SAT Math Randomized Questions - 2 Full Math Practice Tests - Answers and Detailed Explanations at the END

Rewrite \(f(x)\) as \((x - 5)^2\). The minimum of \(f(x)\) is at \(x = 5\), so \(g(x) = f(x - 2)\) shifts this minimum to \(x = 7\).

First, find the measure of angle Q in radians: \( \frac{5\pi}{6} + \frac{\pi}{4} = \frac{14\pi}{12} = \frac{7\pi}{6} \). Then, convert to degrees by multiplying by \( \frac{180}{\pi} \): \( \frac{7\pi}{6} \cdot \frac{180}{\pi} = 210 \) degrees.

Since the decay factor is 0.70 annually, this translates to a 30% decrease per year, so \(q = 30\).

Using \( P = 40,000 \) and \( C = 5,000 \), set up the equation \( 40,000 = 5,000(2)^{5r} \). Dividing both sides by 5,000 gives \( 8 = (2)^{5r} \). Taking the logarithm base 2 of both sides yields \( 3 = 5r \). Therefore, \( r = \frac{3}{5} \).

Calculate the percentages: Candidate A received \( \frac{600}{1000} = 0.6 \), or 60%. Candidate B received \( \frac{400}{1000} = 0.4 \), or 40%. If 10,000 people vote, Candidate A is expected to receive \( 0.6 \times 10000 = 6000 \) votes, and Candidate B \( 0.4 \times 10000 = 4000 \) votes. The difference is \( 6000 - 4000 = 2000 \) votes.

Since \(f(x) = 1.2x\) shows a constant proportional increase, the function is linear and increases with \(x\). Thus, the correct answer is 'Increasing Linear.'

The vertex of the function \( f(x) = \frac{1}{8}(x - 6)^2 + 5 \) occurs at \( (6, 5) \). This represents the minimum height of the kite. The vertex form of a quadratic function, \( a(x - h)^2 + k \), indicates the vertex as \( (h, k) \). Since \( h = 6 \) and \( k = 5 \), the minimum height is 5 feet above the ground.

Let the number of voters who voted in favor of the bill be w. According to the problem, 3 times as many voters voted against, so the number against is 3w. The report states that 18,000 more people voted against, so the equation is 3w = w + 18,000. Solving for w gives 9,000, and the number against is 3(9,000) = 27,000.

For a quadratic in the form \( y = ax^2 + bx + c \), the minimum or maximum value of \( y \) occurs at \( x = \frac{-b}{2a} \). Here, \( a = 1 \) and \( b = -10 \), so \( x = \frac{10}{2} = 5 \).

To find \( a \), set \( f(x) = 0 \) and solve for \( x \): \( 5x - 15 = 0 Rightarrow x = 3 \). The x-intercept is \( (3, 0) \), so \( a = 3 \). The y-intercept \( b \) is found by setting \( x = 0 \): \( f(0) = -15 \), so \( b = -15 \). Then \( a + b = 3 - 15 = -12 \).

The percentage can be calculated using the formula: \( \frac{62.5}{250} \times 100 = 25\% \).

Multiplying the first equation by -2 results in \( -2x + 8y = -6 \), which matches the second equation. Thus, the lines are the same, intersecting at infinitely many points.

To cover the walls twice, double the total area \(w\) to get \(2w\). Since each gallon covers 150 square feet, divide \(2w\) by 150 to get the equation \(P = \frac{w}{75}\).

To find \( a \), set \( q(x) = 0 \): \( -7x + 21 = 0 Rightarrow x = 3 \), so \( a = 3 \). For \( b \), evaluate \( q(0) = 21 \). Thus, \( a + b = 3 + 21 = 24 \).

Data set A is skewed toward lower values, while data set B is skewed toward higher values. Therefore, the mean of Data set A is less than the mean of Data set B.

The function increases by 20%, which means it becomes 1.20 times its previous value at each step. The multiplier is 1.20. Therefore, the correct equation is \( g(x) = 30(1.20)^x \).

Since \( g(2) = 625 \), substitute into \( g(x) = 25 \cdot b^x \) to solve for \( b \): \( 25 \cdot b^2 = 625 \Rightarrow b^2 = 25 \Rightarrow b = 5 \). Now find \( g(\frac{1}{2}) \): \( g(\frac{1}{2}) = 25 \cdot 5^{\frac{1}{2}} = 25 \cdot \sqrt{5} = 25 \sqrt{5} \).

We use the Pythagorean theorem: \( c^2 = 8^2 + 15^2 = 64 + 225 = 289 \). Therefore, \( c = \sqrt{289} = 17 \). The hypotenuse can be written as \( \sqrt{289} \), so \( d = 289 \).

We use the Pythagorean theorem: \( c^2 = 5^2 + 12^2 = 25 + 144 = 169 \). Therefore, \( c = \sqrt{169} = 13 \). Since the length is already in the form \( \sqrt{169} [/latex}, the value of [latex] d \) is 169.

Using the quadratic formula, \(x = \frac{12 \pm \sqrt{144 - 44}}{2} = 6 \pm \sqrt{25}\). So, the solution can be written as \(6 + \sqrt{25}\) and the correct value of \(k\) is 25.

Rearrange and substitute to find that \( y = 10 \).

The change in degrees Fahrenheit is determined by multiplying the temperature increase in kelvins by \( \frac{9}{5} \), since the \( x \) term in the function is being multiplied by this factor. Therefore, the increase in Fahrenheit temperature is \( \frac{9}{5} \times 5.00 = 9.00 \).

For a quadratic equation, the minimum value occurs at \( x = \frac{-b}{2a} \). Here, \( x = \frac{16}{2} = 8 \).

To find the surface area, find the side length first. The volume of the cube is \( V = s^3 \). Given \( V = 729 \), \( s = \sqrt[3]{729} = 9 \). The surface area formula is \( 6s^2 \). Therefore, the surface area is \( 6(9)^2 = 486 \) square units.

The two lines have the same slope but different intercepts, which means they are parallel and do not intersect at any point. Thus, the answer is zero points of intersection.

Substitute \( (2, 2) \) into both inequalities. Both inequalities are satisfied.

Using the points, calculate the slope and equation of the line, then substitute \(x = q + 5\) to find \(b = -55\).

Calculate percentages: Candidate L received \( \frac{600}{1000} = 0.6 \), or 60%. Candidate M received \( \frac{400}{1000} = 0.4 \), or 40%. If 7,500 people vote, Candidate L would receive \( 0.6 \times 7500 = 4500 \) votes, and Candidate M \( 0.4 \times 7500 = 3000 \) votes. The difference is \( 4500 - 3000 = 1500 \) votes.

Let the number of people who voted in favor of the policy be x. According to the problem, 5 times as many people voted against it, so the number against is 5x. The article states that 40,000 more people voted against it, so the equation is 5x = x + 40,000. Solving for x gives 10,000.

To find \( f(0) \), substitute \( 0 \) for \( x \): \( f(0) = 500(0.05)^0 \). Since \( (0.05)^0 = 1 \), the expression becomes \( 500 imes 1 = 500 \).

The lines have the same slope but different intercepts, indicating that they are parallel and do not intersect at any point.

With the slope \( \frac{7}{8} \) and point \( (4, -1) \), apply the point-slope formula to find the equation \( y = \frac{7}{8}x - \frac{23}{8} \).

The leakage rate is 5 gallons per hour. Over one day (24 hours), the volume leaked is 5 * 24 = 120 gallons. Converting to pints (1 gallon = 8 pints), the answer is 960 pints.

The decay factor is 0.90 per three months, equating to a 10% drop per quarter. After one year, this approximates to a 34% annual decay rate, so \(m = 34\).

Since the side lengths of square P are 20 times the side lengths of square Q, the area of square P is \( 20^2 = 400 \) times the area of square Q. Therefore, \( k = 400 \).

The percentage can be calculated using the formula: \( \frac{250}{500} \times 100 = 50\% \).

Rewrite \(f(x)\) in vertex form as \(2(x + 2)^2 - 2\). The minimum of \(f(x)\) is at \(x = -2\), so \(g(x) = f(x + 1)\) shifts this to \(x = -3\).

Substitute \( b = 8 \) into the equation: \( 2.5d + 3.5(8) = 70 \Rightarrow 2.5d + 28 = 70 \Rightarrow 2.5d = 42 \Rightarrow d = 16 \).

To find \( a \), set \( g(x) = 0 \) and solve: \( -2x + 10 = 0 Rightarrow x = 5 \), so \( a = 5 \). For the y-intercept \( b \), set \( x = 0 \): \( g(0) = 10 \). Therefore, \( a + b = 5 + 10 = 15 \).

Since the decay factor is 0.82 per quarter, this gives a 18% decrease each quarter. Calculating this for a year results in approximately a 51% annual decay rate, so \(r = 51\).

To find \( f(0) \), substitute \( 0 \) for \( x \) in the function: \( f(0) = 300(0.2)^0 \). Since any non-zero number raised to the power of \( 0 \) is \( 1 \), the expression becomes \( 300 imes 1 = 300 \).

The function \(q(x) = 1.3x\) represents a proportional increase, showing a steady, linear growth with respect to \(x\). Therefore, 'Increasing Linear' is correct.

Expanding the equation, we obtain \( x^2 + 2x - 10 = 3x^2 - 15x \). Rearranging terms, we get \( -2x^2 + 17x - 10 = 0 \). Using the quadratic formula, the sum of the solutions is given by \( -b/a \), where \( a = -2 \) and \( b = 17 \), which yields \( 17/2 \).

Solving for \( y \) using substitution or elimination, we find that \( y = 30 \).