Most advanced Algebra MCQ of all time with
answer keys
Questions:
1. If \( x^3 + y^3 + z^3 - 3xyz = 1 \) and \( x + y + z = 1 \), what is \( xyz \)?
- A) \(-1\)
- B) \(0\)
- C) \(1\)
- D) \(-1/6\)
2. Given that \( f(x) = x^3 + ax^2 + bx + c \) has roots at \( \alpha, \beta,
\gamma \), what is \( \alpha^2 + \beta^2 + \gamma^2 \) in terms of \( a
\) and \( b \)?
- A) \( a^2 - 2b \)
- B) \( 2a - b \)
- C) \( a^2 + b \)
- D) \( 2a^2 - b \)
3. What is the sum of all integers from 1 to 100 that are not divisible
by 2, 3, or 5?
- A) 1060
- B) 1147
, - C) 1212
- D) 1243
4. If \( p(x) = x^4 - 6x^3 + 13x^2 - 12x + k \) has a double root at \( x = 2
\), what is the value of \( k \)?
- A) \(-4\)
- B) \(3\)
- C) \(-7\)
- D) \(5\)
5. Given the system of equations:
\[
\begin{cases}
x + 2y + 3z = 6 \\
2x + 4y + 6z = 12 \\
x-y+z=2
\end{cases}
\]
How many solutions does this system have?
- A) None
- B) Exactly one
- C) Infinitely many
- D) Two
,6. Solve the equation \( e^x + e^{-x} = 6 \).
- A) \( x = 0 \)
- B) \( x = \ln(3) \)
- C) \( x = \ln(2) \)
- D) \( x = 2 \ln(3) \)
7. What is the general solution to the differential equation \( \frac{dy}
{dx} = y \)?
- A) \( y = e^x \)
- B) \( y = Ce^{x} \)
- C) \( y = C \cdot x \)
- D) \( y = x + C \)
8. Evaluate the limit \( \lim_{x \to 0} \frac{\sin(5x)}{x} \).
- A) 0
- B) 1
- C) 5
- D) Undefined
9. If \( x \) and \( y \) are real numbers satisfying \( x^2 + y^2 + x + y = 1
\), find the maximum value of \( x + y \).
- A) \( \frac{1}{2} \)
, - B) \( 1 \)
- C) \( \frac{3}{2} \)
- D) \( 2 \)
10. Given that \( z \) is a complex number such that \( z + \frac{1}{z} =
2 \cos \theta \), what is \( |z| \)?
- A) \(1\)
- B) \(2 \cos \theta\)
- C) \( \cos \theta\)
- D) \( 2 \)
11. Solve the equation \( \sqrt{4x - 7} + 2 = x \).
- A) \( x = 3 \)
- B) \( x = 4 \)
- C) \( x = 5 \)
- D) No solution
12. Find the radius of convergence for the power series \(
\sum_{n=1}^{\infty} \frac{x^n}{n!} \).
- A) \(0\)
- B) \(1\)
- C) \(e\)
- D) \(\infty\)
answer keys
Questions:
1. If \( x^3 + y^3 + z^3 - 3xyz = 1 \) and \( x + y + z = 1 \), what is \( xyz \)?
- A) \(-1\)
- B) \(0\)
- C) \(1\)
- D) \(-1/6\)
2. Given that \( f(x) = x^3 + ax^2 + bx + c \) has roots at \( \alpha, \beta,
\gamma \), what is \( \alpha^2 + \beta^2 + \gamma^2 \) in terms of \( a
\) and \( b \)?
- A) \( a^2 - 2b \)
- B) \( 2a - b \)
- C) \( a^2 + b \)
- D) \( 2a^2 - b \)
3. What is the sum of all integers from 1 to 100 that are not divisible
by 2, 3, or 5?
- A) 1060
- B) 1147
, - C) 1212
- D) 1243
4. If \( p(x) = x^4 - 6x^3 + 13x^2 - 12x + k \) has a double root at \( x = 2
\), what is the value of \( k \)?
- A) \(-4\)
- B) \(3\)
- C) \(-7\)
- D) \(5\)
5. Given the system of equations:
\[
\begin{cases}
x + 2y + 3z = 6 \\
2x + 4y + 6z = 12 \\
x-y+z=2
\end{cases}
\]
How many solutions does this system have?
- A) None
- B) Exactly one
- C) Infinitely many
- D) Two
,6. Solve the equation \( e^x + e^{-x} = 6 \).
- A) \( x = 0 \)
- B) \( x = \ln(3) \)
- C) \( x = \ln(2) \)
- D) \( x = 2 \ln(3) \)
7. What is the general solution to the differential equation \( \frac{dy}
{dx} = y \)?
- A) \( y = e^x \)
- B) \( y = Ce^{x} \)
- C) \( y = C \cdot x \)
- D) \( y = x + C \)
8. Evaluate the limit \( \lim_{x \to 0} \frac{\sin(5x)}{x} \).
- A) 0
- B) 1
- C) 5
- D) Undefined
9. If \( x \) and \( y \) are real numbers satisfying \( x^2 + y^2 + x + y = 1
\), find the maximum value of \( x + y \).
- A) \( \frac{1}{2} \)
, - B) \( 1 \)
- C) \( \frac{3}{2} \)
- D) \( 2 \)
10. Given that \( z \) is a complex number such that \( z + \frac{1}{z} =
2 \cos \theta \), what is \( |z| \)?
- A) \(1\)
- B) \(2 \cos \theta\)
- C) \( \cos \theta\)
- D) \( 2 \)
11. Solve the equation \( \sqrt{4x - 7} + 2 = x \).
- A) \( x = 3 \)
- B) \( x = 4 \)
- C) \( x = 5 \)
- D) No solution
12. Find the radius of convergence for the power series \(
\sum_{n=1}^{\infty} \frac{x^n}{n!} \).
- A) \(0\)
- B) \(1\)
- C) \(e\)
- D) \(\infty\)
