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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M3 Calculus: Continuity and Differentiation

All the topics in Mathematics: Calculus: Continuity and Differentiation

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.

32 topics shown

#	Topic	Type	Marks	Times asked	Last asked	Status
1	Prove a second-order relation: clear the denominator, differentiate againFind dy/dx, cross-multiply to remove the fraction or square root (e.g. sqrt(1 - x^2) dy/dx = m y, x dy/dx = ...), square if needed, differentiate that equation again and rearrange to the given relation.	Proof	3	4 papers	2025	Asked
2	Prove a second-order relation by direct differentiationFind dy/dx and d2y/dx2 of the given y, substitute both and y into the left side of the relation, and simplify to the right side.	Proof	3	3 papers	2026	Asked
3	Limits used in continuity checks: factorise, rationalise, standard limitsEvaluate the 0/0 limit by cancelling the common factor, rationalising, or reducing to a standard limit, showing the step that removes the 0/0, then give the limit value.	Sum	2	3 papers	2024	Asked
4	Find k so a function is continuous at a pointWrite LHL, RHL and f(a) separately, evaluate each, set them equal as the condition for continuity, and solve for k.	Sum	2	3 papers	2024	Asked
5	Removable discontinuity: identify it and redefine f(a)Show the limit at a exists (LHL = RHL) but f(a) is undefined or not equal to it, name it a removable discontinuity, and state the value of f(a) that makes f continuous.	Sum	2	2 papers	Specimen 2027	Asked
6	Read discontinuities and non-differentiable points from a graphFrom the graph, list x-values with a hole, jump or missing point as discontinuities (naming the removable ones), and list x-values with a corner, a vertical tangent or a discontinuity as non-differentiable points.	Graph	1	2 papers	Specimen 2027	Asked
7	Inverse trig derivative by substitution, e.g. x = tan tChoose the substitution that turns the inside into a single trig ratio (e.g. 2x/(1 + x^2) with x = tan t, or 2^(x+1) 3^x/(1 + 36^x) with 6^x = tan t), simplify to a multiple of t, write t back in x, then differentiate.	Sum	2	2 papers	Specimen 2027	Asked
8	Derivative of one function with respect to anotherCall the two functions u and v of x, find du/dx and dv/dx, and write du/dv = (du/dx)/(dv/dx), simplified (e.g. sin x^3 with respect to x^3).	Sum	2	2 papers	2023	Asked
9	Find constants making a piecewise function differentiableWrite the continuity condition at the point to get one equation, then equate LHD and RHD (derivatives of the two pieces at the point) to get the second, and solve for both constants.	Sum	2	1 paper	Specimen 2027	Asked
10	Simplify logs or surds first, then differentiateUse log laws or rationalising to break the expression into simple terms (e.g. log((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x)) = 2 log(sqrt(1 + x^2) + x)), then differentiate term by term and simplify.	Sum	2	1 paper	Specimen 2027	Asked
11	Second derivative under a change of variableUse the chain rule twice (dy/dz = dy/dx . dx/dz, then d/dz(dy/dx) = d2y/dx2 . dx/dz) to link the two second derivatives and reach the stated relation.	Proof	3	1 paper	Specimen 2027	Asked
12	Differentiability at a point: LHD, RHD; differentiable implies continuousState Lf'(a) = lim h->0 (f(a - h) - f(a))/(-h) and Rf'(a) = lim h->0 (f(a + h) - f(a))/h, that f is differentiable at a when they are equal and finite, and that differentiable implies continuous but not conversely.	Statement	1	1 paper	2026	Asked
13	Assertion-reason or MCQ on continuity and differentiability at a pointTest the assertion by computing LHL, RHL, LHD and RHD at the point, judge the reason against the definition, and choose the option that matches both judgements.	Give reason	1	1 paper	2026	Asked
14	Infinite repeated expressions: x^x^x..., sqrt(x + sqrt(x + ...))Replace the repeated tail by y itself (y = x^y, y = sqrt(x + y), x = e^(y + x)), take log or square, differentiate implicitly, and solve for dy/dx.	Sum	2	1 paper	2026	Asked
15	Find all points of discontinuity of a piecewise functionState each piece is a polynomial or standard function and so continuous inside its interval, then test only the joining points with LHL, RHL and f(a), and list the points of discontinuity.	Sum	1	1 paper	2025	Asked
16	Inverse trig of a trig expression: simplify, then differentiateRewrite the inside as one trig ratio of a shifted or halved angle (e.g. (cos x - sin x)/(cos x + sin x) = tan(pi/4 - x), sqrt((1 + cos x)/(1 - cos x)) = cot(x/2)), cancel the inverse using the given interval, then differentiate the linear result.	Sum	1	1 paper	2025 improvement	Asked
17	Parametric functions: dy/dx and its value at a parameterFind dx/dt and dy/dt separately, write dy/dx = (dy/dt)/(dx/dt), simplify, and substitute the given parameter value, noting when dx/dt = 0 makes dy/dx undefined.	Sum	1	1 paper	2025 improvement	Asked
18	Logarithmic differentiation of variable powers like x^(sin x)Take log of both sides, bring the power down, differentiate implicitly, solve for dy/dx; for a sum like x^x + (sin x)^x set u and v and treat each term separately.	Sum	3	1 paper	2025	Asked
19	Relate a function and its derivative: graph or form, assertion-reasonRead the derivative's graph or formula (linear, constant, sign), state what kind of function f must be, and judge each statement before choosing the option.	Give reason	1	1 paper	2025 improvement	Asked
20	Second derivative of parametric functionsFind dy/dx = (dy/dt)/(dx/dt), then d2y/dx2 = d/dt(dy/dx) . dt/dx, or clear denominators first, and substitute into the given relation.	Proof	3	1 paper	2025 improvement	Asked
21	Standard derivatives of polynomial, trig, exp and log functionsWrite the derivative using the standard results and the sum, product and quotient rules, simplifying to a single final expression.	Statement	1	1 paper	2024	Asked
22	Implicit differentiation: dy/dx from an equation in x, yDifferentiate both sides term by term with respect to x, writing dy/dx after every y-term, collect the dy/dx terms on one side and solve for dy/dx.	Sum	2	1 paper	2020	Asked
23	Continuity and differentiability of \|x - a\| and [x]Rewrite \|x - a\| or [x] piecewise around the point, check LHL = RHL = f(a), then compute LHD and RHD and state whether the function is differentiable there.	Proof	3	1 paper	2019	Asked
24	Standard derivatives of inverse trigonometric functionsWrite the derivative of sin^-1 u, cos^-1 u, tan^-1 u, cot^-1 u, sec^-1 u, cosec^-1 u with the chain rule factor du/dx.	Statement	1	1 paper	2019	Asked
25	Show a piecewise function is continuous but not differentiableShow LHL = RHL = f(a) for continuity, then compute LHD and RHD from the definition with h, show they differ, and conclude f is continuous but not differentiable at a.	Proof	3	1 paper	2018	Asked
26	Chain rule: derivatives of composite functionsName the inner function, differentiate outer then inner, multiply, and simplify (e.g. sin(e^(x^2)), log(cos x), (ax + b)^n).	Sum	1	1 paper	2017	Asked
27	Definition of continuity at a point; algebra of continuous functionsState f is continuous at x = a when lim x->a- f(x) = lim x->a+ f(x) = f(a), and use the facts that sums, differences, products, quotients (denominator not zero) and composites of continuous functions are continuous to judge a given function.	Statement	1	never		Never asked
28	Find two constants for continuity of a three-piece functionAt each joining point write LHL, RHL and f(a), form one equation per point, and solve the pair of equations for both constants.	Sum	3	never		Never asked
29	Exponential and logarithmic functions: laws, domain and rangeState and use the laws a^m a^n = a^(m+n), log(mn) = log m + log n, log(m^n) = n log m, a^x = e^(x log a), log_a x = log x / log a, with domain x > 0 for log x.	Statement	1	never		Never asked
30	Implicit relation: make y explicit, then prove the given dy/dxRearrange the relation (e.g. x sqrt(1 + y) + y sqrt(1 + x) = 0: square, factorise, cancel x - y) to write y as a function of x, then differentiate to reach the stated form.	Proof	3	never		Never asked
31	Logarithmic differentiation of a product or quotient of many factorsTake log, expand into a sum of logs, differentiate each term, multiply back by y and write dy/dx in terms of x.	Sum	2	never		Never asked
32	Second derivative of an explicit function, value at a pointFind dy/dx, differentiate again with product, quotient and chain rules, simplify, and substitute the point if asked.	Sum	2	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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