Algebra Problems Class 10

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?

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?

If 50 choose r = 29 × 43 × where a ∈ I+ is not a multiple of 47, which of the following is true?

Which of the following statements about f(m) is correct?

What characteristic does the integer just below 53 + 7 - 2 × 711 have?

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?

What is the number of ways to divide 200 people into 100 couples?

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?

If $10! = 2^p \cdot 3^q \cdot 5^r \cdot 7^s$, which statement is true?

If Kanchan has two married friends among ten friends and wants to invite five, which scenario is not valid?

What is the maximum value of $x + y$ if all digits must be distinct and that the number is divisible by 3?

Which of the following statements about number 200! and its factorization is correct?

What can be inferred about the prime factorization of 10! based on its components?

Which of these options represents the correct number of distinct ways Kanchan can invite her friends?

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?

What is the correct probability value for obtaining an obtuse triangle from a 5-sided polygon?

In a regular 6-sided polygon, what is the probability of randomly selecting points to form an obtuse triangle?

What is the expression for the probability that the event E1 occurs, given the context of card drawing without replacement?

What is the probability expression for the event E4, where the drawn 12th card is club ‘6’?

Which option correctly states the probability for the event E2, that the 8th card drawn is the third ace?

If three distinct points are chosen at random from a 7-sided polygon, what is the probability that they form an obtuse triangle?

What is the probability of selecting the first king on the 9th draw under the events described?

If the first digit of a natural number N is deleted, resulting in 29, what is the sum of all the digits of N?

How many handshakes occur at a party with n married couples if everyone shakes hands with every person except their spouse?

What is the required condition for the positive integer m in the expression $\sum_{n=1}^{m} (a_{n+2} + b_{n+1} + 2a_{n}) = 2016$?

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?

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'?

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?

What is the value of f(n) where n=4 for the function defining handshakes among n married couples?

If the sequences generated allow digits to differ by 1, how many such sequences can be formed with digits 0 to 4?

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?

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?

Which of the following represents the probability that a randomly selected rational number from (2010, 2011) contains the digit 3 after the decimal point?

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 value of k in the equation $egin{pmatrix} a_{11} + 4 & a_{12} & a_{13} \ a_{21} & a_{22} + 4 & a_{23} + 5\ a_{31} & a_{32} & a_{33} + 4 \ ext{ } & & ext{ } \ a_{11} + 1 & a_{12} & a_{13} \ a_{21} & a_{22} + 1 & a_{23} \ a_{31} & a_{32} & a_{33} + 1 \ ext{ } & & ext{ }} ight] = 0$?

Consider the equation $a^2 + b^2 + c^2 = 2$. What are the possible integer values of a, b, and c?

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 $|A| = 8$ and $|adj(I - 2A)| = K$, what could be a possible alternative expression for K in terms of determinants?

For a rational number selected randomly from (2010, 2011), what is the probability that the last digit after the decimal point is exactly 4?

Which characteristic of matrices A and B is indicated by the condition $tr(AB) - tr(A) imes tr(B) + tr(AB^{-1}) + 2$?

If n is even, what is the probability that three numbers selected from { 1, 3, 3^2, ..., 3^n } form an increasing geometric progression?

Given that $A^3 - 6A^2 + 12A - 81 = 0$, which of the following could represent the relationship between A and its eigenvalues?

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?

What is an implication of having $AA^T = 4I$ for matrix A?

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 $b$?

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?