Reference: A Beautiful Mind by Sylvia Nasar, the life of mathematical genius and Nobel Laureate, John Nash, A Touchstone Book, Published by Simon and Schuster.
…”Geniuses”, the mathematician Paul Halmos wrote, “are of two kinds: the ones who are just like all of us, but very much more so, and the ones who apparently have an extra human spark. We can all run, and some of us can run the mile in less than 4 minutes, but there is nothing that most of us can do that compares with the creation of the Great G-minor Fugue.” Nash’s genius was of that mysterious variety more often associated with music and art than with the oldest of all sciences. It wasn’t that his mind worked faster, that his memory was more retentive or that his power of concentration was greater. The flashes of intuition were non-rational. Like other great mathematical intuitionists — Georg Friedrich Bernhard Riemann, Jules Henri Poincare, Srinivasa Ramanujan —- Nash saw the vision first constructing the laborious proofs long afterwards. But even after he would try to explain some astonishing result, the actual route he had taken remained a mystery to others who tried to follow his reasoning. Donald Newman, a mathematician who knew Nash at MIT in the 1950s, used to say about him that “everyone else would climb a peak by looking for a path somewhere on the mountains. Nash would climb another mountain altogether and from that distant peak would shine a searchlight back onto the first peak.”
Hats off,
Nalin Pithwa.
.
and 
may have a common root.
and 
may have a common linear factor.
be capable of resolution into two rational factors?
equivalent to the product of two linear factors.
always admits of two real linear factors.
and 
have a common root, show that it must be equal to 
or 
.
and 
may have a common linear factor.
can be resolved into linear factors, prove that P must be be one of the roots of the equation 
.
and 
may be respectively divisible by factors of the form 
and 
.
for every real value of x, there is a real value of y, and for every real value of y, there is a real value of x.
, then will x lie between 3 and 6, and y between 1 and 10.
, find the condition that x may be a rational function of y.
can have all numerical values except such as lie between 2 and 6. In other words, find the range of this rational function. (the domain of this rational function is all real numbers except 
quite obviously.
has the same sign as a, except when the roots of the quadratic equation 
will always have the same sign, whatever real value x may have, provided that 
is negative or zero; and if this condition is satisfied, the expression is positive, or negative accordingly as a is positive or negative.
and 
is a parabola. The roots of this 
are the points where the parabola cuts the y axis. Can you find the vertex of this parabola? Compare the graph of the elementary parabola 
, with the graph of 
where 
. Note you will just have to convert the expression 
may be real, if x is real.
may have real roots.
must lie between 1 and 
.
lies between 3 and 
for all real values of x.
, Prove that the rational function 
can have no value between 5 and 9. In other words, prove that the range of the function is 
.
.
are roots of the quadratic equation 
, find the value of (a) 
(b) 
.
be in the ratio p:q, prove that 
admits of all values except such as those that lie between 2n and 2m.
are 
and 
, and those of the equation 
be 
and 
, prove that 
.
will be capable of all values when x is real, provided that p has any real value between 1 and 7. That is, under the conditions on p, we have to show that the given rational function has as its range the full real numbers. (Of course, the domain is real except those values of x for which the denominator is zero).
for any real value of x. (Remarks: this is maxima-minima problem which can be solved with algebra only, calculus is not needed). 
has no real value between b and a.
will not be real, and vice-versa. Prove this.
will be capable of all real values when x is real, if 
and 
have the same sign.
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
(j) 
(k) 
has equal roots, find the values of m.
have equal roots?
have roots equal in magnitude but opposite in sign?
when 
when 
when 
and 
/
are always real.
are the roots of 
(ii) 
and 
.
.
has four real positive roots, prove that (a) 
(b) 
, where equality holds, in each case, if and only if the roots are equal.
be a quadratic polynomial in which a and b are integers. Given any integer n, show that there is an integer M such that 
.
, find 
.
, show that 
show that 
.
.
is defined by 
where 
for 
and 
for 
, find 
; the domain and range of f; and the value of x for which 
given by 
and 
, find (a) 
(b) 
(c) 
and (d) 
and 
if (a) 
and 
(b) 
and 
and 
prove that 
(b) 
(c) 
(d) 
(e) 
(f) 
, when 
(b) 
for 
(c) 
for all 
, find 
, and 
. (b) If 
, find 
, 
, 
, find 
.
(b) 
(c) 
(d) 
, and 
for 
, find x.
, then find the value of x satisfying 
.
and 
be the set of integers. Define 
by 
. Show that f is a function from A to Z. Also, find the range of f.
given by 
for all 
.
defined from A to B where 
and 
and 
. Find 
and 
and 
.
and 
(b) 
and 
.
prove that 
and 
, prove that 
.
and 
find 
given by 
and 
be defined by 
and 
. Find whether the two functions f and g are same, or not same. Justify your answers.
, where 
is a constant. It is a horizontal line on the XY-plane.
. It maps a real value x back to itself. It is a straight line passing through origin at an angle 45 degrees to the positive X axis.
, where set A is domain and set B is co-domain, if further, 
such that 
, then it follows that 
. Sometimes, to prove that a function is injective, we can prove the conrapositive statement of the definition also; that is, 
where 
, then it follows that 
. It might be easier to prove the contrapositive. It would be illuminating to construct your own pictorial examples of such a function. 
such that 
, that is, the images of all the elements of the domain is full of set Y. In other words, in such a case, the range is equal to co-domain. it would be illuminating to construct your own pictorial examples of such a function.
, where n is zero or positive integer only and 
is called a polynomial with real coefficients. Example. 
, where 
, where 
, where 
and 
are polynomial functions of x, defined in a domain, is called a rational function. Such a function can have asymptotes, a term we define later. Example, 
, which is a hyperbola with asymptotes X and Y axes.
when 
and 
, when 
for any 
since the radical sign indicates positive root of a quantity by convention.
, when 
and 
when 
and 
when 
whereas the range is 
.
given by 
where 
is called an exponential function. An exponential function is bijective and its inverse is the natural logarithmic function. (the logarithmic function is difficult to define, though; we will consider the details later). PS: Quiz: Which function has a faster growth rate — exponential or a power function ? Consider various parameters.
. If 
, where 
. (By the way, the logarithmic function is used in the very much loved mp3 music :-))
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
(j) 
(k) 
(l) 
(m) 
(n) 
(o) 
(p) 
as appropriate.
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
. For what real numbers a is the equation true? For what real numbers is it false?
.
…..why ?
….why ?
….why?
….why ?
for any numbers a and b.
and 
, what can you say about x?
(b) 
(c) 
. Express the coordinates of P as functions of the slope of the line joining P to the origin.
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
(j) 
(k) 
(l) 
(b) 
(b) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
(j) 
(k) 
, 
, 
(ii) 
, 
, 
, and 
, find the following: (a) 
(b) 
(c) 
(f) 
(g) 
(h) 
and 
, find the following: (a) 
(b) 
(c) 
(f) 
, 
, and 
(b) 
(c) 
(d) 
(e) 
(f) 
, and 
, find formulas or formulae for the following: (a) 
(b) 
(c) 
(d) 
(e) 
(f) 
, 
, and 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
(k) 
(l) 
(m) 
, 
, find 
, 
, find 
, 
, find 
, 
, find 
, 
(b) 
?
?
for all real x? Give reasons for your answer.
when 
, when 
.
.
=0$
. Note that columns 2 and 3 of the three by three determinant do not change.
as a quadratic in a with b and c as parameters.
, which in turn equals:
are cube roots of unity.
.
.
