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Calculator guides

Casio fx-CG50

A-level Maths & Further Maths

Graphical calculator · organised by the exam question that needs it.

The graphical calculator most A-level students sit the exam with — graphs, the equation solver, numerical calculus, statistics and distributions, organised by the exam question that needs them.

Steps are for the fx-CG50 (they also match the older fx-9750GII/9860GII menus). Keys shown as F1–F6 are the on-screen soft-menu keys along the bottom of the screen.

Graphs

Roots, turning points and intercepts of a curve

When to use: The question shows a graph, or says 'solve graphically', or asks where a curve crosses the axes or for its maximum/minimum.

  1. 1MENU → Graph — Open the Graph app.
  2. 2Type the curve into Y1, then EXE
  3. 3F6 (DRAW) — Plot it. Set the window first with SHIFT F3 (V-Window) if the feature is off-screen.
  4. 4SHIFT F5 (G-Solv) — Opens the graph-solve menu.
  5. 5F1 ROOT · F2 MAX · F3 MIN · F4 Y-ICPT · F5 ISCT — Pick what you need; press → to step to the next root/intersection.

Find the roots of y = x² − 5x + 6.

MENU → GraphY1 = X² − 5X + 6, EXEF6 (DRAW)SHIFT F5 (G-Solv)F1 (ROOT)→ for the next root

Answer: x = 2 and x = 3

  • ISCT (intersection) needs two graphs entered — put the second curve or line in Y2.
  • If a root is missing, widen the V-Window (SHIFT F3) so it is actually on screen.
  • 'Solve graphically' means the graph IS the method — but if it says 'solve algebraically' the calculator answer earns nothing without working.

Plot a parametric curve

When to use: A parametric question gives x and y separately in terms of t and asks for a sketch, an intersection or a check on the curve's shape.

  1. 1MENU → Graph
  2. 2F3 (TYPE) → F3 (Param) — The graph list changes from Y1 to the pair Xt1 and Yt1.
  3. 3Enter Xt1 and Yt1, pressing EXE after each
  4. 4SHIFT F3 (V-Window) → Tθ min / Tθ max / Tθ pitch — Set the parameter interval and step as well as a suitable x-y window.
  5. 5F6 (DRAW) — Use SHIFT F1 (TRACE) to follow the curve and read parameter values.

Sketch x = 3 sin t, y = 3 cos t for 0 ≤ t ≤ 2π.

MENU → GraphF3 (TYPE) → F3 (Param)Xt1 = 3 sin T, Yt1 = 3 cos TTθ min = 0, Tθ max = 2πF6 (DRAW)

Answer: The circle x² + y² = 9, starting at (0, 3) when t = 0.

  • Use the stated parameter interval: a different interval can draw extra or missing parts of the curve.
  • The plot checks the shape, but eliminate the parameter or show exact coordinate working when the question asks for it.

Equations & algebra

Solve a polynomial or simultaneous equations

When to use: Solve a quadratic/cubic/quartic, or a set of simultaneous linear equations, and you only need the answer (or a check).

  1. 1MENU → Equation
  2. 2F1 (Simultaneous) or F2 (Polynomial)
  3. 3Choose the number of unknowns / the degree, then EXE
  4. 4Type each coefficient, pressing EXE after each
  5. 5F1 (SOLVE) — Reads out every root, including complex ones (Further Maths).

Solve x² − 5x + 6 = 0.

MENU → EquationF2 (Polynomial)Degree 2, EXE1 EXE, −5 EXE, 6 EXEF1 (SOLVE)

Answer: x = 3 and x = 2

  • Polynomial mode handles degrees 2–6 and returns complex roots — useful for FM.
  • For a 'solve algebraically'/'show that' question this is a check only; the marks are for the method (factorising, quadratic formula, elimination).

Solve any equation numerically (SolveN)

When to use: An equation that will not factorise — trig, exponential, or mixed — and you need a decimal root (or to check one).

  1. 1MENU → Run-Matrix
  2. 2OPTN → F4 (CALC)
  3. 3Choose SolveN(
  4. 4Type the equation and the variable, e.g. SolveN(2^X = 10, X) — Use = from the CALC/relation menu.
  5. 5EXE — Returns every root it finds in the default range.
  • SolveN lists all roots in range; Solve( finds the one nearest a guess you give.
  • Give a decimal to the accuracy the question asks for — the calculator's full-precision value is not automatically the required rounding.

Calculus

Definite integral / area under a curve

When to use: Evaluate a definite integral, or find the area between a curve and the x-axis over a given interval.

  1. 1MENU → Run-Matrix
  2. 2OPTN → F4 (CALC) → ∫dx(
  3. 3Type ∫dx(expression, lower, upper) — e.g. ∫dx(X², 0, 2)
  4. 4EXE
  5. 5Or, on a drawn graph: SHIFT F5 (G-Solv) → ∫dx — Shades the area and gives its value.

Evaluate the integral of x² from 0 to 2.

MENU → Run-MatrixOPTN → F4 (CALC) → ∫dx(∫dx(X², 0, 2)EXE

Answer: 8/3 ≈ 2.667

  • This gives a number, not the algebra — a 'find the integral' question still needs the integrated expression shown for the method marks.
  • Area below the x-axis comes out negative; split the integral at the roots if the question wants total area.

Gradient at a point (numerical derivative)

When to use: Check the gradient of a curve at a particular x-value, or the value of dy/dx you found by differentiating.

  1. 1MENU → Run-Matrix
  2. 2OPTN → F4 (CALC) → d/dx(
  3. 3Type d/dx(expression, x-value) — e.g. d/dx(X³, 2)
  4. 4EXE

Find the gradient of y = x³ at x = 2.

MENU → Run-MatrixOPTN → F4 (CALC) → d/dx(d/dx(X³, 2)EXE

Answer: 12

  • It gives one gradient value — you still differentiate by hand for a 'find dy/dx' question.
  • Useful to check a normal/tangent gradient before you write the equation of the line.

Statistics

Mean, standard deviation and quartiles

When to use: A data set (or frequency table) and the question asks for the mean, standard deviation, median or quartiles.

  1. 1MENU → Statistics
  2. 2Type the data into List 1 (put frequencies in List 2)
  3. 3F2 (CALC) → F6 (SET) — Set 1Var XList: List1 and 1Var Freq: List2 (or 1) — then EXIT.
  4. 4F2 (CALC) → F1 (1-VAR) — Reads x̄ (mean), σx (population sd), sx (sample sd), Σx, min, Q1, Med, Q3, max.

Find the mean and population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.

MENU → StatisticsEnter the eight values in List 1F2 (CALC) → F6 (SET), XList List1, Freq 1, EXITF2 (CALC) → F1 (1-VAR)

Answer: mean x̄ = 5, population sd σx = 2 (sample sd sx ≈ 2.14)

  • σx is the population standard deviation; sx is the sample one — read the question for which is wanted.
  • For a grouped table, use the mid-interval values in List 1 and the frequencies in List 2, with Freq set to List2.

Regression line and correlation (PMCC)

When to use: Bivariate data where the question asks for the equation of the regression line y = a + bx, or the product-moment correlation coefficient r.

  1. 1MENU → Statistics
  2. 2x-values in List 1, y-values in List 2
  3. 3F2 (CALC) → F6 (SET) — Set 2Var XList: List1, YList: List2 — then EXIT.
  4. 4F2 (CALC) → F3 (REG) → F1 (X) — Linear regression aX + b; the screen also shows r and r².
  • r near ±1 is strong correlation; r near 0 is weak — quote it to the accuracy asked.
  • Correlation is not causation, and don't extrapolate beyond the data range — both are common written-mark traps the calculator can't save you from.

Distributions

Binomial probabilities

When to use: X ~ B(n, p): 'exactly r' uses the probability, 'at most r' or 'fewer than' uses the cumulative version.

  1. 1MENU → Statistics
  2. 2F5 (DIST) → F5 (BINM)
  3. 3F1 (Bpd) for P(X = r), or F2 (Bcd) for P(X ≤ r)
  4. 4Set Data: Variable, then enter x = r, Numtrial = n, p — EXE to compute.

X ~ B(10, 0.5). Find P(X ≤ 3).

MENU → StatisticsF5 (DIST) → F5 (BINM)F2 (Bcd)Variable, x = 3, Numtrial = 10, p = 0.5, EXE

Answer: 0.171875

  • Bpd = exactly r; Bcd = r or fewer. For P(X ≥ r) use 1 − P(X ≤ r − 1).
  • For P(a ≤ X ≤ b) do Bcd(b) − Bcd(a − 1); watch the strict-vs-inclusive wording.

Normal distribution probabilities and inverse normal

When to use: X ~ N(μ, σ²): a probability like P(X < a) or P(a < X < b), or 'find the value exceeded by 5%' (inverse normal).

  1. 1MENU → Statistics
  2. 2F5 (DIST) → F1 (NORM)
  3. 3F2 (Ncd) for a probability, or F3 (InvN) for a value
  4. 4Enter Lower, Upper, σ, μ — For a one-sided tail use −1×10⁹⁹ for −∞ or 1×10⁹⁹ for +∞ (the ×10ˣ / EXP key).

X ~ N(50, 4²). Find P(X < 54).

MENU → StatisticsF5 (DIST) → F1 (NORM)F2 (Ncd)Lower −1×10⁹⁹, Upper 54, σ = 4, μ = 50, EXE

Answer: ≈ 0.8413

  • Enter σ (the standard deviation), not σ² (the variance) — the classic slip.
  • InvN with Tail: Left gives the value below which the given probability lies; switch Tail for 'greater than' problems.

Matrices (Further Maths)

Determinant and inverse of a matrix

When to use: Further Maths: evaluate a determinant, invert a matrix, or multiply matrices for a transformation or a system of equations.

  1. 1MENU → Run-Matrix
  2. 2F3 (▸MAT/VCT) — Define Mat A: set the dimensions, enter each entry, EXIT back to Run-Matrix.
  3. 3For the inverse: MatA, then the x⁻¹ key, EXE
  4. 4For the determinant: OPTN → F2 (MAT/VCT) → Det → MatA, EXE
  • A zero determinant means the matrix is singular (no inverse) — a mapping that collapses area to zero.
  • Show the determinant/adjugate working for a 'find the inverse' method mark; the calculator value confirms it.

Sequences & series

Evaluate a finite sum with Σ

When to use: A sequence or series question gives a term in r and asks for the sum between two integer limits.

  1. 1MENU → Run-Matrix
  2. 2OPTN → F4 (CALC) → F6 (g) → F3 (Σ)
  3. 3Enter Σ(expression, variable, lower limit, upper limit)
  4. 4EXE

Evaluate Σ(3r − 1) from r = 1 to r = 20.

MENU → Run-MatrixOPTN → F4 (CALC) → F6 (g) → F3 (Σ)Σ(3R − 1, R, 1, 20)EXE

Answer: 610

  • Check that the lower and upper limits match the question; an index starting at 0 changes the sum.
  • Use Σ to verify arithmetic, but show the arithmetic/geometric-series formula when method marks are available.

Build a sequence and cumulative-sum table

When to use: A sequence question asks for particular terms, a running total, or the first n for which a sum passes a threshold.

  1. 1MENU → Recursion
  2. 2F3 (TYPE) → F1 (aₙ)
  3. 3Enter the explicit nth-term formula, then EXE
  4. 4SHIFT MENU (SET UP) → Σ Display: On — This adds a cumulative-sum column to the table.
  5. 5F5 (SET) → choose Start and End → F6 (TABLE)

For aₙ = 3 × 2ⁿ⁻¹, find a₆ and the sum of the first six terms.

MENU → RecursionF3 (TYPE) → F1 (aₙ)aₙ = 3 × 2^(n − 1)Σ Display: OnF5 (SET): Start 1, End 6F6 (TABLE)

Answer: a₆ = 96 and S₆ = 189

  • Match Start to the sequence's indexing: a formula beginning at n = 1 should not be tabulated from 0.
  • For a proof or 'show that' question, use the table as a check and write the algebraic argument separately.

Numerical methods

Generate fixed-point iterations

When to use: A numerical-methods question gives an iteration such as xₙ₊₁ = g(xₙ) and asks for several iterates or evidence of convergence.

  1. 1MENU → Recursion
  2. 2F3 (TYPE) → F2 (aₙ₊₁)
  3. 3Enter the recurrence in aₙ, then EXE — Use the on-screen n/aₙ variable menu rather than typing an ordinary letter a.
  4. 4F5 (SET) → set Start, End and a₀
  5. 5F6 (TABLE)

Starting with a₀ = 1, use aₙ₊₁ = cos(aₙ) to find a₅ (radians).

Set Angle: RadianMENU → RecursionF3 (TYPE) → F2 (aₙ₊₁)aₙ₊₁ = cos(aₙ)F5 (SET): Start 0, End 5, a₀ = 1F6 (TABLE)

Answer: a₅ ≈ 0.701369

  • Set radians or degrees to match the question before a trigonometric iteration.
  • Keep the calculator's full stored values between iterations; only round the requested final value.

Complex numbers (Further Maths)

Calculate with complex numbers

When to use: Further Maths: check arithmetic with a + bi, or find a complex number's modulus, argument or conjugate.

  1. 1MENU → Run-Matrix
  2. 2SHIFT MENU (SET UP) → Complex Mode → a+bi — Choose r∠θ instead when a polar-form answer is required.
  3. 3OPTN → F3 (COMPLEX) → F1 (i) — Use the calculator's imaginary-unit symbol, not the variable I.
  4. 4Enter the expression with brackets, then EXE

Evaluate (3 + 4i) ÷ (1 − i) in the form a + bi.

MENU → Run-MatrixComplex Mode: a+biOPTN → F3 (COMPLEX) for i(3 + 4i) ÷ (1 − i)EXE

Answer: −1/2 + 7i/2 (that is, −0.5 + 3.5i)

  • Set the angle unit before finding an argument; the same complex number has degree and radian arguments.
  • Use the result to check your rationalisation, but show the conjugate multiplication when the method is assessed.

Vectors (Further Maths)

Store vectors and find a dot product

When to use: Further Maths or mechanics: check vector arithmetic, a scalar product, perpendicularity or an angle between vectors.

  1. 1MENU → Run-Matrix → F3 (MAT/VCT) → F6 (M⇔V) — This opens the vector editor.
  2. 2Select Vct A → F3 (DIM) → choose the dimension → enter its components
  3. 3Define Vct B in the same way, then EXIT to Run-Matrix
  4. 4OPTN → F2 (MAT/VCT) → F6 (g) → F6 (g) → F2 (DotP)
  5. 5Enter DotP(Vct A, Vct B), then EXE

Find (1, 2, 3) · (4, −1, 2).

Define Vct A = (1, 2, 3)Define Vct B = (4, −1, 2)OPTN → F2 (MAT/VCT) → F6 → F6 → F2 (DotP)DotP(Vct A, Vct B)EXE

Answer: 8

  • Both vectors must have the same dimension for a dot product.
  • A dot product of zero confirms perpendicularity, but write the dot-product equation for the reasoning mark.

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