# 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.

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- **Markdown URL:** https://flashcards.world/flashcards/sets/78cb1a8c-ce47-4767-a371-12d6e2f8476b.md
- **Level:** advanced
- **Languages:** en → en
- **Total Cards:** 32

## Flashcards (32 Cards)

| # | Front (Term / Prompt) | Back (Definition / Answer) | Example | Multiple-Choice Options |
| :--- | :--- | :--- | :--- | :--- |
| 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. |  |  |

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