Тренировочные тесты по математике на английском языке (SET)

Решение задач для поступающих в Назарбаев Университет

Есмагамбетова Лязат Амиргалиевна, Байсуйеуева Жаннат Нурлыбаевна,

учителя математики РСФМСШИ;

Абилева Екатерина, Базарбаева Айдана, выпускницы РСФМСШИ 2014 года

2015 год - год первого выпуска студентов из стен Назарбаев Университета (НУ), которые зарекомендовали себя как выпускники «Лучшей школы РК» в рейтинге НУ. В течение последних пяти лет учащиеся нашей школы усердно готовятся к поступлению в этот вуз, и им это вполне удается. Порядка 50 выпускников РСФМСШИ ежегодно становятся студентами самого престижного вуза страны.

Думаем, что данные задачи будут хорошим подспорьем при подготовке к сдаче экзамена по математике.

SET Sample Test

30 questions, 60 minutes

1. p +=. Find p+q+r.

2. Which of the graphs is shown on the picture?

1. Y=|sinx|

2. Y=sin|x|

3. Y=|sin|x||

4. Y=sinx

5. |y|=sinx

3. cosx=?

4. +++=. Find x.

5. y=sin.

6. If p and q are positive integers with pq=36, then cannot be

1. 

2. 

3. 1

4. 2

5. 9

7. The equation of the perpendicular bisector of the segment joining A(-9,2) to B(3,-4) is …

8. Grades for the test on proofs did not go as well as the teacher had hoped. The mean grade was 68, the median grade was 64, and the standart deviation was 12. The teacher curves the score by raising each score by a total of 6 points. Which of the following statements is true?

1. The new mean is 75.

2. The new mean is 71.

3. The new standart deviation is 7.

D) Several answers are possible.

9. A sphere with diameter 50 cm intersects a plane 14 cm from the center of the sphere. What is the number of square centimeters in the area of the circle formed?

10. Given g(x)= , g(g(x))=?

11. The area of . QE = 48 and QD = 52. To the nearest degree, what is the measure of the largest possible angle of ?

12. Given =c and =2c, a=?

13. The length of the sides of a triangle are 25, 29 and 34. To the nearest tenth of a degree, the measure of the largest angle is…

14. A country commissioner will randomly select 5 people to form a non-partisan committee to look into the issue of country services. If there are 8 Democrats and 6 Republicans to choose from, what is the probability that the Democrats will have the more members than the Republicans on this committee?

15. Given the vectors u=[-5,4] and v=[3,-1], |2u-3v|=?

16. has vertices A(-11,4), B(-3,8) and C(3,-10). The coordinates of the center of the circle circumscribed about are …

17. Each side of the base of a square pyramid is reduced by 20%. By what percent must the height be increased so that the volume of the new pyramid is the same as the volume of the original pyramid?

18. Given =x and =y, ) =?

19. Which of the following statements is true about the function

f(x)=3-2cos()?

1. The graph has an amplitude of 2.

2. The graph is shifted to the right 2.

3. The function satisfies the equation f()=f().

4. Several answers are possible.

20. Which of the following statements is true about the expression -?

1. It has terms if n is even.

2. It has terms if n is odd.

3. The exponent of the last term is always n.

D.Several answers are possible.

21. In three angles A, B and C of a triangle, the measure of angle A exceeds the mesure of angle B by 15 degrees and is twice the measure of angle C. The measures of angles A, B and C are…

22. What is the present age of John's son if 2 years ago it was one-third of John'sage. Take John's present age to be x years.

23. Two square tiles are placed such that they touch each other all along one side. They have areas of 36 sq.in. each. The length of a wire bounding the two tiles will be…

24. By how many degrees does the minute hand of a clock turn in 20 minutes?

25. A man throws a stone up an incline of gradient 1 from the base of the incline. The stone follows a path given by the equation =. What are the coordinates of the point at which the stone hits the incline? Assume the man to be at the origin of the coordinate system.

26. When the sun is above the horizon, how long is the shadow that is cast by a tree 150 feet tall?

27. The mean of fifteen integers is 102. On adding another integer, the mean reduces to 100. What is the new integer added?

28. The probability that the roll of an unbiased cubical die produces a 4 or a 6 is …

29. How many possible five-digit zip codes can be formed from the set {0,1,2...9} such that no code begins with a zero?

30. If the population of a city is increasing by 3 percent every year, and the current population is 300,000, what was the population five years ago?

Answers

1. Answer: 15.

=4+;

P+=4+: p=4, ==: q=6, r=5;

P+q+r=4+6+5=15.

2. Answer: B.

The graph is symmetric with respect to the x-axis, so the function should be even. So, it can be A, B or C, but if the answer is A or C, the graph should lie above the x-axis.

3. Answer:

Cosx=.

4. Answer: 68.

4*=.

5. Answer:2xcos

6. Answer: D.

The factors of 36 are 1,2,3,4,6,9,12,18, and 36. =, =1, and =9. The ratio of the factors cannot be 2.

7. Answer: y+1=2(x+3).

The slope of AB==. The slope of the perpendicular line is 2. The midpoint of AB is (-3,-1), so the equation of the perpendicular bisector is y+1=2(x+3).

8. Answer: A and B.

Adding a constant to a set of data shifts the center of the data but does not alter the spread of the data.

9. Answer: 429

The radius of the sphere is 25. The distance from the center of the sphere to the intersecting plane lies along the perpendicular. Use the Phythagorean Theorem to get +14= or =429. The area of the circle formed by the plane and sphere is , or 429

10. Answer: .

11. Answer: 143.

The area of a triangle is given by the formula A=absin(Substituting the given information, 750=absin(Solving this equation for Q yields that Q=37or 143. Of m=37, then ∠E would be the largest angle of the triangle, and the measure, found using the Law of Cosines, would be 78 , less than 143

12. Answer: .

=c and =2c are equivalent to a= and b===. Therefore, a=.

13. Answer: 77.7

With SSS information, use the Law of Cosines to determine the measure of the largest angle (which is located opposite the longest side). and .

14. Answer: .

There is no indication in the problem that titles are associated with any of the positions on the committee, so this problem represents a combination (order is not important). There are 14 people to choose from, so the number of combinations is =2002. A committee with more Democrats that Republicans could have 5 Democrats and 0 Republicans, 4 D. and 1 R., or 3 D. and 2 R. The number of ways this can happen is ()(+()()+(()=1316. Therefore, the probability that the D. will have more members on the committee that the R. is

15. Answer: .

2u-3v=[2(-5)-3(3), 2(4)-3(-1)]=[-19,11]. |2u-3v|, the distance from the origin to the end of the vector,=.

16. Answer: (-3,-2).

The circumcenter of a triangle is the intersection of the perpendicular bisectors of the sides of the triangle. The slope of AB is , and its midpoints is (-7,6). The equation of the perpendicular bisector to AB is y=-2x-8. The slope of AC is -1, and its midpoint is (-4,-3). The equation of the perpendicular bisector of AC is y=x+1. The intersection of y=-2x-8 and y=x+1 is (-3,-2).

17. Answer: 56.25.

The volume of the original pyramid is . The volume of the new pyramid is , where H is the new height. Set these expressions equal. Solve for H: 1.5625h. The height needs to be increased by 56.25% to keep the volume the same.

18. Answer: 1.

Use the change-of-base formula to rewrite as . Apply the properties of logarithms to simplify the numerator and denominator of the fraction.

19. Answer: A and C.

f(x)=3-2cos()=-2cos( The amplitude of the function is 2, the graph is shifted to the right , and the period is .

20. Answer: A and B.

Both and contain n + 1 terms when they are

expanded. All the odd terms will be positive, and the even

terms will be negative.

If n is even, there will be odd terms andeven terms in the

expansion of . The odd terms will subtract out, leaving terms.

The last terms in the expansions of both binomials, , will subtract out

of the answer.

If n is odd, there will be an equal number of even and odd terms.

The odd terms will subtract out leaving terms in the difference.

21. Answer: ∠A=78, ∠B=63, ∠C=39.

∠A+(∠A-15)+ =180.

22. Answer: .

If John's present age is x years, his age two years ago would be (x-2) years. His son's age two years ago was of his age. So, two years ago John's son age was , and his present age is

23. Answer: 36.

The two square tiles put together as shown in the figure below, form a rectangle with length equal to the sum of the sides of the two squares and the width equal to that of a side of the squares. To find the length of a wire that can bind the two square tiles, we need to find the perimeter of the rectangle.

The perimeter of this rectangle is 2(12+6)=36.

24. Answer: 120.

In 60 minutes, the minute hand of a clock does one complete revolution or rotates by 360 Therefore, in 20 minutes, it will move by( =120.

25. Answer: (4,4).

The incline has a gradient (or slope) of 1, which means that the equation of the line is y=x. Finding the coordinates of the point where the stone hits the incline is the same as finding the point of intersection of the two curves y=x and =4x. Solving the two equations simultaneously, we have x=0 or x=4. Since the value x=0 does not apply in this case, we have x=4, which gives us y=4. Thus, the point at which the stone hits the incline is (4, 4).

26. Answer: 412 feet.

If z is the length of the shadow cast by tree, than tan20=, x==412 ft.

27. Answer: 70.

Let the new added integer be x. Now, if the mean of the 15 numbers before we added the new integer was 102, then we have: =102 ⇒ S=1530, where S is the sum of the fifteen integers. Now, we know that the mean changes to 100 after adding x to the list of the fifteen integers. This gives us =100 ⇒ (S =x)=1,600 ⇒x =1,600- S=70.

28. Answer: .

A roll of an unbiased die has equal probability to produce any of the six possible outcomes (1, 2, 3, 4, 5, or 6). Thus, the probability that 4 or a 6 turns up will be

29. Answer: 90,000 options.

Since no zip codes can begin with a zero, we have nine digits to choose from for the first of the five-digit code. Also, since there is no restriction on repeating the digits, the remaining four digits of the five-digit code can each take 10values. Therefore, we have a total of =90,000 options.

30. Answer: 258,783.

This problem can be solved using the formula of exponential growth, which is A=P, where A = the amount present finally, P= the initial amount, r = the rate of growth, and t = the total time. Thus, for our problem, we have 300,000 = P⇒ P 5 258,783.