What the board asks

We went through the ISC board papers, question by question, and matched each question to a topic. Here is what was asked, how often, and what was never asked.

Mathematics: 309 topics in the 2027 syllabus, across 11 units. The board has asked 255 of them at least once. 54 were never asked in the papers we checked.

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M4 Calculus: Applications of Derivatives

All the topics in Mathematics: Calculus: Applications of Derivatives

Asked topics first, the most asked at the top. Tap a topic to see what to write, the result, the years, and the real board questions.

33 topics shown

#	Topic	Type	Marks	Times asked	Last asked	Status
1	Related rates with one formula: cube, sphere, circle, cylinderWrite the formula linking the two quantities, differentiate both sides with respect to t, substitute the given rate and size, and state the answer with units.	Sum	2	4 papers	2026	Asked
2	Intervals of increase and decrease of a polynomialFind f'(x), factorise, mark the critical points on a number line, test the sign of f'(x) in each interval and write the intervals where f increases and where it decreases.	Sum	2	3 papers	Specimen 2027	Asked
3	Equation of tangent and normal at a given pointFind dy/dx at the point, write the tangent y - y1 = m(x - x1) and the normal y - y1 = (-1/m)(x - x1), and simplify to ax + by + c = 0; handle m = 0 or infinite separately.	Sum	2	3 papers	Specimen 2027	Asked
4	Find critical, stationary or turning points of a functionFind f'(x), solve f'(x) = 0 (and note points where f' does not exist), and give the points as coordinates (x, f(x)).	Sum	2	3 papers	Specimen 2027	Asked
5	Case study on a function model: values, turning points, meaningFrom the given model, compute the values asked (start value, rate, turning points, max/min), then answer each part in the words of the story with units and any comparison to a threshold.	Sum	5	3 papers	Specimen 2027	Asked
6	Points where tangent is parallel or perpendicular to a lineFind dy/dx, set it equal to the required slope (0 for parallel to x-axis, the line's slope, or the slope of a chord), solve for x, find y from the curve and write the points.	Sum	2	3 papers	2025	Asked
7	Prove a solid of given volume or surface is optimalWrite the constraint, express the quantity to optimise in one variable, differentiate, set to zero, show the second derivative sign, and deduce the stated relation (cube, r = h/2, given angle).	Proof	5	3 papers	2023	Asked
8	Related rates word problem: cone, funnel or tank filling/drainingUse the semi-vertical angle or similar triangles to write r in terms of h (or l), express V or S in one variable, differentiate with respect to t, substitute and state the rate with sign and units.	Sum	3	2 papers	Specimen 2027	Asked
9	Related rates on a curve or implicit relationDifferentiate the relation with respect to t, put in the given rate condition (for example dy/dt = k dx/dt), solve for the point or the unknown rate, and check the point lies on the curve.	Sum	1	2 papers	Specimen 2027	Asked
10	Interpret the sign or zero of a derivative in contextDecide true or false for each statement (or prove the claim) by linking the sign of the derivative to increasing/decreasing and dy/dx = 0 to a highest or lowest point, giving one line of reason each.	Give reason	1	2 papers	Specimen 2027	Asked
11	Show a function is increasing or decreasing on an intervalFind f'(x), write it in a form whose sign is obvious (perfect square, sum of squares, positive factors), state f'(x) > 0 (or >= 0 with equality only at isolated points) on the interval, and conclude.	Proof	2	2 papers	Specimen 2027	Asked
12	Local maxima and minima by the first derivative testFind the critical points, test the sign of f'(x) just left and right of each, state local maximum (+ to -), local minimum (- to +) or neither, and give the local values.	Sum	3	2 papers	Specimen 2027	Asked
13	Absolute maximum and minimum on a closed intervalFind the critical points inside [a, b], evaluate f at them and at both end points, and name the greatest value as absolute maximum and the least as absolute minimum with the x where each occurs.	Sum	2	2 papers	Specimen 2027	Asked
14	Read increasing, decreasing or strictly increasing intervals from a graphLook at each stated interval on the graph, decide whether the curve rises, falls or stays flat there, and mark each statement true or false with a reason.	Graph	1	2 papers	2026	Asked
15	Local maxima and minima by the second derivative testFind f'(x) and f''(x), solve f'(x) = 0, evaluate f''(c) at each root, state max (f'' < 0) or min (f'' > 0), and give the local maximum and minimum values.	Sum	3	2 papers	2026	Asked
16	Slope of tangent and normal at a pointDifferentiate (implicitly if needed), substitute the point to get the slope of the tangent, and give the normal slope as -1/m.	Sum	1	2 papers	2025	Asked
17	Word problem: plane figure with given perimeter, maximise areaDraw and label the figure, write the perimeter condition and the area, eliminate one variable, differentiate, set to zero, check with the second derivative, and give the dimensions (and area if asked) with units.	Sum	5	2 papers	2024	Asked
18	Word problem: minimum cost of making or painting a containerUse the fixed volume to express height in terms of the base, write the cost as rate times each surface, differentiate, set to zero, check minimum, and state the dimensions and the minimum cost.	Sum	5	2 papers	2024	Asked
19	Rate of change of a derived quantity by chain ruleExpress the new quantity (cost, area) in terms of the variable whose rate is known, differentiate using the chain rule dC/dt = dC/dh . dh/dt, simplify and answer the question asked about when it is greatest or least.	Sum	2	1 paper	Specimen 2027	Asked
20	Curves cutting at right angles (orthogonality condition)Find the slopes of both curves at the intersection, set m1 m2 = -1, use the curve equations to find the condition or the constant, or prove the curves cut orthogonally.	Sum	1	1 paper	Specimen 2027	Asked
21	Judge claims about extrema: test fails, no local maximumDecide whether the claim holds by checking the sign of f'(x) (always positive means no local extremum) or, when f''(c) = 0, by the first derivative test, and give a one-line reason.	Give reason	1	1 paper	Specimen 2027	Asked
22	Rate of change of one quantity with respect to anotherWrite the quantity as a function of the variable, differentiate, substitute the given value and state the rate with units and meaning.	Sum	1	1 paper	2025 improvement	Asked
23	Find unknown constants given a tangent or normalUse that the point lies on the curve (one equation) and that dy/dx at the point equals the slope of the given line (second equation), and solve the two equations for the constants.	Sum	2	1 paper	2020	Asked
24	Prove a plane figure is extreme under a conditionSet the figure in one variable (angle or side), write the quantity to optimise, differentiate, find the critical value, check max or min, and deduce the stated shape (for example isosceles).	Proof	5	1 paper	2020	Asked
25	Related rates word problem: ladder or shadow (right triangle)Draw and label the figure, write the Pythagoras or similar-triangles relation, differentiate with respect to t, find the missing length at the instant, substitute and state the rate with sign and units.	Sum	3	1 paper	2019	Asked
26	Word problem: nearest point or shortest distanceTake a general point on the line or curve in one variable, write the square of its distance from the fixed point, minimise it with the derivative tests, and state the nearest point and the distance if asked.	Sum	3	1 paper	2019	Asked
27	Word problem: solid inscribed in a sphere or coneDraw the cross-section, link the solid's radius and height to the fixed sphere or cone by Pythagoras or similar triangles, write the volume in one variable, maximise with the derivative tests, and state the height or dimensions.	Sum	5	1 paper	2018	Asked
28	Velocity and acceleration of a particle from s = f(t)Differentiate s to get v = ds/dt and again to get a = dv/dt, substitute the time asked, and state units and direction (sign).	Sum	2	never		Never asked
29	Intervals of increase/decrease for trig, log or exponential functionsFind f'(x), solve f'(x) = 0 in the given interval (for example [0, 2pi]), test the sign in each sub-interval and write the answer in interval notation.	Sum	3	never		Never asked
30	Find the parameter for which a function is monotonicWrite f'(x), state the condition f'(x) >= 0 (or <= 0) for all x in the interval, use the discriminant or the minimum of f'(x), and give the range of the parameter.	Sum	2	never		Never asked
31	Tangent and normal to a curve in parametric formFind dy/dx = (dy/dt)/(dx/dt) at the given parameter, find the point (x, y) from t, and write the tangent and normal equations.	Sum	3	never		Never asked
32	Angle between two curves at their point of intersectionSolve the curves together to find the intersection point, find each tangent slope there, and use tan(theta) = \|(m1 - m2)/(1 + m1 m2)\| to give the acute angle.	Sum	3	never		Never asked
33	Word problem: numbers with given sum, maximise productLet one number be x, write the other from the condition, form the function to optimise, find the critical point, confirm max or min with f'', and state the numbers.	Sum	3	never		Never asked

This list was made from the 2027 syllabus and checked with 10 board papers, including the 2027 specimen paper. It is not yet checked with the textbook.

The 10 topics Mathematics papers ask most

Counted on the board papers from 2018 to the 2027 specimen paper, which every unit has been checked against.

1.Formulate an LPP from a word problemM10 Linear Programming · asked 7 times since 2018
2.Scalar and vector projection of one vector on anotherM8 Vector Algebra · asked 6 times since 2018
3.Mean (expected value) of a random variableM11 Probability · asked 6 times since 2018
4.Find the inverse of an invertible algebraic function and verifyM1 Relations and Functions · asked 5 times since 2018
5.Principal value of expressions and nested inverse trigM1 Relations and Functions · asked 5 times since 2018
6.Evaluate or show zero without expanding, using propertiesM2 Algebra: Matrices and Determinants · asked 5 times since 2018
7.Prove a determinant equals a factorised expression using propertiesM2 Algebra: Matrices and Determinants · asked 5 times since 2018
8.Definite integral of a modulus functionM5 Calculus: Integrals · asked 5 times since 2018
9.Case study or linked parts on lines and planesM9 Three-Dimensional Geometry · asked 5 times since 2018
10.Equations solved with the complementary identitiesM1 Relations and Functions · asked 4 times since 2018

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