Set 10

Explanation

Define the letters according to their alphabetical order:

• $A = 1$
• $B = 2$
• $C = 3$
• ...
• $Z = 26$

Thus, we can convert the letters in the triangles into numbers:

Observe that the first triangle follows a pattern:

The number $20$ corresponds to the letter $T$ (the $20$th letter).

Using the same pattern, the number in the second triangle is:

Substitute the number back to the letter sequence. The $25$th letter is $Y$.

Therefore, the answer is $Y$.

Explanation

From the figure, we can see that the angle vertically opposite to $z^\circ$ is the angle between the two diagonal lines at the top. The measure of that angle is equal to $z^\circ$.

Since the angles on a straight line sum to $180^\circ$, the sum of angle $x^\circ$, vertical angle $z^\circ$, and $y^\circ$ is $180^\circ$:

Given that $x^\circ = y^\circ$, we can substitute $y$ with $x$:

Therefore, the measure of angle $z$ is $180 - 2x$.

Explanation

Let $P(A)$ be the probability that each person gets a prize, then:

Therefore, the probability that a participant gets a prize is $0.0067$.

Explanation

Let's define the members of each set:

We will check each option to see if the number can be formed by multiplying two members of these sets.

Analysis of Options

1. Option A: $-14$

   $$-14 = (-7) \times 2$$

   Here, $-7 \in A$ and $2 \in B$. So, $-14$ is possible.

2. Option B: $-15$

   $$-15 = (-5) \times 3$$

   Here, $-5 \in A$ and $3 \in C$. So, $-15$ is possible.

3. Option C: $-18$

   $$-18 = (-3) \times 6$$

   Here, $-3 \in A$ and $6 \in B$ (also $6 \in C$). So, $-18$ is possible.

4. Option D: $14$

   $$14 = 2 \times 7$$

   Here, $2 \in B$. However, $7$ is a positive odd integer that is not divisible by $3$. This means:

   • $7 \notin A$ (since $A$ contains negative integers).
   • $7 \notin B$ (since $7$ is odd).
   • $7 \notin C$ (since $7$ is not divisible by $3$).

   Since one of the factors ($7$) is not present in sets $A$, $B$, or $C$, $14$ cannot be formed (assuming the pattern involves factors from different sets or positive factors).

5. Option E: $18$

   $$18 = 2 \times 9$$

   Here, $2 \in B$ and $9 \in C$. So, $18$ is possible.

Therefore, the number that is NOT included in the product is $14$.

Explanation

Given the system of inequalities:

We can sum the two main inequalities to find the bound for $x + y$:

It is given that the maximum value of $x + y$ is $5$. Therefore, we substitute this maximum value into the inequality:

Thus, the value of $a$ is:

(We take the positive value since the options are positive).

So, the value of $a$ is $5$.

Explanation

Given:

• Original price of math book = $85,000$
• Price of math book after discount = $28,900$
• Original price of biology book = $40,000$

We need to find the percentage of the price to be paid (price after discount) relative to the original price. Let's find this ratio.

Based on the math book price:

Simplify the fraction:

Or as a percentage:

This means the price on that day is $34\%$ of the original price.

Now, we calculate the price of the biology book using the same percentage:

So, the price of the biology book on that day is $13,600$ rupiah.

Explanation

Given the quadratic function:

The axis of symmetry for a quadratic function $ax^2 + bx + c$ is given by the formula $x = -\frac{b}{2a}$. In this case, $a = k$ and $b = 8$.

Given the axis of symmetry $x = 4$, then:

After finding the value of $k$, we substitute it back into the original function:

To find the maximum value, we substitute the axis of symmetry value $x = 4$ into the function (since the optimum value is always at the axis of symmetry):

Thus, the maximum value of the function is $14$.

Explanation

We will solve the inequality:

Factor the quadratic equation:

The zeros (roots) of the equation are:

We test points on the number line:

• For $x < -3$ (e.g., $-4$): $(-4-2)(-4+3) = (-6)(-1) = 6$ (Positive)
• For $-3 < x < 2$ (e.g., $0$): $(0-2)(0+3) = (-2)(3) = -6$ (Negative)
• For $x > 2$ (e.g., $3$): $(3-2)(3+3) = (1)(6) = 6$ (Positive)

Since the inequality sign is $> 0$ (positive), the solution region is:

Based on the given statements:

1. $x > 2$ (Matches)
2. $-3 < x < 2$ (Does not match)
3. $x < -3$ (Matches)
4. $x > -3$ (Does not match)

Thus, the inequality is satisfied by statements $1$ and $3$.

Explanation

Defined:

• $A =$ event construction finishes on time
• $A^c =$ event construction does not finish on time
• $B =$ event a protest occurs
• $B^c =$ event no protest occurs
• $A|B =$ event construction finishes on time given a protest
• $A|B^c =$ event construction finishes on time given no protest

The probability that construction finishes on time is the sum of the probability of a protest multiplied by the probability of finishing on time given a protest, plus the probability of no protest multiplied by the probability of finishing on time given no protest.

We know that:

Then:

Thus, the probability that the construction finishes on time is $0.52$.

Explanation

We know that:

Then:

Nina travels a distance of $3 \text{ km}$.

The total time taken by Nina is the sum of the time spent running and walking, which is:

Convert to minutes:

Thus, the total time taken by Nina is $60$ minutes.

Explanation

Let Putri's current age be $x$, then Tina's age is $2x$.

In $6$ years:

Given that in $6$ years Putri's age will be $\frac{2}{3}$ of Tina's age, then:

Thus, Putri's current age is $6$ years.

Explanation

Given:

Using inverse proportion:

Thus, they can complete the job in $49.5$ days.

Explanation

Let $a = 1.5454545454...$

Multiply $a$ by $100$, so:

Subtract $a$ from $100a$, so:

Therefore, the fraction form is $\frac{17}{11}$.

Explanation

This sequence consists of two alternating patterns.

First pattern (odd positions):

Second pattern (even positions):

The missing terms are the $5$th term (from the first pattern) and the $6$th term (from the second pattern).

For the $5$th term:

For the $6$th term:

Therefore, the correct values to complete the sequence are $16$ and $-1$.

Explanation

Let $x$ be the number of students who like all three subjects (Mathematics, Physics, and Biology).

We can use the inclusion-exclusion principle for sets:

Given:

Substitute these values into the formula:

Therefore, the number of students who like all three subjects is $5$.

Explanation

Let's define the events as follows:

• $S$ = Sample space (total eggs)
• $D$ = Event of picking a duck egg
• $C$ = Event of picking a chicken egg
• $B$ = Event of picking a broken egg

First, we calculate the total number of eggs in the basket:

Next, we calculate the number of broken eggs:

• Broken duck eggs: $\frac{1}{2} \times 10 = 5$
• Broken chicken eggs: $\frac{1}{4} \times 20 = 5$

The total number of broken eggs ($n(B)$) is:

We are asked to find the probability of picking a duck egg or a broken egg ($D \cup B$). The formula for the probability of the union of two events is:

Where $n(D \cup B)$ can be calculated using the inclusion-exclusion principle:

• $n(D) = 10$ (number of duck eggs)
• $n(B) = 10$ (total number of broken eggs)
• $n(D \cap B)$ is the number of eggs that are both duck eggs and broken. From the calculation above, there are $5$ broken duck eggs.

Thus:

So the probability is:

Therefore, the probability of picking a duck egg or a broken egg is $0.5$.

Explanation

First, let's list the members of set $A$. Set $A$ contains integers between $-3$ and $8$ (exclusive):

Next, set $B$ contains all integers less than or equal to $7$:

The intersection of two sets ($A \cap B$) is the set containing elements that are present in both sets.

Since all members of set $A$ (from $-2$ to $7$) are also contained in set $B$, the intersection is set $A$ itself:

Converting this back to set-builder notation, the elements are integers greater than $-3$ and less than or equal to $7$:

Explanation

First, let's calculate the total exam score for Semester $I$:

Next, we calculate the total exam score for the whole year (Semester $I$ & $II$):

To find the total exam score for Semester $II$, we subtract the Semester $I$ total from the yearly total:

The number of exams in Semester $II$ is the difference between the total exams and Semester $I$ exams:

Now we can calculate the average score for Semester $II$:

Finally, we calculate the increase in the average score from Semester $I$ to Semester $II$:

Therefore, the increase in Hani's mathematics exam average is $0.92$.

Explanation

First, we find the value of $x$ from the first equation:

After finding that $x = 10$, we substitute it into the second equation to find the value of $y$:

Finally, we compare the values of $x$ and $y$:

Since $10 > 8$, then: