Fundamentals
The core facts every aspirant should own — each a titled nugget with a real-world story, the concept in plain words, and a memory trick. Works even when the internet doesn't.
1,200 fundamentals
“A GPS bearing in degrees must be converted to radians before using ω = dθ/dt for angular speed.”
Derivative of sinθ is cosθ only when θ is in radians; in degrees d/dθ sinθ = (π/180)cosθ. The frequent error is applying radian formulas directly with degree values, giving a wrong factor.
Memory trick: d/dθ sinθ = cosθ only in rad.
“When you tilt a flashlight, the beam’s brightness never exceeds the lamp’s max, just like sinθ never exceeds 1.”
For any real θ, -1 ≤ sinθ ≤ 1 and -1 ≤ cosθ ≤ 1 because they are the y- and x-coordinates of a point on the unit circle (radius 1). If algebra yields sinθ>1 or <-1, the previous manipulation is wrong. In JEE, always verify the value lies in this interval before accepting a solution.
Memory trick: |sinθ|,|cosθ| ≤ 1 (unit-circle limit)
“A spinning wheel’s horizontal and vertical shadows always add to the wheel’s full radius squared.”
The identity sin²θ + cos²θ = 1 comes from the Pythagorean theorem on the unit circle: (cosθ)²+(sinθ)²=1. It holds for every real angle. A frequent trap is applying it to expressions that are not exactly sin²θ+cos²θ (e.g., after squaring both sides) without simplifying first.
sin²θ + cos²θ = 1
Memory trick: sin²+cos²=1 on unit circle
“Turning a fan blade first by A then by B (same direction) adds angles; turning opposite (A then –B) subtracts, both follow sin(A±B)=sinA·cosB±cosA·sinB.”
The sine of a sum or difference splits into two products: sin(A±B)=sinA·cosB±cosA·sinB. It comes from rotating the unit-circle point (cosA, sinA) by angle B. In problems the ± sign in the formula always matches the sign between the angles; the second term carries that sign.
sin(A±B) = sinA cosB ± cosA sinB
Memory trick: Match the ± to the angle operation: sin(A±B)=sinA·cosB ± cosA·sinB (second term gets same sign).
“A pizza slice measured by the crust length (radians) tells you the exact arc, while the slice’s angle in degrees just tells how many of 360 parts it is.”
A full circle is 360° = 2π rad. One degree equals π/180 rad. Radians are defined as arc length per unit radius, making calculus formulas simple (d/dx sin x = cos x only in radians). Forgetting conversion adds extra π/180 factors in derivatives or series.
Memory trick: °→rad: ×π/180
“When surveying a triangular plot, you use the sine rule to find a hidden side from two angles, but the cosine rule when you know two sides and the angle between them.”
Sine rule a/sinA = b/sinB = c/sinC works when you know two angles and any side; cosine rule c² = a² + b² − 2ab·cosC works when you know two sides and the included angle, or to find an angle from three sides. Mistake: using sine rule with only side-angle data that fits cosine rule.
Memory trick: Sine ↔ angles, Cosine ↔ sides-angle-included
“A GPS satellite’s signal strength never goes beyond its maximum, just as sinθ never exceeds 1.”
Since sinθ and cosθ are coordinates on the unit circle, their absolute values cannot exceed 1. Any result |sinθ|>1 (or |cosθ|>1) signals an algebraic slip such as squaring both sides or ignoring domain restrictions. Check the range before proceeding to inverse-trig steps.
Memory trick: |sinθ|,|cosθ| ≤ 1 (unit circle)
“Climbing a hill feels like the rise over run you feel; that ratio is the slope m = Δy/Δx = tanθ.”
Slope m measures a line’s steepness as the ratio of vertical change to horizontal change, m=Δy/Δx, which equals tanθ of the inclination angle. It is undefined for vertical lines and zero for horizontal lines. Trap: using Δx/Δy or forgetting sign for decreasing lines.
Memory trick: Slope = rise ÷ run = tanθ
“A skateboard ramp’s steepness is measured by how high it rises for each foot you roll forward, not the opposite.”
For points (x1,y1) and (x2,y2), slope m = (y2−y1)/(x2−x1) = tanθ. Swapping Δx and Δy yields the reciprocal 1/m, which retains the original sign (a negative stays negative). This inversion gives wrong steepness and can falsely suggest a finite value for a vertical line, whose slope is actually undefined.
Memory trick: Slope = rise ÷ run; swap → reciprocal, sign unchanged.
“Parallel railway tracks run side-by-side, so their slopes are identical; a road crossing them at right angles has slopes whose product is –1.”
For any straight line, slope m = Δy/Δx. Two lines are parallel ⇔ their slopes are equal. Two lines are perpendicular ⇔ m1·m2 = –1 (negative reciprocal). Exam trap: many write m1 = m2 for right angles; remember the product rule.
Memory trick: Parallel: same m; Perp: m1·m2 = –1 (negative reciprocal).
“A ladder leaning against a wall looks slanted, but its slope isn’t the same as the floor’s; assuming equal slopes for perpendicular lines is a common slip.”
Students often write m1 = m2 for perpendicular lines, mixing up the rules. Correctly, parallel lines share the same slope; perpendicular lines have slopes whose product is –1. In JEE, the product condition is tested directly, so verify before substituting.
Memory trick: Equal slopes → parallel, not perpendicular.
“Writing a line as y=mx+c is like noting a car’s speed (m) and its starting kilometre marker (c) on a highway map.”
A line can be expressed as y=mx+c (slope-intercept), y−y1=m(x−x1) (point-slope), or using two points (y−y1)/(x−x1) = (y2−y1)/(x2−x1). Choose the form that matches the given data; using the wrong one wastes time and may cause algebra errors.
Memory trick: y=mx+c for slope & intercept; y−y1=m(x−x1) when a point is known.
“If you only know a line’s slope and one point, using the two-point formula is like trying to draw a river with just one bank.”
When only the slope and a single point are given, picking the two-point form forces you to invent a second point, leading to wrong equations. Use point-slope form directly; it avoids unnecessary calculations and reduces sign mistakes.
Memory trick: Match given data → pick matching form.
“Surveyors determine a hill’s steepness by measuring two altitude posts; the rise over run gives the slope m = (y2−y1)/(x2−x1).”
The slope between (x1,y1) and (x2,y2) is m = (y2−y1)/(x2−x1). It measures the line’s steepness. If x1 = x2 the line is vertical and the slope is undefined; JEE may ask to state ‘no slope’ in that case.
m = (y₂−y₁)/(x₂−x₁)
Memory trick: Δy/Δx, only if Δx ≠ 0.
“The shortest walk from a lamp post to a straight road is a perpendicular; its length is d = |ax0+by0+c|/√(a²+b²).”
For a line ax+by+c=0 and a point (x0,y0), the perpendicular distance is d = |ax0+by0+c|/√(a²+b²). The numerator is the line’s left side evaluated at the point, absolute value ensures positivity. Forgetting the absolute value flips sign errors.
d = |ax₀+by₀+c|/√(a²+b²)
Memory trick: d = |ax0+by0+c| / √(a²+b²).
“Two railway tracks that never meet are parallel, while a ladder against a wall is perpendicular, their slopes obey m1=m2 or m1·m2=−1 (0·∞ case).”
Parallel lines share the same slope; vertical lines are parallel even though their slope is undefined. Perpendicular lines obey m1·m2 = –1; for a horizontal (m=0) and vertical (undefined) pair the right-angle condition holds despite the product not being definable. Exam trap: don’t force the –1 product on a 0–∞ case.
Memory trick: Parallel = same m; Perp = m1·m2 = –1 (or 0 & ∞)
“Writing y=mx+c is like setting a car’s speed (m) and its starting mileage (c) before a trip.”
Use y=mx+c when the slope m and y-intercept c are known; use (y−y1)=m(x−x1) when a point (x1,y1) and the slope are given. Both equations describe the same line; avoid substituting a point into y=mx+c unless you first determine c.
Memory trick: Intercept first → y=mx+c; point first → (y−y1)=m(x−x1).
“Two ladders leaning against a wall at the same angle have identical slopes, so they never cross.”
Equal slopes (m1=m2) make lines parallel, not perpendicular. For perpendicular lines the slopes multiply to –1 (or one line is vertical and the other horizontal). The myth confuses “right angle” with “same steepness.”
Memory trick: Parallel = Same; Perp = Product –1.
“Eccentricity e tells how stretched a conic is, like tightening a rubber band from a circle to a parabola and then to a hyperbola.”
Eccentricity e = distance focus-centre / semi-major axis. Values: circle e=0, ellipse 0<e<1, parabola e=1, hyperbola e>1. It is dimensionless and same for all similar conics. Do not assume only hyperbola has e>1; ellipse also has a positive e less than 1.
Memory trick: e: 0-circle, 1-parabola, >1-hyperbola
“A basketball (circle, e=0), an Olympic track (ellipse, e<1), a water-spray arc (parabola, e=1) and a cooling-tower silhouette (hyperbola, e>1).”
Eccentricity e classifies conics: e=0 → circle, 0<e<1 → ellipse, e=1 → parabola, e>1 → hyperbola. It is the constant ratio PF/PD (focus to directrix). Trap: forgetting a circle is a special ellipse (e=0) and that hyperbola always has two separate branches.
Memory trick: e: 0-circle, <1-ellipse, =1-parabola, >1-hyperbola.
“A flashlight’s reflector cross-section is a parabola: the bulb sits at the focus and a straight line behind it is the directrix.”
A conic consists of points P such that PF = e·PD, where F is a fixed point (focus) and D is a fixed line (directrix). The eccentricity e determines the type: e<1 ellipse, e=1 parabola, e>1 hyperbola. Remember that the ratio uses absolute distances; a common trap is to mix up the sign of PD when the point lies on the opposite side of the directrix.
Memory trick: F = point, D = line – point vs line.
“In a satellite dish, the receiver at the focus collects reflected signals; the straight line behind the dish is the directrix.”
A conic is the set of points P with PF/PD = e, where F is a fixed point (focus) and D a fixed line (directrix). For a parabola e=1, PF = PD; for an ellipse e<1, for a hyperbola e>1. The common trap is treating the curve itself as the directrix or swapping point and line, which gives wrong equations.
Memory trick: Focus = point (F), Directrix = line (D) → remember F-point, D-line
“The equation (x−h)²+(y−k)²=r² is the GPS map for any circle, just like latitude-longitude pins its centre.”
The circle’s standard form (x−h)²+(y−k)²=r² gives centre (h,k) and radius r. The parabola’s basic forms are y²=4ax (opens right) and x²=4ay (opens up). Mixing the variables flips the opening direction, a frequent slip in JEE.
Memory trick: y²=4ax opens right; x²=4ay opens up.
“If you write y²=4ax but draw the curve opening upward, you’ve swapped the axis – the equation tells the direction.”
When the squared term is y², the parabola opens horizontally (right for positive a); when it is x², it opens vertically (up for positive a). Using the wrong axis produces an incorrect graph and wrong focus/directrix.
Memory trick: Square y → opens right; square x → opens up.