Geography
There are two diametrically opposite points on the unit circle. A funny incident from life. Coordinates on the sphere

There are two diametrically opposite points on the unit circle. A funny incident from life. Coordinates on the sphere

Final work in MATHEMATICS
Grade 10
April 28, 2017
Option MA00602
(a basic level of)
Completed by: Full name_______________________________________ class ______
Instructions for performing the work
You are given 90 minutes to complete the final math work. Job
includes 15 tasks and consists of two parts.
The answer in the tasks of the first part (1-10) is an integer,
decimal fraction or sequence of numbers. Write your answer in the field
answer in the text of the work.
In task 11 of the second part you need to write down the answer in a special
the field allocated for this.
In tasks 12-14 of the second part you need to write down the solution and answer
in the field provided for this purpose. The answer to task 15 is
function graph.
Each of tasks 5 and 11 is presented in two versions, of which
You only need to select and execute one.
When doing work, you cannot use textbooks, work
notebooks, reference books, calculator.
If necessary, you can use a draft. Entries in draft will not be reviewed or graded.
You can complete tasks in any order, the main thing is to do it correctly
solve as many tasks as possible. We advise you to save time
skip a task that cannot be completed immediately and move on
to the next. If after completing all the work you still have time,
You will be able to return to missed tasks.
We wish you success!

Part 1
In tasks 1–10, give the answer in the form of an integer, decimal or
sequences of numbers. Write your answer in the answer field in the text
work.
1

The price for an electric kettle was increased by 10% and amounted to
1980 rubles. How many rubles did the kettle cost before the price increase?

Oleg and Tolya left school at the same time and went home in the same
Expensive. The boys live in the same house. The figure shows a graph
the movements of each: Oleg - with a solid line, Tolya - with a dotted line. By
the vertical axis shows the distance (in meters), the horizontal axis shows the distance
travel time for each in minutes.

Using the graph, choose the correct statements.
1)
2)
3)

Oleg came home before Tolya.
Three minutes after leaving school, Oleg caught up with Tolya.
Throughout the entire journey the distance between the boys was less
100 meters.
4) In the first six minutes the boys covered the same distance.

Answer: ___________________________

Find the meaning of the expression

π
π
- 2 sin 2.
8
8

Answer: ___________________________
StatGrad 2016−2017 academic year. Publishing online or in print
without the written consent of StatGrad it is prohibited

Mathematics. Grade 10. Option 00602 (basic level)

There are two marked on the unit circle
diametrically opposite points Pα and
Pβ corresponding to rotations through angles α and
β (see figure).
Is it possible to say that:
1) α  β  0
2) cosα  cosβ
3) α  β  2π
4) sin α  sin β  0

Please indicate the numbers in your answer. true statements no spaces, commas and
other additional characters.
Answer: ___________________________
Select and complete only ONE of tasks 5.1 or 5.2.
5.1

The figure shows a graph
function y  f (x) defined on the interval   3;11 .
Find the smallest value
functions on the segment  1; 5 .

Answer: ___________________________
5.2

Solve the equation log 2 4 x5  6.

Answer: ___________________________

StatGrad 2016−2017 academic year. Publishing online or in print
without the written consent of StatGrad it is prohibited

Mathematics. Grade 10. Option 00602 (basic level)

A plane passing through points A, B and C (see.
figure), splits the cube into two polyhedra. One of
it has four sides. How many faces does the second one have?

Answer: ___________________________
7

Choose the numbers of the correct statements.
1)
2)
3)
4)

In space, through a point not lying on a given line, you can
draw a plane that does not intersect a given line, and, moreover, only
one.
An inclined line drawn to a plane forms the same angle with
all straight lines lying in this plane.
A plane can be drawn through any two intersecting lines.
Through a point in space that does not lie on a given line, one can
Draw two straight lines that do not intersect a given line.

In your answer, indicate the numbers of the correct statements without spaces, commas and
other additional characters.
Answer: ___________________________
8

On the poultry farm there are only chickens and ducks, and there are 7 times more chickens than
ducks Find the probability that a randomly selected farm
the bird turns out to be a duck.
Answer: ___________________________

The roof of the canopy is located at an angle of 14
to the horizontal. Distance between two supports
is 400 centimeters. Using the table,
determine how many centimeters one support is
longer than the other.
α
13
14
15
16
17
18
19

Sin α
0,225
0,241
0,258
0,275
0,292
0,309
0,325

Cos α
0,974
0,970
0,965
0,961
0,956
0,951
0,945

Tg α
0,230
0,249
0,267
0,286
0,305
0,324
0,344

Answer: ___________________________
StatGrad 2016−2017 academic year. Publishing online or in print
without the written consent of StatGrad it is prohibited

Mathematics. Grade 10. Option 00602 (basic level)

Find the smallest natural seven-digit number that is divisible by 3,
but not divisible by 6 and each digit of which, starting from the second, is less
previous one.
Answer: ___________________________
Part 2
In task 11, write your answer in the space provided. In tasks
12-14 you need to write down the solution and answer in the specially designated
for this field. The answer to task 15 is the graph of the function.
Select and complete only ONE of the tasks: 11.1 or 11.2.

2
. Write down three different possible values
2
such angles. Give your answer in radians.

Find the smallest natural number, which is greater than log 7 80 .

The cosine of the angle is 

StatGrad 2016−2017 academic year. Publishing online or in print
without the written consent of StatGrad it is prohibited

Mathematics. Grade 10. Option 00602 (basic level)

In triangle ABC, sides AB and BC are marked
points M and K, respectively, so that BM: AB  1: 2, and
BK:BC  2:3. How many times the area of triangle ABC?
greater than the area of triangle MVK?

Choose some pair of numbers a and b so that the inequality ax  b  0
satisfied exactly three of the five points marked in the figure.
-1

StatGrad 2016−2017 academic year. Publishing online or in print
without the written consent of StatGrad it is prohibited

Mathematics. Grade 10. Option 00602 (basic level)

The price of the iron was increased twice by the same percentage. On
how many percent did the price of the iron increase each time if it
the initial cost is 2000 rubles, and the final cost is 3380 rubles?

StatGrad 2016−2017 academic year. Publishing online or in print
without the written consent of StatGrad it is prohibited

Mathematics. Grade 10. Option 00602 (basic level)

The function y  f (x) has the following properties:
1) f (x)  3 x  4 at 2  x  1;
2) f (x)  x  2 at 1  x  0;
3) f (x)  2  2 x at 0  x  2;
4) the function y  f (x) is periodic with period 4.
Draw a graph of this function on the segment  6;4.
y

StatGrad 2016−2017 academic year. Publishing online or in print
without the written consent of StatGrad it is prohibited

I once witnessed a conversation between two applicants:

– When should you add 2πn, and when should you add πn? I just can't remember!

– And I have the same problem.

I just wanted to tell them: “You don’t need to memorize, but understand!”

This article is addressed primarily to high school students and, I hope, will help them solve the simplest trigonometric equations with “understanding”:

Number circle

Along with the concept of a number line, there is also the concept of a number circle. As we know, in a rectangular system coordinates of the circle, s center at the point (0;0) and radius 1, is called unit. Let’s imagine the number line as a thin thread and wind it around this circle: we will attach the origin (point 0) to the “right” point of the unit circle, we will wrap the positive semi-axis counterclockwise, and the negative semi-axis in the direction (Fig. 1). Such a unit circle is called a numerical circle.

Properties of the number circle

Each real number lies on one point on the number circle.
At each point on the number circle there are infinitely many real numbers. Since the length of the unit circle is 2π, the difference between any two numbers at one point on the circle is equal to one of the numbers ±2π; ±4π ; ±6π ; ...

Let's conclude: knowing one of the numbers of point A, we can find all the numbers of point A.

Let's draw the diameter of the AC (Fig. 2). Since x_0 is one of the numbers of point A, then the numbers x_0±π ; x_0±3π; x_0±5π; ... and only they will be the numbers of point C. Let's choose one of these numbers, say, x_0+π, and use it to write down all the numbers of point C: x_C=x_0+π+2πk ,k∈Z. Note that the numbers at points A and C can be combined into one formula: x_(A ; C)=x_0+πk ,k∈Z (for k = 0; ±2; ±4; ... we obtain the numbers of point A, and for k = ±1; ±3; … – numbers of point C).

Let's conclude: knowing one of the numbers at one of the points A or C of the diameter AC, we can find all the numbers at these points.

Two opposite numbers are located on points of the circle that are symmetrical with respect to the abscissa axis.

Let's draw a vertical chord AB (Fig. 2). Since points A and B are symmetrical about the Ox axis, the number -x_0 is located at point B and, therefore, all numbers of point B are given by the formula: x_B=-x_0+2πk ,k∈Z. We write the numbers at points A and B using one formula: x_(A ; B)=±x_0+2πk ,k∈Z. Let us conclude: knowing one of the numbers at one of the points A or B of the vertical chord AB, we can find all the numbers at these points. Consider the horizontal chord AD and let's find the numbers point D (Fig. 2). Since BD is a diameter and the number -x_0 belongs to point B, then -x_0 + π is one of the numbers of point D and, therefore, all the numbers of this point are given by the formula x_D=-x_0+π+2πk ,k∈Z. The numbers at points A and D can be written using one formula: x_(A ; D)=(-1)^k∙x_0+πk ,k∈Z . (for k= 0; ±2; ±4; … we get the numbers of point A, and for k = ±1; ±3; ±5; … – the numbers of point D).

Let's conclude: knowing one of the numbers at one of the points A or D of the horizontal chord AD, we can find all the numbers at these points.

Sixteen main points of the number circle

In practice, the solution to most simplest trigonometric equations associated with sixteen points on the circle (Fig. 3). What are these dots? Red, blue and green dots divide the circle into 12 equal parts. Since the length of the semicircle is π, then the length of the arc A1A2 is π/2, the length of the arc A1B1 is π/6, and the length of the arc A1C1 is π/3.

Now we can indicate one number at a time:

π/3 on C1 and

The vertices of the orange square are the midpoints of the arcs of each quarter, therefore, the length of the arc A1D1 is equal to π/4 and, therefore, π/4 is one of the numbers of point D1. Using the properties of the number circle, we can use formulas to write down all the numbers on all marked points of our circle. The coordinates of these points are also marked in the figure (we will omit the description of their acquisition).

Having learned the above, we now have sufficient preparation to solve special cases (for nine values of the number a) simplest equations.

Solve equations

1)sinx=1⁄(2).

– What is required of us?

– Find all those numbers x whose sine is equal to 1/2.

Let's remember the definition of sine: sinx – ordinate of the point on the number circle on which the number x is located. We have two points on the circle whose ordinate is equal to 1/2. These are the ends of the horizontal chord B1B2. This means that the requirement “solve the equation sinx=1⁄2” is equivalent to the requirement “find all the numbers at point B1 and all the numbers at point B2.”

2)sinx=-√3⁄2 .

We need to find all the numbers at points C4 and C3.

3) sinx=1. On the circle we have only one point with ordinate 1 - point A2 and, therefore, we need to find only all the numbers of this point.

Answer: x=π/2+2πk, k∈Z.

4)sinx=-1 .

Only point A_4 has an ordinate of -1. All the numbers of this point will be the horses of the equation.

Answer: x=-π/2+2πk, k∈Z.

5) sinx=0 .

On the circle we have two points with ordinate 0 - points A1 and A3. You can indicate the numbers at each of the points separately, but given that these points are diametrically opposite, it is better to combine them into one formula: x=πk,k∈Z.

Answer: x=πk ,k∈Z .

6)cosx=√2⁄2 .

Let's remember the definition of cosine: cosx is the abscissa of the point on the number circle on which the number x is located. On the circle we have two points with the abscissa √2⁄2 - the ends of the horizontal chord D1D4. We need to find all the numbers on these points. Let's write them down, combining them into one formula.

Answer: x=±π/4+2πk, k∈Z.

7) cosx=-1⁄2 .

We need to find the numbers at points C_2 and C_3.

Answer: x=±2π/3+2πk , k∈Z .

10) cosx=0 .

Only points A2 and A4 have an abscissa of 0, which means that all the numbers at each of these points will be solutions to the equation.
.

The solutions to the equation of the system are the numbers at points B_3 and B_4. To the cosx inequality<0 удовлетворяют только числа b_3
Answer: x=-5π/6+2πk, k∈Z.

Note that for any admissible value of x, the second factor is positive and, therefore, the equation is equivalent to the system

The solutions to the system equation are the number of points D_2 and D_3. The numbers of point D_2 do not satisfy the inequality sinx≤0.5, but the numbers of point D_3 do.

website, when copying material in full or in part, a link to the source is required.

Question: On a circle, diametrically opposite points A and B and a different point C are chosen. The tangent drawn to the circle at point A and the line BC intersect at point D. Prove that the tangent drawn to the circle at point C bisects the segment A.D. The incircle of triangle ABC touches sides AB and BC at points M and N respectively. A line passes through the midpoint of AC parallel to the line. MN intersects lines BA and BC at points D and E, respectively. Prove that AD=CE.

On the circle, diametrically opposite points A and B and a different point C are chosen. The tangent drawn to the circle at point A and the straight line BC intersect at point D. Prove that the tangent drawn to the circle at point C bisects the segment AD. The incircle of triangle ABC touches sides AB and BC at points M and N respectively. A line passes through the midpoint of AC parallel to the line. MN intersects lines BA and BC at points D and E, respectively. Prove that AD=CE.

There are two diametrically opposite points on the unit circle. A funny incident from life. Coordinates on the sphere

Answers: