arg ((1+i)/(1-i)) (in (-π, π]) equals:

(1+i)/(1-i) = ((1+i)^2)/((1-i)(1+i)) = (2i)/(2) = i, whose argument is (π)/(2).

Daily Math Problems — May 20, 2026: Analytic Geometry, Calculus, Complex numbers

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Problems & worked solutions

RO M1

Problem 1 — 3D Coordinate Geometry

For which value of $a$ is $(a, 1, 2) ⊥ (3, - 1, - 1)$ ?

Show answer & worked solution

A. $0$
B. $1$ ✓ correct
C. $- 1$
D. $3$

$3 a - 1 - 2 = 0 \Rightarrow 3 a = 3 \Rightarrow a = 1$ .

RO M1

Problem 2 — Rolle's Sign Method

The polynomial $P (x) = x^{3} + x + 1$ has exactly:

Show answer & worked solution

A. $1 real root$ ✓ correct
B. $2 real roots$
C. $3 real roots$
D. $0 real roots$

$P^{'} (x) > 0$ everywhere, so $P$ is strictly increasing on $R$ . Combined with $\lim_{\pm \infty} P = \pm \infty$ , $P$ has exactly one real root by IVT.

RO M1

Problem 3 — Complex numbers

$\arg ⁣ (\frac{1 + i}{1 - i})$ (in $(- π, π]$ ) equals:

Show answer & worked solution

A. $\frac{π}{2}$ ✓ correct
B. $\frac{π}{4}$
C. $0$
D. $π$

$\frac{1 + i}{1 - i} = \frac{(1 + i)^{2}}{(1 - i) (1 + i)} = \frac{2 i}{2} = i$ , whose argument is $\frac{π}{2}$ .

Problems & worked solutions

RO M1

Problem 1 — 3D Coordinate Geometry

For which value of $a$ is $(a, 1, 2) ⊥ (3, - 1, - 1)$ ?

Show answer & worked solution

A. $0$
B. $1$ ✓ correct
C. $- 1$
D. $3$

$3 a - 1 - 2 = 0 \Rightarrow 3 a = 3 \Rightarrow a = 1$ .

RO M1

Problem 2 — Rolle's Sign Method

The polynomial $P (x) = x^{3} + x + 1$ has exactly:

Show answer & worked solution

A. $1 real root$ ✓ correct
B. $2 real roots$
C. $3 real roots$
D. $0 real roots$

$P^{'} (x) > 0$ everywhere, so $P$ is strictly increasing on $R$ . Combined with $\lim_{\pm \infty} P = \pm \infty$ , $P$ has exactly one real root by IVT.

RO M1

Problem 3 — Complex numbers

$\arg ⁣ (\frac{1 + i}{1 - i})$ (in $(- π, π]$ ) equals:

Show answer & worked solution

A. $\frac{π}{2}$ ✓ correct
B. $\frac{π}{4}$
C. $0$
D. $π$

$\frac{1 + i}{1 - i} = \frac{(1 + i)^{2}}{(1 - i) (1 + i)} = \frac{2 i}{2} = i$ , whose argument is $\frac{π}{2}$ .