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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 +...
JM
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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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...
EC
Answered by Elliot C. • Maths tutor
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Solve the inequality 6x - 7 + x^2 > 0

Firstly rearrange the quadratic such that the coefficient of x 2 is positive (already done in this example) and the quadratic is in the form of ax 2 +bx + c, then solve for x, like you would solve a regular ...
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Let N be an integer not divisible by 3. Prove N^2 = 3a + 1, where a is an integer

For N to be not divisible by 3, N can either be of the form 3k + 1 (1,4...) or 3k + 2 (2,5...), where k is an integer. The proof can then be done by checking both 3k + 1 and 3k + 2 when N is squared, to see ...
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