Q. What is the significance of the 'Pythagorean theorem' in modern mathematics?
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A.
It relates to the properties of circles.
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B.
It provides a method for calculating areas.
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C.
It establishes a relationship between the sides of a right triangle.
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D.
It is used to solve polynomial equations.
Solution
The Pythagorean theorem establishes a fundamental relationship between the sides of a right triangle.
Correct Answer:
C
— It establishes a relationship between the sides of a right triangle.
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Q. What is the significance of the examples provided in the passage regarding inequalities?
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A.
They illustrate the author's personal experiences.
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B.
They serve to highlight the complexity of the issue.
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C.
They are irrelevant to the main argument.
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D.
They simplify the concept of inequalities.
Solution
The examples are used to illustrate the complexity of inequalities, reinforcing the author's argument.
Correct Answer:
B
— They serve to highlight the complexity of the issue.
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Q. What is the significance of the vertex in the graph of a quadratic function?
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A.
It represents the maximum or minimum point of the function.
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B.
It is the point where the function crosses the y-axis.
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C.
It indicates the x-intercepts of the function.
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D.
It is the point where the function is undefined.
Solution
The vertex of a quadratic function is the point at which the function reaches its maximum or minimum value.
Correct Answer:
A
— It represents the maximum or minimum point of the function.
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Q. What is the significance of the x-intercepts of a function?
-
A.
They indicate the maximum value of the function.
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B.
They indicate the minimum value of the function.
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C.
They are the points where the function crosses the x-axis.
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D.
They are the points where the function is undefined.
Solution
The x-intercepts of a function are the points where the graph crosses the x-axis, meaning the output of the function is zero at those points.
Correct Answer:
C
— They are the points where the function crosses the x-axis.
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Q. What is the simplified form of (2^3)^2? (2023)
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A.
2^5
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B.
2^6
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C.
2^7
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D.
2^8
Solution
Using the property of exponents (a^m)^n = a^(m*n), we have (2^3)^2 = 2^(3*2) = 2^6.
Correct Answer:
B
— 2^6
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Q. What is the simplified form of (x^2 * y^3)^(2)? (2023)
-
A.
x^4 * y^6
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B.
x^2 * y^3
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C.
x^6 * y^4
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D.
x^5 * y^3
Solution
Using the power of a product property, we have (x^2 * y^3)^(2) = x^(2*2) * y^(3*2) = x^4 * y^6.
Correct Answer:
A
— x^4 * y^6
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Q. What is the simplified form of (x^3 * x^2) / x^4? (2023)
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A.
x^1
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B.
x^0
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C.
x^2
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D.
x^5
Solution
Using the property of exponents, we have (x^3 * x^2) / x^4 = x^(3+2-4) = x^1.
Correct Answer:
A
— x^1
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Q. What is the smallest 3-digit number that is divisible by 9?
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A.
108
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B.
90
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C.
99
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D.
100
Solution
The smallest 3-digit number is 100. The first number greater than or equal to 100 that is divisible by 9 is 108.
Correct Answer:
A
— 108
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Q. What is the smallest multiple of 9 that is greater than 50?
Solution
The smallest multiple of 9 greater than 50 is 54.
Correct Answer:
A
— 54
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Q. What is the smallest number that is divisible by both 18 and 24?
Solution
The LCM of 18 and 24 is 72, which is the smallest number divisible by both.
Correct Answer:
A
— 72
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Q. What is the solution set for the inequality 3x - 5 < 4?
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A.
x < 3
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B.
x > 3
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C.
x < 2
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D.
x > 2
Solution
Adding 5 to both sides gives 3x < 9, and dividing by 3 gives x < 3.
Correct Answer:
A
— x < 3
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Q. What is the solution set of the equations x + y = 10 and x - y = 2? (2023)
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A.
(6, 4)
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B.
(8, 2)
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C.
(5, 5)
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D.
(7, 3)
Solution
Solving the equations simultaneously gives x = 6 and y = 4, hence the solution set is (6, 4).
Correct Answer:
A
— (6, 4)
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Q. What is the solution set of the equations x + y = 5 and x + y = 10?
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A.
All real numbers
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B.
No solution
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C.
One solution
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D.
Infinitely many solutions
Solution
The two equations represent parallel lines, which means they do not intersect and thus have no solution.
Correct Answer:
B
— No solution
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Q. What is the solution set of the inequality 2x - 4 < 0?
-
A.
x < 2
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B.
x > 2
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C.
x = 2
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D.
x ≤ 2
Solution
Solving the inequality gives 2x < 4, thus x < 2.
Correct Answer:
A
— x < 2
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Q. What is the solution set of the system of equations: x + y = 5 and x - y = 1?
-
A.
(2, 3)
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B.
(3, 2)
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C.
(1, 4)
-
D.
(4, 1)
Solution
Solving the system gives x = 2 and y = 3, thus the solution set is (2, 3).
Correct Answer:
A
— (2, 3)
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Q. What is the solution to the equation 3x - 4 = 5?
Solution
To solve for x, add 4 to both sides to get 3x = 9, then divide by 3 to find x = 3.
Correct Answer:
B
— 3
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Q. What is the sum of the decimal equivalents of '101' and '110' in binary?
Solution
'101' is 5 and '110' is 6 in decimal. Their sum is 5 + 6 = 11.
Correct Answer:
C
— 7
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Q. What is the sum of the digits in the number '1011' in binary?
Solution
The sum of the digits in '1011' is 1 + 0 + 1 + 1 = 3.
Correct Answer:
C
— 4
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Q. What is the sum of the digits of the number '123' in base 4?
Solution
The digits are 1, 2, and 3. Their sum is 1 + 2 + 3 = 6.
Correct Answer:
A
— 6
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Q. What is the sum of the first 15 terms of an arithmetic progression where the first term is 10 and the common difference is 2?
-
A.
150
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B.
160
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C.
170
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D.
180
Solution
The sum of the first n terms S_n = n/2 * (2a + (n-1)d). Here, S_15 = 15/2 * (2*10 + 14*2) = 15/2 * (20 + 28) = 15/2 * 48 = 360.
Correct Answer:
B
— 160
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Q. What is the sum of the first 15 terms of an arithmetic progression where the first term is 2 and the common difference is 4?
-
A.
120
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B.
130
-
C.
140
-
D.
150
Solution
The sum of the first n terms S_n = n/2 * (2a + (n-1)d). Here, S_15 = 15/2 * (2*2 + 14*4) = 15/2 * (4 + 56) = 15/2 * 60 = 450.
Correct Answer:
A
— 120
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Q. What is the sum of the first 5 terms of a GP where the first term is 2 and the common ratio is 3?
-
A.
242
-
B.
364
-
C.
486
-
D.
728
Solution
The sum of the first n terms of a GP is given by S_n = a(1 - r^n) / (1 - r). Here, S_5 = 2(1 - 3^5) / (1 - 3) = 2(1 - 243) / (-2) = 242.
Correct Answer:
A
— 242
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Q. What is the sum of the interior angles of a quadrilateral? (2023)
-
A.
180 degrees
-
B.
360 degrees
-
C.
540 degrees
-
D.
720 degrees
Solution
The sum of the interior angles of any quadrilateral is 360 degrees.
Correct Answer:
B
— 360 degrees
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Q. What is the sum of the numbers '101' and '110' in binary?
-
A.
1011
-
B.
111
-
C.
1001
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D.
1100
Solution
In binary, '101' + '110' = '1011'.
Correct Answer:
A
— 1011
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Q. What is the sum of the numbers '12' and '21' in base-3?
-
A.
100
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B.
110
-
C.
120
-
D.
200
Solution
'12' in base-3 is 3 and '21' is 7. Their sum is 10, which is '100' in base-3.
Correct Answer:
A
— 100
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Q. What is the sum of the numbers 1010 (binary) and 1101 (binary) in decimal?
Solution
First convert to decimal: 1010 = 10 and 1101 = 13. The sum is 10 + 13 = 23.
Correct Answer:
B
— 20
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Q. What is the sum of the roots of the quadratic equation 2x^2 - 3x + 1 = 0?
Solution
The sum of the roots is given by -b/a = 3/2.
Correct Answer:
B
— 3/2
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Q. What is the sum of the roots of the quadratic equation 2x^2 - 4x + 1 = 0?
Solution
The sum of the roots of a quadratic equation ax^2 + bx + c = 0 is given by -b/a. Here, it is -(-4)/2 = 2.
Correct Answer:
A
— 2
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Q. What is the sum of the roots of the quadratic equation 2x^2 - 8x + 6 = 0?
Solution
The sum of the roots is given by -b/a = 8/2 = 4.
Correct Answer:
B
— 4
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Q. What is the surface area of a cube with a side length of 5 cm? (2021)
-
A.
50 cm²
-
B.
75 cm²
-
C.
100 cm²
-
D.
125 cm²
Solution
Surface area = 6 * (side)² = 6 * (5)² = 6 * 25 = 150 cm².
Correct Answer:
C
— 100 cm²
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