Modern Math for Competitive Exam Success

Modern Math

Binomial Theorem Permutation & Combination Probability Sequences & Series Set Theory Venn Diagrams

Q. If a sequence is defined as a_n = 3n + 1, what is the value of a_7? (2023)

A. 22
B. 21
C. 20
D. 19

Solution

Substituting n = 7 into the formula gives a_7 = 3(7) + 1 = 21 + 1 = 22.

Correct Answer: A — 22

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Q. If a sequence is defined as a_n = 3n + 2, what is the value of a_5?

A. 15
B. 17
C. 20
D. 22

Solution

Substituting n = 5 into the equation gives a_5 = 3(5) + 2 = 15 + 2 = 17.

Correct Answer: B — 17

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Q. If a sequence is defined by the formula a_n = 3n + 2, what is the value of a_7? (2023)

A. 23
B. 20
C. 19
D. 25

Solution

Substituting n = 7 into the formula gives a_7 = 3(7) + 2 = 21 + 2 = 23.

Correct Answer: A — 23

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Q. If a student can choose 2 subjects from 5 available subjects, how many different combinations can be made?

A. 10
B. 15
C. 20
D. 5

Solution

The number of ways to choose 2 subjects from 5 is given by 5C2 = 10.

Correct Answer: A — 10

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Q. If a student can choose 2 subjects from 5 available subjects, how many different combinations of subjects can be chosen?

A. 10
B. 15
C. 20
D. 5

Solution

The number of ways to choose 2 subjects from 5 is 5C2 = 10.

Correct Answer: A — 10

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Q. If a team of 4 is to be selected from 10 players, how many different teams can be formed?

A. 210
B. 120
C. 300
D. 150

Solution

The number of ways to choose 4 players from 10 is given by 10C4 = 210.

Correct Answer: A — 210

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Q. If a team of 4 is to be selected from 8 players, how many different teams can be formed?

A. 70
B. 56
C. 28
D. 12

Solution

The number of ways to choose 4 players from 8 is given by 8C4 = 70.

Correct Answer: B — 56

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Q. If a team of 5 is to be selected from 10 players, how many different teams can be formed?

A. 252
B. 120
C. 210
D. 300

Solution

The number of ways to choose 5 players from 10 is given by 10C5 = 252.

Correct Answer: A — 252

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Q. If set A = {1, 2, 3, 4} and set B = {3, 4, 5, 6}, what is the difference A - B?

A. {1, 2}
B. {3, 4}
C. {5, 6}
D. {}

Solution

The difference A - B includes elements in A that are not in B, which are {1, 2}.

Correct Answer: A — {1, 2}

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Q. If set A = {1, 2, 3, 4} and set B = {3, 4, 5, 6}, what is the intersection of sets A and B? (2023)

A. {1, 2}
B. {3, 4}
C. {5, 6}
D. {1, 2, 5, 6}

Solution

The intersection of sets A and B includes the elements that are common to both sets. Therefore, the intersection is {3, 4}.

Correct Answer: B — {3, 4}

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Q. If set A = {1, 2, 3} and set B = {2, 3, 4}, what is A - B? (2023)

A. {1}
B. {2, 3}
C. {3, 4}
D. {}

Solution

A - B represents the elements in A that are not in B. Thus, A - B = {1}.

Correct Answer: A — {1}

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Q. If set A = {x | x is an even number less than 10} and set B = {x | x is a prime number less than 10}, what is A ∩ B?

A. {2, 4, 6, 8}
B. {2}
C. {2, 3, 5, 7}
D. {2, 3, 5, 7, 4, 6, 8}

Solution

The intersection A ∩ B includes elements that are both even and prime, which is {2}.

Correct Answer: B — {2}

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Q. If set A contains the elements {1, 2, 3, 4} and set B contains the elements {3, 4, 5, 6}, what is the intersection of sets A and B?

A. {1, 2}
B. {3, 4}
C. {5, 6}
D. {1, 2, 3, 4, 5, 6}

Solution

The intersection of sets A and B is the set of elements that are common to both sets. Therefore, the intersection is {3, 4}.

Correct Answer: B — {3, 4}

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Q. If set A contains the numbers {1, 2, 3, 4, 5} and set B contains the numbers {4, 5, 6, 7, 8}, what is the intersection of sets A and B?

A. {1, 2, 3}
B. {4, 5}
C. {6, 7, 8}
D. {1, 2, 3, 4, 5, 6, 7, 8}

Solution

The intersection of sets A and B is the set of elements that are common to both sets, which is {4, 5}.

Correct Answer: B — {4, 5}

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Q. If set C = {x | x is a multiple of 3 and less than 30}, how many elements are in set C?

A. 8
B. 9
C. 10
D. 11

Solution

The multiples of 3 less than 30 are {3, 6, 9, 12, 15, 18, 21, 24, 27}, totaling 9 elements.

Correct Answer: B — 9

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Q. If set P = {1, 2, 3, 4} and set Q = {3, 4, 5, 6}, what is the difference P - Q?

A. {1, 2}
B. {3, 4}
C. {5, 6}
D. {1, 2, 5, 6}

Solution

The difference P - Q includes elements in P that are not in Q, which is {1, 2}.

Correct Answer: A — {1, 2}

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Q. If set P = {x | x is an even number less than 10} and set Q = {x | x is a prime number less than 10}, what is the intersection of sets P and Q?

A. {2, 4, 6, 8}
B. {2, 3, 5, 7}
C. {2}
D. {4, 6, 8}

Solution

The intersection of sets P and Q includes elements that are both even and prime. The only even prime number is 2.

Correct Answer: C — {2}

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Q. If set P = {x | x is an even number less than 10} and set Q = {x | x is a prime number less than 10}, what is the difference P - Q?

A. {2, 4, 6, 8}
B. {4, 6, 8}
C. {2, 6, 8}
D. {2, 4, 6, 8, 3, 5, 7}

Solution

Set P = {2, 4, 6, 8} and set Q = {2, 3, 5, 7}. The difference P - Q = {4, 6, 8}.

Correct Answer: B — {4, 6, 8}

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Q. If set P = {x | x is an even number less than 10} and set Q = {x | x is a prime number less than 10}, what is P ∩ Q?

A. {2, 4, 6, 8}
B. {2, 3, 5, 7}
C. {2}
D. {4, 6, 8}

Solution

The intersection P ∩ Q includes only the even prime number, which is {2}.

Correct Answer: C — {2}

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Q. If set P = {x | x is an even number less than 10} and set Q = {x | x is a prime number less than 10}, what is the union of sets P and Q?

A. {2, 3, 4, 5, 6, 8}
B. {2, 3, 5, 7}
C. {2, 4, 6, 8}
D. {2, 3, 4, 5, 7, 8}

Solution

Set P = {2, 4, 6, 8} and set Q = {2, 3, 5, 7}. The union is {2, 3, 4, 5, 6, 7, 8}.

Correct Answer: D — {2, 3, 4, 5, 7, 8}

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Q. If set R = {1, 2, 3, 4, 5} and set S = {4, 5, 6, 7}, what is the symmetric difference of sets R and S?

A. {1, 2, 3, 6, 7}
B. {4, 5}
C. {1, 2, 3, 4, 5, 6, 7}
D. {6, 7}

Solution

The symmetric difference of sets R and S includes elements that are in either set but not in both. Thus, it is {1, 2, 3, 6, 7}.

Correct Answer: A — {1, 2, 3, 6, 7}

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Q. If set R = {1, 2, 3, 4} and set S = {3, 4, 5, 6}, what is the difference R - S?

A. {1, 2}
B. {3, 4}
C. {5, 6}
D. {}

Solution

The difference R - S includes elements in R that are not in S, which is {1, 2}.

Correct Answer: A — {1, 2}

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Q. If set R = {1, 2, 3, 4} and set S = {3, 4, 5, 6}, what is the symmetric difference of sets R and S?

A. {1, 2, 5, 6}
B. {3, 4}
C. {1, 2, 3, 4, 5, 6}
D. {3, 4, 5}

Solution

The symmetric difference is the set of elements in either set R or set S but not in both, which is {1, 2, 5, 6}.

Correct Answer: A — {1, 2, 5, 6}

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Q. If set X = {a, b, c} and set Y = {b, c, d}, what is the union of sets X and Y?

A. {a, b, c, d}
B. {b, c}
C. {a, b}
D. {c, d}

Solution

The union of sets X and Y includes all unique elements from both sets. Thus, the union is {a, b, c, d}.

Correct Answer: A — {a, b, c, d}

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Q. If the 3rd term of a geometric sequence is 12 and the common ratio is 2, what is the first term? (2023)

A. 3
B. 6
C. 4
D. 8

Solution

The 3rd term is given by ar^2. So, 12 = a(2^2) => a = 12/4 = 3.

Correct Answer: B — 6

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Q. If the 3rd term of an arithmetic sequence is 12 and the 7th term is 24, what is the common difference? (2023)

A. 2
B. 3
C. 4
D. 5

Solution

Let the first term be a and the common difference be d. Then, a + 2d = 12 and a + 6d = 24. Solving these gives d = 3.

Correct Answer: B — 3

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Q. If the Binomial Theorem is applied to (x + 1)^n, what is the sum of the coefficients of the expansion?

A. 0
B. 1
C. n
D. 2^n

Solution

The sum of the coefficients in the expansion of (x + 1)^n is found by substituting x = 1, which gives 2^n.

Correct Answer: D — 2^n

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Q. If the Binomial Theorem is applied to (x + 2)^3, what is the coefficient of x^2?

A. 3
B. 6
C. 12
D. 8

Solution

The coefficient of x^2 in the expansion of (x + 2)^3 is given by C(3, 2) * 2^1 = 3 * 2 = 6.

Correct Answer: C — 12

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Q. If the Binomial Theorem is applied to (x + 2)^4, what is the term containing x^2?

A. 12x^2
B. 24x^2
C. 36x^2
D. 48x^2

Solution

The term containing x^2 is C(4,2) * x^2 * 2^2 = 6 * x^2 * 4 = 24x^2.

Correct Answer: B — 24x^2

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Q. If the Binomial Theorem is applied to (x + y)^4, what is the term containing x^2y^2?

A. 6x^2y^2
B. 4x^2y^2
C. 8x^2y^2
D. 12x^2y^2

Solution

The term containing x^2y^2 in the expansion of (x + y)^4 is given by 4C2 * x^2 * y^2 = 6x^2y^2.

Correct Answer: A — 6x^2y^2

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Modern Math

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