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Questions and Answers
What is the relationship between x, y, and z if the equation $x + 9y + 25z = xyz(15x + 5y + z)$ holds?
What is the relationship between x, y, and z if the equation $x + 9y + 25z = xyz(15x + 5y + z)$ holds?
If $a + b = 7$ and $ab = 12$, what is the value of $a^2 - ab - b^2$?
If $a + b = 7$ and $ab = 12$, what is the value of $a^2 - ab - b^2$?
If $x^2 + 12 = 51$, what is $x$?
If $x^2 + 12 = 51$, what is $x$?
What is the simplified expression of $A^3 - 3A$ if $A = (4 +
oot{15}{3}) + (4 -
oot{15}{3})$?
What is the simplified expression of $A^3 - 3A$ if $A = (4 + oot{15}{3}) + (4 - oot{15}{3})$?
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What is the least value of the expression $x^2 + 4y^2 + 9z^2 - 2x + 8y + 27z + 15$?
What is the least value of the expression $x^2 + 4y^2 + 9z^2 - 2x + 8y + 27z + 15$?
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If $a + b + c = 12$ and $a^2 + b^2 + c^2 = 50$, what is the value of $ab + bc + ca$?
If $a + b + c = 12$ and $a^2 + b^2 + c^2 = 50$, what is the value of $ab + bc + ca$?
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What is the value of $x^3 - rac{1}{x^3}$ if $x = 7$?
What is the value of $x^3 - rac{1}{x^3}$ if $x = 7$?
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If $x + y + z = 10$ and $xy + xz + yz = 35$, what is the value of $xyz$?
If $x + y + z = 10$ and $xy + xz + yz = 35$, what is the value of $xyz$?
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What is the value of $x^3 + y^3 + z^3$ given that $x + y + z = 1$, $xy + yz + zx = -1$, and $xyz = -1$?
What is the value of $x^3 + y^3 + z^3$ given that $x + y + z = 1$, $xy + yz + zx = -1$, and $xyz = -1$?
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What is the result of $( ext{√}4 - ext{√}10 + ext{√}25)( ext{√}2 + ext{√}5)$?
What is the result of $( ext{√}4 - ext{√}10 + ext{√}25)( ext{√}2 + ext{√}5)$?
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If $x + x_1 = 2$, how does the equation $x^2 + rac{1}{x^2} = x^4 + rac{1}{x^4} + rac{1}{x^6}$ relate to the variables x and $x_1$?
If $x + x_1 = 2$, how does the equation $x^2 + rac{1}{x^2} = x^4 + rac{1}{x^4} + rac{1}{x^6}$ relate to the variables x and $x_1$?
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What is the correct expansion of $(x + y - z)^2$?
What is the correct expansion of $(x + y - z)^2$?
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What is the result of $15 + 5 - 31 + 7 - 41$?
What is the result of $15 + 5 - 31 + 7 - 41$?
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If $xy + yz + zx = -1$ and $xyz = -1$, how are these expressions used in polynomial roots?
If $xy + yz + zx = -1$ and $xyz = -1$, how are these expressions used in polynomial roots?
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If $x + y + z = 1$, what can be inferred about the values of $xy$, $yz$, and $zx$?
If $x + y + z = 1$, what can be inferred about the values of $xy$, $yz$, and $zx$?
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Which of the following values could potentially satisfy the condition that $x + y + z = 1$ and $xyz = -1$?
Which of the following values could potentially satisfy the condition that $x + y + z = 1$ and $xyz = -1$?
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Study Notes
Problem Solving in Mathematics
- Relationship between variables x, y, z is analyzed through the equation: (x + 9y + 25z^2 = xyz(15x + 5y + z))
- Multiple relationship options include:
- (A) (x = 3y = 5z)
- (B) (x = 2y = 3z)
- (C) (x = -3y = 5z)
- (D) (x = -3y = -5z)
Algebraic Manipulations
- Given (a + b = 7) and (ab = 12), find (a^2 - ab - b^2).
- Options to consider:
- (A) 12
- (B) 13
- (C) 14
- (D) -5
Function Values
- Solve for (x) in the equation (x^2 + 12 = 51).
- Evaluate the expression (x^3 - \frac{1}{x^3}) with potential values:
- (A) 364
- (B) 365
- (C) 756
- (D) 367
Expression Minimization
- Least value of (x^2 + 4y^2 + 9z^2 - 2x + 8y + 27z + 15) identifies minimum value options:
- (A) 44
- (B) 45
- (C) 46
- (D) 47
Quadratic and Cubic Relationships
- Simplifying ((\sqrt{4} - \sqrt{10} + \sqrt{25})(\sqrt{2} + \sqrt{5})) yields a specific value.
- Set (a + b + c = 12) and (a^2 + b^2 + c^2 = 50) to find (ab + bc + ca).
Polynomial Identities
- For the values (x + y + z = 1), (xy + yz + zx = -1), and (xyz = -1), calculate (x^3 + y^3 + z^3) with possible outcomes:
- (A) -1
- (B) 1
- (C) 2
- (D) -2
Algebraic Expansion
- Expansion of ((x + y - z)^2) provides multiple forms:
- (A) (x^2 + y^2 + z^2 + 2xy + 2yz + 2zx)
- (B) (x^2 + y^2 - z^2 - 2xy + yz + 2zx)
- (C) (x^2 + y^2 + z^2 + 2xy - 2yz - 2zx)
- (D) (x^2 + y^2 - z^2 + 2xy - 2yz - 2zx)
Answer Key Summary
- Problems Q1 through Q10 each have correct answers indicated, providing direct reference for studying solutions and methods.
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Description
Take this quiz to test your knowledge of basic mathematics concepts in preparation for the JEE exam. This quiz covers various topics and relationships involving variables x, y, and z, along with fundamental equations. Challenge yourself and see how prepared you are for the upcoming JEE.
