1.
Factor 6x^{2} + 17x + 12.
Correct Answer
B. (2x+3)(3x+4)
Explanation
The given expression is a quadratic trinomial in the form of ax^2 + bx + c. To factor it, we need to find two binomials whose product equals the original expression. By multiplying the first terms of the binomials (2x and 3x) and the last terms (3 and 4), we can see that they add up to the middle term (17x). Therefore, the correct answer is (2x+3)(3x+4).
2.
Solve the System of Linear Equations:
x + 2y = 9
3x - 4y = - 33
Correct Answer
D. X = -3, y = 6
Explanation
The given system of linear equations is:
x + 2y = 9
3x - 4y = -33
To solve this system, we can use the method of substitution or elimination.
Let's use the method of elimination to solve the system:
Multiply the first equation by 3 and the second equation by 1 to make the coefficients of x in both equations equal:
3(x + 2y) = 3(9)
1(3x - 4y) = 1(-33)
This simplifies to:
3x + 6y = 27
3x - 4y = -33
Now, subtract the second equation from the first equation:
(3x + 6y) - (3x - 4y) = 27 - (-33)
3x + 6y - 3x + 4y = 27 + 33
10y = 60
y = 6
Substitute the value of y back into the first equation:
x + 2(6) = 9
x + 12 = 9
x = -3
Therefore, the solution to the system of linear equations is x = -3 and y = 6.
3.
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 for a line passing through two points can be found using the slope-intercept form, y = mx + b, where m is the slope and b is 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 points. In this case, the slope is (4 - (-10)) / (3 - 1) = 14 / 2 = 7. Plugging in the slope and one of the points into the slope-intercept form, we get y = 7x - 17.
4.
Find the Exterior Angle of a heptagon
Correct Answer
C. 51 degrees
Explanation
The exterior angle of a regular heptagon (a polygon with 7 sides) can be calculated by dividing 360 degrees (the sum of all exterior angles in any polygon) by the number of sides, which in this case is 7. Therefore, the exterior angle of a heptagon is 360/7 = 51 degrees.
5.
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 are proportional. Using the given lengths, we can set up a proportion: AB/DE = BC/EF. Plugging in the values, we have 16/DE = 5/7. Cross-multiplying gives us 7*16 = 5*DE, which simplifies to 112 = 5*DE. Dividing both sides by 5, we get DE = 112/5 = 22.4 inches.
6.
Find the volume of a sphere that has a diameter of 5 meters.
Correct Answer
D. 83.33π 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. Given that the diameter of the sphere is 5 meters, the radius would be half of that, which is 2.5 meters. Plugging this value into the formula, we get V = (4/3)π(5/2)^3 = (4/3)π(125/8) = 83.33π meters^3. Therefore, the correct answer is 83.33π meters^3.
7.
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 AED and angle BED are vertical angles, which means they are congruent. Therefore, angle BED is also 127 degrees. Angle AEC and angle BED are adjacent angles that form a straight line, so their sum is 180 degrees. Therefore, angle BED + angle AEC = 127 degrees + 53 degrees = 180 degrees.
8.
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) is the correct alternative expression for sin(π/3). This is because sin(π/3) can be simplified using the double angle formula for sine: sin(2θ) = 2 sin(θ) cos(θ). In this case, θ is π/6, so sin(π/3) = 2 sin(π/6) cos(π/6). Therefore, the correct answer is 2 sin(π/6) cos(π/6).
9.
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 opposite side to the adjacent side in a right triangle. When the angle is π/2 (90 degrees), the adjacent side becomes zero, resulting in division by zero, which is undefined. Therefore, none of the given options accurately represent the value of tan(π/2).
10.
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 becomes 0. When the denominator is 0, the function is undefined. Therefore, the limit of the function does not exist or there is no limit.
11.
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 specific point can be found by taking the derivative of the equation and evaluating it at that point. In this case, the derivative of y = x^3 is 3x^2. Evaluating this derivative at x = 2 gives us 3(2)^2 = 12. Therefore, the slope of the tangent line to y = x^3 at x = 2 is 12.
12.
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 represents the rate of change of the derivative of that function. In this case, the given function is y = 3x^5 - 4x^4 + 2x^2 + 6. To find the second derivative, we need to take the derivative of the first derivative. The first derivative of y is y' = 15x^4 - 16x^3 + 4x. Taking the derivative of y' gives us the second derivative, which is y'' = 60x^3 - 48x^2 + 4. Therefore, the correct answer is y = 60x^3 - 48x^2 + 4.
13.
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 can use the chain rule. The derivative of sin(u) is cos(u), and the derivative of (2x + 5) with respect to x is 2. Therefore, the derivative of y = sin (2x + 5) is 2 cos (2x + 5).
14.
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 integrate this expression, we can use the power rule of integration. According to the power rule, the integral of x^n is (x^(n+1))/(n+1), where n is any real number except -1. Applying this rule to the given expression, we have the integral of (x+4)^2 as (x^3)/3 + 4x^2 + 16x + c, where c is the constant of integration. Therefore, the correct answer is x^3/3 + 4x^2 + 16x + c.
15.
What is the value of sin(2x) if sin(x)=3/5 and x is in the first quadrant?
Correct Answer
A. 24/25