Set 7

Explanation

We will analyze each statement based on the given graph.

Analysis of Statement 1

Based on the graph, it is clear that the curve intersects the $x$-axis at the point $(4, 0)$.

Thus, statement $(1)$ is correct.

Analysis of Statement 2

Based on the graph, the curve intersects the $y$-axis at two points, namely at $y = 3$ and $y = -5$. So the intersection points are $(0, 3)$ and $(0, -5)$.

Statement $(2)$ states that the intersection points are $(0, -5)$ and $(0, 1)$. Since $(0, 3) \neq (0, 1)$, this statement does not match the graph.

Thus, statement $(2)$ is incorrect.

Analysis of Statement 3

The axis of symmetry for a horizontal parabola can be determined from the midpoint between the two $y$-intercepts (if known).

The axis of symmetry is obtained as the line $y = -1$.

Thus, statement $(3)$ is correct.

Analysis of Statement 4

Since the axis of symmetry is the horizontal line $y = -1$, the statement that the axis of symmetry is $x = -3$ (a vertical line) is clearly wrong.

Thus, statement $(4)$ is incorrect.

Conclusion

The correct statements are statements $(1)$ and $(3)$.

Explanation

Given the function $y = -x^2 - 2x + 8$. We want to find the equation of the tangent line at the point with abscissa $x = -2$.

Finding the Ordinate

Substitute $x = -2$ into the curve equation:

So, the point of tangency is $(-2, 8)$.

Finding the Gradient of the Tangent Line

The gradient of the tangent line $m$ is the value of the first derivative $y'$ at the abscissa of the point of tangency.

Substitute $x = -2$:

Determining the Equation of the Tangent Line

Use the point-slope form for the equation of a line passing through $(x_1, y_1)$ with gradient $m$:

Substitute $x_1 = -2$, $y_1 = 8$, and $m = 2$:

Thus, the equation of the tangent line is $y = 2x + 12$.

Explanation

We will check each number to see if it satisfies the following two conditions:

1. When divided by $5$, the remainder is $4$ ($n \equiv 4 \pmod 5$).
2. When divided by $4$, the remainder is $3$ ($n \equiv 3 \pmod 4$).

Checking Number 19

The number $19$ satisfies both conditions. (Correct)

Checking Number 29

The number $29$ does not satisfy the second condition because the remainder is $1$ when divided by $4$. (Incorrect)

Checking Number 39

The number $39$ satisfies both conditions. (Correct)

Checking Number 49

The number $49$ does not satisfy the second condition because the remainder is $1$ when divided by $4$. (Incorrect)

Conclusion

The numbers that satisfy the conditions are $19$ and $39$ (Statements $1$ and $3$).

Explanation

We will analyze the number of lines of symmetry and rotational symmetries for each plane figure. The requirement is to have at most $2$ lines of symmetry and $2$ rotational symmetries.

Analyzing Right Trapezoid

A right trapezoid has no axis of symmetry and no rotational symmetry of higher order (only order $1$, which is the initial position).

• Lines of symmetry: $0$
• Rotational symmetry: $1$

This figure satisfies the condition (since $0 \le 2$ and $1 \le 2$).

Analyzing Regular Pentagon

A regular pentagon has $5$ equal sides.

• Lines of symmetry: $5$
• Rotational symmetry: $5$

This figure does not satisfy the condition.

Analyzing Equilateral Triangle

An equilateral triangle has $3$ equal sides.

• Lines of symmetry: $3$
• Rotational symmetry: $3$

This figure does not satisfy the condition.

Analyzing Kite (Not a Square)

A kite has one main axis of symmetry.

• Lines of symmetry: $1$
• Rotational symmetry: $1$

This figure satisfies the condition.

Conclusion

The plane figures that satisfy the condition are:

1. Right trapezoid.
2. Kite (not a square).

Thus, there are $2$ plane figures that satisfy the condition.

Explanation

Given the recursive formula for the sequence:

We are given $a_3 = -1$ and $a_5 = 3$. We need to find the value of $a_2$.

First, we find the value of $a_4$ by using the recursive formula and substituting $n = 3$ to relate $a_3$, $a_4$, and $a_5$.

Substitute the values $a_5 = 3$ and $a_3 = -1$:

Next, we find the value of $a_2$. Since we now have the values of $a_3$ and $a_4$, we can use the recursive formula with $n = 2$ to relate $a_2$, $a_3$, and $a_4$.

Substitute the values $a_3 = -1$ and $a_4 = 1$:

Thus, the value of $a_2$ is $-1$.

Explanation

Given the matrix $A = \begin{pmatrix} 2 & x-2 \\ -1 & 1 \end{pmatrix}$. We need to find the value of $x$ if $\det(A) = 8$.

The determinant of a $2 \times 2$ matrix is calculated using the formula $ad - bc$.

We know that $\det(A) = 8$, so:

Thus, the value of $x$ is $8$.

Explanation

We are given the equation $(f(x))^2 = 5f(x) - 4$. We can rearrange it into a quadratic equation in terms of $f(x)$:

This equation is satisfied by $x_1$ and $x_2$. Based on the sum of roots formula for quadratic equations ($x_1 + x_2 = -\frac{b}{a}$), the sum of the values of $f(x)$ satisfying the equation is:

Given $f(x) = x + 1$. We substitute this into the equation above:

Thus, the value of $x_1 + x_2$ is $3$.

Explanation

We have two groups:

• People raising goats: $K = 450$
• People raising chickens: $A = 300$
• Total population: $S = 600$

Maximum Number of People Raising Both

The maximum number of people raising both animals is the size of the smaller group. This happens if all members of the smaller group are also members of the larger group.

Minimum Number of People Raising Both

The minimum number of people raising both animals occurs when the two groups are spread out as much as possible, minimizing their intersection. The formula is:

Substitute the known values:

Sum of Maximum and Minimum

We are asked to find the sum of these maximum and minimum values.

Thus, the sum of the maximum and minimum number of people raising both animals is $450$.

Explanation

We need to form a code with the pattern Letter - Digit - Letter - Digit.

Let the slots for the code be:

Available sets:

• Vowels: $\{A, I, U, E, O\}$ ($5$ letters).
• Odd digits for $D_1$: $\{1, 3, 5, 7, 9\}$ ($5$ digits).
• Choice digits for $D_2$: $\{0, 1, 2, 4, 6\}$ ($5$ digits).

Rules and Constraints:

1. $L_1 \neq A$.
2. $L_2 = O$.
3. No repetition of letters or digits is allowed.

Since $D_1$ must be odd and $D_2$ is chosen from a specific set, there is an overlap at the digit $1$. We need to split the calculation into cases based on the value of $D_2$.

Case 1 If The Second Digit Is 1

If we choose the digit $1$ for position $D_2$:

1. $L_2$ (Second letter): Must be $O$ ($1$ way).
2. $D_2$ (Second digit): Must be $1$ ($1$ way).
3. $L_1$ (First letter): Vowel other than $A$ and other than $O$ (since $O$ is used in $L_2$).
   • Choices: $\{I, U, E\}$ ($3$ ways).
4. $D_1$ (First digit): Odd digit other than $1$ (since $1$ is used in $D_2$).
   • Choices: $\{3, 5, 7, 9\}$ ($4$ ways).

Total ways for Case $1$:

Case 2 If The Second Digit Is Not 1

If we choose a digit other than $1$ for position $D_2$ (which means it is even):

1. $L_2$ (Second letter): Must be $O$ ($1$ way).
2. $D_2$ (Second digit): Chosen from $\{0, 2, 4, 6\}$ ($4$ ways).
3. $L_1$ (First letter): Vowel other than $A$ and other than $O$.
   • Choices: $\{I, U, E\}$ ($3$ ways).
4. $D_1$ (First digit): Odd digit $\{1, 3, 5, 7, 9\}$. Since $D_2$ is even, there is no conflict. All odd digits can be chosen.
   • Choices: $5$ ways.

Total ways for Case $2$:

Total Combinations

Sum the results from both cases:

Therefore, the number of codes that can be formed is $72$.

Explanation

The given ordered data set is $4, 4, a, b, c$. Since the data is already ordered and there are an odd number of values ($5$ values), the median is the $3$rd value, which is $a$.

The largest data value is $c$, and it is $3$ times the median, so:

The average of the five numbers is $8$:

Substitute $c = 3a$ into the equation:

Since the data is ordered, $4 \le a \le b \le c$ holds true. We will find the bounds for the value of $a$.

Upper Bound of a

From the inequality $a \le b$:

Lower Bound of a

From the inequality $b \le c$:

Note also that $a \ge 4$ (from the data order). Since $4.57 > 4$, the stronger condition is $a \ge 4.57$.

Conclusion

The range of values for $a$ is $4.57 \le a \le 6.4$.

The value of $P$ is the median ($a$) and the value of $Q$ is $4$. Since the minimum value of $a$ is $4.57$, which is greater than $4$, it is certain that $P > Q$.

Therefore, the correct relationship is $P > Q$.

Explanation

We will simplify the form of $P$ first. Recall that $4^x = (2^2)^x = (2^x)^2$. Thus, the form $4^x - 1$ is a difference of squares which can be factored into $(2^x - 1)(2^x + 1)$.

Next, let's look at the form of $Q$:

We can see the relationship between $P$ and $Q$ by substituting $P = 2^x + 1$ into the equation for $Q$:

Given $0 < x < 1$. Let's analyze the value of $2^x$ in that range:

• If $x > 0$, then $2^x > 2^0 \implies 2^x > 1$.
• If $x < 1$, then $2^x < 2^1 \implies 2^x < 2$.

Thus, $1 < 2^x < 2$.

Back to the equation $P = Q \cdot 2^x$: Since $2^x > 1$ and $Q$ is positive (because $2^x > 0$), multiplying $Q$ by a number greater than $1$ will result in a value greater than $Q$ itself.

Therefore, the correct relationship is $P > Q$.

Explanation

Given the operation rule:

We are asked to find the value of:

By comparing variable positions, we get:

• $w = 2$
• $x = 3$
• $y = 4$
• $z = -2$

Substitute these values into the formula:

Therefore, the value of the operation is $14$.

Explanation

We are asked to determine the measure of angle $\angle BPC$ in an isosceles triangle $ABC$ with $AC = BC$.

Analysis of Statement 1

Given $\angle CAB = 40^\circ$. Since $\triangle ABC$ is isosceles with $AC = BC$, the base angles are equal:

We can calculate the vertex angle $\angle ACB$:

However, this information does not specify the position of point $P$ on side $AB$. Point $P$ can be anywhere along the segment $AB$. Consequently, the measure of $\angle BPC$ varies depending on the position of $P$. Therefore, statement $(1)$ ALONE is not sufficient to answer the question.

Analysis of Statement 2

Given $PC$ is the angle bisector of triangle $ABC$. In this context, $PC$ is the bisector of angle $C$ intersecting side $AB$ at point $P$.

In an isosceles triangle with $AC = BC$, the angle bisector drawn from the vertex (the angle between the equal sides) to the base has special properties. It is also:

1. An altitude (perpendicular to the base).
2. A median (bisects the base).

Since $PC$ is an altitude, $PC \perp AB$. Thus, the angle formed is a right angle:

We obtain a definite value for $\angle BPC$. Therefore, statement $(2)$ ALONE is sufficient to answer the question.

Conclusion

Statement $(1)$ is not sufficient, while statement $(2)$ is sufficient. Thus, statement $(2)$ ALONE is sufficient, but statement $(1)$ ALONE is not sufficient.

Explanation

We will analyze each statement one by one based on the functions $f(x) = 5 - x$ and $g(x) = 2^x - 1$.

Analysis of Statement 1

Statement: $g(x)$ is a linear line with a gradient of $2$.

The function $g(x) = 2^x - 1$ is an exponential function, not a linear function. A linear function has the general form $y = mx + c$, whereas this is an exponential form. Therefore, statement $(1)$ is incorrect.

Analysis of Statement 2

Statement: $f(x)$ and $g(x)$ intersect at $x > 0$.

To find the intersection point, we equate the two functions:

If we substitute $x = 2$:

Both functions have the same value of $3$ when $x = 2$. Since $2 > 0$, statement $(2)$ is correct.

Analysis of Statement 3

Statement: $f(x)$ is above $g(x)$ for all values of $x$.

Let's check a value $x > 2$, for example $x = 3$:

Here it is seen that $g(3) > f(3)$, which means the curve $g(x)$ is above $f(x)$. Therefore, the statement that $f(x)$ is always above $g(x)$ is incorrect. Statement $(3)$ is incorrect.