Mathematics Proofs and Contradictions

<p>The sum of any four consecutive numbers is always even. (D)</p> Signup and view all the answers

<p>Always even (A)</p> Signup and view all the answers

<p>Always non-negative (A)</p> Signup and view all the answers

<p>The RHS of the equation is an integer, while the LHS is not, as 𝜋 is irrational. (B)</p> Signup and view all the answers

<p>The expression 2 sin 𝑥 cos 𝑥 is always positive for the given domain, contradicting the derived inequality 2 sin 𝑥 cos 𝑥 &lt; 0. (A)</p> Signup and view all the answers

<p>The derived expression for 𝑏, which is rational, contradicts the initial assumption that 𝑏 is irrational. (D)</p> Signup and view all the answers

<p>The equation derived shows that 𝑛 − 𝑚 is equal to 1/𝑎, but the left side is an integer, while the right side is not. (C)</p> Signup and view all the answers

<p>A rational number (C)</p> Signup and view all the answers

<p>Less than 1 (D)</p> Signup and view all the answers

<p>Proof by contradiction (B)</p> Signup and view all the answers

<p>The value of an expression being negative while it must be positive within a specific domain. (B)</p> Signup and view all the answers

<p>Examining different methods for proving mathematical statements (B)</p> Signup and view all the answers

<p>Proof by Induction (D)</p> Signup and view all the answers

<p>Being able to construct logically sound arguments (D)</p> Signup and view all the answers

<p>6 (B)</p> Signup and view all the answers

<p>Because it introduces the fundamental skills required for other topics (C)</p> Signup and view all the answers

<p>All of the above (D)</p> Signup and view all the answers

<p>To provide students with a strong foundation in the key ideas of proof and reasoning (D)</p> Signup and view all the answers

<p>Contact the author via email (A)</p> Signup and view all the answers

<p>To help students prepare for the HSC in Extension 2 Maths. (D)</p> Signup and view all the answers

<p>At the author's website, howardmathematics.com. (D)</p> Signup and view all the answers

<p>A list of commercially published textbooks for the old syllabus. (A)</p> Signup and view all the answers

<p>By difficulty level, with basic, medium, and hard categories. (B)</p> Signup and view all the answers

<p>To create a passion project that benefits both students and teachers. (D)</p> Signup and view all the answers

<p>To provide supplementary material for high-ability students. (D)</p> Signup and view all the answers

<p>To help students prepare for their Trials and the HSC. (C)</p> Signup and view all the answers

<p>The textbook provides detailed explanations of the concepts. (A)</p> Signup and view all the answers

<p>That the square of an even integer is even. (C)</p> Signup and view all the answers

<p>That 2 can be expressed as a fraction of two even integers. (C)</p> Signup and view all the answers

<p>That the statement is false, hence a contradiction arises. (B)</p> Signup and view all the answers

<p>To assume the opposite of what you want to prove and show it leads to an invalid conclusion. (A)</p> Signup and view all the answers

<p>They can be expressed as the ratio of two integers. (A)</p> Signup and view all the answers

<p>Positive integers. (D)</p> Signup and view all the answers

<p>$sin^2 x + cos^2 x = 1$. (C)</p> Signup and view all the answers

<p>They have a common factor of 2. (A)</p> Signup and view all the answers

An advanced mathematics course designed for HSC students.

Higher School Certificate, a credential awarded to students in Australian high schools.

Previous exam questions used to prepare students for the HSC.

Questions organized by difficulty level to aid learning.

Complete answers and explanations provided for all exercises.

Visual aids included to help understand mathematical concepts.

Guidance to simplify answering exam questions.

Additional content for deeper understanding or interest.

The foundation of constructing logically sound proofs in mathematics.

A proof technique that proves a statement by proving its contrapositive.

A method of proof where you assume the opposite of what you want to prove, leading to a contradiction.

Two statements are equivalent if they imply each other.

To demonstrate that a statement is false, typically through a counterexample.

Proving relationships between quantities that demonstrate one is greater or less than another.

A mathematical inequality stating that the arithmetic mean is always greater than or equal to the geometric mean.

The specific terminology and phrasing used to formulate logical statements and proofs.

An integer that is divisible by 2 without a remainder.

A number that can be expressed as a fraction of two integers.

A number that cannot be expressed as a simple fraction; it has an infinite non-repeating decimal expansion.

It shows that if 2 is rational, it leads to both integers being even, contradicting their properties.

A statement that contradicts an assumption or leads to an impossible conclusion, thus proving the original assumption false.

To show that if n is even, then n^2 is also even by contradiction.

In proofs, they explore conditions on sums leading to algebraic consequences about their individual components.

If 4n + 8m = 102 has no integer solutions, then no integers m, n satisfy the equation.

A number that can be expressed as p/q, where p and q are integers with no common factors aside from 1.

Assuming Pi is rational leads to an integer that cannot exist, showing Pi is irrational.

If sin x + cos x < 1, it implies that 2 sin x cos x < 0, which leads to a contradiction.

If a is rational and b is irrational, then a + b must be irrational, as it leads to a contradiction otherwise.

If b and b+1 are both divisible by a, it leads to a contradiction as one term must not be an integer.

If m is irrational, then √m must also be irrational, leading to a contradiction if assumed otherwise.

sin²x + 2sinxcosx + cos²x < 1 indicates a summation that shrinks, leading to contradiction.

Demonstrating that a mathematical statement does not hold true.

A prime number greater than 1 that can only be divided by 1 and itself; 7 is prime.

A set of three positive integers a, b, c that satisfy a² + b² = c².

A number is divisible by 6 if it’s divisible by both 2 and 3.

A number is even if it is divisible by 2; odd if not.

A three-digit number is divisible by 9 if the sum of its digits is divisible by 9.

A number is divisible by 4 if its last two digits form a number that is a multiple of 4.

A number that can be divided by 3 with no remainder.

A number that can be divided by 9 with no remainder.

Numbers that follow each other in order without gaps.

An integer that is exactly divisible by 2.

An integer that is not divisible by 2.

The result of multiplying an even number by itself.

Multiplying two odd integers results in an odd integer.

The total of four numbers in sequence is always even.

The total of three consecutive numbers is always odd.

The result of a number multiplied by itself.