Number Theory
List all prime numbers from 1 to 50.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Solve the equation \[ 2\cos x + 3\sin x = 0 \] for \[ 0 \le x \le 360^\circ. \]
\[ x = 146.3^\circ,\; 326.3^\circ \]
A die is rolled twice. What is the probability of landing a 6 twice in a row?
1/36
Find the integral of \( (1 + 2x)^7 \).
\[ \frac{(1 + 2x)^8}{16} + {C} \]
Find the value of x at the point of intersection of:
5x + 3y = 1, 3y = 21
-4
Find a pair of twin primes (that differ by 2) between 50 and 80.
(59 & 61) or (71 & 73)
The angles \( \alpha \) and \( \beta \) lie between \(0^\circ\) and \(180^\circ\) and are such that \[ \tan(\alpha + \beta) = 2 \quad \text{and} \quad \tan \alpha = 3 \tan \beta. \] Find the possible values of \( \alpha \) and \( \beta \).
𝛼= 45 𝑎𝑛𝑑 𝛽 = 18.4, 𝛼= 108.4 𝑎𝑛𝑑 𝛽 = 135
A team of four has to be selected from 6 boys and 4 girls. How many different ways can a team be selected if at least one boy must be there?
209
Evaluate the integral:
\[ \int x(1 + x^2)^2 \, dx \]
\[ \frac{x^2}{2} + \frac{x^4}{2} + \frac{x^6}{6} + C \]
What is the minimum possible value of the function
f(x) = 3(2x + 5)2 + 2
2
What is the units digit of 3160?
1
Express \[ \cos(4\theta) - 4\cos(2\theta) \] in terms of \[ \sin \theta. \]
\[ 8\sin^4(\theta) - 3 \]
There are three identical boxes each containing two coins.
• Box A: 2 gold coins
• Box B: 2 silver coins
• Box C: 1 gold and 1 silver coin
A box is chosen at random and one coin is drawn. It is gold. What is the probability that the other coin in that box is also gold?
2/3
Evaluate the following integral:
\[ \int_{0}^{\pi/12} \sec(2x)\, dx \]
\[ \frac{\ln(3)}{4} \]
Solve the following system of equations:
\[ \begin{cases} a + b + c = 12 \\ -3a - b + c = -2 \\ 5a - b - c = 0 \end{cases} \]
Find the values of \(a\), \(b\), and \(c\).
(a,b,c) = (2,3,7)
Convert 13 (written in base 10) to base 3.
111
Solve, for \[ 0 \le \theta \le 2\pi, \] the equation \[ \sin 2\theta = 1 + \cos \theta. \]
\[ \theta = \pi,\; \frac{\pi}{2},\; \frac{3\pi}{2} \]
You have a 10 × 10 grid of real numbers. Each number equals the average of its neighbors (up, down, left, right). Edge and corner cells are the average of their existing neighbors. If all four corners are 0, what is the maximum possible sum of all entries?
0
What is the area bounded between the curves
\( y = \sin^2 x \) and \( y = \cos^2 x \)
in the interval \( 0 < x < \frac{3\pi}{4} \)?
3 units2
Find x
\[ \sqrt{x + 2\sqrt{x - 1}} + \sqrt{x - 2\sqrt{x - 1}} = 4 \]
x = 5
Find 72025 mod 13.
8
Solve, for \[ -90^\circ < x < 90^\circ, \] giving answers correct to 1 decimal place, \[ \tan(3x + 20^\circ) = \frac{3}{2}. \]
\[ x = -47.9^\circ,\; 12.1^\circ,\; 72.1^\circ \]
Mr. and Mrs. Zeta want the baby’s monogram (first, middle, last initial) to be in alphabetical order with no repeated letters. How many such monograms are possible?
300
Find the derivative of \[ \sin\!\big(\cos(\ln x)\big). \]
\[ \frac{\sin(\ln x)\,\cos(\cos(\ln x))}{x} \]
Find all ordered pairs \((x, y)\) that satisfy both equations:
\[ \begin{cases} x^2 + xy = 9, \\ y^2 + xy = 16. \end{cases} \]
(1.8, 3.2) & (-1.8, -3.2)