Daily Math · 2026-07-25

Question 1

For the polynomial $P(X) = 2X^3 - 6X^2 + X - 4$ with roots $x_1, x_2, x_3$, the product $x_1 x_2 x_3$ equals:

Accepted Answer

$x_1 x_2 x_3 = -\dfrac{-4}{2} = 2$.

Answer

$-4$

Answer

$-2$

Answer

$2$

Answer

$4$

Question 2

A menu offers $4$ appetizers, $5$ mains, and $3$ desserts. The number of distinct three-course meals is:

Accepted Answer

$4 \cdot 5 \cdot 3 = 60$.

Answer

$12$

Answer

$36$

Answer

$60$

Answer

$120$

Question 3

The volume generated by rotating $f(x) = 2$ on $[0, 5]$ about the $x$-axis is:

Accepted Answer

Cylinder of radius $2$ and height $5$: $V = \pi \cdot 4 \cdot 5 = 20\pi$.

Answer

$5\pi$

Answer

$10\pi$

Answer

$20\pi$

Answer

$40\pi$

Question 4

Find the $3	imes3$ determinant whose entries are all limit values (see problem image).

Accepted Answer

$\bullet$ Evaluate each of the nine limits to fill the matrix:

$$\begin{pmatrix}\sqrt{3}/2 & 2 & 1\ 3 & 4 & 4\ 0 & 3 & 1\end{pmatrix}$$

$\bullet$ Expand along row 1. The cofactor of the $(1,1)$ entry is $4\cdot 1-4\cdot 3=-8$:

$$	frac{\sqrt{3}}{2}\cdot(-8)=-4\sqrt{3}$$

$\bullet$ Cofactors of the $(1,2)$ and $(1,3)$ entries are $3\cdot 1-4\cdot 0=3$ and $3\cdot 3-4\cdot 0=9$:

$$-2\cdot 3+1\cdot 9=3$$

$\bullet$ Add the three contributions:

$$\det=3-4\sqrt{3}$$

Answer

$3-4\sqrt{3}$

Answer

$0$

Answer

$3+4\sqrt{3}$

Answer

$12$

Answer

$-4\sqrt3$

Answer

$3$

Question 5

Using Pascal's identity $\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}$, compute $\binom{7}{3} + \binom{7}{4}$.

Accepted Answer

By Pascal: $\binom{7}{3} + \binom{7}{4} = \binom{8}{4} = \dfrac{8!}{4!\,4!} = 70$.

Answer

$35$

Answer

$56$

Answer

$70$

Answer

$128$

Question 6

Let $f: \mathbb{R} 	o \mathbb{R}$ be defined by $f(x) = \begin{cases} x + 1, & x < 0 \ 2x + 1, & x \ge 0 \end{cases}$. Which statement is correct?

Accepted Answer

**Left branch** ($x < 0$): $f(x) = x + 1 \in (-\infty,\, 1)$, strictly increasing.
**Right branch** ($x \ge 0$): $f(x) = 2x + 1 \in [1,\, +\infty)$, strictly increasing.

The two images $(-\infty, 1)$ and $[1, +\infty)$ are disjoint and together cover all of $\mathbb{R}$, so $f$ is **surjective**. Each branch is strictly increasing, and the ranges don't overlap, so no two distinct $x$-values give the same $f(x)$ — $f$ is **injective**. Hence $f$ is bijective.

(Note: $f(-1) = 0$ and $f(0) = 1$, so distractor A is factually false.)

Answer

$f$ is not injective because $f(-1) = f(0)$

Answer

$f$ is injective but not surjective

Answer

$f$ is surjective but not injective

Answer

$f$ is bijective

Question 7

The solution of $\sqrt{x + 5} - \sqrt{x - 3} = 2$ is:

Accepted Answer

$\sqrt{x + 5} = 2 + \sqrt{x - 3}$. Square: $x + 5 = 4 + 4\sqrt{x - 3} + x - 3 \Rightarrow 4 = 4\sqrt{x - 3} \Rightarrow \sqrt{x - 3} = 1 \Rightarrow x = 4$. Check: $\sqrt{9} - \sqrt{1} = 3 - 1 = 2$ ✓.

Answer

$3$

Answer

$\dfrac{7}{2}$

Answer

$4$

Answer

$5$

Question 8

A square matrix $A$ is invertible iff:

Accepted Answer

Invertibility ⇔ non-zero determinant.

Answer

$\det(A) = 0$

Answer

$A$ is symmetric

Answer

$\det(A) 
e 0$

Answer

all entries are non-zero

Question 9

How many subsets of $\{1, 2, 3, \dots, 10\}$ have an even sum (the empty subset counts, with sum $0$)?

Accepted Answer

Pair each subset $S$ with $S 	riangle \{1\}$ (i.e., toggle the element $1$). This pairing has no fixed points and flips the parity of the sum, so even-sum and odd-sum subsets are equinumerous: $2^{10}/2 = 512$.

Answer

$512$

Answer

$511$

Answer

$256$

Answer

$1024$

Question 10

Given $A=\begin{pmatrix}2&-1&1&-1\1&1&a&1\1&-1&1&b\end{pmatrix}$ with $	ext{rank}(A)=2$, find $(a,b)$.

Accepted Answer

$\bullet$ Eliminate the first column using $2R_2-R_1$ and $2R_3-R_1$:

$$2R_2-R_1=(0,3,2a-1,3)$$

$$2R_3-R_1=(0,-1,1,2b+1)$$

$\bullet$ For rank $2$, these two rows must be proportional. Matching the second components gives ratio $-3$:

$$2a-1=-3\cdot 1\ \Rightarrow\ a=-1$$

$$3=-3(2b+1)\ \Rightarrow\ b=-1$$

$\bullet$ So the answer is:

$$(a,b)=(-1,-1)$$

Answer

$(1,0)$

Answer

$(-1,-1)$

Answer

$(0,1)$

Answer

$(2,-2)$

Answer

$(1,1)$

Answer

$(-1,3)$

Problems & worked solutions

Problem 1 — Polynomials in ℂ

Problem 2 — Permutations & Combinations

Problem 3 — Volumes of Revolution

Problem 4 — Matrices

Problem 5 — Permutations & Combinations

Problem 6 — Functions — General Properties

Problem 7 — Elementary Functions

Problem 8 — Determinants

Problem 9 — Combinatorics

Problem 10 — Matrices