intercept

The x-axis intercept occurs when $y = 0$.

The coordinate point is $\left(\frac{3}{5}, 0\right)$. So, Statement 2 is true.

Statement 3 and 4: Y-axis intercept

The y-axis intercept occurs when $x = 0$.

The coordinate point is $\left(0, -\frac{3}{10}\right)$. So, Statement 3 is false and Statement 4 is true.

Conclusion: The correct statements are (2) and (4).

Explanation

Let the smallest odd number be $a$. Since consecutive odd numbers have a difference of $2$, the three numbers are $a, a + 2, a + 4$.

The sum of these three numbers is given as $1005$.

The largest number is $a + 4$.

Thus, the value of $P = 337$. Given the value of $Q = 401$.

Therefore, the relationship between $P$ and $Q$ is:

Explanation

Given $x = \frac{5}{6}$ and $y = \frac{3}{7}$.

Calculating the value of $P$

The value of $P = \frac{18}{85}$ (less than $1$, approx $0.21$).

Calculating the value of $Q$

The value of $Q = \frac{42}{53}$ (less than $1$, approx $0.79$).

Comparing $P$ and $Q$

It is clear that $0.21 < 0.79$.

Therefore, $P < Q$.

Explanation

Given $\angle BOC = 50^\circ$. Line $AOB$ is a straight line (diameter), so the straight angle is $180^\circ$.

Thus, we can find the measure of $\angle AOC$:

Consider triangle $\triangle AOC$. Since $O$ is the center of the circle, $OA$ and $OC$ are radii of the circle. Therefore, $OA = OC$, so triangle $\triangle AOC$ is an isosceles triangle.

In an isosceles triangle $\triangle AOC$, the base angles are equal, i.e., $\angle OAC = \angle OCA$.

The sum of angles in a triangle is $180^\circ$.

So, the measure of $\angle OAC$ is $25^\circ$.

Explanation

Given the data: $55, a, 73, b$. The average of these numbers is $68$.

The difference between $b$ and $a$ is $22$.

We eliminate equations (1) and (2).

Add both equations:

Substitute $b = 83$ into equation (1):

Thus, the four numbers are $55, 61, 73, 83$. These are already ordered. The median is the middle value of the sorted data. For 4 data points, the median is the average of the 2nd and 3rd values.

So, the median of the four data points is $67$.

Explanation

We verify each statement by converting the fractions into decimals.

Statement: $\frac{3}{7} > \frac{1}{3}$

Statement: $\frac{1}{5} > \frac{2}{9}$

Statement: $\frac{2}{3} > \frac{7}{8}$

Statement: $\frac{3}{5} > \frac{5}{6}$

Statement: $\frac{5}{9} > \frac{3}{4}$

Thus, the correct statement is $\frac{3}{7} > \frac{1}{3}$.

Explanation

The ratio between spoons of tea and spoons of sugar to make a cup of tea is:

We simplify the ratio by multiplying both sides by $2$:

If $2$ spoons of tea are used, we can find the number of spoons of sugar ($a$) using the ratio:

Thus, the value of $P = a = 6$. Given the value of $Q = 8$.

Therefore, the relationship between $P$ and $Q$ is:

Explanation

There are $3$ boxes available. Two marbles will be placed into these boxes, with the condition that one box can only hold one marble.

Let the boxes be numbered 1, 2, 3 (from left to right).

Sample space

The total number of ways to choose 2 boxes from 3 available boxes:

Possible pairs of filled boxes:

• Boxes 2 - Adjacent
• Boxes 3 - Adjacent
• Boxes 3 - Not adjacent

Desired event

The event that marbles are not adjacent ($A$) occurs when the filled boxes are 3.

Probability

The probability that marbles are stored not adjacent to each other:

Thus, the probability that the marbles are stored not adjacent to each other is $\frac{1}{3}$.

Explanation

Given the matrix $\mathbf{P} = \begin{pmatrix} 3 - x & -5 \\ 6 & 3 \end{pmatrix}$.

The formula for the determinant of a $2 \times 2$ matrix:

For matrix $\mathbf{P}$:

Given $\det(\mathbf{P}) = 33$, then:

Thus, the value of $x$ is $2$.

Explanation

Given the function:

We substitute the values $w = 5$, $z = 4$, $x = 3$, and $y = 2$.

Thus, the value of $A(5 \otimes 4 \oslash 3 \oplus 2)$ is $6$.

Explanation

Let's analyze each plane figure based on rotational symmetry and reflectional symmetry.

Explanation

Regular octagon

• Rotational symmetry $8$ (can be rotated at $45^\circ, 90^\circ, 135^\circ, \ldots, 315^\circ$).
• Reflectional symmetry $8$ (through diagonals and midlines).

Rhombus that is not a square

• Rotational symmetry $2$ (initial position and $180^\circ$ rotation).
• Reflectional symmetry $2$ (through both diagonals).

Isosceles triangle that is not equilateral

• Rotational symmetry $1$ (only initial position).
• Reflectional symmetry $1$ (through the height from the apex angle).

Kite that is not a square

• Rotational symmetry $1$ (only initial position).
• Reflectional symmetry $1$ (through one of the diagonals).

Conclusion

There are no figures that have more rotational symmetries than reflectional symmetries. All figures have the same number of rotational symmetries as reflectional symmetries or fewer.

Thus, the answer is $0$.

Explanation

The conditions that must be satisfied by the number $n$:

• If $n$ is divided by $8$, the remainder is $5$.
• If $n$ is divided by $5$, the remainder is $2$.

Let's check each option.

Check value 29

• Divided by $8$: $29 = 8 \times 3 + 5$ (remainder $5$ ✓)
• Divided by $5$: $29 = 5 \times 5 + 4$ (remainder $4$ ✗)

The value $29$ does not satisfy the second condition.

Check value 37

• Divided by $8$: $37 = 8 \times 4 + 5$ (remainder $5$ ✓)
• Divided by $5$: $37 = 5 \times 7 + 2$ (remainder $2$ ✓)

The value $37$ satisfies both conditions.

Check value 44

• Divided by $8$: $44 = 8 \times 5 + 4$ (remainder $4$ ✗)
• Divided by $5$: $44 = 5 \times 8 + 4$ (remainder $4$ ✗)

The value $44$ does not satisfy the conditions.

Check value 77

• Divided by $8$: $77 = 8 \times 9 + 5$ (remainder $5$ ✓)
• Divided by $5$: $77 = 5 \times 15 + 2$ (remainder $2$ ✓)

The value $77$ satisfies both conditions.

Conclusion

The values that satisfy the conditions in the question are (2) and (4).

Explanation

Given:

• $B_3 = 2$
• $B_5 = -1$
• $B_n = B_{n+2} + B_{n-1}$

We will find the value of $B_2$ using the given recursive formula.

Finding the value of B2

Use the formula with $n = 3$.

Substitute the values $B_3 = 2$ and $B_5 = -1$.