Number and Algebra

Round the value of $\frac{2}{3}$ to three significant figures.

Express $0.00001$ as a power of $10$.

Round $8,888$ to one significant figure.

Calculate $(3 \times 10^{-2}) \times (2 \times 10^{-3})$ and express in scientific notation.

Calculate the percentage error if the number $12.345$ is rounded to one decimal place.

In a calculation, the result is $0.00000000000567$. How many zeros are between the decimal point and the first non-zero digit when this is written in standard decimal form?

A marathon is $42.195$ km. Round this distance to the nearest $0.1$ km.

A value $V$ is given by $V = \frac{4}{3}\pi r^3$. If $r = 10.0$ cm is rounded to three significant figures, calculate the difference between the upper and lower bounds of $V$. (Use $\pi = 3.14159$ and round to one decimal place).

Find the result of $(2.5 \times 10^4) \times (4.0 \times 10^{-8})$ and write it as a decimal.

Express $\frac{1}{800}$ in scientific notation.

If $u_n = 10 \cdot (0.5)^n$, find the sum of the first two terms ($u_1 + u_2$).

If the general term of an arithmetic sequence is $u_n = 4n + 3$, find the first term.

What is the sum of the first $5$ terms of the sequence $2, 2, 2, 2, 2$?

Find the common difference for the sequence $-10, -7, -4, \dots$.

In an arithmetic sequence, the fifth term is $10$. Find the sum of the first nine terms, $S_9$.

Find the sum of the first $7$ terms of the geometric sequence $3, -6, 12, -24, ...$.

The sum of the first $k$ odd numbers is given by $\sum_{r=1}^{k} (2r - 1) = 625$. Find the value of $k$.

A sequence is defined by $a_n = n^2 - n + 41$. For which of the following $n$ is $a_n$ not a prime number?

What is the $100$-th term of the sequence $2, 5, 10, 17, 26, \dots$?

Find the value of $k$ for which $\sum_{r=1}^n (2r-1) = kn^2$.

Find the sum of the first 3 terms of an arithmetic sequence with $u_1 = 5$ and $d = 2$.

If $u_1 = 10$ and $u_3 = 20$ in an arithmetic sequence, find the term $u_2$.

In the arithmetic sequence $1, 3, 5, \dots$, which term is equal to 11?

If the sum of the first $n$ terms of a sequence is $S_n = 3n^2 + n$, find the first term $u_1$.

In an arithmetic sequence, $u_1 = 15$ and $d = -2$. Find the value of $n$ for which $S_n = 63$.

The sum of the first $n$ terms of an arithmetic sequence is $S_n = 3n^2 - 2n$. Find the common difference $d$.

An arithmetic sequence has $u_1 = 7$ and $d = -3$. Find the largest value of $n$ such that the sum $S_n$ is positive.

An arithmetic sequence has $u_4 = 10$ and $u_8 = 2$. Find the value of $n$ such that $u_n = -10$.

In an arithmetic sequence, the sum of the first 5 terms is 50 and the sum of the first 10 terms is 200. Find $u_1$.

Find the sum of the first 50 odd positive integers.

Which of the following is a geometric sequence?

Calculate the sum of the first 5 terms of a geometric sequence where $u_1 = 1$ and $r = 0.5$.

Given $u_1 = 7$ and $r = 10$, find the 4th term $u_4$.

If $u_n = 5 \times 2^{n-1}$, find the 5th term $u_5$.

Find the sum $\sum_{k=1}^{6} 10(0.5)^{k-1}$.

A geometric sequence is such that $u_k = 3^{k+1}$. Find the common ratio $r$.

The fourth term of a geometric sequence is 2. The seventh term is 16. Find the tenth term.

The first term of a geometric sequence is $\ln x$ and the second term is $\ln(x^2)$. Find the sum of the first 5 terms in terms of $\ln x$.

In an infinite geometric series, the first term is $a$ and the common ratio is $r$. If the sum to infinity is 4 times the first term, find $r$.

For a geometric sequence $u_n$, $u_1 = 4$ and $r = 0.5$. Find the value of $n$ for which $u_n = \frac{1}{512}$.

In the formula $u_n = u_1 r^{n-1}$, if $u_n$ represents the value of an account after $n-1$ years of 4% growth, what is $r$?

A car bought for 30000 dollars depreciates by 15% each year. What is the value after 1 year?

If a town's population increases from 2000 to 2100 in one year, what is the common ratio $r$?

Calculate the value of $1500(1.03)^4$ to two decimal places.

An account grows from 1000 USD to 1400 USD in 5 years. If the growth is geometric, what will the balance be after another 5 years?

The number of students in a school is 800. If the number of students increases by 4% each year, what is the total number of students after 10 years?

A painting was sold for 12,000 USD in 2010. Its value increased by 6% each year until 2020. What was its value in 2020?

The price of a house grows from 200,000 to 300,000 in 5 years. Find the annual growth rate.

An investment at $6\%$ interest compounded annually is worth 5000 USD after 10 years. Find the initial investment.

Find the total sum of the infinite geometric series $100 + 90 + 81 + ...$

How many terms are in the expansion of $(2x + 3y)^5$?

What is the term independent of $x$ in $(x + 5)^1$?

Evaluate $\binom{11}{10}$.

What is the sum of the powers of $a$ and $b$ in the 5th term of $(a + b)^9$?

What is the middle term of the expansion of $(2x - \frac{1}{2x})^6$?

If the coefficients of $x^2$ and $x^3$ in the expansion of $(1 + ax)^6$ are equal, and $a \neq 0$, find the value of $a$.

Find the coefficient of $x^4$ in the expansion of $(2x^2 - 1)^4$.

The ratio of the coefficient of $x^4$ to $x^2$ in the expansion of $(1+kx)^{10}$ is $3:1$. Find the positive value of $k$.

Find the value of $n$ such that $\binom{n}{n-2} + \binom{n+1}{n-1} = 100$.

In the expansion of $(1+x)^n$, the ratio of the coefficients of $x^2$ and $x^3$ is $3:8$. Find $n$.

Which of the following describes the set of numbers that can be written in the form $\\frac{p}{q}$ where $p, q \\in \\mathbb{Z}$ and $q \\neq 0$?

What is the parity of $n^2 + 3n + 2$ if $n$ is an integer?

To show that the product of two even numbers is a multiple of 4, we multiply $2k$ and $2m$. The result is:

A student claims that $n^2 + 1$ is always even. Which value of $n$ is a counter-example?

Consider the sum of the squares of three consecutive integers $n-1$, $n$, and $n+1$. Let $S = (n-1)^2 + n^2 + (n+1)^2$, where $n \in \mathbb{Z}$. By expanding and simplifying the expression for $S$, which of the following deductions must be correct for all integers $n$?

What is the smallest positive integer $n$ for which $n! + 1$ is NOT a prime number?

Which of the following is equivalent to the statement 'The square of an even number is even'?

Let $n \in \mathbb{Z}$. A student investigates the sum of the squares of four consecutive integers, $S = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2$. The student determines that for all possible values of $n$, the remainder when $S$ is divided by 8 is a constant $r$. Find the value of $r$.

If $n$ is an odd integer, the expression $n^2 - 1$ is always a multiple of:

Which of the following is a factor of $n^4 - 1$ for any integer $n$?

If $a$ and $b$ are integers such that $a$ is a multiple of 3 and $b$ is a multiple of 3, which of the following expressions represents their sum $a + b$ in terms of integers $k$ and $m$?

What is the negation of the statement: '$\\exists x \\in \\mathbb{R}, x^2 + 1 = 0$'?

Which of the following values of $x$ serves as a counterexample to the statement: 'If $x^2 > 4$, then $x > 2$'?

To prove that the expression $n^2 - 1$ is a multiple of 8 for any odd integer $n$, we substitute $n = 2k + 1$ into the expression. What is the simplified form of $(2k + 1)^2 - 1$?

To prove that the sum of three consecutive even integers is a multiple of 6, let the integers be $2n-2, 2n, 2n+2$. The sum is $6n$. This completes the proof because:

A student is proving by exhaustion that $n^2 < 2^n$ for $n \in \{3, 4, 5\}$. Which of the following evaluations shows that the statement is false for one of these values?

The difference between the squares of two consecutive integers, $(n+1)^2 - n^2$, is always equal to:

To prove that $n^2 + n + 2$ is even for all integers $n$, we check two cases. In the case where $n$ is odd ($n = 2k + 1$), the expression becomes:

Find a counter-example to the statement: 'If $x$ is irrational, then $x^2$ is rational.'

A student attempts to prove that the sum of four consecutive integers is never a multiple of 4. Let the integers be $n, n+1, n+2, n+3$. Their sum is:

1. Mathematical induction (HL): practice question 1 for Number and Algebra?

1. Mathematical induction (HL): practice question 2 for Number and Algebra?

1. Mathematical induction (HL): practice question 3 for Number and Algebra?

1. Mathematical induction (HL): practice question 4 for Number and Algebra?

1. Mathematical induction (HL): practice question 5 for Number and Algebra?

1. Mathematical induction (HL): practice question 6 for Number and Algebra?

1. Mathematical induction (HL): practice question 7 for Number and Algebra?

1. Mathematical induction (HL): practice question 8 for Number and Algebra?

1. Mathematical induction (HL): practice question 9 for Number and Algebra?

1. Mathematical induction (HL): practice question 10 for Number and Algebra?

If $x + y + z = 0$ and $x + y = 5$, what is $z$?

Solve for $x$ and $y$ in $x + y = 10$ and $x = y$.

Given $2x - y = 4$, what is $y$ when $x = 0$?

Which system has the solution $(1, 1)$?

Which of the following describes the system $2x + 3y = 5$ and $4x + 6y = 11$?

Given $x - 2y = 3$ and $2x + y = 11$, find the value of $x + y$.

For what value of $k$ will the lines $3x + ky = 7$ and $kx + 3y = 5$ be parallel?

Consider the system of equations: $x + y + z = 6$ $2x + y - z = 1$ $x - y + 2z = 5$ Find the value of $x^2 + y^2 + z^2$.

Solve the system for $x^2$ and $y^2$: $\frac{4}{x^2} + \frac{1}{y^2} = 7$ $\frac{1}{x^2} - \frac{2}{y^2} = -5$ Find the value of $x^2 + y^2$.

Three apples and two bananas cost $2.80$. Five apples and four bananas cost $5.00$. Find the total cost of one apple and one banana.

1. Complex numbers – basic operations (HL): practice question 1 for Number and Algebra?

1. Complex numbers – basic operations (HL): practice question 2 for Number and Algebra?

1. Complex numbers – basic operations (HL): practice question 3 for Number and Algebra?

1. Complex numbers – basic operations (HL): practice question 4 for Number and Algebra?

1. Complex numbers – basic operations (HL): practice question 5 for Number and Algebra?

1. Complex numbers – basic operations (HL): practice question 6 for Number and Algebra?

1. Complex numbers – basic operations (HL): practice question 7 for Number and Algebra?

1. Complex numbers – basic operations (HL): practice question 8 for Number and Algebra?

1. Complex numbers – basic operations (HL): practice question 9 for Number and Algebra?

1. Complex numbers – basic operations (HL): practice question 10 for Number and Algebra?

1. Polynomials over the Complex field (HL): practice question 1 for Number and Algebra?

1. Polynomials over the Complex field (HL): practice question 2 for Number and Algebra?

1. Polynomials over the Complex field (HL): practice question 3 for Number and Algebra?

1. Polynomials over the Complex field (HL): practice question 4 for Number and Algebra?

1. Polynomials over the Complex field (HL): practice question 5 for Number and Algebra?

1. Polynomials over the Complex field (HL): practice question 6 for Number and Algebra?

1. Polynomials over the Complex field (HL): practice question 7 for Number and Algebra?

1. Polynomials over the Complex field (HL): practice question 8 for Number and Algebra?

1. Polynomials over the Complex field (HL): practice question 9 for Number and Algebra?

1. Polynomials over the Complex field (HL): practice question 10 for Number and Algebra?

1. The complex plane (HL): practice question 1 for Number and Algebra?

1. The complex plane (HL): practice question 2 for Number and Algebra?

1. The complex plane (HL): practice question 3 for Number and Algebra?

1. The complex plane (HL): practice question 4 for Number and Algebra?