1.
Factor x^{2} - 2x - 24.
Correct Answer
E. (x-6)(x+4)
Explanation
The given expression can be factored as (x-6)(x+4). This can be determined by finding two numbers that multiply to give -24 and add up to -2, which are -6 and 4. Thus, the correct answer is (x-6)(x+4).
2.
Factor 6x^{2} + 17x + 12.
Correct Answer
B. (3x+4)(2x+3)
Explanation
The given expression is a quadratic trinomial. To factor it, we need to find two binomials whose product equals the given expression. To do this, we can look for two binomials of the form (ax + b)(cx + d) and then multiply them out to see if they match the given expression. In this case, if we choose (3x+4)(2x+3) and multiply them out, we get 6x^2 + 17x + 12, which matches the given expression. Therefore, the correct answer is (3x+4)(2x+3).
3.
Solve the system of linear equations:x + 2y = 93x - 4y = - 33
Correct Answer
D. X = -3, y = 6
Explanation
The given system of linear equations is solved by substituting the value of x in the second equation. By substituting x = -3 in the second equation, we get -3 - 4y = -33. Simplifying further, we get -4y = -30, and solving for y, we get y = 6. Therefore, the solution to the system of linear equations is x = -3 and y = 6.
4.
Find the equation for a line that passes through the two points (3,4) and (1,-10).
Correct Answer
A. Y = 7x - 17
Explanation
The equation of a line can be found using the formula y = mx + b, where m represents the slope of the line and b represents the y-intercept. To find the slope, we can use the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two given points.
Using the points (3,4) and (1,-10), we can calculate the slope as (4 - (-10)) / (3 - 1) = 14/2 = 7.
Now that we have the slope, we can substitute it into the equation y = mx + b, giving us y = 7x + b. To find the y-intercept, we can substitute the coordinates of one of the given points into the equation and solve for b.
Using the point (3,4), we have 4 = 7(3) + b. Simplifying this equation, we get 4 = 21 + b, and solving for b, we find b = -17.
Substituting the values of m and b into the equation y = 7x + b, we get y = 7x - 17.
5.
Find the exterior angle of a heptagon.
Correct Answer
C. 51.4 degrees
Explanation
The exterior angle of a regular heptagon can be found by dividing 360 degrees (the sum of all exterior angles of any polygon) by the number of sides, which in this case is 7. Therefore, the exterior angle of a heptagon is 51.4 degrees.
6.
Triangle ABC is similar to DEF. The length of EF is 7 inches, while the length of BC is 5 inches. If the length of AB is 16 inches, what is the length of DE?
Correct Answer
A. 22.4 inches
Explanation
Since triangle ABC is similar to DEF, the corresponding sides of the triangles are proportional. Therefore, the ratio of the length of AB to DE is equal to the ratio of the length of BC to EF. Using this proportion, we can set up the equation: AB/DE = BC/EF. Plugging in the given values, we have 16/DE = 5/7. Cross-multiplying and solving for DE gives us DE = (16*7)/5 = 112/5 = 22.4 inches.
7.
Find the volume of a sphere that has a diameter of 5 meters.
Correct Answer
D. 20.8π meters^3
Explanation
The volume of a sphere can be calculated using the formula V = (4/3)πr^3, where r is the radius of the sphere. In this case, the diameter of the sphere is given as 5 meters, so the radius would be half of that, which is 2.5 meters. Plugging this value into the formula, we get V = (4/3)π(2.5)^3 = (4/3)π(15.625) = 20.8π meters^3.
8.
Two lines, AB and CD, intersect at point E. If angle AED is 127 degrees, what is angle BED + angle AEC?
Correct Answer
D. 106 degrees
Explanation
Angle BED + angle AEC is equal to the sum of the angles around point E. The sum of the angles around a point is always 360 degrees. Since angle AED is given as 127 degrees, angle BED + angle AEC can be calculated as 360 - 127 = 233 degrees. Therefore, the correct answer is 106 degrees.
9.
Sin(π/3) can be expressed alternatively as
Correct Answer
C. 2 sin(π/6) cos(π/6)
Explanation
The given expression, 2 sin(π/6) cos(π/6), can be simplified using the double angle formula for sine, which states that sin(2θ) = 2sin(θ)cos(θ). In this case, θ is π/6. Therefore, 2 sin(π/6) cos(π/6) is equal to sin(π/3), providing an alternative expression for sin(π/3).
10.
Tan(π/2) is equal to
Correct Answer
E. None of the above.
Explanation
The correct answer is "None of the above" because the value of tan(π/2) is undefined. In trigonometry, the tangent function is defined as the ratio of the sine of an angle to the cosine of the same angle. At π/2 (90 degrees), the cosine of the angle is 0, which means the denominator of the tangent function becomes 0. Division by 0 is undefined, so the value of tan(π/2) does not exist. Therefore, none of the given options accurately represent the value of tan(π/2).
11.
What is the limit of the function y = 1 / (x - 4) as x approaches 4?
Correct Answer
E. No limit
Explanation
As x approaches 4, the denominator of the function (x - 4) approaches 0. Since division by 0 is undefined, the function does not have a limit as x approaches 4. Therefore, the correct answer is "No limit".
12.
What is the slope of the tangent line of y = x^{3} at x = 2?
Correct Answer
B. M = 12
Explanation
The slope of the tangent line to a curve at a given point can be found by taking the derivative of the function at that point. In this case, the derivative of y = x^3 is y' = 3x^2. Evaluating this derivative at x = 2 gives y' = 3(2)^2 = 12. Therefore, the slope of the tangent line to y = x^3 at x = 2 is 12.
13.
What is the second derivative of y = 3x^{5} - 4x^{4} + 2x^{2} + 6?
Correct Answer
E. Y = 60x^3 - 48x^2 + 4
Explanation
The second derivative of a function is found by taking the derivative of the first derivative. In this case, the given function is y = 3x^5 - 4x^4 + 2x^2 + 6. Taking the derivative of this function once will give the first derivative, which is y' = 15x^4 - 16x^3 + 4x. Taking the derivative of y' will give the second derivative, which is y'' = 60x^3 - 48x^2 + 4. Therefore, the correct answer is y = 60x^3 - 48x^2 + 4.
14.
What is the derivative of y = sin (2x + 5)?
Correct Answer
C. 2 cos (2x + 5)
Explanation
The given function is y = sin (2x + 5). To find its derivative, we use the chain rule. The derivative of sin(u) is cos(u), and the derivative of (2x + 5) is 2. Therefore, applying the chain rule, the derivative of y = sin (2x + 5) is 2 cos (2x + 5).
15.
Integrate (x+4)^{2}.
Correct Answer
A. X^3/3 + 4x^2 + 16x + c
Explanation
The given expression is the integral of (x+4)^2. To find the integral, we can use the power rule for integration. The power rule states that the integral of x^n is (x^(n+1))/(n+1), where n is any real number except -1. Applying the power rule, we can find the integral of (x+4)^2 as (x^(2+1))/(2+1) + 4(x^(1+1))/(1+1) + 16(x^(0+1))/(0+1) + c. Simplifying this expression gives us x^3/3 + 4x^2 + 16x + c, which matches the given correct answer.
16.
Integrate 1 / (x^{2} + 3x + 2).