Digital SAT Math Practice Test 6

This is the sixth practice test adapted for the new Digital SAT format. The questions are categorized into the four official testing domains: Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry.

Domain 1: Algebra

Focus: Linear equations, systems of linear equations, and inequalities.

Question 1

A line in the \(xy\)-plane passes through the points \((2,5)\) and \((4,6)\). If the point \((7,k)\) lies on this line, what is the value of \(k\)?

View Solution

Step 1: Calculate the slope (\(m\))

\[ m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{6 - 5}{4 - 2} = \dfrac{1}{2} \]

Step 2: Find the equation of the line

Using point-slope form \(y - y_1 = m(x - x_1)\) with \((2,5)\):

\[ y - 5 = \dfrac{1}{2}(x - 2) \Rightarrow y = \dfrac{1}{2}x + 4 \]

Step 3: Solve for \(k\)

Substitute \(x = 7\) and \(y = k\) into the equation:

\[ k = \dfrac{1}{2}(7) + 4 = 3.5 + 4 = 7.5 \]

Question 2

Pump A can empty a pool in 4 hours, and Pump B can empty the same pool in 6 hours. If both pumps work together at their respective constant rates, how many hours will it take them to empty exactly 50% of the pool?

View Solution

Find the combined rate in "pools per hour":

Pump A rate: \(1/4\). Pump B rate: \(1/6\).

\[ \text{Combined Rate} = \dfrac{1}{4} + \dfrac{1}{6} = \dfrac{3+2}{12} = \dfrac{5}{12} \text{ pools/hr} \]

To find the time \(t\) to complete 50% (\(1/2\)) of the pool:

\[ \dfrac{5}{12}t = \dfrac{1}{2} \Rightarrow t = \dfrac{1}{2} \times \dfrac{12}{5} = 1.2 \text{ hours} \]

Question 3

In an electrical circuit, the equivalent resistance \(R_e\) of two resistors \(R_1\) and \(R_2\) connected in parallel is given by \(\dfrac{1}{R_e} = \dfrac{1}{R_1} + \dfrac{1}{R_2}\). Express \(R_e\) exclusively in terms of \(R_1\) and \(R_2\).

View Solution

Combine the fractions on the right side using a common denominator \(R_1 R_2\):

\[ \dfrac{1}{R_e} = \dfrac{R_2 + R_1}{R_1 R_2} \]

To solve for \(R_e\), take the reciprocal of both sides:

\[ R_e = \dfrac{R_1 R_2}{R_1 + R_2} \]

Domain 2: Advanced Math

Focus: Quadratics, higher-order polynomials, and non-linear equations.

Question 4

The square root of a positive real number plus twice that same number is equal to 10. What is the number?

View Solution

Let the number be \(x\). The equation is \(\sqrt{x} + 2x = 10\).

Isolate the radical: \(\sqrt{x} = 10 - 2x\).

Square both sides: \(x = (10 - 2x)^2 \Rightarrow x = 100 - 40x + 4x^2\).

Standard form: \(4x^2 - 41x + 100 = 0\).

Factor: \((4x - 25)(x - 4) = 0\). Potential solutions are \(x = 6.25\) and \(x = 4\).

Check for extraneous solutions: \(\sqrt{4} + 2(4) = 10\) (True). \(\sqrt{6.25} + 2(6.25) = 15\) (False). The answer is 4.

Question 5

If the polynomial \(P(x) = x^3 - 2x^2 + 3kx + 18\) is divided by \(x - 6\) and the remainder is zero, what is the value of the constant \(k\)?

View Solution

By the Remainder Theorem, \(P(6) = 0\):

\[ (6)^3 - 2(6)^2 + 3k(6) + 18 = 0 \]

\[ 216 - 72 + 18k + 18 = 0 \Rightarrow 162 + 18k = 0 \]

\[ 18k = -162 \Rightarrow k = -9 \]

Question 6

For what values of \(k\) does \(-2|x - 4| - 2 = k + 1\) have exactly two distinct real solutions?

View Solution

Isolate the absolute value: \(-2|x - 4| = k + 3 \Rightarrow |x - 4| = \dfrac{-(k+3)}{2}\).

An absolute value equation \(|A| = B\) has two solutions only if \(B > 0\):

\[ \dfrac{-(k+3)}{2} > 0 \Rightarrow -(k+3) > 0 \Rightarrow k+3 < 0 \Rightarrow k < -3 \]

Domain 3: Problem-Solving and Data Analysis

Question 7

A dealer increases a price by 20%, then increases that new price by 30%. If \(x\) is the original price, what is the final price?

View Solution

First increase: \(x \times 1.20\). Second increase: \((1.20x) \times 1.30\).

\[ \text{Final Price} = 1.56x \]

Question 8

The median of \(\{21, 7, 45, 33, 62, x\}\) is 35. If \(x\) is an integer, what is its value?

View Solution

Arrange knowns: \(7, 21, 33, 45, 62\). With 6 numbers, the median is the average of the 3rd and 4th terms. To get 35, the sum of middle terms must be 70.

If \(x\) is between 33 and 45: \((33 + x)/2 = 35 \Rightarrow 33 + x = 70 \Rightarrow x = 37\).

Question 9

A data set has \(SD = 1.2\). If every value is multiplied by 4, what is the new standard deviation?

View Solution

Scaling every value by a factor \(C\) scales the standard deviation by \(|C|\).

\[ \text{New SD} = 1.2 \times 4 = 4.8 \]

Domain 4: Geometry and Trigonometry

Question 10

Triangle \(BCM\) is a right triangle with hypotenuse \(BC\) where \(B(2,6)\) and \(C(4,8)\). If it is inscribed in a circle, what is the radius?

View Solution

In an inscribed right triangle, the hypotenuse is the diameter.

\[ \text{Diameter } BC = \sqrt{(4-2)^2 + (8-6)^2} = \sqrt{4+4} = \sqrt{8} = 2\sqrt{2} \]

Radius = Diameter / 2 = \(\sqrt{2}\).

Question 11

Shortest distance between \((-2,-2)\) and circumference of \((x - 1)^2 + (y - 2)^2 = 4\)?

View Solution

Center is \((1,2)\), radius is 2. Distance from point to center:

\[ D = \sqrt{(1 - (-2))^2 + (2 - (-2))^2} = \sqrt{3^2 + 4^2} = 5 \]

Shortest distance to edge = Distance to center - Radius = \(5 - 2 = 3\).

Question 12

A right triangle with legs 3 and 4 is rotated \(360^\circ\) around the leg of length 4. What is the volume of the resulting cone?

View Solution

The leg of length 4 is the height (\(h\)), and leg of length 3 is the radius (\(r\)).

\[ V = \dfrac{1}{3}\pi r^2 h = \dfrac{1}{3}\pi (3^2)(4) = 12\pi \]