Math

• What is the first scenario in proving equations?

  Prove an equation is always true for all integers *n*
• What is a common approach to prove an equation?

  Look for patterns or factorization
• How would you prove n*² + *n is even?

  Factorize to n*(n* + 1)
• Why is one term in n*(n* + 1) always even?

  Because one of the terms is always even
• What is the second scenario in proving statements?

  Disprove a statement with a counterexample
• How would you disprove that n*² − *n + 1 is divisible by 3?

  Test small values of *n*
• What is a common scenario in algebra and functions?

  Solve a quadratic equation
• What keywords indicate solving a quadratic equation?

  Keywords like "roots" or "solutions"
• How would you solve x*² − 5x* + 6 = 0?

  Factorize or use the quadratic formula
• What is the quadratic formula?

  x* = −b* ± √(b*² − 4ac) / 2a*
• What is another common scenario in algebra and functions?

  Find transformations of a graph
• What phrases indicate graph transformations?

  "Describe the transformation of y* = *f(*x*)"
• How would you describe the transformation of y* = 3f(*x − 2) + 4?

  Translation 2 units right, stretch by 3, up 4
• What is a common scenario in coordinate geometry?

  Find the equation of a straight line
• What keywords indicate finding the equation of a line?

  "tangent," "normal," "find the line equation"
• How would you find the equation of a line with gradient *m* = −1 through (2, 3)?

  Use y* − *y₁ = m*(x* − *x*₁)
• What is another common scenario in coordinate geometry?

  Solve problems with circles
• What keywords indicate circle problems?

  "radius," "center," "tangents"
• How would you find the equation of a circle with center (3, −2) and radius 5?

  Use (x* − *a)² + (y* − *b)² = *r*²
• What is a common scenario in sequences and series?

  Sum of an arithmetic sequence
• What phrases indicate finding the sum of a sequence?

  "find the sum" or terms defined by a*, *d
• How would you sum the first 10 terms of 3, 5, 7, …?

  Use S*_n = *n/2(a* + *l)
• What is another common scenario in sequences and series?

  Converging geometric series
• What phrase indicates finding the sum to infinity?

  "sum to infinity"
• How would you find *S*_∞ of 1, ½, ¼, …?

  Use S*_∞ = *a/(1 − *r*)
• What is the condition for using the formula S*_∞ = *a/(1 − *r*)?

  |*r*| < 1
• What is a common scenario in trigonometry?

  Solve trigonometric equations
• What keywords indicate solving trigonometric equations?

  "find θ" or "solution in radians"
• How would you solve sin x* = 0.5 for 0 ≤ *x ≤ 2π?

  *x* = arcsin(0.5) = π/6
• How do you find other solutions for sin *x* = 0.5?

  Use the unit circle
• What is another common scenario in trigonometry?

  Triangle problems
• What keywords indicate triangle problems?

  "area," "angle," or "find a side"
• How would you find the area of a triangle with sides 5 and 7, and angle 60°?

  Use Area = ½absin *C*
• What is a common scenario in exponentials and logarithms?

  Solve exponential equations
• What keywords indicate solving exponential equations?

  "solve" and equations with e*^x*
• How would you solve e*^2x* = 8?

  Take natural logarithms, *x* = ½ ln 8
• What is another common scenario in exponentials and logarithms?

  Logarithmic simplification
• What keywords indicate logarithmic simplification?

  "simplify logs"
• How would you simplify log₃(9) + log₃(27)?

  Use |log(ab) = |log a + |log b|
• What is the formula for the magnitude of a vector?

  |a| = √(x*² + *y² + *z*²)