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Use the Binomial Shortcut and Linearise Before You Plot

The mathematics behind physics questions is small but used constantly: the binomial approximation for tiny corrections, the sum and product of roots, combining sine and cosine, composing functions, and choosing variables so a graph is a straight line.

NSEJS · Basic Mathematics Used in Physics and Vectors · 8 linked questions · 100% coverage

Know this first

Algebra, Trigonometry and Approximations for Physics is the central idea in this question pattern. Identify the condition in the stem before applying the method below.

Study deeper: NCERT Science · Basic Mathematics Used in Physics and Vectors

Binomial approximation and quadratic roots

For |x| much less than 1, (1 + x)^n is approximately 1 + n x. So (1 + h/R)^(-2) is about 1 - 2h/R, and (0.996)^(1/4) = (1 - 0.004)^(1/4) is about 1 - 0.001 = 0.999. For a x² + b x + c = 0, the sum of the roots is -b/a and the product is c/a, even when the roots are complex: for 2x² - 3x + 5 = 0 the sum is 3/2 and the product is 5/2.

Combining sine and cosine, and functions of functions

Write a cosθ + b sin

θ = A sin(θ + α) by expanding the right side: A sinα = a and A cosα = b, so A = sqrt(a² + b²) and tan α = a/b

For 3 cosθ + 4 sinθ,

A = 5 and sin α = 3/5, so α = sin⁻¹(3/5)

A function machine is read in the order of the boxes: double the input and add three gives 2x + 3, then taking the square root gives

z = sqrt(2x + 3)

A quantity such as kinetic energy cannot be negative, so K(x) = 9 - x² limits the particle to -3 ≤ x ≤ 3.

Straight-line graphs

To get a straight line, choose axes so the relation has the form

y = m x + c

From f = (1/2π) sqrt(g/ℓ), squaring gives f² = (g / 4π²)(1/ℓ), so f² against 1/ℓ is a straight line through the origin, while f against ℓ is a curve

A parabola through the origin has y proportional to x².

OlympIQ exam lens

✓(1 + x)^n ≈ 1 + n x for small x
✓Quadratic: sum of roots = -b/a, product = c/a
✓a cosθ + b sinθ = A sin(θ + α), A = sqrt(a² + b²), tan α = a/b
✓Physical quantities such as kinetic energy cannot be negative; this limits the allowed range of x
✓Square or invert the relation until it reads y = m x + c

OlympIQ trap

Do not use the binomial shortcut when the correction is not small. For x = 0.5 the approximation 1 + n x is poor, and you must expand exactly; the shortcut is for x of a few percent or less.

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