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Questions and Answers
Let $a$ and $b$ be two positive real numbers. The expression $\sqrt{a^2b} + 2\sqrt{b^5} - 3a\sqrt{b} + 2b\sqrt{b} - b^2\sqrt{b}$ is equal to
Let $a$ and $b$ be two positive real numbers. The expression $\sqrt{a^2b} + 2\sqrt{b^5} - 3a\sqrt{b} + 2b\sqrt{b} - b^2\sqrt{b}$ is equal to
- $\sqrt{b}(2b-2a+b^2)$
- $\sqrt{b}(2b-2a+b^2)$ (correct)
- $2\sqrt{b}(b-a+b^2)$
- $\sqrt{b}(2b+2a+b^2)$
- $2\sqrt{b}(b+a+b^2)$
In a café, they serve 4 types of muffins. It is known that all but 78 are chocolate, all but 80 are blueberry, all but 85 are pistachio. Finally, all but 81 are orange. How many muffins are there in total?
In a café, they serve 4 types of muffins. It is known that all but 78 are chocolate, all but 80 are blueberry, all but 85 are pistachio. Finally, all but 81 are orange. How many muffins are there in total?
- 324
- 81
- The available data are not sufficient to establish this.
- 86
- 108 (correct)
The symmetric point of the point $(-1, -4)$ with respect to the line $x + y = 0$ is
The symmetric point of the point $(-1, -4)$ with respect to the line $x + y = 0$ is
- (-1, 4)
- (-4, -1)
- (1, 4)
- (4, 1) (correct)
- (-4, 1)
The numbers $\sqrt{5}$, $3\sqrt{2}$, $2\sqrt{3}$ are in the following order relation:
The numbers $\sqrt{5}$, $3\sqrt{2}$, $2\sqrt{3}$ are in the following order relation:
The relationship $\sqrt{x^2 + 1} > 1$ is satisfied:
The relationship $\sqrt{x^2 + 1} > 1$ is satisfied:
Let $x_1$ and $x_2$ be real numbers such that $x_1^2 + x_2^2 = 5$ and $x_1 x_2 = 2$. Of which of the following polynomials are $x_1$ and $x_2$ the roots?
Let $x_1$ and $x_2$ be real numbers such that $x_1^2 + x_2^2 = 5$ and $x_1 x_2 = 2$. Of which of the following polynomials are $x_1$ and $x_2$ the roots?
The equation $x^2 + y^2 - 2xy = 0$ defines in the Cartesian plane
The equation $x^2 + y^2 - 2xy = 0$ defines in the Cartesian plane
Consider an arbitrary sequence of 45 consecutive integers. What is the maximum number of multiples of 7 that can be in such a sequence? We can say that:
Consider an arbitrary sequence of 45 consecutive integers. What is the maximum number of multiples of 7 that can be in such a sequence? We can say that:
Simplifying and factoring the expression $[(-a)^2 \times (-a)^3 : (-a)^4]^2 + 5(a^6 : a^4 - 6a)$, where a is a non-zero real number, we obtain
Simplifying and factoring the expression $[(-a)^2 \times (-a)^3 : (-a)^4]^2 + 5(a^6 : a^4 - 6a)$, where a is a non-zero real number, we obtain
The figure shows the parabolas of equations $y = -x^2$ and $y = x^2 + 2x - 4$. The colored plane region is the set
The figure shows the parabolas of equations $y = -x^2$ and $y = x^2 + 2x - 4$. The colored plane region is the set
The equation $\log_5(x^2) = \log_5(x)$ has the following solution:
The equation $\log_5(x^2) = \log_5(x)$ has the following solution:
Let us consider two lines, one with an equation $y = ax + 1$ and the other with an equation $y = bx - 3$. If the two lines meet at the abscissa point $x = 5$, then necessarily
Let us consider two lines, one with an equation $y = ax + 1$ and the other with an equation $y = bx - 3$. If the two lines meet at the abscissa point $x = 5$, then necessarily
Let $n$ be a natural number that, written in base 10, has exactly 14 digits. Then the logarithm $\log_{10} n$
Let $n$ be a natural number that, written in base 10, has exactly 14 digits. Then the logarithm $\log_{10} n$
If an object whose volume is $10 \text{ cm}^3$ is reproduced in a larger format, with the same proportions but in a 3:1 scale with respect to the original object, then the volume of the enlarged object is
If an object whose volume is $10 \text{ cm}^3$ is reproduced in a larger format, with the same proportions but in a 3:1 scale with respect to the original object, then the volume of the enlarged object is
The graph of the function is shown in the figure.
The graph of the function is shown in the figure.
The ellipses of equation $x^2 + 4y^2 = 1$ and $x^2 + 9y^2 = 1$ intersect at:
The ellipses of equation $x^2 + 4y^2 = 1$ and $x^2 + 9y^2 = 1$ intersect at:
Flashcards
Simplifying Radicals
Simplifying Radicals
A simplified form of an algebraic expression involving radicals and positive real numbers.
Symmetric Point
Symmetric Point
To find a corresponding position on the opposite side of a line.
Order Relation
Order Relation
Arranging numbers from smallest to largest.
Satisfied Relationship
Satisfied Relationship
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Polynomial Roots
Polynomial Roots
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Straight Line Equation
Straight Line Equation
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Multiples in Sequence
Multiples in Sequence
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Factoring Expressions
Factoring Expressions
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Equation of Parabolas
Equation of Parabolas
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Logarithmic Equation
Logarithmic Equation
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Line Intersection
Line Intersection
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Logarithm Interval
Logarithm Interval
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Scaling Volume
Scaling Volume
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Graph of Function
Graph of Function
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Ellipses Intersection
Ellipses Intersection
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Study Notes
- Training test completed March 18, 2025, started at 05:05 and finished at 05:34
- Time taken was 29 minutes, 13 seconds
- The score was 875.00/1600.00, which is 55%.
Question 1
- Given two positive real numbers a and b, simplify √a²b + 2√b⁵ - 3a√b + 2b√b – b²√b
- The expression is equal to √b(2b - 2a + b²)
Question 2
- There are 4 types of muffins in a café: chocolate, blueberry, pistachio, and orange
- All muffins except 78 are chocolate, all except 80 are blueberry, all except 85 are pistachio, and all except 81 are orange
- There are 108 muffins in total
Question 3
- Determine the symmetric point of (-1, -4) with respect to the line x + y = 0
- The symmetric point is (4, 1)
Question 4
- Arrange the numbers √5, 3√2, and 2√3 in the correct order
- The correct order is √5 < 2√3 < 3√2
Question 5
- Determine when the relationship √(x² + 1) > 1 is satisfied
- It is satisfied for every real number x
Question 6
- Given real numbers x₁ and x₂ such that x₁ + x₂ = 5 and x₁x₂ = 2, find the polynomial with x₁ and x₂ as roots
- The polynomial is x² - 3x + 2
Question 7
- Determine what the equation x² + y² - 2xy = 0 defines in the Cartesian plane
- It defines a straight line
Question 8
- Consider an arbitrary sequence of 45 consecutive integers
- Determine the maximum number of multiples of 7 that can be in such a sequence
- The maximum number is equal to 7
Question 9
- Simplify and factor the expression [(-a)² × (-a)⁵ ÷ (-a)⁴]² + 5(a⁶ ÷ a⁴ - 6a), given a is a non-zero real number
- The result is 6a(a - 5)
Question 10
- The figure shows the parabolas y = -x² and y = x² + 2x - 4 in a colored plane region
- The colored plane region is {(x, y) : -2 ≤ x ≤ 1, x² + 2x - 4 ≤ y ≤ -x²}
Question 11
- Solve the equation log₅(x²) = log₅(x)
- The solution is x = 1
Question 12
- Consider two lines: y = ax + 1 and y = bx - 3
- If the two lines meet at the abscissa point where x = 5, then a<b
Question 13
- Let n be a natural number that, when written in base 10, has exactly 14 digits
- Find the logarithmlog₁₀ n
- It is a number in the interval [13, 14]
Question 14
- An object with volume 10 cm³ is reproduced in a larger format with a 3:1 scale relative to the original
- The volume of the enlarged object will be 270 cm³
Question 15
- The graph of the function is shown in the figure
- The function is y = 1 - log₂(x + 1)
Question 16
- Determine where the ellipses x² + 4y² = 1 and x² + 9y² = 1 intersect
- They intersect at 2 points
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