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Units of Area and Volume – Questions Answered | Pearson Edexcel International GCSE Maths

These examples practise converting units of area and volume, using powers of 10, writing values in standard form, and applying the pressure formula. Particular care is needed because area conversions are squared and volume conversions are cubed. Question 1(a) – Converting m² to cm² Convert 2.3 m² into cm² . Working Convert 2.3 m² into cm². 1 m = 10² cm 2.3 × (10² cm)² = 2.3 × 10⁴ cm² Answer: 2.3 × 10⁴ cm² Question 1(b) – Converting mm³ to cm³ Convert 400 mm³ into cm³ . Working Convert 400 mm³ into cm³ 10 mm = 1 cm 1 mm = 10⁻¹ cm 400 × (10⁻¹ cm)³ = 400 × 10⁻³ cm³ = 4 × 10² × 10⁻³ cm³ = 4 × 10⁻¹ cm³ Answer: 4 ×...
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Pressure, Force and Area – Pearson Edexcel International GCSE Maths

Pressure, force and area are connected by a simple formula. Once the relationship is understood, the formula can be rearranged depending on which quantity needs to be found. Pressure, Force and Area The basic relationship is: To find pressure: P = F / A To find force: F = P × A To find area: A = F / P What the Symbols Mean P represents pressure. F represents force. A represents area. Units * Pressure is measured in pascals (Pa), where 1 Pa = 1 N/m². * Force is measured in newtons (N). * Area (A) is measured in square metres (m²). Understanding the Formula Pressure describes h...
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Ratio Problems – Pearson Edexcel International GCSE (9–1) Mathematics A Higher Tier

These ratio problems are based on questions from the Revise Pearson Edexcel International GCSE (9–1) Mathematics A – Higher Tier Revision Guide . The solutions below use an algebraic ratio method in which each part of the ratio is represented as a multiple of x . Question 1 – Sharing Money in a Ratio Andre, Becky and Makito share money in the ratio 3 : 6 : 7 . Andre and Becky receive £207 altogether. Work out how much Makito receives. Working A : B : M = 3x : 6x : 7x 3x + 6x = 207 9x = 207 x = 207/9 7 × (207/9) = 161 Makito receives £161. Answer: £161 Question 2(a) – Ages in a Ratio Amir and Petra's ages are in the ratio 3 : 7 . Amir is 9 years old . Work out Petra's age. Wo...
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Deriving the Addition and Subtraction Rules for Fractions from First Principles

Fractions | Algebra | First Principles The familiar rules for adding and subtracting fractions can be derived directly from the properties of real numbers. The key idea is that multiplying a number by 1 does not change its value. By writing 1 in a suitable fractional form, both fractions can be expressed with the same denominator. Assumptions Let a, b, c, d ∈ ℝ, with b ≠ 0 and d ≠ 0. Addition of Two Fractions Begin with two fractions whose denominators are not necessarily equal: a b + c d = a b · 1 + c d · 1 = a b ...
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Deriving the Equation of a Plane Through A Perpendicular to the Vector A

Let A be a non-zero vector in ℝ³, and also regard its endpoint as the point through which the plane passes: A = (a 1 , a 2 , a 3 ) T ≠ (0, 0, 0) T . Let P be an arbitrary point on the plane: P = (x, y, z) T . The displacement from A to P is P − A = (x − a 1 , y − a 2 , z − a 3 ) T . Because the plane is perpendicular to A , the vector A is its normal vector. Every displacement P − A lying in the plane must therefore be perpendicular to A . Perpendicular vectors have a dot product of zero. Hence A · ( P − A ) = 0. Writing this condition in coordinates gives a 1 (x − a 1 ) + a 2 (y − a 2 ) + a 3 (z − a 3 ) = 0. Expanding the brackets: a 1 x − a 1 2 + a 2 y − a 2 2 + a 3 z − a 3 2 = 0. Move the squared terms to the right-hand side: a 1 x + a 2 y + a 3 z = a 1 2 + a 2 2 + a 3 2 . In vector notation, this becomes A · P = A · A = ‖ A ‖ 2 . ...
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Deriving the Outer-Product Matrix BAᵀ from Matrix Multiplication

Let A, B and X be column vectors in ℝ³: A = a₁ a₂ a₃ , B = b₁ b₂ b₃ , X = x y z . BAᵀX = B(AᵀX). Matrix multiplication is associative, so AᵀX may be evaluated first. Since Aᵀ is a 1 × 3 row matrix and X is a 3 × 1 column matrix, their product is a scalar: AᵀX = a₁x + a₂y + a₃z. Therefore, BAᵀX = b₁ b₂ b₃ (a₁x + a₂y + a₃z). The quantity in parentheses is a scalar, so it multiplies every component of B: BAᵀX = b₁a₁x + b₁a₂y + b₁a₃z b₂a₁x + b₂a₂y + b₂a₃z b₃a₁x + b₃a₂y + b₃a₃z . Collect the coefficients of x, y and z into a matrix multiplying X: BAᵀX = b₁a₁ b₁a₂ b₁a₃ b₂a₁ b₂a₂ b...
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Constructing the [v]ₓ Matrix from Skew-Symmetry and the Null Space

Let v = p q r ≠ 0 0 0 The objective is to construct the standard 3 × 3 matrix [v]ₓ associated with v, without assuming its entries in advance. Begin with two requirements: Kᵀ = −K Kv = 0. The first condition requires skew-symmetry. The second requires v to lie in the null space of K. 1. Write the general skew-symmetric matrix Since Kᵀ = −K, the diagonal entries must be zero and entries reflected across the diagonal must have opposite signs. Therefore, K = 0 α β −α 0 γ −β −γ 0 At this stage, α, β and γ are unknown. 2. Require Kv = 0 0 α β −α 0 γ −β −γ 0 p q r ...
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