PRE CALCULUS Advanced Topics 3 — Questions and Answers
Question 1: If the discriminant of a quadratic equation is negative, how many real solutions does it have?
- No real solutions (Correct answer)
- Exactly one real solution
- Two distinct real solutions
- Infinitely many solutions
Correct answer: No real solutions
A negative discriminant (b²−4ac < 0) means the roots are complex, so there are no real solutions.
Question 2: What is the magnitude of the vector v = ⟨−3, 4⟩?
- 5 (Correct answer)
- 7
- 1
- √7
Correct answer: 5
|v| = √((-3)²+4²) = √(9+16) = √25 = 5.
Question 3: Which identity is equivalent to sin²x + cos²x?
- 1 (Correct answer)
- 0
- 2sin(x)cos(x)
- cos(2x)
Correct answer: 1
The Pythagorean identity states sin²x + cos²x = 1 for all x.
Question 4: What is the inverse function of f(x) = 2x − 6?
- f⁻¹(x) = (x+6)/2 (Correct answer)
- f⁻¹(x) = (x−6)/2
- f⁻¹(x) = 2x+6
- f⁻¹(x) = x/2 − 6
Correct answer: f⁻¹(x) = (x+6)/2
Replace f(x) with y, swap x and y to get x=2y−6, then solve: y=(x+6)/2.
Question 5: Which of the following represents a one-to-one function?
- f(x) = x³ (Correct answer)
- f(x) = x²
- f(x) = |x|
- f(x) = sin(x)
Correct answer: f(x) = x³
f(x)=x³ is strictly increasing and passes the horizontal line test, so it is one-to-one.
Question 6: What is the solution to the equation e^(2x) = 7?
- x = ln(7)/2 (Correct answer)
- x = 7/e
- x = ln(7²)
- x = 2/ln(7)
Correct answer: x = ln(7)/2
Taking the natural log of both sides: 2x = ln7, so x = ln7/2.
Question 7: Which formula gives the number of combinations of n things taken r at a time?
- n! / (r!(n−r)!) (Correct answer)
- n! / (n−r)!
- n! / r!
- r! / (n!(n−r)!)
Correct answer: n! / (r!(n−r)!)
The combination formula is C(n,r) = n! / (r!(n−r)!), which counts selections without regard to order.
PRE CALCULUS Advanced Topics 3
If the discriminant of a quadratic equation is negative, how many real solutions does it have?