Let λ ≠ 0 be a real number. Let α, β be the roots of the equation \$14x^2 - 31x + 3λ\$ = 0 and α, γ be the roots of the equation \$35x^2 - 53x + 4λ\$ = 0. Then \$(3α) / β\$ and \$(4α) / γ\$ are the roots of the equation.

A) \$7x^2 - 245x + 250\$ = 0

B) \$49x^2 - 245x + 250\$ = 0

C) \$49x^2 + 245x + 250\$ = 0

D) \$7x^2 + 245x - 250\$ = 0

Let α be the root of the equation \$1 + x^2 + x^4\$ = 0. Then, the value of \$α^1011 + α^2022 - α^3033\$ is equal to.

A) α

B) 1 + α

C) 1

D) 1 + 2α

If the quadratic equation \$"x"^2 + "px" + "q" = 0\$ has roots α and 𝛽, \$α^2 + β^2 = 7\$ and α + β = 3 , find the values of p and q.

A) p = -3 and q = -1

B) p = 3 and q = 1

C) p = 3 and q = -1

D) p = -3 and q = 1

The sum of all the solutions of the equation \$8^2x - 16 * 8^x + 48 = 0\$ is:

A) \$1 + log_8(6)\$

B) \$2 + log_8(5)\$

C) \$1 + log_8(7)\$

D) \$1 + log_7(9)\$

If α, β are the roots of the equation, \$x^2 - x - 1 = 0\$ and \$S_n = 2023α^n + 2024β^n\$, then:

A) \$S_12 = S_11 + S_10\$

B) \$2S_12 = S_11 + S_10\$

C) \$S_11 = S_10 + S_12\$

D) \$2S_1 = S_12 + S_10\$

Let α, β be the roots of the equation \$x^2 - sqrt2x + 2 = 0\$. Then \$α^14 + β^14\$ is equal to.

A) -128

B) -64

C) \$-64sqrt2\$

D) \$-128sqrt2\$

A quadratic equation has one root as \$3 + sqrt5\$. Find the quadratic equation.

A) \$x^2 - 6x + 4 = 0\$

B) \$2x^2 - 6x + 4 = 0\$

C) \$x^2 - 5x + 4 = 0\$

D) \$3x^2 - 6x + 4 = 0\$

The equation \$"e"^4"x" + 8"e"^3"x" + 13"e"^2"x" - 8"e"^"x" + 1 = 0\$, x ∈ R has:

A) Two solutions and only one of them is negative.

B) Two solutions and both are negative.

C) Four solutions two of which are negative.

D) no solution.

Factor the quadratic equation \$x^2 - 6x + 8\$.

A) x = 4 and x = -2

B) x = -4 and x = 2

C) x = -4 and x = -2

D) x = 4 and x = 2

Solve the quadratic equation \$x^2 - 3x - 4 = 0\$ by factoring.

A) x = -4 and x = -1

B) x = -4 and x = 1

C) x = 4 and x = 1

D) x = 4 and x = -1