problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given sets $A=\{x\left|\right.-2 \lt x \lt 4\}$ and $B=\{x\left|\right.x \gt -1\}$, then $A$⋂$B=\left(\ \ \right)$
A: $(0,4)$
B: $(-1,4)$
C: $\left[-1,4\right)$
D: $(-1,4]$ | B |
Given that the period of the function $f(x)= \sqrt {3}\sin ωx\cos ωx+ \sin ^{2}ωx- \frac {1}{2}$ is $\pi$.
(1) Find the expression for $f(x)$;
(2) Find the maximum and minimum values of $f(x)$ when $x \in [0, \frac{\pi}{2}]$. | f(x)_{\min}=f(0)=- \frac{1}{2} |
For example, as shown in Figure $24-2$, a tunnel is designed for four lanes in both directions, with a total lane width of 22 meters, and the vehicle height limit is 4.5 meters. The total length of the tunnel is 25 kilometers, and the arch of the tunnel is approximately a half-ellipse shape.
(1) If the maximum arch hei... | 33.3 |
A moving circle C is located inside the parabola $x^2=4y$ (where $x^2 \leq 4y$) and passes through the vertex of the parabola. The maximum value of the circumference of circle C is ( )
A: $\pi$
B: $2\pi$
C: $4\pi$
D: $16\pi$ | C |
2. The number of triangles with integer side lengths and a perimeter of 20 is $\qquad$ . | 8 |
4. Let quadrilateral $A B C D$ be inscribed. The rays $A B$ and $D C$ intersect at point K. It turns out that points $B, D$, and the midpoints $M$ and $N$ of segments $A C$ and $K C$ lie on the same circle. What values can the angle $A D C$ take?
$ is $D$, if $∀x∈D,xf\left(x\right)≧f\left(x\right)$, then the function $f(x)$ is called an "$L$ type function".
(1) Determine whether the functions $y={e}^{x}$ and $y=\mathrm{ln}x$ are "$L$ type functions", and explain the reasons;
(2) Suppose the function $f(x)=(x+1)\mat... | 0<a≦{e}^{2} |
A common cone's lateral surface area is in the ratio of $25: 32$ to its total surface area. What is the surface area and volume of the cone if the height $m=96 \mathrm{~cm}$? | 3584\pi\mathrm{}^{2},25088\pi\mathrm{}^{3} |
9. (16 points) Let $a, b, c > 0$, and $a+b+c=3$. Find the maximum value of $a^{2} b+b^{2} c+c^{2} a+a b c$.
| 4 |
[ Pythagorean Theorem (direct and inverse). ] Complex [ The ratio of the areas of triangles with a common base or a common height] Class
In trapezoid $A B C D$, the lateral side $A D$ is perpendicular to the bases and equals $9, C D=12$, and the segment $A O$, where $O-$ is the point of intersection of the diagonals o... | \frac{108}{5} |
Given vectors $\overrightarrow{OA}=(1,-3)$, $\overrightarrow{OB}=(2,-1)$, $\overrightarrow{OC}=(k+1,k-2)$, if points $A$, $B$, and $C$ cannot form a triangle, then $k$ is equal to ( )
A: $-2$
B: $\frac{1}{2}$
C: $1$
D: $-1$ | C: 1 |
Which of the following statements are correct? ( )
① The least squares method refers to the method of summing up each deviation as the total deviation and minimizing it;
② The least squares method refers to the method of taking the sum of the squares of each deviation as the total deviation and minimizing it;
③ Lin... | \text{B} |
41. "Square" family. One person had 9 children, all of whom were born at equal intervals, and the sum of the squares of their ages equaled the square of his own age. How many full years old was each child? | 2,5,8,11,14,17,20,23,26 |
Task B-1.1. How many zeros does the number $N=x^{3}+3 x^{2}-9 x-27$ end with, if $x=999997$? | 12 |
【Question 19】
There are 2014 apples in three baskets. If the same number of apples is taken from each basket, the second basket will have 49 apples left, and the number of apples left in the first basket is twice the number of apples left in the third basket. How many apples were originally in the third basket? $\qquad... | 655 |
Original: $f\left(\frac{2000}{2001}\right)+f\left(\frac{1999}{2001}\right)+\cdots+f\left(\frac{1}{2001}\right)$. | 1000 |
12) In a triangle, for every pair of consecutive sides, the two perpendicular bisectors of the sides and the angle bisector of the angle formed by the two sides intersect at the same point. We can state that:
(A) there is no triangle with this property, (B) the triangle is equilateral, (C) the triangle has an angle of... | B |
The relationship between the graph of the function $y=f(x)$ and the graph of $y=f(-x)$ is ( )
A: Symmetric about the x-axis
B: Symmetric about the y-axis
C: Symmetric about the origin
D: Symmetric about the line $y=x$ | \text{B} |
## Task 1 - 080611
a) The sum of the areas of a rectangular surface and a square surface is $3000 \mathrm{~m}^{2}$. The side of the square and one side of the rectangle have a length of $30 \mathrm{~m}$ each.
a) How long is the other side of the rectangle?
b) Draw both areas to scale $1: 2000$! | 70 |
If the geometric sequence $\{a_n\}$ satisfies $a_2 + a_4 = 20$ and $a_3 + a_5 = 40$, then $a_5 + a_7 = \_\_\_\_\_\_$. | 160 |
Given $a=\log_{3}4$, $b=(\frac{1}{5})^{0}$, $c=\log_{\frac{1}{3}}10$, the correct relationship among them is $(\ \ \ )$
A: $a > b > c$
B: $b > a > c$
C: $a > c > b$
D: $c > a > b$ | A |
3.357. $\cos \frac{3 \pi}{5} \cos \frac{6 \pi}{5}$. | \frac{1}{4} |
There are 7 different products, among which there are 4 defective ones and 3 non-defective ones. If one product is tested at a time until all 4 defective ones are identified, the number of all testing methods where the third defective product is exactly identified on the 4th test is $\boxed{1080}$. | 1080 |
Problem 8.6. For quadrilateral $ABCD$, it is known that $AB=BD, \angle ABD=\angle DBC, \angle BCD=90^{\circ}$. A point $E$ is marked on segment $BC$ such that $AD=DE$. What is the length of segment $BD$, if it is known that $BE=7, EC=5$?
 = 15x^5 - 24x^4 + 33x^3 - 42x^2 + 51x$. Calculate the value of $f(2)$ using the Horner's method (秦九韶算法).
A: 147
B: 294
C: 699
D: 1398 | \text{B} |
Solve the equation $[x] \cdot \{x\} = 1991x$. (Here $[x]$ denotes the greatest integer less than or equal to $x$, and $\{x\}=x-[x]$.) | 0 \text{ or } -\frac{1}{1992} |
## Task 1 - 280821
A cargo ship needs exactly 12 days for a shipping route from port $A$ to port $B$. A tanker travels this route in the opposite direction and takes 15 days for it.
The cargo ship departs from port $A$ 6 days later than the tanker from port $B$.
a) How many days after the departure of the cargo ship... | 4 |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{1} \frac{x^{4} \cdot d x}{\left(2-x^{2}\right)^{3 / 2}}
$$ | \frac{5}{2}-\frac{3\pi}{4} |
Let $P$ be a given quadratic polynomial. Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that $$f(x+y)=f(x)+f(y)\text{ and } f(P(x))=f(x)\text{ for all }x,y\in\mathbb{R}.$$ | f(x) = 0 |
6.169. $\sqrt{x}+\frac{2 x+1}{x+2}=2$.
6.169. $\sqrt{x}+\frac{2 x+1}{x+2}=2$. | 1 |
Numbers 0 and 2013 are written at two opposite vertices of a cube. Some real numbers are to be written at the remaining 6 vertices of the cube. On each edge of the cube the difference between the numbers at its endpoints is written. When is the sum of squares of the numbers written on the edges minimal? | \left\{\frac{2 \cdot 2013}{5}, \frac{2 \cdot 2013}{5}, \frac{2 \cdot 2013}{5}, \frac{3 \cdot 2013}{5}, \frac{3 \cdot 2013}{5}, \frac{3 \cdot 2013}{5}\right\} |
Question 208: In 1, 2, ..., 2011, how many numbers can be expressed in the form $[\mathrm{x}[\mathrm{x}]]$, where $[\mathrm{x}]$ denotes the greatest integer not exceeding x.
In 1, 2, ..., 2011, how many numbers can be expressed in the form $[\mathrm{x}[\mathrm{x}]]$, here $[\mathrm{x}]$ represents the greatest intege... | 990 |
Given that $\sin\alpha = \frac{1}{3}$ and $\alpha$ is an angle in the second quadrant, find the value of $\tan(\pi - \alpha)$. | \frac{\sqrt{2}}{4} |
If for any $x\in R$, $2x+2\leqslant ax^{2}+bx+c\leqslant 2x^{2}-2x+4$ always holds, then the maximum value of $ab$ is ______. | \frac{1}{2} |
Given a sequence of positive terms $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, for $n \geqslant 2$, we have $(a_n - S_{n-1})^2 = S_n S_{n-1}$, and $a_1 = 1$. Let $b_n = \log_2 \dfrac{a_{n+1}}{3}$, find the minimum value of $\dfrac{b_1 + b_2 + \ldots + b_n + 34}{n+1}$. | 9 |
8. Given $a_{k}$ as the number of integer terms in $\log _{2} k, \log _{3} k, \cdots, \log _{2018} k$. Then $\sum_{k=1}^{2018} a_{k}=$ $\qquad$ . | 4102 |
Given the function $f(x)=\sin (ωx+φ)(ω > 0,0 < φ < \dfrac {π}{2})$ whose graph passes through the point $(0, \dfrac {1}{2})$, and the distance between two adjacent symmetry axes is $\dfrac {π}{2}$.
(1) Find the analytical expression of the function $f(x)$ and its monotonically increasing interval in $[0,π]$;
(2) In $\t... | \sqrt {6} |
13.194. The front wheel of a moving model makes 6 more revolutions than the rear wheel over a distance of 120 m. If the circumference of the front wheel is increased by $1 / 4$ of its length, and the circumference of the rear wheel is increased by $1 / 5$ of its length, then over the same distance, the front wheel will... | 4 |
4. Figure 1 is a triangular number table, from top to bottom called the first row, second row, …. It is known that each “ ” in the triangular number table is the reciprocal of a positive integer, and is equal to the sum of the two numbers directly below it. If the number in the first row is $\frac{1}{1}$, then the numb... | C |
## Problem Statement
Calculate the lengths of the arcs of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=5(t-\sin t) \\
y=5(1-\cos t)
\end{array}\right. \\
& 0 \leq t \leq \pi
\end{aligned}
$$ | 20 |
Given that for any $x \in (0, 1]$, the value of the function $f(x) = x|x-a|-2$ is always negative, then the range of $a$ is \_\_\_\_\_\_. | -1 < a < 3 |
Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+{y}^{2}=1(a＞1)$ with eccentricity $e=\frac{\sqrt{3}}{2}$, where points $M$ and $N$ are two distinct moving points on the ellipse.$(1)$ Find the standard equation of the ellipse $C$;$(2)$ If the line $MN$ passes through the point $Q\left(0,2\right)$ and $\overrightarrow{OM}⋅\o... | (-\frac{\sqrt{3}}{6}, -\frac{1}{8}) \cup (\frac{1}{8}, \frac{\sqrt{3}}{6}) |
Simplify first, then find the value of $4x^{2}y-[6xy-3\left(4xy-2\right)-x^{2}y]+1$, where $|x+1|+\left(y-2\right)^{2}=0$. | -7 |
6. Circle $k_{1}$ with a radius of 8 cm lies inside circle $k$. Both circles intersect circle $k_{2}$ with a radius of 15 cm, as shown in the figure. What is the radius of $k$ if the shaded area inside $k$ but outside $k_{1}$ is equal to the total shaded area inside $k_{2}$?
^{2}=\right| z-\left.1\right|^{2}\right\}$, then
A. $M=\{$ pure imaginary numbers $\}$
B. $M=\{$ real numbers $\}$
C. $\{$ real numbers $\} \subsetneq M \subsetneq\{$ complex numbers $\}$
D. $M=\{$ complex numbers $\}$ | B |
In right triangle $\triangle ABC$, the length of the hypotenuse $BC$ is 5. A circle with radius 2 and center $O$ at the midpoint of $BC$ intersects $BC$ at points $P$ and $Q$. Calculate $|AP|^2+|AQ|^2+|PQ|^2$. | \frac{73}{2} |
2. Given that the origin is inside the ellipse $k^{2} x^{2}+y^{2}-4 k x+2 k y+k^{2}-1=0$, the range of the parameter $k$ is
A. $|k|>1$
B. $|k| \neq 1$
C. $-1<k<1$
D. $0<|k|<1$ | 0<|k|<1 |
In a math quiz consisting of 20 questions, each correct answer scores 5 points, no answer scores 0 points, and a wrong answer scores -2 points. If Xiao Li's score in this quiz is a prime number, then the maximum number of questions Xiao Li could have answered correctly is ___. | 17 |
There exist unique positive integers $x$ and $y$ that satisfy the equation $x^2 + 84x + 2008 = y^2$. Find $x + y$. | 80 |
23. In the canteen for lunch, 4 first-course dishes, 5 second-course dishes, and 3 desserts are prepared for selection. In how many ways can one lunch be composed? | 60 |
Which of the following calculations is correct?
A: $\sqrt{2}×\sqrt{3}=\sqrt{6}$
B: $\sqrt{2}+\sqrt{3}=\sqrt{5}$
C: $(\sqrt{2})^{2}=2\sqrt{2}$
D: $2+\sqrt{2}=2\sqrt{2}$ | A |
A six-sided die has the numbers one to six on its sides. What is the probability of rolling a five?
(A) $\frac{2}{6}$
(B) $\frac{1}{6}$
(C) $\frac{5}{6}$
(D) $\frac{3}{6}$
(E) $\frac{4}{6}$ | \frac{1}{6} |
3. Find all integers greater than 1 by which a cancellation can be made in some fraction of the form
$$
\frac{3 p-q}{5 p+2 q}
$$
where $p$ and $q$ are mutually prime integers.
| 11 |
Which of the following conclusions are correct?
A: If random variables $\xi$ and $\eta$ satisfy $\eta = 2\xi + 1$, then $D(\eta) = 2D(\xi) + 1$
B: If the random variable $\xi$ follows $N(3, \sigma^2)$, and $P(\xi < 6) = 0.84$, then $P(3 < \xi < 6) = 0.34$
C: The closer the absolute value of the linear correlation co... | BC |
If $\left(m-2\right)^{0}$ is undefined, then the value of $m$ is ______. | m = 2 |
Let's determine the coefficients $C$ and $D$ in the equation
$$
x^{4}+4 x^{3}-34 x^{2}+C x+D=0
$$
such that the roots of the equation form an arithmetic sequence. | C=-76,D=105 |
## problem statement
Write the equation of the plane passing through point $A$ and perpendicular to the vector $\overrightarrow{B C}$.
$A(1 ; 0 ;-6)$
$B(-7 ; 2 ; 1)$
$C(-9 ; 6 ; 1)$ | -x+2y+1=0 |
Given $U=\mathbb{R}$, and $A=\{x|-4<x<4\}$, $B=\{x|x\leq1 \text{ or } x\geq3\}$, find
(1) $A\cap B$;
(2) $\complement_U (A\cup B)$ | \emptyset |
Reflect a triangle's one median over the internal angle bisector from the same vertex.
a) In what ratio does each reflection divide the opposite side?
b) Is it true that the three reflections lie on a straight line? | \frac{^{2}}{b^{2}} |
Two candles of the same height are lighted at the same time. The first is consumed in $4$ hours and the second in $3$ hours.
Assuming that each candle burns at a constant rate, in how many hours after being lighted was the first candle twice the height of the second?
$\textbf{(A)}\ \frac{3}{4}\qquad\textbf{(B)}\ 1\fr... | 2\frac{2}{5} |
2. Let the function $f(x)=x^{2}+a x+b$, for any $a, b \in \mathbf{R}$, there always exists $t \in[0,4]$, such that $|f(t)| \geqslant m$ holds, then the maximum value of the real number $m$ is $\qquad$. | 2 |
## 34. Alsatian Village
The main street of the village (it is called Main) is perpendicular to General de Gaulle Street; they intersect at the Town Hall Square. The Catholic cathedral faces the main street, while the Protestant one faces General de Gaulle Street. The school is located on the street connecting these tw... | 0.002 |
Given that the sum of the first $n$ terms of an arithmetic sequence $\{a\_n\}$ is $S\_n$, if $\overrightarrow{OB}=a\_3\overrightarrow{OA}+a\_2017\overrightarrow{OC}$, and points $A$, $B$, $C$ are collinear ($O$ is a point outside this line), find $S\_2019$. | \frac{2019}{2} |
Given the set $M=\{0,1,2\}$, and $N=\{x|x^{2}-3x+2 > 0\}$, determine $M∩(∁\_RN)=(\ \ \ )$
A: $\{1\}$
B: $\{2\}$
C: $\{0,1\}$
D: $\{1,2\}$ | D |
II. Answer Questions (Total 56 points)
9. (16 points) Given a tetrahedron $P-ABC$ with each edge of the base being 1 and the height being $\sqrt{2}$, the center of its inscribed sphere is $O$, and the radius is $r$. Find the area of the planar region within the base $ABC$ consisting of points that are no more than $2r$... | \frac{1}{4}+\frac{\pi}{24} |
7. Given that the ellipse $C$ passes through the point $M(1,2)$, with two foci at $(0, \pm \sqrt{6})$, and $O$ is the origin, a line $l$ parallel to $OM$ intersects the ellipse $C$ at points $A$ and $B$. Then the maximum value of the area of $\triangle OAB$ is $\qquad$. | 2 |
7.067. $2,5^{\frac{4+\sqrt{9-x}}{\sqrt{9-x}}} \cdot 0,4^{1-\sqrt{9-x}}=5^{10} \cdot 0,1^{5}$. | -7,8 |
2.1. The numbers 7, 8, 9, 10, 11 are arranged in a row in some order. It turned out that the sum of the first three of them is 26, and the sum of the last three is 30. Determine the number standing in the middle.
 | 180 |
12. Let $F_{1}$ and $F_{2}$ be the two foci of the ellipse $C$, and $AB$ be a chord of the ellipse passing through point $F_{2}$. In $\triangle F_{1} A B$,
$$
\left|F_{1} A\right|=3,|A B|=4,\left|B F_{1}\right|=5 \text {. }
$$
Then $\tan \angle F_{2} F_{1} B=$ $\qquad$ | \frac{1}{7} |
[Chain (continuous) fractions]
Solve the equation in positive integers
$$
1-\frac{1}{2+\frac{1}{3+\ldots+\frac{1}{n-1+\frac{1}{n}}}}=\frac{1}{x_{1}+\frac{1}{x_{2}+\ldots \ddots+\frac{1}{x_{n-1}+\frac{1}{x_{n}}}}}
$$ | x_{1}=1,x_{2}=1,x_{3}=3,x_{4}=4,\ldots,x_{n}=n |
In the sequence $\{a_n\}$, $a_1=2$, $a_{n+1}=3a_n$ $(n\in\mathbb{N}^*)$, then $a_4=$ ______. | 54 |
Given $f(x) = x^2 - 2x + 1$, then $f(-x) =$ （ ）
A: $-x^2 + 2x - 1$
B: $x^2 - 2x + 1$
C: $x^2 + 2x + 1$
D: $-x^2 + 2x + 1$ | \text{C} |
A regular quadrilateral pyramid has a circumscribed sphere whose center coincides with the center of the inscribed sphere. What is the angle between two adjacent side edges? | 45 |
3. Given that $AB$ is a chord of the circle $\odot O$ with radius 1, and the length of $AB$ is the positive root of the equation $x^{2}+x-1=0$. Then the degree of $\angle AOB$ is $\qquad$ . | 36^{\circ} |
Let the line \\(x=m\\) intersect the graphs of the functions \\(f(x)=x^2\\) and \\(g(x)=\ln x\\) at points \\(P\\) and \\(Q\\) respectively. Find the value of \\(m\\) when the distance \\(|PQ|\\) is minimized. | \dfrac{\sqrt{2}}{2} |
2. What is the greatest possible value of the expression $x y - x^{3} y - x y^{3}$, where $x$, $y$ are positive real numbers? For which $x$, $y$ is this value achieved?
(Mária Dományová, Patrik Bak) | \frac{1}{8} |
9. Given the sequence $\left\{a_{n}\right\}$ with the sum of the first $n$ terms being $S_{n}, a_{1}=\frac{1}{5}$, and for any positive integers $m, n$, it holds that $a_{m+n}=a_{m} a_{n}$. If $S_{n}<a$ for any $n \in \mathbf{Z}_{+}$, then the minimum value of the real number $a$ is | \frac{1}{4} |
A sowing plot for rye has a rectangular shape. As part of the restructuring of the collective farm lands, one side of the plot was increased by $20\%$, and the other side was reduced by $20\%$. Will the rye yield change as a result, and if so, by how much? | 4 |
78***. a) Someone wrote $n$ letters and sealed them in envelopes without writing the addresses beforehand. After that, he no longer knew which letter was in which envelope, and therefore wrote $n$ addresses on the envelopes at random. What is the probability that at least one of the recipients will receive the letter i... | 1-\frac{1}{e} |
24. A polynomial of degree 10 has three distinct roots. What is the maximum number of zero coefficients it can have?
(A. Khryabrov) | 9 |
The lengths of three small wooden sticks in each group are as follows. Which group can form a triangle?
A: $9$, $6$, $13$
B: $6$, $8$, $16$
C: $18$, $9$, $8$
D: $3$, $5$, $9$ | A |
171. Find $\varphi(360)$. | 96 |
II. (20 points) Find all positive decimal integers $n$ that satisfy the following conditions: $n$ is an $a$-digit number, and $a^{a}=n$.
| 1,16777216,387420489 |
Given a positive integer $n(n \geqslant 3)$. Try to find the largest constant $\lambda(n)$, such that for any $a_{1}, a_{2}, \cdots, a_{n} \in \mathbf{R}_{+}$, we have
$$
\prod_{i=1}^{n}\left(a_{i}^{2}+n-1\right) \geqslant \lambda(n)\left(\sum_{i=1}^{n} a_{i}\right)^{2} .
$$ | n^{n-2} |
If $\frac{x}{x-1} = \frac{y^2 + 2y - 1}{y^2 + 2y - 2},$ then $x$ equals
$\text{(A)} \quad y^2 + 2y - 1$
$\text{(B)} \quad y^2 + 2y - 2$
$\text{(C)} \quad y^2 + 2y + 2$
$\text{(D)} \quad y^2 + 2y + 1$
$\text{(E)} \quad -y^2 - 2y + 1$ | y^2+2y-1 |
Example 1. Find the curve passing through the point $M_{0}(1,4)$ and having the property that the segment of any of its tangents, enclosed between the coordinate axes, is bisected at the point of tangency. | xy=4 |
Task 4. (20 points) A two-meter gas pipe has rusted in two places. Determine the probability that all three resulting sections can be used as offsets for gas stoves, if according to regulations, the stove should not be located closer than 50 cm to the main gas pipe.
# | \frac{1}{16} |
Problem 5.3. The figure shows a plan of the road system of a certain city. In this city, there are 8 straight streets, and 11 intersections are named with Latin letters $A, B, C, \ldots, J, K$.
Three police officers need to be placed at some intersections so that at least one police officer is on each of the 8 streets... | B,G,H |
I3.1 Suppose that $a=\cos ^{4} \theta-\sin ^{4} \theta-2 \cos ^{2} \theta$, find the value of $a$.
I3.2 If $x^{y}=3$ and $b=x^{3 y}+10 a$, find the value of $b$.
I3.3 If there is (are) $c$ positive integer(s) $n$ such that $\frac{n+b}{n-7}$ is also a positive integer, find the value of $c$.
I3.4 Suppose that $d=\log _... | -1,17,8,18 |
# Problem 3.
One side of the parallelogram is $\sqrt{3}$ times larger than the other side. One diagonal of the parallelogram is $\sqrt{7}$ times larger than the other diagonal. How many times larger is one angle of the parallelogram than the other angle? | 5 |
8.5. Vasya wrote a set of different natural numbers on the board, each of which does not exceed 2023. It turned out that for any two written numbers $a$ and $b$, the number $a+b$ is not divisible by $a-b$. What is the maximum number of numbers Vasya could have written? | 675 |
Given a complex number $z=1+ai$ satisfies the condition $|z|<2$, find the range of the real number $a$. | (-\sqrt{3}, \sqrt{3}) |
Given \\(\alpha \in (0, \frac{\pi}{2})\\) and \\(\beta \in (\frac{\pi}{2}, \pi)\\) with \\(\sin(\alpha + \beta) = \frac{3}{5}\\) and \\(\cos \beta = -\frac{5}{13}\\), find the value of \\(\sin \alpha\\). | \frac{33}{65} |
If the simplest quadratic radical $\sqrt{3a-8}$ and $\sqrt{17-2a}$ are of the same type, then a correct value for $a$ is ( )
A: $1$
B: $3$
C: $5$
D: $7$ | C |
G6.3 Find $d$, if
$$
d=\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{1994}\right)\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{1995}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{1995}\right)\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{1994}\right)
$$ | \frac{1}{1995} |
In the Cartesian coordinate system $xOy$, with $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C_{1}$ is $\rho\sin (\theta+ \frac {\pi}{4})= \frac { \sqrt {2}}{2}a$, and the parametric equation of curve $C_{2}$ is $\begin{cases} x=-1+\cos \theta \\ y=-1+\sin \theta\end{... | -2- \sqrt {2} < a < -2+ \sqrt {2} |
$(1)$ Given the general formula of the sequence $\{a_{n}\}$ as $a_{n}=\frac{{2^n+1}}{{4^n}}$, find the sum of the first $n$ terms $S_{n}$ of $\{a_{n}\}$;$(2)$ Given the general formula of the sequence $\{a_{n}\}$ as $a_{n}=3n-1$, find the value of ${a}_{1}^{2}-{a}_{2}^{2}+{a}_{3}^{2}-{a}_{4}^{2}+\cdots +{a}_{19}^{2}-{a... | -1830 |
11. In the complex plane, non-zero complex numbers $z_{1}, z_{2}$ lie on a circle centered at $i$ with a radius of 1, the real part of $\bar{z}_{1} \cdot z_{2}$ is zero, and the principal value of the argument of $z_{1}$ is $\frac{\pi}{6}$, find $z_{2}$. | z_{2}=-\frac{\sqrt{3}}{2}+\frac{3}{2}i |
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