Sec 3 A-Math Surds topical questions

Simplify √72.

Simplify √45 + √20.

Expand and simplify (4 + √2)(4 − √2).

Rationalise the denominator of 5/√10, writing your answer in simplest form.

Solve the equation √x = 7.

Let E = (√3 + 2)².

1. (a)Expand and simplify E.[2]
2. (b)Explain why E is irrational.[2]

Rationalise the denominator of 12/(√7 − 3), giving your answer in simplest form.

Solve the equation √(2x + 1) = x − 1.

1. (a)Show that x satisfies x² − 4x = 0.[1]
2. (b)Find both candidates for x.[2]
3. (c)Check the candidates and state the solution of the original equation.[2]

Express (√7 − √2)² in the form a + b√14, where a and b are integers.

A right-angled triangle has legs of length √8 cm and √12 cm.

1. (a)Show that the hypotenuse is 2√5 cm.[2]
2. (b)Find the exact area of the triangle.[2]
3. (c)State whether the area is rational, and justify your answer.[1]

Solve the equation x − √(x + 3) = 3.

1. (a)Rearrange and show that x satisfies x² − 7x + 6 = 0.[2]
2. (b)Find both candidates for x.[1]
3. (c)Check the candidates and state the final solution.[2]

A rectangle has sides (√5 + 1) cm and (√5 − 1) cm.

1. (a)Show that the area is exactly 4 cm².[1]
2. (b)Show that the perimeter is 4√5 cm.[2]
3. (c)A square has the same area as the rectangle. State the side length of the square, and comment on which shape has the larger perimeter.[2]

Express (2√3 − 1)/(√3 + 1) in the form a + b√3, where a and b are rational.

Let r = √2 + 1.

1. (a)Show that 1/(√2 − 1) = √2 + 1.[2]
2. (b)Write 1/(√2 + 1) in the form a + b√2.[1]
3. (c)Hence find the exact value of (√2 + 1)² + (√2 − 1)².[2]

An equilateral triangle has sides of length 2√3 cm.

1. (a)Show that its height is 3 cm.[2]
2. (b)Find the exact area of the triangle.[2]
3. (c)Show that the perimeter is greater than 10 cm, using √3 ≈ 1.732.[2]