Let the numbers be x and y. We have x + y = 12 and xy = 20. These are roots of the quadratic equation t^2 - 12t + 20 = 0. Factoring gives (t - 10)(t - 2) = 0, so the numbers are 10 and 2. The larger number is 10.
22
Determine the quadratic equation that has roots equal to 3 and 4.
A quadratic equation with roots r1 and r2 can be expressed as (x - r1)(x - r2) = 0. Substituting the given roots 3 and 4, we get (x - 3)(x - 4) = 0. Expanding this expression results in x^2 - 4x - 3x + 12 = 0, which simplifies to x^2 - 7x + 12 = 0.
23
How is the mathematical function defined by f(x) = ax^2 + bx + c classified?
A function in the form f(x) = ax^2 + bx + c is a polynomial of degree two, where 'a' is a non-zero coefficient. Because the highest exponent of the variable x is 2, this type of mathematical expression is formally classified as a quadratic function.
24
Determine the value of m that makes the expression x^2 + 4x + m a perfect square trinomial.
A quadratic expression x^2 + bx + m is a perfect square if m = (b/2)^2. Here, b = 4. Calculating (4/2)^2 gives 2^2 = 4. Thus, x^2 + 4x + 4 is equivalent to (x + 2)^2, which is a perfect square.
25
What is the simplified value of the trigonometric expression sin^2 θ + cos^2 θ?
The expression sin^2 θ + cos^2 θ is the fundamental Pythagorean trigonometric identity. For any angle θ, the sum of the square of the sine and the square of the cosine is always equal to 1. This identity is derived from the unit circle definition where x^2 + y^2 = 1, with x = cos θ and y = sin θ. Thus, option D is the correct answer.
26
If both cot θ and cosec θ are positive, in which quadrant does the angle θ lie?
In the Cartesian coordinate system, all trigonometric functions are positive in the first quadrant. Specifically, cot θ = cos θ / sin θ and cosec θ = 1 / sin θ. Since both sin θ and cos θ are positive in the first quadrant, both cot θ and cosec θ must also be positive, confirming that θ lies in the first quadrant.
27
A father's current age is three times his son's age. Five years ago, the father was four times as old as his son. What is the son's current age?
Let the son's age be x and the father's age be 3x. Five years ago, their ages were x-5 and 3x-5. According to the problem, 3x-5 = 4(x-5). Solving this equation: 3x-5 = 4x-20, which simplifies to x = 15.
28
Ten years ago, a father's age was three times his son's age. Ten years from now, the father's age will be twice his son's age. Determine the ratio of their current ages.
Let the son's age 10 years ago be x and the father's be 3x. Currently, they are x+10 and 3x+10. In 10 years, they will be x+20 and 3x+20. Given 3x+20 = 2(x+20), we find x=20. Thus, current ages are 30 and 70. The ratio 70:30 simplifies to 7:3.
29
Among five friends A, B, C, D, and E, A is shorter than B but taller than E. C is the tallest, and D is shorter than B but taller than A. Who is positioned such that two people are taller and two are shorter than them?
By analyzing the height relationships: A < B, E < A, D < B, and A < D. Combining these, we get the order E < A < D < B < C. In this sequence, D is in the middle, with two people shorter (E and A) and two people taller (B and C). Therefore, D is the correct answer.
30
A girl is 18 years younger than her mother. If the sum of their ages will be 54 in six years, what is the girl's current age?
Let the girl's current age be g. Her mother's age is g + 18. In six years, their ages will be g + 6 and g + 24. Setting the sum to 54: (g + 6) + (g + 24) = 54, which simplifies to 2g + 30 = 54. Solving for g gives 2g = 24, so g = 12.
