How do you prove that three points A, B, and C are collinear (lie on a straight line) using GCSE vector geometry?

1) Find the vector expressions for AB and BC (or AC) in terms of the base vectors a and b and factorise fully; 2) Show that AB is a scalar multiple of BC (i.e., AB = k × BC), which proves AB and BC are parallel; 3) State that since they are parallel and share the common point B, the points A, B, and C are collinear.

How do you solve a quadratic inequality such as x² − 2x − 8 0?

1) Factorise to find the critical values (roots): (x − 4)(x + 2) = 0 → roots at x = −2 and x = 4; 2) Sketch the U-shaped parabola. For x² − 2x − 8 0 (above the x-axis), the solution is two separate regions: x 4.

How do you complete the square for x² + bx + c, and how do you read off the turning point of the parabola y = (x + p)² + q?

x² + bx + c = (x + b/2)² − (b/2)² + c. For y = (x + p)² + q, the minimum turning point (vertex) has coordinates (−p, q) and the line of symmetry is x = −p.

How do you rationalise the denominator of a surd fraction of the form a / (b + √c)? Rationalise 6 / (3 − √3).

Multiply the numerator and denominator by the conjugate (b − √c) to use the difference of two squares: (b + √c)(b − √c) = b² − c. For 6 / (3 − √3), multiply top and bottom by (3 + √3): [6(3 + √3)] / (9 − 3) = [6(3 + √3)] / 6 = 3 + √3.

How do you construct and interpret a Histogram for grouped continuous data with unequal class widths?

On a histogram, the vertical y-axis is Frequency Density, where Frequency Density = Frequency ÷ Class Width. The frequency of any class interval is represented by the AREA of the bar (Frequency = Frequency Density × Class Width), NOT the height.

How do you calculate the Upper and Lower Bounds of a division a / b or subtraction a − b when measurements are rounded?

First find Upper Bound (UB) and Lower Bound (LB) of each number (± half the rounding unit). Upper Bound of (a / b) = UB(a) ÷ LB(b); Lower Bound of (a / b) = LB(a) ÷ UB(b). Upper Bound of (a − b) = UB(a) − LB(b); Lower Bound of (a − b) = LB(a) − UB(b).

State the probability formulas for mutually exclusive events, independent events, and conditional probability without replacement.

Mutually exclusive events (cannot happen together): P(A or B) = P(A) + P(B). Independent events: P(A and B) = P(A) × P(B). Conditional probability: P(A given B) = P(A ∩ B) / P(B); in tree diagrams without replacement, reduce both the denominator and the relevant numerator by 1 on the second branch.

State the graph transformations of y = f(x) for: f(x) + a, f(x + a), −f(x), and f(−x).

y = f(x) + a: vertical translation by vector (0, a). y = f(x + a): horizontal translation by vector (−a, 0) (shifts left by a). y = −f(x): reflection in the x-axis. y = f(−x): reflection in the y-axis.

How do you use iterative formulas (e.g., x_{n+1} = ∛(10 − 2x_n)) and why does a sign change between f(a) and f(b) prove a root lies in (a, b)?

If f(x) is continuous and f(a) and f(b) have opposite signs (one positive, one negative), the graph must cross the x-axis (f(x) = 0) at least once between x = a and x = b. For iteration, enter x_0 into the calculator, press '=', then type the formula using the 'ANS' key to generate x_1, x_2, x_3 until values converge to the required decimal places.

How do you set up algebraic equations for Direct Proportion and Inverse Proportion?

If y is directly proportional to x (y ∝ x), write y = kx. If y is directly proportional to x² (y ∝ x²), write y = kx². If y is inversely proportional to x (y ∝ 1/x), write y = k / x. Substitute the given pair of values to find the constant of proportionality k.

State the 3 GCSE Circle Theorems involving tangents and chords, including the Alternate Segment Theorem.

1) The angle between a tangent and a radius at the point of contact is 90°; 2) Tangents drawn from an external point to a circle are equal in length; 3) Alternate Segment Theorem: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

In GCSE Higher Maths, how do you estimate the instantaneous rate of change (gradient) and distance travelled (area) from a curved speed–time graph?

To estimate the instantaneous acceleration at time t, draw a tangent to the curve at t and calculate its gradient (Δy / Δx). To estimate the distance travelled under a curve, divide the area into strips of equal width h and sum the areas of the trapezia using Area = ½(a + b)h (tell whether it is an overestimate or underestimate based on concavity).

How do you write an algebraic proof that the sum of the squares of any two consecutive odd numbers is always 2 more than a multiple of 8?

Let the two consecutive odd numbers be (2n − 1) and (2n + 1), where n is an integer. Expand and add their squares: (2n − 1)² + (2n + 1)² = (4n² − 4n + 1) + (4n² + 4n + 1) = 8n² + 2. Since 8n² is a multiple of 8 for all integers n, 8n² + 2 is always 2 more than a multiple of 8.

State the exact trigonometric values (without a calculator) for sin, cos, and tan of 0°, 30°, 45°, 60°, and 90°.

sin: 0°→0, 30°→1/2, 45°→√2/2, 60°→√3/2, 90°→1. cos: 0°→1, 30°→√3/2, 45°→√2/2, 60°→1/2, 90°→0. tan: 0°→0, 30°→1/√3 (or √3/3), 45°→1, 60°→√3, 90°→undefined.

State the Difference of Two Squares identity and factorise 9x² − 25y².

a² − b² = (a + b)(a − b). Therefore, 9x² − 25y² = (3x + 5y)(3x − 5y).

Describe the algebraic method for solving simultaneous equations where one is linear (y = 2x − 1) and one is quadratic (x² + y² = 10).

1) Rearrange the linear equation to make y (or x) the subject; 2) Substitute the linear expression into the quadratic equation: x² + (2x − 1)² = 10 → 5x² − 4x − 9 = 0; 3) Factorise or use the quadratic formula to find both x-values: (5x − 9)(x + 1) = 0 → x = 9/5 or x = −1; 4) Substitute each x-value back into the linear equation to find the corresponding y-values.

For mathematically similar 3D shapes with linear scale factor k, what are the area and volume scale factors?

If the ratio of corresponding lengths is 1 : k (linear scale factor = k), then the ratio of corresponding surface areas is 1 : k² (area scale factor = k²), and the ratio of volumes (or masses of uniform density) is 1 : k³ (volume scale factor = k³).

State the formulas for Arc Length and Sector Area of a circle with radius r and angle θ (in degrees).

Arc Length = (θ / 360°) × 2πr. Area of a Sector = (θ / 360°) × πr².

State the Quadratic Formula for solving ax² + bx + c = 0 and explain what the discriminant (b² − 4ac) tells you.

x = [−b ± √(b² − 4ac)] / (2a). If b² − 4ac > 0, there are 2 distinct real roots; if b² − 4ac = 0, there is 1 repeated root; if b² − 4ac < 0, there are no real roots.

State the formula for the nth term of a linear (arithmetic) sequence and the method for finding the nth term of a quadratic sequence.

Linear sequence: nth term = dn + (a − d), where d is the common first difference and a is the first term. Quadratic sequence (an² + bn + c): halve the constant second difference to find 'a', subtract an² from each term of the original sequence, and find the linear nth term (bn + c) of the remaining sequence.

State the 4 core GCSE Circle Theorems involving angles at the centre, angles in a semicircle, angles in the same segment, and cyclic quadrilaterals.

1) The angle subtended at the centre of a circle is twice the angle subtended at the circumference by the same arc; 2) The angle in a semicircle (subtended by a diameter) is always 90°; 3) Angles in the same segment are equal; 4) Opposite angles in a cyclic quadrilateral add up to 180°.

How do the gradients of parallel and perpendicular lines relate in coordinate geometry, and what is the equation of a circle centred at the origin?

Parallel lines have equal gradients (m1 = m2). Perpendicular lines have gradients whose product is −1 (m1 × m2 = −1, so m2 = −1 / m1). The equation of a circle with centre (0, 0) and radius r is x² + y² = r². The tangent to a circle at point P is always perpendicular to the radius OP.

State the fractional and negative laws of indices and evaluate 27^(-2/3).

a^(-n) = 1 / a^n and a^(m/n) = (ⁿ√a)^m. Therefore, 27^(-2/3) = 1 / (³√27)² = 1 / 3² = 1/9.

State the formulas for the Interior and Exterior angles of an n-sided polygon.

Sum of exterior angles of ANY convex polygon = 360° (each exterior angle of a regular n-gon = 360° / n). Sum of interior angles = (n − 2) × 180°. At any vertex, Interior angle + Exterior angle = 180°.

Given a function f(x), how do you find the composite function fg(x) and the inverse function f⁻¹(x)?

For fg(x), substitute the entire expression for g(x) in place of x inside f(x): fg(x) = f(g(x)). To find the inverse f⁻¹(x): write y = f(x), swap x and y, rearrange algebraically to make y the subject, and write as f⁻¹(x).

What are the four standard triangle congruence conditions (SSS, SAS, ASA, RHS)?

1) SSS (Side-Side-Side): all 3 corresponding sides are equal; 2) SAS (Side-Angle-Side): 2 sides and the included angle between them are equal; 3) ASA / AAS (Angle-Side-Angle): 2 angles and a corresponding side are equal; 4) RHS (Right-angle-Hypotenuse-Side): both triangles have a 90° angle, equal hypotenuses, and one other equal side.

How does the Capture-Recapture method (Petersen estimator) estimate the total population size N of animals in a habitat?

Capture a sample of M animals, mark them harmlessly, and release them. Later capture a second sample of C animals and count how many (R) are marked. Assuming no births/deaths/migration and random mixing: R / C = M / N, so Total Population N = (M × C) ÷ R.

What are the percentage multiplier formulas for Compound Interest/Growth, Depreciation, and Reverse Percentages?

Compound Growth/Depreciation over n years at r%: Final Amount = Initial × (1 ± r/100)^n. Reverse Percentage (finding original value before an r% increase/decrease): Original Value = Final Value ÷ (1 ± r/100).

State the surface area and volume formulas for a Sphere, a Cone, and a Cylinder.

Cylinder: Volume = πr²h, Curved Surface Area = 2πrh, Total SA = 2πr² + 2πrh. Sphere: Volume = (4/3)πr³, Surface Area = 4πr². Cone: Volume = (1/3)πr²h, Curved Surface Area = πrl (where l is slant height), Total SA = πr² + πrl.

How do you convert the recurring decimal 0.272727... (0.2̇7̇) into a simplified fraction using algebra?

Let x = 0.272727... Since 2 digits repeat, multiply by 100: 100x = 27.272727... Subtract x from 100x: 99x = 27 → x = 27/99 = 3/11.

State the Sine Rule, the Cosine Rule, and the Trigonometric Area of a Triangle formula for any non-right-angled triangle.

Sine Rule: a / sin A = b / sin B = c / sin C (for sides) or sin A / a = sin B / b (for angles). Cosine Rule: a² = b² + c² − 2bc cos A (or cos A = (b² + c² − a²) / 2bc). Area of a Triangle: Area = ½ab sin C.

How do you plot a Cumulative Frequency curve and use it to construct a Box Plot?

Plot cumulative frequencies against the UPPER boundary of each class interval and join with a smooth curve. Read off the Median (Q2 at 50% of N), Lower Quartile (Q1 at 25% of N), and Upper Quartile (Q3 at 75% of N). Interquartile Range (IQR) = Q3 − Q1. Plot the box plot using Minimum, Q1, Median, Q3, and Maximum.

GCSE Mathematics (Higher Tier): Algebra, Quadratics, Surds, Circle Theorems, Trigonometry, Vectors & Probability

Grade 7–9 GCSE and IGCSE Higher Tier Mathematics deck covering laws of indices, surd rationalisation, completing the square, quadratic formula, function notation, circle theorems, sine/cosine rules, vector geometry proofs, histograms, and conditional probability.

Level: advanced
Languages: en → en
Cards: 32
Markdown: GCSE Mathematics (Higher Tier): Algebra, Quadratics, Surds, Circle Theorems, Trigonometry, Vectors & Probability (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	How do you prove that three points A, B, and C are collinear (lie on a straight line) using GCSE vector geometry?	1) Find the vector expressions for AB and BC (or AC) in terms of the base vectors a and b and factorise fully; 2) Show that AB is a scalar multiple of BC (i.e., AB = k × BC), which proves AB and BC are parallel; 3) State that since they are parallel and share the common point B, the points A, B, and C are collinear.
2	How do you solve a quadratic inequality such as x² − 2x − 8 < 0 vs x² − 2x − 8 > 0?	1) Factorise to find the critical values (roots): (x − 4)(x + 2) = 0 → roots at x = −2 and x = 4; 2) Sketch the U-shaped parabola. For x² − 2x − 8 < 0 (below the x-axis), the solution is a single connected region: −2 < x < 4. For x² − 2x − 8 > 0 (above the x-axis), the solution is two separate regions: x < −2 or x > 4.
3	How do you complete the square for x² + bx + c, and how do you read off the turning point of the parabola y = (x + p)² + q?	x² + bx + c = (x + b/2)² − (b/2)² + c. For y = (x + p)² + q, the minimum turning point (vertex) has coordinates (−p, q) and the line of symmetry is x = −p.
4	How do you rationalise the denominator of a surd fraction of the form a / (b + √c)? Rationalise 6 / (3 − √3).	Multiply the numerator and denominator by the conjugate (b − √c) to use the difference of two squares: (b + √c)(b − √c) = b² − c. For 6 / (3 − √3), multiply top and bottom by (3 + √3): [6(3 + √3)] / (9 − 3) = [6(3 + √3)] / 6 = 3 + √3.
5	How do you construct and interpret a Histogram for grouped continuous data with unequal class widths?	On a histogram, the vertical y-axis is Frequency Density, where Frequency Density = Frequency ÷ Class Width. The frequency of any class interval is represented by the AREA of the bar (Frequency = Frequency Density × Class Width), NOT the height.
6	How do you calculate the Upper and Lower Bounds of a division a / b or subtraction a − b when measurements are rounded?	First find Upper Bound (UB) and Lower Bound (LB) of each number (± half the rounding unit). Upper Bound of (a / b) = UB(a) ÷ LB(b); Lower Bound of (a / b) = LB(a) ÷ UB(b). Upper Bound of (a − b) = UB(a) − LB(b); Lower Bound of (a − b) = LB(a) − UB(b).
7	State the probability formulas for mutually exclusive events, independent events, and conditional probability without replacement.	Mutually exclusive events (cannot happen together): P(A or B) = P(A) + P(B). Independent events: P(A and B) = P(A) × P(B). Conditional probability: P(A given B) = P(A ∩ B) / P(B); in tree diagrams without replacement, reduce both the denominator and the relevant numerator by 1 on the second branch.
8	State the graph transformations of y = f(x) for: f(x) + a, f(x + a), −f(x), and f(−x).	y = f(x) + a: vertical translation by vector (0, a). y = f(x + a): horizontal translation by vector (−a, 0) (shifts left by a). y = −f(x): reflection in the x-axis. y = f(−x): reflection in the y-axis.
9	How do you use iterative formulas (e.g., x_{n+1} = ∛(10 − 2x_n)) and why does a sign change between f(a) and f(b) prove a root lies in (a, b)?	If f(x) is continuous and f(a) and f(b) have opposite signs (one positive, one negative), the graph must cross the x-axis (f(x) = 0) at least once between x = a and x = b. For iteration, enter x_0 into the calculator, press '=', then type the formula using the 'ANS' key to generate x_1, x_2, x_3 until values converge to the required decimal places.
10	How do you set up algebraic equations for Direct Proportion and Inverse Proportion?	If y is directly proportional to x (y ∝ x), write y = kx. If y is directly proportional to x² (y ∝ x²), write y = kx². If y is inversely proportional to x (y ∝ 1/x), write y = k / x. Substitute the given pair of values to find the constant of proportionality k.
11	State the 3 GCSE Circle Theorems involving tangents and chords, including the Alternate Segment Theorem.	1) The angle between a tangent and a radius at the point of contact is 90°; 2) Tangents drawn from an external point to a circle are equal in length; 3) Alternate Segment Theorem: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
12	In GCSE Higher Maths, how do you estimate the instantaneous rate of change (gradient) and distance travelled (area) from a curved speed–time graph?	To estimate the instantaneous acceleration at time t, draw a tangent to the curve at t and calculate its gradient (Δy / Δx). To estimate the distance travelled under a curve, divide the area into strips of equal width h and sum the areas of the trapezia using Area = ½(a + b)h (tell whether it is an overestimate or underestimate based on concavity).
13	How do you write an algebraic proof that the sum of the squares of any two consecutive odd numbers is always 2 more than a multiple of 8?	Let the two consecutive odd numbers be (2n − 1) and (2n + 1), where n is an integer. Expand and add their squares: (2n − 1)² + (2n + 1)² = (4n² − 4n + 1) + (4n² + 4n + 1) = 8n² + 2. Since 8n² is a multiple of 8 for all integers n, 8n² + 2 is always 2 more than a multiple of 8.
14	State the exact trigonometric values (without a calculator) for sin, cos, and tan of 0°, 30°, 45°, 60°, and 90°.	sin: 0°→0, 30°→1/2, 45°→√2/2, 60°→√3/2, 90°→1. cos: 0°→1, 30°→√3/2, 45°→√2/2, 60°→1/2, 90°→0. tan: 0°→0, 30°→1/√3 (or √3/3), 45°→1, 60°→√3, 90°→undefined.
15	State the Difference of Two Squares identity and factorise 9x² − 25y².	a² − b² = (a + b)(a − b). Therefore, 9x² − 25y² = (3x + 5y)(3x − 5y).
16	Describe the algebraic method for solving simultaneous equations where one is linear (y = 2x − 1) and one is quadratic (x² + y² = 10).	1) Rearrange the linear equation to make y (or x) the subject; 2) Substitute the linear expression into the quadratic equation: x² + (2x − 1)² = 10 → 5x² − 4x − 9 = 0; 3) Factorise or use the quadratic formula to find both x-values: (5x − 9)(x + 1) = 0 → x = 9/5 or x = −1; 4) Substitute each x-value back into the linear equation to find the corresponding y-values.
17	For mathematically similar 3D shapes with linear scale factor k, what are the area and volume scale factors?	If the ratio of corresponding lengths is 1 : k (linear scale factor = k), then the ratio of corresponding surface areas is 1 : k² (area scale factor = k²), and the ratio of volumes (or masses of uniform density) is 1 : k³ (volume scale factor = k³).
18	State the formulas for Arc Length and Sector Area of a circle with radius r and angle θ (in degrees).	Arc Length = (θ / 360°) × 2πr. Area of a Sector = (θ / 360°) × πr².
19	State the Quadratic Formula for solving ax² + bx + c = 0 and explain what the discriminant (b² − 4ac) tells you.	x = [−b ± √(b² − 4ac)] / (2a). If b² − 4ac > 0, there are 2 distinct real roots; if b² − 4ac = 0, there is 1 repeated root; if b² − 4ac < 0, there are no real roots.
20	State the formula for the nth term of a linear (arithmetic) sequence and the method for finding the nth term of a quadratic sequence.	Linear sequence: nth term = dn + (a − d), where d is the common first difference and a is the first term. Quadratic sequence (an² + bn + c): halve the constant second difference to find 'a', subtract an² from each term of the original sequence, and find the linear nth term (bn + c) of the remaining sequence.
21	State the 4 core GCSE Circle Theorems involving angles at the centre, angles in a semicircle, angles in the same segment, and cyclic quadrilaterals.	1) The angle subtended at the centre of a circle is twice the angle subtended at the circumference by the same arc; 2) The angle in a semicircle (subtended by a diameter) is always 90°; 3) Angles in the same segment are equal; 4) Opposite angles in a cyclic quadrilateral add up to 180°.
22	How do the gradients of parallel and perpendicular lines relate in coordinate geometry, and what is the equation of a circle centred at the origin?	Parallel lines have equal gradients (m1 = m2). Perpendicular lines have gradients whose product is −1 (m1 × m2 = −1, so m2 = −1 / m1). The equation of a circle with centre (0, 0) and radius r is x² + y² = r². The tangent to a circle at point P is always perpendicular to the radius OP.
23	State the fractional and negative laws of indices and evaluate 27^(-2/3).	a^(-n) = 1 / a^n and a^(m/n) = (ⁿ√a)^m. Therefore, 27^(-2/3) = 1 / (³√27)² = 1 / 3² = 1/9.
24	State the formulas for the Interior and Exterior angles of an n-sided polygon.	Sum of exterior angles of ANY convex polygon = 360° (each exterior angle of a regular n-gon = 360° / n). Sum of interior angles = (n − 2) × 180°. At any vertex, Interior angle + Exterior angle = 180°.
25	Given a function f(x), how do you find the composite function fg(x) and the inverse function f⁻¹(x)?	For fg(x), substitute the entire expression for g(x) in place of x inside f(x): fg(x) = f(g(x)). To find the inverse f⁻¹(x): write y = f(x), swap x and y, rearrange algebraically to make y the subject, and write as f⁻¹(x).
26	What are the four standard triangle congruence conditions (SSS, SAS, ASA, RHS)?	1) SSS (Side-Side-Side): all 3 corresponding sides are equal; 2) SAS (Side-Angle-Side): 2 sides and the included angle between them are equal; 3) ASA / AAS (Angle-Side-Angle): 2 angles and a corresponding side are equal; 4) RHS (Right-angle-Hypotenuse-Side): both triangles have a 90° angle, equal hypotenuses, and one other equal side.
27	How does the Capture-Recapture method (Petersen estimator) estimate the total population size N of animals in a habitat?	Capture a sample of M animals, mark them harmlessly, and release them. Later capture a second sample of C animals and count how many (R) are marked. Assuming no births/deaths/migration and random mixing: R / C = M / N, so Total Population N = (M × C) ÷ R.
28	What are the percentage multiplier formulas for Compound Interest/Growth, Depreciation, and Reverse Percentages?	Compound Growth/Depreciation over n years at r%: Final Amount = Initial × (1 ± r/100)^n. Reverse Percentage (finding original value before an r% increase/decrease): Original Value = Final Value ÷ (1 ± r/100).
29	State the surface area and volume formulas for a Sphere, a Cone, and a Cylinder.	Cylinder: Volume = πr²h, Curved Surface Area = 2πrh, Total SA = 2πr² + 2πrh. Sphere: Volume = (4/3)πr³, Surface Area = 4πr². Cone: Volume = (1/3)πr²h, Curved Surface Area = πrl (where l is slant height), Total SA = πr² + πrl.
30	How do you convert the recurring decimal 0.272727... (0.2̇7̇) into a simplified fraction using algebra?	Let x = 0.272727... Since 2 digits repeat, multiply by 100: 100x = 27.272727... Subtract x from 100x: 99x = 27 → x = 27/99 = 3/11.
31	State the Sine Rule, the Cosine Rule, and the Trigonometric Area of a Triangle formula for any non-right-angled triangle.	Sine Rule: a / sin A = b / sin B = c / sin C (for sides) or sin A / a = sin B / b (for angles). Cosine Rule: a² = b² + c² − 2bc cos A (or cos A = (b² + c² − a²) / 2bc). Area of a Triangle: Area = ½ab sin C.
32	How do you plot a Cumulative Frequency curve and use it to construct a Box Plot?	Plot cumulative frequencies against the UPPER boundary of each class interval and join with a smooth curve. Read off the Median (Q2 at 50% of N), Lower Quartile (Q1 at 25% of N), and Upper Quartile (Q3 at 75% of N). Interquartile Range (IQR) = Q3 − Q1. Plot the box plot using Minimum, Q1, Median, Q3, and Maximum.