Set 1

Explanation

Given: $a = \frac{1}{2}$; $b = 2$, $c = 1$

Explanation

To simplify $\frac{3\sqrt{3} + \sqrt{7}}{\sqrt{7} - 2\sqrt{3}}$, we rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is $\sqrt{7} + 2\sqrt{3}$

Explanation

To solve this problem, we use logarithm properties. First, simplify each part by converting the logarithm forms and using change of base.

Explanation

To solve this inequality, we need to factor it first.

Let $u = 9^x$, so the inequality becomes

Factor the quadratic equation

Substitute back $u = 9^x$

Find the values of $x$ when the expression equals zero

1. $9^x = 1 \Rightarrow 9^x = 9^0 \Rightarrow x = 0$
2. $9^x = 9 \Rightarrow 9^x = 9^1 \Rightarrow x = 1$

Using a number line, we can determine the regions that satisfy the inequality

For $x < 0$, both factors are negative, so the result is positive

For $0 < x < 1$, one factor is positive and one is negative, so the result is negative

For $x > 1$, both factors are positive, so the result is positive

Therefore, the solution set is $\{x < 0 \text{ or } x > 1, x \in \mathbb{R}\}$

Explanation

Given the quadratic equation $x^2 + ax - 4 = 0$ with roots $p$ and $q$

Based on Vieta's formulas, we know that

If $p^2 - 2pq + q^2 = 8a$, then we can simplify the left side

Substitute the values $p + q = -a$ and $pq = -4$

Factor the quadratic equation

Therefore, the value of $a$ that satisfies is $4$

Explanation

To analyze the function $H(t) = A \cos(Bt) + C$, we need to determine three parameters based on the given data.

Amplitude

Amplitude $A$ is half the difference between the maximum and minimum values of the function

Therefore, $|A| = 4$. Since the cosine function starts from the maximum value, we can take $A = 4$ (positive).

Vertical Shift

The vertical shift $C$ is the midpoint between the maximum and minimum values, or the average of the maximum and minimum values

Therefore, $C = 6$

Period and Value of B

The period of the function $\cos(Bt)$ is $\frac{2\pi}{B}$

Given that the tidal period is $12$ hours

Since the tidal period is typically positive, we take $B = \frac{\pi}{6}$ (positive).

Thus, the parameter values are $A = 4$, $B = \frac{\pi}{6}$, and $C = 6$

The most appropriate answer choice is $A = 4, B = \frac{\pi}{6}, C = 6$

Explanation

Let the price of $1$ book $= x$, $1$ pen $= y$, $1$ pencil $= z$

We obtain the following equations

From equation $\text{(iii)}$, we get

Substitute $z$ into equation $\text{(ii)}$

Substitute $z$ and $y$ into equation $\text{(i)}$

From $x = 3{,}500$, we can find $z$

Dina buys $2$ pens and $2$ pencils, so the amount to be paid is

Therefore, Dina must pay $\text{Rp}10{,}000$

Explanation

Let capsule $= x$, tablet $= y$, with $x \geq 0$ and $y \geq 0$

We obtain the mathematical model

Finding the Intersection Point

If both line equations are eliminated, we get the intersection point

Solution Region Graph

From the inequalities, we obtain the graph

Determining Minimum Cost

The minimum cost function for capsules and tablets is $f(x, y) = 1{,}000x + 800y$

Values at each corner point

Therefore, the minimum cost that must be spent to meet the calcium and iron needs of the toddler is $\text{Rp}12{,}000$

Explanation

Given $f(x) = 2x^2 - 3$ and $g(x) = 3x - 1$

The composite function $(f \circ g)(x)$ is $f(g(x))$

Substitute function $g(x)$ into function $f(x)$

Expand $(3x - 1)^2$ using the formula $(a - b)^2 = a^2 - 2ab + b^2$

Therefore, the composite function $(f \circ g)(x) = 18x^2 - 12x - 1$

Explanation

Given $f(x) = \frac{2x - 5}{x - 4}$ with $x \neq 4$ and $g(x) = 3x + 8$

Determining the Composite Function

Calculate the composite function $(f \circ g)(x) = f(g(x))$

Determining the Inverse Function

Let $y = \frac{6x + 11}{3x + 4}$, then solve for $x$ in terms of $y$

Replace $y$ with $x$ to get the inverse function

The condition is $3x - 6 \neq 0 \Rightarrow x \neq 2$

Therefore, the inverse of $(f \circ g)(x)$ is $\frac{11 - 4x}{3x - 6}$ with $x \neq 2$

Explanation

Scale Transformation Matrix

Scale matrix $(T_1)$ with factor $2$ on the x-axis and $0.5$ on the y-axis

Shear Transformation Matrix

Shear matrix along the x-axis with factor $1$ $(T_2)$

Composition Matrix

The composition matrix $(T_2 \circ T_1)$ is $M_2 \times M_1$

Calculating Initial Area

A square with side $2$ units has an initial area of

Determining Area Change Scale Factor

The determinant of the composition matrix $M_2 \cdot M_1$ gives the area change scale factor

Area After Transformation

The area after transformation is calculated by

Although scale and shear transformations occurred, the area of the square remains the same because the determinant of the composition matrix is $1$

Therefore, the area of the square after both transformations is $4$ square units

Explanation

Given $P(x) = x^3 - ax^2 - bx + 12$ with factors $(x - 1)$ and $(x + 3)$

Finding Values of a and b

If the factor is $x = 1$, then $P(1) = 0$

If the factor is $x = -3$, then $P(-3) = 0$

Eliminate equations $\text{(i)}$ and $\text{(ii)}$

Substitute the value of $a$

Therefore $P(x) = x^3 - 2x^2 - 11x + 12$

Finding Other Factors Using Horner

Other factors of $P(x)$ are found using Horner with $(x - 1) \Rightarrow x = 1$

The division result is $S(x) = x^2 - x - 12$

Factor $x^2 - x - 12$

Therefore, the factor besides $(x - 1)$ and $(x + 3)$ is $(x - 4)$

Determining Sum of Roots

Since $x_1 < x_2 < x_3$, then

Therefore, the value of $x_1 + x_2 + x_3 = 2$

Explanation

Given matrices $A = \begin{pmatrix} 3 & y \\ 5 & -1 \end{pmatrix}$, $B = \begin{pmatrix} x & 5 \\ -3 & 6 \end{pmatrix}$, and $C = \begin{pmatrix} -3 & -1 \\ y & 9 \end{pmatrix}$

Matrix Addition

Calculate $A + B + C$

Add corresponding elements

Simplify

Finding Values of x and y

From element in first row, first column

From element in first row, second column

Calculating Final Result

Substitute values $x = 2$ and $y = 4$

Therefore, the value of $x + 2xy + y = 22$