Use examples and counterexamples.

Consider the statement: "For all integers $n \geq 1$, the expression $n^2 + n + 41$ is prime." Which of the following is the smallest counterexample to this statement?

A student proposes the following statement: "For all complex numbers z and w, if |z + w| = |z| + |w|, then arg(z) = arg(w)." Refer to the stimulus showing two complex numbers plotted on an Argand diagram. (a) Verify that the two complex numbers shown satisfy the condition |z + w| = |z| + |w|. (2 marks) (b) The proposition states that the condition |z + w| = |z| + |w| is sufficient to guarantee arg(z) = arg(w). Determine whether this claim is true or false. Provide a counterexample using specific complex numbers in Cartesian form, or explain why no counterexample exists. (3 marks)

Consider the statement: "If $f(x)$ is continuous on $[a, b]$ and $\int_a^b f(x)\,dx = 0$, then $f(x) = 0$ for all $x \in [a, b]$." Which diagram provides the simplest counterexample to this statement?

Consider the statement: "For all real numbers $a$ and $b$, if $a^2 = b^2$, then $a = b$." (a) Use a counterexample to show that this statement is false. (1 mark) (b) Modify the original statement by adding a single condition so that it becomes true. (1 mark) (c) Use an example to verify that your modified statement is true. (1 mark)

A student proposes the following statement: "For all non-zero complex numbers z and w, if |z + w| = |z| + |w|, then arg(z) = arg(w)." Refer to the stimulus. (a) Using the complex numbers provided in the stimulus, verify that the condition |z + w| = |z| + |w| holds. (2 marks) (b) Show that arg(z) = arg(w) for the complex numbers in the stimulus. (1 mark) (c) Consider whether the student's proposition can be disproved. Either provide a counterexample with specific non-zero complex numbers in Cartesian form where |z + w| = |z| + |w| holds but arg(z) ≠ arg(w), or explain why no such counterexample exists. (2 marks)

A student proposes the following statement: "For all complex numbers z and w, if |z + w| = |z| + |w|, then arg(z) = arg(w)." Refer to the stimulus showing an Argand diagram. (a) Using the example shown in the diagram, verify that the condition |z + w| = |z| + |w| holds for the given complex numbers z and w. (2 marks) (b) Determine whether the converse statement is true: "For all complex numbers z and w, if arg(z) = arg(w), then |z + w| = |z| + |w|." Provide either a proof or a counterexample with specific complex numbers in Cartesian form. (3 marks)

Consider the statement: "If $f(x)$ is continuous on $[a, b]$ and $\int_a^b f(x)\,dx = 0$, then $f(x) = 0$ for all $x \in [a, b]$." Which diagram provides the simplest counterexample to this statement (a continuous function that changes sign exactly once)?

A student proposes the following statement: "For all complex numbers z and w, if |z + w| = |z| + |w|, then arg(z) = arg(w)." Refer to the stimulus showing three pairs of complex numbers. (a) From the stimulus, identify which pair(s) satisfy the condition |z + w| = |z| + |w|. Show all calculations to verify your answer. (2 marks) (b) Determine whether the student's proposition is true. If true, provide a brief justification. If false, provide a counterexample using specific complex numbers from the stimulus or otherwise. (3 marks)

Consider the statement: "If \(f(x)\) is continuous on \([a, b]\), then \(\int_a^b f(x)\,dx = 0\) implies \(f(x) = 0\) for all \(x \in [a, b]\)." Which diagram provides the simplest counterexample to this statement, showing a function that crosses the \(x\)-axis exactly once within the interval?

A student claims: "If $f(x)$ is continuous on $[a, b]$ and $\int_a^b f(x)\,dx = 0$, then $f(x) = 0$ for all $x \in [a, b]$." Which diagram provides the simplest counterexample to this claim?

Consider the statement: "For all positive integers $n$, the expression $n^2 + n + 41$ produces a prime number." (a) Use $n = 1$ and $n = 2$ to determine whether these values support the statement. (1 mark) (b) Use a counterexample to show that the statement is false. (1 mark) (c) Determine the smallest positive integer value of $n$ for which the statement fails. Show evidence of your calculation. (1 mark)

Consider the statement: "If \( f(x) \) is continuous on \([a, b]\) and \(\int_a^b f(x)\,dx = 0\), then \( f(x) = 0 \) for all \( x \in [a, b] \)." Which diagram provides a counterexample to this statement by showing a continuous function that oscillates symmetrically about the \(x\)-axis?

Consider the proposition: "For all real numbers $a$ and $b$, if $a^2 = b^2$, then $a = b$." (a) Determine whether this proposition is true or false by evaluating it using an appropriate example or counterexample. (2 marks) (b) Consider the modified proposition: "For all complex numbers $z$ and $w$, if $|z| = |w|$, then $z = w$." Evaluate the reasonableness of this proposition using an appropriate counterexample from the Argand plane. (3 marks)