CHSE Odisha Elective Mathematics common mistakes
CHSE Odisha Elective Mathematics · Understand the traps and how to avoid them.
Simplifies sin⁻¹(sin 5π/6) to 5π/6, or tan⁻¹(tan 3π/4) to 3π/4, without checking the principal-value range.
The identity sin⁻¹(sin x)=x is true only when x lies in [-π/2, π/2]; candidates memorize the identity and apply it to any angle, ignoring that inverse trigonometric functions return a unique principal value.
What to do instead
Before simplifying, write the range of the inverse function. For sin⁻¹(sin θ), reduce θ to an angle α in [-π/2, π/2] with the same sine (e.g. 5π/6 gives π/6); for cos⁻¹ use [0, π]; for tan⁻¹ use (-π/2, π/2). Check the quadrant and sign explicitly.
Writes |2A|=2|A| for a 3×3 matrix A, or expands (A+B)² as A²+2AB+B².
Determinant is multilinear in rows and columns, so a scalar multiplying every entry of an n×n matrix contributes kⁿ; matrix multiplication is not commutative, so cross terms AB and BA cannot be merged.
What to do instead
State the order n before using |kA|=kⁿ|A|. For matrix algebra, keep AB and BA separate unless commutativity is given; use (A+B)²=A²+AB+BA+B². Check determinant row-operation effects: swapping rows changes sign, adding a multiple leaves it unchanged.
Computes ∫₀² x/(x²−1) dx by directly applying ln|x²−1|/2 and substituting 0 and 2, even though x=1 lies inside the interval.
The integrand is undefined at x=1, so the integral is improper; a blind antiderivative hides the singularity and can produce a finite-looking but invalid answer.
What to do instead
Before integrating a definite integral, check continuity on the whole interval. Split at x=1 and write two limits; if either limit diverges, state that the integral diverges. For ordinary integrals, include +C and use ln|x| where appropriate.
Tests continuity at x=a by calculating only f(a), or by finding left- and right-hand limits but not comparing them with f(a).
Continuity requires all three pieces: the two-sided limit exists and equals the function value; many candidates treat continuity as a one-sided or value-only condition.
What to do instead
Write the definition lim_{x→a⁻} f(x)=lim_{x→a⁺} f(x)=f(a). Compute both one-sided limits, state whether they are equal, and then check equality with f(a). For piecewise functions, evaluate the correct branch at the boundary.
Uses ⁿP_r instead of ⁿC_r when forming a committee, or adds separate cases for 'at least one' without using the complement.
The candidate matches a formula to words like 'select' or 'arrange' without deciding whether order matters; 'at least one' is often much shorter as total minus none.
What to do instead
First decide whether order matters: selection → ⁿC_r, arrangement → ⁿP_r. For 'at least one', compute 1 − P(none) or total − none. For identical objects, divide by factorials of repeats; for circular arrangements, fix one object and arrange the rest.
Finds stationary points of a continuous function on [a,b], evaluates f only at those points, and calls the largest value the maximum.
On a closed interval the absolute maximum or minimum can occur at an endpoint; the second derivative test can also be inconclusive when f''=0.
What to do instead
List all critical points in (a,b) plus the endpoints a and b. Evaluate f at every listed point and compare. Use the first-derivative sign change for local extrema; if f''=0, fall back to sign change or a table.
Solves dy/dx = x, writes y=x²/2, and omits the arbitrary constant; or solves a second-order equation with only one constant.
Integration introduces one constant per integration, and the order of a differential equation equals the number of independent arbitrary constants in its general solution.
What to do instead
Integrate step by step and add +C after each integration. Count constants: first order → one, second order → two. Use initial conditions only after writing the general solution; when dividing by a variable, check for lost singular solutions.
Finds the angle between a line and a plane by taking cos θ = |a·n|/(|a||n|), where a is the line direction and n is the plane normal.
The dot product gives the angle between the line and the normal, not between the line and the plane; these angles are complementary.
What to do instead
Use sin θ = |a·n|/(|a||n|) for the line-plane angle, or compute the line-normal angle φ and take θ=90°−φ. Also keep direction ratios and direction cosines distinct: cosines are normalized by the vector's length.
Solves an LPP by testing only intersections of pairs of constraint lines, without checking which intersections lie in the feasible region or including axis intercepts.
The corner-point method requires all vertices of the feasible polygon, including intercepts on the axes; some line intersections are outside the feasible region.
What to do instead
Draw the feasible region, shade it, and list every vertex: axis intercepts and feasible line intersections. Evaluate the objective at each vertex. If the region is unbounded, check whether the objective can improve indefinitely; do not ignore non-negativity constraints.
Uses P(A∩B)=P(A)P(B) for mutually exclusive events, or P(A∪B)=P(A)+P(B) for independent events.
Mutually exclusive means A∩B=∅; independent means P(A∩B)=P(A)P(B). The words are confused, and the formulas are memorized without their conditions.
What to do instead
Write definitions first. Mutually exclusive: P(A∩B)=0, so P(A∪B)=P(A)+P(B). Independent: P(A∩B)=P(A)P(B), so P(A∪B)=P(A)+P(B)−P(A)P(B). For conditional probability use P(A|B)=P(A∩B)/P(B) with P(B)>0.