Surds

The diagram shows a shape formed by a semicircle joined to a sector of a circle, BAC, along a common length AB.

The area of the sector BAC is 16π cm².

Work out the total area of the shape. Give your answer in terms of π. You must show your working.

Simplify.

Work out the following, giving each answer as a fraction in its simplest form.

You must show each step in your working.

You are given this identity.

Find the value of a and the value of b. You must show each step in your working.

A sphere has a radius of 3 cm.

Calculate the volume of the sphere. Give your answer in terms of π in its simplest form. [The volume V of a sphere with radius r is V = r³.]

The diagram shows a triangle, ABC, with perpendicular height BD.

BC = 12 cm, angle BCD = 30° and angle BAD = 45°.

Work out the length of BD.

Work out the exact length of AB. Give your answer in its simplest form.

Simplify.

Work out.

The diagram shows triangle ABC.

Find the area of the triangle.

The diagram shows a shaded shape made by removing sector OAB from sector OCD. Both sectors have an angle of 60°. The radius, OA, of the smaller sector is r cm. The ratio of radius OA to radius OC is 2: 3.

Work out, in terms of π and r, the total length of arc AB and arc CD. Give your answer in its simplest form. You must show your working.

Find the equation of the circle.

AOB is a sector of a circle, centre O and radius 10 cm.

The area of the sector is 40π cm².

Work out the perimeter of the sector. Give your answer in the form a + b π, where a and b are integers. You must show your working.

AOB is a sector of a circle, centre O.

The area of the sector is 8 cm².

Work out the exact value of the radius, r cm.

Work out.

Simplify.

Simplify fully by rationalising the denominator.

Write down the value of sin 45°.

Evaluate.₁

Work out.

Give your answer as a mixed number.

Isaac and Maya eat part of a pizza. Isaac eats of the pizza. Maya then eats of the remaining pizza. What fraction of the original pizza is left?

AOB is a sector of a circle, centre O and radius 6 cm. The length of arc AB is 5 π cm.

Find the area of the sector. Give your answer in terms of π.

Here is a sequence.

Work out the next term.

Find the n th term.

The shape below is formed from two semicircles and a straight line.

The radius of the large semicircle is 8 cm. The radius of the small semicircle is t cm.

Find an expression, in terms of t, for the exact perimeter of the shaded shape.

Find the exact coordinates of the two intersections of the line y = 2x and the circle x² + y² = 30

(........,........) and (........,........)

The diagram shows two right-angled triangles ABD and BCD, sharing a common side BD. AD = 10 cm, BC = 12 cm and angle DBC = 60°.

Work out the length of AB.

In the following equation, n is an integer greater than 1.

Find k when n = 7.

Find n when k = 64.

Show that can be written in the form a + b

Work out.

Simplify fully.

There are two errors in Sam’s method for finding the value of 64 shown below.

Find the cube root of 64 and then multiply by 2. The cube root of 64 is 4 and then 4 x 2 = 8. The negative power makes the answer negative so answer equals –8. Describe these errors and then give the correct value of 64