# A modest thought on CBSE paper of XII Mathematics 2016

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 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.

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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.

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:

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)

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/)

‘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/)

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:

https://drive.google.com/file/d/0B2U2iudPu5ssYWstRmhUZHg0WmM/view?usp=sharing

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

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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.