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SQU Aptitude Academic Aptitude and Quantitative Reasoning

Core mathematical logic, algebraic identities, probability rules, and academic English structural rules for Sultan Qaboos University foundation tests.

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Difference of Two Squares Identity
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1
Difference of Two Squares Identity
a^2 - b^2 = (a - b)(a + b). Useful for rapid algebraic factorization and simplifying polynomial expressions.
2
Quadratic Formula and the Discriminant (Delta)
x = (-b +/- sqrt(b^2 - 4ac)) / (2a). If Delta = b^2 - 4ac > 0: two distinct real roots; if Delta = 0: one repeated real root; if Delta < 0: two complex roots.
3
Slope of a Line and Slope-Intercept Form
Slope m = (y2 - y1) / (x2 - x1); Equation: y = mx + c, where m is gradient and c is y-intercept. Two lines are perpendicular if m1 m2 = -1.
4
Distance Formula Between Points (x1, y1) and (x2, y2)
d = sqrt((x2 - x1)^2 + (y2 - y1)^2), derived directly from the Pythagorean theorem.
5
Arithmetic Progression: n-th Term and Sum of First n Terms
n-th term: a_n = a1 + (n - 1)d. Sum: S_n = (n / 2) [2a1 + (n - 1)d] = (n / 2) * (a1 + a_n).
6
Geometric Progression: n-th Term and Sum of Finite Sequence
n-th term: a_n = a1 r^(n - 1). Sum: S_n = a1 (1 - r^n) / (1 - r) for common ratio r != 1.
7
Sum to Infinity of an Infinite Geometric Series
S_infinity = a1 / (1 - r), valid if and only if the common ratio satisfies |r| < 1.
8
Laws of Exponents (Multiplication, Division, Power)
x^a x^b = x^(a + b); x^a / x^b = x^(a - b); (x^a)^b = x^(a b); x^(-a) = 1 / x^a; x^(1/n) = n-th root of x.
9
Logarithm Fundamental Change of Base Rule
log_b(x) = ln(x) / ln(b) = log_k(x) / log_k(b). Product: log(xy) = log(x) + log(y); Quotient: log(x/y) = log(x) - log(y).
10
Compound Interest Formula (Annual and Continuous)
Periodic: A = P (1 + r/n)^(nt); Continuous compounding: A = P e^(rt), where P is principal, r is annual rate, and t is time in years.
11
Addition Rule for Probability of Two Events
P(A union B) = P(A) + P(B) - P(A intersect B). If events A and B are mutually exclusive, P(A intersect B) = 0.
12
Conditional Probability and Bayes' Rule
P(A | B) = P(A intersect B) / P(B). For independent events: P(A | B) = P(A) and P(A intersect B) = P(A) * P(B).
13
Permutations vs Combinations Formulas
Permutations (order matters): nPr = n! / (n - r)!. Combinations (order does not matter): nCr = n! / [r! * (n - r)!].
14
Mean, Median, and Mode Definitions in Statistics
Mean = Sigma x / n; Median = middle value of ordered observations (average of two middle values if n is even); Mode = most frequently occurring value.
15
Pythagorean Theorem and Special Right Triangles
a^2 + b^2 = c^2. 45-45-90 triangle side ratios: 1 : 1 : sqrt(2). 30-60-90 triangle side ratios: 1 : sqrt(3) : 2.
16
Subject-Verb Agreement with Collective Nouns
In standard academic English, collective nouns (team, committee, faculty) take a singular verb when acting as a unified whole, and plural when individual members act independently.
17
Dangling and Misplaced Modifiers in Academic Writing
A modifier must immediately precede or follow the specific noun it describes. An introductory participial phrase must modify the grammatical subject of the independent clause.
18
Conditionals: Second vs Third Conditional Structures
Second (hypothetical present): If + Past Simple, would + base verb ('If I studied, I would pass'). Third (unreal past): If + Past Perfect, would have + past participle ('If I had studied, I would have passed').
19
Semicolon vs Colon Usage in Formal Writing
Semicolon connects two closely related independent clauses without a coordinating conjunction. Colon introduces an explanation, quote, or list following a complete independent clause.
20
Active Voice vs Passive Voice Structure
Active: Subject performs action ('The researcher analyzed the data'). Passive: Subject receives action ('The data were analyzed by the researcher'). Passive uses form of 'to be' + past participle.
21
Parallel Structure (Parallelism) Rule
Elements in a list or series joined by coordinating conjunctions must share the same grammatical form (e.g., all gerunds: 'reading, writing, and editing', not 'reading, writing, and to edit').
22
Relative Pronouns: 'Who' vs 'Whom' vs 'Which' vs 'That'
'Who' acts as subject of a clause; 'Whom' acts as object; 'That' introduces restrictive (essential) clauses without commas; 'Which' introduces non-restrictive (extra info) clauses set off by commas.
23
Subjunctive Mood in Formal Academic Demands and Recommendations
Verbs like require, recommend, suggest, and insist take a that-clause with base verb form (bare infinitive): 'The university requires that every candidate submit (not submits) valid credentials.'
24
Run-on Sentences and Comma Splices
A comma splice occurs when two independent clauses are joined solely with a comma. Correct with: period, semicolon, or comma + coordinating conjunction (for, and, nor, but, or, yet, so).
25
Transitions of Contrast: 'However', 'Although', 'Despite'
'Although' is a subordinating conjunction introducing a dependent clause; 'Despite' is a preposition followed by a noun/gerund; 'However' is a conjunctive adverb requiring punctuation (semicolon/period before, comma after).

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