I am not a teacher of Mathematics in any school and I don’t teach
in any institute either, nor do I have any coaching institute. This is has been
written by me simply to share my views as a common man on the gradual and
perpetual deterioration in studies of Mathematics. Basically certain segments of this paper are addressed to students of
XI-XII (with Mathematics) and hence I have divided this article in two parts,
one, about the CBSE paper of XII Mathematics, 2016, and two, about the
exquisiteness and elegance of mathematics.
.
There was a big hue and cry on the issue of
the questions in the CBSE paper of XII Mathematics (held on 14th
March 2016) being very difficult. On 20th March 2016, Hindi
newspaper दैनिक भास्कर published a big write-up with the caption “तीन घंटे में तो टीचर भी हल
नहीं कर सकते गणित का ऐसा पर्चा” on how the students suffered miserably due to the paper not only
being lengthy but also being out of course.
I went through all the 26 questions of the
Maths paper and solved it.
Strange is: The paper is not only from within
the syllabus, it is one of the easiest of those of preceding years. And I have
tried to look into why and how a teacher (a real teacher) of mathematics cannot
solve such a simple paper in three hours.
Stranger is: Almost
80-90 percent of students of XII (PCM) are attending Maths coaching in highly
prestigious institutes, and claiming the paper to be tough.
Strangest is: To understand what the student
are studying and what they are being taught in their schools and in their
coaching. I feel surprised at the (low ??) level of art of teaching of Mathematics at school and at coaching institutes when I find that many students
of Maths XI class are not able to answer some very basic questions, like: why
do we need to introduce radian in place of degree for measuring angle and what
is one radian?, and why do we have a chapter in XI Maths with title
‘Trigonometric Functions”, and not “Trigonometry” only?, and What is the
difference between Relation and Function?, and many more.
Further to my astonishment, in many
supplementary books of Mathematics I see the most tedious methods of solving
even the easiest problems. In one (so very famous??) supplementary book of
Maths XI, I found that the author was explaining the two forms of ellipse (One
having major axis along X axis and another having major axis along Y axis) in
most ridicules way, where this book says: always make ‘a’ and ‘b’ the
denominators of X and Y respectively for both the type of ellipse. This means
for first type of ellipse ‘a’ shall represent major axis, and for second type
of ellipse ‘a’ shall represent minor axis.
What rubbish? This makes even the prudent
student scare of problems related with conic section because he will have to
remember two different sets of formulae for two types of ellipse which he would
not have done, had he would have taken ‘a’ always a major axis (This is what is
precisely done by NCERT). Similar problem arises in case of Hyperbola if the
student goes by the opinion of author of that supplementary book.
In the same book for XII Maths, in Relations
and Functions, the author defines two functions, f and g; and asks the students
to work out ‘fog’ and ‘gof’ and writes in brackets ‘if they exist’. This
question gives the impression that ‘fog’ or ‘gof’ may either exist or may not
exist, which otherwise means that occurrence of ‘fog’ (or ‘gof’) is one of two
mutually exclusive events, i.e. occurring or not occurring.
Preposterous. Isn’t it? For, for any given
two functions ‘f’ and ‘g’, the ‘fog’ or ‘gof’ may exist for segmented
intervals, and may not exist for many other segmented intervals. For example,
if ‘f’ is sin x, and ‘g’ is
log x; then ‘fog’ is ‘sin log x’ and ‘gof’ is ‘log sin x’. Obviously, ‘fog’
i.e. ‘sin log x’ exists for all x belonging to positive real numbers but does
not exist for negative real numbers. Similarly, ‘gof’ i.e. ‘log sin x’ exists
for segmented intervals like, {0,π},
{2π,3π},
{4π,5π},
{6π,7π},….,
{2nπ,(2n+1)π}
but does not exists for {π,2π}, {3π,4π}, {5π,6π},….,
{(2n-1)π, 2nπ}. There are question galore in this (these)
supplementary book(s) where the छात्र मात्र अपना सिर धुन सकता है.
I find that almost all school teachers of
Maths are heavily banked upon making use of supplementary books in the class.
Why? The reason is clear. This activity does not promote the (mathematical)
intellect of the student; rather promote the sale of the book.
Hence, if by any chance, the paper of XII
Maths is set in such a way which demands some basic understandings of concepts
of mathematics, the students – nay – teachers implore the (so called)
difficulty level and total google is flooded with news items of how the
students have been metal-tortured by this paper, and a debate has been asked
for in the Parliament, the HRD Minister has been requested to intervene and so
on. Some instances of this hue and cry are given below:
1.
People's view on CBSE Class 12 Mathematics paper 2016
(http://indiatoday.intoday.in/education/story/cbse/1/622550.html)
2. CBSE
Class 12 Maths exam 2016: Remedial steps promised to soften 'very tough' paper
blow (http://indiatoday.intoday.in/education/story/cbse-maths-paper-remedial-steps/1/622518.html)
3. CBSE
Class 12 Maths exam 2016: Board sets up committee of subject experts (http://indiatoday.intoday.in/education/story/cbse-class-12-maths-subject-expert/1/622400.html)
4.
CBSE Maths paper: Govt favours inquiry, Board denies question
paper leak (http://indianexpress.com/article/education/cbse-class-12-maths-paper-leak-govt-favours-inquiry/)
5.
‘Tough’ Class XII paper: Maths test concerns could be due to changes in
curriculum, says CBSE (http://indianexpress.com/article/education/tough-class-xii-paper-maths-test-concerns-could-be-due-to-changes-in-curriculum-says-cbse/)
6.
CBSE class XII students
in tears after math exam (http://www.thehindu.com/news/cities/chennai/cbse-class-xii-students-in-tears-after-math-exam/article8354049.ece)
The furor is multi-dimensional. There are
statements of saddened students from all across the country, statement of
teachers cursing CBSE, but never looking into their own selves as to what
quality of study they have bestowed upon their students, and what education
they are imparting. A phobia of Mathematics is engulfing all students and not
only coaching institutes are practicing billowing of this phobia in order that
more and more students come to them; the supplementary books are solving the
problems in lengthier way.
I come to the paper. I have in my hand the Set
3 of the Mathematics paper (CBSE, XII, 2016). This question paper is available in
public domain at
For the first six questions of one mark each,
the public outcry (created more by media) is much about their being very
lengthy. Strange, the first three questions are from three questions are from
Vector and 3D Geometry, and are so very easy. Each question demands only 2-3
lines from a student.
Question 1 is based on a very simple formula
(NCERT Maths II, 10.5.3, Page 438}.
Question No. 2 is just a matter of simple
understanding. Any cross product of two vectors gives a vector perpendicular to
the two vectors, and hence we have two such vectors equal and opposite. In
question 3, if the student knows how to put an equation of a plane in intercept
form, and how to change the equation of a plane from Cartesian form to Vector
form, there is nothing typical in this question.
Question 4 is a matter of two lines as it is
(x+3)(2x) – (-2)(-3x) = 8. Question 5 is a matter of just one matrix, but the
student must remember that in case of some X = AB (where X, A and B are
matrices of compatible order for multiplication), the elementary column
operations are applied on second matrix of RHS (NCERT Maths, I, Page 92,
3.8.1).
Though Question 6 appears to be from
Matrices, it is virtually a question involving ‘Fundamental principle of
counting’ which the students learnt in XI in Permutations and Combinations {A
similar question is given in NCERT Maths I, Exercise 3.1, Question 10}.
Hence, prima facie दैनिक भास्कर का यह कथन
ठीक नहीं है कि एक अंक वाले सवालों को हल करने में 5-6 मिनट का समय लगा.
Similarly, in Section B, all questions (from Question
nos. 7 to 19) are very simple. Both the options in first question (Question no.
7) are from Definite Integral and in option 1, the use of IV property of
Definite Integral makes it solvable in 5-6 lines. Question 8 is from
Probability and does need some brain storming, but first option is quite easy.
Question 9 is such an easy question when the student puts x2=t, and
solves it by applying partial fractions. Question 10 is easy where the
derivative is to be worked out using parametric coordinates. Solution to
question 11, pertaining to Three D Geometry, is a matter of hardly 5-6 lines as
when a straight line crosses XZ plane, the Y coordinate of point of cross is
equal to zero. Question 12 of Indefinite Integral is very easy. Question 13 has
come from Tangent and Normal and is an easy question which is solved once the
derivative of the function of the curve is taken at the given point.
Question 14 is easy one from Matrices and
Determinants, where two simultaneous equations in two variables are formed, and
solved. Question 15 is a homogeneous differential equation, and is very easy to
solve in 5-6 lines by putting y=vx, and then differentiating with reference to
‘x’. Question 16 is from Vectors, and first part is easily done by using
parallelogram rule of summation of two vectors, hardly a matter of 4-5 lines;
and second part uses the formula of area of a parallelogram equal to product of
diagonals divided by 2. Easy.
The Question 17 is from Inverse Trigonometric
Functions. The option one of this question is a very easy question, and is solved in
hardly 6-7 lines by taking sin inverse(x-1) = sin inverse(x) - Cos inverse(x),
and taking sin of both the sides, and applying sin(A-B) = sinAcosB - cosAsinB. The
second option is again not difficult and is done the same way the first option
is solved. The first option of question 18 is from differential calculus and is
done by putting y = u+v where u and v are two given functions of ‘x’, and
derivative of y is obtained by first taking log of u and v separately and differentiating
with respect to ‘x.’, really a very easy question (NCERT Maths Part I, Exercise 5.5,
question 9}.
But the second option of this question, which is question of second order
differentiation, is one of the easiest questions and takes hardly 4-5 lines to
solve it completely. What
more do you require from a question of Mathematics, which fetches you 4 marks
out of 4 by writing simply 4-5 lines. And it is virtually the same question as
given in NCERT (Maths Part I, Exercise 5.7, question 13}.
And finally question number 19 from Section B is so very simple.
In fact, it is a solved example in NCERT, II (Example 25, page 415). A student
must understand that any circle in II quadrant will have negative x coordinate
and positive y coordinate for the centre of the circle, and the radius of
family of this type of circles will be equal to the numerical value of either
of the coordinate. Hence, this family of circle can be represented by (x+a)
square + (y-a) square = ‘a’ square, where (-a, a) are the coordinates of centre
and ‘a’ is the radius. The question is very easy.
Similarly, in Section C (from question no. 20
to 26), again, not all questions are very difficult, and are from within the
syllabus. The first question (question no. 20) is from Areas of the curves
using Definite Integral. This question divides the total region into three
segments, and three areas, using definite integral, have to be worked out. The
question number 21 is from Relations and Functions and is a traditional
question, and is not difficult.
The first option of question 22 is from
Determinant and it seems (when you look at it) to be a very lengthy question.
But it is a question of hardly 5 variations of the given determinant after
applying two elementary column operations, i.e. C2 becomes C2-C1, and C3
becomes C3-C1. Finally you get A=B=C, which proves that the triangle ABC is
isosceles. What more do you expect from a question to get 6 out of 6 by solving
a question in 6 variations of the given determinant. The second option is from Matrices and
Determinants and forms three simultaneous linear equations in three variables
which are then solved using Matrices (X = A inverse B, where X, A and B are
matrices of compatible order for multiplication). This is an easy question but its
solution is much lengthier than that of the option 1.
Question 23 is from Linear Programming and is
a kind of diet problem. It is a traditional question and easy one. The student must take care in plotting the
graph.
Question 24 is from 3D Geometry, though, in
the question, the point P and the plane have been given in Vector form. The
question is easily solvable once the Vector form is converted into Cartesian
form. Question 25 has two options, from Applications of Derivatives (Maxima and
Minima) and answers to both the questions are lengthy, but the option 2 is
simpler. Question 26 is from Probability and is very easy.
I repeat that I am not a teacher of Mathematics in any school and
I don’t teach in any institute either, nor do I have any coaching institute.
But, to vindicate my point of view that the paper was not that tough that was
media-hyped, I myself have solved certain questions of this paper. A pdf file
containing 26 pages and having solutions (as done by me) to Question Nos. 7
(first option), 8 (second option), 9, 10, 11, 12, 13, 15, 16, 17 (both options),
18 (second option), 19, 20, 22 (first option), 24, 25 (second option) and 26 is
available in public domain at https://drive.google.com/file/d/0B2U2iudPu5ssSW5ONGllYkExNkU/view?usp=sharing.
I have solved these questions simply to convey the effortlessness
these questions are carrying vis-à-vis the hardship that was cried for.
About the gradual loss of exquisiteness and
elegance of mathematics.
I find that the standard of teaching
mathematics (particularly in most of the schools, nay, public school) is
diminishing. The teachers don’t do any brain-storming, or any R&D in
providing newer ways of solving problems. They are teaching mathematics simply
for the sake of teaching and not for letting the students savor the classical
aspect of mathematics. I tell you my point of view. I developed a knack for
Maths which culminated in true love for mathematics, not because I was very
good at Maths in my school or college days, but because I was taught Maths by
some of best brains of the subject. Though they
were not IITians, they had that inkling which made the most boring subject i.e.
Maths most romantic to me. These teachers included those of Hindu Inter College
Amroha, those of Meerut College, Meerut (Some big names I feel proud of paying
my gratitude include Mr. M L Khanna, Mr. A R Vashishtha, Mr. S N Goel, Mr. B D
Gupta, Mr. S D Sharma and many more). All these stalwarts are known not only
for the classical techniques of teaching Mathematics but also for the books
they wrote and which I still possess in my library, which again I am proud of.
I remember, in eighties, when me and many
friends of mine were doing their XII with PCM in UP, there was no occurrence of
suicides of students due to
Mathematics or Physics or Chemistry. The syllabus of Mathematics was vast,
really immense. We had 7 books of Mathematics in XI-XII and the Board Paper of
XII covered the syllabus for both the years and there were three papers for
Mathematics. क्या शानदार दिन थे. There were no coaching institutes and if
there were some private tutors for Maths and Physics, our parents could not
afford. And we enjoyed Mathematics.
The coaching came into
vogue in Nineties. And since then, Mathematics
started losing its lure. The teachers in school started failing in invoking
interest in students for Mathematics and they finally submitted themselves to a
large market of supplementary books and became a puppet in the hands of
coaching institutes. And this resulted into what? This resulted into a
perpetual destandardizing, debunking and destabilizing of government schools
where, once upon a time, the invaluable treasure of teachers of Mathematics and
Physics was preserved, nourished, refined and cultured. I remember, when I
joined Meerut College, Meerut for B.Sc (PCM) after doing XII from Amroha (UP),
the Government Inter College, Meerut (GIC, in short) had a very big name, fame
and charisma. I remember, apart from some of the outsider students like me,
most of the students were from GIC and they were superb in all three subjects,
i.e. Physics, Chemistry and Mathematics.
I learn कि जीआईसी तो
ज़मींदोज़ हुआ जाता है और उसके स्थान पर अनेकों पब्लिक स्कूल्स की इमारतें खड़ी
हो गई हैं. और कमोबेश यह स्थिति हर शहर के सरकारी स्कूलों की है and I lament.
Doing Mathematics is fun, thrill and excitement. I
don’t know why the teachers don’t make themselves belong to the subject. Why
don’t they open before the students the euphoria Mathematics possesses? It is
only Mathematics that helped human race to achieve greater insights into the
enigma of nature. Could we expect happening of various discoveries of Physics, both
Mechanical (Newtonian) and Quantum, without deploying the means of differential
and integral calculus in general or of concepts of infinite limit in
particular? If you look at the NCERT books of Physics (and of Chemistry too)
for XI and XII, you will see that more or less every discovery is associated
with use of Mathematics. But the most astonishing and most divine feature of
Mathematics is to reach same result in solving any problem while using
different methods of solving. For example, if you have to find the area of a
triangle, the two most elementary and traditional but different methods
which are learnt in early classes are ‘half of base into height’ or ‘square root
of {s(s-a)(s-b)(s-c)} where s is ‘half of a+b+c’ and where a, b, c are the
three sides of the given triangle. Later, one learns use of trigonometry in
finding out area of a given triangle provided one side and one angle is given.
You get the same result. Later in XII class, you learn two different methods.
These are: use of integral calculus (area of curves) and use of determinants. You
get the same results.
Similarly, in XII Maths, the three dimensional geometry is taught
using vectors, while the two are entirely two different branches of pure
mathematics. The fact is that in their
modern form, the vectors were discovered late in the 19th century when Josiah Willard Gibbs (US) and Oliver Heaviside (Britain)
independently developed vector analysis to
express the new laws of electromagnetism discovered
by the Scottish physicist James Clerk Maxwell,
while, the concepts of 2D geometry and of 3D
Geometry are much older. Further, see the example of simultaneous linear
equations (say in three variables). You solve it by traditional method of
replacement or by use of matrices and determinants (XII class), you get the
same result.
Hence, enjoy Mathematics, never be scared of it. Calculus
(particularly differential calculus) becomes too tough to grasp if the student
(or the teacher) sidetracks its understanding through graphs. Make preparations
right from the beginning under certain strategy, especially for six marks
questions which are seven in numbers. At least three chapters which don’t
require any previous knowledge of something or prerequisites and which give you
at least three questions of 6 marks are: Linear Programming (LP), Probability
and Matrices & Determinants. LP in XII class is very easy, as in XI class
you have already learnt to draw the graph of the problem. In XII class, you
have to frame the problem, draw the graph and solve it. So take these three
chapters from very beginning and do as much questions as possible. Apart from
this, remember that the first chapter (Relations and Functions) may be a bit
difficult if you don’t have proper ideas of relations. This is a chapter which
should be taken care of regularly throughout the year. You can always expect a
six marks question from this.
Always learn the
basics of everything in Mathematics. And if you have selected something in
Mathematics, do everything of that something. There are no shortcuts in
Mathematics. Sharpening of Mathematical inkling will make you understand other
subjects in a more analytical way. Further, Mathematics gives you an edge over
other candidates in any competitive exam. Remember, it is the Mathematics only
which disconnects Maths students from non-Maths students in scoring marks and in
giving an extra edge in any competitive exam.
God Himself has
created integers, Calculus is another Avatar of God, and hence here is an
appeal to all students of Mathematics. Do Mathematics the way you worship God. You
will get results, better results and best outcomes from within you.
Regards
.. Sanjay Mohan Bhatnagar (Bhopal)
anukriti.maths@yahoo.in
8871503701
