Q. In a survey, 45 people like apples, 30 like oranges, and 15 like both. How many people like only oranges?
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Solution
The number of people who like only oranges is 30 - 15 = 15.
Correct Answer: B — 30
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Q. In a survey, 50 people like apples, 40 like bananas, and 20 like both. How many people like only bananas?
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Solution
The number of people who like only bananas is calculated as: Only Bananas = Total Bananas - Both = 40 - 20 = 20.
Correct Answer: B — 30
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Q. In a survey, the average age of a group of people is 30 years. If one person aged 40 leaves the group, what will be the new average age if the group originally had 10 people? (2023)
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Solution
New total age = (30 × 10) - 40 = 260. New average = 260 / 9 = 28.89, which rounds to 29.
Correct Answer: B — 29
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Q. In a survey, the average age of a group of people is 40 years. If one person aged 60 leaves the group, what will be the new average age if the group originally had 10 members? (2023)
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Solution
Total age = 40 × 10 = 400. New total age = 400 - 60 = 340. New average = 340 / 9 = 37.78, which rounds to 38.
Correct Answer: A — 38
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Q. In a system of linear equations, what does it mean if the equations are dependent?
A.
They have exactly one solution.
B.
They have infinitely many solutions.
C.
They have no solutions.
D.
They are inconsistent.
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Solution
Dependent equations represent the same line, leading to infinitely many solutions.
Correct Answer: B — They have infinitely many solutions.
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Q. In a system of linear equations, what does it mean if the equations are inconsistent?
A.
There is exactly one solution.
B.
There are infinitely many solutions.
C.
There is no solution.
D.
The equations are dependent.
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Solution
Inconsistent equations do not intersect, meaning there is no solution.
Correct Answer: C — There is no solution.
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Q. In a triangle, if one angle is 90 degrees, what can be inferred about the other two angles?
A.
They are both acute angles.
B.
One is obtuse and the other is acute.
C.
They are both obtuse angles.
D.
They are equal.
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Solution
In a right triangle, the sum of the other two angles must be 90 degrees, making them both acute.
Correct Answer: A — They are both acute angles.
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Q. In a triangle, if one angle is twice the size of another angle, and the third angle is 30 degrees less than the largest angle, what is the measure of the smallest angle?
A.
30 degrees
B.
45 degrees
C.
60 degrees
D.
75 degrees
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Solution
Let the smallest angle be x. Then the second angle is 2x and the largest angle is 2x - 30. The sum of angles in a triangle is 180 degrees. Therefore, x + 2x + (2x - 30) = 180. Solving this gives x = 30 degrees.
Correct Answer: A — 30 degrees
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Q. In a triangle, if the lengths of two sides are 7 cm and 10 cm, which of the following could be the length of the third side?
A.
3 cm
B.
15 cm
C.
5 cm
D.
17 cm
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Solution
According to the triangle inequality theorem, the length of the third side must be less than the sum and greater than the difference of the other two sides. Therefore, the third side must be greater than 3 cm and less than 17 cm.
Correct Answer: A — 3 cm
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Q. In an arithmetic progression, if the 3rd term is 15 and the 6th term is 24, what is the common difference?
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Solution
Let the first term be a and the common difference be d. From the equations a + 2d = 15 and a + 5d = 24, we can solve for d, which gives d = 3.
Correct Answer: B — 4
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Q. In an arithmetic progression, if the 4th term is 20 and the 7th term is 26, what is the first term?
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Solution
Let the first term be a and the common difference be d. From the equations a + 3d = 20 and a + 6d = 26, we can solve for a and find it to be 12.
Correct Answer: B — 12
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Q. In an arithmetic progression, if the 5th term is 20 and the 10th term is 35, what is the first term?
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Solution
Let the first term be a and the common difference be d. From the given terms, we have a + 4d = 20 and a + 9d = 35. Solving these gives a = 10.
Correct Answer: B — 10
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Q. In an arithmetic progression, if the first term is 12 and the last term is 48, and there are 10 terms, what is the common difference?
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Solution
Using the formula for the last term, 48 = 12 + (10-1)d. Solving gives d = 4.
Correct Answer: A — 4
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Q. In an arithmetic progression, if the first term is 5 and the common difference is 3, what is the 10th term?
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Solution
The nth term of an AP is given by a + (n-1)d. Here, a = 5, d = 3, and n = 10. So, the 10th term = 5 + (10-1) * 3 = 5 + 27 = 32.
Correct Answer: A — 32
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Q. In an arithmetic progression, if the sum of the first 10 terms is 250, what is the first term if the common difference is 5?
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Solution
Using the formula S_n = n/2 * (2a + (n-1)d), we can substitute n = 10 and d = 5 to find a = 20.
Correct Answer: B — 20
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Q. In converting the hexadecimal number 1A to decimal, what is the result?
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Solution
The hexadecimal number 1A converts to decimal as 1*16^1 + 10*16^0 = 26.
Correct Answer: A — 26
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Q. In how many different ways can the letters of the word 'MATH' be arranged?
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Solution
The word 'MATH' has 4 distinct letters. The number of arrangements is 4! = 24.
Correct Answer: B — 24
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Q. In how many ways can 3 students be selected from a class of 8 to represent in a competition?
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Solution
The number of ways to choose 3 students from 8 is given by 8C3 = 56.
Correct Answer: A — 56
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Q. In how many ways can 4 different books be arranged on a shelf?
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Solution
The number of arrangements of 4 different books is 4! = 24.
Correct Answer: B — 24
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Q. In how many years will a sum of money triple itself at a compound interest rate of 10% per annum?
A.
10 years
B.
12 years
C.
15 years
D.
20 years
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Solution
Using the formula A = P(1 + r)^n, we set A = 3P and solve for n, which gives approximately 12 years.
Correct Answer: B — 12 years
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Q. In modular arithmetic, what is the multiplicative inverse of 3 modulo 11?
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Solution
The multiplicative inverse of 3 mod 11 is 4, since (3 * 4) mod 11 = 12 mod 11 = 1.
Correct Answer: B — 7
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Q. In modular arithmetic, which of the following is a valid operation?
A.
Adding two numbers and taking mod
B.
Subtracting two numbers and taking mod
C.
Multiplying two numbers and taking mod
D.
All of the above
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Solution
All operations (addition, subtraction, multiplication) are valid in modular arithmetic.
Correct Answer: D — All of the above
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Q. In polynomial long division, what is the first step when dividing 4x^3 + 2x^2 - x by 2x?
A.
Divide the leading term of the dividend by the leading term of the divisor.
B.
Multiply the divisor by the leading term of the dividend.
C.
Subtract the product from the dividend.
D.
Write down the remainder.
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Solution
The first step in polynomial long division is to divide the leading term of the dividend by the leading term of the divisor.
Correct Answer: A — Divide the leading term of the dividend by the leading term of the divisor.
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Q. In polynomial long division, what is the first step when dividing 4x^3 + 2x^2 - x by 2x + 1?
A.
Multiply the divisor by the leading term of the dividend.
B.
Subtract the product from the dividend.
C.
Identify the degree of both polynomials.
D.
Write the remainder.
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Solution
The first step in polynomial long division is to multiply the divisor by the leading term of the dividend.
Correct Answer: A — Multiply the divisor by the leading term of the dividend.
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Q. In probability theory, what does the term 'independent events' mean?
A.
Events that cannot occur at the same time.
B.
Events where the outcome of one does not affect the other.
C.
Events that are mutually exclusive.
D.
Events that have the same probability of occurring.
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Solution
Independent events are defined as events where the outcome of one does not affect the outcome of the other, a key concept in probability theory.
Correct Answer: B — Events where the outcome of one does not affect the other.
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Q. In the context of algebra, which of the following statements best describes the relationship between variables and constants?
A.
Variables are fixed values while constants can change.
B.
Constants are fixed values while variables can change.
C.
Both variables and constants can change.
D.
Neither variables nor constants can change.
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Solution
In algebra, constants are fixed values that do not change, while variables represent values that can vary.
Correct Answer: B — Constants are fixed values while variables can change.
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Q. In the context of functions and graphs, which of the following statements best describes a linear function?
A.
A function that has a constant rate of change and can be represented by a straight line.
B.
A function that varies exponentially and is represented by a curve.
C.
A function that has multiple outputs for a single input.
D.
A function that is defined only for positive integers.
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Solution
A linear function is characterized by a constant rate of change, which means that its graph is a straight line.
Correct Answer: A — A function that has a constant rate of change and can be represented by a straight line.
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Q. In the context of functions and graphs, which of the following statements best describes a quadratic function?
A.
It is a linear function with a constant slope.
B.
It is a polynomial function of degree two.
C.
It is a function that can only take positive values.
D.
It is a function that has a single output for every input.
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Solution
A quadratic function is defined as a polynomial function of degree two, typically represented in the form f(x) = ax^2 + bx + c.
Correct Answer: B — It is a polynomial function of degree two.
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Q. In the context of functions, what does the term 'asymptote' refer to?
A.
A line that the graph approaches but never touches.
B.
A point where the graph intersects the x-axis.
C.
A maximum or minimum point on the graph.
D.
A point of discontinuity in the graph.
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Solution
An asymptote is a line that a graph approaches as it heads towards infinity but does not intersect.
Correct Answer: A — A line that the graph approaches but never touches.
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Q. In the context of functions, what does the term 'domain' refer to?
A.
The set of all possible output values.
B.
The set of all possible input values.
C.
The maximum value of the function.
D.
The minimum value of the function.
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Solution
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
Correct Answer: B — The set of all possible input values.
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