Lead City International School Mathematics Mid-Term Examination 2025/2026

Examination Overview and Logistics

The Lead City International School located in Jericho, Ibadan, conducts the Mid Term Examination for the 2025/2026 Session during the Third Term. The subject of this specific assessment is Mathematics. For students taking this examination, the allotted time for completion is exactly $1\,hour$. The examination paper features a specific instructional requirement: candidates must answer Question 1, which is compulsory, and then select and complete any other 3 questions from the remaining options provided on the sheet. The document also includes administrative fields for the student's name (recorded as Mosaderi in the sample), a box number listed as $1267$, and a telephone prefix of $08$. There is a clear instruction stating "Do not write here" in designated administrative margins.

Modular Arithmetic and Binary Operations

The first section of the exam focuses on binary operations acting upon a specific finite set. The operation, denoted by the symbol $A$, is defined on the set $T = \{2, 3, 5, 7\}$. The mathematical rule governing this operation is given by the formula $xAy = (x + y + xy) \pmod{8}$. This requires the student to perform the addition and multiplication of two elements $x$ and $y$, sum the results, and then find the remainder when that sum is divided by $8$.

The assessment requires two specific tasks related to this operation. First, the student must construct a complete modulo $8$ table for the operation $A$ on the set $T$. This involve calculating every possible pairing within the set: $(2A2, 2A3, 2A5, 2A7, 3A2, \dots, 7A7)$. Second, the table is to be used as a reference to solve specific expressions. These include finding the result of the nested operation $2A(5A7)$ and solving the algebraic equation $2An = 5A7$ to determine the value of the unknown variable $n$.

Surds Simplification and Rational Expressions

Question 2 addresses the simplification of surds and algebraic fractions. In the first part, students are tasked with simplifying the expression $\frac{15}{\sqrt{3}} + \sqrt{108} + \sqrt{432}$. The goal is to provide a final answer in the simplified surd form $a\sqrt{b}$, where both $a$ and $b$ are stipulated to be positive integers. This requires rationalizing the denominator of the first term and factoring the radicands of the second and third terms to identify perfect squares.

The second part of this section involves the division and simplification of complex algebraic rational expressions. The expression provided is $\frac{2n^2 - 3n - 2}{n^2 - 1} \div …$ but looking at the layout, it is structured as the division of $\frac{2n^2 - 3n - 2}{n^2 - 1}$ by the fraction $\frac{2n^2 + n + 3}{n^2 - 4}$. Simplifying this requires the factorization of quadratic trinomials and the application of the difference of two squares identity, specifically for the denominators $n^2 - 1$ and $n^2 - 4$.

Algebraic Substitutions and Functional Terms

Question 3a and Question 4a involve representing one set of variables in terms of another through substitution. In Question 3a, variables are defined as $x = \frac{2m}{1 - m^2}$ and $y = \frac{2m}{1 + m}$. The objective is to express the specific calculation $2x - y$ entirely in terms of $m$ and to reduce the result to its simplest form. This requires finding common denominators and performing polynomial expansion and subtraction.

Similarly, Question 4a provides definitions for $p$ and $q$ such that $p = \frac{2u}{1 - u}$ and $q = \frac{1 + u}{1 - u}$. Students are required to express the ratio $\frac{p + q}{p - q}$ in terms of the variable $u$. This test of algebraic manipulation involves substituting the fractional definitions of $p$ and $q$ into the ratio and simplifying the resulting complex fraction by canceling common terms in the numerator and denominator.

Word Problems and Fractional Equality

Question 3b presents a word problem involving the properties of a fraction. The problem states that when a specific fraction is reduced to its lowest terms, its value is equal to $\frac{2}{3}$. The problem provides a secondary condition: if the numerator of this fraction is doubled, the resulting value is $34$ greater than the denominator. Students are required to translate these verbal conditions into a system of equations to determine the original value of the fraction.

Trigonometry and Geometric Applications

The final portions of the examination apply algebraic concepts to trigonometry and geometry. Question 4b defines two variables based on trigonometric tangents: $m = \tan(30^\circ)$ and $n = \tan(45^\circ)$. Without the use of a calculator, the student must simplify the expression $\frac{m - n}{mn}$. This requires knowledge of exact trigonometric values, where $\tan(30^\circ) = \frac{1}{\sqrt{3}}$ and $\tan(45^\circ) = 1$, and subsequent surd simplification.

Question 5 involves evaluating expressions and calculating area. Part (a) asks to evaluate the expression $(2x^2 - 2x)$ given the specific value for $x$ as $x = (1 - \sqrt{2})^2$. This involves squaring a binomial containing a surd and then substituting it into a quadratic expression. Part (b) focuses on a rectangle with a given length of $4\,cm$ and a diagonal of $8\,cm$. The student must find the area of this rectangle, which requires using the Pythagorean theorem to find the width ($w^2 + 4^2 = 8^2$) and reporting the final area in surd form ($\text{Area} = \text{length} \times \text{width}$).