Assam ASSEB Higher Secondary Mathematics common mistakes
Assam ASSEB Higher Secondary Mathematics · Understand the traps and how to avoid them.
When a relation is given as a set of ordered pairs, marks it as a function whenever every first element appears, without checking whether any first element repeats with a different second element.
The candidate equates 'defined for each input' with 'function' and does not apply the one-output rule to discrete ordered pairs.
What to do instead
List the first coordinates separately; if any first coordinate repeats with a different second coordinate, the relation is not a function. Then write the domain as the set of first coordinates and the range as the set of second coordinates.
Evaluates expressions such as sin^{-1}(sin x) or cos^{-1}(cos x) by cancelling the inverse and returning x even when x lies outside the principal range.
The candidate treats inverse trigonometric functions as ordinary inverse operations and ignores the restricted ranges that make them functions.
What to do instead
Always reduce the angle to the principal branch first: sin^{-1} has range [-π/2, π/2], cos^{-1} has range [0, π], and tan^{-1} has range (-π/2, π/2). Check the quadrant on the unit circle before writing the final angle.
Computes arg(z) using tan^{-1}(y/x) and reports the first-quadrant angle even when z lies in the second or third quadrant.
The formula ignores the signs of x and y, and a calculator returns only the principal arctangent value.
What to do instead
Plot the complex number or inspect the signs of its real and imaginary parts. Choose the quadrant, then add or subtract π as needed so that the argument lies in the convention required by the question, usually (-π, π] or [0, 2π).
Multiplies matrices entry by entry, reverses the order when the question asks for BA rather than AB, or assumes AB = BA without checking dimensions and order.
The candidate carries over scalar multiplication habits and does not treat matrix multiplication as a row-by-column operation with a required order.
What to do instead
Write the dimensions of each matrix before multiplying and check that the inner dimensions match. Compute each entry as a row of the first matrix times a column of the second, and remember that AB and BA may not both exist and are not generally equal.
In a binomial expansion, writes the general term with the exponents of a and b swapped, or gives the whole term when the question asks only for the coefficient.
The candidate memorises a formula without tracking which exponent increases with r, and conflates a term with its numerical coefficient.
What to do instead
Use T_{r+1} = nCr a^{n-r} b^r and identify clearly which quantity is a and which is b. If a coefficient is requested, isolate the constant factor after substituting the required r and exclude the variable powers.
At a piecewise-defined point, checks only that f(c) equals one branch value, or assumes continuity automatically implies differentiability, without computing left-hand and right-hand limits and derivatives.
The candidate is used to single-formula functions and misremembers the relationship between continuity and differentiability.
What to do instead
For continuity at x = c, compute the left-hand limit, right-hand limit, and f(c), and require all three to be equal. For differentiability, compute the left-hand derivative and right-hand derivative and require them to be equal; continuity alone is not enough.
Evaluates a definite integral after substitution but substitutes the original x-limits into the new u-antiderivative, or omits the constant of integration in an indefinite integral.
The candidate focuses on finding the antiderivative and forgets that changing the variable also changes the limits; the constant is treated as optional notation.
What to do instead
When substituting u = g(x), either change the limits to u-values before integrating or back-substitute to x before applying the original limits. For indefinite integrals, always include + C unless the question explicitly asks for a particular antiderivative.
Finds the angle between a line and a plane by substituting the line direction and plane normal directly into the dot-product cosine formula, or uses direction ratios as if they were direction cosines.
The candidate confuses the angle between two vectors with the angle between a line and a plane, and does not normalise vectors before using cosine or distance formulas.
What to do instead
For angle θ between a line with direction d and a plane with normal n, use sin θ = |d·n|/(|d||n|). For direction cosines, divide each direction ratio by the vector's magnitude; never use raw direction ratios as cosines.
Shades the same side of every linear inequality, for example always choosing the side containing the origin, and tests intersection points that are not actually feasible vertices of the region.
The candidate applies a shading rule mechanically without substituting a test point into each inequality, and does not recheck whether solved intersection points satisfy all constraints.
What to do instead
Substitute a convenient test point such as (0,0) into each inequality separately and shade only the side that makes it true. Find every corner point by solving pairs of boundary lines, then keep only points that satisfy all constraints before evaluating the objective function.
Uses P(A ∩ B) = P(A)P(B) for any two events, or adds P(A) + P(B) for events that can occur together, and describes mutually exclusive events as independent.
The candidate memorises probability formulas without their conditions and confuses 'cannot occur together' with 'occurrence of one does not affect the other'.
What to do instead
Use P(A ∩ B) = 0 for mutually exclusive events and P(A ∩ B) = P(A)P(B) only for independent events. For any two events use P(A ∪ B) = P(A) + P(B) - P(A ∩ B), and for conditional probability use P(A|B) = P(A ∩ B)/P(B) with P(B) > 0.