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TURIN Math and Logic Sample Test
1.
The circle c has equation
𝑥
2
+ 𝑦
2
= 1 and the circle d has equation(𝑥 − 2)
2
+ 𝑦
2
= 9.
One of the following statements is correct. Which one?
a) They intersect in one point and
c
is
contained d
b)
They intersect in two distinct points
c)
They have no points of intersections
d)
They intersect in four distinct points
e) They intersect in one point and d is contained in c
2. The inequality
1 + sin 3𝑥 ≥ 2 is verified:
a) For all real x
b) Only if
𝑥 = 2𝑘𝜋 (k is an integer)
c) If
𝑥 =
𝜋
6
+
2𝑘𝜋
3
(k is an integer)
d) For no real value of x
e) Only if
𝑥 =
𝜋
2
+
2𝑘𝜋
3
(k is an integer)
3. Among the following second degree equations which one has two real, distinct solution r
and s such that
𝑟 = 8 − 𝑠
a)
8𝑥
2
− 𝑥 = 0
b)
𝑥
2
− 8𝑥 + 16 = 0
c)
𝑥
2
− 6𝑥 + 8 = 0
d)
𝑥
2
− 8𝑥 + 15 = 0
e)
𝑥
2
− 8𝑥 + 20 = 0
4. How can you insert the + (plus) or the - (minus) sign in the following sequence of
numbers: 10 11 12 13 14 in order to have +10 as a result? (an example: if I insert + - in
the sequence 5 6 7 the result is 5+6-7=+4, if I insert - + the result is 5-6+7=+6)
a) + + - -
b) - + + -
c) - + - +
d) - - - +
e) - - + +
5. The equation
log
3
(2 − 𝑥) = 3 has the solution:
a)
𝑥 = −log
2
3
b)
𝑥 = −1
c)
𝑥 = −25
d)
𝑥 = 3
3+log
3
2
e)
𝑥 = −log
3
2
6. The set of real numbers that satisfy the inequality
(𝑥
2
+ 5)(𝑥 + 2) ≤ 0, from the
geometrical point of view, is composed by
a) One half line
b) Two segments
c) Two half lines
d) One segment and one half line
e) One segment
7. The measure of the angle
𝛼 is 3 radians; then we can say that:
a)
sin 𝛼 and cos 𝛼 are both negative
b)
cos 𝛼 < 0
c)
sin 𝛼 < 0
d)
cos 𝛼 > 0
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e) The measure of the
𝛼 in degrees is180
°
bigger than
8. The integer numbers a,b,c,d and e satisfy the relations:
𝑐 = 𝑏 + 2
𝑑 = 𝑒 + 1
𝑒 = 𝑎 − 2
𝑎 = 𝑏 + 4
One of the following statements is true: which one?
a)
𝑏 > 𝑑
b)
𝑐 = 𝑒
c) The order of numbers is the same as the alphabetical order
(𝑎 < 𝑏 < 𝑐 < 𝑑 < 𝑒)
d)
𝑑 < 𝑎 < 𝑒
e)
𝑐 > 𝑒
9. If
𝑎 > 0 which of the following statements is true:
a)
log
4
𝑎 =
1
2
log
2
𝑎
b)
log
4
𝑎 = 2 log
2
𝑎
c) All the other statements are false
d)
log
4
𝑎 = √log
2
𝑎
e)
log
4
𝑎 = (log
2
𝑎)
2
10. Every morning professor Anders Celsius looks at the outdoor thermometer before leaving
home. If the temperature is not below
15° C, he won’t wear his overcoat.
This means that
a) If he doesn’t wear his overcoat, then the temperature is not below
15° C
b) If he wears his overcoat, then the temperature is below
15° C
c) If the temperature is not below
15° C, then he can go out with or without his overcoat
indifferently
d) If the temperature is below
15° C, then he doesn’t wear his overcoat
e) If he doesn’t wear his overcoat, the temperature is below
15° C
11. Tell which number fits in the sequence 76845 67845 68745 68475 … choosing among
a) 87654
b) 78564
c) 45678
d) 65487
e) 68457
12. The equation
𝑥
10
+ 3𝑥
5
= 10
a) Has one (and only one) real solution
b) Has two (and only two) real solutions
c) Has no real solutions
d) Has six distinct real solutions
e) Can not be solved, because its degree is too high
13. The equation
3
2𝑥
− 12 ∙ 𝑥 = 0
a) Has the solutions:
𝑥
1
= −1; 𝑥
2
= 1
b) Has the solutions:
𝑥
1
= 1; 𝑥
2
= 2
c) Has the solutions:
𝑥
1
= 3; 𝑥
2
= 9
d) Has no solutions
e) Has the solutions:
𝑥
1
= 0; 𝑥
2
= 1; 𝑥
3
= 2
14. The number
𝑥 =
((9
.3
)
2
)
3
((11
2
)
3
)
2
a)
9
8
11
7
b)
(
9
11
)
18
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c)
(
9
11
)
12
∙ 9
6
d)
(
9
11
)
6
e)
(
9
11
)
12
15. In a car race
• Mona surpasses Mary
• Megan is surpassed by Mary
• Margaret surpasses Mona
Then, the correct order of arrival is:
a) Margaret – Mona – Mary – Megan
b) Mona – Mary – Megan – Margaret
c) Megan – Mona – Mary – Margaret
d) Mary – Megan – Mona – Margaret
e) Margaret – Mary – Megan – Mona
16. The ages of three brothers are consecutive numbers. The difference of the square of the
age of the eldest and the square of the age of the youngest is 100. How is the youngest
brother old?
a) 28 years
b) 18 years
c) 26 years
d) 25 years
e) 24 years
17. The 22 students of the course of Hystory og Italian Opera must write a short paper about
at least two composers; they have a better grade if they choose to write a paper about one
more composer. Among the students, 17 choose Verdi, 15 choose Rossini and 15
Donizetti. How many students have chosen to write three papers?
a) 3 students
b) It is impossible to answer this question
c) Nobody has chosen three composers
d) 15 students
e) 2 students
18. The real number √
7√7
3
∙ √
3
7
4
3
+ 2√3
3
eqauals to:
a)
7√3
3
b) √
3
3
+ √3
3
c)
√2 ∙ √3
6
d)
21√21
e)
√3
2
3
19. Consider the numbers
2
−3
, 3
−3
, 2
−2
and
3
−2
; which is the correct way of ordering them
in ascending order (from the greatest to the smallest)?
a)
3
−3
> 2
−3
> 3
−2
> 2
−2
b)
2
−2
> 2
−3
> 3
−2
> 3
−3
c)
3
−3
> 3
−3
> 2
−2
> 2
−3
d)
2
−3
> 2
−2
> 3
−3
> 3
−2
e)
3
−3
> 3
−3
> 2
−3
> 2
−2
20. A high school is made by 25 pupils, of which 11 cycle, 13 swim and 8 go skiing. There is
no one practicing all three sports. In yesterday’s test the sportsmen (those practicing at
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least one sport) went pretty well: they all got 8 or 9 (out of 10) as a mark. There were 9
pupils who got less than 8. We can therefore deduce that:
a) One sportsmen might have gotten a 10
b) Four cyclists can also ski
c) Three cyclists only can ski as well
d) Someone in the class got 10
e) Every cyclists can also ski
21. The equation
𝑥
4
− 3𝑥
2
+ 2 = 0
a) Has at least three solutions with the same sign
b) Has two (and only two) real solutions
c) Has no real solutions
d) Has only negative solutions
e) Has four real solutions
22. The line r has equation
2𝑥 + 2𝑦 = 1 and the lines has equation 𝑦 = 7 − 𝑥; we can
affirm that the two lines
a) Intersect in one point and they are not perpendicular
b) Coincide
c) Are parallel and distinct
d) Intersect in two (and only two) points
e) Are perpendicular
23. Laureen is talking with Humphrey “if the weather is good on Sunday I will go on horse
races, I will also bring my friend Sam with me, if he feels OK”. On Sunday afternoon
Humphrey meets Loureen in town. What is true?
a) It is good weather, but Sam is not well
b) It is raining
c) It is raining and Sam is not well
d) Sam is not well
e) It is good weather, but Sam did not want to go the horse races
24. A point P(x,y) of the Cartesian plane belongs to the first or to the third quadrant
(excluding coordinate axes) if and only if its coordinates satisfy the relation:
a)
𝑥 > 0 𝑎𝑛𝑑 𝑦 > 0
b)
𝑥 + 𝑦 > 0
c)
𝑥
𝑦
< 0
d)
𝑥 > 0 𝑜𝑟 𝑦 > 0
e)
𝑥𝑦 > 0
25. The polynomial
𝑃(𝑥) = 𝑥
3
− 3𝑥
2
+ 𝑘𝑥 − 12 (k is a real parameter) is divisible by 𝑥 − 2
a) If
𝑘 = ±8
b) If k is even
c) If
𝑘 = 8
d) If
𝑘 = 5
e) For no values of k
26. Tell which number fits in the sequence 33 17 9 5 … choosing among
a) 1
b) 4
c) 3
d) 2
e) 0
27. A real number x satisfies the inequality
𝑥+2
𝑥−2
< 3 If and only if:
a)
𝑥 < 2 𝑜𝑟 𝑥 > 4
b)
2 < 𝑥 < 4
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c)
2 < 𝑥 ≤ 4
d)
𝑥 ≤ 2
e)
4 ≤ 𝑥
28. The polynomial P(x) is divisible by
𝑥
2
and the polynomial Q(x) is divisible by (
𝑥
2
− 1).
From these informations, we can affirm that the polynomial
𝑃
2
(𝑥)𝑄(𝑥)
a) Is divisible by
𝑃 − 𝑄
b) Is a multiple of
𝑥
5
c) Is divisible by
𝑥
5
− 𝑥
4
d) Is a multiple of
𝑥 − 2
e) Is divisible by
(𝑥 + 1)
2
29. PQR is a right – angled triangle (the right angle has vertex R) and RH is the altitude
drawn from R to the hypotenuse PQ. Denoting by
𝑃𝑄 the measure of the segment PQ, we
can say that:
a)
𝑃𝐻
2
+ 𝐻𝑅
2
= 𝑃𝑅
2
b)
𝑃𝑄
2
− 𝑅𝐻
2
= 𝑄𝑅
2
c)
𝑃𝐻
2
+ 𝐻𝑄
2
= 𝑃𝑄
2
d)
𝑃𝑅
2
− 𝑄𝑅
2
= 𝑃𝑄
2
e)
𝐻𝑅
2
+ 𝑄𝑅
2
= 𝐻𝑄
2
30. A common divisor of the monomials
4𝑎
2
𝑏
5
, 2𝑎
2
𝑏
3
𝑐, 8𝑎
3
𝑏
2
a)
4𝑎
2
𝑏
b)
2𝑎𝑏
2
𝑐
c)
64𝑎
7
𝑏
10
𝑐
d)
𝑎𝑏
3
e)
𝑏𝑎
2
31. Discussing their personal finances, four friends (Feruza, Gulzoda, Laziza and Nigora)
state that:
• Feruza has less money than Laziza
• Gulzoda has less money than Laziza
• Laziza has more money than Nigora
• Gulzoda has more money than Feruza
Then, which of the following statements is NOT NECESSARILY correct?
a) Nigora has less money than Laziza
b) Feruza is the poorest among the friends
c) The richest among the friends is Laziza
d) The alphabetical order of the names is not the same as the (increasing order) of the
money owned
e) Gulzoda is not poorest
32. The circle C has a diameter of 4 m; the square S has a side of 2 dm and is completely
contained in C. The area of the region contained in C and lying outside S is:
a) (
20
2
𝜋 − 2
2
)𝑑𝑚
2
b) (
20
2
𝜋 − 2
2
)𝑚
2
c)
(20 − 2)
2
𝜋𝑑𝑚
2
d) 0
e) (
4
2
𝜋 − 2
2
)𝑑𝑚
2
33. An investor buys shares with value of 100,000 Euro. The next month he reads in a
newspaper that his shares have increased their value by 20%. Another month passes and
his shares were reduced in value by 20%. He then decides to sell his shares.
The total value is:
a) 100,000 Euro
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b) 96,000 Euro
c) 80,000 Euro
d) 120,000 Euro
e) 92,000 Euro
34. The product of 40 integer numbers is positive. From this information we can deduce that
it is necessarily true that:
a) The number of positive factors is either zero or an even number
b) All factors are positive
c) 20 factors are positive and 20 factors are negative
d) 2 factors are positive and 38 factors are negative
e) At least two factors are negative
35. Starting with the number
30
19
, we divide it by 3 and then we divide the result by
10
5
.
At the end we have the number:
a)
30
14
b)
2
14
∙ 3
18
∙ 5
14
c) The number is too big and I cannot compute it
d)
30
19
∙ 10
14
e)
10
14
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