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0 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Compute the value of $\tan 15^\circ$.<|eot_id|><|start_header_id|>assistant<|end_header... |
1 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a right triangle with side lengths $2\sqrt3$, $5$, and $\sqrt{37}$, an equilatera... |
2 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a matrix $\mathbf{P}$ that projects vectors onto a line $\ell$ passing through th... |
3 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a matrix equation:$\begin{pmatrix} 2 & 3 & -1 \\ 0 & 4 & 5 \\ 4 & 0 & -2 \end{pma... |
4 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a vector $\begin{pmatrix} a \\ 7 \end{pmatrix}$ and another vector $\begin{pmatri... |
5 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a vector $\begin{pmatrix} 2 \\ y \\ -5 \end{pmatrix}$ and its projection onto ano... |
6 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
In triangle $ABC$, $AB = AC = 10$ and $BC = 12$. Point $D$ lies strictly between $A$ an... |
7 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a point $A = (1,8,4)$ and a line passing through points $B = (0,-1,3)$ and $C = (... |
8 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a set of vectors $\mathbf{v}$ such that the projection of $\mathbf{v}$ onto the v... |
9 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find $\begin{pmatrix} 1 \\ 4 \\ -6 \end{pmatrix} + \begin{pmatrix} 2 \\ -1 \\ 3 \end{pm... |
10 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find all values of $x$, where $0 \le x \le 2 \pi$, that satisfy the equation:\[\sin^2 x... |
11 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find all values of $x$, where $0 \le x \le 2 \pi$, that satisfy the equation:\[\sin^2 x... |
12 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given:- Points $A = (0 ,0 ,0)$, $B = (1, 0, 0)$, $C = (0, 2, 0)$, and $D = (0, 0, 3)$.-... |
13 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find the value of $\cos^{-1} \frac{1}{2}$, and express the answer in radians.<|eot_id|>... |
14 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given:- Angle $x$ lies in the third quadrant- $\cos x = -\frac{20}{29}$Find the value o... |
15 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a line parameterized by the vector equation:\[\mathbf{v} = \begin{pmatrix} 1 + t ... |
16 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
For real numbers $t \neq 0,$ the point\[(x,y) = \left( t + \frac{1}{t}, t - \frac{1}{t}... |
17 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Consider a cube of cheese defined by the set:\[C = \{(x,y,z) : 0 \le x, y, z \le 1\}\]T... |
18 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Consider a cube of cheese defined by the set:\[C = \{(x,y,z) : 0 \le x, y, z \le 1\}\]T... |
19 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $\theta$ be an angle such that $\tan \theta = 2$. Simplify the following expression... |
20 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Simplify the following expression:\[\sin 70^\circ \cos 50^\circ + \sin 260^\circ \cos 2... |
21 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find the $2 \times 2$ matrix $\mathbf{M}$ such that\[\mathbf{M} \begin{pmatrix} 1 \\ 0 ... |
22 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
In triangle $ABC$, $\angle C$ is a right angle and $CB > CA$. Point $D$ is located on $... |
23 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
In triangle $ABC$, $\angle C$ is a right angle and $CB > CA$. Point $D$ is located on $... |
24 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given positive real numbers $x$, $y$, and $z$, none of which are equal to 1, find the d... |
25 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find the imaginary part of the complex number $(\cos12^\circ+i\sin12^\circ+\cos48^\circ... |
26 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Compute the value of the following expression:\[ \frac {\sum_{n = 1}^{44} \cos n^\circ}... |
27 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find the smallest positive integer $n$ such that the complex number $(\cos 84^\circ + i... |
28 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given two vectors $\mathbf{a}$ and $\mathbf{b}$ with magnitudes $\|\mathbf{a}\| = 3$ an... |
29 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given two spheres in 3D space:- Sphere 1: Centered at $(3,-5,7)$ with radius $5 \sqrt{5... |
30 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a reflection that takes the vector $\begin{pmatrix} 5 \\ 0 \end{pmatrix}$ to $\be... |
31 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given two matrices $\begin{pmatrix} 1 & 5 \\ -2 & 4 \end{pmatrix}$ and $\begin{pmatrix}... |
32 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $a = e^{4 \pi i/5}$ and $b = e^{2 \pi i/17}$. Compute the value of the product:$$(a... |
33 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Consider the following three planes in 3D space:\begin{align*}-x + cy + bz &= 0 \\cx - ... |
34 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given that\begin{align*}\cos x + \cos y + \cos z &= 0, \\\sin x + \sin y + \sin z &= 0,... |
35 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a 3x3 matrix with elements $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$, and $i$, such ... |
36 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a 3x3 matrix with elements $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$, and $i$, such ... |
37 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find the cubic polynomial in $x$ with integer coefficients that has $\cos 20^\circ$ as ... |
38 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Simplify the expression $\sin 17^\circ + \sin 43^\circ$.The answer should be entered as... |
39 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $\ell$ be the line in space through the points $(0,0,1)$ and $(1,1,1)$. The goal is... |
40 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
If $\det \mathbf{A} = 4$ and $\det \mathbf{B} = -5,$ then find $\det (\mathbf{A} \mathb... |
41 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $a$ and $b$ be two 12th roots of unity, not necessarily distinct. The problem is to... |
42 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $a$ and $b$ be two 12th roots of unity, not necessarily distinct. The problem is to... |
43 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given two vectors $\mathbf{a}$ and $\mathbf{b}$ with the following properties:- $\|\mat... |
44 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
A graph is defined in polar coordinates by the equation:$r = \cos \theta + \frac{1}{2}$... |
45 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
The problem asks to find the least period of the composite function $\cos(\sin(x))$, gi... |
46 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
The parametric curve $(x,y) = (\sin t, \sin 2t)$ is plotted below:[asy]unitsize(2 cm);p... |
47 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given two complex numbers $z$ and $w$, where $|z| = 5$ and $|w| = 2$, find the minimum ... |
48 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
For real numbers $t$, the point $(x,y)$ is plotted according to the following equations... |
49 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find the distance from the point $P(2,1,0)$ to the plane $2x + y + 2z + 5 = 0$.<|eot_id... |
50 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given that $\tan^{-1} x + \tan^{-1} y = \frac{\pi}{4}$, compute the value of $xy + x + ... |
51 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Consider a curve parameterized by the following equations:\begin{align*}x &= t^3 + 7 \\... |
52 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Compute the sum of the following two matrices:$\begin{pmatrix} 5 \\ -4 \end{pmatrix} + ... |
53 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given two vectors $\mathbf{a}$ and $\mathbf{b}$ with the following properties:- $\|\mat... |
54 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Solve the equation:\[2 \sin \theta (\sin 2 \theta + \sin 4 \theta + \sin 6 \theta + \do... |
55 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given real numbers $a$ and $b$, for every positive number $x$, the following identity h... |
56 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Evaluate the following determinant:\[\begin{vmatrix} x & \sin \theta & \cos \theta \\ -... |
57 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find the degree measure of the least positive angle $\theta$ for which\[\tan \theta = \... |
58 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find the curve defined by the equation\[r^2 \cos 2 \theta = 4.\]The options are:(A) Lin... |
59 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Consider the system of linear equations:\begin{align*}x + ky - z &= 0, \\kx - y - z &= ... |
60 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $\mathbf{a}$ be a unit vector. The problem is to compute the sum of the squares of ... |
61 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
A translation of the plane takes the complex number $5 + 5i$ to $6 + 8i$. The problem i... |
62 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
In the right triangle $FGH$, where $\angle H = 90^\circ$, the lengths of the sides are ... |
63 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given the polar coordinates of three points $A$, $B$, and $C$ as follows:- $A: (2,70^\c... |
64 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
A particle moves along a circular path defined by the parametric equations:\begin{align... |
65 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a $2 \times 2$ matrix $\mathbf{A}$ with $\det \mathbf{A} = 2$, find the determina... |
66 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given:- Points $O$, $A$, and $B$ satisfy $\overrightarrow{OA} = \begin{pmatrix} 1 \\ 2 ... |
67 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
The problem provides a graph of the function $y = a \csc bx$ for some positive constant... |
68 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Consider an equiangular hexagon $ABCDEF$ with side lengths $AB=CD=EF=1$ and $BC=DE=FA=r... |
69 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a unit cube with two opposite corners labeled as $A$ and $B$, and a third vertex ... |
70 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given the vertices of a tetrahedron $A = (0,1,2),$ $B = (3,0,1),$ $C = (4,3,6),$ and $D... |
71 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given two vectors $\mathbf{a}$ and $\mathbf{b}$ with $\|\mathbf{a}\| = 3$ and $\|\mathb... |
72 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given the equation:\[\frac{\sin^2 3A}{\sin^2 A} - \frac{\cos^2 3A}{\cos^2 A} = 2,\]find... |
73 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $\mathbf{M}$ be a $2 \times 2$ matrix with real entries:\[\mathbf{M} = \begin{pmatr... |
74 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
In triangle $ABC$, the tangent of angle $CAB$ is given as $\frac{22}{7}$. The altitude ... |
75 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Compute the value of $\arccos (-1)$, and express the answer in radians.<|eot_id|><|star... |
76 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Evaluate the expression:$\sin(1998^\circ+237^\circ)\sin(1998^\circ-1653^\circ)$<|eot_id... |
77 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $f(x) = (\arccos x)^2 + (\arcsin x)^2$, where all functions are in radians. The tas... |
78 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given two right triangles with a common hypotenuse of length 25, find the sum of the ba... |
79 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a vector $\mathbf{v}$, find the matrix $\mathbf{P}$ such that $\mathbf{P} \mathbf... |
80 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a complex number $4 - \sqrt{3} + (-1 - 4 \sqrt{3})i$, find the resulting complex ... |
81 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
In a three-dimensional coordinate system with origin $O$, points $A$, $B$, and $C$ are ... |
82 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find the integer $n$ such that $-90 \le n \le 90$ and $\sin n^\circ = \sin 1021^\circ$.... |
83 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Compute the value of the following expression:\[\tan 20^\circ + \tan 40^\circ + \sqrt{3... |
84 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $\bold{a}$ and $\bold{b}$ be three-dimensional vectors. The problem states that:\[(... |
85 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $\mathbf{M} = \begin{pmatrix} 1 & -4 \\ 1 & 2 \end{pmatrix}.$ Find constants $a$ a... |
86 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Compute the value of $\tan 22.5^\circ$.<|eot_id|><|start_header_id|>assistant<|end_head... |
87 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Compute the value of $\tan 22.5^\circ$.<|eot_id|><|start_header_id|>assistant<|end_head... |
88 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given two lines parameterized by the following equations:Line 1: $\begin{pmatrix} 2 - 3... |
89 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a $2 \times 2$ matrix $\mathbf{M}$, find the values of its entries such that:$$\m... |
90 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a $2 \times 2$ matrix $\mathbf{M}$, find the values of its entries such that:$$\m... |
91 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find the matrix that corresponds to rotating about the origin by an angle of $45^\circ$... |
92 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a point with rectangular coordinates $(2,-1,-2)$ and spherical coordinates $(\rho... |
93 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find all real numbers $x$ between 0 and 360 such that:$$\sqrt{3} \cos 10^\circ = \cos 4... |
94 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find all the solutions to the equation:\[\arctan \frac{1}{x} + \arctan \frac{1}{x + 2} ... |
95 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find the matrix $\mathbf{M}$ with real entries such that:$$\mathbf{M}^3 = \begin{pmatri... |
96 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $ABC$ be a triangle with $\angle A = 45^\circ$. Let $P$ be a point on side $\overli... |
97 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given that $\tan \theta = 7$, find the value of $\tan 2 \theta$.<|eot_id|><|start_heade... |
98 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
A line is parameterized by the following equation:\[\begin{pmatrix} x \\ y \end{pmatrix... |
99 | <|begin_of_text|><|start_header_id|>system<|end_header_id|>You are a expert mathematician, solve the problems accordingly
using relevant mathematical techniques<|eot_id|><|begin_of_text|><|start_header_id|>user<|end_header_id|>
Given a point $P=(1,2,3)$, the problem is to find the coordinates of the point $S$ afte... |
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