sample_index int64 17 251k | problem large_stringlengths 14 1.29k | ground_truth large_stringlengths 1 269 | partial_solution large_stringlengths 62 41k | empirical_success_rate float64 0 1 | chop_fraction float64 0.2 0.8 | difficulty_bucket int64 0 4 | llama8b_solve_rate float64 0 1 | num_continuations_intervened int64 0 0 |
|---|---|---|---|---|---|---|---|---|
17 | The remainder of $3^{20}$ divided by 5 is __________. | 1 | Okay, let's see. I need to find the remainder when 3^20 is divided by 5. Hmm, modular arithmetic, right? So 3^20 mod 5. I remember that there's something called Fermat's little theorem, but maybe I can do it with patterns of remainders.
First, let's list | 1 | 0.2 | 4 | 1 | 0 |
54 | If the equation $\frac{x^{2}}{k-3} + \frac{y^{2}}{k+3} = 1$ represents a hyperbola with foci on the $y$-axis, then the range of the real number $k$ is $\_\_\_\_\_\_\_.$ | (-3, 3) | Okay, let's try to solve this problem. The question is about finding the range of k such that the equation (x²)/(k-3) + (y²)/(k+3) = 1 represents a hyperbola with foci on the y-axis. Hmm, first, I need to remember the standard forms of hyperbolas.
Wait, hyperbolas have two forms: one where the x-term is positive and ... | 1 | 0.2 | 1 | 0.203125 | 0 |
123 | Given sets $A = \{x\,|\,x < -1 \text{ or } 2 \leq x < 3\}$ and $B = \{x\,|\,-2 \leq x < 4\}$, find the union $A \cup B$. | \\{x | x < 4\\} | Okay, let's try to solve this problem. So, we have two sets A and B, and we need to find their union. First, I need to remember what the union of two sets means. The union A ∪ B is all the elements that are in A, in B, or in both. So, I should probably write out what each set looks like on a number line and then see wh... | 0 | 0.2 | 2 | 0.421875 | 0 |
136 | Given $x, y \in R^{+}$, and they satisfy the equation $\frac{x}{3} + \frac{y}{4} = 1$, find the maximum value of $xy$. | 3 | Okay, let's see. I need to find the maximum value of xy given that x and y are positive real numbers and they satisfy x/3 + y/4 = 1. Hmm, so this is a constrained optimization problem. I remember that for these kinds of problems, you can use substitution or maybe the AM-GM inequality. Let me try substitution first beca... | 1 | 0.2 | 1 | 0.234375 | 0 |
174 | Evaluate: $$\frac {1}{2- \sqrt {3}}$$\-π<sup>0</sup>\-2cos30°= \_\_\_\_\_\_ ． | 1 | Okay, let's try to solve this problem. Hmm, the problem is to evaluate 1 divided by (2 minus the square root of 3) minus pi to the power of 0 minus 2 times cosine of 30 degrees. Alright, let's break this down step by step.
First, I need to handle each part separately. Let's start with the first term: 1/(2 - √3). I rem... | 1 | 0.2 | 1 | 0.21875 | 0 |
345 | Given the function $f(x) = x^2 + |x - 2|$, find the value of $f(1)$. | 2 | Okay, let's see. I need to find f(1) for the function f(x) = x² + |x - 2|. Hmm, absolute value functions can sometimes be tricky because they depend on whether the inside is positive or negative. But since I just need to plu | 1 | 0.2 | 4 | 0.984375 | 0 |
352 | Use the method of successive subtraction to find the greatest common divisor of 459 and 357. | 51 | Okay, let's see. I need to find the greatest common divisor (GCD) of 459 and 357 using the method of successive subtraction. Hmm, I remember that the method of successive subtraction involves subtracting the smaller number from the larger one repeatedly until both numbers become equal. That common value is the GCD. Let... | 1 | 0.2 | 3 | 0.796875 | 0 |
644 | Calculate: ${(-2)}^{3}-{(2002-\pi)}^{0}+{(-\frac{1}{3})}^{-2}-|-5|$. | -5 | Okay, let's see. I need to calculate this expression: (-2)³ - (2002 - π)⁰ + (-1/3)⁻² - |-5|. Hmm, I should break it down term by term. Let's start with each part.
First term: (-2)³. That's negative 2 cubed. So, (-2) * (-2) * (-2). Let's compute that. (-2) squared is 4, then times (-2) is -8. So the first term is -8.
... | 1 | 0.2 | 3 | 0.796875 | 0 |
705 | Given the function $f(x)=ax^{3}+3x^{2}+2$, determine the value of $a$ if ${f}'(-1)=-12$. | -2 | Okay, let's see. I need to find the value of a for the function f(x) = ax³ + 3x² + 2, given that the derivative at x = -1 is -12. Hmm, right. First, I remember that to find the derivative of a function, I need to apply the power rule. The power rule says t | 1 | 0.2 | 3 | 0.71875 | 0 |
744 | If the speed of an airplane flying against the wind is $x$ kilometers per hour, and the wind speed is $y$ kilometers per hour, then the speed of the airplane flying with the wind is ______ kilometers per hour. | x + y | Okay, let's try to figure out this problem. So the question is about the speed of an airplane when it's flying with the wind, given that its speed against the wind is x km/h and the wind speed is y km/h. Hmm.
First, I need to remember how wind affects an airplane's speed. When the plane is flying against the wind, the... | 0 | 0.2 | 2 | 0.515625 | 0 |
756 | If $a$, $b$, and $c$ are the sides of a right triangle, with $c$ being the hypotenuse, then $a^{2}+b^{2}=c^{2}$, which is known as the Pythagorean theorem. Now, let's extend this theorem to solid geometry: In the tetrahedron $O-ABC$, where $\angle AOB=\angle BOC=\angle COA=90^{\circ}$, $S$ is the area of the face oppos... | S^{2}= S_{1}^{2}+ S_{2}^{2}+ S_{3}^{2} | Okay, let's try to solve this problem. First, I need to recall what a tetrahedron is. A tetrahedron has four triangular faces, and in this case, O is one vertex, and A, B, C are the other three. The problem says that the angles at O are all 90 degrees, so angle AOB, BOC, and COA are all right angles. That means OA, OB,... | 1 | 0.2 | 1 | 0.34375 | 0 |
789 | Evaluate the integral $\int_{1}^{2} (x- \frac{1}{x})dx$. | 1-\ln 2 | Okay, let's see. I need to evaluate the integral from 1 to 2 of (x minus 1 over x) dx. Hmm, integrals. I remember that to integrate a function, you find the antiderivative and then plug in the limits. So first, I should find the antiderivative of x minus 1/x.
Let me recall the basic integration rul | 0 | 0.2 | 1 | 0.28125 | 0 |
880 | Determine the equation of the tangent line to the function $f(x) = x^3 + 1$ at point $P(1, m)$. | 3x - y - 1 = 0 | Okay, let's see. I need to find the equation of the tangent line to the function f(x) = x³ + 1 at the point P(1, m). Hmm, first, I should figure out what m is because the point is given as (1, m). Since the function is f(x) = x³ + 1, when x is 1, f(1) should be 1³ + 1, which is 2. So m | 1 | 0.2 | 4 | 0.9375 | 0 |
947 | Given the set $A={x|x > 5}$ and set $B={x|x > a}$, if the proposition "$x\in A$" is a sufficient but not necessary condition for the proposition "$x\in B$", then the range of values for the real number $a$ is _______. | a < 5 | Okay, let's try to figure out this problem. So, we have two sets: A is all x greater than 5, and B is all x greater than a. The question says that "x is in A" is a sufficient but not necessary condition for "x is in B". We need to find the range of a.
First, I need to remember what sufficient and necessary conditions ... | 1 | 0.2 | 1 | 0.296875 | 0 |
1,028 | Given vectors $\overrightarrow{a}=(x,3)$ and $\overrightarrow{b}=(-1,y-1)$, and $\overrightarrow{a}+2\overrightarrow{b}=(0,1)$, find the value of $|\overrightarrow{a}+\overrightarrow{b}|$. | \sqrt{5} | Okay, let's see. I need to solve this problem where I have two vectors, a and b. Vector a is (x, 3) and vector b is (-1, y - 1). Then they say that a plus 2b equals (0, 1). I need to find the magnitude of a plus b.
First, I remember that when you add vectors, you add their corresponding components. So | 1 | 0.2 | 2 | 0.546875 | 0 |
1,084 | Let complex number $z$ satisfy $(1+i)z=-3+i$ (where $i$ is the imaginary unit), then $|z|=$_______. | \sqrt{5} | Okay, let's see. I need to find the modulus of the complex number z that satisfies (1 + i)z = -3 + i. Hmm, modulus... right, the modulus of a product is the product of the moduli. So maybe I can find |z| without even finding z itself? Let me think.
First, the equation is (1 + i)z = -3 + i. If I take the modulus of bot... | 1 | 0.2 | 3 | 0.71875 | 0 |
1,101 | Given: $m^{2}=n+2$, $n^{2}=m+2$ ($m \neq n$), find the value of $m^{3}-2mn+n^{3}$. | -2 | Okay, let's try to solve this problem. So we have two equations: m squared equals n plus 2, and n squared equals m plus 2, and m is not equal to n. We need to find the value of m cubed minus 2mn plus n cubed. Hmm, that sounds a bit complicated, but maybe we can use the given equations to simplify it.
First, I remember... | 1 | 0.2 | 0 | 0.03125 | 0 |
1,155 | Given the matrix $$M= \begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix}$$, find the equation of the line obtained by applying the transformation corresponding to $M$ to the line $x+y+2=0$. | x+2y+2=0 | Okay, let's see. I need to find the equation of the line after applying the matrix M to the line x + y + 2 = 0. Hmm, first, I remember that when you apply a linear transformation to a line, the new line can be found by taking a general point on the original line, transforming it with M, and then finding the equation of... | 0 | 0.2 | 1 | 0.234375 | 0 |
1,196 | Calculate: $(π-3.14)^0+|-\sqrt{3}|-(\frac{1}{2})^{-1}-sin60°$. | -1 + \frac{\sqrt{3}}{2} | Okay, let's try to solve this problem step by step. The question is to calculate: (π - 3.14)^0 + | -√3 | - (1/2)^(-1) - sin60°. Hmm, I need to remember the rules for exponents, absolute values, and trigonometric functions. Let's break it down term by term.
First term: (π - 3.14)^0. Any non-zero number raised to the 0 ... | 0.9 | 0.2 | 4 | 0.84375 | 0 |
1,227 | Observe the number array below, the 20th number in the 20th row is ___.
1
2 3 4
5 6 7 8 9
11 12 13 14 15 16
18 19 20 21 22 23 24 25
… … … … … … … … … | 381 | Okay, let's try to figure out this problem. So the question is asking for the 20th number in the 20th row of this number array. Hmm, first I need to understand how the array is structured.
Looking at the given numbers:
Row 1: 1
Row 2: 2, 3, 4
Row 3: 5, 6, 7, 8, 9
Row 4: 11, 12, 13, 14, 15, 16
Row 5: 18, 19, 20, 2... | 0 | 0.2 | 0 | 0.03125 | 0 |
1,331 | If the one-variable quadratic equation $\left(k-1\right)x^{2}+4x+1=0$ has two distinct real roots with respect to $x$, then the range of values for $k$ is ____. | k < 5 \text{ and } k \neq 1 | Okay, let's see. I need to find the range of values for k such that the quadratic equation (k - 1)x² + 4x + 1 = 0 has two distinct real roots. Hmm, right. For a quadratic equation to have two distinct real roots, the discriminant must be positive. But first, I should remember that it's a quadratic equation, so the coef... | 0 | 0.2 | 1 | 0.203125 | 0 |
1,334 | Given that function $f(x)$ is an even function defined on $R$, $f(2)=0$ when $x > 0$, and $\frac{{xf}{{{{'}}}}(x)-f(x)}{x^{2}}{ < }0$. Find the solution set of the inequality ${xf}(x){ < }0$. | \{x|-2 < x < 0\; or\; x > 2\} | Okay, let's try to solve this problem step by step. First, let's recall what's given. The function f(x) is even, so f(-x) = f(x) for all x. Also, f(2) = 0 when x > 0. Wait, no, that might be a typo. It says "f(2) = 0 when x > 0", but f(2) is a specific value, so maybe it's "f(2) = 0" (since x=2 is positive). That makes... | 0 | 0.2 | 0 | 0.03125 | 0 |
1,338 | Given the set $A=\{x\in\mathbb{R} \,|\, |x-55|\leqslant \frac{11}{2}\}$, the largest integer in set $A$ is ______. | 60 | Okay, let's see. I need to find the largest integer in set A, which is defined as all real numbers x such that |x - 55| is less than or equal to 11/2. Hmm, absolute value inequalities. I remember that |x - a| ≤ b means | 1 | 0.2 | 2 | 0.59375 | 0 |
1,373 | Given the function $f(x) = \begin{cases} (a-3)x - 3, & x \leqslant 1 \\ \log_{a}^{x}, & x > 1 \end{cases}$, if $f(x)$ is monotonically increasing on $\mathbb{R}$, then the range of the real number $a$ is $\_\_\_\_\_\_\_.$ | 3 < a \leqslant 6 | Okay, let's try to figure out this problem. So, we have a piecewise function f(x) that's defined as (a-3)x - 3 when x is less than or equal to 1, and log base a of x when x is greater than 1. The question is asking for the range of real numbers a such that f(x) is monotonically increasing on all real numbers. Hmm, mono... | 0.9 | 0.2 | 0 | 0.015625 | 0 |
1,378 | (The full score of this question is 14 points)
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively. The equation $b(x^2 + 1) + c(x^2 - 1) - 2ax = 0$ has two equal real roots, and $\sin C \cos A - \cos C \sin A = 0$. Determine the shape of $\triangle ABC$. | isosceles right-angled triangle | Okay, let's try to solve this problem step by step. First, let's list out what we know.
We have a triangle ABC with sides a, b, c opposite to angles A, B, C respectively. The equation given is \( b(x^2 + 1) + c(x^2 - 1) - 2ax = 0 \), which has two equal real roots. Also, we have \( \sin C \cos A - \cos C \sin A = 0 \... | 0 | 0.2 | 0 | 0.046875 | 0 |
1,438 | Let the universal set $U=\mathbb{R}$, $A=\{x|x<1\}$, and $B=\{x|x<0\}$. Then, the complement of $A\cup B$ in $U$ is. | \{x|x\geq1\} | Okay, let's try to figure out this problem. So, the question is about finding the complement of A union B in the universal set U, which is all real numbers.
First, I need to remember what A and B are. A is the set of all real numbers less than 1, so A = {x | x < 1}. B is the set of all real numbers less than 0, so B ... | 0 | 0.2 | 3 | 0.71875 | 0 |
1,502 | Given a complex number $z=(3+i)^2$ (where $i$ is the imaginary unit), find the magnitude $|z|$. | 10 | Okay, let's see. I need to find the magnitude of z, which is (3 + i) squared. Hmm, first, maybe I should remember how to compute the magnitude of a complex number. I think the magnitude of a complex number a + bi is sqrt(a² + b²). But here z is (3 + i) s | 1 | 0.2 | 4 | 0.96875 | 0 |
1,534 | Given vectors $\overrightarrow {a}$=(-2, 1, 3) and $\overrightarrow {b}$=(-1, 2, 1), if $\overrightarrow {a}$ is orthogonal to ($\overrightarrow {a}$-$λ \overrightarrow {b}$), find the value of the real number $λ$. | 2 | Okay, let's try to solve this problem. Hmm, so we have two vectors, a and b. The problem says that vector a is orthogonal to (a - λb), and we need to find λ. Alright, first, what does it mean for two vectors to be orthogonal? I remember that if two vectors are orthogonal, their dot product is zero. So, that means the d... | 1 | 0.2 | 3 | 0.65625 | 0 |
1,598 | Given the parabola $y^{2}=2px\left(p \gt 0\right)$ with focus $F$, a line passing through point $F$ intersects the parabola at points $A$ and $B$, where $|{AB}|=5\sqrt{2}$, and the midpoint of $AB$ has a $y$-coordinate of $\sqrt{2}$. Find the value of $p$. | p=2\sqrt{2} \text{ or } p=\frac{\sqrt{2}}{2} | Okay, let's try to solve this problem step by step. First, I need to recall some properties of the parabola \( y^2 = 2px \) where \( p > 0 \). The focus \( F \) of this parabola should be at \( \left( \frac{p}{2}, 0 \right) \) because for a parabola \( y^2 = 4ax \), the focus is at \( (a, 0) \), so here \( 4a = 2p \) i... | 0.9 | 0.2 | 0 | 0.03125 | 0 |
1,710 | Point P is a point outside the plane of triangle ABC, with PO perpendicular to plane ABC at foot O. If PA, PB, and PC are pairwise perpendicular, then point O is the ______ of triangle ABC. | orthocenter | Okay, let's try to figure out this problem. So, we have a point P outside the plane of triangle ABC. PO is perpendicular to the plane ABC, with O being the foot of that perpendicular. Also, PA, PB, and PC are pairwise perpendicular. We need to find out what O is in triangle ABC.
Hmm. Pairwise perpendicular... So PA is... | 1 | 0.2 | 1 | 0.203125 | 0 |
1,711 | Given the functions $y=\cos x$ and $y=\sin(2x+\varphi)$ ($0 \leq \varphi < \pi$), their graphs intersect at a point where the x-coordinate is $\frac{\pi}{3}$. Find the value of $\varphi$. | \frac{\pi}{6} | Okay, let's try to solve this problem. So, the question is about finding the value of φ such that the graphs of y = cos x and y = sin(2x + φ) intersect at x = π/3. Hmm, right. When two graphs intersect at a point, their y-values must be equal at that x-coordinate. So, I need to set cos(π/3) equal to sin(2*(π/3) + φ) an... | 1 | 0.2 | 0 | 0.109375 | 0 |
1,784 | Find the standard equation of the hyperbola with the center at the origin, a real axis length of 10, and an imaginary axis length of 6. | B: \frac{x^2}{25} - \frac{y^2}{9} = 1 \text{ or } \frac{y^2}{25} - \frac{x^2}{9} = 1 | Okay, let's see. I need to find the standard equation of a hyperbola with center at the origin, real axis length 10, and imaginary axis length 6. Hmm, first, I should remember the standard form of a hyperbola. Since the center is at the origin, it's either horizontal or vertical. The problem mentions the real axis and ... | 0 | 0.2 | 0 | 0.03125 | 0 |
1,874 | From the scatter plot, the points are scattered from the bottom left corner to the top right corner. This kind of correlation between two variables is called __________. | positive correlation | Okay, let's see. The problem is about a scatter plot where the points are scattered from the bottom left to the top right. Hmm, I need to remember the types of correlation.
So, correlation can be positive or nega | 1 | 0.2 | 3 | 0.75 | 0 |
1,913 | The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$, with a non-zero common difference, is $S_n$. If $a_4$ is the geometric mean of $a_3$ and $a_7$, and $S_8=32$, then $S_{10}$ equals ______. | 60 | Okay, let's try to solve this problem step by step. First, let's recall what an arithmetic sequence is. An arithmetic sequence has a common difference, let's call it d (which is non-zero, as given). The nth term of an arithmetic sequence is given by a_n = a_1 + (n-1)d, where a_1 is the first term. The sum of the first ... | 1 | 0.2 | 0 | 0.125 | 0 |
1,938 | Given the function $f(x) = x^3 + a \log_3 x$, if $f(2) = 6$, find $f(\frac{1}{2})$. | \frac{17}{8} | Okay, let's try to solve this problem. So, the function is f(x) = x³ + a log base 3 of x. They tell us that f(2) = 6, and we need to find f(1/2). Hmm, first I need to figure out what a is, right? Because we don't know a yet. Let's start by plugging in x = 2 into the function.
So f(2) = (2)³ + a log₃(2). That should eq... | 1 | 0.2 | 3 | 0.640625 | 0 |
2,021 | Given that the positive term geometric sequence $\{a_{n}\}$ satisfies $\log_{2}a_{n+2} - \log_{2}a_{n} = 2$, and $a_{3} = 8$, find the sum of the first $n$ terms of the sequence, denoted as $S_{n}=$\_\_\_\_\_\_. | S_{n} = 2^{n+1} - 2 | Okay, let's try to solve this problem step by step. First, let's recall what a geometric sequence is. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio, usually denoted as r. So the nth term of a geometric sequence is a_n = a_1 * r^(n-1), ... | 1 | 0.2 | 2 | 0.46875 | 0 |
2,039 | Calculate the result of $a^{3}\div a^{2}$ is ______. | a | Okay, let's see. I need to figure out what a cubed divided by a squared is. Hmm, so the problem is a³ ÷ a². I remember from algebra th | 1 | 0.2 | 4 | 0.984375 | 0 |
2,085 | Calculate: $\frac{{x-2}}{{{x^2}-2x+1}}÷\frac{x}{{x-1}}+\frac{1}{{{x^2}-x}}$. | \frac{1}{x} | Okay, let's try to solve this problem. The question is to calculate: (x-2)/(x² - 2x + 1) divided by (x)/(x-1) plus 1/(x² - x). Hmm, first I need to remember how to divide fractions. Dividing by a fraction is the same as multiplying by its reciprocal, right? So maybe I can rewrite the division part as multiplication.
F... | 1 | 0.2 | 3 | 0.6875 | 0 |
2,104 | If the one-variable quadratic equation $x^{2}-3x-k=0$ has one root as $1$, then the value of $k$ is ____. | -2 | Okay, let's see. I need to find the value of k in the quadratic equation x² - 3x - k = 0, given that one of the roots is 1. Hmm, right. Since 1 is a root, that me | 1 | 0.2 | 3 | 0.6875 | 0 |
2,105 | Simplify first and then evaluate: $[\left(xy+2\right)\left(xy-2\right)-2x^{2}y^{2}+4]\div xy$, where $x=2$, $y=3$. | -6 | Okay, let's try to solve this problem step by step. First, the problem says to simplify first and then evaluate where x=2 and y=3. So I need to simplify the expression [ (xy + 2)(xy - 2) - 2x²y² + 4 ] divided by xy.
Hmm, let's start by looking at the part (xy + 2)(xy - 2). That looks like a difference of squares beca... | 1 | 0.2 | 2 | 0.5625 | 0 |
2,265 | Given the complex number $z$ that satisfies $z(1+i)=2-4i$, find $z=$ _____. | -1-3i | Okay, let's see. I need to find the complex number z that satisfies the equation z times (1 + i) equals 2 minus 4i. Hmm, so first, I remember that to solve for z, I should divide both sides by (1 + i). That makes sense because if z*(1 + i) = 2 - 4i, then z = (2 - 4i)/(1 + i).
Now, I need to simplify that fraction. To... | 1 | 0.2 | 4 | 0.8125 | 0 |
2,395 | Given $0<x<\frac{1}{3}$, the maximum value of $x(1-3x)$ is ______. | \frac{1}{12} | Okay, let's see. I need to find the maximum value of the function f(x) = x(1 - 3x) where 0 < x < 1/3. Hmm, first, maybe I should expand this to make it easier to handle. So f(x) = x - 3x². That's a quadratic function in terms of x. Quadratic functions have their maximum or minimum at the vertex | 1 | 0.2 | 2 | 0.546875 | 0 |
2,418 | An instrument consists of one part $A$ and two parts $B$. A factory has 50 workers, each worker can produce either 40 parts of $A$ or 120 parts of $B$. How should the workers be allocated to produce matching sets of parts $A$ and $B$ each day? | 30 \text{ workers produce part } A, \text{ and } 20 \text{ workers produce part } B. | Okay, let's try to figure out this problem. So, the question is about a factory with 50 workers. Each worker can make either 40 parts of A or 120 parts of B. They need to produce matching sets, which I guess means each set has one part A and two parts B. So, the goal is to have the right number of A and B parts so that... | 0 | 0.2 | 0 | 0.046875 | 0 |
2,561 | The distance between two parallel lines $x+y-1=0$ and $2x+2y+1=0$ is \_\_\_\_\_. | \frac{3\sqrt{2}}{4} | Okay, let's see. I need to find the distance between two parallel lines: x + y - 1 = 0 and 2x + 2y + 1 = 0. Hmm, first, I remember that the distance between two parallel lines can be calculated using a formula. But wait, the lines have to be in the same form, right? Like both should be in Ax + By + C1 = 0 and Ax + By +... | 1 | 0.2 | 2 | 0.53125 | 0 |
2,699 | Find the equation of the line that passes through the intersection point of the two lines $2x+y+2=0$ and $3x+4y-2=0$, and is perpendicular to the line $3x-2y+4=0$. | 2x+3y-2=0 | Okay, let's try to figure out this problem step by step. First, the problem says we need to find the equation of a line that passes through the intersection point of two given lines and is perpendicular to another given line.
So, first step: find the intersection point of 2x + y + 2 = 0 and 3x + 4y - 2 = 0. To find t... | 1 | 0.2 | 3 | 0.75 | 0 |
2,833 | Given that \\(z\\) is a complex number, both \\(z+2i\\) and \\( \dfrac{z}{2-i} \\) are real numbers, where \\(i\\) is the imaginary unit. Find the complex number \\(z\\). | 4-2i | Okay, let's try to solve this problem. Hmm, so we need to find a complex number z such that both z + 2i and z divided by (2 - i) are real numbers. Alright, first, I remember that if a complex number is real, its imaginary part must be zero. So maybe I can express z in terms of its real and imaginary parts and set up eq... | 1 | 0.2 | 2 | 0.40625 | 0 |
2,855 | Compute: ($\sqrt {3}$ - 3)$^0$ - 2$^{-1}$ = \_\_\_\_\_\_. | \frac{1}{2} | Okay, let's see. I need to compute (√3 - 3)^0 minus 2 to the power of -1. Hmm, first, I remember that any | 1 | 0.2 | 4 | 0.96875 | 0 |
2,995 | Given \\(\alpha\\) and \\(\beta\\) are in the interval \\((0,2\pi)\\), and \\(\alpha\\) is symmetric to \\(\beta\\) with respect to the \\(x\\)-axis, then \\(\alpha+\beta=\)________. | 2\pi | Okay, let's try to figure out this problem. So the question is: Given α and β are in the interval (0, 2π), and α is symmetric to β with respect to the x-axis, then what is α + β?
Hmm, symmetry with respect to the x-axis. I remember that reflecting a point over the x-axis changes the sign of the y-coordinate. So if we ... | 1 | 0.2 | 0 | 0.09375 | 0 |
3,042 | "x > 1" is a sufficient but not necessary condition for "x > a". What is the range of $a$? | a < 1 | Okay, let's try to figure out this problem. The question is: "x > 1" is a sufficient but not necessary condition for "x > a". We need to find the range of a. Hmm, so first, I need to remember what sufficient and necessary conditions mean.
A sufficient condition means that if the first statement is true, then the secon... | 1 | 0.2 | 3 | 0.78125 | 0 |
3,058 | Given that $a, b > 0$, $2^a = 3^b = m$, and $a, ab, b$ form an arithmetic sequence, find $m$. | \sqrt{6} | Okay, let's try to solve this problem. So, the problem says that a and b are positive numbers, 2^a equals 3^b equals m, and a, ab, b form an arithmetic sequence. We need to find m. Hmm, arithmetic sequence means that the difference between consecutive terms is the same. So, since it's a, ab, b, that should mean that 2a... | 1 | 0.2 | 0 | 0 | 0 |
3,320 | Find the angle of inclination for the line represented by the parametric equations \[\begin{cases} \left.\begin{matrix}x=3+t\cos230 ° \\ y=-1+t\sin230 °\end{matrix}\right.\end{cases}\] where \(t\) is a parameter. | 50° | Okay, let's see. I need to find the angle of inclination for this line given by parametric equations. Hmm, angle of inclination is the angle that the line makes with the positive x-axis, right? So it's usually between 0° and 180°, and the slope of the line is the tangent of that angle.
The parametric equations are x =... | 1 | 0.2 | 1 | 0.3125 | 0 |
3,330 | If the function $f(x) = \ln(x^2 + ax + 1)$ is an even function, then the value of the real number $a$ is ____. | 0 | Okay, let's try to figure out this problem. The question is asking for the value of a such that f(x) = ln(x² + a x + 1) is an even function. Hmm, first, I need to remember what an even function is. An even function satisfies f(-x) = f(x) for all x in its domain. So, I should start by writing down what f(-x) wo | 1 | 0.2 | 4 | 0.921875 | 0 |
3,370 | The central angle of a regular nonagon is ______ degrees. | 40^{\circ} | Okay, so I need to find the central angle of a regular nonagon. Hmm, let me think. First, what's a regular nonagon? It's a nine-sided polygon where all sides are equal and all internal angles are equal. The central angle... that's th | 1 | 0.2 | 2 | 0.515625 | 0 |
3,419 | Given the parabola $y^{2}=2px(p > 0)$, the coordinates of its focus are $\left( \dfrac {1}{4},0\right)$, then $p=$ ______. | \dfrac {1}{2} | Okay, let's see. I need to find the value of p for the parabola y² = 2px where the focus is (1/4, 0). Hmm, I remember that for a parabola that opens to the right or left, the standard form is y² = 4ax, where the focus is at (a, 0). Wait, but here the equation is y² = 2px. So maybe I need to compare these two.
First, l... | 1 | 0.2 | 3 | 0.78125 | 0 |
3,456 | Determine the range of the function $y= \frac {3x+2}{x+1}(x\geqslant 2)$. | [\frac {8}{3},3) | Okay, let's try to figure out the range of the function y = (3x + 2)/(x + 1) where x is at least 2. Hmm, so first, maybe I should simplify the function to make it easier to handle. Let's see, if I do polynomial division or rewrite the numerator in terms of the denominator. The denominator is x + 1, | 1 | 0.2 | 0 | 0.03125 | 0 |
3,515 | Factorize: $(x^2+4)^2-16x^2$. | (x+2)^2(x-2)^2 | Okay, let's see. I need to factorize (x² + 4)² - 16x². Hmm, that looks like a difference of squares. Because it's something squared minus another square. Let me recall that a difference of squares factors into (a - b)(a + b). So first, I should identify what a and b are here.
The expression is (x² + 4)² minus (4x)², r... | 1 | 0.2 | 3 | 0.75 | 0 |
3,522 | Given the function $f(x)=a^{x}(a > 0,a\neq 1)$, and $f(2)=4$, then the explicit formula of the function $f(x)$ is $f(x)=$ ______. | f(x)=2^{x} | Okay, let's see. I need to find the explicit formula for the function f(x) = a^x, where a is greater than 0 and not equal to 1. They also told me that f(2) = 4. Hmm, so I need to figure out what | 1 | 0.2 | 4 | 1 | 0 |
3,603 | If $n \in \mathbb{N}^*$, and $n$ is an odd number, then the remainder obtained when dividing $6^{n}+C_{n}^{1} \cdot 6^{n-1}+C_{n}^{2} \cdot 6^{n-2}+...+C_{n}^{n-1} \cdot 6$ by $8$ is $\_\_\_\_\_\_\_\_.$ | 6 | Okay, let's try to solve this problem. The question is: If n is a positive odd integer, what's the remainder when 6^n + C(n,1)6^(n-1) + C(n,2)6^(n-2) + ... + C(n,n-1)6 is divided by 8?
First, I need to recall what that sum is. It looks like the binomial expansion of (6 + 1)^n, but wait, the binomial theorem says (a +... | 1 | 0.2 | 0 | 0.140625 | 0 |
3,612 | Given a set of data $x_1$, $x_2$, $x_3$ with an average of 8, what is the average of another set of data $2x_1 - 1$, $2x_2 - 1$, $2x_3 - 1$? | 15 | Okay, let's see. I need to find the average of the new set of data: 2x₁ - 1, 2x₂ - 1, 2x₃ - 1. The original set has an average of 8. Hmm.
First, I remember that the average of a set of numbers is the sum divided by how many number | 1 | 0.2 | 4 | 0.984375 | 0 |
3,646 | Given that the supplement of an angle is three times its complement, find the measure of the angle in degrees. | 45^{\circ} | Okay, let's see. I need to solve this problem where the supplement of an angle is three times its complement. Hmm, first, I should remember what supplements and complements are.
So, the supplement of an angle is 180 degrees minus the angle, | 1 | 0.2 | 4 | 0.859375 | 0 |
3,651 | $(4a^2-b^2) \div (b-2a) = \_\_\_\_\_\_$. | -2a-b | Okay, let's see. I need to solve (4a² - b²) divided by (b - 2a). Hmm, first, I remember that 4a² - b² is a difference of squares. Because 4a² is (2a)² and b² is (b)². So the numerator factors into (2a - b)(2a + b). Right? Because (x - y)(x + y) = x² - y². So here x is 2a and y is b, so (2a - b)(2 | 1 | 0.2 | 3 | 0.703125 | 0 |
3,678 | Let $a= \int_{1}^{2}2xdx$, then the constant term in the expansion of $(ax- \frac{1}{x})^{6}$ is ______. | -540 | Okay, let's try to solve this problem step by step. First, I need to find the value of a, which is the integral from 1 to 2 of 2x dx. Then, once I have a, I need to find the constant term in the expansion of (ax - 1/x)^6.
Starting with the integral. The integral of 2x with respect to x is straightforward. The antider... | 1 | 0.2 | 2 | 0.4375 | 0 |
3,872 | Remove all perfect squares and perfect cubes from the set $A=\{n|n\leq10000, n\in\mathbb{N}^*\}$. Then, arrange the remaining elements in ascending order to form a sequence. The number 2014 is the $\boxed{\text{nth}}$ element of this sequence. | 1961\text{th} | Okay, let's try to figure out this problem. So, we need to remove all perfect squares and perfect cubes from the set of natural numbers up to 10,000, and then find where 2014 is in the resulting sequence. Hmm, right.
First, I remember that when you remove numbers that are perfect squares or cubes, you have to be caref... | 1 | 0.2 | 0 | 0.015625 | 0 |
3,968 | Given the function $y=a^{x-m}+n-3$ ($a>0$ and $a\neq1$), if the graph of the function always passes through the point $(3,2)$, find the value of $m+n$. | 7 | Okay, let's try to solve this problem. So, the function is y equals a to the power of (x minus m) plus n minus 3, and it's given that the graph always passes through the point (3, 2). We need to find m plus n. Hmm, right.
First, I remember that exponential functions like a^x usually pass through (0,1) because a^0 is ... | 1 | 0.2 | 1 | 0.28125 | 0 |
4,026 | For curve C, the polar equation is $\rho = 2$, and for line $l$, the parametric equation is
$$
\begin{cases}
x = t \\
y = -t + 1
\end{cases}
$$
Find the length of the chord AB formed by the intersection of line $l$ with curve C. | \sqrt{14} | Okay, let's see. I need to find the length of the chord AB where line l intersects curve C. First, I should figure out what curve C is. The polar equation is ρ = 2. Oh, right, in polar coordinates, ρ is the distance from the origin, so ρ = 2 is a circle with radius 2 centered at the origin. That makes sense.
Then the ... | 1 | 0.2 | 2 | 0.421875 | 0 |
4,070 | The minimal positive period of the function $f(x)=\dfrac{\sin 4x}{1+\cos 4x}$ is __________. | \dfrac{\pi}{2} | Okay, let's try to figure out the minimal positive period of the function f(x) = sin(4x)/(1 + cos(4x)). Hmm, first I remember that the period of a function is the smallest positive number T such that f(x + T) = f(x) for all x. So I need to find the smallest T where this holds.
First, maybe I can simplify the function ... | 1 | 0.2 | 2 | 0.46875 | 0 |
4,094 | A hospital has 5 internal medicine doctors and 6 surgeons. Now, 4 doctors need to be dispatched to participate in a disaster relief medical team. If it is required that there must be people from both internal medicine and surgery, then there are \_\_\_\_\_ ways of selection (answer with a number). | 310 | Okay, let's see. I need to figure out how many ways there are to select 4 doctors from 5 internal medicine doctors and 6 surgeons, with the condition that there must be at least one from each specialty. Hmm.
First, maybe I should remember that when you have to have at least one from each group, it's often easier to ca... | 1 | 0.2 | 3 | 0.671875 | 0 |
4,167 | A father and his son are skating in a circular skating rink. When they skate in the same direction, the father occasionally catches up with the son. However, when they skate in opposite directions, their encounters are five times more frequent. Therefore, the father's skating speed is ___ times the son's speed. | \frac{3}{2} | Okay, let's try to figure out this problem. So, we have a father and his son skating in a circular rink. When they go the same direction, the father catches up with the son sometimes. But when they go opposite directions, they meet five times more often. We need to find how many times faster the father is than the son.... | 1 | 0.2 | 2 | 0.53125 | 0 |
4,184 | Given that statement p: the equation $x^2 - 2x + \frac{1}{2}m = 0$ has real roots, and statement q: the equation $\frac{x^2}{m+3} + \frac{y^2}{4} = 1$ represents an ellipse with foci on the x-axis, if both p and q are true, find the range of values for the real number $m$. | m \in (1, 2] | Okay, let's try to solve this problem step by step. First, I need to remember what the problem is asking. We have two statements, p and q, both of which are true, and we need to find the range of m.
Starting with statement p: the equation x² - 2x + (1/2)m = 0 has real roots. For a quadratic equation to have real root... | 1 | 0.2 | 1 | 0.359375 | 0 |
4,196 | Given $f\left(\frac{1}{x}\right)=\frac{1}{x+1}$, find the analytical expression of $f\left(x\right)$. | \frac{x}{1+x} \quad (x \neq -1, x \neq 0) | Okay, let's see. I need to find the analytical expression of f(x) given that f(1/x) = 1/(x + 1). Hmm, so they told me what f does when the input is 1/x, and I need to figure out what f does in general.
First, maybe I should | 0 | 0.2 | 3 | 0.78125 | 0 |
4,332 | Observe the following expressions: $1+ \frac{1}{{2}^{2}} < \frac{3}{2},1+ \frac{1}{{2}^{2}}+ \frac{1}{{3}^{2}} < \frac{5}{3},1+ \frac{1}{{2}^{2}}+ \frac{1}{{3}^{2}}+ \frac{1}{{4}^{2}} < \frac{7}{4}\cdots $. Based on these expressions, what can we infer about the value of $1+\frac{1}{{{2}^{2}}}+\frac{1}{{{3}^{2}}}+\frac... | \frac{4029}{2015} | Okay, let's try to figure out this problem. So, the question is about these expressions that start with 1 plus the sum of reciprocals of squares up to some number, and they're all less than a fraction where the numerator is 2n-1 and the denominator is n. Wait, let me check the examples given.
First expression: 1 + 1/2... | 1 | 0.2 | 0 | 0.109375 | 0 |
4,353 | Given that the axis of symmetry equation of the parabola $y=ax^2$ is $y=-2$, the value of the real number $a$ is $\underline{\hspace{1cm}}$. | \frac{1}{8} | Okay, let's try to figure out this problem. So the question is about a parabola given by y = a x squared, and we know its axis of symmetry is y = -2. We need to find the value of a. Hmm, wait a second. I remember that the standard form of a parabola that opens up or down is y = a(x - h)^2 + k, where (h, k) is the verte... | 0.6 | 0.2 | 0 | 0.078125 | 0 |
4,642 | Insert a number between $3$ and an unknown number to form an arithmetic sequence. If the middle term is reduced by $6$, it becomes a geometric sequence. Then, this unknown number is \_\_\_\_\_\_. | 3 \text{ or } 27 | Okay, let's try to figure out this problem. Hmm, so the problem says: Insert a number between 3 and an unknown number to form an arithmetic sequence. If the middle term is reduced by 6, it becomes a geometric sequence. Then, find the unknown number.
First, let me parse this. So we have two numbers: 3 and an unknown nu... | 1 | 0.2 | 1 | 0.203125 | 0 |
4,666 | Person A and person B each select 2 courses from 4 available courses. The number of ways in which at least 1 of the courses selected by A and B are not the same is _______. | 30 | Okay, let's try to figure out this problem. So, the question is: Person A and Person B each select 2 courses from 4 available courses. We need to find the number of ways in which at least 1 of the courses selected by A and B are not the same. Hmm, that sounds a bit tricky, but maybe I can approach it step by step.
Fir... | 1 | 0.2 | 3 | 0.71875 | 0 |
4,674 | Factorize $-a^{2}+4b^{2}=\_\_\_\_\_\_.$ | (2b + a)(2b - a) | Okay, let's see. I need to factorize -a² + 4b². Hmm, factorizing expressions. First, maybe I should rearrange the terms to make it look more familiar. Like, usually, we have something like x² - y², which factors into (x - y)(x + y). But here it's -a² + 4b². Let me write t | 1 | 0.2 | 3 | 0.765625 | 0 |
4,695 | Given that the binomial coefficient of only the sixth term in the expansion of $(x^{3}+ \frac{1}{x^{2}})^{n}$ is the maximum, find the term in the expansion that does not contain $x$. | T_{7} = C_{10}^{6} = 210 | Okay, let's try to solve this problem step by step. First, I need to recall some basics about binomial expansions. The problem is about the expansion of (x³ + 1/x²)ⁿ. We're told that the binomial coefficient of only the sixth term is the maximum. We need to find the term without x.
First, let's remember that in a bin... | 1 | 0.2 | 0 | 0.09375 | 0 |
4,721 | In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$, respectively. Given that $\tan A= \frac{1}{3}$, $\tan B= \frac{1}{2}$, and $a=1$, find the value of $b$. | b = \sqrt{2} | Okay, let's try to solve this problem. So, we have triangle ABC, with sides a, b, c opposite to angles A, B, C respectively. We know tan A is 1/3, tan B is 1/2, and a is 1. We need to find b.
First, I remember that in triangles, the Law of Sines might be useful here because it relates the sides and the sines of the a... | 1 | 0.2 | 1 | 0.28125 | 0 |
4,726 | Find the equation of the tangent line to the curve $y= \frac {1}{x}$ at the point A(1, 1). | x + y - 2 = 0 | Okay, let's see. I need to find the equation of the tangent line to the curve y = 1/x at the point A(1, 1). Hmm, right. So first, I remember that the equation of a tangent line at a point (x₀, y₀) on a curve is given by y - y₀ = m(x - x₀), where m is the slope of the tangent line at that point.
So, the first thing I ... | 1 | 0.2 | 4 | 0.9375 | 0 |
4,750 | In a certain high school physical examination for seniors, the heights (in centimeters) of 12 students are $173$, $174$, $166$, $172$, $170$, $165$, $165$, $168$, $164$, $173$, $175$, $178$. Find the upper quartile of this data set. | 173.5 | Okay, let's try to figure out the upper quartile for this data set. First, I need to remember what the upper quartile is. From what I recall, quartiles divide the data into four equal parts. The upper quartile (Q3) is the median of the upper half of the data. But I should make sure I do this step by step.
First, I thi... | 1 | 0.2 | 1 | 0.375 | 0 |
4,901 | Point $P$ on the number line represents the number $-3$. The number represented by a point that is $4$ units away from point $P$ is ______. | -7 \text{ or } 1 | Okay, let's try to figure out this problem. So, point P is on the number line at -3. We need to find the number that's 4 units away from P. Hmm, right. On a number line, distance is absolute value, so if you go 4 units to the right or 4 units to the left from P, those should be the two points.
First, let's recall that... | 0 | 0.2 | 4 | 0.84375 | 0 |
4,910 | A lidless water tank with a volume of 256 cubic units has a square base. The length of its base side that minimizes the material used is ______. | 8 | Okay, let's try to figure out this problem. So, the question is about a lidless water tank with a volume of 256 cubic units and a square base. We need to find the length of the base side that minimizes the material used. Hmm, right. Since it's a lidless tank, the material used would be the surface area of the base plus... | 1 | 0.2 | 2 | 0.59375 | 0 |
5,001 | Given the function $f(x) = x^2 - 3x + m$, $g(x) = 2x^2 - 4x$, if $f(x) \geq g(x)$ holds exactly for $x \in [-1, 2]$, then the value of the real number $m$ is ______. | 2 | Okay, let's try to solve this problem. So, we have two functions: f(x) = x² - 3x + m and g(x) = 2x² - 4x. The problem says that f(x) is greater than or equal to g(x) exactly when x is in the interval [-1, 2]. We need to find the value of m.
First, I should probably set up the inequality f(x) ≥ g(x) and see what that s... | 1 | 0.2 | 2 | 0.421875 | 0 |
5,054 | Express "$3$ times $x$ plus $2$ is greater than $10$" in inequality form: ____. | 3x + 2 > 10 | Okay, let's see. I need to express "3 times x plus 2 is greater than 10" as an inequality. Hmm, first, I should break down the sentence.
So, "3 times x" would be 3x. Then "plus 2" | 1 | 0.2 | 3 | 0.71875 | 0 |
5,064 | $\sqrt{8}+\sqrt{2}=\_\_\_\_\_\_.$ | 3\sqrt{2} | Okay, let's see. I need to solve √8 + √2. Hmm, how do I add these square roots? I remember that square roots can sometimes be simplified. Let me start by simplifying each term.
Firs | 1 | 0.2 | 4 | 1 | 0 |
5,100 | Given that $\left\{\begin{array}{l}x=2\\ y=1\end{array}\right.$ is a solution of the system of linear equations $\left\{\begin{array}{l}mx+ny=8\\ nx-my=1\end{array}\right.$, then the arithmetic square root of $2m-n$ is ______. | 2 | Okay, let's try to solve this problem. Hmm, so we have a system of linear equations, and we know that x=2 and y=1 is a solution. That means when we plug x=2 and y=1 into both equations, they should hold true. Right? So I need to substitute x and y into the equations and solve for m and n. Then once I have m and n, I ca... | 1 | 0.2 | 2 | 0.5625 | 0 |
5,139 | The solution set of the inequality $1+ \sqrt{3}\tan x \geq 0$ is \_\_\_\_\_. | \left\{ x \mid -\frac{\pi}{6} + k\pi \leq x < \frac{\pi}{2} + k\pi, k \in \mathbb{Z} \right\} | Okay, let's try to solve this inequality: 1 + √3 tan x ≥ 0. Hmm, so first I need to remember that tan x has a period of π, so the solution should repeat every π. Let me start by isolating tan x. If I subtract 1 from both sides, I get √3 tan x ≥ -1. Then, divide both sides by √3. That gives tan x ≥ -1/√3.
Wait, I shou... | 0 | 0.2 | 0 | 0.109375 | 0 |
5,220 | The length of the real axis of a hyperbola is $m$, and the distance from a point $P$ on this hyperbola to the right focus is also $m$. The distance from point $P$ to the left focus of this hyperbola is. | 2m | Okay, let's try to solve this problem. Hmm, first, I need to remember some properties of hyperbolas. The problem says the length of the real axis is m. Wait, the real axis of a hyperbola is the transverse axis, right? So the length of the transverse axis is 2a, where a is the semi-transverse axis. So if the length is m... | 1 | 0.2 | 0 | 0.078125 | 0 |
5,248 | Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. Then, the solution set of the inequality $\lfloor x \rfloor^2 - 5\lfloor x \rfloor - 36 \leq 0$ is. | \{x | -4 \leq x < 10\} | Okay, let's try to solve this problem. The question is about finding the solution set of the inequality ⌊x⌋² - 5⌊x⌋ - 36 ≤ 0. Hmm, first, I remember that ⌊x⌋ is the floor function, which gives the greatest integer less than or equal to x. So, if I let n = ⌊x⌋, then n is an integer, and x is in [n, n+1). That might be a... | 0 | 0.2 | 1 | 0.21875 | 0 |
5,252 | Given space vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=3$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{π}{3}$. If $({2\overrightarrow{a}-\overrightarrow{b}})⊥({λ\overrightarrow{a}+2\overrightarrow{b}})$, then the real n... | \frac{6}{5} | Okay, let's try to solve this problem. So, we have two vectors, a and b. The magnitudes are 2 and 3, and the angle between them is π/3. We need to find λ such that (2a - b) is perpendicular to (λa + 2b).
First, I remember that if two vectors are perpendicular, their dot product is zero. So, the key here is to take th... | 1 | 0.2 | 1 | 0.34375 | 0 |
5,261 | Given the function $f(x)=2\ln x+\frac{8}{x}-m$ has a local minimum value of $2$, find the value of $m$. | 4\ln 2 | Okay, let's try to solve this problem. The function is f(x) = 2 ln x + 8/x - m, and we need to find m such that it has a local minimum value of 2. Hmm, first, I remember that to find local minima, we need to take the derivative and set it to zero. Let's start by finding the derivative of f(x).
The derivative of 2 ln x... | 1 | 0.2 | 2 | 0.4375 | 0 |
5,305 | Given the function $f(x)=f'( \frac{\pi}{2})\sin x+\cos x$, then $f'( \frac{\pi}{4})=$ __________. | - \sqrt{2} | Okay, let's try to solve this problem. Hmm, the function is given as f(x) equals f prime of pi over 2 times sin x plus cos x. And we need to find f prime of pi over 4. Alright, so first, I need to find the derivative of f(x) because we have f'(pi/2) in the equation. Let me write that down.
So, f(x) = f'(π/2) * sin x +... | 1 | 0.2 | 1 | 0.375 | 0 |
5,361 | If $(x^2+1)(x-2)^9 = a_0 + a_1x + a_2x^2 + \dots + a_{11}x^{11}$, find the value of $a_1 + a_2 + a_3 + \dots + a_{11}$. | 510 | Okay, let's try to figure out this problem. So, the question is: If (x² + 1)(x - 2)⁹ equals a₀ + a₁x + a₂x² + ... + a₁₁x¹¹, find the value of a₁ + a₂ + ... + a₁₁. Hmm, right. I need to find the sum of all the coefficients from a₁ up to a₁₁.
First, I remember that when you have a polynomial like P(x) = a₀ + a₁x + ... ... | 1 | 0.2 | 0 | 0 | 0 |
5,382 | Given that the perimeter of a sector is $8 cm$ and its area is $4 cm^2$, find the central angle of the sector in radians. | 2 \text{ radians} | Okay, let's see. I need to find the central angle of a sector when the perimeter is 8 cm and the area is 4 cm². Hmm, first, I should remember the formulas for the perimeter and area of a sector.
The perimeter of a sector is the sum of the two radii and the arc length. So, perimeter P = 2r + s, where s is the arc leng... | 1 | 0.2 | 1 | 0.34375 | 0 |
5,449 | If proposition p states: There exists some $x \in \mathbb{R}$ such that $x^2+x-1 \geq 0$, then the negation of p, denoted $\neg p$, is: ____________. | \forall x \in \mathbb{R}, x^2+x-1 < 0 | Okay, let's see. I need to find the negation of the proposition p, which says "There exists some real number x such that x squared plus x minus 1 is greater than or equal to zero." Hmm, negation of an existential statement... I remember that the negation of "there exists" is "for all," right? Like, if it's saying there... | 1 | 0.2 | 2 | 0.59375 | 0 |
5,488 | In △ABC, the sides opposite to angles A, B, C are a, b, c, respectively. If 2sin<sup>2</sup>A + c(sinC - sinA) = 2sin<sup>2</sup>B, and the area S of △ABC is $$\frac {1}{4}$$abc, then angle B = _____. | \frac {\pi}{3} | Okay, let's try to solve this problem step by step. First, let's recall some basic triangle properties and trigonometric identities that might be helpful here. The problem gives us an equation involving sines of angles and the area of the triangle.
The problem states: In △ABC, sides a, b, c are opposite angles A, B, ... | 0.1 | 0.2 | 0 | 0.0625 | 0 |
5,509 | Given the universe $U=\mathbb{R}$, and $A=\{x \mid |x-1| > 2\}$, $B=\{x \mid x^2 -6x + 8 < 0\}$, find $(U-A) \cap B$. | \{x \mid 2 < x \leq 3\} | Okay, let's try to solve this problem step by step. First, I need to understand what each set is. The universe U is all real numbers, so we're dealing with the real line.
Starting with set A: A is defined as {x | |x - 1| > 2}. I remember that the absolute value inequality |x - a| > b (where b > 0) translates to x < a... | 0 | 0.2 | 0 | 0.0625 | 0 |
5,542 | Given $\tan \alpha = 2$, find the value of $3\sin^2 \alpha + 5\sin \alpha \cos \alpha - 2\cos^2 \alpha$. | 4 | Okay, let's see. I need to find the value of 3 sin²α + 5 sinα cosα - 2 cos²α given that tanα = 2. Hmm, tanα is 2, so that means sinα/cosα = 2, right? So sinα = 2 cosα. Maybe I can express everything in terms of cosα or sinα.
First, I remember that for these kinds of problems, dividing numerator and denominator by cos... | 1 | 0.2 | 2 | 0.578125 | 0 |
5,714 | If a pair of corresponding sides of two similar triangles are $16$ and $32$ respectively, and the difference between their perimeters is $36$, then the perimeter of the smaller triangle is ______. | 36 | Okay, let's try to solve this problem. Hmm, so we have two similar triangles. The problem says that a pair of corresponding sides are 16 and 32. I need to find the perimeter of the smaller triangle, given that the difference between their perimeters is 36.
First, since the triangles are similar, their sides are propor... | 1 | 0.2 | 3 | 0.75 | 0 |
5,770 | In an isosceles triangle, if the lengths of two sides are $3$ and $6$, then the perimeter of this triangle is ____. | 15 | Okay, let's see. I need to find the perimeter of an isosceles triangle where two sides are 3 and 6. Hmm, isosceles means two sides are equal. So the triangle could have either two sides of 3 or two sides of 6.
First, let's check | 1 | 0.2 | 2 | 0.515625 | 0 |
5,813 | The H1N1 influenza is highly contagious, with an infection rate of 20% in the population. There are two vaccines, I and II. Vaccine I was administered to 8 healthy individuals, and none of them got infected. Vaccine II was administered to 25 healthy individuals, and one of them got infected. Which vaccine do you think ... | vaccine II is more effective | Okay, let's try to figure out which vaccine is more effective here. So the problem is about H1N1 influenza with a 20% infection rate in the population. We have two vaccines: Vaccine I and Vaccine II.
First, Vaccine I was given to 8 healthy people, and none got infected. So that's 0 out of 8. Vaccine II was given to 25... | 0 | 0.2 | 0 | 0.0625 | 0 |
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