COMMON ERRORS DONE BY STUDENTS
Class 12, Mathematics: Common
errors or common mistakes done by students while solving the questions in the examination, Common mistakes done by the students of class XII while solving the mathematical problems.
Chapter 1
Relations and Functions
Let A = { 1, 2, 3}. Check whether R = { (1, 2), (2, 1), (1, 1), (1, 3)} is symmetric or not.
Common Error : Here, (1, 2) ∈ R , (2,1) ∈ R. So, it is symmetric.
Correction : The student thinks that only an ordered pair is to check for Symmetric. So, he
forgets to check for (1, 3).
If A = {1, 2, 3}, check whether R = {(1, 1), (1, 2), (2, 1)} is transitive or not.
Common Error : (1,2) ∈ R , (2,1) ∈ R ⇒ that (1,1) ∈ R. So it is transitive.
Correction : Here the students forget to see for (2, 1) ∈ R , (1, 2) ∈ R implies (2, 2) ∈ R or not
Show that relation R in the set A ={1, 2, 3, 4, 5 } given by R = {(a, b) : a  b is even }, is an equivalence relation.
Common Error : Student forget to write ± while proving transitive
Correction :
Transitive: Let (a, b) ∈ R and (b, c) ∈ R, where a, b, c ∈ A
⇒ a  b is even and b  c is even
⇒ a  b = ± 2k_{1} and b  c = ± 2k_{2}
On adding above equations
a  c = ±2k_{1} ± 2k_{2 }= ±2 (k_{1 }+ 2k_{2})
⇒ a  c = 2(k_{1}+2k_{2})
⇒ a  c is even
⇒ (a, c) ∈ R, so R is transitive.
Chapter  2
INVERSE TRIGONOMETRIC FUNCTIONS
Students must know the principal value branch of the Inverse Trigonometric Functions
Functions

Domain
(Value of
x)

Range
(Principal Value Branch)
(Value of
y)

y = sin^{1}x

[1, 1]


y = cos^{1}x

[1, 1]


y = tan^{1}x

R


y = cot^{1}x

R


y = sec^{1}x

R(1, 1)


y = cosec^{1}x

R(1, 1)


GENERAL DISCUSSION:
Note
Trigonometric Substitution in Algebraic Expressions
S. No.

Expression

Substitution

1


x = tan Ó¨ or x = cot Ó¨

2


x = a tan Ó¨ or x = a cot Ó¨ 
3


x = sin Ó¨ or x = cos Ó¨ 
4


x = a sin Ó¨ or x = a cos Ó¨

5


x = a sec Ó¨ or x = a cosec Ó¨

6


x^{2} = cos 2Ó¨

7


x^{2} = a^{2}cos 2Ó¨

8


x = cos 2Ó¨

9


x = a cos 2Ó¨

Chapter  4
Determinants
Common error : Finding the adjoint of a matrix. Students find cofactors without taking proper sign
Students don't take the transpose of a matrix made from the cofactors.
Remedy:
Teachers must tell students to
Find cofactors by using proper sign and insist them to take transpose of a matrix made from the cofactors by giving more questions for practice
Common Error:
It is wrong answer
Remedy
For expanding a determinant we must use sign as follows
Correct Solution is = 2(2+18)+1(06)+4(01)= 30
Common Error : While using the matrix method of finding the area of triangle
Find the value of k, such that area of triangle with vertices (3, k), (2, 2), (4, 1) is 3 square units.Common Error
3(21)  k(2+4)+1(2+8) = 6
36k+10 = 6
6k = 613
6k = 7
k=7/6 (Wrong Value )
Remedy
Here area of triangle = 3 square unit
So the values of determinant can be 3 or 3
So we must take both the cases
3(21)  k(2+4)+1(2+8) = ± 6
36k+10 = ±6
Either 6k = 613 or 6k = 6  13
Either 6k = 7 or 6k = 19
Either k=7/6 or k = 19/6
So k = { 7/6, 19/6 } (Correct Values)
Prove
that A^{2}  5A + 7I = O, and hence find A^{1}Common Error: For finding the value of A^{1} , some students apply the formula
In this question using above formula is a wrong working
Remedy: Here first of all we prove that A^{2}  5A + 7I = O
Now multiply on both side by A^{1} we get
A^{1} ( A^{2}  5A + 7I) = A^{1}.O
(A^{1}A)A 5(A^{1}A) + 7 A^{1}I = O
IA  5I + 7A^{1} = O
Now we substitute the value of A and I to get A^{1}
Common Mistakes while solving the linear equations
Find the product of matrices A and B and hence solve the following system of equations
x  y = 3, 2x + 3y + 4z = 17, y + 2z = 7
Common Error: In this problem some of the students opts the following procedure
Step 1: Find AB
Step 2: Find A^{1} by applying the following formula
Step 3: Find the value of x, y and z
Remedy: Correct procedure is
Step 1: Find the product AB
Step 2: Find A^{1} by using AB = 6I
Multiply on both side by A^{1} we get
(A^{1}A)B = 6(A^{1}I)
IB = 6 A^{1} or B = 6 A^{1}
Now given equations can be written in matrix form as AX = C where
Step 3: Find the value of X by using the formula
X = A^{1}C
⇒ x = 2, y = 1, z = 4
Chapter 5
Continuity and Differentiability
Common Error: Solving Implicit functions when y is a explicit function of x
For Example: y^{5} = x^{2}
When students differentiating this function w. r. t. x, on both side, then sometimes students by mistake or in rush or unknowingly forget to write the term dy/dx to the L.H.S.
Students simply write here
y^{5} = x^{2}
Differentiating both side w.r.t x we get
5y^{4} = 2x (Which is wrong)
Remedy: y^{5} = x^{2}
Differentiating both side w.r.t x we get
Students should use chain rule here.Common Error: Finding second derivative when parametric function is given
NCERT Book Page No. 192 Q. No. 17
If x = a(cost + t sint) and y = a(sint  t cost) then find d^{2}y/dx^{2}
x = a(cost + t sint)
Differentiating w.r.t x we get
dx/dt = at cost ........... (1)
y = a(sin t  t cost)
Differentiating w.r.t x we get
dy/dt = at sint
Again differentiating w.r.t x we get
Remedy
From Eqn. (1) dx/dt = at cost
Putting this value above we get
Common Error: Improper use of log during logarithmic differentiation. Students generally use wrong log properties.
For Example : Log (a + b) = Log a + Log b
(This is not a property and is a wrong perception made by the students)
Remedy
Students should know the correct formula for logarithmic differentiation
Correct formula is Log (ab) = Log a + Log b
Common Error: For Example:
y = sin x^{x }+
x^{x }
Taking log on both side we get
Log y = Log (sin x^{x }+ x^{x} )
Log y = Log ((sin x^{x }) + Log (x^{x}) (Error)
Remedy
y = sin x^{x }+ x^{x}
Here teacher should insist the students to take y = u + v
Where u = sin x^{x } and v = x^{x}
Chapter 6
Application of Derivatives
Common Error: Incorrect identifications of intervals after obtaining the critical points.
Remedy: After obtaining the critical points it is better to draw the table for identifying the sign of f '(x) in different intervals or students may use Baby Curve Method.
Common Error: Incorrect sign of f ' (x) to identify the increasing / decreasing functions.
Remedy: Correctly observing the sign of f '(x) in the interval before deciding the nature of the function.
Common Error: Incorrect factorization of an algebraic function to obtain the critical point.
Remedy: Students should Revise the question after solving.
Common Error: Finding intervals for Trigonometric Functions involving multiples and submultiples of angles.
Remedy: Here students should recapitulate and make use of the general solutions of Trigonometric Functions.
Chapter 7
Integration
Common Error: Solving Definite Integral Problems by using Substitution Method.
While making the substitution students forget to change the limit and committed mistake
For Example:
Putting x^{2 }+ 3 = t so 2xdx = dt
Remedy:
While making the substitution students should remember to change the limit also
Putting x^{2 }+ 3 = t so 2xdx = dt
If x = 0 then t = 3 and if x = 1, then t = 4
Common Error: Dropping the absolute value when integrating
Remedy: Recall that the following formula
Here absolute value bars on the argument are required. It is certainly true that on occasion they can be dropped after the integration is done. But in most of the cases they are required.
Let us take the following two examples
In first case x^{2 }+ 10 is always > zero. So here absolute bars are not required
But in second case x^{2 } 10 may be + ve,  ve or 0. Its value depends upon the value of x. As the domain of log function is (0, ∞) so x^{2 } 10 should always be positive. So it is compulsory to apply here absolute bars.
Common Error: In indefinite integral students forget to write constant.
Remedy: Student must learn to write constant in indefinite integral.
Common Error: Some time students write more than one constants when they do integration of terms in one problem separately.
Remedy: This is not wrong but it may create problem in finding particular result. So students should write constant once at last of the final answer. Otherwise they should combined all constants.
Chapter 9
Differential Equations
Common Error: In finding the degree of a differential equation when it is not defined.
Remedy: Students should learn, " when a differential equation is not a polynomial in derivatives its degree is not defined".
Students should have more practice and understanding of questions in which degree is not defined.
Common Error: Difference between General solution and Particular solution .
Remedy: Here students should learn General solution contains an arbitrary constant where as Particular solution contains particular value in place of arbitrary constant.
Chapter 10
Vector Algebra
Colinear Vectors
Show that the points are collinear.
Common Error : Here students commonly use the following concept
Three points are collinear if
Remedy: Correct form is
Three points are collinear if
Triangle inequality : For any two vectors and Common Mistake :
Correct Form :
is true only when points are collinear.
Equality of two vectors: Find the value of x, y and z so that and are equal vectors. Where
Common Error: Here students may use the concept
Correct Form: If two vectors are equal then their components are equal
⇒ x = 2, y = 2, z = 1
Chapter 11
Three Dimensional Geometry
Common Mistake : Students start using the cartesian form of equation of line without converting them into standard form.
Direction ratios of the line are 2, 7, 2 ( wrong)
Remedy:
Direction ratios of the line are 1, 7, 2 (correct)
So students should learn and understand the standard form of equation of line
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