Daily Math · 2026-06-07

Question 1

In $	riangle ABC$, $a = 5$, $b = 6$, $c = 7$. The inradius $r$ is closest to:

Accepted Answer

$s = 9$. By Heron, Area $= \sqrt{9 \cdot 4 \cdot 3 \cdot 2} = 6\sqrt{6} \approx 14.70$. So $r \approx \dfrac{14.70}{9} \approx 1.63$.

Answer

$1.4$

Answer

$1.6$

Answer

$2.4$

Answer

$3.0$

Question 2

In the expansion of $\!\left(\sqrt{x} + \dfrac{1}{x}\right)^{6}$, the rational (i.e. integer-power-of-$x$) terms count is:

Accepted Answer

The exponent $\dfrac{6 - 3k}{2}$ is an integer when $k$ is even: $k \in \{0, 2, 4, 6\}$ — four values.

Answer

$1$

Answer

$2$

Answer

$3$

Answer

$4$

Question 3

By Rolle, the equation $x^4 - 6x^2 + 8 = 0$ has how many distinct real solutions?

Accepted Answer

$y^2 - 6y + 8 = 0 \Rightarrow y \in \{2, 4\}$. Then $x^2 = 2 \Rightarrow x = \pm\sqrt{2}$ and $x^2 = 4 \Rightarrow x = \pm 2$. Four distinct real solutions.

Answer

$0$

Answer

$1$

Answer

$2$

Answer

$4$

Problems & worked solutions

Problem 1 — Solving Triangles

Problem 2 — Binomial Theorem

Problem 3 — Rolle's Sign Method