Stem Entrance Exam Trivia

The common ratio between any two consecutive terms in a geometric sequence is found by dividing one term by the previous term. In this case, dividing -12 by 3 gives us -4. Therefore, -4 is the common ratio between any two consecutive terms in the given sequence.

The given arithmetic sequence starts with 2 and increases by 5/4 each time. To find the next term, we add 5/4 to the previous term 23/4, which gives us 28/4 or 7. Therefore, the next term in the sequence is 7.

The sum of all even integers from 21 to 51 can be calculated by finding the average of the first and last even numbers in the range (22 and 50) and multiplying it by the number of terms. Calculating the number of terms = (50 - 22) / 2 + 1 = 15. The average of 22 and 50 is 36, so multiplying it by 15 gives us a sum of 540.

When (x+9) divides (x^2 - 81), it means that (x^2 - 81) can be expressed as a product of (x+9) and another factor. To find this factor, we can use the method of polynomial long division. By dividing (x^2 - 81) by (x+9), we get a quotient of (x-9). Therefore, the correct answer is X-9.

In synthetic division, the number of coefficients written in the first row is equal to the degree of the polynomial plus one. Since the polynomial in this question is a 7th-degree polynomial, there will be 8 coefficients written in the first row of the synthetic division procedure.

To find the value of P(-2), we substitute -2 for x in the expression P(x). Plugging in -2, we get P(-2) = 7(-2)^4 - 5(-2)^3 + 2(-2)^2 - 3. Simplifying this expression, we get P(-2) = 7(16) + 5(8) + 2(4) - 3 = 112 + 40 + 8 - 3 = 157. Therefore, the value of P(-2) is 157.

The given polynomial equation has roots -1, 1, and 4. To find the correct equation, we need to check which equation gives us these roots when plugged in. By substituting -1, 1, and 4 into each equation, we find that only P(x) = x3 – 4x2 – x + 4 gives us the desired roots. Therefore, P(x) = x3 – 4x2 – x + 4 is the correct equation.

To find the roots of the quadratic equation b^2 - 5b - 36 = 0, we can use the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, our equation is b^2 - 5b - 36 = 0. So, we have:
a = 1 b = -5 c = -36
Plugging these values into the quadratic formula, we get:
b = [5 ± √((-5)^2 - 4(1)(-36))] / 2(1) b = [5 ± √(25 + 144)] / 2 b = [5 ± √169] / 2 b = [5 ± 13] / 2
Now, we have two possible roots:
b = (5 + 13) / 2 = 18 / 2 = 9
b = (5 - 13) / 2 = -8 / 2 = -4
Therefore, the roots of the quadratic equation b^2 - 5b - 36 = 0 are 9 and -4.

The area of a square is calculated by multiplying the length of one side by itself. In this case, the side of the square is given as 12 cm, so the area of the square is 12 cm * 12 cm = 144 cm^2. The area of a rectangle is calculated by multiplying its length by its width. Since the rectangle and the square have the same area, we can set up the equation 144 cm^2 = length of rectangle * 8 cm. Solving for the length of the rectangle, we find that it is 18 cm.

The measure of angle F in square FATE is given as 4x + 30. To find the value of x, we can equate this expression to the given answer choices and solve for x. By substituting 15 degrees for x in the expression 4x + 30, we get 4(15) + 30 = 60 + 30 = 90 degrees. Since 90 degrees is not one of the answer choices, we can conclude that the given answer of 15 degrees is incorrect. Therefore, the correct answer cannot be determined from the information provided.

The given polynomial function P(x) = xn is true when N is a non-negative number because the exponent, represented by N, must be a non-negative integer or zero. This is because raising a number to a negative exponent or a non-integer exponent would result in a non-polynomial function. Therefore, the given polynomial function is only true when N is a non-negative number.

To find the computed profit when 250 meters are being sold, we can substitute the value of x = 250 into the given mathematical model:
P(x) = 3x^2 + 6x - 120,000
So,
P(250) = 3(250)^2 + 6(250) - 120,000
Now, let's compute:
P(250) = 3(62500) + 1500 - 120,000 = 187500 + 1500 - 120,000 = 189000 - 120,000 = 69,000
Therefore, the computed profit when 250 meters are being sold is $69,000.

The function P(x) = (2x^2 - 5x + 2) (x+3) will cross the x-axis thrice because it is a quadratic function with a leading coefficient of 2, which means it opens upwards. The quadratic factor (2x^2 - 5x + 2) will have two x-intercepts, and the linear factor (x+3) will have one x-intercept. When multiplied together, these factors will result in a function that crosses the x-axis three times.

The correct answer is D. The distance between two points whose coordinates are (x1, y1) and (x2, y2) can be determined using the mathematical model known as the distance formula. This formula is derived from the Pythagorean theorem and states that the distance between two points in a Cartesian plane is equal to the square root of the sum of the squares of the differences between their x-coordinates and y-coordinates.

To calculate C(8, 6), we use the combination formula:

C(n, k) = n! / (k!(n - k)!)

Plug in the values:

C(8, 6) = 8! / (6!(8 - 6)!)

Calculate the factorials:

8! = 8 * 7 * 6!
6! = 6 * 5 * 4 * 3 * 2 * 1

Now calculate:

C(8, 6) = (8 * 7 * 6!) / (6! * 2!)

Simplify:

C(8, 6) = (8 * 7) / (2 * 1)

C(8, 6) = 56 / 2

C(8, 6) = 28

So, the answer is:

C) C(8, 6) = 28

The expression 10!/(4!7!) can be simplified as follows:
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 4! = 4 × 3 × 2 × 1 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1
Substituting these values into the expression:
10!/(4!7!) = (10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / [(4 × 3 × 2 × 1) × (7 × 6 × 5 × 4 × 3 × 2 × 1)]
Many terms cancel out in the numerator and the denominator:
(10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / [(4 × 3 × 2 × 1) × (7 × 6 × 5 × 4 × 3 × 2 × 1)] = (10 × 9 × 8) / (4 × 3 × 2) = (720) / (24) = 30
So, the value of the expression 10!/(4!7!) is 30.

The equation of a circle with center (h, k) and radius r is given by (x - h)^2 + (y - k)^2 = r^2. In this question, the center of the circle is given as (3, 2) and a point on the circle is (9, 10). By substituting these values into the equation, we get (x - 3)^2 + (y - 2)^2 = 100, which matches the given answer.