Pearson Edexcel GCSE (9-1) Mathematics Higher Tier Flashcards

Comprehensive flashcards covering key problems, solutions, formulas, and mark scheme abbreviations from Pearson Edexcel GCSE (9-1) Mathematics Higher Tier Papers 1H (Non-Calculator) and 2H (Calculator).

What do the mark scheme abbreviations M, P, A, C, B, and oe stand for in Pearson Edexcel GCSE Mathematics marking?

M = method mark, P = process mark, A = accuracy mark, C = communication mark, B = unconditional accuracy mark, oe = or equivalent.

What do the mark scheme abbreviations cao, ft, sc, dep, indep, awrt, and isw stand for in Pearson Edexcel GCSE Mathematics marking?

cao = correct answer only, ft = follow through, sc = special case, dep = dependent, indep = independent, awrt = answer which rounds to, isw = ignore subsequent working.

How do you express $500$ as a product of powers of its prime factors?

$$500 = 2^2 \times 5^3$$

What is the result of working out $1\frac{3}{5} + 2\frac{1}{4}$ expressed as a mixed number?

$$3\frac{17}{20}$$

How do you show that $2\frac{2}{3} \div 6 = \frac{4}{9}$?

Convert $2\frac{2}{3}$ to the improper fraction $\frac{8}{3}$, then divide by $6$ (multiply by $\frac{1}{6}$) to obtain $\frac{8}{18}$, which simplifies to $\frac{4}{9}$.

How do you simplify $(2^{-5} \times 2^8)^2$ as a power of $2$?

First simplify inside parentheses: $2^{-5+8} = 2^3$. Then apply the outer power: $(2^3)^2 = 2^{3 \times 2} = 2^6$.

What is the product of $0.004 \times 0.32$?

$0.00128$ (or $1.28 \times 10^{-3}$)

A car factory surveyed $80$ people on car models: A ($23$), B ($15$), C ($30$), D ($12$). If the factory makes $40000$ cars next year, how many model B cars should it make?

$$\frac{15}{80} \times 40000 = 7500\,\text{model B cars}$$

Given the ratios $a : b = 1 : 3$ and $b : c = 6 : 5$, what is the combined ratio $a : b : c$?

Multiply $a : b$ by $2$ to get $a : b = 2 : 6$. Since $b : c = 6 : 5$, the combined ratio is $2 : 6 : 5$.

If $a : b : c = 2 : 6 : 5$, express $a$ as a fraction of the total of the three numbers $a$, $b$, and $c$.

$$\frac{2}{2 + 6 + 5} = \frac{2}{13}$$

Given $n = 2m$ and $p = 5n$, what is the ratio $m : p$ in its simplest form?

Since $p = 5 \times (2m) = 10m$, the ratio $m : p = m : 10m = 1 : 10$.

A storage tank exerts a force of $10000\,\text{newtons}$ on a base area of $4\,\text{m} \times 2\,\text{m}$. What is the pressure on the ground?

$$\text{pressure} = \frac{\text{force}}{\text{area}} = \frac{10000}{4 \times 2} = \frac{10000}{8} = 1250\,\text{newtons/m}^2$$

Two numbers $m$ and $n$ are such that $m$ is a multiple of $5$, $n$ is an even number, and the HCF of $m$ and $n$ is $7$. What is a possible pair of values for $m$ and $n$?

$m = 35$ and $n = 14$ (or $m = 35$ and $n = 28$)

What are the values of $y$ for $y = 6x - x^3$ when $x = -2$, $x = -1$, $x = 0$, and $x = 1$?

For $x = -2$, $y = -4$; for $x = -1$, $y = -5$; for $x = 0$, $y = 0$; for $x = 1$, $y = 5$.

A biased 5-sided spinner spun $40$ times yields frequencies: score 1: $6$, score 2: $8$, score 3: $9$, score 4: $7$, score 5: $10$. What is the estimated probability of getting a score of 5 on two consecutive spins?

Probability of 5 on one spin is $\frac{10}{40} = \frac{1}{4}$. For two spins: $\left(\frac{1}{4}\right)^2 = \frac{1}{16}$.

What single transformation maps shape P onto shape Q shown on the grid?

Enlargement, scale factor $\frac{1}{3}$, centre $(0, 2)$

What are the solutions to the simultaneous equations $5x + 2y = 11$ and $4x + 3y = 6$?

$x = 3$, $y = -2$

If $p$ is inversely proportional to $t$ and $p = 5$ when $t = 20$, what is the missing value of $t$ when $p = 1$ and the missing value of $p$ when $t = 25$?

Since $p \times t = 100$: when $p = 1$, $t = 100$; when $t = 25$, $p = 4$.

A sector of a circle with radius $18\,\text{cm}$ has an arc length of $4\pi\,\text{cm}$. What is the angle $x^\circ$ of the sector?

Using $\text{arc length} = \frac{x}{360} \times 2\pi r$: $4\pi = \frac{x}{360} \times 36\pi \implies x = 40^\circ$.

How do you evaluate $\left(\frac{8}{27}\right)^{\frac{4}{3}}$?

$$\left(\sqrt[3]{\frac{8}{27}}\right)^4 = \left(\frac{2}{3}\right)^4 = \frac{16}{81}$$

What circle theorem proves that angle $ABC = \frac{1}{2}x^\circ$ when $A$ and $B$ are on a circle with centre $O$, $DBC$ is a tangent at $B$, and angle $AOB = x^\circ$?

The tangent to a circle is perpendicular to the radius (or angle at centre is twice angle at circumference / alternate segment theorem).

What is the exact solution to $\frac{1}{x} - \frac{1}{x + 1} = 4$ in the form $a \pm b\sqrt{2}$?

$$-\frac{1}{2} \pm \frac{1}{2}\sqrt{2}$$

Alfie has $11$ cards ($3$ blue, $7$ green, $1$ white). What is the probability that he takes $2$ cards of different colours at random without replacement?

$\frac{62}{110}$ (or $\frac{31}{55}$)

What are the coordinates of the minimum point P on the curve $y = \cos(x^\circ)$ for $0^\circ \le x^\circ \le 360^\circ$?

$$(180, -1)$$

If a solid sphere of radius $r$ and a solid cone of radius $r$ and height $h$ have equal volumes, what is the ratio $r : h$ in simplest form?

$$\frac{4}{3}\pi r^3 = \frac{1}{3}\pi r^2 h \implies 4r = h \implies r : h = 1 : 4$$

What is the coordinate of the outlier in the rainfall versus sunshine scatter graph?

$$(2, 1)$$

What is the $n\text{th}$ term expression for the arithmetic sequence $7, 13, 19, 25, 31$?

$$6n + 1$$

How do you calculate the side $x$ in a right triangle with hypotenuse $14.5\,\text{cm}$ and angle $53^\circ$ adjacent to $x$?

$$x = 14.5 \times \cos(53^\circ) \approx 8.73\,\text{cm}$$

Ella invests $£7000$ for $2\,\text{years}$ with compound interest rates of $3\%$ in year 1 and $1.5\%$ in year 2. What is the value of her investment at the end of $2\,\text{years}$?

$$7000 \times 1.03 \times 1.015 = £7318.15$$

What are the y-intercept and turning point of the quadratic graph $y = x^2 - 6x + 4$?

y-intercept is $4$ (or $(0, 4)$), and turning point is $(3, -5)$.

Chanda buys a necklace for $£120$ and sells it for $£135$. What is her percentage profit?

$$\frac{135 - 120}{120} \times 100 = 12.5\%$$

Are the lines $y = \frac{1}{2}x - 6$ and $6y = 3x + 7$ parallel?

Yes, because dividing $6y = 3x + 7$ by $6$ yields $y = \frac{1}{2}x + \frac{7}{6}$, showing both lines have a gradient of $\frac{1}{2}$.

A car's value depreciated by $23\%$ in year 1 and $19\%$ in year 2 to $£10914.75$. What was its original value?

$$\frac{10914.75}{(1 - 0.23) \times (1 - 0.19)} = \frac{10914.75}{0.77 \times 0.81} = £17500$$

What is the distance between point $A(-7, 6)$ and point $B(8, -5)$ on a centimetre grid to 1 decimal place?

$$\text{AB} = \sqrt{(8 - (-7))^2 + (-5 - 6)^2} = \sqrt{15^2 + (-11)^2} = \sqrt{346} \approx 18.6\,\text{cm}$$

How do you prove algebraically that $1.06\dot{2}$ equals $1\frac{14}{225}$?

Let $x = 1.06222...$. Then $1000x = 1062.222...$ and $100x = 106.222...$. Subtracting gives $900x = 956 \implies x = \frac{956}{900} = 1\frac{14}{225}$.

What is the area of a triangle with sides $11.2\,\text{cm}$ and $4.3\,\text{cm}$ and an included angle of $118^\circ$ to 3 significant figures?

$$\text{Area} = \frac{1}{2} \times 11.2 \times 4.3 \times \sin(118^\circ) \approx 21.3\,\text{cm}^2$$

What are the solutions to the quadratic equation $6x^2 + 5x - 6 = 0$?

Factorising gives $(2x + 3)(3x - 2) = 0$, so $x = -\frac{3}{2}$ or $x = \frac{2}{3}$.

Given $f(x) = \sqrt[3]{x}$, $g(x) = 2x + 3$, and $h(x) = f(g(x))$, what is $h^{-1}(x)$?

$h(x) = \sqrt[3]{2x + 3}$. Setting $y = \sqrt[3]{2x + 3} \implies y^3 = 2x + 3 \implies h^{-1}(x) = \frac{x^3 - 3}{2}$.

Find the equation of the tangent line to the circle $x^2 + y^2 = 12.25$ at point $P(2.1, 2.8)$ in the form $ax + by = c$ with integer coefficients.

Gradient of radius = $\frac{2.8}{2.1} = \frac{4}{3}$, so gradient of tangent = $-\frac{3}{4}$. Substituting $P(2.1, 2.8)$ into $y - 2.8 = -\frac{3}{4}(x - 2.1)$ gives $6x + 8y = 35$.