Maharashtra HSC Mathematics & Statistics - Commerce (MSBSHSE) common mistakes
Maharashtra HSC Mathematics & Statistics - Commerce (MSBSHSE) · Understand the traps and how to avoid them.
When negating a compound statement, the candidate writes 'not p and not q' for the negation of 'p and q' instead of 'not p or not q', and likewise writes 'not p or not q' for the negation of 'p or q'.
The candidate treats negation as a word-by-word prefix operation, pushing 'not' onto each component while keeping the connective unchanged, because in everyday speech 'not both' and 'neither' are used loosely. De Morgan's laws are memorised as formulas but not applied as a structural rewrite, and the candidate does not check the result against a truth table.
What to do instead
Before writing the answer, build a two-row or four-row truth table for p and q, write the truth values of the original statement, then flip every value. Practise the swap rule aloud: 'negation flips the connective and negates each part'. Verify with one concrete instance (e.g. p = 'price rises', q = 'demand falls') to see that 'it is false that both happened' does not mean 'neither happened'.
The candidate computes the adjoint of a matrix as the transpose of the cofactor matrix but then divides by the determinant without checking that the determinant is nonzero, or writes A⁻¹ = adj A · |A| instead of adj A / |A|.
The formula A⁻¹ = (1/|A|) adj A is recalled as a single visual chunk and the position of the determinant is misremembered under exam pressure. The candidate also skips the singularity check because it feels like an extra step, and the question may not explicitly ask whether the inverse exists.
What to do instead
Adopt a fixed three-step routine: compute |A|, state explicitly 'since |A| ≠ 0, A⁻¹ exists', then write A⁻¹ = adj A / |A|. Sanity-check by multiplying A · A⁻¹ mentally for one entry - if it does not give 1 on the diagonal, the determinant is in the wrong place. For 2×2 matrices, use the shortcut A⁻¹ = (1/(ad − bc)) [[d, −b], [−c, a]] and verify it against the cofactor method once.
When differentiating a product or quotient, the candidate differentiates only one factor - writing d/dx(uv) = u · dv/dx, or d/dx(u/v) = (du/dx)/v - instead of applying the product or quotient rule in full.
The candidate pattern-matches to the simple power rule, where a constant multiplier passes through untouched, and carries that habit into situations where both factors depend on x. The rules are often recalled as words rather than as a written template, so the second term is dropped mid-derivation.
What to do instead
Write the rule template on the rough sheet before substituting: (uv)' = u'v + uv' and (u/v)' = (u'v − uv')/v². Assign u and v explicitly, list u', v', then assemble. For quotients, always square the denominator - a missing v² is the single most common arithmetic slip. Cross-check a product answer by expanding first when the factors are polynomials, since both routes must agree.
In a maxima–minima word problem, the candidate finds the stationary point by setting the derivative to zero and immediately declares it the maximum (or minimum) without evaluating the second derivative or checking the sign change, and sometimes answers with the x-value instead of the required maximum profit or minimum cost.
The candidate conflates 'stationary point' with 'optimum' and treats the second-derivative test as optional decoration. Because the word problem ends with 'find the maximum profit', the candidate stops as soon as a number appears, forgetting that the question asks for the function value, not the input.
What to do instead
Use a fixed four-line layout: (i) define the function and its variable with units, (ii) set f′(x) = 0 and solve, (iii) compute f″(x) at each root and state the sign with its conclusion, (iv) substitute back to give the maximum/minimum value with units. Circle the final demanded quantity in the question before starting so the last line answers exactly that.
The candidate omits the constant of integration when evaluating an indefinite integral, or drops the limits and the evaluation bracket when the question is a definite integral, leaving the answer as an antiderivative rather than a number.
Indefinite and definite integrals look almost identical in notation, and the candidate works mechanically from the integrand without re-reading whether limits are present. The '+ C' is treated as a formality rather than part of the answer, so it is the first thing forgotten under time pressure.
What to do instead
Before integrating, copy the integral sign and any limits onto the answer line. If limits exist, write the bracket [F(x)] from lower to upper and subtract; the answer must be a number. If no limits exist, write '+ C' immediately after the integral sign as a placeholder, then fill in the antiderivative in front of it.
When finding the area between a curve and an axis, the candidate integrates the function over the whole interval in one go even though the curve crosses the axis inside that interval, so the positive and negative regions cancel and the reported area is too small.
The candidate treats area as identical to the definite integral, forgetting that the integral gives signed area. Because the crossing point is not visible without a sketch and the question gives only the interval, the candidate never suspects a sign change.
What to do instead
Always sketch the curve and mark where it meets the axis by solving f(x) = 0 within the interval. Split the integral at each crossing, take the absolute value of each piece, and add. State the split explicitly, e.g. ∫ from a to c plus ∫ from c to b with signs adjusted, so the marker can see the reasoning.
When solving a differential equation by separating variables, the candidate divides by a variable factor without considering that it may be zero, loses the solution obtained when that factor equals zero, and forgets to apply the initial condition until after the general solution is written in a form that cannot easily absorb it.
Separation is taught as an algebraic shuffle, so division feels harmless. The singular solution is never mentioned in routine practice, and the constant is added at the end rather than carried through, making the initial condition awkward to substitute.
What to do instead
Note any value of y for which the divisor vanishes and check whether it satisfies the original equation as a constant solution. Add the constant immediately after integrating, then substitute the initial condition into that line to fix C before rearranging. Finish by writing y explicitly if the question asks for y in terms of x.
The candidate applies the binomial distribution to a situation where the number of trials is not fixed or the trials are not independent, and applies the Poisson distribution using λ equal to the number of trials rather than the mean rate over the stated interval.
Both distributions are associated with 'counting successes', so the candidate selects by keyword rather than by checking conditions. For Poisson, λ is often read off the wrong quantity - for example using n instead of np, or forgetting to rescale λ when the interval changes from per hour to per day.
What to do instead
Run a condition checklist before choosing: fixed n, two outcomes, constant p, independent trials → binomial; counts in a fixed interval with a known average rate → Poisson. For Poisson, write λ with its interval in words (e.g. λ = 3 defects per hour) and rescale explicitly if the question asks about a different interval. Verify that np is moderate before using Poisson as an approximation to binomial.
When computing the mean and variance of a random variable, the candidate calculates E(X²) and reports it as the variance, or computes Var(X) = E(X²) − [E(X)]² but squares the already-squared term, and in grouped data uses class values without multiplying by the given probabilities or frequencies.
The two expectations E(X) and E(X²) are computed in adjacent columns and the candidate grabs the wrong one at the final step. The variance formula is remembered as a shape rather than as 'mean of square minus square of mean', so the bracket placement slips.
What to do instead
Build a table with columns x, P(x), xP(x), x²P(x), and total each column. Write Var(X) = E(X²) − (E(X))² with the bracket around E(X) only, then substitute the two totals. Check that variance is non-negative; a negative result signals a squared-term error. Report standard deviation as the positive square root only when asked.
In index-number questions, the candidate uses the current-year quantities as weights when the question specifies a Laspeyres (base-year-weighted) index, or mixes the price and quantity relatives when computing a weighted aggregative index, and in time-series questions fits a trend but reports the moving average against the wrong time period.
Laspeyres and Paasche formulas differ only in which year's quantities appear, so they are visually confusable and often memorised as one formula. For moving averages, centring shifts the value to the middle of the window, and the candidate labels it with the first period of the window instead.
What to do instead
Write the formula with a verbal tag before substituting: 'Laspeyres uses base-year quantities p₀q₀ in the denominator; Paasche uses current-year p₀q₁'. For moving averages, state the window and mark the centred position explicitly (for a 4-period window, place the average between periods 2 and 3, then centre again). Cross-check that the index equals 100 when current prices equal base prices.