A surd such as √7 represents which kind of number?
Number And Operation
Question · Easy
Cruise the loop: a log is an exponent
Short
Which of the following is a rational number?
Simplify √50, leaving the answer in the form a√b where b is square-free.
Simplify √72.
Which statement is true?
Find x
Express √27 + √12 in the form a√3, where a is an integer.
Question · Medium
Simplify √20 × √45.
Rationalise the denominator of 1/√5.
Rationalise the denominator of 3 / (√7 − 2).
Simplify √48 − √12 + √3.
Express (√18 + √8) / √2 in simplest form.
Given that (8 − √18) / √2 = a + b√2, where a and b are integers, find a and b.
Question · Hard
Rationalise (√5 + 1) / (√5 − 1) and simplify fully.
Make x the subject of the formula y = 3x + 7.
Algebra
Which of the following is NOT the equation of a straight line?
Evaluate 3^4 × 3^2.
Evaluate log_10 1000.
Solve 2x + 5 = 17.
Express 8^{2/3} as an integer.
Make r the subject of A = π r^2, taking r > 0.
Simplify log 8 + log 3 − log 6 as the logarithm of a single number.
Solve 5^{x+1} = 125.
Solve x^2 − 5x − 14 = 0.
Express 27^{-2/3} as a fraction in simplest form.
Find the remainder when f(x) = x^3 − 4x + 5 is divided by (x − 2).
Solve the inequality 3x − 7 > 2x + 5, where x is real.
If 2^x = 16√2, find x.
Solve the simultaneous equations 2x + y = 11 and 3x − y = 4.
Solve (x + 1)/3 − (x − 2)/2 = 1.
Solve log_2 (x + 3) = 3.
Make v the subject of the formula 1/u + 1/v = 1/f.
Solve 2^{2x} − 5 · 2^x + 4 = 0.
Solve log_3 (x − 1) + log_3 (x + 1) = 1.
The quadratic equation 2x^2 − k x + 8 = 0 has equal roots. Find the possible values of k.
The function f(x) = (x − 3)(x + 2) + k leaves a remainder of 7 when divided by (x − 1). Find k.
Convert 3.5 km to metres.
Measurements
A rectangle measures 12 cm by 8 cm. What is its area?
A circle has radius 7 cm. Using π = 22/7, find its circumference.
Convert 2.4 m^2 to cm^2.
Find the volume of a cube with side length 5 cm.
A cylinder has radius 3 cm and height 10 cm. Find its volume in terms of π.
A trapezium has parallel sides 8 cm and 14 cm and perpendicular height 5 cm. What is its area?
Convert 72 km/h to metres per second.
A triangle has base 14 cm and perpendicular height 9 cm. Find its area.
A pyramid has a rectangular base 6 cm by 4 cm and a perpendicular height of 9 cm. Find its volume.
A circle of radius 4 cm is cut from the interior of a square of side 10 cm. Find the remaining area in terms of π.
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