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Questions and Answers
What is the possible value of n when a1, a2, and a3 are in arithmetic progression?
What is the possible value of n when a1, a2, and a3 are in arithmetic progression?
For which value of x is the 6th term in the expansion equal to 84?
For which value of x is the 6th term in the expansion equal to 84?
What is true about the function f(m) if f(m) = ∑(30 choose i)(m choose i)?
What is true about the function f(m) if f(m) = ∑(30 choose i)(m choose i)?
What is the nature of the integer just below (53 + 7 - 2 × 711) in relation to its prime factors?
What is the nature of the integer just below (53 + 7 - 2 × 711) in relation to its prime factors?
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If 50 choose r = 29 × 43 × where a ∈ I+ is not a multiple of 47, which of the following is true?
If 50 choose r = 29 × 43 × where a ∈ I+ is not a multiple of 47, which of the following is true?
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Which of the following statements about f(m) is correct?
Which of the following statements about f(m) is correct?
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What characteristic does the integer just below 53 + 7 - 2 × 711 have?
What characteristic does the integer just below 53 + 7 - 2 × 711 have?
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In the context of the equation 50 choose r = 29 × 43 × where a is not a multiple of 47, what can be inferred about a?
In the context of the equation 50 choose r = 29 × 43 × where a is not a multiple of 47, what can be inferred about a?
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What is the number of ways to divide 200 people into 100 couples?
What is the number of ways to divide 200 people into 100 couples?
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For the seven-digit number made from the digits 8, 7, 6, 4, 3, x, and y to be divisible by 3, which of the following is correct?
For the seven-digit number made from the digits 8, 7, 6, 4, 3, x, and y to be divisible by 3, which of the following is correct?
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If $10! = 2^p \cdot 3^q \cdot 5^r \cdot 7^s$, which statement is true?
If $10! = 2^p \cdot 3^q \cdot 5^r \cdot 7^s$, which statement is true?
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If Kanchan has two married friends among ten friends and wants to invite five, which scenario is not valid?
If Kanchan has two married friends among ten friends and wants to invite five, which scenario is not valid?
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What is the maximum value of $x + y$ if all digits must be distinct and that the number is divisible by 3?
What is the maximum value of $x + y$ if all digits must be distinct and that the number is divisible by 3?
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Which of the following statements about number 200! and its factorization is correct?
Which of the following statements about number 200! and its factorization is correct?
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What can be inferred about the prime factorization of 10! based on its components?
What can be inferred about the prime factorization of 10! based on its components?
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Which of these options represents the correct number of distinct ways Kanchan can invite her friends?
Which of these options represents the correct number of distinct ways Kanchan can invite her friends?
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If a ball is drawn randomly and replaced along with 5 additional balls of the same color, what is the probability that the next ball drawn is red if the total number of red balls is 5?
If a ball is drawn randomly and replaced along with 5 additional balls of the same color, what is the probability that the next ball drawn is red if the total number of red balls is 5?
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What is the correct probability value for obtaining an obtuse triangle from a 5-sided polygon?
What is the correct probability value for obtaining an obtuse triangle from a 5-sided polygon?
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In a regular 6-sided polygon, what is the probability of randomly selecting points to form an obtuse triangle?
In a regular 6-sided polygon, what is the probability of randomly selecting points to form an obtuse triangle?
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What is the expression for the probability that the event E1 occurs, given the context of card drawing without replacement?
What is the expression for the probability that the event E1 occurs, given the context of card drawing without replacement?
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What is the probability expression for the event E4, where the drawn 12th card is club ‘6’?
What is the probability expression for the event E4, where the drawn 12th card is club ‘6’?
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Which option correctly states the probability for the event E2, that the 8th card drawn is the third ace?
Which option correctly states the probability for the event E2, that the 8th card drawn is the third ace?
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If three distinct points are chosen at random from a 7-sided polygon, what is the probability that they form an obtuse triangle?
If three distinct points are chosen at random from a 7-sided polygon, what is the probability that they form an obtuse triangle?
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What is the probability of selecting the first king on the 9th draw under the events described?
What is the probability of selecting the first king on the 9th draw under the events described?
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If the first digit of a natural number N is deleted, resulting in 29, what is the sum of all the digits of N?
If the first digit of a natural number N is deleted, resulting in 29, what is the sum of all the digits of N?
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How many handshakes occur at a party with n married couples if everyone shakes hands with every person except their spouse?
How many handshakes occur at a party with n married couples if everyone shakes hands with every person except their spouse?
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What is the required condition for the positive integer m in the expression $\sum_{n=1}^{m} (a_{n+2} + b_{n+1} + 2a_{n}) = 2016$?
What is the required condition for the positive integer m in the expression $\sum_{n=1}^{m} (a_{n+2} + b_{n+1} + 2a_{n}) = 2016$?
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Using the digits 0, 1, 2, 3, and 4, what value of k represents the number of ten digit sequences where the difference between two consecutive digits is 1?
Using the digits 0, 1, 2, 3, and 4, what value of k represents the number of ten digit sequences where the difference between two consecutive digits is 1?
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What is the probability of drawing 4 slips from a box containing numbers 1 to 10 (each number appears on 4 slips) such that 2 slips have number 'a' and the other 2 slips have number 'b'?
What is the probability of drawing 4 slips from a box containing numbers 1 to 10 (each number appears on 4 slips) such that 2 slips have number 'a' and the other 2 slips have number 'b'?
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Given the probability of getting at least one head from tossing a coin n times is greater than the probability of getting at least two tails, what should be the minimum value of n?
Given the probability of getting at least one head from tossing a coin n times is greater than the probability of getting at least two tails, what should be the minimum value of n?
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What is the value of f(n) where n=4 for the function defining handshakes among n married couples?
What is the value of f(n) where n=4 for the function defining handshakes among n married couples?
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If the sequences generated allow digits to differ by 1, how many such sequences can be formed with digits 0 to 4?
If the sequences generated allow digits to differ by 1, how many such sequences can be formed with digits 0 to 4?
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What is the probability of randomly selecting three numbers from the set { 1, 3, 3^2, ..., 3^n } that form an increasing geometric progression if n is odd?
What is the probability of randomly selecting three numbers from the set { 1, 3, 3^2, ..., 3^n } that form an increasing geometric progression if n is odd?
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From the rational numbers in the interval (2010, 2011) with non-zero digits in decreasing order, what is the probability that a randomly selected number has exactly seven digits after the decimal point?
From the rational numbers in the interval (2010, 2011) with non-zero digits in decreasing order, what is the probability that a randomly selected number has exactly seven digits after the decimal point?
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Which of the following represents the probability that a randomly selected rational number from (2010, 2011) contains the digit 3 after the decimal point?
Which of the following represents the probability that a randomly selected rational number from (2010, 2011) contains the digit 3 after the decimal point?
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What is the probability that the last digit of a randomly selected rational number after the decimal point is at least 4?
What is the probability that the last digit of a randomly selected rational number after the decimal point is at least 4?
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What is the value of k in the equation $egin{pmatrix} a_{11} + 4 & a_{12} & a_{13} \ a_{21} & a_{22} + 4 & a_{23} + 5\ a_{31} & a_{32} & a_{33} + 4 \ ext{ } & & ext{ } \ a_{11} + 1 & a_{12} & a_{13} \ a_{21} & a_{22} + 1 & a_{23} \ a_{31} & a_{32} & a_{33} + 1 \ ext{ } & & ext{ }}
ight] = 0$?
What is the value of k in the equation $egin{pmatrix} a_{11} + 4 & a_{12} & a_{13} \ a_{21} & a_{22} + 4 & a_{23} + 5\ a_{31} & a_{32} & a_{33} + 4 \ ext{ } & & ext{ } \ a_{11} + 1 & a_{12} & a_{13} \ a_{21} & a_{22} + 1 & a_{23} \ a_{31} & a_{32} & a_{33} + 1 \ ext{ } & & ext{ }} ight] = 0$?
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Consider the equation $a^2 + b^2 + c^2 = 2$. What are the possible integer values of a, b, and c?
Consider the equation $a^2 + b^2 + c^2 = 2$. What are the possible integer values of a, b, and c?
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If A is defined as $A = \begin{pmatrix} a & b & c \ b & c & a \ c & a & b \end{pmatrix}$, what form can the determinant of A take given $a^2 + b^2 + c^2 = 2$?
If A is defined as $A = \begin{pmatrix} a & b & c \ b & c & a \ c & a & b \end{pmatrix}$, what form can the determinant of A take given $a^2 + b^2 + c^2 = 2$?
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If $|A| = 8$ and $|adj(I - 2A)| = K$, what could be a possible alternative expression for K in terms of determinants?
If $|A| = 8$ and $|adj(I - 2A)| = K$, what could be a possible alternative expression for K in terms of determinants?
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For a rational number selected randomly from (2010, 2011), what is the probability that the last digit after the decimal point is exactly 4?
For a rational number selected randomly from (2010, 2011), what is the probability that the last digit after the decimal point is exactly 4?
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Which characteristic of matrices A and B is indicated by the condition $tr(AB) - tr(A) imes tr(B) + tr(AB^{-1}) + 2$?
Which characteristic of matrices A and B is indicated by the condition $tr(AB) - tr(A) imes tr(B) + tr(AB^{-1}) + 2$?
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If n is even, what is the probability that three numbers selected from { 1, 3, 3^2, ..., 3^n } form an increasing geometric progression?
If n is even, what is the probability that three numbers selected from { 1, 3, 3^2, ..., 3^n } form an increasing geometric progression?
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Given that $A^3 - 6A^2 + 12A - 81 = 0$, which of the following could represent the relationship between A and its eigenvalues?
Given that $A^3 - 6A^2 + 12A - 81 = 0$, which of the following could represent the relationship between A and its eigenvalues?
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If matrices A and B are both non-singular and satisfy the condition $AB = BA$, which of the following is a potential property they share?
If matrices A and B are both non-singular and satisfy the condition $AB = BA$, which of the following is a potential property they share?
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What is an implication of having $AA^T = 4I$ for matrix A?
What is an implication of having $AA^T = 4I$ for matrix A?
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For the ordered triplet $(eta, eta, eta)$ constructed from $x + y + 3z = 1$ and $x - y + z = 1$, which of the following statements is true about the values of $a$ and $b$?
For the ordered triplet $(eta, eta, eta)$ constructed from $x + y + 3z = 1$ and $x - y + z = 1$, which of the following statements is true about the values of $a$ and $b$?
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In the expansion involving sums, such as $r = \sum_{r=1}^{k} 2 {r}\binom{n}{k-1}$, what type of combinatorial identity does it represent?
In the expansion involving sums, such as $r = \sum_{r=1}^{k} 2 {r}\binom{n}{k-1}$, what type of combinatorial identity does it represent?
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Study Notes
Algebra Problems
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Arithmetic Progression (AP): If a1, a2, and a3 are in arithmetic progression, possible values of n are given in the options (a-d). Determine the correct answer.
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Expansion of Logarithms: The problem involves finding x where the 6th term in a logarithmic expansion is 84. The expression is complex, involving nested logarithms with terms (9 + 7 + ...).
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Summations and Combinations: A function f(m) is defined as a summation of combinations. The possible maximum value and sum of f(m) values relate to combinations C and Pascal's Triangle. It also involves possible values of f(m) being divisible by 50.
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Integer Values: An expression involving the sum of 53th and 7th powers minus some factor of 711 is evaluated. The integer result's factors and properties are to be determined.
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Combination Expressions: A combination involving 50C5r / nCr has factored constants. Analyze the properties related to primes to find which condition related to r is true.
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Dividing into Couples: Finding the number of ways 200 people can be divided into 100 couples. The expression involve factorials and combinations.
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Divisibility by 3: A seven-digit number using 8, 7, 6, 4, 3, x, and y digits is divisible by 3. Find possible ranges for sums, products, and differences involving x and y, given their digits are distinct.
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Factorials: 10! is expressed as the product of primes (2p, 3q, 5r, 7s). Determine values of exponents and related properties like divisors.
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Invitations: Kanchan has 10 friends, 2 are married. She wants to invite five. If married couples don't want to be invited separately, determine the ways she can invite five people.
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Probability with Urn and Colors: A Probability problem regarding drawing balls from an urn related to colors (red) and number of balls. Find possible value(s) of m+n.
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Geometric Progression: Probabilities are calculated for random selection of three points from a circle forming an obtuse-angle triangle. Values are specific for different numbers of sides (5,8,6,7)
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Probability with Cards: Calculate probabilities related to sequence of draws from a deck of cards (hearts, aces, kings, clubs). Relevant combinations are calculated for each.
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Probability of Selection: A problem about selecting three numbers in geometric progression from a set. The probability is considered a function of n, odd or even.
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Digit Arrangement: A problem related to rational numbers within a specific interval, analyzing digit values after the decimal points. The probabilities of digit composition are evaluated.
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Matrices: Problems related to matrices, matrix operations (multiplication, transpose & inverse), and determinants. Complex matrix equations are solved, and values are determined.
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Algebraic Equations: Problems involving equations (sum of variables and coefficients) with unknown values. A minimum value needs to be evaluated or identified by solving simultaneous algebraic equations.
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Combinations of Remainders: Evaluate a sum related to remainders of a sequence of numbers when divided by 3. Values depend upon remainders from integer values.
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Digit Sequences: The problem involves finding the number of ten-digit sequences using specific digits (0, 1, 2, 3, 4) with consecutive differences of specific values.
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Probability with Slips: A probability problem using a set with numbers. Combination issues related to a set of chosen slips by picking out two numbers in specific relations. Relevant combination values are evaluated in probability calculation.
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Probability and Coin Toss: Problems related to tossing a coin and the possible outcome that a specific number of heads and tails might occur. Evaluating probabilities of obtaining heads and/or tails with conditions.
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Geometric Progressions Evaluate the probability that randomly chosen triples are in geometric progression.
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Description
Test your knowledge of various algebra concepts including arithmetic progression, logarithmic expansions, and combinatorial functions. This quiz covers a range of problems designed to challenge your understanding of algebraic principles, from summations to integer values. Perfect for students preparing for their math exams!
