Algebra

Equations, inequalities, and functions.

  1. Given the system: a+bc=2021 ab+c=2020 find the value of a,b and c. (a,b,c ∈ Z⁺)
  2. x²-2x+3=0 has roots α, β. Let α³-3 α²+5 α -2 = φ and β³- β²+ β +5= χ Then find (φ + χ)+ (φ χ)
  3. If 2 logₓ7+ log₇ₓ7_+3 log₄₉ₓ7=0 has two solutions α and β then find α ^((-3)/(4))+ β⁻² such that both α ^((-3)/(4)), β⁻² ∈ Z⁺.
  4. Let f(x) a quadratic polynomial with leading coeffient 1 such that f(0)=p and p ≠ 0. f(1)= 1/3. If equation f(x)=0 and f ∘ f ∘ f ∘ f(x)=0 have a common root then find f(-3)
  5. If | 2- | x-2026 | |<2 has the solution x ∈ (a,b) and x ≠ c then find ((c+4)/(2030)) · a +8
  6. P(x)=x⁶-x⁵-x³-x²-x Q(x)=x⁴-x³-x²-1. If α, β, γ and δ be the roots of Q(x)=0, then find P(α)+P(β)+P(γ)+P(δ)
  7. The value of 6 + log _(3/2) ((1)/(3√2) √ 4 - (1)/(3√2) √ 4 - (1)/(3√2) √(4 - …)) is:
  8. Find the solution set of (Π _(r=1)¹⁰⁰(x-r)ʳ) sin ((π x)/(100)) log _e(x²-100x+901) ≥ 0; x ∈ [0,100]
  9. If Σ _(k=1)ᴺ (2k+1)/((k²+k)²) =0.9999, then find the value of N.
  10. Find the number of ordered pairs (x, y), where both x and y are integers between -100 and 100 inclusive, such that 12x² - xy - 6y² = 0.
  11. Consider a quadratic equation x²-4x-n=0 where n ∈ N and n ∈ [10,200]. Find the total number of possible values of n that give the equation integer roots.