xNEW HORIZON PU COLLEGE

II PUC MATHS PROJECT DETAILS

GROUP 1
1. Define sin-1x ,draw the graph of sin x and sin -1 x
2. Define cosec-1 x, draw the graph of cosec x and cosec-1 x
3. Illustrate by giving examples, a function which is 1-1 need not be onto
and a function which is onto need not be 1-1
4. Write an assignment on the life history and contribution to
mathematics by one Indian and one foreign Mathematician
Inverse sine one of the inverse trigonometric The inverse sine function
or Sin-1 takes the ratio, Opposite Side / Hypotenuse Side and produces
angle θ. Sin inverse is denoted by sin-1
Sinx graph

Sin inverse graph

Cos inverse
In a right-angled triangle, the cosine function is defined as the
ratio of the length of base or adjacent side of the triangle(adjacent
to angle) to that of the hypotenuse(the longest side) of the triangle.
The Inverse Cosine function is the inverse of the Cosine function
and is used to obtain the value of angles for a right-angled
triangle.

Cos inverse graph

Cos graph
, a function which is 1-1 need not be onto

(i) Let function f:N→N, given by f(x)=2x

Calculate f(x1):
⇒ f(x1)=2x1
Now, calculate f(x2):
⇒ f(x2)=2x2
Now, f(x1)=f(x2)
⇒ 2x1=2x2
⇒ x1=x2
Hence, if f(x1)=f(x2), x1=x2 the function f is one−one.
Now, f(x)=2x
Let f(x)=y, such that y∈N
⇒ 2x=y
⇒ x=2y
If y=1
x=21=0.5, which is not possible as x∈N
Hence, f is not onto.
So, the function f:N→N, given by f(x)=2x, is one-one but not onto.

a function which is onto need not be 1-1

Here, f(x)=f(1)=1 and
⇒ f(x)=f(2)=1
Since, different elements 1,2 have same image 1,
∴ f is not one-one.
Let f(x)=y, such that y∈N
Here, y is a natural number and for every y, there is a value
of x which is natural number.

Hence f is onto.
So, the function f:N→N, given by f(1)=f(2)=1 is not one-one but
onto.
the life history and contribution to mathematics by Indian
Srinivasa Ramanujan, the mathematical genius, came to be recognized only
posthumously for his incredible contribution to the world of Mathematics.
Leaving this world at the young age of 32, Srinivasa Ramanujan (1887-
1920) contributed a great deal to mathematics that only a few could
overtake in their lifetime.

Born in Erode (Tamil Nadu), Ramanujan demonstrated that he had an

exceptional intuitive grasp of mathematics at a very young age. He began
developing his theories in mathematics and published his first paper in
1911. Infact, he was the second Indian to be included as a Fellow of the
Royal Society 9a fellowship of the world’s most respected and famous
scientists) in 1918.

The field of number theory in mathematics was enriched with his intuitive
research and his vast contribution. Every year, Srinivasa Ramanujan’s
birth anniversary on December 22 is commemorated as
National Mathematics Day.

A wizard of intuition

Ramanujan has been recognized as one of the greatest mathematicians of

his time. Surprisingly, he never had any formal training in mathematics.
Most of his mathematics discoveries were based on sheer intuition, and
most of them were proved to be right much later. GH Hardy, a famous
British Mathematician, mentored him at Cambridge and encouraged
Ramanujan to publish his findings in several papers.

Inspiring legacy

The Indian mathematician had few opportunities during his lifetime to

showcase his talents. Still, his passion for giving his best to mathematics did
not hold him back from leaving back his legacy for the world to marvel at.
Ramanujan died at the age of 32 after contracting tuberculosis. But he has
left behind a legacy that continues to inspire mathematicians to this day.
Ramanujan’s contributions to mathematics

 Ramanujan compiled around 3,900 results consisting of equations

and identities. One of his most treasured findings was his infinite
series for pi. This series forms the basis of many algorithms we use
today. He gave several fascinating formulas to calculate the digits of
pi in many unconventional ways.
 He discovered a long list of new ideas to solve many challenging
mathematical problems, which gave a significant impetus to the
development of game theory. His contribution to game theory is
purely based on intuition and natural talent and remains unrivalled
to this day.
 He elaborately described the mock theta function, which is a concept
in the realm of modular form in mathematics. Considered an enigma
till sometime back, it is now recognized as holomorphic parts of mass
forms.
 One of Ramanujan’s notebooks was discovered by George Andrews
in 1976 in the library at Trinity College. Later the contents of this
notebook were published as a book.
 1729 is known as the Ramanujan number. It is the sum of the cubes
of two numbers 10 and 9. For instance, 1729 results from adding
1000 (the cube of 10) and 729 (the cube of 9). This is the smallest
number that can be expressed in two different ways as it is the sum of
these two cubes. Interestingly, 1729 is a natural number following
1728 and preceding 1730.
 Ramanujan’s contributions stretch across mathematics fields,
including complex analysis, number theory, infinite series, and
continued fractions.

Ramanujan’s other notable contributions include hypergeometric series,

the Riemann series, the elliptic integrals, the theory of divergent series, and
the functional equations of the zeta function.
Life histroy of ramanujam
Born in 1887, Ramanujan’s life, as said by Sri Aurobindo, was a “rags to
mathematical riches” life story. His geniuses of the 20th century are still
giving shape to 21st-century mathematics.
Discussed below is the history, achievements, contributions, etc. of
Ramanujan’s life journey.
Birth –
  Srinivasa Ramanujan was born on 22nd December 1887 in the south
Indian town of Tamil Nad, named Erode.
 His father, Kuppuswamy Srinivasa Iyengar worked as a clerk in a
saree shop and his mother, Komalatamma was a housewife.
 Since a very early age, he had a keen interest in mathematics and had
already become a child prodigy
Srinivasa Ramanujan Education –

 He attained his early education and schooling from Madras, where

he was enrolled in a local school
 His love for mathematics had grown at a very young age and was
mostly self-taught
 He was a promising student and had won many academic prizes in
high school
 But his love for mathematics proved to be a disadvantage when he
reached college. As he continued to excel in only one subject and kept
failing in all others. This resulted in him dropping out of college
 However, he continued to work on his collection of mathematical
theorems, ideologies and concepts until he got his final breakthrough
Final Break Through –

 S. Ramanujam did not keep all his discoveries to himself but

continued to send his works to International mathematicians
 In 1912, he was appointed at the position of clerk in the Madras Post
Trust Office, where the manager, S.N. Aiyar encouraged him to
reach out to G.H. Hardy, a famous mathematician at the Cambridge
University
 In 1913, he had sent the famous letter to Hardy, in which he had
attached 120 theorems as a sample of his work
 Hardy along with another mathematician at Cambridge,
J.E.Littlewood analysed his work and concluded it to be a work of
true genius
 It was after this that his journey and recognition as one of the
greatest mathematicians had started
Death –

 In 1919, Ramanujan’s health had started to deteriorate, after which

he decided to move back to India
 After his return in 1920, his health further worsened and he died at
the age of just 32 years