problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Express the complex number $z=\frac{1+\sin \theta+\mathrm{i} \cos \theta}{1+\sin \theta-\mathrm{i} \cos \theta}$ in polar form as $\qquad$ | \cos(\frac{\pi}{2}-\theta)+i\sin(\frac{\pi}{2}-\theta) |
9.103. For what values of $m$ are the roots of the equation $4 x^{2}-(3 m+1) x-m-2=0$ contained in the interval between -1 and 2? | \in(-\frac{3}{2};\frac{12}{7}) |
Given triangle $\triangle ABC$, where the angles $A$, $B$, $C$ have corresponding opposite sides $a$, $b$, $c$. If $C={120}^{{}^{\circ }}$, $a=2b$, then $\tan A=$___. | \frac{\sqrt{3}}{2} |
Given the function $f(x)=x^3+ax^2+bx+1$, its derivative $f'(x)$ satisfies $f'(1)=2a$, $f'(2)=-b$, where $a$ and $b$ are constants.
(I) Find the values of $a$ and $b$;
(II) Let $g(x)=f'(x)e^{-x}$, find the extreme values of the function $g(x)$. | 15e^{-3} |
5. Let $\frac{x}{x^{2}-p x+1}=1$. Then the value of $\frac{x^{3}}{x^{6}-p^{3} x^{3}+1}$ is ( )
(A) 1
(B) $\frac{1}{3 p^{2}-2}$
(C) $\frac{1}{p^{3}+3^{2}}$
(D) $\frac{1}{3 p^{2}+1}$ | B |
18) How many angles greater than $90^{\circ}$ can a quadrilateral (non-intersecting) have?
(A) It always has at least one
(B) it has at most one
(C) it has at most two
(D) it has at most three
(E) it can have four. | D |
Given the function $f(x)=\ln x+x^{2}$.
(1) If the function $g(x)=f(x)-ax$ is an increasing function in its domain, find the range of values for the real number $a$.
(2) Under the condition of (1) and $a > 1$, let $h(x)=e^{3x}-3ae^{x}$, $x\in[0,\ln 2]$, find the minimum value of $h(x)$. | -2(a)^{ \frac {3}{2}} |
5. Each of the diagonals of the inscribed quadrilateral $A B C D$ is the bisector of one of the angles from which it is drawn, and also divides the second angle in the ratio $2: 3$. What can the angle $\angle A$ of this quadrilateral be? Express the answer in degrees. List all possible options. If necessary, round the ... | 80,100,\frac{1080}{11},\frac{900}{11} |
In the rectangular coordinate system, the coordinates of the point P(3, 4, 5) symmetric about the yOz plane are ( ).
A: (-3, 4, 5)
B: (-3, -4, 5)
C: (3, -4, -5)
D: (-3, 4, -5) | \text{A} |
Given $27^2 = a^6$, then $a^{-2} =$ ? | \frac{1}{9} |
Given the function $y=x^2-2x+3$ has a maximum value of 3 and a minimum value of 2 on the closed interval $[0,m]$, then the range of $m$ is ( )
A: $[1,+\infty)$
B: $[0,2]$
C: $[1,2]$
D: $(-\infty,2]$ | C |
## Task 4 - 100924
A regular right triangular pyramid is a pyramid whose base is an equilateral triangle and whose foot of the height coincides with the centroid of the base. In the regular pyramid with vertices $A, B, C, D$ and apex $D$, the inclination angle between each of the three side faces and the base is $60^{... | \frac{\sqrt{3}}{24}^{3} |
# Problem 5
The bisector of a right triangle divides a leg into segments of lengths 25 cm and 24 cm. Find the length of the circumscribed circle around this triangle. | 175\pi |
Find all functions $f: \mathbb{R} \longrightarrow \mathbb{R}$ such that $f(0)=0$, and for all $x, y \in \mathbb{R}$,
$$
(x-y)\left(f\left(f(x)^{2}\right)-f\left(f(y)^{2}\right)\right)=(f(x)+f(y))(f(x)-f(y))^{2}
$$ | f(x)=cx |
In the diagram, $M$ is the midpoint of $Y Z, \angle X M Z=30^{\circ}$, and $\angle X Y Z=15^{\circ}$. The measure of $\angle X Z Y$ is
(A) $75^{\circ}$
(B) $65^{\circ}$
(C) $60^{\circ}$
(D) $80^{\circ}$
(E) $85^{\circ}$
$ satisfies
$$
f(x)=\left\{\begin{array}{ll}
x-3, & x \geqslant 1000 ; \\
f(f(x+5)), & x<1000 .
\end{array}\right.
$$
Then $f(84)=$ . $\qquad$ | 997 |
The sum of the three smallest distinct divisors of some number $A$ is 8. How many zeros can the number $A$ end with?
# | 1 |
5. Drop perpendiculars $D D_{1}, D D_{2}, D D_{3}$ from point $D$ to the planes $S B C$, $S A C$, and $S A B$ respectively. Let $D D_{1}=x, D D_{2}=y, D D_{3}=z$. According to the condition, we form the system of equations
$$
\left\{\begin{array}{l}
y^{2}+z^{2}=5 \\
x^{2}+z^{2}=13 \\
x^{2}+y^{2}=10
\end{array}\right.
... | 27 |
In a table tennis team of 5 players, which includes 2 veteran players and 3 new players, we need to select 3 players to be ranked as No. 1, No. 2, and No. 3 for a team competition. The selection must ensure that among the 3 chosen players, there is at least 1 veteran player and among players No. 1 and No. 2, there is a... | 48 |
(7) There are 14 small balls of the same size and shape, 7 of which are red and 7 are white. They are placed in two boxes, A and B, where box A contains 4 red balls and 3 white balls, and box B contains 3 red balls and 4 white balls. Now, a ball is randomly drawn from box A and placed into box B, then a ball is randoml... | \frac{25}{8} |
2. In $\triangle A B C$, it is known that $\angle C=100^{\circ}$, points $P$ and $Q$ are both on side $A B$, such that $A P=B C, B Q=A C$, the midpoints of segments $A B$, $C P$, and $C Q$ are $M$, $N$, and $K$ respectively. Find $\angle N M K$. | 40^{\circ} |
4. Snow White entered a room where there were 30 chairs around a round table. Some of the chairs were occupied by dwarfs. It turned out that Snow White could not sit down without having someone next to her. What is the minimum number of dwarfs that could have been at the table? (Explain how the dwarfs should have been ... | 10 |
(5 points) (2011•Jiangxi Simulation) Given proposition p: "For all $x \in [0,1]$, $a \geq e^x$", and proposition q: "There exists $x \in \mathbb{R}$, such that $x^2 + 4x + a = 0$", if the proposition "p ∧ q" is true, then the range of the real number $a$ is ( )
A: $[e,4]$
B: $[1,4]$
C: $(4,+\infty)$
D: $(-\infty,1... | A |
$F\_1$, $F\_2$ are the left and right foci of the hyperbola $C$: $\frac{x^2}{9} - \frac{y^2}{4} = 1$. Point $M$ is on the hyperbola and $\angle F\_1 M F\_2 = 60^{\circ}$, then $S_{\triangle F\_1 M F\_2} =$ _____ . | 4 \sqrt{3} |
3. Let the function $f(x, y)$ satisfy:
(1) $f(x, x)=x$;
(2) $f(k x, k y)=k f(x, y)$;
(3) $f\left(x_{1}+x_{2}, y_{1}+y_{2}\right)=f\left(x_{1}, y_{1}\right)+f\left(x_{2}, y_{2}\right)$;
(4) $f(x, y)=f\left(y, \frac{x+y}{2}\right)$.
Then $f(x, y)=$ . $\qquad$ | \frac{x}{3}+\frac{2 y}{3} |
Three, (50 points) A positive integer is called "simple" if it does not have any square factors greater than 1. Determine how many numbers in $1,2, \cdots, 2010$ are simple.
| 1221 |
What is the value of $\frac{15-3^{2}}{3}$ ?
(A) 2
(B) 4
(C) 12
(D) 48
(E) 3 | 2 |
}[}{[}$
The median to the hypotenuse
Pythagorean Theorem (direct and inverse)
Mean proportionals in a right triangle
In a right triangle $A B C\left(\angle C=90^{\circ}\right)$, the height $C D$ and the median $C E$ are drawn. The areas of triangles $A B C$ and $C D E$ are 10 and 3, respectively. Find $A B$. | AB=\sqrt{50} |
Reading material:($1$) When simplifying and operating square roots, sometimes we encounter expressions like $\frac{2}{{\sqrt{3}+1}}$, in fact, we can further simplify it: $\frac{2}{{\sqrt{3}+1}}=\frac{{2×({\sqrt{3}-1})}}{{({\sqrt{3}+1})({\sqrt{3}-1})}}=\frac{{2×({\sqrt{3}-1})}}{{{{({\sqrt{3}})}^2}-{1^2}}}=\sqrt{3}-1$. ... | -\frac{{\sqrt{6}}}{3} |
145. The Partitioned Corner. One of the corners of a rectangular room is partitioned off using two identical screens, each 4 meters long. How should the screens be placed to maximize the area of the partitioned section?
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 8(\sqrt{2}+1) |
175. On the curve $y=x^{2}-3 x+5$, find the point where the ordinate $y$ increases 5 times faster than the abscissa $x$. | (4,9) |
Which of the following fractions is closest to 0 ?
(A) $\frac{1}{2}$
(B) $\frac{1}{8}$
(C) $\frac{1}{3}$
(D) $\frac{1}{6}$
(E) $\frac{1}{9}$ | \frac{1}{9} |
The degree measures of the angles of a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle. | 143 |
5. Given a numerical riddle: TETA + BETA = GAMMA. (Different letters correspond to different digits, the same letters correspond to the same digits.) Find all its solutions and prove that there are no others. | 4940+5940=10880 |
Point P(8, 1) bisects a chord of the hyperbola $x^2-4y^2=4$. The equation of the line containing this chord is __________. | 2x-y-15=0 |
Define an operation "$\odot$" between planar vectors as follows: for any $a$ $(=$ $m$, $n$$)$, $b$ $(=$ $p$, $q$$)$, let $a$ $\odot$ $b$ $=$ $mq$ $-$$ np$. Which of the following statements is incorrect?
A: If $a$ and $b$ are collinear, then $a$ $\odot$ $b$ $=$ $0$
B: $a$ $\odot$ $b$ $=$ $b$ $\odot$ $a$
C: For any $\l... | B |
IMO 1965 Problem B1 Find all sets of four real numbers such that the sum of any one and the product of the other three is 2. | 1,1,1,1or3,-1,-1,-1 |
## 125. Math Puzzle 10/75
"What time is it?" Marie-Luise was asked. She jokingly replies: "Until the end of the day, twice two fifths of what has already elapsed since its beginning remain." What time was it at that moment? | 13:20 |
28. How many standard sheets of paper will be required to print all numbers from 1 to 1000000 inclusive, if printing on one side of the sheet? Each sheet has 30 lines with 60 characters each, and there are intervals between numbers, each equivalent to two characters. | 4383 |
The front view of a cylinder is a rectangle with an area of 6. Its lateral area is ( )
A: $8\pi$
B: $6\pi$
C: $4\pi$
D: $3\pi$ | B |
7.212. $\frac{\log _{4 \sqrt{x}} 2}{\log _{2 x} 2}+\log _{2 x} 2 \cdot \log _{1 / 2} 2 x=0$. | 4 |
6.2. Mathematicians Andrey, Boris, and Viktor were solving problems from an olympiad. First, Andrey solved several problems, then Boris solved a third of the remaining problems. After this, a third of the problems remained unsolved, which Viktor completed. What part of all the problems did Andrey solve? | \frac{1}{2} |
4. Calculate $\sec \frac{2 \pi}{9}+\sec \frac{4 \pi}{9}+\sec \frac{6 \pi}{9}+\sec \frac{8 \pi}{9}$. | 4 |
Given the sets A = {x | -1 < x < 1} and B = {x | 0 < x < 2}, find A ∪ B = ( ).
A: {x | -1 < x < 1}
B: {x | -1 < x < 2}
C: {x | 0 < x < 2}
D: {x | 0 < x < 1} | \text{B} |
What are the last two digits of $7^{9^{9^{9}}}$? | 7 |
19. For the convex quadrilateral $A B C D$, the lengths of the sides $A B, B C, C D$, and $D A$ are known, in order, to be $4, 9, 6$, and $11 \, \text{cm}$. Indicated with $E$ and $F$ the midpoints of the sides $A B$ and $C D$, it is also known that the area of the quadrilateral $B E D F$ is $18 \, \text{cm}^2$. By how... | 36 |
In the Cartesian coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=1+t\\ y=\sqrt{3}t\end{array}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis. The polar coordinate equation of ... | \frac{4}{3} |
2. A parallelepiped is composed of white and black unit cubes in a checkerboard pattern. It is known that the number of black cubes is $1 \frac{12}{13} \%$ more than the number of white cubes. Find the surface area of the parallelepiped, given that each side of the parallelepiped is greater than 1. | 142 |
Determine all pairs $(n, k)$ of non-negative integers such that
$$
2023+2^{n}=k^{2}
$$ | (1,45) |
Given the system of linear equations in two variables $x$ and $y$: $\left\{\begin{array}{l}2x+y=1+2m\\ x+2y=2-m\end{array}\right.$, where the solution satisfies the inequality $x+y \gt 0$. Find the range of real numbers for $m$. If the inequality $\left(2m+1\right)x-2m \lt 1$ has a solution $x \gt 1$ under the conditio... | -2, -1 |
* Let $[\tan x]$ denote the greatest integer not exceeding the real number $\tan x$. Then the solution to the equation $[\tan x]=2 \cos ^{2} x$ is $\qquad$ | \frac{\pi}{4}+k\pi,k\in{Z} |
Task 1. Represent in the form of an irreducible fraction
$$
6 \frac{7}{2015} \times 4 \frac{5}{2016}-7 \frac{2008}{2015} \times 2 \frac{2011}{2016}-7 \times \frac{7}{2015}
$$ | \frac{5}{144} |
Given $\lg 6 = a$ and $\lg 12 = b$, express $\lg 24$ in terms of $a$ and $b$. | 2b - a |
For any positive integer $n$, let $D_{n}$ be the set of all positive divisors of $n$, and $f_{i}(n)$ be the number of elements in the set
$$
F_{i}(n)=\left\{a \in D_{n} \mid a \equiv i(\bmod 4)\right\}
$$
Find the smallest positive integer $m$ such that
$$
2 f_{1}(m)-f_{2}(m)=2017 \text {. }{ }^{[1]}
$$
(14th China So... | 2\times5^{2016} |
Let the function $f(x)=\sin(2x+\varphi)$ $(|\varphi| < \frac{\pi}{2})$ be shifted to the left by $\frac{\pi}{3}$ units to obtain a new function that is an odd function. Then, $\varphi=$ \_\_\_\_\_\_. | \frac{\pi}{3} |
Calculate without a calculator $1001^{2}-999^{2}$. | 4000 |
678. The number 43 is written in the septenary (base-7) numeral system. In which system is it written with the same digits but in reverse order? | 9 |
What is the sum of all the solutions of $x = \left|2x-|60-2x|\right|$?
$\textbf{(A)}\ 32 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 92 \qquad \textbf{(D)}\ 120 \qquad \textbf{(E)}\ 124$ | \textbf{(C)}\92 |
(5 points) The order of the three numbers 20.3, 0.32, and $\log{0.32}$ is ()
A: $0.32 < \log{0.32} < 20.3$
B: $0.32 < 20.3 < \log{0.32}$
C: $\log{0.32} < 20.3 < 0.32$
D: $\log{0.32} < 0.32 < 20.3$ | D |
Which of the following four deductive reasoning processes is in the form of a syllogism and is correct? ( )
A: Major premise: Non-repeating infinite decimals are irrational numbers; Minor premise: $\pi$ is an irrational number; Conclusion: $\pi$ is a non-repeating infinite decimal.
B: Major premise: Non-repeating ... | \text{B} |
12. The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by $4, 5, 6$ is between ( ).
(A) 2 to 19
(B) 20 to 39
(C) 40 to 59
(D) 60 to 79
(E) 80 to 124 | 61 |
If \\(a{{x}^{2}}+ax+a+3 > 0\\) holds true for all real numbers \\(x\\), then the range of the real number \\(a\\) is \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_. | [0,+\infty) |
The negation of the universal proposition: For all $x \in \mathbb{R}$, $x^2 > 0$ is ( )
A: For all $x \in \mathbb{R}$, $x^2 \leq 0$
B: There exists an $x \in \mathbb{R}$, $x^2 > 0$
C: There exists an $x \in \mathbb{R}$, $x^2 < 0$
D: There exists an $x \in \mathbb{R}$, $x^2 \leq 0$ | D |
In a certain school, there are 5 classes in the first grade, 8 classes in the second grade, and 3 classes in the third grade. A round-robin basketball tournament is held between classes within each grade. The total number of games that need to be played is ______. | 41 |
If the line $y=-2x+1$ is shifted $2$ units to the left, the resulting equation of the line is:
A: $y=-2x-3$
B: $y=-2x-1$
C: $y=-2x+3$
D: $y=-2x+5$ | A |
Joana must place three chips on a $5 \times 5$ board, where the cells are numbered from 1 to 25, with one chip in each cell. In how many ways can she do this, if:
a) The 3 chips are of different colors?
b) The 3 chips are identical?
c) The chips are of different colors and no two can be in the same row or column? | 2,300 |
$15 A B C$ is a right-angled triangle with $\angle B A C=90^{\circ}$. A square is constructed on the side $A B$ and $B C$ as shown. The area of the square $A B D E$ is $8 \mathrm{~cm}^{2}$ and the area of the square $B C F G$ is $26 \mathrm{~cm}^{2}$. Find the area of triangle $D B G$ in $\mathrm{cm}^{2}$. | 6 |
Let $a$ and $b$ be two distinct roots of the polynomial $X^{3}+3 X^{2}+X+1$. Calculate $a^{2} b+a b^{2}+3 a b$. | 1 |
In a senior high school class, there are two study groups, Group A and Group B, each with 10 students. Group A has 4 female students and 6 male students; Group B has 6 female students and 4 male students. Now, stratified sampling is used to randomly select 2 students from each group for a study situation survey. Calcul... | \dfrac{31}{75} |
(The alphabet with 26 letters) How many words are there
1. of 5 letters?
2. of 5 different letters
3. of 5 letters but without the same letter twice in a row? | 26^5,26\times25\times24\times23\times22,26\times25^4 |
8. As shown in the figure, the perimeter of square $A B C D$ is $40 \mathrm{~m}$. Two people, Jia and Yi, start from $A$ and $B$ respectively at the same time and walk along the edges of the square. Jia walks $35 \mathrm{~m}$ per minute in a counterclockwise direction, and Yi walks $30 \mathrm{~m}$ per minute in a cloc... | (6,10) |
Calculate:$(1)\sqrt{64}-\sqrt[3]{64}$;$(2)\sqrt{18}-\sqrt{8}-|1-\sqrt{2}|$. | 1 |
1379. Find the curve passing through the point $(2 ; 3)$ and having the property that the segment of any of its tangents, enclosed between the coordinate axes, is bisected at the point of tangency. | x6 |
1. Consider the set $A=\{100,101,102, \ldots, 200\}$.
a) Determine the number of elements in $A$ that are divisible by 10.
b) Determine the elements of $A$ that, when divided by 21 give a remainder of 12, and when divided by 28 give a remainder of 5. | 117 |
3. Solve the equation $3 \sqrt{(x+1)^{2}}+\sqrt{x^{2}-4 x+4}=|1+6 x|-4|x-1|$. | x\in[1,2] |
4-15 Given that $\operatorname{tg} \alpha$ and $\operatorname{tg} \beta$ are the roots of $x^{2}+p x+q=0$, try to find
$$
\sin ^{2}(\alpha+\beta)+p \sin (\alpha+\beta) \cos (\alpha+\beta)+q \cos ^{2}(\alpha+\beta) \text { }
$$
value. | q |
Example 8 (2005 Romanian Mathematical Olympiad) Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, satisfying the following two conditions:
(1) $x[f(x+1)-f(x)]=f(x)$, for all $x, y \in \mathbf{R}$;
(2) $|f(x)-f(y)| \leqslant|x-y|$, for all $x, y \in \mathbf{R}$. | f(x)=kx,wherek\in{R}|k|\leqslant1 |
The slope of the line $y=\frac{π}{2}$ is ( ).
A: $180^{\circ}$
B: $0^{\circ}$
C: $90^{\circ}$
D: Does not exist | B |
Given the function $f(x)=\begin{cases} & \frac{1}{2}x-\frac{1}{2}, & x < 1 \\ & \ln x, & x\geqslant 1 \end{cases}$, if $f(f(a))=\ln f(a)$, then the range of the real number $a$ is
A: $(-\infty,e)$
B: $[e,+\infty)$
C: $\left[\frac{3}{2e},e\right]$
D: $(2,e]$ | \text{B} |
Given $\sin\alpha = -\frac{5}{13}$, and $\tan\alpha > 0$, then $\cos\alpha = \_\_\_\_\_\_$. | -\frac{12}{13} |
Condition of the problem
Find the point of intersection of the line and the plane.
$\frac{x-3}{-1}=\frac{y-4}{5}=\frac{z-4}{2}$
$7 x+y+4 z-47=0$ | (2;9;6) |
A certain brand of packaged rice sold in a shopping mall has a weight (unit: kg) that follows a normal distribution X～N(10, 0.1²). Find the probability that a randomly selected bag of this rice weighs between 9.8 kg and 10.2 kg. | 0.9544 |
If real numbers $x$ and $y$ satisfy $\left(x+i\right)\left(3+yi\right)=2+4i$, then $xy=\left(\ \ \right)$
A: $-1$
B: $1$
C: $3$
D: $-3$ | \text{B} |
3. In the coordinate plane $x O y$, an integer coordinate grid is drawn. The segment ( $p$ ) in this plane is defined by
$$
(p) \quad 7 x-3 y-5=0, \quad 0 \leq x \leq 100
$$
Determine the number of squares of this grid that contain points of the segment ( p). | 301 |
A vegetable vendor purchased cucumbers and potatoes from the vegetable wholesale market for a total of $114$ yuan, and brought a total of $40kg$ of vegetables to the market. The wholesale and retail prices of cucumbers and potatoes on that day are shown in the table below:
| Item | Wholesale Price (yuan/kg) | Retail... | 76 |
In the geometric sequence $\{a_{n}\}$, given that $a_{3}+a_{5}=5$ and $a_{2}a_{6}=4$, then $a_3^2+a_5^2=\left(\ \ \right)$
A: $20$
B: $17$
C: $16$
D: $15$ | B |
4. Simplify $\sqrt{13+4 \sqrt{3}}+\sqrt{13-4 \sqrt{3}}$. | 4\sqrt{3} |
In the rectangular coordinate system $(xOy)$, there is a circle $C_{1}$: $(x+\sqrt{3})^{2}+y^{2}=4$, and a curve $C_{2}$ with parametric equations $\begin{cases} x=2+2\cos \theta \\ y=2\sin \theta \\ \end{cases} (\theta \text{ is the parameter})$. Establish a polar coordinate system with $O$ as the pole and the positiv... | \dfrac{3}{2} |
7.20 Find $\log _{\sqrt{3}} \sqrt[6]{a}$, if it is known that $\log _{a} 27=b$.
Translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly. | \frac{1}{b} |
Let the function $f(x) = ax^3 + 2$. If $f'(－1) = 3$, then $a = \boxed{\text{\_\_\_\_\_\_\_\_}}$. | 1 |
6. Let the function $f(x)=(2 a-b) x+2 b-8$ $(b \leqslant 6)$, when $x \in[-4,2]$, $f(x) \geqslant 0$ always holds. Then the range of $\frac{a^{4}-2 b^{4}}{a^{3} b}$ is $\qquad$ . | \left[-\frac{161}{3},-\frac{39071}{4116}\right] |
Example 1 If $x, y$ are positive numbers, then the minimum value of $\left(x+\frac{1}{y}\right)^{2}+\left(y+\frac{1}{2 x}\right)^{2}$ is
A. $4 \sqrt{2}$
B. $2+2 \sqrt{2}$
C. $3+2 \sqrt{2}$
D. $6 \frac{1}{4}$ | 3+2\sqrt{2} |
The sum of the surface areas of all red blood cells in a normal adult body is equivalent to $2000$ times the body surface area. If the body surface area of an adult is $1800 cm^{2}$, then the sum of the surface areas of all red blood cells in the body can be expressed in scientific notation as ( ).
A: $36\times 10^{5}... | B |
Find the coordinates of the projection of point P(-1, 3, -4) onto the yOz coordinate plane. | (0, 3, -4) |
Given that the diameter of a new type of flu virus is $0.000000815$ meters, express it in scientific notation as ______ meters. | 8.15 \times 10^{-7} |
3. What is $\sqrt{2004 \cdot 2002 \cdot 1998 \cdot 1996+36}$ ? | 3999990 |
In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C_{1}$ is $\rho\cos \theta=4$.
(Ⅰ) Point $M$ is a moving point on curve $C_{1}$, and point $P$ is on the line segment $OM$ and satisfies $|OM|\cdot|OP|=16$. Find the C... | 2+ \sqrt{3} |
7. The dial of a watch lies in a plane. The 12 numbers 1, 2, $\cdots$, 12 are evenly distributed on the circumference of a circle with radius $\frac{\sqrt{2}}{2}$. The vector from number $i$ to number $(i+1)$ is denoted as $\boldsymbol{t}_{i} \boldsymbol{t}_{i+1}$. Then $\boldsymbol{t}_{1} \boldsymbol{t}_{2} \cdot \bol... | 6 \sqrt{3}-9 |
2.2.1. A covered football field of rectangular shape with a length of 90 m and a width of 60 m is being designed, which should be illuminated by four spotlights, each hanging at some point on the ceiling. Each spotlight illuminates a circle, the radius of which is equal to the height at which the spotlight is hanging. ... | 27.1 |
1. $1+3+5+7+\cdots+97+99-2014=$ | 486 |
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