Class 11 · Chapter 1
Basic Mathematics & Vectors
Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.
Basic Mathematics & Vectors Short Notes
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Condensed revision points for Basic Mathematics & Vectors — for quick recall before exams, not a substitute for the full notes.
Angles & Trigonometry
- θ(rad) = θ(degree) × π/180. 1 rad ≈ 57.3°.
- sinθ = opp/hyp, cosθ = adj/hyp, tanθ = opp/adj; cosecθ=1/sinθ, secθ=1/cosθ, cotθ=1/tanθ.
- sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.
- ASTC rule: All positive in Q1, only Sin (& cosec) in Q2, only Tan (& cot) in Q3, only Cos (& sec) in Q4.
- sin30°=½, sin45°=1/√2, sin60°=√3/2 (cosine mirrors this; tan = sin/cos).
- Small-angle (θ in rad, θ<5°): sinθ≈θ, tanθ≈θ, cosθ≈1.
Calculus Essentials
- d/dx(xⁿ) = nxⁿ⁻¹; d/dx(c)=0; product rule: d/dx(uv)=u v'+v u'.
- d/dx(sinx)=cosx, d/dx(cosx)=−sinx, d/dx(eˣ)=eˣ, d/dx(lnx)=1/x.
- Stationary point: dy/dx=0. Minimum if d²y/dx²>0; maximum if d²y/dx²<0.
- ∫xⁿ dx = xⁿ⁺¹/(n+1)+c (n≠−1); ∫(1/x)dx = lnx+c.
- ∫sinx dx = −cosx+c; ∫cosx dx = sinx+c; ∫eˣ dx = eˣ+c.
- Definite integral ∫ₐᵇ f(x)dx = area under curve from a to b = F(b)−F(a).
- Average value of y over [a,b] = (∫ₐᵇ y dx)/(b−a).
Algebra & Series
- Quadratic roots: x = [−b ± √(b²−4ac)]/2a. Sum = −b/a, product = c/a.
- Real & distinct roots if b²−4ac>0; equal if =0; imaginary if <0.
- Binomial approximation: (1+x)ⁿ ≈ 1+nx for |x|≪1.
- AP: aₙ=a+(n−1)d; Sₙ=(n/2)[2a+(n−1)d].
- GP: aₙ=arⁿ⁻¹; Sₙ=a(1−rⁿ)/(1−r); S∞=a/(1−r) for |r|<1.
Mensuration Quick Reference
- Circle: A=πr², circumference=2πr.
- Sphere: surface area=4πr², volume=(4/3)πr³.
- Cylinder: curved surface area=2πrl, volume=πr²l.
- Cone: volume=(1/3)πr²h.
- Ellipse: area=πab.
Vectors — Basics
- Scalar: magnitude only (mass, time, speed). Vector: magnitude + direction (displacement, velocity, force).
- Unit vector  = A/|A|, magnitude exactly 1, marks direction only.
- Null vector: zero magnitude, direction undefined.
- Parallel vectors: angle 0°. Antiparallel: angle 180°. Collinear: angle 0° or 180°.
- Polar vectors (force, displacement) have a point of application; axial vectors (torque, angular velocity) point along the rotation axis via the right-hand rule.
Vector Addition & Subtraction
- R = √(A²+B²+2AB cosθ); direction tanα = B sinθ/(A+B cosθ).
- R_max = A+B (θ=0°); R_min = |A−B| (θ=180°).
- Equal-magnitude vectors a, a, separated by θ: resultant = 2a cos(θ/2), along the bisector.
- Resultant of two UNEQUAL vectors can never be zero.
- A−B = A+(−B); |A−B| = √(A²+B²−2AB cosθ).
Vector Resolution & Components
- Plane: Aₓ=A cosα, A_y=A sinα; A=√(Aₓ²+A_y²); tanα=A_y/Aₓ.
- Space: A=Aₓî+A_yĵ+A_zk̂; A²=Aₓ²+A_y²+A_z².
- Direction cosines: cosα=Aₓ/A, cosβ=A_y/A, cosγ=A_z/A; cos²α+cos²β+cos²γ=1.
Dot & Cross Product
- A·B = AB cosθ = AₓBₓ+A_yB_y+A_zB_z. Work W=F·S=FS cosθ.
- A·B > 0 if θ<90°, = 0 if θ=90° (perpendicular), < 0 if θ>90°.
- î·î=ĵ·ĵ=k̂·k̂=1; î·ĵ=ĵ·k̂=k̂·î=0.
- A×B = AB sinθ n̂, direction by right-hand rule. Torque τ=r×F; angular momentum L=r×p.
- A×B = −(B×A); zero only when vectors are parallel/antiparallel (θ=0° or 180°).
- î×ĵ=k̂, ĵ×k̂=î, k̂×î=ĵ (reversing any pair flips the sign).
Common Exam Traps
- θ must be in radians for the small-angle approximation and for calculus formulas like d/dx(sinθ)=cosθ — degrees give a wrong numerical answer.
- Resultant of two vectors of UNEQUAL magnitude can never be zero — only an equal-and-opposite pair cancels.
- A·B = 0 means the vectors are perpendicular, NOT that either vector itself is zero.
- Cross product order matters: A×B = −(B×A) — swapping flips the sign, not the magnitude.
- dy/dx = 0 only locates a turning point — always check the sign of d²y/dx² before calling it a maximum or minimum.
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