BSEB Inter Mathematics common mistakes
BSEB Inter Mathematics · Understand the traps and how to avoid them.
When finding the inverse of a function, students swap x and y but then solve for y algebraically and stop, forgetting to state the domain restriction and range of the inverse, or they declare f⁻¹(x) exists without first checking that f is one-one and onto on the stated domain.
Textbooks present the swap-and-solve algorithm as a mechanical recipe, so students treat invertibility as automatic. They also conflate the domain of f with the domain of f⁻¹, and forget that a function like f(x)=x² is invertible only after restricting the domain to [0,∞).
What to do instead
Before any algebra, write the checklist: (1) prove f is one-one (assume f(a)=f(b) and derive a=b, or show f is strictly monotonic), (2) show f maps onto its codomain on the given domain, (3) swap and solve, (4) explicitly write domain and range of f⁻¹ and verify range(f⁻¹)=domain(f). Practise with restricted-domain quadratics and with f(x)=(ax+b)/(cx+d).
Applying principal-value ranges incorrectly in inverse trigonometric problems: writing sin⁻¹(sin θ)=θ for any θ, or using sin⁻¹(-x)=-sin⁻¹x but then applying the same even-style simplification to cos⁻¹(-x) or tan⁻¹(-x) without the correct sign rules.
Students memorise the identities for sin⁻¹ only and generalise them to all six inverse functions. They also forget that sin⁻¹(sin θ)=θ holds only when θ lies in [-π/2, π/2], so they substitute large angles directly.
What to do instead
Write out all six principal ranges on one card and check every answer against them. For sin⁻¹(sin θ) with θ outside [-π/2,π/2], reduce θ using periodicity and symmetry first (e.g. sin⁻¹(sin 5π/6)=sin⁻¹(1/2)=π/6). Memorise the sign rules as a table: sin⁻¹, tan⁻¹, cosec⁻¹ are odd; cos⁻¹, sec⁻¹, cot⁻¹ satisfy f(-x)=π-f(x). Verify each with a quick numerical substitution.
In matrix problems, students compute A⁻¹ using the adjoint method but forget the 1/|A| factor, or they check invertibility by computing |A| and then proceed even after getting |A|=0. They also multiply matrices in the wrong order when solving AX=B, writing X=BA⁻¹ instead of X=A⁻¹B.
The adjoint formula is long and the 1/|A| scalar is easy to drop at the final step. Non-commutativity of matrix multiplication is abstract, so students treat A⁻¹B and BA⁻¹ as interchangeable.
What to do instead
Adopt a fixed template: compute |A| first and write 'if |A|=0, stop, no inverse exists'. Then compute cofactors, transpose to adjoint, and write the answer as (1/|A|)·adj(A) with the scalar visibly in front. For AX=B always left-multiply both sides by A⁻¹ and show the step X=A⁻¹B explicitly; test with a 2×2 numerical example where A⁻¹B≠BA⁻¹ to make the point concrete.
Testing continuity and differentiability at a point by only checking the left-hand and right-hand limits of f, without separately checking that the left-hand and right-hand derivatives exist and are equal, or by differentiating the two pieces and comparing the formulas without evaluating at the join point.
Continuity and differentiability are taught together, and students collapse them into one check. They also assume that if the two pieces are made of standard functions, differentiability follows automatically from continuity.
What to do instead
Use a three-line audit at every join point x=a: (1) LHL=RHL=f(a) for continuity, (2) compute f'(a⁻)=lim_{h→0}[f(a-h)-f(a)]/(-h) and f'(a⁺) similarly from the definition, (3) require these one-sided derivatives to be equal and finite. Then solve for the unknown constants from both conditions in order. Drill with piecewise functions containing |x-a| or modulus terms, where continuity holds but differentiability fails.
In maxima–minima and optimisation word problems, students find critical points by setting f'(x)=0 and then declare the largest critical value the maximum, without checking endpoints of the interval or verifying the second derivative, and without converting the word problem's constraint into an equation in one variable.
Classroom examples often have a single interior critical point, so the pattern 'set derivative to zero, that's the answer' gets over-learned. Endpoint behaviour and constrained optimisation are treated as afterthoughts.
What to do instead
Follow a written procedure every time: identify the quantity to optimise and express it in one variable using the constraint, state the feasible domain including endpoints, find f'(x)=0, test each critical point with the second derivative (or a sign chart of f'), and compare with endpoint values. For word problems, explicitly write 'Let x = …, constraint: …, maximise: …' before differentiating. Include at least one problem where the maximum occurs at an endpoint.
Omitting the constant of integration, or forgetting the modulus inside logarithmic integrals, e.g. writing ∫dx/x = log x instead of log|x|+C, and misapplying integration by parts by choosing u and dv in the wrong order or dropping the minus sign in ∫u dv = uv − ∫v du.
The constant is treated as cosmetic and is dropped under exam time pressure. The modulus is skipped because textbooks sometimes abbreviate log|x| as log x. Integration by parts requires choosing u by the ILATE priority, a step students skip.
What to do instead
Make '+C' and '|·|' part of the written form of every integral, not an addition - write them in the same stroke as the answer. For by-parts, write the ILATE order (Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential) above the problem and label u and dv before starting, then re-derive the sign by differentiating your answer back. Verify every indefinite integral by differentiating the result.
When finding the area between curves or the area under a curve, students integrate without checking where the curves intersect, add signed integrals instead of splitting the region, or set up the integral with respect to x when the region is more naturally described with respect to y, producing wrong limits.
Area is introduced with simple curves above the x-axis, so students assume the integrand is always positive and the limits are always the given x-values. They do not sketch, so they miss intersection points and regions that cross the axis.
What to do instead
Always sketch the region first and mark intersection points by solving f(x)=g(x). Split the interval wherever the upper and lower curves swap or the curve crosses the x-axis, and take absolute values of each piece. If vertical strips require splitting but horizontal strips do not, switch to integrating with respect to y. Sanity-check the sign and rough magnitude of the area before finalising.
Solving differential equations by separating variables but losing the constant of integration until the end (or merging it incorrectly), and mishandling the arbitrary constant in linear equations by forgetting the integrating factor e^{∫P dx} or by omitting the constant when integrating P.
The constant is treated as a final decoration rather than a step-by-step object, and the integrating-factor method has several nested integrals where the constant is easy to drop. Students also forget that the solution of a first-order equation needs one arbitrary constant.
What to do instead
Introduce the constant at the moment of the first integration and carry it visibly through each line. For linear equations dy/dx+Py=Q, write the integrating factor as IF=e^{∫P dx} with no constant inside the exponent (absorb it later), multiply both sides, and integrate the right side as ∫Q·IF dx + C. Finish by applying any initial condition to fix C and state the particular solution.
In vector and 3-D geometry problems, students confuse the direction ratios of a line with the direction cosines, compute the angle between a line and a plane as the angle between the line's direction vector and the plane's normal (instead of its complement), and use the wrong formula for the shortest distance between skew lines.
Direction ratios and direction cosines look similar and differ only by normalisation, and the angle conventions for line–line, line–plane, and plane–plane are easily mixed. The skew-line distance formula is long and rarely practised enough.
What to do instead
Keep a formula sheet with the angle rules stated geometrically: line–line uses cos θ=(d₁·d₂)/(|d₁||d₂|); line–plane uses sin θ=(d·n)/(|d||n|); plane–plane uses cos θ=(n₁·n₂)/(|n₁||n₂|). Normalise direction ratios to cosines only when the problem asks for cosines. For skew lines, write both lines in vector form and apply the scalar triple product formula, checking that the lines are indeed skew (not parallel, not intersecting) before using it.
In linear programming, students graph the constraints but test only the corner points they happen to find, ignoring the feasible region's unboundedness, or they forget that the objective function's optimum on a bounded feasible region occurs at a vertex and instead test interior points or the origin.
The corner-point method is memorised as a procedure without the underlying theorem, so students do not know why vertices suffice or what to do when the region is unbounded. Graphing errors also cause vertices to be missed.
What to do instead
Draw the feasible region carefully, shade it, and label every vertex with its exact coordinates found by solving pairs of constraint equations. State the corner-point theorem: for a bounded feasible region the optimum of a linear objective occurs at a vertex. For unbounded regions, check whether the objective can be increased indefinitely along the open direction; if so, report that no maximum exists. Tabulate Z at each vertex and state the final answer with the coordinates and the value.