Daily Math · 2026-07-14

Question 1

Given $\sin(x+y)+\sin(x-y)=\dfrac{3}{2}$ and $\cos(x-y)+\cos(y-x)=\sqrt{3}$, find $	an\!\dfrac{x+y}{2}$.

Accepted Answer

$\bullet$ Cosine is even, so $\cos(y-x) = \cos(x-y)$:

$$2\cos(x-y) = \sqrt{3} \;\Rightarrow\; \cos(x-y) = \frac{\sqrt{3}}{2}$$

$\bullet$ So $x - y = 	frac{\pi}{6}$ (principal value).

$\bullet$ Substitute into the first equation:

$$\sin(x+y) + \sin	frac{\pi}{6} = 	frac{3}{2}$$

$$\sin(x+y) = 1 \;\Rightarrow\; x + y = 	frac{\pi}{2}$$

$\bullet$ Compute the tangent:

$$	an\frac{x+y}{2} = 	an\frac{\pi}{4} = 1$$

Answer

$\dfrac{1}{\sqrt{3}}$

Answer

$\sqrt{3}$

Answer

$0$

Answer

$1$

Answer

$\dfrac{1}{2}$

Answer

$\dfrac{\sqrt{3}}{2}$

Question 2

Which statement is logically equivalent to "if $n^2$ is even, then $n$ is even"?

Accepted Answer

The contrapositive of "$n^2$ even $\Rightarrow$ $n$ even" is "$n$ odd $\Rightarrow$ $n^2$ odd". A statement and its contrapositive always have the same truth value.

Answer

If $n$ is even, then $n^2$ is even

Answer

If $n$ is odd, then $n^2$ is odd

Answer

If $n^2$ is odd, then $n$ is even

Answer

$n$ is even if and only if $n^2$ is odd

Question 3

The slope of the line through $A(1, 2)$ and $B(4, 11)$ is:

Accepted Answer

$m = \dfrac{11 - 2}{4 - 1} = \dfrac{9}{3} = 3$.

Answer

$\dfrac{1}{3}$

Answer

$2$

Answer

$3$

Answer

$9$

Question 4

The set of points $(x, y)$ equidistant from $A(0, 0)$ and $B(4, 0)$ is the line:

Accepted Answer

Set $\sqrt{x^2 + y^2} = \sqrt{(x - 4)^2 + y^2}$. Squaring: $x^2 = (x - 4)^2 = x^2 - 8x + 16$, so $8x = 16$, $x = 2$.

Answer

$y = 2$

Answer

$y = 0$

Answer

$x = 2$

Answer

$x = 0$

Question 5

A biased coin shows heads with probability $\dfrac{1}{3}$. In $4$ tosses, the probability of getting exactly $2$ heads is:

Accepted Answer

$P = \binom{4}{2}\!\left(\dfrac{1}{3}\right)^2 \!\left(\dfrac{2}{3}\right)^2 = 6 \cdot \dfrac{1}{9} \cdot \dfrac{4}{9} = \dfrac{24}{81} = \dfrac{8}{27}$.

Answer

$\dfrac{4}{27}$

Answer

$\dfrac{6}{81}$

Answer

$\dfrac{8}{27}$

Answer

$\dfrac{16}{81}$

Question 6

For every $x \in [-1, 1]$, the value of $\arcsin x + \arccos x$ is:

Accepted Answer

For any $x \in [-1, 1]$, $\arcsin x + \arccos x = \dfrac{\pi}{2}$ (a standard identity, since $\arccos x = \dfrac{\pi}{2} - \arcsin x$).

Answer

$0$

Answer

$\dfrac{\pi}{4}$

Answer

$\dfrac{\pi}{2}$

Answer

$\pi$

Question 7

The middle term of the expansion of $(x + 1)^8$ has coefficient:

Accepted Answer

Middle coefficient: $\binom{8}{4} = 70$.

Answer

$28$

Answer

$56$

Answer

$70$

Answer

$128$

Question 8

The value of $	an\!\left(\arccos\dfrac{2}{3}ight)$ is:

Accepted Answer

$\sin^2	heta = 1 - \dfrac{4}{9} = \dfrac{5}{9}$, so $\sin	heta = \dfrac{\sqrt{5}}{3}$. Hence $	an	heta = \dfrac{\sin	heta}{\cos	heta} = \dfrac{\sqrt{5}/3}{2/3} = \dfrac{\sqrt{5}}{2}$.

Answer

$\dfrac{\sqrt{5}}{3}$

Answer

$\dfrac{2}{\sqrt{5}}$

Answer

$\dfrac{\sqrt{5}}{2}$

Answer

$\sqrt{5}$

Question 9

The map $\varphi: (\mathbb{R}, +) 	o ((0, \infty), \cdot)$ defined by $\varphi(x) = e^x$ is:

Accepted Answer

$e^{a + b} = e^a e^b$ ✓ (homomorphism). $e^x$ is strictly increasing (injective) and surjective onto $(0, \infty)$. Hence an isomorphism. (Inverse: $\ln$.)

Answer

not a homomorphism

Answer

a non-injective homomorphism

Answer

injective but not surjective

Answer

an isomorphism

Question 10

Let $f : \mathbb{R} 	o \mathbb{R}$ be continuous with $f(x+y) = f(x) + f(y)$ for all real $x, y$ and $f(1) = 3$. Then $f(5)$ equals:

Accepted Answer

Cauchy's equation with continuity gives $f(x) = cx$ for some constant $c$. Since $f(1) = 3$, $c = 3$, so $f(5) = 15$.

Answer

$15$

Answer

$8$

Answer

$243$

Answer

$5$

Problems & worked solutions

Problem 1 — Trigonometry

Problem 2 — Logic & Induction

Problem 3 — Linear Function

Problem 4 — Distances & Areas

Problem 5 — Probability

Problem 6 — Inverse Trigonometric Functions

Problem 7 — Binomial Theorem

Problem 8 — Inverse Trigonometric Functions

Problem 9 — Homomorphisms

Problem 10 — Algebra