Detailed reasoning must be shown in this question.
A researcher studying fluid dynamics determines that for any two positive flow rates uuu and vvv in a bifurcated pipe, the 'effective velocity' VeV_eVe and the 'total sum' SSS are defined as:
Ve=u3+v3u2+v2andS=u+v V_e = \frac{u^3 + v^3}{u^2 + v^2} \quad \text{and} \quad S = u + v Ve=u2+v2u3+v3andS=u+vUse proof by contradiction to show that Ve<SV_e < SVe<S for all positive real numbers uuu and vvv.
By providing a counter-example, show that the statement in part (a) is not true for all real numbers uuu and vvv where u2+v2≠0u^2 + v^2 \neq 0u2+v2=0.
368 exam-style questions on Edexcel A Level Maths Algebraic Methods, covering 1.1 Proof by Contradiction, 1.2 Algebraic Fractions, 1.3 Partial Fractions, 1.4 Repeated Factors, and 1.5 Algebraic Division. Each one has a worked solution and a mark scheme showing where the marks go.