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Number and Alphabet Series · Key Concepts
Arithmetic progression
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Number and Alphabet Series — Logical Reasoning · 8 cards
- Arithmetic progression
- Each term increases or decreases by a constant difference (e.g., 3, 7, 11, 15 → difference +4).
- Geometric progression
- Each term is multiplied or divided by a constant ratio (e.g., 2, 6, 18, 54 → ratio ×3).
- Square/cube sequences
- Terms follow n², n³ or related patterns (e.g., 1, 4, 9, 16, 25 are squares of 1, 2, 3, 4, 5).
- Prime number series
- Sequence of consecutive primes (2, 3, 5, 7, 11, 13...).
- Alternating patterns
- Two interleaved sub-series, each with its own rule (e.g., odd positions increase by 3, even positions double).
- Alphabet series
- Letters advance by fixed positions (A+3=D), or follow patterns like vowels, consonants, or reverse alphabet coding (A↔Z, B↔Y...).
- Mixed alphanumeric
- Separate the letter and number components, identify each pattern independently, then combine.
- Two-step or nested patterns
- The differences themselves form a series (e.g., differences are 2, 4, 6, 8...).
Analogy and Classification — Logical Reasoning · 7 cards
- Analogy
- means similarity or correspondence between two pairs. If A relates to B in a certain way, then C relates to D in the same way. Format: A : B :: C : ?
- Classification
- means grouping items by a common property, then identifying the element that doesn't belong. Three items share a trait; one does not.
- Number-based patterns
- include arithmetic operations (sum, difference, product), divisibility rules, squares/cubes, prime/composite classification and place-value manipulation (reverse, digit sum).
- Alphabet-based patterns
- rely on letter positions (A=1, B=2...Z=26), vowel/consonant classification, alphabetical order and letter shifts (next letter, skip patterns).
- Word-based classification
- uses semantic categories (fruits, animals, metals), part-of-speech grouping (noun, verb) or functional relationships (tool-action, animal-habitat).
- Figure-based patterns
- involve shape properties (number of sides, symmetry), shading/rotation, element count (dots, lines) and spatial transformations (mirror, rotation by 90°).
- Systematic elimination
- is your best friend: check each option against the identified rule; discard mismatches immediately.
Coding-Decoding — Logical Reasoning · 8 cards
- Letter Position Value
- Each letter has a position in the alphabet—A=1, B=2, C=3, ... Z=26. Many codes involve adding or subtracting from these positions.
- Reverse Alphabet
- The reverse alphabet maps A↔Z, B↔Y, C↔X, etc. A common trick is to replace each letter with its opposite-end partner.
- Shift/Caesar Cipher
- Move every letter forward or backward by a fixed number of steps (e.g. shift +3 means A→D, B→E, C→F).
- Arithmetic Operations on Positions
- Add, subtract, multiply or divide the position values and convert the result back to a letter (e.g. if position value exceeds 26, apply modulo 26 or wrap around).
- Symbol or Number Substitution
- Letters are replaced by symbols or numbers according to a given key, or each letter is assigned a fixed digit.
- Mixed Rules
- Some problems combine two operations—first reverse, then shift; or first shift, then swap vowels and consonants.
- Word-Level Coding
- The entire word is treated as a unit—count letters, sum position values or rearrange by a rule (e.g. reverse the word, then shift each letter).
- Number Coding
- Digits in a number are rearranged, incremented or replaced by their complements (e.g. 9−digit for each digit).
Mathematical Operations — Logical Reasoning · 6 cards
- Symbol substitution
- Familiar operators (+, −, ×, ÷) are redefined. Always use the new meanings given in the question; ignore what the symbol normally means.
- BODMAS priority
- Even when symbols are swapped, BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction) usually still applies unless the question explicitly says otherwise. Apply the substituted operation in BODMAS order.
- Operator-based logic problems
- Some questions define new symbols (e.g. a @ b = 2a + b). You must compute by plugging values into the defined formula.
- "Which statement is correct?"
- questions: Test each answer choice by substituting and computing. Only one will satisfy the equation.
- Nested operations
- When expressions have multiple operations, work inside-out (brackets first) and left-to-right for equal-precedence operations after substitution.
- Comparative problems
- Questions may ask "if 12 ÷ 3 @ 4 = X, find X" where each symbol has a new meaning. Translate, then calculate.
Direction Sense Test — Logical Reasoning · 6 cards
- Compass Directions
- North (N), South (S), East (E), West (W) are the four cardinal directions. Intermediate directions are North-East (NE), North-West (NW), South-East (SE), South-West (SW), each making 45° angles with cardinal directions.
- Coordinate Model
- Treat the starting point as origin (0, 0). North is positive y-direction, South is negative y, East is positive x-direction, West is negative x. Every movement updates your coordinates.
- Displacement vs Distance
- Total distance traveled is the sum of all movement lengths (scalar). Displacement is the straight-line distance from start to finish (vector) — always calculated using Pythagoras theorem for perpendicular movements.
- Turn Instructions
- "Turn left" or "turn right" means rotate 90° from your current facing direction, not absolute compass direction. If facing North and you turn right, you now face East. If facing South and you turn left, you now face East.
- Net Movement
- After all movements, calculate net northward/southward displacement (sum all N movements minus all S movements) and net eastward/westward displacement (sum all E movements minus all W movements). This gives you the two legs of a right triangle.
- Final Direction
- The direction from starting point to final position is determined by the signs of net x and net y displacements — positive x and positive y means North-East sector, negative x and positive y means North-West, etc.
Blood Relations — Logical Reasoning · 8 cards
- Reference Person
- The individual from whose perspective relationships are described. In "A is B's father," B is the reference. In pointing problems, the speaker is usually the reference.
- Generation Counting
- Parents and children differ by one generation, grandparents and grandchildren by two. Siblings are in the same generation. Track generation gaps to avoid confusing nephew with son or uncle with brother.
- Gender Determination
- Many relationships specify gender (father, sister, son), but some are ambiguous (parent, sibling, child). Use context clues—possessive forms like "his brother" imply the brother is male, while "her brother" tells you the reference person is female.
- Paternal vs. Maternal Relations
- Father's side is paternal (paternal uncle = father's brother), mother's side is maternal (maternal uncle = mother's brother). "Uncle" alone is ambiguous—check whether it's specified as father's or mother's sibling.
- Only Child / Only Son / Only Daughter
- "Only" means there are no other siblings of that type. "Only son" means no other sons exist, but daughters may exist. "Only child" means no siblings at all.
- Pointing Problems
- "Pointing to a photograph, A says 'He is my...'" requires you to work backward from A's statement. Identify A's relationship to the person in the photo, then deduce who that person is relative to A or another named person.
- Sibling Terminology
- Brother/sister are direct siblings. Cousin is the child of your parent's sibling. Terms like "brother-in-law" or "sister-in-law" introduce marriage links—spouse's sibling or sibling's spouse.
- "Of" and "To" Relationships
- "Son of X" means X is the parent. "Father to Y" means Y is the child. The preposition indicates direction of relationship.
Ranking and Arrangement — Logical Reasoning · 7 cards
- Ranking from top and bottom
- If someone is nth from top and mth from bottom in a queue, total people = n + m − 1 (subtract 1 to avoid counting that person twice).
- Linear arrangement
- People or objects seated in a straight row, facing a direction (usually north). "Left" and "right" are relative to the person's viewpoint when facing that direction.
- Circular arrangement
- Seating around a table—clockwise/anticlockwise matters, and there's no absolute "first" position. If all face center, "left" means clockwise neighbor; if facing outward, "left" means anticlockwise neighbor.
- Fixed reference point
- Always anchor your diagram to a fixed person or position mentioned in the problem. Place them first, then position others relative to them.
- Immediate vs. positional neighbors
- "Next to" means immediately adjacent (one seat away). "Two places to the left" means one person in between.
- Overlapping constraints
- Some problems give multiple clues that must all be satisfied simultaneously. Draw and redraw until all conditions fit, or use elimination on answer choices.
- Rank overlap formula
- If A is pth from start and qth from end in a line, and B is rth from start and sth from end, you can find people between A and B or total count using addition/subtraction logic carefully.
Mirror and Water Images — Logical Reasoning · 2 cards
- Mirror Image
- means reflection across a vertical axis (imagine a mirror held vertically to the right or left of the object). Left and right swap, but top and bottom stay the same.
- Water Image
- means reflection across a horizontal axis (imagine seeing the object's reflection in still water below it). Top and bottom swap, but left and right stay the same.
Paper Folding and Cutting — Logical Reasoning · 8 cards
- Fold creates symmetry
- Every fold acts as a mirror line. A cut on the folded side will appear on both sides of the fold line when unfolded.
- Layers multiply holes
- If paper is folded once, a single punch creates 2 holes (one on each half). Two folds create up to 4 layers, so one punch creates 4 holes.
- Order of folds matters
- The sequence in which folds are made affects the final symmetry axes. Always process folds in the given order.
- Punch position is critical
- A hole near a fold line creates holes close together when unfolded. A hole far from fold lines creates widely spaced holes.
- Common fold types
- Horizontal fold (top to bottom or bottom to top), vertical fold (left to right or right to left), diagonal fold (corner to corner).
- Mental reconstruction
- After unfolding, every hole punched appears in reflected positions across each fold line. Use symmetry to place each hole.
- No rotation after unfolding
- The unfolded paper returns to its original orientation. Do not rotate the final figure unless the question explicitly asks for it.
- Edge cuts and corner cuts
- Cuts at edges or corners produce distinctive patterns (half-circles, quarter-circles) when unfolded.
Cubes and Dice — Logical Reasoning · 8 cards
- Standard dice convention
- A standard die has faces numbered 1 to 6 such that opposite faces always sum to 7 (1-6, 2-5, 3-4). This is the most common assumption unless stated otherwise.
- Cube colouring structure
- When a cube is painted and cut into smaller cubes, only the outermost layer gets paint. Interior cubes remain unpainted. The number of painted faces on a small cube depends on its position: corner (3 faces), edge (2 faces), face-centre (1 face), or interior (0 faces).
- Corner-edge-face formula
- For an n×n×n cube, there are 8 corners (always), 12(n−2) edge cubes, 6(n−2)² face-centre cubes, and (n−2)³ interior cubes. These formulas let you count without drawing.
- Dice net/unfolding
- When a dice is shown in different orientations, track one face and use the right-hand rule or elimination to deduce opposite faces. Two views of the same dice always maintain consistent internal relationships.
- Visibility principle in stacks
- In a stack of cubes, only outer layer cubes are visible from any single viewpoint. Hidden cubes must be inferred from the structure—count total cubes in the stack, then subtract visible ones if needed.
- Rotation vs reflection
- When comparing two dice images, ensure you're accounting for rotation (same dice, different angle) versus reflection (mirror image). Standard dice problems assume rotation only.
- Layer-by-layer counting
- For complex stacks or painted cubes, break the structure into layers or categories. Count systematically—don't jump around or you'll double-count or miss cubes.
- Process of elimination
- If you know which faces are adjacent to a given face, the opposite face is the one not shown in any adjacent position. This is faster than trying to visualize the entire 3D model.
Matrix and Pattern Completion — Logical Reasoning · 8 cards
- Grid-based logic
- In 3×3 matrices, the pattern typically operates row-wise (all rows follow the same rule), column-wise (all columns follow the same rule), or both. Always check complete rows and columns first before analyzing diagonals.
- Figure transformation rules
- Elements may rotate (90°, 180°, clockwise/anticlockwise), flip horizontally or vertically, increase/decrease in size, change shading (white → gray → black), or add/remove internal features like dots, lines, or corners.
- Element progression
- Shapes may cycle through a sequence (circle → square → triangle → circle), positions may shift systematically (top-left → center → bottom-right), or the number of elements may follow arithmetic patterns (+1, +2, or ×2 per cell).
- Combined operations
- A single cell may result from combining features of other cells in the same row or column. Example: Cell 3 = union of shapes from Cell 1 and Cell 2, or Cell 3 contains only features common to both Cell 1 and Cell 2.
- Number and symbol patterns
- When matrices contain numbers or symbols, apply arithmetic operations (addition, subtraction, multiplication across rows/columns) or look for differences, ratios, or modular patterns.
- Repeating sequences
- Linear patterns present a sequence with regular repetition or a step-by-step rule. Identify the repeat unit or the increment/decrement rule, then predict the next term.
- Elimination strategy
- If the pattern isn't immediately obvious, eliminate answer choices that clearly violate any observed partial rule. Often two options can be discarded quickly, leaving a 50-50 guess if time is short.
- Symmetry and balance
- Some matrices maintain symmetry (central cell reflects properties of surrounding cells) or balance (sum of features in row 1 = row 2 = row 3). Check for these when standard row/column rules don't apply.
Number Systems — Mathematical Reasoning · 8 cards
- Natural Numbers (N)
- Counting numbers {1, 2, 3, ...}. Whole Numbers (W) = N ∪ {0}. Integers (Z) include negative numbers too.
- Rational Numbers (Q)
- Any number expressible as p/q where p, q are integers and q ≠ 0. Includes terminating and repeating decimals.
- Irrational Numbers
- Real numbers that cannot be written as p/q. Their decimal expansions are non-terminating and non-repeating (e.g., √2, π, e).
- Real Numbers (R)
- The union of all rational and irrational numbers. Every point on the number line represents a real number.
- Terminating Decimals
- Rational numbers whose denominators (in simplest form) have only 2 and/or 5 as prime factors. Example: 7/8 = 0.875 terminates.
- Non-Terminating Repeating Decimals
- Rational numbers whose denominators contain prime factors other than 2 or 5. Example: 1/3 = 0.333... repeats.
- Laws of Exponents for Real Numbers
- For any positive real numbers a, b and rational exponents m, n: a^m × a^n = a^(m+n), (a^m)^n = a^(mn), a^m / a^n = a^(m−n), (ab)^n = a^n b^n, a^0 = 1, a^(−n) = 1/a^n, a^(1/n) = nth root of a.
- Rationalization
- Multiplying numerator and denominator by the conjugate to eliminate radicals from the denominator. For 1/(√a + √b), multiply by (√a − √b)/(√a − √b).
Polynomials — Mathematical Reasoning · 7 cards
- Polynomial definition
- An algebraic expression of the form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ ≠ 0, n is a non-negative integer, and coefficients are real numbers.
- Degree of a polynomial
- The highest power of the variable with a non-zero coefficient. A constant (non-zero) has degree 0; zero polynomial has no defined degree.
- Zero (root) of a polynomial
- A value α such that p(α) = 0. A polynomial of degree n has at most n real zeroes.
- Remainder Theorem
- When polynomial p(x) is divided by (x − a), the remainder is p(a). This lets you find remainders without performing long division.
- Factor Theorem
- (x − a) is a factor of p(x) if and only if p(a) = 0. This is the converse logic used to factorize polynomials.
- Relationship between zeroes and coefficients
- For a quadratic ax² + bx + c, if α and β are zeroes, then α + β = −b/a and αβ = c/a. Similar relations exist for cubic polynomials.
- Algebraic identities
- Pre-memorized expansion formulas like (a + b)² = a² + 2ab + b², (a − b)(a + b) = a² − b², and (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca speed up factorization and simplification.
Coordinate Geometry — Mathematical Reasoning · 8 cards
- Cartesian Plane
- A two-dimensional plane defined by perpendicular x-axis (horizontal) and y-axis (vertical) intersecting at origin O(0,0). Any point is represented as an ordered pair (x, y) where x is the abscissa and y is the ordinate.
- Quadrants
- The plane divides into four quadrants — I (+,+), II (−,+), III (−,−), IV (+,−). The signs of coordinates determine which quadrant a point lies in. Points on axes belong to no quadrant.
- Distance Formula
- The distance between two points A(x₁, y₁) and B(x₂, y₂) is given by d = √[(x₂−x₁)² + (y₁−y₂)²]. This comes from the Pythagorean theorem applied to the right triangle formed by the points.
- Section Formula (Internal Division)
- If point P divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in ratio m:n, then P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).
- Midpoint Formula
- Special case of section formula when m:n = 1:1. Midpoint M = ((x₁+x₂)/2, (y₁+y₂)/2). This is the arithmetic mean of the coordinates.
- Section Formula (External Division)
- When P divides AB externally in ratio m:n, P = ((mx₂−nx₁)/(m−n), (my₂−ny₁)/(m−n)). Note the subtraction in both numerator and denominator.
- Collinearity Test
- Three points A, B, C are collinear if the distance AB + BC = AC (or any cyclic permutation). Alternatively, area of triangle ABC equals zero.
- Origin and Axes
- Distance from origin to point (x, y) is √(x²+y²). Distance of a point from x-axis is |y| and from y-axis is |x|.
Linear Equations in Two Variables — Mathematical Reasoning · 8 cards
- Standard Form
- Any linear equation can be written as ax + by + c = 0. Every solution (x, y) of this equation corresponds to a point on a straight line when plotted on the Cartesian plane.
- Graphical Solution
- The solution of two linear equations is the point where their graphs intersect. If lines are parallel (no intersection), there's no solution; if they coincide (overlap completely), infinitely many solutions exist.
- Condition for Solutions
- For equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, compare ratios: a₁/a₂ ≠ b₁/b₂ gives unique solution; a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives no solution; a₁/a₂ = b₁/b₂ = c₁/c₂ gives infinite solutions.
- Substitution Method
- Express one variable in terms of the other from one equation, then substitute into the second equation to get a single-variable equation. Most efficient when one coefficient is already 1 or when one equation is simpler.
- Elimination Method
- Multiply equations by suitable numbers to make coefficients of one variable equal (or opposite), then add or subtract equations to eliminate that variable. Best when coefficients can be easily matched.
- Cross-Multiplication Method
- Direct formula-based approach where x/(b₁c₂ - b₂c₁) = y/(c₁a₂ - c₂a₁) = 1/(a₁b₂ - a₂b₁). Fastest for equations already in standard form with no zero coefficients.
- Word Problem Strategy
- Identify two unknowns, form two equations from given conditions, solve using any method, then verify if the solution makes practical sense in context.
- Consistency Check
- After solving, always substitute your answer back into both original equations to verify correctness—critical for avoiding sign errors and calculation mistakes.
Introduction to Euclid's Geometry — Mathematical Reasoning · 6 cards
- Axiom vs Postulate
- Euclid distinguished axioms (universal truths applying to all sciences, like "things equal to the same thing are equal to each other") from postulates (geometric-specific assumptions, like "a straight line may be drawn between any two points"). Modern usage often treats them interchangeably as "axioms."
- Undefined Terms
- Point, line and plane are now taken as undefined (Euclid did define them, but mathematicians do not accept his definitions); they are accepted intuitively. A point has no dimension, a line is breadthless length extending infinitely in both directions, and a plane is a flat surface extending infinitely in all …
- Definitions vs Axioms
- A definition assigns meaning to a term (e.g., "a circle is the locus of points equidistant from a center"), while an axiom is an assumed truth used to prove other statements. Definitions are not proved; axioms are not defined.
- Deductive Proof Structure
- Euclidean geometry proceeds by logical deduction: start with axioms/postulates, apply definitions, and use previously proved theorems to establish new results. Each statement in a proof must be justified.
- Equivalence Relations
- Several axioms establish transitive, symmetric and reflexive properties of equality, which underpin substitution and algebraic manipulation in geometry.
- Playfair's Axiom
- The modern replacement for Euclid's fifth postulate states "through a point not on a line, exactly one parallel to that line can be drawn." This is equivalent to the parallel postulate and easier to understand.
Lines and Angles — Mathematical Reasoning · 5 cards
- Complementary angles
- sum to 90° (think: they complete a right angle). Supplementary angles sum to 180° (they form a straight line). These appear in both isolated pairs and within larger figures.
- Adjacent angles
- share a common vertex and arm, with no overlap. A linear pair is two adjacent angles whose non-common arms form a straight line—they're always supplementary.
- Vertically opposite angles
- form when two lines intersect. They're always equal. This is one of the fastest angle deductions you can make.
- Angle sum property of a triangle
- (180°) often combines with line-angle problems. An exterior angle of a triangle equals the sum of the two non-adjacent interior angles—a powerful shortcut.
- Bisectors
- (angle or line) create equal parts. If a transversal bisects an angle formed by parallel lines, it often creates congruent or supplementary relationships useful for multi-step solutions.
Triangles — Mathematical Reasoning · 8 cards
- Congruent triangles
- are identical in shape and size. All corresponding sides and angles are equal. Symbol: △ABC ≅ △PQR means AB = PQ, BC = QR, CA = RP and all corresponding angles match.
- Four congruence criteria
- SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), and RHS (right angle, hypotenuse, one side). No AAA criterion—similar triangles can have different sizes.
- CPCT principle
- Corresponding Parts of Congruent Triangles are equal. After proving congruence, you can state any corresponding element is equal without further proof.
- Isosceles triangle
- Two sides equal implies two base angles equal. Conversely, two angles equal implies opposite sides equal. The altitude from the apex bisects the base and is perpendicular to it.
- Equilateral triangle
- All sides equal, all angles 60°. Every altitude is also a median, angle bisector, and perpendicular bisector.
- Triangle inequality theorem
- Sum of any two sides must exceed the third side. For sides a, b, c: a + b > c, b + c > a, c + a > b. Also, difference of two sides is less than the third: |a − b| < c.
- Angle-side relationship
- In any triangle, the side opposite the largest angle is longest; the side opposite the smallest angle is shortest. This extends to inequalities: if angle A > angle B, then side opposite A (BC) > side opposite B (AC).
- Exterior angle theorem
- An exterior angle equals the sum of the two non-adjacent interior angles. The exterior angle is greater than either remote interior angle.
Quadrilaterals — Mathematical Reasoning · 8 cards
- Quadrilateral basics
- A closed figure with four sides, four vertices and four angles. Sum of all interior angles is always 360°. Quadrilaterals are classified by their side lengths, angle measures and diagonal properties.
- Parallelogram
- Opposite sides are parallel and equal. Opposite angles are equal. Consecutive angles are supplementary (sum = 180°). Diagonals bisect each other but are not necessarily equal or perpendicular.
- Rectangle
- A parallelogram with all four angles equal to 90°. Diagonals are equal in length and bisect each other, but they are not perpendicular unless the rectangle is a square.
- Rhombus
- A parallelogram with all four sides equal. Diagonals are perpendicular bisectors of each other. Opposite angles are equal, and diagonals bisect the vertex angles.
- Square
- A rectangle with all sides equal, or equivalently a rhombus with all angles 90°. Diagonals are equal, perpendicular and bisect each other at right angles. Each diagonal bisects opposite angles into 45° each.
- Trapezium
- Exactly one pair of opposite sides is parallel. The parallel sides are called bases. Mid-segment (line joining mid-points of non-parallel sides) is parallel to bases and equals half the sum of the bases.
- Kite
- Two pairs of consecutive sides are equal. One diagonal is the perpendicular bisector of the other. The angles between unequal sides are equal.
- Mid-point theorem for quadrilaterals
- The figure formed by joining the mid-points of the sides of any quadrilateral is always a parallelogram. If the original quadrilateral has perpendicular diagonals, the mid-point figure is a rectangle; if it has equal diagonals, the mid-point figure is a rhombus.
Areas of Parallelograms and Triangles — Mathematical Reasoning · 7 cards
- Equal bases, equal heights
- If two parallelograms (or two triangles) have equal bases and equal heights, their areas are equal. The height is the perpendicular distance between the parallel lines containing the base and opposite side.
- Same base, same parallels
- When two figures share the same base and their opposite vertices (or sides) lie on a line parallel to the base, they have the same height, hence equal areas.
- Parallelograms on the same base between same parallels have equal area
- All parallelograms with base AB and opposite sides on a line parallel to AB have identical area = AB × h, where h is the distance between the parallels.
- Triangles on the same base between same parallels have equal area
- All triangles with base AB and third vertex on a line parallel to AB have identical area = ½ × AB × h.
- A triangle is half a parallelogram
- A triangle and a parallelogram on the same base and between the same parallels satisfy Area(triangle) = ½ Area(parallelogram).
- Median divides a triangle into two equal areas
- The median from any vertex splits the triangle into two smaller triangles of equal area because both have the same height from that vertex and equal bases (half the original base each).
- Area additivity
- If a figure is divided into non-overlapping parts, total area = sum of parts. Use this with equal-area theorems to solve complex problems.
Circles — Mathematical Reasoning · 8 cards
- Equal chords equidistant from center
- Chords equal in length lie at equal perpendicular distances from the circle's center, and conversely, chords equidistant from the center are equal in length.
- Perpendicular from center bisects chord
- A line from the center perpendicular to a chord always bisects that chord; this creates right triangles useful for applying Pythagoras.
- Angles subtended by an arc
- An arc subtends twice the angle at the center compared to any point on the remaining part of the circle (angle at center = 2 × angle at circumference).
- Angles in the same segment
- All angles subtended by the same arc at different points on the circle (in the same segment) are equal.
- Angle in a semicircle
- Any angle subtended by a diameter at a point on the circle is a right angle (90°).
- Cyclic quadrilateral property
- A quadrilateral is cyclic (all four vertices lie on a circle) if and only if the sum of each pair of opposite angles equals 180°.
- Tangent perpendicular to radius
- A tangent to a circle is always perpendicular to the radius at the point of contact.
- Equal tangents from external point
- Two tangents drawn from the same external point to a circle are equal in length and subtend equal angles with the line joining that point to the center.
Constructions — Mathematical Reasoning · 8 cards
- Compass and straight-edge only
- All constructions must be done without measuring angles or lengths. The compass transfers distances; the straight-edge connects points. No protractor or marked ruler allowed.
- Arc intersections create key points
- Most constructions rely on drawing arcs from two different centers. Where arcs intersect defines new points with specific geometric properties (equidistant from centers, for example).
- Angle bisector divides an angle into two equal parts
- Constructed by drawing equal arcs from the angle vertex, then arcs from where those arcs cut the arms, creating an intersection that lies on the bisector.
- Perpendicular bisector of a segment
- Every point on this line is equidistant from the segment's endpoints. Constructed by arcs of equal radius from both endpoints; their intersections determine the perpendicular bisector.
- Triangle construction uses congruence criteria
- SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), RHS (right angle, hypotenuse, one side). Each criterion has a specific construction sequence.
- Tangent from external point
- A tangent is perpendicular to the radius at the point of contact. Constructing tangents requires finding the perpendicular from the external point to the radius, often using a semicircle on the line joining the point to the center.
- 60° and 30° angles
- Easily constructed using equilateral triangle properties (compass set to radius = side length) and bisecting 60° to get 30°.
- 90° angle (perpendicular)
- Constructed by replicating the perpendicular bisector method at a point on a line or from a point to a line.
Heron's Formula — Mathematical Reasoning · 6 cards
- Semi-perimeter (s)
- Half the perimeter of the triangle. For a triangle with sides a, b, c: s = (a + b + c)/2. This value is central to Heron's Formula and must be calculated first.
- Heron's Formula
- Area = √[s(s − a)(s − b)(s − c)], where s is the semi-perimeter and a, b, c are the three sides. Works for any triangle — scalene, isosceles, or equilateral.
- Units consistency
- All three sides must be in the same unit (cm, m, etc.). The area will be in square units of that measurement.
- Quadrilateral decomposition
- Any quadrilateral can be split into two triangles by drawing one diagonal. Find the area of each triangle separately using Heron's Formula, then add them.
- Triangle inequality check
- Before applying the formula, verify that a + b > c, b + c > a, and c + a > b. If any inequality fails, those sides cannot form a triangle.
- Validation with known formulas
- For right triangles, check your Heron result against (1/2) × base × height. For equilateral triangles with side a, Heron gives (√3/4)a², matching the standard formula.
Surface Areas and Volumes — Mathematical Reasoning · 8 cards
- Lateral vs. Total Surface Area
- Lateral surface area (LSA or CSA—curved surface area) covers only the sides, excluding top and bottom bases. Total surface area (TSA) includes all exposed faces. For a cylinder, CSA = 2πrh; TSA = 2πr(r + h).
- Volume as Capacity
- Volume measures the space inside a solid, directly translating to capacity for containers. 1 cubic metre = 1000 litres, and 1 cubic centimetre = 1 millilitre—critical for word problems involving water tanks or paint coverage.
- Composite Solids
- Real IMO problems present combinations like a tent (cone + cylinder), a capsule (cylinder + two hemispheres), or a toy (cone on hemisphere). Calculate each part separately, then sum for total volume; for surface area, exclude contact regions.
- Dimension Relationships
- Radius (r), diameter (d = 2r), height (h), and slant height (l) are interconnected. For a cone, l² = r² + h² by Pythagoras. Always check which measurements the problem provides versus what the formula needs.
- Unit Conversions
- Problems may mix units—radius in cm, height in m, asking answer in litres. Convert everything to one unit system before plugging into formulas. 1 m = 100 cm; 1 m³ = 10⁶ cm³.
- Cost and Material Problems
- If painting costs ₹5 per m² and TSA is 120 m², total cost = 5 × 120 = ₹600. When thickness is involved (hollow cylinders, walls), treat as the difference of two volumes: outer solid minus inner cavity.
- Sphere vs. Hemisphere
- A hemisphere is exactly half a sphere by volume, but its TSA includes the flat circular base. Hemisphere TSA = 2πr² (curved) + πr² (base) = 3πr².
- Melting and Recasting
- When a solid is melted and recast into a different shape, volume remains constant but surface area changes. Set volume₁ = volume₂ and solve for the unknown dimension.
Statistics — Mathematical Reasoning · 7 cards
- Ungrouped data
- consists of individual observations listed separately, while grouped data organizes observations into class intervals with their frequencies. Ungrouped data is easier to handle but becomes unwieldy for large datasets.
- Mean (arithmetic average)
- represents the sum of all observations divided by their count. For grouped data, we use class marks (midpoints) multiplied by frequencies, making it sensitive to extreme values.
- Median
- is the middle value when data is arranged in ascending or descending order, making it resistant to outliers. For even number of observations, median equals the average of the two middle values.
- Mode
- is the observation or class interval with the highest frequency. A dataset can be unimodal (one mode), bimodal (two modes) or multimodal, while some datasets have no mode if all frequencies are equal.
- Class mark (midpoint)
- for any class interval equals (lower limit + upper limit)/2 and serves as the representative value for that entire class in grouped data calculations.
- Cumulative frequency
- is the running total of frequencies up to a particular class, essential for finding the median in grouped data and constructing cumulative frequency curves (ogives).
- Range
- measures data spread as the difference between maximum and minimum values, providing a simple indicator of variability alongside central tendency measures.
Probability — Mathematical Reasoning · 8 cards
- Random experiment
- Any action whose outcome cannot be predicted with certainty (e.g., tossing a coin, rolling a die).
- Sample space (S)
- The set of all possible outcomes. For a die, S = {1, 2, 3, 4, 5, 6}; for two coins, S = {HH, HT, TH, TT}.
- Event (E)
- A subset of the sample space. Example: getting an even number on a die is E = {2, 4, 6}.
- Theoretical probability
- P(E) = (Number of favourable outcomes) / (Total number of equally likely outcomes). All outcomes must be equally likely for this formula to work.
- Experimental probability
- P(E) = (Number of times event occurred) / (Total number of trials). Based on observed data or repeated experiments.
- Certain event
- Probability = 1 (it always happens).
- Impossible event
- Probability = 0 (it never happens).
- Range of probability
- For any event E, 0 ≤ P(E) ≤ 1. Probabilities can be expressed as fractions, decimals, or percentages.
Real Numbers (Class 10) — Mathematical Reasoning · 8 cards
- Euclid's Division Lemma
- For any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that a = bq + r, where 0 ≤ r < b. This is the foundation of the division algorithm and HCF computation.
- Euclid's Division Algorithm
- To find HCF of two numbers, repeatedly apply the lemma: divide the larger by the smaller, replace the larger with the smaller and the smaller with the remainder, until remainder becomes zero. The last non-zero remainder is the HCF.
- Fundamental Theorem of Arithmetic
- Every composite number can be expressed as a product of primes in exactly one way (ignoring order). This uniqueness property is central to solving HCF, LCM and divisibility problems.
- Prime Factorization Method
- Express numbers as products of prime powers (e.g. 72 = 2³ × 3²). HCF is the product of lowest powers of common primes; LCM is the product of highest powers of all primes present.
- Rational Numbers
- Numbers expressible as p/q where p, q are integers and q ≠ 0. Their decimal expansions either terminate or repeat periodically.
- Irrational Numbers
- Numbers that cannot be expressed as p/q. Their decimal expansions are non-terminating and non-repeating. Examples: √2, √3, π, e.
- Proving Irrationality
- Use contradiction — assume the number is rational (p/q in lowest terms), derive that both p and q share a common factor, contradicting the assumption that p/q is in lowest terms.
- Operations on Irrationals
- Sum or product of rational and irrational is irrational (for the product, the rational must be non-zero: 0 × √2 = 0). Sum of two irrationals may be rational (√2 + (-√2) = 0) or irrational (√2 + √3).
Quadratic Equations (Class 10) — Mathematical Reasoning · 8 cards
- Standard form
- A quadratic equation must be written as ax² + bx + c = 0 (a ≠ 0) before solving. If a = 0, it becomes linear, not quadratic.
- Roots or zeroes
- The values of x that satisfy the equation. A quadratic can have at most two real roots, though they may be equal or complex.
- Factorization method
- Express ax² + bx + c as a product of two linear factors. Works cleanly only when roots are rational. Look for two numbers that multiply to ac and add to b.
- Completing the square
- Transform the equation into (x + p)² = q form. This method reveals the vertex of the parabola and works for all quadratics, though it's algebraically intensive.
- Quadratic formula
- x = (–b ± √(b² – 4ac)) / 2a. Universal method that works for any quadratic. Memorize this formula—it's your fallback for difficult factorizations.
- Discriminant Δ
- Defined as Δ = b² – 4ac. It determines root nature: Δ > 0 gives two distinct real roots, Δ = 0 gives two equal real roots (repeated root), Δ < 0 gives no real roots (complex roots, beyond Class 10 scope).
- Sum and product of roots
- If α and β are roots, then α + β = –b/a and αβ = c/a. Use these relations to form equations from given roots or verify solutions quickly.
- Word problems translation
- "Consecutive integers", "length exceeds breadth by 5", "sum of a number and its reciprocal" all translate to quadratic setups. After solving, reject negative or non-physical solutions.
Arithmetic Progressions (Class 10) — Mathematical Reasoning · 7 cards
- Arithmetic Progression definition
- A sequence where the difference between consecutive terms is constant. Example: 3, 7, 11, 15, … has common difference d = 4.
- Common difference (d)
- Found by subtracting any term from the next term: d = a₂ − a₁. The sign of d determines if the AP is increasing (d > 0), decreasing (d < 0), or constant (d = 0).
- First term (a)
- The starting value of the sequence, often denoted as a or a₁. Every AP formula requires knowing the first term.
- nth term formula
- Gives the value of any term in the sequence directly without listing all previous terms. Saves time in competitive exams.
- Sum of n terms formula
- Computes the total of the first n terms efficiently. Two versions exist—use the one that fits the given information.
- Word problems
- Typically describe patterns (salary increments, seating rows, stacking objects) that form an AP. Identify a, d, and n from the context, then apply formulas.
- Finite AP
- Has a specific number of terms. Problems often ask for the last term or the sum of all terms in such sequences.
Introduction to Trigonometry (Class 10) — Mathematical Reasoning · 11 cards
- Trigonometric Ratios
- In a right triangle with angle θ, the six ratios relate the sides: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent, and their reciprocals cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
- Domain Restrictions
- Trigonometric ratios are defined only when denominators are non-zero; for example, tan 90° and sec 90° are undefined because cos 90° = 0.
- Complementary Angles
- Two angles are complementary if they sum to 90°. Key relationships: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ, and vice versa for reciprocals.
- Pythagorean Identities
- These stem from Pythagoras theorem applied to the unit circle: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ. These identities are essential for simplification and proof problems.
- Standard Angle Values
- Memorize exact values for 0°, 30°, 45°, 60°, 90° for all six ratios. For instance, sin 30° = 1/2, cos 45° = 1/√2, tan 60° = √3, cosec 90° = 1.
- Heights and Distances
- Real-world problems involve angle of elevation (looking up) or angle of depression (looking down). The key is to identify the right triangle, label sides relative to the angle, and choose the appropriate trigonometric ratio to solve for the unknown.
- Ratio Relationships
- Remember that tan θ = sin θ/cos θ and cot θ = cos θ/sin θ. These help convert expressions between different ratios during simplification.
- Sign Convention
- In Class 10 scope, all angles are acute (0° to 90°) and all defined trigonometric ratios are non-negative (zero only at 0° or 90°), simplifying computations and avoiding sign errors.
- sin
- 0°: 0 · 30°: 1/2 · 45°: 1/√2 · 60°: √3/2 · 90°: 1
- cos
- 0°: 1 · 30°: √3/2 · 45°: 1/√2 · 60°: 1/2 · 90°: 0
- tan
- 0°: 0 · 30°: 1/√3 · 45°: 1 · 60°: √3 · 90°: undefined
Percentage, Profit and Loss — Everyday Mathematics · 6 cards
- Percentage
- means "per hundred" and is denoted by %. To convert a fraction to percentage, multiply by 100. To convert percentage to fraction, divide by 100.
- Cost Price (CP)
- is the price at which an article is purchased. Selling Price (SP) is the price at which it is sold. Profit or loss is always calculated on CP unless stated otherwise.
- Profit
- occurs when SP > CP. Profit = SP − CP. Loss occurs when SP < CP. Loss = CP − SP.
- Marked Price (MP)
- is the printed or list price on an article. Discount is the reduction offered on MP. Selling Price after discount: SP = MP − Discount.
- Successive percentages
- are not additive. If two discounts or changes are applied one after another, you must apply them sequentially, not by adding percentages. For example, 10% discount followed by 20% discount is not 30% discount.
- Overhead expenses
- or additional costs increase the effective CP. Always add these to the purchase price before calculating profit or loss.
Time, Speed and Distance — Everyday Mathematics · 8 cards
- Core Formula
- Distance (D) = Speed (S) × Time (T). Any one can be found if the other two are known. Speed = D/T, Time = D/S.
- Unit Conversion
- Always match units. 1 km/h = 5/18 m/s; to convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.
- Relative Speed (Same Direction)
- When two objects move in the same direction, their relative speed is the difference: S₁ − S₂.
- Relative Speed (Opposite Direction)
- When two objects move towards each other, their relative speed is the sum: S₁ + S₂.
- Average Speed
- Average speed ≠ arithmetic mean of speeds. Use Average Speed = Total Distance / Total Time. Never average two speeds directly unless the times are equal.
- Trains Crossing
- When a train crosses a platform, pole, or another train, add relevant lengths to the distance. Crossing a pole uses train length; crossing a platform uses train length + platform length; crossing another train uses sum of both train lengths.
- Boats and Streams
- Downstream speed = (speed in still water) + (stream speed). Upstream speed = (speed in still water) − (stream speed). Speed in still water = (downstream + upstream)/2. Stream speed = (downstream − upstream)/2.
- Meeting and Crossing
- If two objects start simultaneously from two points and move towards each other, time to meet = (distance between them) / (sum of speeds).
Time and Work — Everyday Mathematics · 8 cards
- Work is one unit
- Treat the entire task as 1 job. If A finishes in 10 days, A's one-day work = 1/10 of the job.
- Rate = Work / Time
- A person's efficiency or work rate is the fraction of the job completed per unit time.
- Combined work rate
- When A and B work together, their joint rate = (A's rate) + (B's rate). Time to finish together = 1 / (combined rate).
- Man-days concept
- Total work = (number of workers) × (days worked). Use this to scale problems when the number of workers changes.
- Pipes and cisterns
- Inlet pipes fill (positive rate), outlet pipes drain (negative rate). Net rate = sum of all pipe rates with correct signs.
- Work left behind
- If workers complete part of a job and leave, calculate remaining work = 1 − (work done), then apply the rate of remaining workers to finish.
- Negative time trap
- If you compute a negative time, check whether you've accidentally reversed filling/draining or miscounted the combined rate.
- Efficiency comparison
- If A is twice as efficient as B, A's rate is double B's rate, so A takes half the time B takes.
Simple and Compound Interest — Everyday Mathematics · 7 cards
- Simple Interest (SI)
- is calculated only on the original principal for the entire period. The interest earned each year remains constant.
- Compound Interest (CI)
- is calculated on the principal plus accumulated interest from previous periods. Money "grows on itself," leading to exponential growth.
- Amount (A)
- is the total sum returned at the end of the period: Amount = Principal + Interest.
- Rate (R)
- is always expressed as a percentage per annum unless stated otherwise (half-yearly, quarterly compounding changes the effective rate).
- Time (T)
- is measured in years unless specified otherwise. For half-yearly compounding, double the time and halve the rate; for quarterly, quadruple the time and quarter the rate.
- Instalment problems
- require working backward: the present value of each instalment (discounted back to the start) must sum to the principal borrowed.
- Difference between CI and SI
- over the same period on the same principal at the same rate increases with time and can be used to find unknowns.
Higher Order Mathematical Thinking — Achievers Section · 8 cards
- Multi-Step Reasoning
- HOTS problems rarely solve in one step. You must chain together 3–5 intermediate results, where each step's conclusion becomes the next step's starting point. Practice identifying what you need to prove and work backwards to find the path.
- Proof Construction
- HOTS questions may ask you to judge whether a statement is always true. Learn proof techniques: direct proof (assume hypothesis, derive conclusion), proof by contradiction (assume opposite, find absurdity), and proof by cases (divide into exhaustive scenarios and prove each).
- Concept Integration
- These problems blend topics—for example, using properties of arithmetic progressions within a geometry problem, or applying divisibility rules while solving an algebraic equation. Stay alert for cross-topic connections.
- Pattern Recognition and Generalization
- You might be given examples for n = 1, 2, 3 and asked to prove a formula for all n. Develop the skill to spot recursive patterns, generate hypotheses, and verify them systematically.
- Counterexample Construction
- When asked to determine if a statement is true or false, sometimes the best approach is finding a single counterexample that breaks the claim. This saves time over attempting a full proof.
- Optimization and Extrema
- Questions asking for maximum/minimum values, best arrangements, or optimal strategies require understanding inequalities (AM-GM, triangle inequality) and testing boundary conditions.
- Logical Equivalence and Contrapositive
- Understanding "if P then Q" is equivalent to "if not Q then not P" often simplifies proofs. Practice rephrasing statements in contrapositive form.
- Exhaustive Case Analysis
- Some problems require checking all possible scenarios. Learn to partition cases systematically without overlap or omission—for instance, checking even/odd cases, or considering all factor pairs of a number.
Problem Solving — Achievers Section · 8 cards
- Cross-topic integration
- Problems intentionally combine 2–4 syllabus areas (e.g., coordinate geometry + arithmetic progressions, or circles + number patterns). You must recognize which tools apply to which part of the problem.
- Multi-step reasoning
- Solutions typically require 3–5 logical steps. Each step uses the output of the previous one. Missing or miscomputing one intermediate value cascades into a wrong final answer.
- Real-world contexts
- Many problems are framed as practical scenarios (construction layouts, race timings, mixture problems, price optimization) requiring you to mathematically model the situation before solving.
- Strategic guessing points
- When stuck, identify which intermediate result you can verify independently. Sometimes working backward from answer options reveals the solution pathway.
- Time management
- If you're not making progress after 2 minutes, mark for review and move on. Returning with fresh eyes often helps.
- Hidden patterns
- Some problems appear complex but contain elegant shortcuts — recognizing symmetry, special triangles, or number properties can reduce a 5-step problem to 2 steps.
- Verification mindset
- Always check if your answer satisfies the original problem conditions. Many problems are designed so common errors produce "reasonable-looking" wrong answers.
- Unit awareness
- Pay attention to unit consistency (cm vs. m, hours vs. minutes, rupees vs. paise). Unit errors are the most preventable mistakes in this section.