Algebra · SS2

Binary operations

New operations defined by a rule, and their properties: closure, identity and inverse.

Before this: Expressions, formulae & change of subject
Leads to: Binary operations

Learn

1Working out an operationWhat a binary operation is, putting numbers into its rule in the right order, brackets first, two operations in one question, and solving equations that use an operation.Done14 min
2Properties and tablesClosure, commutativity and associativity, the identity element and inverses, from a rule and from a table, and building and using an operation table in modular arithmetic.Done17 min

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Answer binary operations questions one at a time. You'll know straight away whether you're right, with hints and the working step by step.

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Past questions (33)

From 1992–2019: 8 WAEC, 25 JAMB. Theory questions first, then objective, newest first.

Theory (8)

WAEC 2019 · Paper 2 · Q1Draw a table for multiplication $\otimes$ in modulo 8 on the set $T = \{2, 3, 5, 7\}$ .
WAEC 2018 · Paper 2 · Q13The points $R(3, -6)$ , $S(6, -2)$ and $T(p, q)$ are on the $xy$ -plane. If $\frac13\overrightarrow{OR} + \overrightarrow{OS} + \overrightarrow{OT} = \overrightarrow{RS}$ , …
WAEC 2017 · Paper 2 · Q13The operation $(*)$ is defined on the set of real numbers, $\mathbb R$ , by $x * y = \dfrac{x + y}{2}$ , $x, y \in \mathbb R$ . (i) Evaluate $3 * \frac25$ . …
WAEC 2017 · Paper 2 · Q13An operation $*$ is defined by $x * y = x + y + 2xy$ , $x, y \in \mathbb R$ . (i) Calculate $(2 * 3) * 5$ . (ii) Find the truth set of $(x * 7) = (x * 5) * 2$ .
WAEC 2016 · Paper 2 · Q13Find the equation of a straight line which passes through the point $(2, -3)$ and is parallel to the line $2x + y = 6$ .
WAEC 2015 · Paper 2 · Q11Make $m$ the subject of the relation $h = \dfrac{mt}{d(m + p)}$ .
WAEC 2014 · Paper 2 · Q13Two functions $f$ and $g$ are defined by $f : x \to 2x^2 - 1$ and $g : x \to 3x + 2$ , where $x$ is a real number. (i) If $f(x - 1) - 7 = 0$ , …
WAEC 2014 · Paper 2 · Q8Copy and complete the table for multiplication $\otimes$ modulo 11 on the set $\{1, 5, 9, 10\}$ .

Objective (25)

JAMB 2017 · UTME (set 2) · Q30The operation $*$ on the set $\mathbb{R}$ of real numbers is defined by $x * y = 3x + 2y - 1$ . Find $3 * (-1)$ .
JAMB 2016 · UTME · Q29| $\otimes$ | $p$ | $q$ | $r$ | $s$ |
JAMB 2014 · UTME · Q21A binary operation $*$ is defined by $x * y = xy$ . If $x * x = 12 - x$ , find the possible values of $x$ .
JAMB 2013 · UTME · Q21If a binary operation $*$ is defined by $x * y = x + 2y$ , find $2 * (3 * 4)$ .
JAMB 2012 · UTME · Q21The binary operation $*$ is defined on the set of integers such that $p * q = pq + p - q$ . Find $2 * (3 * 4)$ .
JAMB 2012 · UTME · Q22A binary operation on the set of real numbers is defined by $m * n = \frac{mn}{2}$ for all $m, n \in \mathbb R$ . If the identity element is 2, …
JAMB 2011 · UTME · Q23A binary operation $\oplus$ on real numbers is defined by $x \oplus y = xy + x + y$ . Find the value of $3 \oplus \left(-\frac23\right)$ .
JAMB 2010 · UTME · Q23If $x * y = x + y^2$ , find the value of $(2 * 3) * 5$ .
JAMB 2004 · UME · Q21If the operation $*$ on the set of integers is defined by $p * q = \sqrt{pq}$ , find the value of $4 * (8 * 32)$ .
JAMB 2002 · UME · Q34The binary operation $*$ is defined on the set of integers by $p * q = pq + p + q$ . Find $2 * (3 * 4)$ .
JAMB 2001 · UME · Q16An operation $*$ is defined on the set of real numbers by $a * b = a + b + 1$ . If the identity element is $-1$ , find the inverse of the element …
JAMB 2001 · UME · Q17The identity element with respect to the multiplication shown in the table is
JAMB 2000 · UME · Q12A binary operation $*$ is defined by $a * b = a^b$ . If $a * 2 = 2 - a$ , find the possible values of $a$ .
JAMB 2000 · UME · Q14Find the inverse of $p$ under the binary operation $p * q = p + q - pq$ , where $p$ and $q$ are real numbers and zero is the identity.
JAMB 1999 · UME · Q14If $m * n = \frac mn - \frac nm$ for real $m$ , $n$ , evaluate $-3 * 4$ .
JAMB 1998 · UME · Q21The identity element with respect to the multiplication shown in the table is
JAMB 1998 · UME · Q22The binary operation $*$ is defined by $x * y = xy - y - x$ for all real values of $x$ and $y$ . If $x * 3 = 2 * x$ , find $x$ .
JAMB 1997 · UME · Q21Two binary operations $*$ and $\oplus$ are defined by $m * n = mn - n - 1$ and $m \oplus n = mn + n - 2$ for all real numbers $m$ , $n$ . Find …
JAMB 1997 · UME · Q22If $x * y = x + y - xy$ , find $x$ when $(x * 2) + (x * 3) = 68$ .
JAMB 1995 · UME · Q21Which of the following binary operations is commutative in the set of integers?
JAMB 1995 · UME · Q22If $a * b = +\sqrt{ab}$ , evaluate $2 * (12 * 27)$ .
JAMB 1994 · UME · Q22A binary operation $*$ is defined on the set of all positive integers by $a * b = ab$ . Which of the following properties does NOT hold?
JAMB 1994 · UME · Q23The table shows multiplication modulo 10 on the set $S = \{2, 4, 6, 8\}$ . Find the inverse of 2.
JAMB 1993 · UME · Q24A binary operation $*$ is defined on the set of real numbers by $x * y = x^y$ for all real values of $x$ and $y$ . If $x * 2 = x$ , find the possible …
JAMB 1992 · UME · Q21If the binary operation $*$ is defined by $m * n = mn + m + n$ for any real numbers $m$ and $n$ , find the identity element under this operation.