Timed ESAT diagnostic

A metal sample has mass 3.6 kg and volume 450 cm³. What is its density in kg m⁻³?

A 1:50 scale model of a rectangular panel has area 320 cm². What is the area of the actual panel in square metres?

Seven revision cards have target times of 2, 3, 4, 7, 8, 9 and 12 minutes. Two cards are selected at random without replacement. What is the probability that their target times add to a multiple of 5?

In a model, response time T seconds is inversely proportional to the square root of power P watts. When P = 9, T = 20. What value of P gives T = 12?

A speed–time graph for a cart is formed by straight-line segments through (0, 0), (4, 8), (7, 8) and (10, 2), where time is in seconds and speed is in m/s. How far does the cart travel in 10 seconds?

A cuboid has a 3 m by 4 m rectangular base and is 5 m high. A cable joins one bottom corner to the opposite top corner. What angle does the cable make with the base?

A histogram of students' completion times has frequency density 1.5 for 0 ≤ t < 10 minutes, 0.6 for 10 ≤ t < 30, and 0.9 for 30 ≤ t < 40. What proportion of the students are in the interval 10 ≤ t < 30?

A right-angled triangle has perpendicular sides of length (√45 + √20) cm and (√45 − √20) cm. What is its exact area?

A tank's starting volume is recorded as 50 L, correct to the nearest 5 L. The volume used is recorded as 18 L, correct to the nearest litre. What is the lower bound for the volume left in the tank?

For x ≠ −3, solve (x² + 5x + 6)/(x + 3) < 7.

Cumulative frequency information for 80 continuous measurements shows that 17 measurements are less than 20 and 65 are less than 40. How many measurements satisfy 20 ≤ x < 40?

Evaluate 125^(2/3) × 25^(−1/2) ÷ 5⁻¹.

A data logger has 4.8 × 10⁶ bytes free. It stores a file of 7.5 × 10⁵ bytes, then deletes a temporary file of 3 × 10⁴ bytes. How many bytes are now free?

For P ≠ 3, which expression makes q the subject of P = 3q/(q + r)?

The nth stage of a tile pattern starts as a square with n + 2 tiles along each side. The four corner tiles are then removed. Which expression gives the number of tiles in the nth stage?

Points A(−3, 4) and B(1, −2) lie on a line. A second line passes through C(4, 2) and is perpendicular to AB. At what x-coordinate does the second line cross the x-axis?

The point P is enlarged by scale factor 2 about centre (1, −1), then translated by the vector (−3, 4). Its final image is (4, 7). What are the coordinates of P?

Relative to origin O, distinct points A and B have position vectors a and b. Point C has position vector 3b − 2a. Which statement must be true?

A circle has diameter AB. Points C and D lie on the circle on the same side of AB, and D lies on the minor arc BC. Given ∠BAC = 32° and ∠DAB = 18°, find ∠CBD.

A container initially holds 320 litres of liquid. At the end of each hour, its contents are reduced by 25% of the amount at the start of that hour. How much liquid must be poured in immediately after the end of the third hour to restore the container to 320 litres?

Two bus routes have these ordered waiting times, in minutes. Route P: 4, 7, 8, 10, 12, 13, 16. Route Q: 2, 8, 9, 10, 11, 12, 18. Which statement is correct?

A cooling system contains 50 litres of fluid. It loses fluid at 1.25 litres per minute for 24 minutes, then at 0.75 litres per minute for 16 minutes. How much fluid remains?

Points A(2, 1) and B(8, 7) are fixed. Point P lies on the x-axis and is equidistant from A and B. What is the x-coordinate of P?

A quality check records 80 components. From Supplier A, 30 pass and 10 fail. From Supplier B, 24 pass and 16 fail. A component is known to have failed. What is the probability that it came from Supplier B?

Three status lights flash every 18 seconds, 24 seconds and 30 seconds. They flash together now. After how many seconds will they next flash together?

A 400 g alloy contains copper and zinc in the ratio 7:3 by mass. Pure zinc is added until the ratio of copper to zinc is 7:5. What mass of zinc was added?

A random test generator returns each whole number from 1 to 12 with equal probability. It is run 180 times. How many outputs are expected to be factors of 12 greater than 1?

The line y = 2x + c is a tangent to the circle x² + y² = 20. What are the possible values of c?

What is the sum of all real solutions of 9ˣ − 4(3ˣ) + 3 = 0?

An arithmetic series has first term 7 and common difference 4. What is the least value of n for which the sum of the first n terms exceeds 500?

A cylindrical container is open at the top and has volume 27π cm³. What is its minimum possible total surface area?

What is the area of the finite region enclosed between the curves y = x² and y = 2x − x²?

What is the coefficient of x³ in the expansion of (2 + x/2)⁶?

A ball is thrown vertically upwards from ground level at 20 m s⁻¹. Taking g = 10 m s⁻² and ignoring air resistance, at what time is it first at a height of 15 m?

A 6 Ω resistor and a 3 Ω resistor are connected in parallel, and that combination is connected in series with a 4 Ω resistor across a 12 V supply of negligible internal resistance. What is the current through the 6 Ω resistor?

Calcium carbonate decomposes completely on heating: CaCO₃ → CaO + CO₂. What mass of calcium carbonate produces 2.2 g of carbon dioxide? (Ar: C = 12, O = 16, Ca = 40)

How many structural isomers have the molecular formula C₄H₉Br?

In guinea pigs, black coat (B) is dominant to white (b). Two heterozygous black guinea pigs are crossed. What fraction of the black offspring are expected to be heterozygous?

A plant cell has a solute potential of −800 kPa and a pressure potential of +300 kPa. It is placed in a solution with a water potential of −400 kPa. What happens?