The solution to a system of two linear equations is the point of intersection of their graphs.
Correct Answer
True
The graphical solution to a system of linear equations is indeed the coordinates of the point where their lines intersect.
Question 2
Multiplying both sides of an inequality by a negative number does not change the direction of the inequality sign.
Correct Answer
False
When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed.
Question 3
A linear equation whose graph is a horizontal line has a slope of zero.
Correct Answer
True
Horizontal lines have a constant y-value and no change in y over x, resulting in a slope of zero.
Question 4
The y-intercept of the line given by y = 2x + 3 is (3, 0).
Correct Answer
False
The y-intercept occurs when x=0. For y = 2x + 3, when x=0, y=3, so the y-intercept is (0, 3).
Question 5
If a system of two linear equations has no solution, their graphs are parallel and distinct lines.
Correct Answer
True
A system with no solution means the lines never intersect, which occurs when they are parallel and separate.
Question 6
The graph of y = -x + 3 has a positive slope.
Correct Answer
False
In the equation y = mx + b, 'm' represents the slope. For y = -x + 3, the slope is -1, which is a negative slope.
Question 7
If a car travels 300 miles in 5 hours, its average speed is 60 miles per hour.
Correct Answer
True
Average speed is calculated as total distance divided by total time: 300 miles / 5 hours = 60 miles per hour.
Question 8
If a price increases by 10% and then decreases by 10%, the final price is the same as the original price.
Correct Answer
False
The 10% decrease is applied to the new, higher price, not the original price, resulting in a final price that is lower than the original. For example, $100 + 10% = $110, then $110 - 10% ($11) = $99.
Question 9
The median is a measure of center that is less affected by extreme outliers than the mean.
Correct Answer
True
The median is the middle value in an ordered data set, so extreme values do not disproportionately pull it in one direction, unlike the mean.
Question 10
A small standard deviation indicates that data points are widely spread out from the mean.
Correct Answer
False
A small standard deviation indicates that data points are clustered closely around the mean, meaning they are not widely spread out.
Question 11
In a proportional relationship, the ratio of the two quantities is constant.
Correct Answer
True
The defining characteristic of a proportional relationship is that the ratio of the dependent variable to the independent variable is always constant.
Question 12
The graph of a quadratic function is a parabola.
Correct Answer
True
All quadratic functions of the form y = ax^2 + bx + c (where a is not 0) have a graph that is a parabola.
Question 13
The expression (x^2)^3 is equivalent to x^5.
Correct Answer
False
According to the exponent rule (a^m)^n = a^(m*n), (x^2)^3 = x^(2*3) = x^6, not x^5.
Question 14
For a quadratic equation ax^2 + bx + c = 0, if the discriminant (b^2 - 4ac) is negative, there are no real solutions.
Correct Answer
True
A negative discriminant indicates that the quadratic formula would involve the square root of a negative number, resulting in no real solutions.
Question 15
To solve x^2 - 9 = 0, one must always use the quadratic formula.
Correct Answer
False
This equation can also be solved by factoring (x-3)(x+3)=0 or by isolating x^2 (x^2=9) and taking the square root (x=±3).
Question 16
If f(x) = x^2 + 1, then f(a+1) = (a+1)^2 + 1.
Correct Answer
True
To evaluate f(a+1), substitute (a+1) for every instance of x in the function definition, so f(a+1) = (a+1)^2 + 1.
Question 17
The sum of the roots of the quadratic equation x^2 + 5x + 6 = 0 is 5.
Correct Answer
False
For a quadratic equation ax^2 + bx + c = 0, the sum of the roots is -b/a. Here, a=1 and b=5, so the sum is -5/1 = -5.
Question 18
The sum of the interior angles of any triangle is 180 degrees.
Correct Answer
True
This is a fundamental theorem in Euclidean geometry for all triangles.
Question 19
If two angles are supplementary, they are both acute.
Correct Answer
False
Supplementary angles sum to 180 degrees. If one angle is acute (less than 90 degrees), the other must be obtuse (greater than 90 degrees), unless both are 90 degrees.
Question 20
In a right triangle, the sine of an acute angle is the ratio of the length of the opposite side to the length of the hypotenuse.
Correct Answer
True
This is the definition of the sine function in right-triangle trigonometry (SOH in SOH CAH TOA).
Question 21
The area of a circle with radius r is given by 2πr.
Correct Answer
False
The formula 2πr represents the circumference of a circle. The area of a circle with radius r is given by πr^2.
Question 22
For parallel lines intersected by a transversal, corresponding angles are congruent.
Correct Answer
True
When a transversal intersects two parallel lines, corresponding angles are located in the same relative position at each intersection and are always equal in measure.