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UOB Aptitude Academic Aptitude and Quantitative Reasoning

Essential mathematical rules, algebraic formulas, geometric theorems, and academic English principles for University of Bahrain entrance examinations.

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Quadratic Formula for Solving ax^2 + bx + c = 0
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1
Quadratic Formula for Solving ax^2 + bx + c = 0
x = [-b +- sqrt(b^2 - 4ac)] / (2a); discriminant D = b^2 - 4ac determines number and nature of roots.
2
Difference of Two Squares and Perfect Square Trinomial Identities
a^2 - b^2 = (a - b)(a + b); (a +- b)^2 = a^2 +- 2ab + b^2.
3
Exponent Rules for Powers with Common Base
a^m a^n = a^(m+n); a^m / a^n = a^(m-n); (a^m)^n = a^(mn); a^(-n) = 1 / a^n; a^0 = 1 (a != 0).
4
Logarithmic Product, Quotient, and Power Rules
log(xy) = log(x) + log(y); log(x/y) = log(x) - log(y); log(x^k) = k * log(x).
5
Arithmetic Sequence nth Term and Sum of n Terms Formulas
a_n = a_1 + (n - 1)d; S_n = (n / 2) [2a_1 + (n - 1)d] = (n / 2) * [a_1 + a_n].
6
Geometric Sequence nth Term and Finite Sum Formulas
a_n = a_1 r^(n - 1); S_n = a_1 (1 - r^n) / (1 - r) for common ratio r != 1.
7
Infinite Geometric Series Sum Formula
S_inf = a_1 / (1 - r), valid if and only if |r| < 1.
8
Slope-Intercept Form and Point-Slope Form of a Linear Equation
Slope-intercept: y = mx + b. Point-slope: y - y_1 = m(x - x_1), where m = (y_2 - y_1) / (x_2 - x_1).
9
Conditions for Parallel and Perpendicular Lines
Parallel lines have equal slopes (m_1 = m_2). Perpendicular lines have negative reciprocal slopes (m_1 * m_2 = -1).
10
Distance and Midpoint Formulas Between Points (x1, y1) and (x2, y2)
Distance d = sqrt[(x_2 - x_1)^2 + (y_2 - y_1)^2]; Midpoint M = ((x_1 + x_2)/2, (y_1 + y_2)/2).
11
Fundamental Trigonometric Pythagorean Identities
sin^2(x) + cos^2(x) = 1; 1 + tan^2(x) = sec^2(x); 1 + cot^2(x) = csc^2(x).
12
Sine and Cosine Double-Angle Identities
sin(2x) = 2sin(x)cos(x); cos(2x) = cos^2(x) - sin^2(x) = 2cos^2(x) - 1 = 1 - 2sin^2(x).
13
Right Triangle Trigonometric Ratios (SOH CAH TOA)
sin(theta) = Opp / Hyp; cos(theta) = Adj / Hyp; tan(theta) = Opp / Adj.
14
Combined Work Formula for Two Entities Working Together
1 / T_total = 1 / T_1 + 1 / T_2, yielding T_total = (T_1 * T_2) / (T_1 + T_2).
15
Average Speed for a Round Trip with Speeds v1 and v2
Harmonic mean: v_avg = (2 v_1 v_2) / (v_1 + v_2), where distance in each direction is identical.
16
Permutations vs Combinations Formulas
Permutations (order matters): P(n, r) = n! / (n - r)!. Combinations (order irrelevant): C(n, r) = n! / [r!(n - r)!].
17
Probability of Union of Two Events (Addition Rule)
P(A or B) = P(A) + P(B) - P(A and B); if mutually exclusive, P(A and B) = 0.
18
Probability of Intersection of Independent Events (Multiplication Rule)
P(A and B) = P(A) P(B); for conditional events: P(A and B) = P(A) P(B|A).
19
Standard Normal Distribution Empirical Rule (68-95-99.7)
68% of data falls within 1 standard deviation, 95% within 2 SDs, and 99.7% within 3 SDs of the mean.
20
Area and Circumference of a Circle with Radius r
Area = pi r^2; Circumference = 2 pi r = pi d.
21
Volume and Surface Area of a Right Circular Cylinder
Volume = pi r^2 h; Total Surface Area = 2 pi r h + 2 pi * r^2.
22
Subject-Verb Agreement with Collective Nouns
Collective nouns (team, committee, jury) take a singular verb when acting as a unified unit, but plural when individuals act separately.
23
Recognizing and Correcting Dangling Modifiers
An introductory modifying phrase must immediately precede the actual subject performing the action.
24
Parallel Structure in Coordinated Clauses and Series
Items joined by coordinating conjunctions (and, but, or) or correlative conjunctions (neither/nor) must share identical grammatical form.
25
Third Conditional Grammar Structure for Unreal Past Situations
If + past perfect (had + past participle), would have + past participle (e.g., 'If she had studied, she would have passed').

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