IIT JAM MS common mistakes
IIT JAM MS · Understand the traps and how to avoid them.
In multiple-choice questions on limits, continuity, and differentiability, the candidate evaluates only the one-sided expression given and picks the option matching that value, without checking the other side or the value at the point, so a function that is discontinuous or non-differentiable at the join is declared continuous/differentiable.
JAM Mathematics repeatedly tests piecewise-defined functions at the join point, and coaching-style practice conditions candidates to compute a limit mechanically. The habit of 'compute and match' survives because most textbook exercises are smooth functions where the shortcut works.
What to do instead
Adopt a fixed three-line protocol for every piecewise or absolute-value question: left limit, right limit, value at the point; then continuity, then equality of left and right derivatives (or differentiability via the definition). Before marking an option, write the three quantities in the margin. If any two differ, the function fails there regardless of how familiar the formula looks.
When asked for the radius or interval of convergence of a power series, the candidate applies the ratio test, gets a limit L in x, sets L < 1, and reports the resulting open interval as the answer, omitting the endpoint check entirely.
Ratio-test computation is drilled heavily and endpoint analysis is treated as an optional extra. Under time pressure the candidate stops at the first clean inequality and never substitutes the endpoints back into the series, where the answer usually changes from open to closed or half-open.
What to do instead
Treat 'find the interval of convergence' as a two-stage question that is not finished until both endpoints are classified. After obtaining |x - a| < R, substitute x = a - R and x = a + R separately and name the test used at each (p-series, alternating series, comparison). Only then choose the option; if the options differ only in bracket type, the endpoints are the whole question.
In linear algebra, the candidate computes eigenvalues correctly but then answers questions about diagonalisability, rank, or the number of linearly independent eigenvectors by counting distinct eigenvalues, concluding that n distinct eigenvalues are needed or that repeated eigenvalues automatically mean failure.
The theorem 'distinct eigenvalues imply diagonalisable' is remembered in the wrong direction, and the converse is assumed. The candidate never computes the geometric multiplicity (nullity of A - λI) for a repeated eigenvalue, which is the actual test.
What to do instead
For any repeated eigenvalue, explicitly compute dim ker(A - λI) by row-reducing A - λI and counting free variables. State the criterion in writing: diagonalisable iff the sum of geometric multiplicities equals n, equivalently each algebraic multiplicity equals its geometric multiplicity. Also distinguish rank from the count of nonzero eigenvalues, since rank depends on the Jordan structure at eigenvalue zero.
On maxima/minima and Lagrange-multiplier problems, the candidate finds stationary points, evaluates the objective there, and reports the largest value as the maximum, without checking boundary behaviour or the constraint's endpoints, and without verifying the second-order or bordered-Hessian condition.
Single-variable calculus training makes 'set derivative to zero and compare values' feel complete. In two variables and under constraints, the boundary and the nature of each critical point carry most of the marks, and candidates have often never practised the bordered Hessian.
What to do instead
Use a checklist: interior critical points, then each boundary segment or constraint endpoint, then the second-order test at each candidate. For constrained problems, form the bordered Hessian and evaluate its relevant leading minors, or parametrise the constraint when it is a circle or line. Compare all candidate values in a table before selecting an option.
In group theory, the candidate answers questions about whether a subset is a subgroup, or about the order of an element, by testing closure only, or by assuming that any subset containing the identity and closed under the operation is a subgroup, forgetting inverses or forgetting that the ambient group may be non-abelian.
Closure is the most visible axiom and the one most practised. Inverse and associativity checks feel trivial, and candidates do not habitually construct a counterexample by taking an inverse of a specific element and checking membership.
What to do instead
For every subgroup claim, run the one-step test: non-empty, and for all a, b in H, ab^{-1} in H. When a claim looks true, try the smallest non-abelian group (S3 or the quaternions) and a specific element whose inverse or square falls outside the set. For order questions, compute powers explicitly rather than guessing from the order of the ambient group.
When computing a probability from a joint, marginal, or conditional distribution, the candidate reads the conditional probability P(A|B) off the joint table directly, or uses P(A)P(B) for events that are not independent, without normalising by the conditioning event.
Joint tables present P(A and B) as the visible cell, and the division by P(B) is easy to skip. Independence is over-applied because it simplifies arithmetic, and the candidate does not check whether the problem states or implies dependence.
What to do instead
Write the definition P(A|B) = P(A and B)/P(B) before substituting any numbers, and compute P(B) by summing the relevant row or column. Test independence explicitly by comparing P(A and B) with P(A)P(B) rather than assuming it. For continuous variables, always identify the support and integrate the joint density over the correct region before dividing.
The candidate confuses the probability density with the probability, reporting f(x) as the answer to 'what is P(X = x)' for a continuous variable, or integrating a density over the wrong limits because the support was not written down.
Discrete habits carry over: for a discrete variable P(X = x) is the mass, and the same notation is used for densities. The support of a transformed or truncated distribution is often implicit and the candidate never states it.
What to do instead
Before any continuous-probability computation, write the density and its support on one line, and note that P(X = x) = 0 for continuous X. For transformations, use the change-of-variable formula with the Jacobian and derive the new support from the old one by mapping the endpoints. Check that the density integrates to 1 over the stated support as a sanity test.
On estimation questions, the candidate computes a sample mean or variance and declares it the UMVUE or the maximum-likelihood estimator without checking unbiasedness, sufficiency, or whether the estimator actually maximises the likelihood (for example stopping at a stationary point that is a minimum, or ignoring the parameter space boundary).
Estimation theory questions look like computation questions, so candidates substitute numbers and stop. The definitions of unbiasedness, sufficiency via the factorisation theorem, and the regularity conditions for the Cramer-Rao bound are learned as separate facts rather than as a procedure to apply.
What to do instead
For each estimator, verify E(estimator) = parameter by direct integration or summation, and check sufficiency with the factorisation theorem, writing the joint density as g(T, parameter) h(x). For MLE, take the log-likelihood, differentiate, solve, and check the second derivative or the boundary; report the parameter space restriction when the stationary point lies outside it.
In hypothesis testing, the candidate computes a test statistic and p-value but states the conclusion in terms of accepting the null hypothesis, or swaps the roles of Type I and Type II error and of the significance level and power when the question asks for them numerically.
Everyday language treats 'not rejected' as 'proved true', and the two error types are symmetric in name but not in definition. Under exam pressure the candidate recalls 'alpha is the error' without recalling which error, and power is often confused with the significance level.
What to do instead
Memorise the pair as a sentence: alpha = P(reject H0 | H0 true), beta = P(fail to reject H0 | H1 true), power = 1 - beta. When a numerical question appears, draw the two distributions and shade the relevant tail before computing. Phrase every conclusion as 'reject H0' or 'fail to reject H0 at level alpha', never 'accept H0'.
In regression and correlation, the candidate interprets the correlation coefficient as the slope of the regression line, or claims that a high R-squared proves a causal relationship or that the slope equals r, and reports the regression of y on x when the question asks for x on y.
The formulas for slope, correlation, and R-squared are visually similar and are memorised as a cluster. The asymmetry of regression (two different lines depending on which variable is regressed) is easy to overlook, and the causal reading of R-squared is a common informal habit.
What to do instead
Keep the identities explicit: slope of y on x is r * (s_y / s_x), and R-squared equals r-squared only in simple linear regression with an intercept. Before computing, underline which variable is the response. When interpreting, state that correlation measures linear association only and that a fitted relationship does not establish causation; check the residual pattern for non-linearity.