Linear equation solving measures the ability to manipulate expressions, isolate variables, handle structure (fractions, parameters, distribution), and determine when equations have zero, one, or infinitely many solutions.
– Fractions, parentheses, coefficients
– “Which value of k makes…”
– “No solution / infinitely many solutions”
– Both sides linear in x
– Expressions like: 5(x−2)=3x+8
1) Distribute all parentheses
2) Clear fractions (multiply by LCD)
3) Combine like terms
4) Move all x terms to one side
5) Move constants to the other
6) Solve for x
7) Check for domain issues / extraneous results
Deep Learning: Why These Steps Work
Distribute → simplify the structure: Linear equations rely on balancing operations. Distribution preserves equality while removing complexity.
Clearing fractions first:
Removing denominators transforms the problem into a simple integer-based equation, reducing mistakes.
One x-term rule:
Any linear equation can be rewritten as:
“something times x equals something else.”
No solution / infinite solutions:
– No solution → false statement (e.g., 5=3)
– Infinite solutions → identity (e.g., 4x−8=4(x−2))
✔ Clear fractions immediately — biggest accuracy booster ✔ For parameter questions, compare coefficients ✔ When DSAT gives expressions like “5 − 2(x−3) = 3x + k”, focus on structure, not numbers ✔ If coefficients of x match but constants differ → no solution
– Sign errors after distribution
– Forgetting to multiply every term when clearing fractions
– Treating expressions like terms (e.g., 2(x+3) ≠ 2x+3)
– Ignoring domain restrictions when denominators contain variables
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify equation type, see the structure (linear vs not), spot parentheses/fractions.
Level 2 — Execution:
Perform distribution, combine like terms, isolate x correctly.
Level 3 — Insight:
Make instant decisions:
– Should I clear fractions?
– Will this have no solution?
– Which k-value creates an identity?
Pattern Examples (Common DSAT Forms)
Template 1: Solve: a(bx + c) = d(ex + f)
Template 2: “For what value of k does the equation have no solution?” (Compare coefficients of x and constants.)
Template 3:
Fraction-based linear equation:
$\frac{x-3}{4} + \frac{2x}{3} = 5$
(Multiply by 12 immediately.)
This question style measures the ability to understand, interpret, and construct linear relationships from graphs, tables, or real-world descriptions. Students must recognize what slope and intercept represent in context and convert between representations.
– Words indicating rate: “every”, “per”, “each”, “for every x increase”
– Words indicating initial value: “starts at”, “initially”, “at time t = 0”
– Graphs with constant slope
– Tables with constant difference in y-values
– Modeling phrases: “x represents…”, “y represents…”
1) Identify what x and y represent (units matter!)
2) Extract slope: – Graph → rise/run – Table → Δy/Δx – Text → rate words (“per”, “each”, “every”)
3) Extract intercept: – Graph → y-axis crossing – Table → value at x = 0 – Text → starting value / initial amount
4) Construct linear model: $y = mx + b$
5) Interpret what slope and intercept mean in the real-world context
6) Answer exactly what the question asks (DSAT traps this!)
Deep Learning: What Slope & Intercept Really Mean
Slope as “constant rate”:
Slope is not just rise/run — it is the real-world “speed” of change.
Example: “The temperature increases 3 degrees per hour” → slope = 3.
Intercept as “initial condition”:
Intercept tells you where the story begins.
Example: “You have $15 initially and earn $2 per hour” → model = y = 2x + 15.
Graph interpretation:
The shape of a linear graph is always a straight line → constant rate.
Flat line → slope 0 → no change.
Tables:
If Δy is constant when Δx is constant → linear relationship.
✔ “Starting at…” almost always means the y-intercept
✔ “Every…” or “per…” = slope
✔ Never calculate slope using approximate graph points — DSAT usually hides exact coordinates in table/text
✔ If table x-values jump irregularly, compute slope with Δy/Δx, not difference in rows ✔ If graph value at x=0 is off-screen → use table or text clues instead
– Using wrong axes (mixing x and y meanings)
– Misreading negative slope as positive (or vice versa)
– Thinking “initial value” = first row of table (only true if x=0 there)
– Computing slope with inconsistent x-intervals
– Assuming linearity when data does NOT have constant rate
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify slope, intercept, rate, table/graph patterns.
Level 2 — Execution:
Compute slope, extract intercept, build y=mx+b.
Level 3 — Insight:
Understand the meaning:
– What does slope represent?
– What does intercept represent?
– What does a negative slope mean in context?
Pattern Examples (Common DSAT Forms)
Pattern 1 — Text-based Model:
“A taxi charges $4 initially and $0.30 per mile.”
→ y = 0.30x + 4
Pattern 2 — Table Interpretation:
x: 0, 3, 6, 9
y: 10, 22, 34, 46
→ Δy = +12 when Δx = +3 → slope = 4 → y = 4x + 10
Pattern 3 — Graph Interpretation:
Identify two clean points → compute slope → read intercept.
Pattern 4 — Rate Comparison:
“Which line shows a faster rate of change?”
→ Compare slopes.
Linear word problems convert real-world descriptions into mathematical models. Students must identify the changing quantity, the rate of change, the initial value, and build a linear equation that describes the situation.
– Keywords: “per”, “each”, “every”, “starting at”, “after t hours”, “initially”
– Situations describing cost, speed, money, temperature, population
– Tables describing growth with constant rate
– Problems asking for predictions: “How many after x?”
– “x represents …” and “y represents …” explicitly defined
1) Identify variables: – What is x? – What is y? – What are their units?
2) Extract the rate of change: – “per”, “each”, “every” → slope m
3) Extract the initial value: – “starts at”, “initially”, “when x=0” → intercept b
4) Build the model: $y = mx + b$
5) Answer the specific question: – plug in x – solve for y – interpret meaning if required
Deep Learning: How Word Problems Become Linear Models
1. Identify the story structure:
Every linear word problem has the form:
“Start with something → increase or decrease at a constant rate.”
2. Rate = slope:
If something changes “per hour”, “per item”, “per mile”, this is ALWAYS the slope.
3. Initial value = intercept:
Anything that happens BEFORE the repeated change is applied is the intercept.
4. Context matters:
Example:
“You burn 600 calories at rest and 8 calories per minute while exercising.”
→ Rest calories = intercept
→ Burn rate = slope
→ Model: $C = 8t + 600$
5. Interpretation is the real test:
DSAT often asks what slope/intercept mean in context.
✔ Identify units immediately — wrong units = wrong slope
✔ Combine constant and variable parts correctly (fixed fee + per item fee)
✔ If the table has irregular x-spacings, always compute Δy/Δx
✔ If text gives two points, build slope from them
✔ “After t hours/days” usually means slope is applied after initial condition
– Confusing initial value with first data point (x must be 0!)
– Using totals instead of per-unit rates
– Forgetting that slope can be negative in decreasing scenarios
– Combining fixed and variable costs incorrectly
– Treating time as y and measurement as x (axis swap trap)
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify keywords (“per”, “starting at”), find the story pattern.
Level 2 — Execution:
Extract slope/intercept, build the equation $y = mx + b$.
Level 3 — Insight:
Interpret slope/intercept, compare situations, predict outcomes.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Cost Function:
“A gym charges a $25 signup fee and $12 per month.”
→ y = 12x + 25
Pattern 2 — Movement/Speed:
“A car starts 30 miles from the city and drives toward it at 55 mph.”
→ distance = 30 − 55t
Pattern 3 — Growth/Decrease:
“A tank contains 800 liters of water and drains at 12 L/min.”
→ V = 800 − 12t
Pattern 4 — Two data points:
“At 2 hours, depth is 14 cm; at 6 hours, depth is 26 cm.”
Compute slope → build model.
Linear systems measure the ability to find the intersection point of two lines. Students must interpret what solutions represent and select the correct solving method depending on the structure of the equations.
– Two linear equations with two variables (x, y)
– Questions involving “intersection”, “same value”, “solution to both”
– Structure suggesting substitution or elimination
– DSAT-specific: asks for 2x + y, x – y, x + y (expression evaluation)
– Parallel lines (no solution) or identical lines (infinitely many solutions)
1) Identify relationship: Are the lines parallel, identical, or intersecting?
2) Select method:
– Substitution: one equation solved for a variable – Elimination: coefficients align easily
3) Execute:
– Substitute or eliminate – Solve for x – Substitute to get y
4) Check the structure:
Many DSAT questions ask for expressions such as x+y, not individual variables.
5) Interpret meaning:
In real-world problems, the solution represents where two conditions are equal.
Deep Learning: When Substitution vs Elimination?
Use substitution when:
– A variable is already isolated (x = … or y = …)
– One equation is simple (x – 3 = y)
Use elimination when:
– Coefficients are aligned (2x + y and –2x + 3y)
– A quick multiplier creates cancellation
Geometric Interpretation:
Solving a system means finding the point (x, y) where both equations are true →
the lines intersect.
Parallel lines: no solution
Same line: infinitely many solutions
✔ If DSAT asks for x+y or 2x–3y → DO NOT solve for x and y separately unless necessary
✔ Multiply both equations only if it creates a clean elimination
✔ Always align equations vertically before eliminating
✔ Recognize parallelism quickly: same slope → different intercept
✔ In real-world DSAT tasks, system solution = where cost/time/amount becomes equal
– Solving for x and y when only an expression was required
– Forgetting to distribute negative signs during elimination
– Interchanging x and y values when plugging into the expression
– Not recognizing when systems have no solution
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify system structure and best solving strategy.
Level 2 — Execution:
Solve the system using elimination/substitution.
Level 3 — Insight:
Interpret geometric meaning, detect special cases, evaluate expressions efficiently.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Clean Elimination:
3x + 2y = 14
–3x + y = –5
Pattern 2 — Substitution-Friendly:
y = 4x – 3 and 2x + y = 11
Pattern 3 — Expression Evaluation:
Solve system → find x + y (DSAT loves this)
Pattern 4 — Parallel Lines:
y = 2x + 1 and y = 2x – 5 → no solution
Pattern 5 — Real-World System:
“Car A and Car B travel… When are they at same position?”
Absolute value equations measure the understanding that absolute value represents distance on the number line. DSAT uses this not only algebraically, but also conceptually (e.g., distance between points, difference between quantities).
– Vertical bars: |x − a| or |expression|
– Language like “distance”, “difference”, “away from”, “how far”
– Situations with symmetry
– Isolated absolute value equals a number (|A| = B)
– Nested absolute values (DSAT occasionally uses this to test structure)
For any equation of the form |A| = B:
✔ If B < 0 → No solution
✔ If B = 0 → A = 0 only
✔ If B > 0 → Solve two equations: 1) A = B 2) A = –B
Then check for extraneous results if there were domain restrictions.
Deep Learning: Absolute Value = Distance
1. |x − a| = d means:
“x is distance d away from a.”
→ Solutions are symmetric around a.
2. Graph Interpretation:
The graph of y = |x − a| is a V-shape with vertex at (a, 0).
Solving |x − a| = d is finding where the graph hits height d.
3. Nested Absolute Values:
For expressions like | |x−3| − 2 | = 5,
work inside → outside.
4. Distance Between Points:
|x − 8| = |x − 2| describes the midpoint of 2 numbers.
✔ Convert word phrases like “distance from 6 is 3” → |x − 6| = 3
✔ If DSAT gives two absolute values, compare the critical points
✔ Always check B: negative → no solution (common DSAT trick)
✔ Be alert: DSAT often asks for the sum of solutions or the greater solution
✔ Symmetry saves time: solutions are a ± d
– Forgetting to solve both branches (A = B and A = –B)
– Missing the fact that B must be ≥ 0
– Solving |A| = B, but giving only one solution
– Misinterpreting distance (thinking direction matters — it doesn’t!)
– Incorrectly solving nested absolute values
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify the absolute value structure and check if B is negative.
Level 2 — Execution:
Solve both branches and verify.
Level 3 — Insight:
Interpret distance geometrically; handle nested or applied contexts.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Standard:
|x − 5| = 3 → x = 8 or x = 2
Pattern 2 — No Solution:
|2x + 1| = –4
Pattern 3 — Set-up from words:
“x is 4 units from 10” → |x − 10| = 4
Pattern 4 — Nested:
| |x − 3| − 2 | = 5
Pattern 5 — Distance comparison:
|x − 8| = |x − 2| → midpoint → x = 5
Inequalities measure a student’s ability to understand directional relationships, solve multi-step comparisons, interpret compound ranges, and translate number-line diagrams into algebraic intervals. DSAT tests structure recognition, not long computation.
– Symbols: <, ≤, >, ≥
– Verbal cues: “at least”, “no more than”, “fewer than”, “greater than”
– Number line diagrams with open/closed circles
– Compound phrases: “between… and…”
– Inequalities requiring flipping (multiplying/dividing negative)
1) Simplify both sides (distribute, combine like terms)
2) Move variable terms to one side
3) Isolate variable: – If you multiply/divide by a negative → **flip inequality**
4) Convert final expression into the correct number-line or interval form
Deep Learning: Inequality Direction & Meaning
1. WHY flipping happens:
Multiplying by –1 reflects the number line across 0.
Reflection reverses order, so inequality direction must flip.
2. Compound Inequalities:
a < x < b means x is between a and b (open interval).
a ≤ x ≤ b means x is between including endpoints.
3. Number-Line Interpretation:
– Open circle = strict (< or >)
– Closed circle = inclusive (≤ or ≥)
– Arrow direction shows infinite extension
4. Interval Notation:
(a, b) → open interval
[a, b] → closed interval
(a, b] → a excluded, b included
[a, b) → a included, b excluded
5. Words-to-symbol mapping:
“At least” → ≥
“No more than” → ≤
“More than” → >
“Less than” → <
✔ Always check if you multiplied/divided by a negative
✔ When seeing number-line graphs, match open/closed circles carefully
✔ DSAT often reverses compound inequalities — rewrite cleanly
✔ If the inequality is written like 8 > 2x + 4, rewrite it as 2x + 4 < 8 first
✔ Pay attention to “including” vs “not including” wording
– Forgetting to flip inequality when multiplying by a negative
– Misinterpreting number-line symbols (open vs closed)
– Reversing compound inequality order
– Thinking “and” means union (it means intersection!)
– Incorrectly interpreting word phrases
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify symbols, understand keyword meanings, read number lines.
Level 2 — Execution:
Solve algebraically, flip appropriately, rewrite compound forms.
Level 3 — Insight:
Interpret context, convert between number-line ↔ interval ↔ algebraic forms.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Basic:
Solve: 3x – 7 ≤ 11
Pattern 2 — Flip Required:
–2(3 – x) > 8
Pattern 3 — Compound:
2 < 3x + 1 ≤ 11
Pattern 4 — Word Interpretation:
“x is at least 5 and no more than 12” → 5 ≤ x ≤ 12
Pattern 5 — Number-Line:
Open circle at –3, closed circle at 4, shading between
→ –3 < x ≤ 4
Quadratic equations test the ability to solve, analyze, and interpret polynomial functions of degree 2. DSAT focuses on recognizing structure, selecting the most efficient method, and interpreting roots in context.
– Equations of the form ax² + bx + c = 0
– Words: “maximum”, “minimum”, “roots”, “zeros”, “intercepts”
– Parabolic graphs
– Factored forms (x − r₁)(x − r₂)
– Vertex forms a(x − h)² + k
– “One solution”, “two solutions”, “no real solutions” → discriminant
1) Check structure:
– Factored form → read solutions directly – Vertex form → identify max/min quickly – Standard form → use discriminant or quadratic formula
2) Choose method:
✔ Factoring (if factorable) ✔ Quadratic formula (always works) ✔ Completing the square (for transformations)
3) Find roots / vertex / symmetry:
– Vertex: $x = -\frac{b}{2a}$ – Axis of symmetry: same as above – Discriminant: $b^2 – 4ac$ tells number of real roots
4) Interpret meaning:
DSAT often asks what a root or vertex represents in a real-world problem.
Deep Learning: Choosing the Best Form
1. Standard Form (ax² + bx + c):
– Best for discriminant
– Best for quadratic formula
– c = y-intercept
2. Factored Form (a(x − r₁)(x − r₂)):
– Best for finding roots
– Shows symmetry: axis = (r₁ + r₂)/2
3. Vertex Form (a(x − h)² + k):
– Best for max/min
– Vertex = (h, k)
4. Discriminant Meaning:
– D > 0 → 2 real roots
– D = 0 → 1 real root
– D < 0 → no real roots
5. Graph Interpretation:
Opening direction = sign of a
Width/stretch = |a|
Vertex location → maximum or minimum
✔ Check if factoring is obvious → DSAT often gives clean roots
✔ If question asks “how many solutions?”, jump straight to discriminant
✔ For max/min → use vertex form or compute h = −b/(2a)
✔ DSAT likes to hide roots in tables/graphs — look for axis of symmetry
✔ Context interpretation: roots are when the quantity becomes zero (profit, height, etc.)
– Using quadratic formula unnecessarily – Forgetting ± in the formula – Thinking vertex is always a maximum (only true if a < 0) – Misinterpreting discriminant – Incorrectly identifying intercepts on graphs
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify the form and the key features (roots, axis, vertex).
Level 2 — Execution:
Solve using factoring, formula, or structure selection.
Level 3 — Insight:
Interpret vertex/roots in real-world context; compare quadratics.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Clean Roots:
(x − 3)(x + 5) = 0 → roots: 3, –5
Pattern 2 — Standard Form:
2x² − 7x + 3 = 0 → use discriminant or factoring
Pattern 3 — Exactly One Solution:
b² − 4ac = 0 questions
Pattern 4 — Vertex Problems:
Maximum height/minimum cost
Pattern 5 — Graph/Table Interpretation:
Identify vertex, find axis, extract roots from symmetry
DSAT does not expect heavy computation here; it tests your ability to read a quadratic function’s structure and immediately predict its graph’s shape, direction, vertex location, and number of solutions. This is one of the most “concept-heavy” topics in DSAT algebra.
– Vertex form: a(x − h)² + k
– Standard form: ax² + bx + c
– Factored form: a(x − r₁)(x − r₂)
– Words: “opens upward/downward”, “vertex”, “axis of symmetry”, “horizontal shift”, “vertical shift”, “stretch”, “compression”
1) Identify form:
– If in vertex form → read vertex & direction immediately – If in factored form → read roots and find axis – If in standard form → extract a, b, c and compute vertex
2) Determine opening direction:
– a > 0 → opens upward (minimum) – a < 0 → opens downward (maximum)
3) Determine stretch/compression:
– |a| > 1 → narrower – |a| < 1 → wider
4) Determine vertex:
– Vertex form: (h, k) – Standard form: $x = -\frac{b}{2a}$ then plug back in
5) Determine roots (# of solutions):
– Factored: roots visible – Vertex form: check if vertex is above/below x-axis – Standard: use discriminant sign
Deep Learning: How to Read Quadratic Behavior Instantly
1. Vertex Form = Graph Blueprint
a(x − h)² + k tells you:
– Vertex: (h, k)
– Opening: sign of a
– Width: |a|
2. Factored Form = Root Blueprint
a(x − r₁)(x − r₂):
– Roots: r₁ and r₂
– Axis: (r₁ + r₂)/2
– Opening: sign of a
3. Standard Form = Behavior Blueprint
ax² + bx + c gives:
– c = y-intercept
– Vertex x-value: −b/(2a)
– Discriminant tells # of real solutions
4. Transformation Map:
– (x − h) shifts right by h
– (x + h) shifts left by h
– +k shifts up
– −k shifts down
5. Symmetry Insight:
Parabolas are symmetric around the axis.
If you know one point, you know its mirror partner.
✔ You almost never need to compute roots — read structure instead
✔ Vertex above x-axis + opening upward → no real roots
✔ Vertex below x-axis + opening downward → no real roots ✔ Factored form → fastest for root count / root identification
✔ Always check sign of a first — it tells half the story ✔ DSAT loves comparing quadratics: “Which graph matches…?”
– Confusing h-sign: (x − 3) → shift right 3 – Thinking vertex is always max (only if a < 0) – Forgetting that symmetric points share y-values – Misreading a negative outside as horizontal reflection – Confusing roots with y-intercept
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify form & read basic features (direction, vertex, roots).
Level 2 — Execution:
Predict shift/stretch accurately; find vertex & axis of symmetry.
Level 3 — Insight:
Compare quadratics; match algebraic forms to graphs/tables instantly.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Vertex Shift:
f(x) = 2(x + 4)² − 3
→ Shift left 4, down 3, narrower, opens upward
Pattern 2 — Root Behavior:
g(x) = −(x − 2)(x − 8)
→ Opens downward, roots at 2 & 8, vertex between
Pattern 3 — Number of Solutions:
Check vertex height relative to x-axis
Pattern 4 — Compare Two Quadratics:
Which opens wider? Compare |a| values.
Pattern 5 — Graph Match:
a(x − h)² + k → direct blueprint for matching.
Exponential functions describe quantities that change by a constant percentage or multiplier per step. DSAT tests whether students can detect this pattern, build the correct exponential model, and interpret growth/decay in tables, graphs, and real-life descriptions.
– Words: “grows by”, “increases by”, “decreases by”, “each year”, “per month”
– Table with constant ratio (not constant difference!)
– Modeling format: y = abˣ or y = a(1 ± r)ᵗ
– Curved graph increasing or decreasing
– Multiplicative change instead of additive
1) Identify the multiplier:
– Increase r% → multiplier = (1 + r) – Decrease r% → multiplier = (1 – r) – From table → ratio = next ÷ previous
2) Identify initial value:
– y-value when x = 0 – First row of table if x = 0 – Starting amount from the text
3) Build the model:
$y = a \cdot b^x$ where – a = initial amount – b = multiplier
4) Interpret behavior:
– b > 1 → exponential growth – 0 < b < 1 → exponential decay
5) Apply as required:
– Predict future value – Compare growth/decay rates – Determine percentage from multiplier
Deep Learning: How Exponential Really Works
1. Percent ≠ Multiplier:
Students often confuse “20% increase” with “×20”.
True multiplier is 1.20.
2. Constant Ratio:
A table is exponential if:
(value₂ ÷ value₁) = (value₃ ÷ value₂) = constant.
3. Growth vs Linear:
Linear → constant addition
Exponential → constant multiplication
(This is the easiest way DSAT distinguishes the two.)
4. Doubling Time:
If b = 2 → quantity doubles each step.
If b = 1.05 → ~5% per step.
5. Halving Time:
If b = 0.5 → halves each step.
If b = 0.9 → decreases 10% each step.
6. Multiplier to Percent:
If b = 1.08 → r = 8%
If b = 0.72 → r = –28%
✔ Always check if difference is constant (linear) or ratio is constant (exponential)
✔ If you see doubling/tripling → automatically exponential
✔ If table jumps irregularly in x, compute multiplier using (Δy based on steps)
✔ DSAT likes to hide the initial value — find the row where x = 0
✔ Exponential graphs never straighten — curvature means multiplier logic
– Using addition instead of multiplication for percent changes
– Misinterpreting “decreases by 30%” as multiplier = 0.30 (correct multiplier = 0.70)
– Misreading tables and thinking ratio must be integer
– Assuming constant difference means exponential (it is linear!)
– Forgetting that negative exponents represent backward steps
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify exponential behavior from keywords/graphs/tables.
Level 2 — Execution:
Build exponential model y = abˣ from multiplier and initial value.
Level 3 — Insight:
Interpret growth/decay, compare rates, convert between percent and multiplier.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Percent Increasing:
“Population grows by 12% each year”
→ multiplier = 1.12
→ y = a(1.12)ᵗ
Pattern 2 — Percent Decreasing:
“Value decreases by 18% annually”
→ multiplier = 0.82
→ y = a(0.82)ᵗ
Pattern 3 — Table to Model:
If values go 50 → 75 → 112.5 → 168.75
→ ratio = 1.5
→ y = 50(1.5)ˣ
Pattern 4 — Backwards Steps:
If b = 1.2 → one step backward = divide by 1.2
→ multiply by (1/1.2)
Pattern 5 — Comparing Growth:
Larger multiplier → faster growth.
Function composition tests the ability to follow an input through multiple transformations. DSAT focuses on the order of application, interpreting compositions from graphs/tables, and evaluating composite expressions quickly.
– f(g(x)), g(f(x)), h(f(g(x)))
– Two functions defined by formulas, graphs, or tables
– Questions like “What is f(g(3))?” or “Which expression is equivalent to…?”
– Input-output chains
– Nested parentheses indicating sequential actions
1) Start from the inside:
For f(g(x)), evaluate g(x) first.
2) Treat output as new input:
Whatever g(x) equals becomes the input of f.
3) Follow transformation sequence:
Function composition is just doing two transformations in order.
4) If graph or table is used:
– Find g(x) from graph/table – Plug that value into f’s graph/table
5) For formula-based questions:
Replace x in f(x) with g(x). Example: f(x) = 2x + 1 g(x) = x² − 3 → f(g(x)) = 2(x² − 3) + 1
Deep Learning: How to Think Like DSAT Wants
1. Composition = Pipeline:
Think of f(g(x)) as a machine with two steps:
Step 1: Apply g
Step 2: Apply f
Order matters.
2. Graph Interpretation:
For f(g(2)):
– Go to x = 2 on g’s graph → get g(2)
– Take that y-value → use it as x-input on f’s graph
3. Table Interpretation:
Composition from tables is purely input-output chaining.
4. Variable vs Constant Input:
f(g(3)) means evaluate a number
f(g(x)) means build an expression
5. Transformation View:
Composition is equivalent to stacking transformations of x.
Example: g doubles, f adds 5 → f(g(x)) = 2x + 5
✔ Always evaluate inside first → g then f
✔ When using graphs: follow vertical → horizontal → vertical jumps
✔ For tables: track input → output → new input exactly
✔ DSAT often hides a missing value — check the table range
✔ Write quick substitution structure instead of rewriting everything
– Doing f(x) first instead of g(x)
– Misreading table values (wrong row/column)
– Mixing input and output roles
– Plugging x into f instead of g(x)
– Treating f(g(x)) same as g(f(x)) (they are almost never the same!)
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify function composition structure and order.
Level 2 — Execution:
Compute f(g(x)) from formulas, tables, or graphs.
Level 3 — Insight:
Compare compositions, build expressions, interpret functional flow.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Numeric Composition:
f(g(4)) from table or graph
Pattern 2 — Symbolic Composition:
f(x) = 3x − 1, g(x) = x² + 2
→ f(g(x)) = 3(x² + 2) − 1
Pattern 3 — Input-Output Chain:
g(3) = 7, f(7) = −2 → f(g(3)) = −2
Pattern 4 — Graph-Based:
Read g(x) from graph A → use as input in graph B.
Pattern 5 — Expression Matching:
“Which expression equals f(g(x))?”
→ Substitute carefully.
Rational and radical equations measure a student’s ability to:
- identify domain restrictions,
- eliminate denominators or radicals correctly,
- check for extraneous solutions,
- analyze structure without heavy computation.
– Variable in the denominator (rational)
– Square roots, cube roots (radical)
– Equations requiring clearing denominators
– Squaring both sides
– Checking final answers in original equation
– Word problems involving rate = distance/time
STEP 1 — Identify Restrictions
Rational: denominator ≠ 0 Radical: radicand ≥ 0 for even roots
STEP 2 — Clear the Structure
Rational: multiply both sides by LCD Radical: isolate radical and square both sides
STEP 3 — Solve the Resulting Equation
Solve linear/quadratic form that arises after clearing.
STEP 4 — Check Solutions (VERY IMPORTANT)
– Plug back to check for extraneous values – Reject any solution violating domain or creating false equality
STEP 5 — Interpret
Apply to context if it’s a real-world question.
Deep Learning: Why Extraneous Solutions Happen
1. Squaring both sides can create false solutions:
If you square both sides, information about sign is lost.
Example: √(x+4) = –3 has no solution,
but squaring gives x+4 = 9 → x=5 (false solution!)
2. Multiplying by LCD introduces validity issues:
If a variable makes a denominator zero, it must be removed —
DSAT loves to hide this trap.
3. Radical isolation prevents mistakes:
Always isolate the radical BEFORE squaring → prevents mixed/extra terms.
4. Rational equations often produce quadratic forms:
Students forget to check both solutions in original denominator.
✔ Write domain restrictions BEFORE solving — DSAT designs traps around this
✔ When clearing denominators, multiply by LCD once, not multiple times
✔ For radicals: isolate → square → isolate again if needed
✔ ALWAYS check your final answers in the original equation ✔ If DSAT gives a multiple-choice solution set, extraneous answers almost always appear
– Forgetting denominator restrictions (x ≠ value)
– Accepting extraneous solutions after squaring
– Squaring too early (before isolating radical)
– Multiplying by wrong LCD
– Treating odd/even roots the same (square root requires radicand ≥ 0)
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify rational/radical structure and domain restrictions.
Level 2 — Execution:
Clear denominators or radicals correctly, solve resulting equation.
Level 3 — Insight:
Check extraneous roots, reason about domain, connect structure to context.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Rational Equation Clearing:
2/(x−3) = 5 → multiply both sides by (x−3)
Pattern 2 — Radical Equation:
√(2x+5) = x−1 → isolate → square → check
Pattern 3 — Domain Only:
“For what values is the expression defined?”
Pattern 4 — Mixed Back-to-Back Radicals:
√(x−4) + √(x−1) = 7
Pattern 5 — Extraneous Root Trap:
Solve, then plug back — DSAT often removes exactly one value.
DSAT tests polynomial structure reasoning rather than long algebra. The core relationship is:
root r → factor (x − r)
This single rule allows quick identification of polynomial factors, zeros, and behavior without heavy computation.
– “Which of the following is a factor?”
– “Given that r is a root…”
– “f(a) = 0 implies…”
– Matching polynomials to roots
– Questions involving sign switches (x + 3 ↔ root = −3)
STEP 1 — Identify the root:
If f(r) = 0, r is a root.
STEP 2 — Convert root to factor:
root r → factor (x − r)
Example: root –4 → factor (x + 4)
STEP 3 — Match polynomial structure:
Compare given polynomials with required factors.
STEP 4 — If multiple roots are given:
Combine into factored polynomial: roots 2 and –3 → a(x − 2)(x + 3)
STEP 5 — Use sign logic carefully:
Factor opposite sign of root.
Deep Learning: Why the Root ↔ Factor Rule Works
1. Factor Theorem:
A polynomial f(x) has a factor (x − r) if and only if f(r) = 0.
This connects evaluation (plugging in) directly with symbolic structure.
2. Polynomials store information in their factors:
Factored form immediately gives zeros, graph crossing points, multiplicity, and behavior.
3. DSAT rarely requires full expansion:
It rewards students who analyze structure instead of expanding and simplifying.
4. Sign pattern is everything:
Students often forget:
root r → factor (x − r)
Example: root −5 → factor (x + 5), not (x − 5)
✔ If the question gives a root, immediately write the factor
✔ If the question gives a factor, immediately write the root
✔ Never expand unless absolutely necessary
✔ For “Which polynomial could be f(x)?” → check sign first, degree second
✔ If a root has multiplicity → graph touches instead of crosses
– Wrong sign: root 3 → many students choose (x + 3)
– Forgetting a leading coefficient “a”
– Mistaking root pairs for factors
– Expanding needlessly and wasting time
– Believing all polynomials must be degree ≥ number of roots (they must — DSAT checks this!)
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify root-factor relationship and sign rules.
Level 2 — Execution:
Build polynomials from roots or identify correct factors.
Level 3 — Insight:
Understand how factors affect graph shape, multiplicity, and DSAT reasoning.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Direct Root → Factor:
If 4 is a root → (x − 4) must be a factor.
Pattern 2 — Factor Given → Root Extraction:
(x + 7) is a factor → root = −7
Pattern 3 — Build a Polynomial:
Roots 1 and –2 → a(x − 1)(x + 2)
Pattern 4 — Evaluate at Root:
If f(3) = 0 → 3 is root → factor (x − 3)
Pattern 5 — Wrong Sign Trap:
DSAT often includes (x + r) and (x − r) options.
Function transformations describe how changes to a function’s equation affect its graph. DSAT tests whether students can interpret shifts, reflections, and stretches without plotting every point. This is a core reasoning skill.
– g(x) = f(x − h) + k
– Vertical/horizontal shifts
– “Which transformation maps f to g?”
– Descriptions like “moved 3 units right”
– “Reflected across the x-axis / y-axis”
– “Vertically stretched by factor a”
Every transformation fits the template:
g(x) = a · f(b(x − h)) + k
1) Horizontal Shift (x − h)
– h > 0 → shift right h units
– h < 0 → shift left |h| units
2) Vertical Shift (k)
– k > 0 → shift up
– k < 0 → shift down
3) Vertical Stretch/Compression (a)
– |a| > 1 → stretch (taller)
– 0 < |a| < 1 → compression (shorter)
– a < 0 → reflect across x-axis
4) Horizontal Stretch/Compression (b)
– |b| > 1 → horizontal compression
– 0 < |b| < 1 → horizontal stretch
– b < 0 → reflect across y-axis
5) Combined Transformations:
Apply order from inside → outside.
Deep Learning: Understanding Transformations Without Memorizing
1. Inside vs Outside logic:
– Changes inside parentheses affect x-values (horizontal)
– Changes outside affect y-values (vertical)
2. Input Mapping View:
g(x) applies a transformation to the input before sending it to f.
3. “Opposite direction” rule for horizontal shifts:
x − 3 moves right
x + 3 moves left
because the input is altered before f processes it.
4. Scaling logic:
Vertical scaling changes height.
Horizontal scaling changes width.
5. Reflection intuition:
Negative factors flip graphs — vertical or horizontal depending on position.
✔ Look for shifts first → fastest identification
✔ Horizontal changes use opposite sign → x − h means “right h”
✔ Check for reflections by sign of a or b
✔ If graphs are given: track key landmark (vertex, intercept, peak)
✔ Don’t expand the function — transformations come from structure
– Mixing up horizontal and vertical shifts
– Treating inside as “same direction” (wrong!)
– Forgetting reflections
– Confusing compression vs stretch
– Using point-by-point graphing (time waste)
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify shift, stretch, and reflection signals in an equation.
Level 2 — Execution:
Correctly determine direction and magnitude.
Level 3 — Insight:
Read transformations from structure instantly; map graphs mentally.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Horizontal Shift:
g(x) = f(x − 5) → shift right 5 units
Pattern 2 — Vertical Shift:
g(x) = f(x) + 4 → shift up 4 units
Pattern 3 — Reflection:
g(x) = −f(x) → reflect over x-axis
Pattern 4 — Scaling:
g(x) = 3f(x) → vertical stretch by 3
Pattern 5 — Combined:
g(x) = 2f(3(x + 2)) − 1
– right/left → (x + 2) = left 2
– horizontal compression by 3
– vertical stretch by 2
– down 1
Ratios and percents measure proportional reasoning. DSAT tests speed in interpreting ratios, converting them into multipliers, and applying percent change rules without long calculations.
– “Increase by r%”
– “x is y% of z”
– “The ratio of A to B is…”
– “Total remains constant”
– “Part–whole relationship”
– “Percent change over two steps”
– “What is the new value?”
– “Unit ratio / unit rate”
STEP 1 — Convert every percent to a multiplier:
Increase r% → ×(1 + r) Decrease r% → ×(1 − r)
STEP 2 — Identify whole vs part:
Whole × percent = part
STEP 3 — Use ratio interpretation:
A : B = k : m → A/B = k/m → A = (k/m)B
STEP 4 — For compound percent changes:
Use multipliers sequentially:
Example: increase 20% then 10% → ×1.20 ×1.10
STEP 5 — For percent of percent:
“x is y% of z” → x = (y/100)z
Deep Learning: Why Multipliers Work
1. Percent change is multiplicative, not additive:
A 20% increase means multiply by 1.20, not “add 20”.
2. Compound changes stack:
DSAT loves two-step percent changes.
20% increase + 30% increase → ×1.20 ×1.30 (not 50%).
3. Ratios define a fixed relationship:
If A:B = 2:3, then A = (2/3)B always.
4. Units matter:
A ratio describes *how many per how many* — not total quantities.
5. Percent decrease > percent increase asymmetry:
DSAT tests this subtle idea.
Example: +25% then –20% → final not original.
✔ Always convert to multiplier first
✔ Identify the “whole” — most percent errors come from wrong base
✔ Ratio → treat as fraction immediately
✔ For compound changes: multiply, never add percentages
✔ If a question is multi-step: track the multiplier chain
– Thinking percent change is additive
– Mixing up “percent of” vs “percent change”
– Using wrong whole (DSAT’s favorite trick!)
– Reversing ratios (A:B but student uses B:A)
– Forgetting that percentages define multiplication
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify percent/ratio structure and the “whole”.
Level 2 — Execution:
Apply multipliers, solve proportions, compute percent changes.
Level 3 — Insight:
Track multi-step changes and interpret ratio relationships instantly.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Percent Increase:
“Increased by 18%” → ×1.18
Pattern 2 — Percent of Percent:
“x is 40% of y” → x = 0.40y
Pattern 3 — Compound Percent:
“Increase 20%, then decrease 10%” → ×1.20 ×0.90
Pattern 4 — Ratio Interpretation:
A:B = 3:5 → A = (3/5)B
Pattern 5 — Percent Change Formula:
change/original × 100
Unit conversions measure a student’s ability to use dimensional logic. DSAT focuses on whether students can set up a correct factor-chain, cancel units properly, and interpret real-world measurement contexts.
– “Convert … to …”
– “How many units of … are in …?”
– Word problems with mixed units
– Rates like miles/hour, dollars/pound, liters/minute
– Conversions hidden inside a percent or ratio question
STEP 1 — Write the original number with its unit:
Example: 5 miles
STEP 2 — Multiply by a conversion fraction whose denominator is the unit you want to cancel:
Example: miles → feet 1 mile = 5280 feet
5 miles × (5280 feet / 1 mile)
STEP 3 — Cancel units (the magic of factor-chain):
“mile” cancels top/bottom → left with feet
STEP 4 — Continue chaining until final unit is reached:
miles → feet → inches → centimeters → etc.
STEP 5 — Interpret in context (DSAT loves this):
If it’s a rate, convert numerator and denominator correctly.
Deep Learning: Why Factor-Chain Always Works
1. Units behave like algebraic variables:
You can cancel them exactly like symbols.
ft × (12 in / 1 ft) → “ft” cancels.
2. Proper setup prevents mistakes:
Students often guess “multiply or divide?”
Factor-chain eliminates guessing.
3. DSAT intentionally mixes units:
Example: gallons/min & miles/hour in the same question.
Only dimensional reasoning solves it correctly.
4. Compound units convert top and bottom separately:
mi/hr → ft/sec
Convert miles → feet, then hours → seconds.
5. Scale logic is multiplicative:
All unit conversions are multiplication — never addition.
✔ Always start by writing the unit you want to cancel
✔ Treat conversion ratios as fractions = 1 in disguise ✔ Convert numerator and denominator separately for rates
✔ If unsure: build the factor-chain and let the units guide you ✔ The last remaining unit is the unit of your answer
– Multiplying/dividing randomly instead of factor-chain
– Converting numerator but forgetting the denominator in rates
– Using wrong conversion direction
– Assuming conversions are additive (they are not) – Incorrect cancellation (DSAT puts similar-looking units on purpose)
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify units, rate structure, and needed conversion factors.
Level 2 — Execution:
Build correct factor-chain and cancel units properly.
Level 3 — Insight:
Interpret conversions inside multi-step modeling problems.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Simple Length Conversion:
7 ft → inches
7 × (12 in / 1 ft)
Pattern 2 — Weight Conversion:
3 lb → ounces
3 × (16 oz / 1 lb)
Pattern 3 — Compound Units (Rates):
30 mi/hr → ft/sec
30 × (5280 ft / 1 mi) × (1 hr / 3600 sec)
Pattern 4 — Real-World Model:
“A pump delivers 2 gallons per minute.
How many liters per hour?”
Pattern 5 — Multi-step Hidden Conversion:
Percent × unit conversion inside ratio problem.
Data tables test a student’s ability to extract values, combine categories, and compute overall statistics. Weighted mean questions measure whether a student understands that each group contributes proportionally to the total, not equally.
– Tables with categories (Class A, Class B…)
– Frequency, score, total, or average columns
– “Overall average” prompts
– “Combined group average”
– “Missing value in table”
– Multi-step totals: total = average × count
STEP 1 — Identify counts (weights):
Weighted mean formula: Total Value / Total Count
STEP 2 — Convert each average into total value:
For a group: total = average × number of items
STEP 3 — Sum all totals:
overall total = sum of all group totals
STEP 4 — Sum all counts:
overall count = sum of all group sizes
STEP 5 — Compute weighted mean:
weighted mean = overall total ÷ overall count
STEP 6 — Interpret the table correctly:
Pay attention to units (scores, dollars, minutes, etc.)
Deep Learning: Why Weighted Mean Works
1. Averages hide totals:
Average = total ÷ count →
Total = average × count
To combine groups, we must recover their totals.
2. DSAT’s favorite trap:
Students often take average of averages — this is almost always wrong.
3. Weighted mean represents proportional contribution:
Larger groups influence the final average more.
4. Missing-value puzzles:
If overall average and group averages are known,
you can recover missing group size or missing group total.
5. Units define operations:
Example:
minutes/student, dollars/item, miles/hour
DSAT mixes units intentionally to test consistency.
✔ NEVER average averages — compute totals first
✔ Always rewrite average into “total = avg × count”
✔ Combine totals before dividing
✔ If a group size is missing, use equation setup from total
✔ Be careful: “category contribution” depends on weight
– Averaging group averages directly (wrong!)
– Mixing up total value vs total count
– Using wrong units (DSAT designs this deliberately)
– Forgetting to reconstruct group totals
– Missing-value tables incorrectly solved
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify averages, totals, and frequencies in tables.
Level 2 — Execution:
Convert averages to totals, combine, divide.
Level 3 — Insight:
Solve multi-step weighted mean problems, interpret missing values.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Two Groups Combine:
Group A avg = 80, count = 20
Group B avg = 70, count = 10
overall avg = (80×20 + 70×10) ÷ 30
Pattern 2 — Missing Group Size:
overall total known → group size unknown.
Pattern 3 — Weighted Category Contribution:
“Which category contributed most to total?”
Pattern 4 — Frequency Table:
value × frequency → contribution to total
Pattern 5 — Table Value Extraction:
DSAT hides necessary numbers in neighboring columns.
Geometry questions assess a student’s ability to interpret shapes, apply fundamental formulas, and reason about scale factors. DSAT does not require complicated geometry — instead, it tests:
- relationships between dimensions, area, and volume
- circle geometry (area, circumference, arc length)
- triangle structure (height, area, right-triangle reasoning)
- composite shapes and proportional reasoning
– Circle, radius, diameter, sector, arc
– “Area of the shaded region”
– “Volume when scaled”
– “Relationship between sides”
– Right triangles and Pythagorean triples
– Diagrams with labeled lengths / angles
– Composite figures (rectangle + semicircle, etc.)
1) Identify the primitive shape:
– Rectangle, triangle, circle, cylinder, etc.
2) Apply the correct formula:
Area of triangle: (1/2)bh Area of circle: πr² Circumference: 2πr Volume of cylinder: πr²h
3) Break composite shapes into parts:
area_total = sum(parts) − removed sections
4) Use scale factor logic:
If lengths scale by k: – area scales by k² – volume scales by k³
5) Interpret diagram carefully:
DSAT often gives unnecessary measurements — look for the key one.
Deep Learning: Why Geometry Formulas Work
1. Triangle area (½bh) comes from rectangle decomposition:
A triangle occupies half the area of a rectangle with same base & height.
2. Circle area (πr²) represents “radius stretching” space:
The radius squared shows that doubling r quadruples the area.
3. Volume formulas generalize area:
A cylinder is “a circle extruded through height h”.
4. Scale factor logic is central:
When dimensions scale by k:
– lengths ×k
– areas ×k²
– volumes ×k³
DSAT uses this constantly.
5. Composite shapes reveal structure:
Most DSAT geometry can be reduced to simple shapes + subtraction.
✔ Identify radius vs diameter — DSAT hides this a lot
✔ Always draw height in triangles (perpendicular!)
✔ For composite shapes: break → compute → sum → subtract
✔ Use scale-factor shortcuts to avoid long calculations
✔ For circle sectors: use proportion (θ/360) × (full area or circumference)
– Using diameter instead of radius
– Forgetting height in triangle area
– Adding instead of subtracting removed regions
– Not applying the k² / k³ scaling laws
– Misreading diagram arrows (length vs height)
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify shapes, formulas, and required dimensions.
Level 2 — Execution:
Apply formulas, compute areas/volumes, combine shapes.
Level 3 — Insight:
Use scaling logic and structural reasoning to solve quickly.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Circle Basics:
Find area or circumference using r or d.
Pattern 2 — Shaded Region:
Composite area = outer − inner.
Pattern 3 — Scale Factors:
“If radius doubles, area becomes…?” → ×4
Pattern 4 — Triangles:
Use (½)bh and Pythagorean relationships.
Pattern 5 — Volume Change:
If height triples and radius halves → new volume = (½)² × 3 = ¾ original.
Right triangle trigonometry measures the ability to use trig ratios (sin, cos, tan), identify opposite/adjacent sides, and apply special triangle relationships. DSAT focuses on structure recognition, not memorized computation.
– Right triangle diagram
– Angle given → missing side asked
– References to “height”, “slope”, “angle of elevation/depression”
– Special triangles (30–60–90 or 45–45–90)
– Tangent in real-world problems (“rise/run”)
1) Label sides relative to the angle:
Opposite = across from angle Adjacent = next to angle Hypotenuse = opposite right angle
2) Choose correct trig ratio:
sinθ = opp/hyp cosθ = adj/hyp tanθ = opp/adj
3) Set up proportion directly:
trig ratio = known side / unknown side Solve by cross-multiplying.
4) Use special triangles when applicable:
30–60–90 → x, x√3, 2x 45–45–90 → x, x, x√2
5) Interpret real-world angles:
angle of elevation = angle from horizontal up angle of depression = angle from horizontal down
Deep Learning: Why Trig Ratios Make Sense
1. All right triangles with the same angle are similar:
Therefore ratios (opp/hyp, adj/hyp, opp/adj) stay constant — this
is why trig works.
2. Tangent is slope:
tanθ = rise/run
DSAT loves using slope interpretation to avoid trig calculator work.
3. Special triangles encode pure geometry:
30–60–90 and 45–45–90 triangles do not require trig — they are exact.
4. Hypotenuse is always the longest side:
Helps quickly check for setup mistakes.
5. Complementary angles swap sine and cosine:
sin(θ) = cos(90° − θ)
cos(θ) = sin(90° − θ)
✔ Always draw/label the triangle before choosing a ratio
✔ For elevation/depression: angles share the same horizontal line
✔ tan = slope is the fastest tool for many DSAT modeling questions
✔ Special triangles save huge time — no trig needed
✔ If two sides are known, use Pythagorean theorem first
– Using wrong “opposite/adjacent” relative to the angle
– Forgetting that tangent has no hypotenuse
– Mixing altitude with side length
– Misreading diagram scale (DSAT does NOT draw to scale)
– Applying trig to non-right triangles (NOT needed on DSAT)
Level 1–2–3 Mastery Map
Level 1 — Recognition:
Identify right triangle + correct trig ratio.
Level 2 — Execution:
Apply SOHCAHTOA or special triangles to find missing sides.
Level 3 — Insight:
Translate real-world problems into triangles and use slope/tangent logic.
Pattern Examples (Common DSAT Forms)
Pattern 1 — Simple Trig Ratio:
Find missing side using sin, cos, or tan.
Pattern 2 — Special Triangle Recognition:
30–60–90 or 45–45–90 shortcuts.
Pattern 3 — Height Problems:
tanθ = height / horizontal distance
Pattern 4 — Complementary Angles:
sin(30°) = cos(60°)
Pattern 5 — Pythagorean + Trig:
Find one side with Pythagoras → use trig for the rest.