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Category Archives: algebra
A problem of log, GP and HP…
Question: If and , pyrove that:
Solution: This is same as proving: y is Harmonic Mean (HM) of x and z;
That is, to prove that is the same as the proof for :
Now, it is given that —– I
and —– II
Let say. By definition of logarithm,
; ;
; ; .
Now let us see what happens to the following two algebraic entities, namely, and ;
Now, …call this III
Now,
Hence, ….equation IV
but it is also given that …see equation II
Hence,
Take log of above both sides w.r.t. base N:
So, above is equivalent to
But now see relations III and IV:
Hence,
Hence,
Hence, as desired.
Regards,
Nalin Pithwa
How to find square root of a binomial quadratic surd
Assume ;
Hence,
If then, , , ,
And, if simultaneously, the values of x, y, z thus found satisfy , we shall have obtained the required root.
Example:
Find the square root of .
Solution:
Clearly, we can’t have anything like
We will have to try the following options:
.
Only the last option will work as we now show:
So, once again, assume that
Hence,
Put , , ;
by multiplication, ; that is ; so it follows that : , , .
And, since, these values satisfy the equation , the required root is .
That is all, for now,
Regards,
Nalin Pithwa
IITJEE Foundation Maths : Tutorial Problems III
 When , find the value of
 Solve the equations: (a) and (b)
 Simplify: (a) (b)
 Find the square root of :
 In a cricket match, the extras in the first innings are onesixteenth of the score, and in the second innings the extras are onetwelfth of the score. The grand total is 296, of which 21 are extras; find the score in each innings.
 Find the value of
 Find the value of : .
 Resolve into factors: (i) (ii)
 Reduce to lowest terms:
 Solve the following equations: (a) ; (b) , , ; (c)
 Simplify:
 A purse of rupees is divided amongst three persons, the first receiving half of them and one more, the second half of the remainder and one more, and the third six. Find the number of rupees the purse contained.
 If , find the value of
 Find the LCM of
 Find the square root of (i) ; (ii)
 Simplify
 Solve the equations: (i) (ii)
 A sum of money is to be divided among a number of persons; if Rs. 8 is given to each there will be Rs. 3 short, and if Rs. 7.50 is given to each there will be Rs. 2 over; find the number of persons.
Regards,
Nalin Pithwa
Two cute problems in HP : IITJEE Foundations\Mains, pre RMO
Problem 1:
If are in AP, show that are in HP.
Proof 1:
Note that a straight forward proof is not so easy.
Below is a nice clever solution:
By adding to each term, we see that:
are in AP.
that is, are in AP.
Dividing each term by .
are in AP.
that is, are in HP.
QED.
Problem 2:
If the terms of an AP are in GP, show that are in GP.
Proof 2:
Once again a straight forward proof is not at all easy.
Below is a “bingo” sort of proof 🙂
With the usual notation, we have
Hence, each of the ratios is equal to
which in turn is equal to
Hence, are in GP.
Cheers,
Nalin Pithwa
Binomial Theorem Tutorial problems I: IITJEE mains practice
I. Expand up to 5 terms the following expressions:
II. Write down and simplify:
 The 8th term of
 The 11th term of
 The 16th term of
 The 6th term of
 The term of
 The term of
 The term of
 The term of
 The 14th term of
 The 7th term of
Regards,
Nalin Pithwa
Theory of Quadratic Equations: Part III: Tutorial practice problems: IITJEE Mains and preRMO
Problem 1:
Find the condition that a quadratic function of x and y may be resolved into two linear factors. For instance, a general form of such a function would be : .
Problem 2:
Find the condition that the equations and may have a common root.
Using the above result, find the condition that the two quadratic functions and may have a common linear factor.
Problem 3:
For what values of m will the expression be capable of resolution into two rational factors?
Problem 4:
Find the values of m which will make equivalent to the product of two linear factors.
Problem 5:
Show that the expression always admits of two real linear factors.
Problem 6:
If the equations and have a common root, show that it must be equal to or .
Problem 7:
Find the condition that the expression and may have a common linear factor.
Problem 8:
If the expression can be resolved into linear factors, prove that P must be be one of the roots of the equation .
Problem 9:
Find the condition that the expressions and may be respectively divisible by factors of the form and .
Problem 10:
Prove that the equation 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.
Problem 11:
If x and y are two real quantities connected by the equation , then will x lie between 3 and 6, and y between 1 and 10.
Problem 11:
If , find the condition that x may be a rational function of y.
More later,
Regards,
Nalin Pithwa.
Theory of Quadratic Equations: part II: tutorial problems: IITJEE Mains, preRMO
Problem 1:
If x is a real number, prove that the rational function 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.
Problem 2:
For all real values of x, prove that the quadratic function has the same sign as a, except when the roots of the quadratic equation are real and unequal, and x has a value lying between them. This is a very useful famous classic result.
Remarks:
a) From your proof, you can conclude the following also: The expression 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.
b) From your proof, and using the above conclusion, you can also conclude the following: Conversely, in order that the expression may be always positive, must be negative or zero; and, a must be positive; and, in order that may be always negative, must be negative or zero, and a must be negative.
Further Remarks:
Please note that the function , where 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 and further with the graph of the general parabola . Note you will just have to convert the expression to a perfect square form.
Problem 3:
Find the limits between which a must lie in order that the rational function may be real, if x is real.
Problem 4:
Determine the limits between which n must lie in order that the equation may have real roots.
Problem 5:
If x be real, prove that must lie between 1 and .
Problem 6:
Prove that the range of the rational function lies between 3 and for all real values of x.
Problem 7:
If , Prove that the rational function can have no value between 5 and 9. In other words, prove that the range of the function is .
Problem 8:
Find the equation whose roots are .
Problem 9:
If are roots of the quadratic equation , find the value of (a) (b) .
Problem 10:
If the roots of be in the ratio p:q, prove that
Problem 11:
If x be real, the expression admits of all values except such as those that lie between 2n and 2m.
Problem 12:
If the roots of the equation are and , and those of the equation be and , prove that .
Problem 13:
Prove that the rational function 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).
Problem 14:
Find the greatest value of for any real value of x. (Remarks: this is maximaminima problem which can be solved with algebra only, calculus is not needed).
Problem 15:
Show that if x is real, the expression has no real value between b and a.
Problem 16:
If the roots of be possible (real) and different, then the roots of will not be real, and viceversa. Prove this.
Problem 17:
Prove that the rational function will be capable of all real values when x is real, if and have the same sign.
Cheers,
Nalin Pithwa
Theory of Quadratic Equations: Tutorial problems : Part I: IITJEE Mains, preRMO
I) Form the equations whose roots are:
a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k)
II) Prove that the roots of the following equations are real:
i)
ii)
III) If the equation has equal roots, find the values of m.
IV) For what values of m will the equation have equal roots?
V) For what value of m will the equation have roots equal in magnitude but opposite in sign?
VI) Prove that the roots of the following equations are rational:
(i)
(ii)
VII) If are the roots of the equation , find the values of
(i)
(ii)
(iii)
VIII) Find the value of:
(a) when
(b) when
(c) when
IX) If and are the roots of form the equation whose roots are and /
X) Prove that the roots of are always real.
XI) If are the roots of , find the value of (i) (ii)
XII) Find the condition that one root of shall be n times the other.
XIII) If are the roots of form the equation whose roots are and .
XIV) Form the equation whose roots are the squares of the sum and of the differences of the roots of .
XV) Discuss the signs of the roots of the equation
XVI) If a, b and c are odd integers, prove that the roots of the equation cannot be rational numbers.
XVII) Given that the equation has four real positive roots, prove that (a) (b) , where equality holds, in each case, if and only if the roots are equal.
XVIII) Let be a quadratic polynomial in which a and b are integers. Given any integer n, show that there is an integer M such that .
Cheers,
Nalin Pithwa.
Set theory, relations, functions: preliminaries: Part V
Types of functions: (please plot as many functions as possible from the list below; as suggested in an earlier blog, please use a TI graphing calculator or GeoGebra freeware graphing software):
 Constant function: A function given by , where is a constant. It is a horizontal line on the XYplane.
 Identity function: A function given by . 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.
 Oneone or injective function: If different inputs give rise to different outputs, the function is said to be injective or oneone. That is, if , where set A is domain and set B is codomain, 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.
 Onto or surjective: If a function is given by 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 codomain. it would be illuminating to construct your own pictorial examples of such a function.
 Bijective function or oneone onto correspondence: A function which is both oneone and onto is called a bijective function. (It is both injective and surjective). Only a bijective function will have a welldefined inverse function. Think why! This is the reason why inverse circular functions (that is, inverse trigonometric functions have their domains restricted to socalled principal values).
 Polynomial function: A function of the form , where n is zero or positive integer only and is called a polynomial with real coefficients. Example. , where , is called a quadratic function in x. (this is a general parabola).
 Rational function: The function of the type , 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.
 Absolute value function: Let be given by when and , when for any . Note that since the radical sign indicates positive root of a quantity by convention.
 Signum function: Let where , when and when and when . Such a function is called the signum function. (If you can, discuss the continuity and differentiability of the signum function). Clearly, the domain of this function is full whereas the range is .
 In part III of the blog series, we have already defined the floor function and the ceiling function. Further properties of these functions are summarized (and some with proofs in the following wikipedia links): (once again, if you can, discuss the continuity and differentiablity of the floor and ceiling functions): https://en.wikipedia.org/wiki/Floor_and_ceiling_functions
 Exponential function: A function 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.
 Logarithmic function: Let a be a positive real number with . If , where , then y is called the logarithm of x with base a and we write it as . (By the way, the logarithmic function is used in the very much loved mp3 music :))
Regards,
Nalin Pithwa