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SL 1.6—Simple proof - IB Questionbank | RevisionDojo

Practice SL 1.6—Simple proof with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SL & HLPaper 1

Prove that the product of an even integer and any other integer is an even integer.

[3]

Question 2

SL & HLPaper 1

Prove that the sum of any two odd integers is an even integer.

[3]

Question 3

SL & HLPaper 1

Given the functions $f(x) = 3x - 2$ and $g(x) = \frac{x+2}{3}$ .

Prove that $f(x)$ and $g(x)$ are inverse functions of each other.

[5]

Question 4

SL & HLPaper 1

Prove that if the quadratic equation $x^2 + (p-5)x + 7 = 0$ has real and distinct roots, then $p < 5 - 2\sqrt{7}$ or $p > 5 + 2\sqrt{7}$ .

[7]

Question 5

SL & HLPaper 1

Prove that for all real numbers $a$ and $b$ , $a^2 + b^2 + 1 \ge ab + a + b$ .

[5]

Question 6

SL & HLPaper 1

Prove that for any real number $x$ , $x(x+2) - (x+1)^2 \equiv -1$ .

[3]

Question 7

SL & HLPaper 1

The first three terms of an arithmetic progression are $k-1$ , $k+2$ , and $k^2-1$ .

Show that $k$ satisfies the equation $k^2 - k - 6 = 0$ .

[3]

Hence, find the possible values of $k$ and, for each value, state the common difference of the arithmetic progression.

[4]

Question 8

SL & HLPaper 1

Prove that if $x, y, z$ are positive real numbers such that $x^a y^b z^c = 1$ , then $\frac{a}{\log_x k} + \frac{b}{\log_y k} + \frac{c}{\log_z k} = 0$ for any positive real number $k \ne 1$ .

[5]

Question 9

SL & HLPaper 1

Prove that for any positive real numbers $a$ and $b$ ,

$\frac{1}{a} - \frac{1}{a+b} \equiv \frac{b}{a(a+b)}$

[3]

Question 10

SLPaper 1

In the context of computer science, we often analyze algorithms that involve whole numbers. One interesting property of whole numbers is how their squares can be categorized based on their divisibility by 4.

Prove that the square of any whole number can be expressed in the form ${4t}$ or ${4t + 1}$ , where ${t}$ is an integer.

[3]

SL 1.6—Simple proof Questionbank