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Questions and Answers
What can the equation log 𝑒(𝑥) + log 𝑒(1 + 𝑥) = 0 be written as?
What can the equation log 𝑒(𝑥) + log 𝑒(1 + 𝑥) = 0 be written as?
- 𝑥 = 1
- 𝑥 = 0
- None of these (correct)
- 𝑥 = -1
What is the value of 𝛼 in the equation 2𝑥^2 + 3𝑥 + 1 = 6𝑥 + 𝛼^2 + 1 = 0 if the product of the roots of the equation 2𝑥^2 + 10 = 0 is −𝛼?
What is the value of 𝛼 in the equation 2𝑥^2 + 3𝑥 + 1 = 6𝑥 + 𝛼^2 + 1 = 0 if the product of the roots of the equation 2𝑥^2 + 10 = 0 is −𝛼?
- -1 (correct)
- 0
- -2
- 1
If the roots of the equation 𝑥^2 − 8𝑥 + (𝑎^2 − 6𝑎) = 0 are real, then the equation can be written as:
If the roots of the equation 𝑥^2 − 8𝑥 + (𝑎^2 − 6𝑎) = 0 are real, then the equation can be written as:
- 𝑥^2 + 𝑥 − 𝑒 = 0
- 𝑥 + 𝑥𝑒 − 𝑒 = 0
- 𝑥 + 𝑥 + 1 = 0
- 𝑥^2 + 𝑥 − 1 = 0 (correct)
{𝑥 ∈ 𝑅: |𝑥 − 2| = 𝑥^2} =
{𝑥 ∈ 𝑅: |𝑥 − 2| = 𝑥^2} =
If 𝑃(𝑥) = 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 and 𝑄(𝑥) = −𝑎𝑥^2 + 𝑑𝑥 + 𝑐, where 𝑎𝑐 ≠ 0, what is the nature of their product 𝑃(𝑥) ⋅ 𝑄(𝑥)?
If 𝑃(𝑥) = 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 and 𝑄(𝑥) = −𝑎𝑥^2 + 𝑑𝑥 + 𝑐, where 𝑎𝑐 ≠ 0, what is the nature of their product 𝑃(𝑥) ⋅ 𝑄(𝑥)?
What are the roots of the equation (𝑥 − 𝑎)(𝑥 − 𝑏) + (𝑥 − 𝑏)(𝑥 − 𝑐) + (𝑥 − 𝑐)(𝑥 − 𝑎) = 0 always?
What are the roots of the equation (𝑥 − 𝑎)(𝑥 − 𝑏) + (𝑥 − 𝑏)(𝑥 − 𝑐) + (𝑥 − 𝑐)(𝑥 − 𝑎) = 0 always?
If the roots of the equation $2𝑥^2 + 3(𝜆 − 2)𝑥 + (a) = 0$ are equal in magnitude but opposite in sign, then $𝜆 =$
If the roots of the equation $2𝑥^2 + 3(𝜆 − 2)𝑥 + (a) = 0$ are equal in magnitude but opposite in sign, then $𝜆 =$
The expression $𝑥^2 + 2𝑏𝑥 + 𝑐$ has a positive value then 𝜆 = if
The expression $𝑥^2 + 2𝑏𝑥 + 𝑐$ has a positive value then 𝜆 = if
If the roots of the equation $(𝑝^2 + 𝑞^2)𝑥^2 - 2𝑞(𝑝 + 𝑟)𝑥 + (𝑞^2 + 𝑟^2) = 0$ be real and equal, then $𝑝, 𝑞, 𝑟$ will be in
If the roots of the equation $(𝑝^2 + 𝑞^2)𝑥^2 - 2𝑞(𝑝 + 𝑟)𝑥 + (𝑞^2 + 𝑟^2) = 0$ be real and equal, then $𝑝, 𝑞, 𝑟$ will be in
If the roots of the equation $(𝑎 + 1)𝑥^2 + (2𝑎 + 3)𝑥 + (3𝑎 + 4) = 0$ be equal to 2, then the sum of roots is
If the roots of the equation $(𝑎 + 1)𝑥^2 + (2𝑎 + 3)𝑥 + (3𝑎 + 4) = 0$ be equal to 2, then the sum of roots is
If one root of $𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0$ is the square of the other, then the value of $𝑏^3 + 𝑎𝑐^2 + 𝑎^2 𝑐$ is
If one root of $𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0$ is the square of the other, then the value of $𝑏^3 + 𝑎𝑐^2 + 𝑎^2 𝑐$ is
Let 𝛼 and 𝛽 be the roots of the equation $4𝑥^2 + 3𝑥 + 1 = 0$. If the product of the roots is $2$, then 𝛼 + 𝛽 is
Let 𝛼 and 𝛽 be the roots of the equation $4𝑥^2 + 3𝑥 + 1 = 0$. If the product of the roots is $2$, then 𝛼 + 𝛽 is
If the sum of the roots of the equation $𝜆𝑥^2 + 2𝑥 + 30 = 0$ be equal to their product, then 𝜆 is
If the sum of the roots of the equation $𝜆𝑥^2 + 2𝑥 + 30 = 0$ be equal to their product, then 𝜆 is
The roots of the given equation $2(𝑎^2 + 𝑏^2)𝑥^2 + 𝑏𝑑)𝑡 + (𝑐^2 + 𝑑^2) = 0$ are equal, then
The roots of the given equation $2(𝑎^2 + 𝑏^2)𝑥^2 + 𝑏𝑑)𝑡 + (𝑐^2 + 𝑑^2) = 0$ are equal, then
If 𝛼 and 𝛽 are the roots of the equation $𝑥^2 + 6𝑥 + 𝜆 = 0$ and $3𝛼 + 2𝛽 = −20$, then 𝜆 is
If 𝛼 and 𝛽 are the roots of the equation $𝑥^2 + 6𝑥 + 𝜆 = 0$ and $3𝛼 + 2𝛽 = −20$, then 𝜆 is
If one root of 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0 be equal to 0, then if $−𝑏 = 0$, then $(𝛼 + 1)(𝛽 + 1)$ is equal to
If one root of 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0 be equal to 0, then if $−𝑏 = 0$, then $(𝛼 + 1)(𝛽 + 1)$ is equal to
If 2 + 𝑖√3 is a root of the equation $𝑥 + 𝑝𝑥 + 𝑞 = 0$, where 𝑝 and 𝑞 are real, then (𝑝, 𝑞) =
If 2 + 𝑖√3 is a root of the equation $𝑥 + 𝑝𝑥 + 𝑞 = 0$, where 𝑝 and 𝑞 are real, then (𝑝, 𝑞) =
If the difference of the roots of $x^2 − p x + 8 = 0$ be 3 , then the value of p is
If the difference of the roots of $x^2 − p x + 8 = 0$ be 3 , then the value of p is
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Study Notes
Logarithmic Equations
- The equation log 𝑒(𝑥) + log 𝑒(1 + 𝑥) = 0 can be rewritten as a different form.
Quadratic Equations
- The value of 𝛼 in the equation 2𝑥^2 + 3𝑥 + 1 = 6𝑥 + 𝛼^2 + 1 = 0 can be found if the product of the roots of the equation 2𝑥^2 + 10 = 0 is −𝛼.
- If the roots of the equation 𝑥^2 − 8𝑥 + (𝑎^2 − 6𝑎) = 0 are real, then the equation can be written as {𝑥 ∈ 𝑅: |𝑥 − 2| = 𝑥^2}.
- The roots of the equation (𝑥 − 𝑎)(𝑥 − 𝑏) + (𝑥 − 𝑏)(𝑥 − 𝑐) + (𝑥 − 𝑐)(𝑥 − 𝑎) = 0 are always 𝑎, 𝑏, and 𝑐.
- If the roots of the equation $2𝑥^2 + 3(𝜆 − 2)𝑥 + (a) = 0$ are equal in magnitude but opposite in sign, then $𝜆 = 4$.
- The expression $𝑥^2 + 2𝑏𝑥 + 𝑐$ has a positive value if 𝜆 = 1.
- If the roots of the equation $(𝑝^2 + 𝑞^2)𝑥^2 - 2𝑞(𝑝 + 𝑟)𝑥 + (𝑞^2 + 𝑟^2) = 0$ are real and equal, then $𝑝, 𝑞, 𝑟$ will be in A.P.
- If the roots of the equation $(𝑎 + 1)𝑥^2 + (2𝑎 + 3)𝑥 + (3𝑎 + 4) = 0$ are equal to 2, then the sum of roots is 4.
- If one root of $𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0$ is the square of the other, then the value of $𝑏^3 + 𝑎𝑐^2 + 𝑎^2 𝑐$ is 0.
- If the product of the roots is $2$, then 𝛼 + 𝛽 = -3/2.
- If the sum of the roots of the equation $𝜆𝑥^2 + 2𝑥 + 30 = 0$ be equal to their product, then 𝜆 is 1.
- The roots of the given equation $2(𝑎^2 + 𝑏^2)𝑥^2 + 𝑏𝑑)𝑡 + (𝑐^2 + 𝑑^2) = 0$ are equal, then $𝑏𝑑 = 0$.
- If 𝛼 and 𝛽 are the roots of the equation $𝑥^2 + 6𝑥 + 𝜆 = 0$ and $3𝛼 + 2𝛽 = −20$, then 𝜆 = 14.
- If one root of 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0 be equal to 0, then if $−𝑏 = 0$, then $(𝛼 + 1)(𝛽 + 1)$ is equal to 1.
- If 2 + 𝑖√3 is a root of the equation $𝑥 + 𝑝𝑥 + 𝑞 = 0$, where 𝑝 and 𝑞 are real, then (𝑝, 𝑞) = (−1, 7).
- If the difference of the roots of $x^2 − p x + 8 = 0$ be 3, then the value of p is 7.
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