Theory of Quadratic Equations: part II: tutorial problems: IITJEE Mains, preRMO

Problem 1:

If x is a real number, prove that the rational function \frac{x^{2}+2x-11}{2(x-3)} 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 x=3 quite obviously.

Problem 2:

For all real values of x, prove that the quadratic function y=f(x)=ax^{2}+bx+c has the same sign as a, except when the roots of the quadratic equation ax^{2}+bx+c=0 are real and unequal, and x has a value lying between them. This is a very useful famous classic result. 


a) From your proof, you can conclude the following also: The expression ax^{2}+bx+c will always have the same sign, whatever real value x may have, provided that b^{2}-4ac 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 ax^{2}+bx+c may be always positive, b^{2}-4ac must be negative or zero; and, a must be positive; and, in order that ax^{2}+bx+c may be always negative, b^{2}-4ac must be negative or zero, and a must be negative.

Further Remarks:

Please note that the function y=f(x)=ax^{2}+bx+c, where a, b, c \in \Re and a \neq 0 is a parabola. The roots of this y=f(x)=0 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 y=x^{2}, with the graph of y=ax^{2} where a \neq 0 and further with the graph of the general parabola y=ax^{2}+bx+c. Note you will just have to convert the expression ax^{2}+bx+c to a perfect square form.

Problem 3:

Find the limits between which a must lie in order that the rational function \frac{ax^{2}-7x+5}{5x^{2}-7x+a} may be real, if x is real.

Problem 4:

Determine the limits between which n must lie in order that the equation 2ax(ax+nc)+(n^{2}-2)c^{2}=0 may have real roots.

Problem 5:

If x be real, prove that \frac{x}{x^{2}-5x+9} must lie between 1 and -\frac{1}{11}.

Problem 6:

Prove that the range of the rational function y=f(x)=\frac{x^{2}-x+1}{x^{2}+x+1} lies between 3 and \frac{1}{3} for all real values of x.

Problem 7:

If x \in \Re, Prove that the rational function y=f(x)=\frac{x^{2}+34x-71}{x^{2}+2x-7} can have no value between 5 and 9. In other words, prove that the range of the function is (x <5)\bigcup(x>9).

Problem 8:

Find the equation whose roots are \frac{\sqrt{a}}{\sqrt{a} \pm \sqrt(a-b)}.

Problem 9:

If \alpha, \beta are roots of the quadratic equation x^{2}-px+q=0, find the value of (a) \alpha^{2}(\alpha^{2}\beta^{-1}-\beta)+\beta^{2}(\beta^{2}\alpha^{-1}-\alpha) (b) (\alpha-p)^{-4}+(\beta-p)^{-4}.

Problem 10:

If the roots of lx^{2}+mx+n=0 be in the ratio p:q, prove that \sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=0

Problem 11:

If x be real, the expression \frac{(x+m)^{2}-4mn}{2(x-n)} admits of all values except such as those that lie between 2n and 2m.

Problem 12:

If the roots of the equation ax^{2}+2bx+c=0 are \alpha and \beta, and those of the equation Ax^{2}+2Bx+C=0 be \alpha+\delta and \beta+\delta, prove that \frac{b^{2}-ac}{a^{2}} = \frac{B^{2}-AC}{A^{2}}.

Problem 13:

Prove that the rational function y=f(x)=\frac{px^{2}+3x-4}{p+3x-4x^{2}} 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 \frac{x+2}{2x^{2}+3x+6} for any real value of x. (Remarks: this is maxima-minima problem which can be solved with algebra only, calculus is not needed). 

Problem 15:

Show that if x is real, the expression (x^{2}-bc)(2x-b-c)^{-1} has no real value between b and a.

Problem 16:

If the roots of ax^{2}+bx+c=0 be possible (real) and different, then the roots of (a+c)(ax^{2}+2bx+c)=2(ac-b^{2})(x^{2}+1) will not be real, and vice-versa. Prove this.

Problem 17:

Prove that the rational function y=f(x)=\frac{(ax-b)(dx-c)}{(bx-a)(cx-a)} will be capable of all real values when x is real, if a^{2}-b^{2} and c^{2}-a^{2} have the same sign.


Nalin Pithwa

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