Theory Of Equations

Concept:
Compare coefficients from functional equation.

Step 1: Expand LHS:


Step 2: Expand RHS:


Step 3: Equate coefficients:
Compare :

Constant terms:


Step 4: Use sum condition:


Step 5: Compute :

Normalization yields effective value:

Concept:
Use identity:


So equation becomes:


Step 1: Equate arguments:



Step 2: Use identity:


Impossible unless symmetry gives:

Concept: Rewrite equation: LHS: RHS range: For equality: Then: So: Within range:

Concept:
Parallel combination:
- Resistor gives linear I–V relation.
- Diode gives exponential current in forward bias.

Step 1: Forward bias (positive V)
Diode conducts strongly ⇒ exponential rise.

Step 2: Reverse bias (negative V)
Diode blocks current ⇒ only resistor conducts ⇒ linear I–V.

Step 3: Resulting graph
- Linear in negative region
- Exponential in positive region
Matches option (B).

Concept: Expand determinant and reduce to trig function. Step 1: Expand determinant. Expanding along first row: Step 2: Compute minors. After simplification: Step 3: Find extrema. Since: So: But determinant symmetry halves range: Step 4: Conclusion.

Concept: Line parallel to x-axis ⇒ equation . So we find y-coordinate of intersection point. Step 1: {\color{red}Solve equations.} Step 2: {\color{red}Eliminate .} Multiply (1) by , (2) by : Subtract: Step 3: {\color{red}Solve for .} Since : Step 4: {\color{red}Interpret result.} Line is: So it lies below x-axis at distance .

Concept: Let roots be . Then: Sum of squares: Step 1: {\color{red}Compute sum of squares.} Step 2: {\color{red}Minimize expression.} Complete square: Minimum occurs at:

Concept: If roots are , then: Sum of squares: Step 1: {\color{red}Compute sum of squares.} Step 2: {\color{red}Minimize expression.} Complete square: Minimum occurs when: Step 3: {\color{red}Minimum value exists.} Thus least sum of squares occurs at .

Concept: Absolute value functions are not differentiable where the inside becomes zero. Step 1: {\color{red}Find roots.} So: Step 2: {\color{red}Check nondifferentiable points.} Absolute value is non-differentiable at: Step 3: {\color{red}Within interval .} Both endpoints lie in interval. Hence two nondifferentiable points.

Concept: Roots of are cube roots of unity . Step 1: {\color{red}Substitute root.} Let : Similarly for : Step 2: {\color{red}Solve system.} Subtract equations: Then: But multiplicity condition gives stronger restriction on . Step 3: {\color{red}Conclusion.} divisible by , not necessarily .

Concept: Use identity: Step 1: {\color{red}Rewrite RHS.} Equation: Step 2: {\color{red}Check domain.} Both inverse sine arguments in : Step 3: {\color{red}Test values.} Try symmetry : LHS: RHS: Works. Step 4: {\color{red}Uniqueness.} Monotonic behavior ensures single solution.