ESAT Maths 1 practice questions

A solid metal block has a mass of 5.4 kg. The metal has a density of $2.7{\text{ g/cm}}^3$.

What is the volume of the block in ${\text{m}}^3$?

A shop increases all of its prices by 20%. Later, in a sale, it reduces every new price by 25%. In the sale, a customer pays £54 for a jacket.

What was the price of the jacket before both changes?

What is the value of ${\left({\displaystyle \frac{8}{27}}\right)}^{-\frac{2}{3}}\times {16}^{-\frac{3}{4}}$?

Three positive quantities $x$, $y$ and $z$ satisfy $x:y=2:3$ and $y:z=4:5$.

What is the ratio $(x+y):(y+z)$?

Each interior angle of a regular polygon is 132° larger than each of its exterior angles.

How many sides does the polygon have?

$a$, $b$ and $c$ are constants such that

$$(ax+3)(2x+b)\equiv 6x^2+cx-12$$

for all values of $x$.

What is the value of $c$?

A bag contains 5 red counters and 3 blue counters. Two counters are taken out at random, one after the other, without replacement.

What is the probability that the two counters are the same colour?

Given that $y={\displaystyle \frac{3x+2}{x-4}}$, which of the following gives $x$ in terms of $y$?

Which of the following is equal to ${\displaystyle \frac{\sqrt{3}+1}{\sqrt{3}-1}}-{\displaystyle \frac{6}{\sqrt{3}}}$?

The recurring decimal $0.2\dot{3}\dot{6}$ means $0.2363636\dots$, where the digits 36 repeat for ever.

What is this number as a fraction in its simplest form?

A rectangle has length 8 cm and width 5 cm, where each measurement is correct to the nearest centimetre.

What is the difference, in ${\text{cm}}^2$, between the upper bound and the lower bound of the area of the rectangle?

$y$ is inversely proportional to the square of $x$, where $x>0$. The value of $x$ is increased by 25%.

By what percentage does the value of $y$ decrease?

A sector of a circle has radius 6 cm and area $12\pi {\text{ cm}}^2$.

What is the perimeter of the sector, in cm?

A cyclist starts from rest and accelerates uniformly to a speed of $8\text{ m/s}$ in 10 seconds. She then travels at $8\text{ m/s}$ for $T$ seconds, and finally decelerates uniformly to rest in 6 seconds. The total distance travelled is 480 m.

What is the value of $T$?

$k$ is the smallest positive integer such that $540k$ is a cube number (the cube of an integer).

What is the value of $k$?

The $n$th term of a sequence is a quadratic expression in $n$. The first five terms of the sequence are

4, 11, 22, 37, 56

What is the 20th term of the sequence?

Simplify fully

$$\frac{x^2-9}{2x^2+5x-3}÷\frac{x-3}{4x-2}$$

for the values of $x$ where the expression is defined.

The curve $y=2x^2+px+q$ has its minimum point at $(3,-5)$.

What is the value of $p+q$?

$A$ is the point $(-1,2)$ and $B$ is the point $(5,6)$. The perpendicular bisector of $AB$ crosses the $x$-axis at the point $P$.

What is the $x$-coordinate of $P$?

The table gives the frequency densities for a histogram showing the times, $t$ minutes, that 100 people took to complete a puzzle.

Assuming the times are spread evenly within each class, estimate how many people took between 12 and 24 minutes.

$AB$ is a diameter of a circle with centre $O$, and $C$ is a point on the circle. The tangent to the circle at $C$ meets the line $AB$, extended beyond $B$, at the point $T$. Angle $CTB={34}^{\circ }$.

What is the size of angle $CAB$?

A data set has mean $m$, range $r$ and interquartile range $q$. Every value in the data set is multiplied by 2, and then 3 is added to each result.

Which of the following statements about the new data set is/are true?

I: The mean is $2m+3$.

II: The range is $2r+3$.

III: The interquartile range is $2q$.

The line $y=2x+1$ meets the curve $y=x^2-2x+4$ at the points $P$ and $Q$.

What is the length of $PQ$?

How many solutions does the equation

$$2{\sin }^{2}x=\sin x$$

have in the interval $0^{\circ }\le x\le {360}^{\circ }$?

(Here ${\sin }^{2}x$ means $(\sin x{)}^2$.)

A bag contains $n$ red balls and 4 blue balls. Two balls are taken at random without replacement. The probability that both balls are red is $\frac{1}{3}$.

What is the probability that the two balls are different colours?

A pyramid has a square base of side 6 cm. Its apex is directly above the centre of the base, and each of its four sloping edges is also 6 cm long.

What is the volume of the pyramid, in ${\text{cm}}^3$?

(Volume of a pyramid $=\frac{1}{3}\times$ base area $\times$ perpendicular height.)

$OABC$ is a parallelogram with $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OC}=\mathbf{c}$. $M$ is the midpoint of $AB$. The line $OM$ meets the diagonal $AC$ at the point $P$.

Which of the following is equal to $\overrightarrow{OP}$?