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Curves C1 and C2 have equations y= ln(4x-7)+18 and y= a(x^2 +b)^1/2 respectively, where a and b are positive constants. The point P lies on both curves and has x-coordinate 2. It is given that the gradient of C1 at P is equal to the gradient of C2 at P.

y= ln(4x-7)+18 y= a(x^2 +b)^1/2 At x=2 dy/dx = dy/dx and y =y At x =2 y = ln(8-7) +18 y = ln 1 +18 y =18 At x = 2 18=a(4 +b)^1/2 18/(4+b)^1/2= a y=ln(4x-7)+18dy/dx= 4/(4x-7) At x= 2 dy/dx= 4/8-7= 4 y=a(x^2 +...
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Answered by Jordan M. • Maths tutor
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Calculate the value of the definite integral (x^3 + 3x + 2) with limits x=2 and x=1

a) Integrate the given expression using integration laws we have learnt to give [(x^4)/4 + (3(x^2))/2 + 2x ] and you do not need a +c constant as we have limits.b) Substitute the limits into the equation we ...
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What is osmosis?

Osmosis is the passive diffusion of water, which therefore, doesn't require any energy to occur. Typically water will move from an area of high concentration to an area of low concentration across a semi-per...
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Answered by Matthew W. • Biology tutor
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Name the key differences between Eukaryotic and Prokaryotic cells?

Eukaryotic cells are typically belonging to multicellular organisms, with exceptions of algae and amoebae. These cells always have multiple linear chromosomes (genetic material) and contain a nucleus (housin...
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Answered by Matthew W. • Biology tutor
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A hemisphere is placed on top of an upside down cone. The cone has height 9cm and the hemisphere has radius 3cm. The total volume of this composite solid is x cm^3. Calculate the value of x, leaving your answer in terms of π.

To work out the total volume of the composite solid, we need the volumes of both the cone and the hemisphere. GCSE Maths students are expected to know these respective formulae; volume of a sphere = ⁴/₃πr³ a...
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Answered by Elliot C. • Maths tutor
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