Surds

The circle C has equation

Find

the coordinates of the centre of C,

the exact radius of C, giving your answer as a simplified surd.

The line l has equation y = 3x + k where k is a constant.

Given that l is a tangent to C,

find the possible values of k, giving your answers as simplified surds.

Simplify

Solve the equation

Give your answer in the form ab where a and b are integers.

Simplify

Express 27 in the form kV3, wherek is an integer. (4)

Figure 2.

Find the coordinates ofA and the coordinates of B. (6)

Find the distance AB in the form r \2 wherer is a rational number. (3)

Express

in the form a + b 2, where a and b are integers.

Express

in the form c 5, where c is an integer.

Write

in the form k √x, where k and x are integers.

A sequence of positive numbers is defined by

Find a₂ and a₃, leaving your answers in surd form.

Find an equation of the line joining A (7, 4) and B (2, 0), giving your answer in the form ax + by + c =0, where a, b and c are integers.

Find the length of AB, leaving your answer in surd form.

The point C has coordinates (2, t), where t > 0, and AC = AB.

Find the value of t.

Find the area of triangle ABC.

On the axes below sketch the graphs of

y = x (4 – x)

y = x² (7 – x) showing clearly the coordinates of the points where the curves cross the coordinate axes.

Show that the x-coordinates of the points of intersection of

are given by the solutions to the equation x(x² – 8x + 4) = 0

The point A lies on both of the curves and the x and y coordinates of A are both positive.

Find the exact coordinates of A, leaving your answer in the form (p q√3, r + s√3), where p, q, r and s are integers.

Expand and simplify (7 + √5)(3 – √5).

Express in the form a b5, where a and b are integers.

Simplify

8 5 2 5

Given that 32 √ 2 = 2^a, find the value of a.

₂

Show that f (x) = 9x Ax B, where A and B are constants to be found.

Find f ' (x).

Evaluate f ' (9).

Expand and simplify (√7 + 2)(√7 – 2).

Simplify

giving your answer in the form a + b√3, where a and b are integers.