Determinants

Verify that a cartesian equation of Π₁ is x - y + 2z = 3.

For some real constant a, cartesian equations of planes Π₂ and Π₃ are

Π₂: x - 3z = 1 Π₃: ax - y - z = 4

By considering a suitable matrix, show that Π₁, Π₂ and Π₃ intersect at a single point for all

values of a except a = 2.

Use the matrix from part (b) to find the coordinates of the point of intersection of Π₁, Π₂

and Π₃ in the case where a = 3.

In the case where a = 2, determine the geometrical arrangement of Π₁, Π₂ and Π₃.

In this question you must show detailed reasoning.

The surface S is given by the equation z = 4x²y - 3xy² - 5x for all real values of x and y.

The point P (1, 1, - 4) lies on S.

Find the equation of the tangent plane to S at P.

The point Q is a stationary point of S with positive x - and y-coordinates.

Find, in exact form, the coordinates of Q.

Find the value of the determinant of H, the Hessian matrix of S, at Q.

Deduce the nature of the stationary point Q.

Find N² in terms of a.

Find det N in terms of a.

Describe geometrically the single transformation consisting of T_A followed by T_B.

By considering the transformation T_A, determine the matrix A⁴²³.

The transformation T_C is represented by the matrix C, where

The region R is defined by the set of points (x, y) satisfying the inequality x² + y² ⩽ 36.

The region R′ is defined as the image of R under T_C.

Find the exact area of the region R ′.

Sketch the region R ′, specifying all the points where the boundary of R ′ intersects the

coordinate axes.

The surface S has equation z = x sin y + for x > 0 and 0 < y < π.

Determine, as a function of x and y, the determinant of H, the Hessian matrix of S.

Given that S has just one stationary point, P, use the answer to part (a) to deduce the nature

of P.

The coordinates of P are (α, β, γ).

Show that β satisfies the equation β + tan β = 0.

Define the transformation represented by A.

Show that the area of any object shape is invariant under the transformation represented by A.