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A matrix M has eigenvectors (3,1,0) (2,8,2) (1,1,6) with corresponding eigenvalues 1, 6, 2 respectively. Write an invertible matrix P and diagonal matrix D such that M=PD(P^-1), hence calculate M^5.

Without even knowing M, the candidate can calculate M^5. This will follow from the fact that P is the matrix consisting of the eigenvectors of M as columns, and D will have the eigenvalues (in matching co...

Answered by Cameron B. • Maths tutor

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A smooth 4g marble is held at rest on a smooth plane which is fixed at 30 degrees to a horizontal table. The marble is released from rest - what speed is the marble travelling at 5 seconds after being released? Let g = 9.8ms^2

This question involves forces and kinematics. First begin by drawing the situation and identifying the forces acting on the marble. The weight of the marble acts vertically downwards and is equal to the m...

Answered by Maya R. • Maths tutor

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Differentiate the function; f(x)=1/((5-2x^3)^2)

We know from the properties of basic indices that a^-x=1/a^x, so 1/((5-2x³)²=(5-2x³)^-2 where in this case, a=5-2x³and x=2. Then t...

Answered by Benjamin K. • Maths tutor

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Differentiate (2^x)(5x^2+5x)^2.

This is a relatively difficult equation to differentiate as there are various parts to consider.Firstly, we will let u=2^x and v=(5x^2+5x)^2 in the product rule. Then the differential of u is (2^x)ln(2). ...

Answered by George H. • Maths tutor

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Determine the tangent to the curve y = sin^2(x)/x at the point, x = pi/2. Leave your answer in the form ax+by+c=0

First determine the y coordinate of the curve at the given x coordinate, in this case, y = sin²(pi/2)/(pi/2) = 1²/(pi/2) = 2/piDifferentiate the function with respect to x to determi...

Answered by Thomas B. • Maths tutor

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