BSEH Class 12 Maths common mistakes
BSEH Class 12 Maths · Understand the traps and how to avoid them.
While checking whether a relation is reflexive, symmetric or transitive, the student tests only one or two example ordered pairs instead of working from the definition for all elements of the set.
Textbook examples often use small sets, so students generalise from a single case and forget that the property must hold for every element or pair in the set.
What to do instead
For each property, write the formal condition (e.g., for all a in A, (a,a) ∈ R) and then verify it by checking every element or by constructing a general argument. Use counterexamples deliberately: to disprove symmetry, find one pair (a,b) in R with (b,a) not in R.
When simplifying expressions involving inverse trigonometric functions, the student ignores the principal-value ranges and writes sin⁻¹(sin θ) = θ for any θ, or cancels inverse functions without checking the domain.
The inverse functions are introduced with restricted domains, but practice often focuses on simple values; students then treat the inverse as a true inverse on all real numbers.
What to do instead
Memorise the principal ranges (sin⁻¹: [-π/2, π/2], cos⁻¹: [0, π], tan⁻¹: (-π/2, π/2), etc.). Before simplifying, reduce the angle to the principal range using identities such as sin⁻¹(sin θ) = θ only if θ ∈ [-π/2, π/2]; otherwise adjust by π or 2π as appropriate.
When multiplying matrices, the student multiplies corresponding entries (Hadamard product) rather than taking row-by-column dot products, or assumes AB = BA.
Element-wise operations are familiar from vectors and ordinary algebra; matrix multiplication is a new rule that is easy to misapply under time pressure.
What to do instead
Always write the product with explicit row and column indices: (AB)ij = Σ aik bkj. Practise with small non-square matrices where the mismatch is obvious. Check order: if A is m×n and B is p×q, AB is defined only when n = p, and the result is m×q. Never assume commutativity; test with a simple 2×2 counterexample.
When computing a determinant, the student expands along a row or column but forgets the alternating sign pattern, or applies row operations without adjusting the determinant value.
The cofactor sign (-1)^{i+j} is easily lost, and the effect of row operations (adding a multiple of one row to another leaves the determinant unchanged, but swapping rows changes sign, and multiplying a row by k multiplies the determinant by k) is often memorised incompletely.
What to do instead
Write the sign pattern explicitly before expanding, or use the formula with cofactors. When using row operations, note each operation's effect: only adding a multiple of a row to another leaves the determinant unchanged; swapping two rows multiplies by -1; multiplying a row by k multiplies by k. Verify with a 3×3 numerical example.
When testing continuity or differentiability at a point, the student checks only that the left-hand and right-hand limits of the function are equal, but does not check that they equal f(a), or does not check the derivative from both sides.
The definition of continuity requires three conditions, and differentiability requires the equality of left and right derivatives; students often conflate continuity with differentiability or omit the function value.
What to do instead
For continuity at x = a, explicitly compute lim_{x→a⁻} f(x), lim_{x→a⁺} f(x), and f(a), and state that all three are equal. For differentiability, compute the left derivative and right derivative using the limit definition (or the derivative formulas on each side) and check they are equal. Remember: differentiability implies continuity, but not conversely.
When finding maxima or minima of a function on a closed interval, the student sets f′(x)=0, finds critical points, and then chooses the largest/smallest function value among those points without checking the endpoints.
Many textbook problems ask for local extrema, so students forget that absolute extrema on a closed interval must also be checked at the endpoints.
What to do instead
Use a checklist: (1) find all critical points in the open interval, (2) evaluate f at each critical point, (3) evaluate f at both endpoints, (4) compare all values. For word problems, also check the physical domain and whether the critical point is a maximum or minimum using the first or second derivative test.
When evaluating a definite integral by substitution, the student changes the variable in the integrand but forgets to change the limits of integration, or substitutes back and then evaluates at the original limits.
The substitution method is often practised with indefinite integrals, where limits are absent; students then carry over the habit of back-substituting.
What to do instead
When using u = g(x), also compute the new limits u(a) and u(b) and write the integral entirely in terms of u. Alternatively, find the antiderivative in terms of x and then apply the original limits. Never mix variables and limits. Practise with a few examples where the substitution is nonlinear, e.g., ∫₀¹ x e^{x²} dx.
When finding the area between two curves, the student integrates the difference of the functions without first determining which curve is on top, or without finding all intersection points that bound the region.
The formula ∫(upper - lower) dx requires identifying the upper and lower functions; students often assume a fixed order or miss intersection points that split the region.
What to do instead
Sketch the curves (or analyse their difference) to find intersection points and determine which function is greater on each subinterval. Set up the integral as ∫(top - bottom) dx over each relevant interval. If the curves cross, split the integral at the crossing point and use absolute difference or change the order.
When solving a differential equation, the student separates variables but omits the constant of integration, or loses a solution when dividing by a variable expression that could be zero.
The constant is often forgotten in the rush, and dividing by a factor like y or cos y can discard singular solutions that are valid.
What to do instead
Always include +C immediately after integrating. Before dividing by an expression, check whether it can be zero; if so, treat that case separately as a possible solution. For initial value problems, use the initial condition to determine C. Verify the final solution by differentiation.
In vector algebra and three-dimensional geometry, the student confuses the direction ratios of a line with the direction cosines, or uses the wrong formula for the angle between two lines or the distance from a point to a plane.
Direction ratios (a,b,c) and direction cosines (l,m,n) are related but not identical; students often normalise incorrectly or forget to normalise at all. Similarly, the formula for the angle between lines uses the dot product of direction vectors, while the distance from a point to a plane requires the normal vector.
What to do instead
Write down the key formulas: direction cosines are l = a/√(a²+b²+c²), etc.; the angle θ between lines with direction vectors d₁ and d₂ satisfies cos θ = |d₁·d₂|/(|d₁||d₂|); the distance from point (x₁,y₁,z₁) to plane ax+by+cz+d=0 is |ax₁+by₁+cz₁+d|/√(a²+b²+c²). When a problem gives direction ratios, normalise before using them as cosines.